shenlab.jpshenlab.jp/content/files/phd_thesis/PhDthesis_JYZhang.pdf · Feedback Control of Discontinuous Dynamical Systems with Application to Gasoline Engines by Jiangyan Zhang A

Dissertation for the Doctoral Degree in Engineering

Feedback Control of Discontinuous Dynamical

Systems with Application to Gasoline Engines

Jiangyan Zhang

Doctoral Program in Science and Technology

Sophia University

January 2011

Feedback Control of Discontinuous Dynamical

Systems with Application to Gasoline Engines

by

Jiangyan Zhang

A dissertation submitted in partial fulfillment

of the requirements for the degree of

Doctor of Philosophy

(Engineering and Applied Sciences)

in the Graduate School of Sophia University

2010

Dissertation Committee:

Professor Tielong Shen, Chair

Professor Yasuhiko Mutoh

Professor Riccardo Marino, University of Rome “Tor Vergata”, Italy

Associate Professor Tetsushi Sasagawa

Associate Professor Takashi Suzuki

Contents

Notation iii

1 Introduction 1

1.1 State of the Art . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.1.1 Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.1.2 Feedback Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.1.3 Control of Gasoline Engines . . . . . . . . . . . . . . . . . . . . . . . 7

1.2 Outline of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.3 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2 Feedback Control Method of Discontinuous Dynamical Systems 11

2.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.1.1 Filippov Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.1.2 Stability Analysis under Filippov Solution . . . . . . . . . . . . . . . 15

2.1.3 L2-gain Analysis under Filippov Solution . . . . . . . . . . . . . . . 24

2.2 Lyapunov-based Feedback Stabilizing Control Design . . . . . . . . . . . . . 26

2.2.1 Stabilization Design . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.2.2 Piecewise Continuous Linear Systems . . . . . . . . . . . . . . . . . 36

2.3 L2-gain Control Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

2.3.1 Hamilton-Jacobi Inequality Characterization . . . . . . . . . . . . . 39

2.3.2 L2-gain Controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

2.4 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

3 Analysis and Control of Piecewise Continuous Time-delay Systems 43

3.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.1.1 Extended Filippov Solution . . . . . . . . . . . . . . . . . . . . . . . 44

3.1.2 Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

3.1.3 L2-gain Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

i

ii Contents

3.2 Feedback Stabilizing Control . . . . . . . . . . . . . . . . . . . . . . . . . . 523.2.1 Stabilization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 523.2.2 Adaptive Stabilization . . . . . . . . . . . . . . . . . . . . . . . . . . 59

3.3 L2-gain Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 613.3.1 Hamilton-Jacobi Inequality-like Characterization . . . . . . . . . . . 623.3.2 L2-gain Controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . 633.3.3 Piecewise Continuous Linear Systems . . . . . . . . . . . . . . . . . 65


4 Model-based Control of IC Engines 73

4.1 Control-oriented Engine Model . . . . . . . . . . . . . . . . . . . . . . . . . 734.1.1 Mean-value Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 744.1.2 Model Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

4.2 Speed Tracking Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 824.2.1 Feedback Control with Delay Compensation . . . . . . . . . . . . . . 824.2.2 Feedback Control without Delay Compensation . . . . . . . . . . . . 90


5 Model-based Starting Control of SI Engines 99

5.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 995.2 Statement of the Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1005.3 Control Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1035.4 Validation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

5.4.1 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1085.4.2 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . 112


A Basic Mathematical Tools 121

Bibliography 124

Acknowledgements 135

Notation

Rn n-dimensional Euclidean spaceRn×m n × m dimensional real matricesa.e. Almost everywheres.t. Such thatco Convex hullco Convex closureμ(N ) Lebesgue measure of set N‖ · ‖ Euclidean norm∂N Boundary of set N

sgn(x) Sign function: sgn(x) =

⎧⎪⎨⎪⎩1, x > 00, x = 0

−1, x < 0∇f Gradient of differentiable function f : Rn → R

∂Cf Clarke’s generalized gradient of function f : Rn → R

< ·, · > Inner product of vectorsLfV Lie derivative of V along the vector field f

Cr Space of continuous function: Cr = {φ | φ : [−r, 0] → Rn}, (r > 0)Cm

r Space of continuous function: Cmr = {φ | φ : [−r, 0] → Rm}, (r > 0)

‖ · ‖c Norm of Cr space: ‖ φ ‖c= sup−r≤τ≤0{‖ φ(τ) ‖}

iii

Chapter 1

Introduction

In engineering practice, large numbers of phenomena can be represented by discontinuousfunctions. Typical examples refer to the static friction in mechanical systems [72, 88, 104,107], the interactions in robot manipulators [22], transmissions and torque managementstrategies generated from the perspective of optimization in hybrid electrical vehicles [37],etc. For the requirements of performance improvement in practical control applications,discontinuities are often employed in the designed systems, for example, the systems withswitching model and switching control [66], and sliding mode control [103]. On the otherhand, time-delay phenomena arise in lots of practical systems, for example the systemsgenerated by manufacturing processes; the information transmissions for feedback controlalways involve time delay; for applications, time delay is often used to model many practi-cal systems (see [35], [83] and references therein). For these systems, discontinuities arisein the differential equations or functional differential equations relevant to time-delay sys-tems that determine the dynamics of the systems or the closed-loop systems with feedbackcontrol. Gasoline engine with multi-cylinders is a benchmark example that possesses theabove characteristics from the point of engine control. According to the physical mecha-nism of a four-stroke spark ignition (SI) engine (see details introduced in Chapter 2), exactdescriptions of the system dynamic behavior will generate a hybrid system that consistsof continuous time modeling and event-based discrete ones. Moreover, the SI engines arewith different dynamics under different operation modes depending on the engine speed,external loads, the environment, etc. On the other hand, in automotive control engineer-ing, mean-value engine models, which ignore the characteristics of individual cylindersand capture the average features of engine physics, have come to be accepted as controloriented models (see [25], [36], [43], [69], [96], etc.). Due to stroke or cycle delays, themodeling always includes delay time, such as the well-known intake-to-power delay.

This thesis investigates analysis and control design problems of discontinuous dynami-cal systems represented by differential equations and functional differential equations withdiscontinuous right-hand side. The basic property for investigations of system control cen-ters the solution of the associated differential equations or functional differential equations.

1

2 Chapter 1. Introduction

Conventional idea of solution, that requires at least the right-hand side to be continuous,is not suitable to describe the behavior of discontinuous dynamical systems. Hence, ageneralized solution is an essential foundation to discuss the analysis and control designproblems. As we can see from what follows, there have been a number of contributionsusing mathematical investigations with respect to differential equations and functional dif-ferential equations with discontinuous right-hand side. Around the generalized notion ofsolution, theoretical analysis of discontinuous systems from the viewpoint of system controlhave attracted significant attention, especially in the recent decade. Furthermore, effortsto use various theoretical tools to provide solutions for synthesis problems have made somegains in achieving performance improvement in designing control systems, where most in-terests are in extending classical frameworks with respect to smooth dynamical systemsto discontinuous ones. Meanwhile, taking the inherent time delay of gasoline engine sys-tems into account, model-based design is investigated to solve engine control problems.In the investigations, discontinuity is involved in the functional differential equations thatrepresent the dynamics of the engine system and/or the closed-loop control systems.

The remainder of this chapter includes a review of previous works on analysis and con-trol design of discontinuous dynamical systems, and a summary of the main contributionsof this thesis.

1.1 State of the Art

System dynamics is usually represented by an ordinary differential equation as

x(t) = f(t, x) (1.1)

and for time-delay systems, dynamics is usually represented by a functional differentialequation as

x(t) = fd(t, xt) (1.2)

where x ∈ Rn denotes the vector of the state variables, f : R×Rn → Rn is a vector field,xt ∈ Cr with xt(τ) = x(t − τ), τ ∈ [0, r] (r > 0), and fd : R × Cr → Rn is a functional.

In the field of mathematics, the differential equation (1.1) and functional differentialequation (1.2) with discontinuous right-hand side are not new topics. Regarding thenotion of a solution of the system (1.1), two kinds of definitions have been proposedas reported in [21, 89]. A natural extension of the conventional notion of solution thatcan be consistent with discontinuous differential equations is the Caratheodory solution.However, due to the limitation of Caratheodory condition, the Caratheodory solutiondoes not exist in many cases when dealing with the concerned discontinuous systems(see [22] for a detailed discussion with corresponding examples). On the other hand,notions of a generalized solution, which attracted significant attention in control theory,are well defined for various objectives. Two ways are mainly followed to define generalizedsolutions. The first one consists in defining approximate solutions by means of particular

1.1. State of the Art 3

algorithms, referring to Euler [18], Hermes [39] and sampling solutions [57]. The otherway is to use the theory of differential inclusions, which involves set-valued mappings 1

formulated with respect to the concerned differential equations. The Krasovskii solution[39] and the Filippov solution [31, 32] follow this method. In fact, more than eight notionsof generalized solution have been proposed to characterize the behavior of discontinuousdynamical systems [22, 31]. Correspondingly, the fundamental issues, such as existence,uniqueness, continuous dependence on initial condition are also investigated by manyliteratures mentioned above. Moreover, comparisons among the generalized solutions areexamined, and more details with interesting examples can be found in the tutorial paper[22] and references therein.

The above available solution definitions are given with respect to differential equations,however, for the case of functional differential equations in the field of time-delay systems,the literatures are not very rich to give systematic investigations of solution related issues.One can find the investigations of time-delay system (1.2) in an early report [42] and [41]with respect to the Caratheodory solution of functional differential equation and the prop-erties in terms of existence, uniqueness and dependence of initial conditions. It is notedthat functional differential inclusion, which can deal with functional differential equationsis actually a general type of differential inclusions. In recent years, mathematicians havebeen intensively studying the existence condition of solution for functional differential in-clusion. The publication [48] provides a result for the functional differential inclusion thatis upper semicontinuous with convex values, which is further investigated in [99] whereone can find a discussion with respect to the Caratheodory solution. Moreover, there aremany contributions to cases of functional differential inclusions, such as the nonconvexcase [11]; see [68] for more results and references. Meanwhile, these contributions focuson providing the uniqueness condition.

This thesis focuses on discussing discontinuous systems in the sense of generalized so-lution associated with differential inclusion and functional differential inclusion, specially,in the sense of Filippov solution. Filippov [31, 32] gives a solution definition of differentialequations with discontinuous right-hand side and the corresponding properties in termsof existence, uniqueness, continuous dependence on initial conditions and the right-handside. We indicate the following significant points of applying Filippov solution to discon-tinuous systems [104]: the Filippov solution which allows exclusion of the set of measurezero usually exists; it is quite general, and is very useful in many engineering problems,since it is close to the true trajectories of the system [80]; another significant contributionof Filippov solution is the characterization of the behavior for differential equations withpiecewise continuous right-hand side, i.e. the sliding motion on the surfaces where thevector fields are not continuous.

1See Definition A.2 in Appendix A for the definition of set-valued mappings.


1.1.1 Stability Analysis

It should be remarked first that generalized solutions of discontinuous systems is not ingeneral unique. Hence there are two distinct terms to characterize the stability: strongstability in the sense that the property holds for all solutions from an initial conditionand oppositely, weak stability when only one or some particular solutions from an initialcondition is concerned [7, 20, 22].

In control engineering, Lyapunov methods are widely used to deal with analysis andsynthesis issues (see [56] and references therein for examples). However, due to the lack ofunique solution of discontinuous systems, the conventional methods do not admit straight-forward extensions to discontinuous systems. In this respect, the interests of the significantresults for discontinuous systems are the extensions of Lyapunov stability theory and in-variance principle. Most of the results, such as the ones in [4, 7, 18, 19, 76, 80, 88, 104, 105]to name a few, are developed under Filippov solution due to its proper formulation de-scribed previously. In other words, the extensions are established within the frameworkof differential inclusions.

Early results for stability analysis problem of discontinuous systems are provided in[3], [80] with nonsmooth Lyapunov functions. It is provided with generalized directionalderivative in the sense of Dini (see [20]), which actually cannot be calculated when it iscomposite with solutions of differential equations. More extensively, there are two reportedapproaches in dealing with stability analysis problems of discontinuous systems. One isby means of nonsmooth Lyapunov functions which is considered to be natural tools fornonsmooth dynamical systems. The other is by applying smooth Lyapunov functions asthe conventional case, but is extended with respect to differential inclusions. Most of theresults in the former case are developed on the basis of the nonsmooth analysis theory byClarke [17, 18], specifically, generalized gradient for upper semicontinuous functions andproximal subdifferentials for lower semicontinuous functions (see [22] for more details).The facts in the calculus for them applied to nonsmooth analysis have been approvedeffective. Shevitz and Paden [88] present a systematic analysis framework originally byusing nonsmooth Lyapunov functions, with Clarke’s generalized gradient; the key of thedeveloped tools is the generalized Lie derivative approach [56] to discontinuous systems;more precisely, these available theoretical tools are established with nonsmooth Lyapunovfunctions satisfying the regularity condition. Bacciotti and Ceragioli in [4], see also [7],[20], propose an alternative Lyapunov method. A modified result of [88] is further proposedin [104]. Several motivated examples that cannot be shown stable by smooth Lyapunovfunction are presented in [4, 20] to illustrate the applications of the established nonsmoothanalysis tools. Moreover, instead of employing regular functions, the authors in [5] proposesemiconcave functions to investigate the stability of discontinuous systems, and indicatethat regularity and semiconcavity have a common feature, namely, nonpathological. Themain results in [19] provide the property of asymptotic stability with respect to differentialinclusion evaluated by smooth Lyapunov functions. In particular, Wu, et al. [105] andOrlov Y.V. [76] indicate that smooth Lyapunov functions are sufficient in many applica-


tions encountered in discontinuous systems, and correspondingly in [105], conditions forconstructing smooth Lyapunov functions are discussed.

The study of invariance principle deduced in terms of discontinuous systems can befound in [4, 76, 85, 88]. The main results are presented with respect to differential inclusion.In [88], the conventional LaSalle invariance principle is generalized by nonsmooth functionsto discontinuous systems with unique solution in the involved invariant set, while [76]presents more general results with no assumption of uniqueness of the solution. A commentfor comparison is given in [4] to three versions of the available results ([4, 85, 88]). Onecan find in [107] an application of the invariance principle of [88] to analyze mechanicalsystems with dry friction.

Focusing on the efforts to the systems under arbitrary switching signals, conditionsto study the stability of such systems have been significantly explored, especially for thesystems in linear situation, see [65], [66], [92]. In this situation, the asymptotic stabilityof the switched system can be achieved by showing that the individual systems associatedto the considered system possess a common Lyapunov function.

There are additional investigations to the stability of discontinuous systems. Condi-tions for semistability and finite-time stability are discussed in [49, 76]. A fundamentalframework to deal with L2-gain analysis is presented in [8]. The dissipativity is reportedrecently in [50].

With respect to time-delay systems, Lyapunov condition is given by using Lyapunov-Krasovskii functional instead of Lyapunov function. Over the past years, there have beenextensive efforts focused on the stability criteria of time-delay systems (1.2) with con-tinuous right-hand side, see [41], [59], [60], [61], [108], etc. As mentioned previously,discontinuities are often involved in a wide range of dynamical systems, such as hybridsystems. There has been much interest in the study of time-delay systems in the field ofhybrid systems [67, 97]. Hence, it is significant to pay attention to discontinuous time-delay systems from the viewpoint of control theory. The literatures on introducing ageneral framework to analyze time-delay systems with discontinuity are not rich. In [81],extended Lyapunov-Krasovskii theorem is investigated in the sense of Caratheodory so-lution of functional differential equation, and similar result is proposed in [76]. Thereare few attentions paid to give stability conditions applied to functional differential inclu-sion. An invariance principle is presented in [15] for the situation of unique solution offunctional differential inclusions. Exponential stability is discussed in [98] for functionaldifferential inclusions with linear structure. The investigations of differential inclusionbased theoretical framework provide the basic motivations to pay attention to functionaldifferential inclusions in analysis of discontinuous time-delay systems. In this thesis, onthe basis of the study of functional differential inclusion in mathematics, a generalizationof the Filippov framework in terms of differential inclusion, Filippov solution and stabilitytheory is proposed with functional differential inclusion-based approach.


1.1.2 Feedback Control

Although there has been much analytical work, the literature reporting control designproblems of discontinuous systems is not very rich. There are several recent results con-cerning the feedback control of classes of discontinuous systems. A state-based switchingcontrol method is given in [6] to stabilize a pair of linear switched systems. In [71], aLgV structure-based switching control approach is established to deal with stabilizationproblems of discontinuous systems; the resulting closed-loop system is shown asymptoti-cally stable and is analyzed in the sense of Filippov solution; moreover, the authors alsopresent an adaptive design scheme in [73] for a class of nonlinear systems with unknownparameters that linearly depend on a discontinuous function; the design approaches areapplied to mechanical systems with kinds of discontinuous uncertainties [72, 74]. Consider-ing a benchmark discontinuous system, the dynamical system of rotor, an observer designstrategy is developed in [29]. In [44], a state feedback stabilization method is presentedfor a class of cascade nonlinear systems with discontinuous connections. Moreover, it isshown in [45] that the design approach can be extended to adaptive stabilization whenthe systems involve bounded uncertainties.

It is well known that the Lyapunov recursive design approach is an approved approachin designing feedback controllers for various control objectives [55]. Systematic descrip-tions of this approach with respect to discontinuous systems are not clearly presented inavailable literatures. An extension is demonstrated in a recent proceeding [86] where itrelies on the assumption that the virtual control laws obtained in each step of the recur-sive design procedure are continuous and differentiable except the final one. Under thesame assumption, an extended backstepping design approach is given in [106] for a classof switched nonlinear systems which restricts the situation of an existing differentiablecontrol law for stabilizing each subsystem; although the techniques are with respect toswitched systems under arbitrary switching signals, it is not clear that the methods canbe extended to solve feedback stabilization problems for the discontinuous systems underconsideration. In [102], the authors try to design discontinuous controllers recursively inthe sense of Filippov solution; however, the application of Clarke’s generalized gradientdoes not seem to be feasible to demonstrate the design approach.

Moreover, several design approaches employing discontinuous feedback control lawsare presented for achieving desired purposes which are guaranteed in the sense of Filippovsolution for the implemented systems. Stabilization problem for kinds of systems whichcannot be asymptotically stabilized by smooth feedback control laws are discussed in[12, 21]. The design idea in [70] gives a sliding mode control from the viewpoint of Filippovsolution. For more flexibility to design control systems, the authors of [126] provide a novelsliding mode control method associated with nonsmooth switching surfaces. In discussingsliding mode control, we should mention that there are two approaches by Filipoov [31]and Utkin [103], respectively, to define the solution of the system with respect to thecontrol induced switching surfaces. It is shown that in the case in which the dependenceof the dynamics on the control input is linear, then the two definitions are equivalent,


otherwise the choice of one of the two solutions can always influence the stability results[27].

We should mention that in order to address the control problems, discontinuous func-tions are usually approximated by smooth ones. For example, the reference [28] introducesa method which gives a differentiable function for nonsmooth design. And in the earlystage, the “Softer” control [33] is a typical example of this design technique. This approachapparently limits the attainable performance of the control system.

In the control theory community, a lot of challenging problems are encountered toprovide synthesis tools for systems with time delay, since time delay in the systems oftencauses undesired performances including oscillations and instability [35, 58, 83]. Thereare continuous interests in developing control design approaches for time-delay systems,for instance, robust stabilization [13, 53], adaptive control [52, 54], sliding mode control[51], etc. Focusing on functional differential inclusion, this thesis deals with control designproblems of discontinuous time-delay systems.

1.1.3 Control of Gasoline Engines

In automotive engineering, model-based development offers prospects for engine controltechniques, although there are many difficulties due to the complexity of the system asmentioned above. In this thesis, we focus on discussing applications of the presentedtheoretical tools to model-based engine speed control problems. Engine speed controlis a classical issue in automotive control applications. The performance of engine speedhas significant impact on the vehicle design attributes such as comfort, emission, fueleconomy, etc., especially during transitional operations [77, 78, 101]. In the community ofcontrol engineering, this has led to many approaches to tackle the speed control problemon the basis of mean-value models. As is well-known, one of the main characteristicsof the engine system is that it involves the intake-to-power delay, which was ignored inmany earlier works. As mentioned on the influence of time delay in dynamical systems, thisdelay characteristic should be considered properly in investigating engine control problems[47]. Over the years, several speed control methods that take the intake-to-power strokedelay into account have been proposed. A sliding mode control method is introducedin [26] where an approximation approach is given to deal with the involved delay statein the dynamical system, and the validation of the control algorithm is shown by bothsimulation and experimental results. In [38] and [82], the controllers are constructedapplying the design techniques for linear systems to the linearized engine model, andpresented simulation results verify the the theoretical analysis. The authors in [95] givea delay compensation control law of spark advance input, and then propose a variablestructure control method of throttle combining with parameter adaption technique. Onthe other hand, the engine system, as an active benchmark example, has been used toassess the control design methods for time-delay systems, see [10], [64].

Based on the available contributions on engine modeling and according to the modeling


investigation by the engine test bench, it is shown that additional discontinuities withrespect to different engine operation models improve the modeling effectiveness to performa model-based development for some concerned engine control problems, especially theones that attempt to get better transient performance. Taking the intake-to-power delayinto account explicitly, this thesis proposes a model-based speed tracking control schemeby means of the given theoretical tools for time-delay systems.

1.2 Outline of the Thesis

The main contents consist of four chapters.

In Chapter 2, a review of Filippov solution and essential elements of Clark’s gener-alized gradient are presented. Generalized theoretical tools in terms of stability analysisand L2-gain are given for investigating discontinuous systems. Lyapunov-based feedbackdesign approaches are presented to solve stabilization and L2-gain synthesis problems fora general class of nonlinear discontinuous systems. Special attention is paid to show aLyapunov recursive design approach when nonsmooth candidate of Lyapunov functionsare encountered.

Chapter 3 is devoted to deal with analysis and synthesis of discontinuous time-delaysystems from an extended viewpoint within the Filippov framework. Formulations ofFilippov solution and associated basic principles with respect to functional differential in-clusion are deduced. Stability conditions are provided with extended Lyapunov-Krasovskiistability theory and invariance principle. Feedback design approaches are developed forstabilization and L2-gain synthesis problems for nonlinear, piecewise continuous lineardiscontinuous time-delay systems.

In Chapter 4, the application to model-based control of gasoline engines is consid-ered. Taking the inherent time delay of the engine system into account, control problemof achieving a target engine speed is treated by proposing switched modeling in wideoperation mode of engine.

Chapter 5 describes starting control problem of gasoline engines. This has been a chal-lenging problem in automotive control engineering, and has been proposed as a benchmarkproblem by SICE Research Committee on Advanced Powertrain Control Theory in 2006.To achieve pre-specified design specifications by the SICE benchmark problem, a model-based control scheme is presented. Experimental investigations indicate that theoreticaltools of discontinuous dynamical systems are potential to improve the control algorithmsfor better performance.

Finally, Appendix A collects several useful concepts and mathematical tools in thefield of discontinuous dynamical systems.

1.3. Contributions 9

1.3 Contributions

The main contributions of this thesis are as follows.

New conditions of stability and L2-gain analysis (Chapter 2). In spite of significantrecent efforts to analyse discontinuous systems, a new stability analysis tool is presented byusing nonsmooth functions to play the role of Lyapunov functions. In contrast to availablemethods in the literatures, the presented result does not require the Lyapunov function tobe regular, and is more general to be applied to design desired feedback control laws fordiscontinuous systems. Moreover, conditions are initially provided to deal with L2-gainissues in discontinuous systems.

A general, systematic design procedure for stabilization of nonlinear discontinuous andpiecewise continuous linear systems (Chapter 2). Considering a general class of nonlineardiscontinuous systems, much work including stabilization, disturbance attenuation, etc.remain as challenging problems under the current framework. Classical feedback controlapproaches need to be carefully extended to be applicable to discontinuous systems. Onthe basis of the generalized theoretical tools, Lyapunov-based feedback stabilizing designapproaches are presented for a class of nonlinear discontinuous systems and piecewise con-tinuous linear systems as a special case. In particular, Lyapunov recursive design approachis effectively extended by using nonsmooth candidate of Lyapunov functions. This designapproach characterizes the essential points in the case that nonsmooth feedback controlaction is employed during the recursive design procedures. The main results have beenpresented in [119].

L2-gain synthesis (Chapter 2). Hamilton-Jacobi inequality (HJI) characterization isformulated with respect to differential inclusions first. Then, the presented control idea ofstabilization is extended to solve the L2-gain synthesis problems for discontinuous systemswith forced inputs.

Extended Filippov framework to handle discontinuous time-delay systems (Chapter 3).To analyze and synthesize discontinuous time-delay systems, the framework of differentialinclusion in the sense of Filippov is extended to functional differential inclusion. Based onthe extension, the concept of Filippov solution is introduced for discontinuous time-delaysystems; useful tools along the line with respect to Filippov solution are presented in termsof computing the functional differential inclusion; the theoretical characterization of thesliding motion in the sense of Filippov solution is given for piecewise continuous time-delaysystems; moreover, the Lyapunov-Krasovskii theorem and the invariance principle are alsoestablished in terms of functional differential inclusions. In particular, we propose someuseful tools to determine the stability of discontinuous time-delay systems; these toolsare provided using a Lyapunov-Krasovskii functional with certain structure that combinesa Lipschitz continuous function with a differentiable functional. On the other hand, aconcept is given to characterize the L2-gain property of discontinuous time-delay systems,and analysis tools with respect to strong asymptotic stability with L2-gain are provided.The main results have been published in [120] and [122].


Feedback stabilization of discontinuous nonlinear time-delay systems (Chapter 3). Byusing the presented theoretical tools, a feedback design approach is provided such thatthe closed-loop systems are strongly asymptotically stable, and it is also extended withadaptive control technique to deal with systems with uncertain parameters. (See [120].)

L2-gain synthesis of discontinuous time-delay systems (Chapter 3). A sufficient con-dition is established with a HJI-like functional partial differential inequality to deal withcontrol problems that concern the L2-gain constraint and strong asymptotic stability ofthe system with zero input. With this condition, a state-feedback control law is finallypresented for a class of nonlinear discontinuous time-delay systems. The application ofthe proposed method to piecewise continuous linear time-delay systems is also discussed.(See [120].)

Gasoline engine control (Chapter 4). Two speed control schemes are proposed based onmean-value engine models including the intake-to-power delay. First, with a delay compen-sation technique by spark advance input, a Lyapunov-based control scheme is presented totackle speed tracking control problems. Feedback control laws are constructed with simplestructure by taking the physics of engine system into the feedback design The validationsof the control laws are demonstrated by simulations and experiments, respectively. Then,mean-value models with different parameters are used to characterize engine dynamics inwide operation modes. A switched feedback control algorithm of throttle opening input ispresented, and the asymptotic convergency of the resulting control system is achieved byselecting proper feedback gains. The gain conditions are obtained based on the functionaldifferential inclusion based stability condition. The theoretical analysis is verified by thenumerical simulation results. The results of the delay compensation-based design havebeen published in [121].

Gasoline engine control (Chapter 5). A solution to the SICE benchmark problem ispresented with the purpose of improving the transient performance of the starting enginespeed. To guarantee constrained combustion condition, a control scheme for provingindividual fuel injection commands (six commands for the SI engines under consideration)is proposed by using a dual sampling rate system: the cycle-based fuel injection commandis individually adjusted for each cylinder by using a TDC (top dead center)-based air chargeestimation. A coordinated switching control law between spark advance and throttleoperation is proposed to achieve the desired performance for speed regulation. The speederror convergence of the closed-loop system is proved for simplified, second-order modelwith intake-to-power delay. The performance of the proposed control scheme is testedusing an industrial engine simulator and an engine test bench, respectively. The designideas have been presented in the publications [116], [123] and [124].

Chapter 2

Feedback Control Method ofDiscontinuous Dynamical Systems

This chapter investigates feedback design problems for the following discontinuous dynam-ical systems

x(t) = f(x) + g(x)u, t ≥ t0 (2.1)

with x(t0) = x0, where x ∈ Rn is the state, u ∈ Rm is the control input, and thevector fields f and g are discontinuous in the state x. In the sense of Filippov solution, afeedback design method is developed for the stabilization and L2-gain synthesis problems.Combining the Filippov solution with the related nonsmooth analysis tools, the mainresults give extensions of the conventional methods under smooth situation [56, 93].

This chapter starts with Section 2.1 that gives a brief review of Filippov solution withthe analysis theory in terms of Lyapunov stability and L2-gain performance. In Section2.2, state feedback stabilization problem is discussed for several cases of the dynamicalsystems. Then, the design approach is extended to the L2-gain synthesis problem inSection 2.3. Section 2.4 gives a discussion of the presented results.

2.1 Preliminaries

Consider the dynamical systems represented by the differential equation

x(t) = f(x), t ≥ t0 (2.2)

with x(t0) = x0, where x ∈ Rn and the vector field f : Rn → Rn is discontinuous in x andis locally bounded.

11

12 Chapter 2. Feedback Control Method of Discontinuous Dynamical Systems

2.1.1 Filippov Solution

Definition 2.1.[31]

A vector function x(t) defined on [t0, t1] (t1 > t0) is called a Filippovsolution of equation (2.2) if it is absolutely continuous1 and

x ∈ K[f ](x), a.e. t ∈ [t0, t1] (2.3)

whereK[f ](x) =

⋂δ>0

⋂μ(N )=0

cof (B(x, δ) −N ) (2.4)

with B(x, δ) = {y ∈ Rn | ‖y − x‖ < δ}.

Expression (2.4) is called the Filippov set-valued mapping. It should be noted that theFilippov solution of differential equation (2.2) is represented by a solution of the differentialinclusion 2.3.

For the calculation of Filippov set-valued mapping, the following equivalent definitionof Filippov set-valued mapping (2.4) is usually used [80]

K[f ](x) = co{

limi→∞

f(xi) | xi → x, xi /∈ Nf ∪N}

(2.5)

where N ∈ Rn is any set of measure zero (μ(N ) = 0), and Nf ∈ Rn is the set ofmeasure zero (μ(Nf ) = 0) where f is discontinuous. Useful results are summarized inAppendix A, which include the consistency, sum rule, product rule, chain rule and matrixtransformation rule2 reported in [22, 80], for calculating the Filippov set-valued mapping.

Piecewise Continuous Vector Field

It is well-known that most of the dynamics of the encountered discontinuous systems aredetermined by piecewise continuous vector fields, which are continuous everywhere expectat a smooth surface of the state space [22]. The following gives some essential statementsof Filippov’s contributions to the case of system (2.2) with piecewise continuous vectorfield3.

Consider the discontinuous dynamical system (2.2) with piecewise continuous vectorfield f described by

x(t) = fi(x), x ∈ Ri, i = 1, · · · , n, t ≥ t0 (2.6)

1See Definition A.1 in Appendix A for the definition of absolutely continuous functions and a fewillustrations considered to be helpful.

2Throughout the thesis, for an operator m : Rn → Rm×n, and a Filippov set-valued operator K : Rn →B(Rn), the calculation m(·) · K[ ](·) means the set {ξ(·) ∈ Rm | ξ(·) = m(·) · γ(·), γ(·) ∈ K[ ](·)}.

3A general definition of piecewise continuous vector field can be found in [22].

2.1. Preliminaries 13

Ri

Si

Rj

f-(x)

f+(x)

K[f](x)f

+

N(x)

x

N

f-

N(x)

Fig. 2.1: A sketch of piecewise continuous vector field.

where fi : Rn → Rn is continuous. The cells Ri (i = 1, · · · , n) are partitions of the statespace Rn such that ∪i=n

i=1Ri = Rn, Ri ∩ Rj = ∅, i = j. For any x ∈ ∂Ri, there existsj = i such that fi(x) = fj(x). Denote Sf = ∪i=n

i=1∂Ri which is the set of points where f isdiscontinuous and is measure zero. Moreover, any Si ∈ Sf (i.e. any ∂Ri) is smooth.

Let Si = ∂Ri = ∂Rj and N be the normal vector to the surface Si. Assume that thepositive direction of the normal vector is from Rj to Ri. f−(x) and f+(x) denote thelimiting vectors as any x ∈ Si is approached from Rj and Ri, respectively, i.e.

f−(x) = limz→x−

fj(z), f+(x) = limz→x+

fi(z), x ∈ Si

Let f−N (x) and f+

N (x) denote the projections of f−(x) and f+(x) on the normal vector Nat x. A sketch of the above description on piecewise continuous vector field is given inFig. 2.1.

Strict illustration is given when at a moment t = t1, a solution x(t1) is at Si and cannotleave it at once. In other words, there is a sliding motion along the surface Si.

Proposition 2.1. Consider system (2.6). Let a vector function x(t) be absolutely con-tinuous. Suppose that for t ∈ [t1, t2] (t2 > t1), x(t) ∈ Si, f−

N (x) ≥ 0, f+N (x) ≤ 0 and

f−N (x) − f+

N (x) > 0. Then x(t) is a solution of equation (2.6), if and only if the x(t)satisfies the following differential equation

x = f0(x) = λ(x)f+(x) + (1 − λ(x))f−(x), a.e. t ∈ [t1, t2] (2.7)

where

λ(x) =f−

N (x)f−

N (x) − f+N (x)

In fact, there are three situations on the behavior of Filippov solution with respectto the smooth surface where the vector field is discontinuous, namely, sliding motion,crossing, never intersect (See lemma 3, lemma 8 and lemma 9 in [31]). For more detaildiscussions see [22, 74, 87, 127] and references therein.


Remark 2.1. The Filippov set-valued mapping with respect to piecewise continuousvector field f is given by the following expression

K[f ](x) = co{

limi→∞

f(xi) | xi → x, xi /∈ Sf

}(2.8)

Take the following example to demonstrate the above descriptions on Filippov solution.

Example 2.1. Consider a discontinuous system given by[x1(t)

x2(t)

]= f(x) =

[−sgn(x1) + x2

−x1 − sgn(x2)

](2.9)

The vector field of the system is piecewise continuous and can be represented in the formof (2.6) with

f1(x) =

[−1 + x2

−x1 − 1

], f2(x) =

[−1−x1

], f3(x) =

[−1 + x2

−x1 + 1

], f4(x) =

[x2

1

]

f5(x) =

[1 + x2

−x1 + 1

], f6(x) =

[1

−x1

], f7(x) =

[1 + x2

−x1 − 1

], f8(x) =

[x2

−1

], f9(x) =

[00

]

and

R1 = {(x1, x2) ∈ R2 | x1 > 0, x2 > 0}, R2 = {(x1, x2) ∈ R2 | x1 > 0, x2 = 0},R3 = {(x1, x2) ∈ R2 | x1 > 0, x2 < 0}, R4 = {(x1, x2) ∈ R2 | x1 = 0, x2 < 0},R5 = {(x1, x2) ∈ R2 | x1 < 0, x2 < 0}, R6 = {(x1, x2) ∈ R2 | x1 < 0, x2 = 0},R7 = {(x1, x2) ∈ R2 | x1 < 0, x2 > 0}, R8 = {(x1, x2) ∈ R2 | x1 = 0, x2 > 0},R9 = {(x1, x2) ∈ R2 | x1 = 0, x2 = 0}

Then, according to (2.8) and Proposition A.1, the Filippov set-valued mapping can becalculated as follows

K[f ](x) =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

{−sgn(x1) + x2} × {−x1 − sgn(x2)}, x ∈ R1,3,5,7

co

{[−sgn(x1) + x2

−x1 − 1

],

[−sgn(x1) + x2

−x1 + 1

]}, x ∈ R2,6

co

{[−1 + x2

−x1 − sgn(x2)

],

[1 + x2

−x1 − sgn(x2)

]}, x ∈ R4,8

co

{[−1−1

],

[1−1

],

[−11

],

[11

]}, x ∈ R9


Moreover, according to the notation Si = ∂Ri = ∂Rj , we have

Sf = {S1 = x1 = 0, S2 = x2 = 0}

Furthermore, with respect to the surface S1, the normal vector field is N = [1 0]T and thelimiting vectors can be obtained as

f+(x(t)) = limx1→0+

f1(x) = limx1→0+

f3(x) =

[−1 + x2

−sgn(x2)

]

f−(x(t)) = limx1→0−

f5(x) = limx1→0−

f7(x) =

[1 + x2

−sgn(x2)

]

Then, we have {f+

N (x(t)) = NT f+(x(t)) = −1 + x2 ≤ 0, if x2 ≤ 1

f−N (x(t)) = NT f−(x(t)) = 1 + x2 ≥ 0 if x2 ≥ −1

and f+N (x(t)) − f−

N (x(t)) > 0. According to Proposition 2.1, on the surface S1 and x2 ∈[−1, 1], an absolutely continuous function x(t) that satisfies

with λ(x) =f−

N (x)f−

N (x) − f+N (x)

=1 + x1

2,

x(t) = f0(x) = λ(x)f+(x) + (1 − λ(x))f−(x) =

[0

−sgn(x2)

], a.e. t

is Filippov solution of system (2.9). Furthermore, by same demonstration, on the surfaceS2 and x1 ∈ [−1, 1], an absolutely continuous function x(t) that satisfies

with λ(x) =f−

N (x)f−

N (x) − f+N (x)

=1 − x1

2,

x(t) = f0(x) = λ(x)f+(x) + (1 − λ(x))f−(x) =

[−sgn(x1)

0

], a.e. t

is Filippov solution of system (2.9). Fig. 2.2 shows the vector fields of the system (2.9).

2.1.2 Stability Analysis under Filippov Solution

To analyze the stability of Filippov solution, Lyapunov second method and invarianceprinciple are still effective as the mentions in Chapter 1. This section reviews the ex-tended results for discontinuous systems. Meanwhile, a new result is introduced thatplays essential role in the Lyapunov-based feedback design in the thesis.


-1.5 -1 -0.5 0 0.5 1 1.5

-1.5

-1

-0.5

0

0.5

1

1.5

x1

x2

Fig. 2.2: Vector fields of system (2.9).

Consider system (2.2). If the vector field f is locally bounded, then Filippov set-valuedmapping (2.4) is upper semi-continuous [109] and have nonempty, compact and convexvalues. These facts guarantee that there exists at least one solution of differential inclusion(2.3)(see [20], [22]).

A point x∗ ∈ Rn is an equilibrium of differential inclusion (2.3), if 0 ∈ K[f ](x∗)[22]. Without loss of generality, suppose that 0 ∈ K[f ](0), which means that the stabilityinvestigation of any Filippov solution of system (2.2) is reduced to study the trivial solutionx = 0 of differential inclusion (2.3).

Proposition 2.2.[104]

Suppose that for any initial condition x0, the Filippov solution xof system (2.2) is unique. If there exists a continuous differentiable, positive-definitivefunction V : Rn → R such that

V (x) ≤ 0 (2.10)

Then, system (2.2) is stable at the origin4. Moreover, if there exists a class K function5

W (x) such thatV (x) ≤ −W (x) (2.11)

then, system (2.2) is asymptotically stable at the origin.

4Throughout this thesis, the system being stable (asymptotically stable) at the origin is equivalent tothat the trivial solution x = 0 of the system is stable (asymptotically stable). Refer to [56] for the stabilitydefinition of the trivial solution.

5See [56] for a definition of class K function.


The above proposition gives a natural extension of the standard Lyapunov stabilitytheorem [56] under the assumption that differential equation (2.3) has unique Filippovsolution. Uniqueness condition on Filippov solution can be found in [22, 31]. Indeed,Filippov solution of a discontinuous dynamical system is not unique in general. See thefollowing example.

Example 2.2.[22]

Consider differential equation (2.2) with vector field f : R → R, i.e.

x(t) = f(x) = −sgn(x) (2.12)

The Filippov differential inclusion is

K[f ](x) =

⎧⎨⎩1, x > 0[−1, 1], x = 0−1, x < 0

It is easy to confirm that there are three Filippov solutions starting from x0 = 0, i.e.

x(t)1 = −t, x(t)2 = 0 and x(t)3 = t

Therefore, we pay attention to the stability associated with all the Filippov solutionsstarting from an initial condition, i.e. characterized by strong stability.

On the other hand, it should be noted that Proposition 2.2 provides the stability condi-tion using continuous differentiable Lyapunov function. It is well-known that the essentialof Lyapunov second method relies on investigating the monotone decreasing evolution ofa candidate Lyapunov function, which is continuous differentiable, along the solution, andthe monotonicity is determined with the gradient of the Lyapunov function and the vectorfield of the dynamical systems. With respect to differential inclusion, well-extended toolsfor stability analysis are established by a candidate Lyapunov function which is generallynot differentiable. In this situation, the generalized gradient, specifically Clarke’s gener-alized gradient [16, 17, 18], is shown feasible and employed to establish the conditions forstability analysis with respect to discontinuous dynamical systems [20, 88].

Clarke’s Generalized Gradient

Definitions and related explication summarized below are based on [16, 17, 18].

First, Clarke presents the generalized theory of gradient with corresponding conse-quences in terms of Lipschitz continuous functions.

Definition 2.2. Let f : Rn → R be locally Lipschitz continuous. The generalized direc-tional derivative at x in the v direction is given by

fo(x; v) = lim supy→xh→0

f(y + hv) − f(y)h

(2.13)


Definition 2.3. Let f : Rn → R be locally Lipschitz continuous. The generalized gradientof f at x is defined by

∂Cf(x) = co{

limi→∞

∇f(xi) | xi → x, xi /∈ Nf

}(2.14)

where ∇f denotes the gradient of f where it exists and Nf denotes the set of points wheref is not differentiable, and equivalently

[17]

∂fC(x) = {ξ ∈ Rn | f◦(x; v) ≥ < v, ξ >,∀v ∈ Rn} (2.15)

By (2.15), generalized gradient ∂Cf(x) of a Lipschitz continuous function f is anonempty, compact and convex subset of Rn [17].

Computing Clarke’s generalized gradient can according presented results includingdilation rule, sum rule, product rule, quotient rule and chain rule (see details from [17,18, 22]). Moreover, extension of Definition 2.3 of generalized gradient is also proposedby Clarke with respect to functions that are not necessary Lipschitz continuous. Thisextension is by means of geometric conceptions, tangents, normals and epigraph shown asfollows

[17].

Let Ω be a nonempty closed subset of Rn, then,

• the distance function dΩ related to Ω, which is a Lipschitze continuous function, isdefined by

dΩ(x) = min {‖x − c‖ | c ∈ Ω} (2.16)

• the tangent cone TΩ(x) to Ω at a point x in Ω is defined by

TΩ(x) = {v ∈ Rn | d◦Ω(x; v) = 0} (2.17)

• the normal cone NΩ(x) to Ω at x is defined by

NΩ(x) = {ξ ∈ Rn | < ξ, v > ≤ 0,∀v ∈ TΩ(x)} (2.18)

• the epigraph of a function f : Ω → R, denoted as epif , is the set

epif = {(x, r) ∈ Ω × R | f(x) ≤ r} (2.19)

Definition 2.4. Let f : Ω → R∪{+∞, −∞}, Ω ⊂ Rn, be finite at x ∈ Ω. The generalizedgradient of f at x is defined by

∂Cf(x) = {ξ | (ξ,−1) ∈ Nepif (x, f(x))} (2.20)

Equation (2.20) as a definition of ∂Cf(x) is an extended result, rather than a derivedone. When f is not necessarily Lipschitz continuous near x, ∂Cf(x) may not be compact,and may be empty. The following examples in Table 2.1 associated with a vector fieldf : R → R being smooth, Lipschitz continuous, continuous and discontinuous, respectively,illustrate Clarke’s generalized gradient according to Definition 2.4. Fig. 2.3 shows theimages of the epif and the tangent cones and normal cones of the four kinds of functionsin Table 2.1.


Table 2.1: Examples of Clark’s generalized gradient

f(x) at x = 0 ∂Cf(0)

I x2 Smooth {0}II |x| Lipschitz [-1,1]

III(a) |x| 12 Continuous (−∞,∞)

III(b) −|x| 12 Continuous {−∞,∞} (∅)

IV(a)

⎧⎨⎩1, x > 0

−1, x ≤ 0Discontinuous [0,∞)

IV(b) sgn(x) Discontinuous ∞ (∅)

Extended Stability Results

Consider the extended stability conditions for discontinuous dynamical systems (2.2)6. Inparticular, the results show how to evaluate the monotonicity of nonsmooth Lyapunovfunctions along the Filippov solution.

The following lemma shows a basic tool in developing the extended stability results.

Lemma 2.1.[20]

Let V : Rn → R be Lipschitz continuous and function ψ : R → Rn beabsolutely continuous. Define a set-valued time derivative of function V (ψ(t)) as7

˙V (ψ(t)) =

{ζ ∈ R | ζ = pT ψ, p ∈ ∂CV (ψ)

}(2.21)

thenV (ψ(t)) ∈ ˙

V (ψ(t)), a.e. t (2.22)

where ψ denotes

ψ(t) = limh→0

ψ(t + h) − ψ(t)h

, a.e. t

Definition 2.5. For closed sets Ωa ∈ Rn and Ωb ∈ Rn, denote

M(Ωa, Ωb) =⋃

φ∈Ωa

{ξ ∈ R | ξ = pT φ, p ∈ Ωb}

6System (2.2) is said to be strongly stable at the origin if for all ε > 0, there exists δ > 0 such that foreach initial condition ‖x0‖ < δ, all the Filippov solutions ‖x(t)‖ < ε, ∀t ≥ t0; is strongly asymptoticallystable at the origin if it is strongly stable and all the Filippov solutions limt→∞ x(t) = 0 as t → ∞.

7Function V is Lipschitz continuous and ψ is absolutely continuous, then the composite function V ◦ψis absolutely continuous and is differentiable a.e. [20].


0 x

f(x)

epi f

0 x

f(x)

epi f

I II

0 x

f(x)

epi f

epi f

f(x)

0

0 x

f(x)

epi f

0 x

f(x)

epi f

III(a) IV(a)

III(b) IV(b)

x

Nepi f(0,0)

Tepi f(0,0)

Nepi f(0,0)

Tepi f(0,0)

Nepi f(0,0)

Tepi f(0,0)

Nepi f(0,-1)

Tepi f(0,-1)

Nepi f(0,0)

Tepi f(0,0)

Nepi f(0,0)

Tepi f(0,0)

Fig. 2.3: Tangent cones and normal cones of four kinds of functions in Table 2.1.


Moreover for a given C ∈ R,

M(Ωa, Ωb) ≤ C ⇔ ξ ≤ C, ∀ξ ∈ M(Ωa, Ωb)

Theorem 2.1. Consider system (2.2). Let V : Rn → R be a locally Lipschitz continuousand positive-definite function. If the following condition holds

M (K[f ](x), ∂CV (x)) ≤ 0 (2.23)

then system (2.2) is strongly stable at the origin, in addition if there exists a functionW (x) ∈ K such that

M (K[f ](x), ∂CV (x)) ≤ −W (x) (2.24)

then system (2.2) is strongly asymptotically stable at the origin.

Proof. If (2.23) holds, then along any Filippov solution x(t) of system (2.2), V (x(t)) ≤ 0,a.e. t ≥ t0 by Lemma 2.1. Hence, the strong stability follows by identical demonstrationof the standard Lyapunov stability theorem with counterpart relations being true almosteverywhere instead of everywhere. Likewise, the system is strongly asymptotically stableby the condition (2.24).

Theorem 2.2. (Invariance Principle) Let Ω ⊂ Rn be a compact set such that eachFilippov solution of system (2.2) starting in Ω is unique and remains in Ω for all t ≥ t0.Let V : Rn → R be Lipschitz continuous such that

M (K[f ](x), ∂CV (x)) ≤ 0 (2.25)

Define set D asD = {x ∈ Ω | 0 ∈ M (K[f ](x), ∂CV (x))} (2.26)

Then all the Filippov solutions in Ω converges to the largest invariant set8 M in the closureof D.

Proof. By Lemma 2.1, the proof is similar to the counterpart proof of LaSalle invarianceprinciple [56] and is omitted here.

Usually, one can find a candidate of Lyapunov function V : Rn → R which is regular9.By the property of regular functions, the following conditions are proposed [20, 88].

Proposition 2.3. Let x(t), t ≥ t0 be a Filippov solution of system (2.2) and V : Rn → Rbe a Lipschitz continuous and regular function. Then,

V (x(t)) ∈ V (x), a.e. t ≥ t0 (2.27)

8A set Ω is said to be an invariant set with respect to system (2.2) if for any initial condition x0 ∈ Ω,all the Filippov solutions x(t) ∈ Ω, ∀t.

9See Definition A.3 in Appendix A for the definition of regular functions and a few helpful comments.


and

V (x) ⊆ ˙V (x) (2.28)

where the set-valued derivatives of V are defined respectively by

V (x) ={ζ ∈ R | ∃v ∈ K[f ](x) s.t. ζ = pT v, ∀p ∈ ∂VC(x)

}(2.29)

˙V (x) =

⋂p∈∂VC(x)

pTK[f ](x) (2.30)

There are extended versions (theorem 1 in [22], and theorem 3.1) of Lyapunov stabil-ity theorem, and LaSalle invariance principle (theorem 3.2 in [88]) based on Proposition2.3. The following example illustrates the application of these stability conditions to thediscontinuous dynamical systems10.

Example 2.3. Consider the dynamical system

x = f(x1, x2) =

[−sgn(−x1 + x2) − sgn(x1 + x2)

−sgn(−x1 + x2) + sgn(x1 + x2)

](2.31)

Fig. 2.4 shows its phase portrait. The associated Filippov set-valued mapping is

K[f ](x) =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

[−sgn(−x1 + x2) − sgn(x1 + x2)−sgn(−x1 + x2) + sgn(x1 + x2)

], x1 = x2 = 0

co

{[−1 − sgn(x1 + x2)1 + sgn(x1 + x2)

],

[1 − sgn(x1 + x2)1 + sgn(x1 + x2)

]}, x1 = x2 = 0

co

{[−sgn(−x1 + x2) − 1−sgn(−x1 + x2) − 1

],

[−sgn(−x1 + x2) + 1−sgn(−x1 + x2) + 1

]}, x1 = −x2 = 0

co

{[−1−1

],

[1−1

],

[−11

],

[11

]}, x1 = x2 = 0

Choose a candidate of Lyapunov function given by

V (x) = |x1 + x2| + |x1 − x2|

10See [20] for other examples like the discontinuous dynamical system (2.31) with regular Lyapunovfunctions.


which is regular in R2. The generalized gradient is

∂CV (x) =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

[sgn(x1 + x2) + sgn(x1 − x2)sgn(x1 + x2) − sgn(x1 − x2)

], x1 = x2 = 0

co

{[sgn(x1 + x2) − 1sgn(x1 + x2) − 1

],

[sgn(x1 + x2) + 1sgn(x1 + x2) + 1

]}, x1 = x2 = 0

co

{[−1 + sgn(x1 + x2)−1 − sgn(x1 + x2)

],

[1 + sgn(x1 + x2)1 − sgn(x1 + x2)

]}, x1 = −x2 = 0

co

{[−1−1

],

[1−1

],

[−11

],

[11

]}, x1 = x2 = 0

One has from (2.29) and (2.30) that

V (x) = ˙V (x) = 0, ∀(x1, x2) ∈ R2

It is clear that for all (x1, x2) ∈ Rn, V (x) ≤ 0 and ˙V (x) ≤ 0. Hence, by theorem 1 in [22]

and theorem 3.1 in [88]), system (2.12) is strongly stable at the origin.

-3 -2 -1 0 1 2 3 -3

-2

-1

0

1

2

3

x1

x2

Fig. 2.4: Phase portrait of Example 2.3.

Note that by using proper regular functions, Lyapunov stability results are less con-servative compared to Theorem 2.1 and Theorem 2.2. However, non-regular function, for


example, the following function (2.32) (see Fig. 2.5 for the level curve), is also possibleto serve as Lyapunov function for the system (2.2). Related example with non-regularLyapunov function to evaluate a designed feedback control system can be found in [5].In other words, the above mentioned results established with set-valued derivatives of aregular function V (x) are not applicable in this case.

V (x) =12x2

1 +12

(x2 + 2h(x1))2 (2.32)

with

h(x1) =

⎧⎨⎩x1, |x1| ≤ 1

1x1

, |x1| > 1

-10 -8 -6 -4 -2 0 2 4 6 8 10 -10

-8

-6

-4

-2

0

2

4

6

8

10

x2

x1

Fig. 2.5: Level curves of function (2.32).

2.1.3 L2-gain Analysis under Filippov Solution

In this section the attention is restricted to the L2-gain analysis with respect to system(2.2) with forced input and an associated output, i.e. consider the following system{

x(t) = f(x) + gw(x)w(t)

y(t) = h(x), t ≥ 0(2.33)

with x(0) = x0, where x ∈ Rn denotes the state, w : R → Rm is the input which iscontinuous in t and y ∈ Rp is the output. Vector fields f : Rn → Rn and gw : Rn → Rn×m

are discontinuous in x and locally bounded, and h : Rn → Rp with h(0) = 0 is continuous.


Denote F (t, x) = f(x)+gw(x)w(t). Consider the solution of system (2.33) in the senseof Filippov. Without loss of generality, suppose that 0 ∈ K[F ](t, 0). We focus on theL2-gain performance of the system from input w to output y.

As is well-known, L2-gain is an induced norm from the spaces of input and outputsignals. In the sense of practice, we consider the signals truncated with a sufficiently largenumber T > 0, i.e.

L2[0, T ] ={

w(t) ∈ Rm∣∣∣ (∫ T

0 ‖w(t)‖2dt) 1

2< ∞

}(2.34)

An input signal w ∈ L2[0, T ] is called admissible, if the corresponding output y belongsto L2[0, T ] along all the Filippov solutions of system (2.33) with respect to any initialcondition.

The following definition extends the classical conception of L2-gain given in [91] to thediscontinuous dynamical systems in the sense of Filippov solution.

Definition 2.6. Let γ > 0. Consider that the initial condition x0 = 0. System (2.33) issaid to have L2-gain less than or equal to γ if∫ T

0‖h(x(t))‖2dt ≤ γ2

∫ T

0‖w(t)‖2dt (2.35)

holds along all the Filippov solutions x(t) with respect to any admissible input w, whereT > 0 is a sufficient large number.

On the other hand, by the relationship between the classical L2-gain definition andthe definition of dissipativity as reported in [93], the following definition is introduced tocharacter the L2-gain of the concerned dynamical system.

Definition 2.7. Let γ > 0. System (2.33) is said to have L2-gain less than or equal to γif there exists a continuous function V : Rn → R such that

V (x) ≥ 0, ∀x ∈ Rn, V (0) = 0 (2.36)

and for any initial condition x0

V (x(t)) ≤ V (x0) +∫ t

0

12[γ2‖w(s)‖2 − ‖h(x(s))‖2

]ds, ∀t ≥ 0 (2.37)

holds along all Filippov solutions x(t) with respect to any admissible input w.

Based on the extended stability results, Theorem 2.1 and Theorem 2.2, the followingtheorem is presented to deal with stability analysis associated with L2-gain performanceof the discontinuous dynamical system.

Definition 2.8. System (2.33) is called zero-state observable if all the Filippov solutionsx(t) that satisfy y(t) ≡ 0 converge to zero when w(t) = 0.


Theorem 2.3. Consider system (2.33). Suppose that the Filippov solution with w = 0starting in Ωh = {x ∈ Rn | h(x) = 0} is unique. If there exists a positive-definite functionV : Rn → R that is locally Lipschitz continuous for all x ∈ Rn such that

M(K[F ](t, x), ∂CV (x)) ≤ 12[γ2‖w‖2 − ‖h(x)‖2

], ∀t ≥ 0 (2.38)

then system (2.33)

(i) has L2-gain less than or equal to γ;

(ii) is strongly asymptotically stable at the origin with w = 0 if in addition it is zero-stateobservable.

Proof. According to Lemma 2.1 and the Definition 2.5, we have that along any Filippovsolution x(t) of system (2.33),

V (x(t)) ∈ ˙V (x(t)) ⊆ M(K[F ](t, x), ∂CV (x)), a.e. t ≥ 0

hence, condition (2.38) implies that

V (x(t)) ≤ 12[γ2‖w‖2 − ‖h(x)‖2

], a.e. t ≥ 0 (2.39)

Then, it is straightforward to prove (i) by integrating the above inequality. We now show(ii). When w = 0, one has from condition (2.38) that

M(K[F ](x), ∂CV (x)) ≤ −12‖h(x)‖2 (2.40)

Hence, the system is strong stable at the origin by the Theorem 2.1. Moreover, for anyx(t) that satisfies h(x) = 0 and x(t) ∈ K[F ](x), a.e. t ≥ 0, x(t) converge to the origin,since system (2.33) is zero-state observable. In other words, set Ωh is positively invariantset with respect to system (2.33) with zero input. Furthermore, condition (2.40) impliesthat the following set

ΩV = {x ∈ Rn | M(K[F ](x), ∂CV (x)) = 0} ⊆ Ωh

is invariant set. Therefore, all the Filippov solution x(t) will converge to the set Ωh andfinally converge to the origin by the invariant principle, Theorem 2.2, and one now canconcludes that system (2.33) with w = 0 is strongly asymptotically stable at the origin.

2.2 Lyapunov-based Feedback Stabilizing Control Design

With the preliminary in Section 2.1.1, we now address the main issue of this chapter. Wewill present a Lyapunov-based approach to design feedback stabilization control laws fordynamical system (2.1) with discontinuous vector field.

2.2. Lyapunov-based Feedback Stabilizing Control Design 27

First, a state feedback controller is designed for a nonlinear system (2.1) with matcheddiscontinuity. Strong asymptotic stability of the closed-loop system is achieved by meansof smooth Lyapunov functions. Due to the discontinuities, it is natural to result in non-differentiable or even discontinuous control laws. Then, design approaches are given toobtain a smooth or Lipschitz continuous at least control laws by picking particular char-acterizations in several kinds of the considered discontinuous dynamical systems. Further-more, the characterization of using Lyapunov recursive design approach to discontinuousdynamical systems is illustrated in detail as the other purpose of this section. An exten-sion is also deduced for the interests in sliding mode control. Finally, the main results isapplied to linear piecewise continuous systems.

2.2.1 Stabilization Design

Consider a nonlinear system (2.1) in the following form

x(t) = f(x) + d(x)δ(x) + g(x)v, t ≥ 0 (2.41)

with x(0) = x0, where x ∈ Rn and v ∈ R are the state and the control input, respectively.δ : Rn → R denotes a discontinuous function in x. f(x), d(x) and g(x) are Lipschitzcontinuous in x. Moreover, the free system has static trivial solution, i.e. 0 ∈ K[F ](0),where F (x) = f(x) + d(x)δ(x).

The stabilization problems considered in this section are as follows.

Problem 2.1. For system (2.41), find a feedback control law

v = Φ(x) (2.42)

with 0 ∈ K[Φ](0)such that the closed-loop system is strongly asymptotically stable at theorigin.

Furthermore, suppose that the control input v of system (2.41) is provided by anintegrator, i.e.

v = u (2.43)

Problem 2.2. For system (2.41) with (2.43), find a feedback control law

u = α(x, v) (2.44)

with 0 ∈ K[α](0) such that the closed-loop system with (2.44) is strongly asymptoticallystable at the origin.

Actually, it can be noted that Problem 2.2 deals with integrator backstepping designapproach to the system (2.41) with (2.43).


On the basis of the presented analysis tools in section 2.1.2, we first show a solutionto Problem 2.1 to system (2.41) that satisfies the following matching condition

d(x) = g(x)η(x) (2.45)

where η(x) : Rn → R is Lipschitz continuous function in x.

The Filippov solution x(t) of the closed-loop system (2.41) with (2.42) satisfies thefollowing differential inclusion

x(t) ∈ K[F ](x), a.e. t ≥ 0 (2.46)

where F (x) = f(x) + d(x)δ(x) + g(x)Φ(x). Without loss of generality, the assumption ismade:

Assumption 2.1. For the dynamical systems x = f(x), which is the nominal systemof system (2.41), there exists a continuous differentiable, positive-definitive function V1 :Rn → R and W (x) ∈ K such that

LfV1(x) ≤ −W (x), ∀x ∈ Rn (2.47)

Then, the following theorem provides a solution to the Problem 2.1.

Theorem 2.4. Under Assumption 2.1, system (2.41) is strongly asymptotically stable atthe origin by the feedback control law (2.42) given with

Φ(x) = −k1sign(LgV1(x))|η(x)|γ∗δ (x) (2.48)

where k1 ≥ 1 is any constant and

γ∗δ (x) = max

ς∈K[δ](x){|ς|} (2.49)

Proof. Choose V1(x) as a candidate of Lyapunov function for the closed-loop system. Since

V1(x) is continuous differentiable, for a Filippov solution x(t) which satisfies (2.46), ˙V 1(x)

from Lemma 2.1 reduces to a singleton. According to the calculation rule (PropositionA.1), we have K[F ](x) ⊂ f(x) + d(x)K[δ](x) + g(x)K[Φ](x). Then by Definition 2.5, wehave that

M(K[F ](x), ∂CV1(x)) =∇V T1 (x)K[F ](x)

⊂LfV1(x) + LdV1(x)K[δ](x) + LgV1(x)K[Φ](x)(2.50)

One can obtain from the feedback control law (2.48) that

LgV1(x)K[Φ](x) = −k1|LgV1(x)||η(x)|K[γ∗δ ](x) (2.51)

Substituting (2.51) into (2.50) results in

M(K[F ](x), ∂CV1(x)

) ⊂ LfV1(x) + LhV1(x)K[δ](x) − k1|LgV1(x)||η(x)|K[γ∗δ ](x)


Then Assumption 2.1 together with the condition (2.45) guarantees that for any ξV1 ∈M(K[F ](x), ∂CV1(x)), we have

ξV1 ≤ −W (x) + |LgV1||η(x)|γ∗δ (x) − k1|LgV1||η(x)|γ∗

δ (x) (2.52)

whereγ∗

δ (x) = maxς∈K[γ∗

δ ](x){ς} ≥ 0

By the fact, γ∗δ (x) ≤ γ∗

δ (x), inequality (2.52) reduces to

ξV1 ≤ −W (x) − (k1 − 1)|LgV1||η(x)|γ∗δ (x), ∀ξV1 ∈ M(K[F ](x), ∂CV1(x)) (2.53)

Hence by Theorem 2.1, the closed-loop system is strongly asymptotically stable at theorigin.

Theorem 2.4 provides a general feedback design approach to stabilize a discontinuousdynamical system. It should be observed that for the discontinuous system (2.41), thefeedback control law (2.48) is not continuous in general. Often, one can encounter thesystems under consideration have special properties that may allow proper designs toachieve the control purpose. The following two theorems give two results of the abovestatement.

Theorem 2.5. Suppose system (2.41) with the following condition,

H1: the function γ∗δ (x) defined by (2.49) is Lipschitz continuous, and there exist a con-

tinuous differentiable, positive-definite function V1 : Rn → R and W (x) ∈ K suchthat

LfV1(x) +12η2(x) ≤ −W (x), ∀x ∈ Rn (2.54)

Then system (2.41) is strongly asymptotically stable at the origin by the feedback controllaw (2.42) given with

Φ(x) = −k1LgV1(x) (γ∗δ (x))2 (2.55)

where k1 ≥ 1/2 is any constant.

Proof. Choose V1(x) as the candidate of Lyapunov function for the closed-loop system.The condition γ∗

δ (x) is Lipschitz continuous in H1 indicates that the feedback control lawΦ(x) given by (2.55) is Lipschitz continuous. In this case, we have

M(K[F ](x), ∂CV1(x)) = LfV1(x) + LhV1(x)K[δ](x) + LgV1(x)Φ(x) (2.56)

Then with condition (2.45), we have that for any ξV1 ∈ M(K[F ](x), ∂CV1(x)), the followinginequality can be obtained.

ξV1 ≤LfV1(x) + |LgV1||η(x)|γ∗δ (x) + LgV1(x)Φ(x)

≤LfV1(x) +12[(LgV1(x)γ∗

δ (x))2 + η2(x)]+ LgV1(x)Φ(x)

(2.57)


Substituting Φ(x) by (2.55) into the above inequality and taking into account the condition(2.54) in H1 yield

ξV1 ≤ −W (x) −(

k1 − 12

)[LgV1(x)γ∗

δ (x)]2 , ∀ξV1 ∈ M(K[F ](x), ∂CV1(x)) (2.58)

Hence the strong asymptotic stability at the origin of the closed-loop system follows byTheorem 2.1.

Theorem 2.6. Suppose system (2.41) with the following condition,

H2: there exist a continuous differentiable, positive-definite function V1 : Rn → R andW (x) ∈ K such that

LfV1(x) +12

(γ∗δ (x))2 ≤ −W (x), ∀x ∈ Rn (2.59)

Then system (2.41) is strongly asymptotically stable at the origin by the feedback controllaw (2.42) with

Φ(x) = −k1LgV1(x)η2(x) (2.60)

where k ≥ 1/2 is any constant.

Proof. It is clear that the feedback control law (2.60) is designed to be directly Lipschitzcontinuous. The proof is proceed similarly to Theorem 2.5. It is given briefly as follows.

Take V1(x) as the candidate Lyapunov function. For any ξV1 ∈ M(K[F ](x), ∂CV1(x))(given by (2.56)), the following inequality can be deduced.

ξV1 ≤ LfV1(x) +12[(LgV1(x)η(x))2 + (γ∗

δ (x))2]+ LgV1(x)Φ(x) (2.61)

Substituting Φ(x) by (2.60) into the above inequality and taking into account thecondition (2.59) in H2 yield

ξV1 ≤ −W (x) −(

k1 − 12

)[LgV1(x)η(x)]2 , ∀ξV1 ∈ M(K[F ](x), ∂CV1(x)) (2.62)

One can conclude that the closed-loop system is strongly asymptotically stable at theorigin by Theorem 2.1.

The above results actually give the LgV -type formula applied to discontinuous dy-namical system. Moreover, the condition 0 ∈ K[Φ](0) is a natural conclusion, sincesgn(LgV1(0)) = 0 in the feedback control law (2.48).

Now we address Problem 2.2. On the basis of the feedback control law (2.48) byTheorem 2.4, we show how to solve the Problem by using a Lyapunov recursive designapproach. In this case, for the class of system (2.41) associated with (2.43), the followingassumption is made.


Assumption 2.2. 11 For system (2.41), the function defined by (2.49) with respect tothe discontinuous function δ(x) is Lipschitz continuous for all x ∈ Rn.

Combining Theorem 2.4 and Assumption 2.2, we have the following result.

Theorem 2.7. Consider system (2.41) with (2.43) that satisfies Assumption 2.1 andAssumption 2.2. Let Ωg = {x ∈ Rn | LgV1(x) = 0} and Ω1 = {x ∈ Rn | η(x)γ∗

δ (x) = 0}.Suppose that for all the solutions x(t) of (2.46), Ωg ⊆ Ω1. Let the feedback control law(2.44) be given by

u = α(x, v) =−LgV1(x) − k2(v − Φ(x)) − k3k1sgn(v − Φ(x))(p∗η(x)γ∗

δ (x) + p∗δ(x)|η(x)|)H(x)(2.63)

then, the closed-loop system is strongly asymptotically stable at the origin, i.e. for anygiven initial condition (x0, v0), the system trajectories x(t) → 0, v(t) → 0 as t → ∞,where H(x) = ‖f(x) + d(x)δ(x) + g(x)v‖, k2 > 0 and k3 ≥ 1 are any constants, and

p∗η(x) = maxpη∈∂C |η(x)|

{‖pη‖}, p∗δ(x) = maxpδ∈∂Cγ∗

δ (x){‖pδ‖} (2.64)

Proof. The associated differential inclusion of the closed-loop system is[xv

]∈ K[Fc](x, v) =

[f(x) + d(x)K[δ](x) + v

K[α](x, v)

], a.e. t ≥ 0 (2.65)

First, it is clear by Theorem 2.4 that the x-subsystem of (2.65) is strongly stabilizedby the feedback control law (2.48).

Then, following the procedure in the proof of Theorem 2.4, we analyze the behavior ofsystem (2.65) with a Lyapunov function constructed as

V2(x, v) = V1(x) +12

(v − Φ(x))2 (2.66)

Obviously, the Lyapunov function (2.66) is not smooth due to the involved feedback controllaw Φ(x) to the x-subsystem, and V2(x, v) cannot be calculated in the conventional sense.On the other hand, Φ(x) is Lipschitz continuous for all x ∈ Rn from Assumption 2.2 andthe condition Ωg ⊆ Ω1. We start the proof from investigating how to describe Φ(x) alongthe solutions of (2.65).

According to Lemma 2.1, with respect to any Filippov solution x(t), we have that

Φ(x) ∈ ˙Φ(x) ={ζφ ∈ R | ζφ = pT

φ x, pφ ∈ ∂CΦ(x)}

, a.e. t ≥ 0 (2.67)

11It can be noted that for a function δ(x), Assumption 2.2 is automatically satisfied if it satisfies| lim

x→x−i

δ(x)| = | limx→x+

iδ(x)|, xi ∈ Sδ, where Sδ denotes the set of points where δ(x) is discontinuous.


with ∂CΦ(x) which by Definition 2.3 and the computation rule of Clarke’s generalizedgradient can be calculated as

∂CΦ(x) ⊂ {pφ(x) ∈ Rn | pφ(x) = −k1

(pη(x)γ∗

δ (x) + pδ(x)|η(x)|),pη(x) ∈ ∂C |η(x)|, pδ(x) ∈ ∂Cγ∗

δ (x)} (2.68)

which gives that for any ζφ ∈ ˙Φ(x),

ζφ ≤ k1

(p∗η(x)γ∗

δ (x) + p∗δ(x)|η(x)|)H(x) (2.69)

Moreover, Lyapunov function (2.66) is Lipschitz continuous due to the function Φ(x).In this case, we have that for any Filippov solution (x(t), v(t)) of the closed-loop system,

˙V 2(x, v) =

{ζV2 ∈ R | ζV2 = [∇V T

1 (x) − pTφ (x)(v − Φ(x))]x + (v − Φ(x)) v,

pφ(x) ∈ ∂CΦ(x)} (2.70)

With the feedback control law (2.63), one obtains that

(v − Φ(x)) v = (v − Φ(x)) u

=− (v − Φ(x)) LgV1(x) − k2 (v − Φ(x))2 − kz2k1| (v − Φ(x)) |(p∗η(x)γ∗

δ (x) + p∗δ(x)|η(x)|)H(x)

(2.71)

With the argument of Theorem 2.4, combining (2.53), (2.69) and (2.71), it can be deduced

from (2.70) that for any ζV2 ∈ ˙V 2(x, v),

ζV2 ≤−W (x) − k2 (v − Φ(x))2 − (k1 − 1)|LgV1(x)||η(x)|γ∗δ (x)

−(k3 − 1)k1| (v − Φ(x)) | (p∗η(x)γ∗δ (x) + p∗δ(x)|η(x)|)H(x)

(2.72)

Inequality (2.72) implies that

∃W (x, v) ∈ K, s.t. ζV2 ≤ −W (x, v), ∀ζV2 ∈ ˙V 2(x, v) (2.73)

Finally, the above argument covers the fact that ξV2 ≤ −W (x, v), ∀ξV2 ∈ M(K[F ](x, v),∂CV2(x, v)), since the condition (2.73) is true for any Filippov solution of the closed-loopsystem (2.65). This completes the proof by Theorem 2.1.

Remark 2.2. It is well-known that one introduces a change of state variable

v = v − Φ(x) (2.74)

for backstepping design when the feedback law Φ(x) is obtained such that the system (2.41)is strong asymptotically stable. Then, the design follows always by taking a Lyapunovfunction candidate constructed as

V2(x, v) = V1(x) +12v2 (2.75)


which is smooth. However, unlike the counterpart in the situation of smooth systems, onecannot evaluate the Lyapunov function candidate (2.75) in the solution of the closed-loopsystem (2.65), since we investigate the system behavior in the sense of Filippov solutionand it is clear that the differential inclusion with respect to variable v cannot be achievedimmediately for the considered discontinuous system (2.41) with (2.43). Hence, instead of(2.75), (2.66) is taken as the candidate of Lyapunov function to prove Theorem 2.7.

With the same Lyapunov function (2.66), the following extended result can be obtainedfor the considered discontinuous systems12.

Theorem 2.8. Consider system (2.41) with (2.43) that satisfies Assumption 2.1 andAssumption 2.2, and suppose that condition Ωg ⊆ Ω1 (Defined in Theorem 2.4) holds.Then the following feedback control law

u = −k4sgn(v)[|LgV1(x)| + k1

(p∗η(x)γ∗

δ (x) + p∗δ(x)|η(x)|)H(x)]

(2.76)

with any constant k4 > 1 guarantees the strong asymptotic stability of system (2.41), (2.43)with (2.76) at the origin, and renders the trajectory of the system along a smooth surfaceS(x, v) = 0, (x, v) ∈ Ωs = {(x, v) ∈ Rn+1 | v(x, v) = 0}, i.e. a sliding motion.

Proof. According to the proof of Theorem 2.7, it can be derived straightforwardly that fora Filippov solution (x(t), v(t)) of the closed-loop system, we have

˙v(x, v) ∈ Ωv ={

ζv ∈ R | ζv = u − ζφ, ζφ ∈ ˙Φ(x)}

, a.e. t ≥ 0 (2.77)

The set-valued derivative (2.70) of the Lyapunov function (2.66) means that Ωs is an in-

variant set with respect to any v(x, v) satisfying the condition (2.77), since 0 ∈ ˙V 2(x, v),

∀(x, v) ∈ Ωs. Furthermore, note that the vector field of the closed-loop system is dis-continuous on the surface S(x, v) = 0 which divides the state space into the domainsΩ+

s = {(x, v) ∈ Rn+1 | v(x, v) > 0} and Ω−s = {(x, v) ∈ Rn+1 | v(x, v) < 0}. For the

(x, v) on the surface S(x, v) = 0, the normal vector is Ns(x, v) = [0 1]T , and the projec-tions of the limiting vectors F+

c (x, v) and F−c (x, v) reached S(x, v) = 0 from ω+

s and ω−s ,

respectively are ⎧⎪⎨⎪⎩F+

Ns(x, v) = NT

s (x, v)F+(x, v) = limv(x,v)→0+

ζv

F−Ns

(x, v) = NTs (x, v)F−(x, v) = lim

v(x,v)→0−ζv

With the control law (2.76) and by (2.69), it is clear that for all ζv ∈ Ωv, limv→0+ ζv < 0 andlimv→0+ ζv > 0. This means that F+

Ns(x, v) < 0, F−

Ns(x, v) > 0 and F−

Ns(x, v)−F+

Ns(x, v) >

0. Therefore, on the surface S(x, v) = 0, sliding motion occurs by Proposition 2.1. More-over on the surface S(x, v) = 0, x(t) → 0 as t → ∞ by Theorem 2.4. Furthermore,v(t) → 0 as t → ∞ by the Invariance Principle, Theorem 2.2, i.e. M = {(x, v) ∈ Rn+1 |x = 0, v = 0} is the largest invariant set in Ωs.

12The above presented result is inspired by the Lyapunov recursive design approach which is in terms ofsliding model control introduced in [90], and covers the original method in dealing with control problemswith respect to the smooth systems.


The solutions presented above to the Problem 2.2 are with respect to the Theorem 2.4.It is easy to confirm that a solution can be achieved by means of the results of Theorem2.5 and Theorem 2.6, respectively.

Remark 2.3. Theorem 2.4 and Theorem 2.7 present a Lyapunov recursive design ap-proach for the considered discontinuous systems by using a smooth Lyapunov functionV1(x) and a Lipschitz continuous Lyapunov function V2(x, v). Moreover, it can be observedthat by means of the nonsmooth analysis tool, Clark’s generalized gradient, the conven-tional Lyapunov recursive design approach can be extended to deal with the feedbackstabilization problems of discontinuous dynamical systems, although nonsmooth feedbackcontrol action is employed during the recursive design procedures.

Remark 2.4. It should be pointed out that the proposed Lyapunov recursive design ap-proach benefits from finding a Lipschitz continuous feedback control action Φ(x), to whichthe Clark’s generalized gradient is computable and proper to be applied to system stabil-ity analysis and synthesis. Otherwise, by the extended computation rule of generalizedgradient Definition 2.4, ∂CΦ(x) of function Φ(x) that is non-Lipschitz at x may not becompact or may be empty.In this case the feedback control law (2.44) will fail to guaranteethe sufficient condition of the stability theorem, Theorem 2.1.

Remark 2.5. Indeed, one step recursive design is performed in the above. To proceed, thecontrol law (2.44) should be Lipschitz continuous which however is usually discontinuous.On the other hand, it is straightforward that the system (2.41) with (2.43) can be stronglyasymptotically stabilized by the feedback control law 2.44 with a minor modification givenbelow.

u =−LgV1(x) − k2v − k3k1sgn(v)(p∗η(x)γ∗

δ (x) + p∗δ(x)|η(x)|)(‖f(x)‖ + ‖d(x)‖γ∗δ (x) + ‖g(x)v‖) (2.78)

which is more conservative actually. Moreover, the control law (2.78) is Lipschitz continu-ous if, similar to the condition Ωg ⊆ Ω1, the condition Ωv ⊆ Ω2 holds, where Ωv = {(x, v) ∈Rn+1 | v(x, v) = 0} and Ω2 = {(x, v) ∈ Rn+1 | (p∗η(x)γ∗

δ (x) + p∗δ(x)|η(x)|)(‖f(x)‖ +‖d(x)‖γ∗

δ (x) + ‖g(x)v‖) = 0}.Example 2.4. Consider the following discontinuous system{

x = −x + (x2 + x)sgn(x − 1) + v

v = u(2.79)

which accordance with the structure of system (2.41) and (2.43) with notations

f(x) = −x, d(x) = x2 + x, δ(x) = sgn(x − 1), g(x) = 1, η(x) = x2 + x

A Lyapunov function V1(x) = 1/2x2 satisfies the condition of Assumption 2.1. Then,according to Theorem 2.4, we design the following feedback control law for system (2.79)

Φ(x) = −k1sgn(x)|x2 + x| (2.80)


with γ∗δ (x) = 1, a smooth function, such that the x-subsystem is strongly asymptotically

stable. Furthermore, we can validate that the feedback control action (2.80) is Lipschitzcontinuous for all x ∈ R, since Ωg = {0} ⊂ Ω1 = {0, −1}. Moreover, it can be computedthat

∂CΦ(x) =

{[−1, 1], x = −1

2x + 1, othersTherefore, from Theorem 2.7, we design the following desired feedback control law

u = −x−k2 (v − Φ(x))−k3k1sgn (v − Φ(x)) |2x+1| · |−x+(x2 +x)sgn(x−1)+v| (2.81)

and from Theorem 2.8, the sliding mode control law can be designed as

u = −k4sgn (v − Φ(x))[|x| + k1|2x + 1| · | − x + (x2 + x)sgn(x − 1) + v|] (2.82)

where p∗η(x) = |2x + 1| and p∗δ(x) = 0 from (2.64).

The system is simulated by Simulink with fixed step 0.001. Choosing k1 = 1.2, k2 = 2and k3 = 1, Fig. 2.6 shows the trajectory starting from (x0, v0) = (2, 0) and the curveof the feedback control law (2.81). The responses indicate that the proposed feedbackcontrol law (2.81) guarantees that the system trajectories converge to the origin. For thesimulation under the control law (2.82), we choose k1 = 1.5, k4 = 1.1 and the initialconditions (x0, v0) = (1, 0), (x0, v0) = (−0.5, 1.5) and (x0, v0) = (0.5,−2.5), respectively.The phase portrait shown in Fig. 2.7 validates the sliding motion and convergency of theclosed-loop system. �

0 0.5 1 1.5 2 2.5 3-10

-5

0

5

0 0.5 1 1.5 2 2.5 3-60

-40

-20

0

20

40

Time [s]

Time [s]

u

x

v

Fig. 2.6: Simulation result of system (2.79) under control law (2.81).


-0.5

0

0.5

1

-3-2

-10

12

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

3

3.5

vx

v

Fig. 2.7: Phase portrait of system (2.79) under control law (2.82).

2.2.2 Piecewise Continuous Linear Systems

In this section, we focus on the piecewise continuous linear systems. As a special case ofnonlinear system (2.1), consider that the system dynamics is given by

x = Aix + Bv, Ri, i = 1, · · · , n (2.83)

Meanwhile, consider that the system (2.83) is also cascaded with the following system

v = u (2.84)

For discontinuous dynamical system in the form

x = Aix (2.85)

it is known that the strong stability is guaranteed if all the vector fields Aix have acommon Lyapunov function [66, 92], which in this case, is a Lyapunov function of anyconvex combination of the Filippov differential inclusion

K[Aix] = {�Ax | �Ax = Σni=1αiAix, αi ≥ 0, Σn

i=1αi = 1}

Consider that for any i ∈ {1, · · · , n}, system (2.83) can be represented by

Aix = Ax + Aδix + Bv, x ∈ Ri (2.86)


with matching conditionAδi

x = BGTδi

x (2.87)

where Gδi∈ Rn. The following will show that the feedback design procedure for system

(2.41) with (2.43) will generate a state-based switching control law u = α(x, v) by means ofa Lipschitz continuous Lyapunov function such that system (2.83) with (2.84) is stronglyasymptotically stable at the origin.

First, for the discontinuous linear system (2.83), we may find a quadratic commonLyapunov function V1 = 1/2xT Px with positive-definite matrix P . Similarly, assume thatthere exist positive-definite matrixes P and Q such that

AT P + PA + Q ≤ 0 (2.88)

By Theorem 2.4, a feedback control law v = Φ(x) with

Φ(x) = −k1sgn(xT PB)G∗δ(x), G∗

δ(x) = maxi∈{1,··· ,n}

{|GTδi

x|} (2.89)

exists that guarantees the strong stability of the closed-loop system (2.83) with (2.84).

Noted that the control law Φ(x) given by (2.89) is immediately Lipschitz continuous.Hence with (2.84), Theorem 2.7 gives the following control law

u = α(x, v) = −xT PB − k2v − k3k1sgnvp∗g(x)‖Aix + Bv‖, x ∈ Ri (2.90)

wherep∗g(x) = max

pg∈∂CG∗δ(x)

{‖pg‖}

such that the discontinuous linear system (2.83), (2.84) with (2.90) is strongly asymptoti-cally stable at the origin by the Lyapunov function V2(x, z) = 1/2xT Px + 1/2(v −Φ(x))2.

In fact, it is easy to confirm that in the case of linear piecewise continuous system, theγ∗

δ (x) = G∗δ(x) is Lipschitz continuous. Hence, by Theorem 2.5, feedback control law v for

a linear system (2.83) can be designed with

Φ(x) = −k1xT PB(G∗

δ(x))2 (2.91)

Theorem 2.9. For the system (2.83) with (2.84), if condition (2.88) is satisfied, then theclosed-loop system with feedback control law (2.90) is strongly asymptotically stable at theorigin.

Remark 2.6. It should be noted that the selected matrix A with respect to system (2.83)effects the above design result. In other words, an additional freedom of degree exists tothe resulting closed-loop system.

The following example illustrates Theorem 2.9.


Example 2.5. Consider the following linear piecewise system⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

x = x + v, x > 2

x = −x + v, x = 2

x = −2x + v, x < 2

v = u

(2.92)

The system can be represented by (2.86) and (2.87) with

A = −1, Aδ1 = Gδ1 = 2, Aδ2 = Gδ2 = 0, Aδ3 = Gδ3 = −1

Take V1(x) = 1/2x2. The following feedback control law can stabilize the system to (0, 0)asymptotically

u = α(x, v) =

⎧⎪⎪⎨⎪⎪⎩−x − k2 (v − Φ(x)) − 2k3k1sgn(v − Φ(x))|x + v|, x > 2

−x − k2 (v − Φ(x)) − 2k3k1sgn(v − Φ(x))| − x + v|, x = 2

−x − k2 (v − Φ(x)) − 2k3k1sgn(v − Φ(x))| − 2x + v|, x < 2

withΦ(x) = −k1sgn(x)|2x| = −2k1x

where G∗δ(x) = |2x| and p∗g(x) = 2. Furthermore, it is evident that according to Remark

2.5, a feedback stabilization control law for system (2.92) can be designed as

u = −x − k2 (v − Φ(x)) − 4k3k1sgn(v − Φ(x))(|x| + |v|) (2.93)

which is smooth everywhere except for the origin. Moreover, taking A = −1/2 for example,one can demonstrates Remark 2.6. �

2.3 L2-gain Control Design

This section addresses the feedback design problem with respect to L2-gain performancefor the discontinuous dynamical systems. Consider dynamical systems described by{

x(t) = f(x) + gw(x)w(t) + g(x)u,

y(t) = h(x), t ≥ 0(2.94)

where x ∈ Rn denotes the state, w : R → Rm is the disturbance input which is continuousin t, and u ∈ R is the control input.

For system (2.94), the L2-gain associated feedback design problem is formulated asfollows.

2.3. L2-gain Control Design 39

Problem 2.3. (L2-gain Synthesis Problem) For a given γ > 0, find, if exists, a statefeedback control law

u = α(x) (2.95)

such that the closed-loop system is strongly asymptotically stable at the origin when w = 0,and has L2-gain less than or equal to γ from w to y, i.e. by Definition 2.6, for any T ≥ 0,when x(0) = 0, the following inequality holds.∫ T

0‖h(x(t))‖2dt ≤ γ2

∫ T

0‖w(t)‖2dt, ∀w ∈ L2[0, T ]

2.3.1 Hamilton-Jacobi Inequality Characterization

It is known that for system (2.33), if the mappings f(x) and gw(x) are smooth, thenthere exists a continuous differentiable function V (x) ≥ 0 satisfies the following HJI is asufficient condition that system (2.2) has L2-gain less than or equal to γ.

LfV (x) +1

2γ2‖LgωV (x)‖2 +

12‖h(x)‖2 ≤ 0 (2.96)

Most of the feedback design approaches are established based on this essential equivalencebetween L2-gain performance and the HJI (2.96). Fortunately, solutions of the HJI canusually play the role of Lyapunov function for evaluating the stability of correspondingunforced systems under appropriate conditions [91, 93].

The main results of this section is given by generalizing the classical approaches bymeans of the condition of Theorem 2.3.

Theorem 2.10. Suppose that system (2.33) is zero-stable observable. If for a given γ > 0,there exist a Lipschitz continuous, positive-definitive function V : Rn → R such that

L∗FV (x) +

12γ2

[L∗QV (x)

]2 +12‖h(x)‖2 ≤ 0 (2.97)

holds, then system (2.33) is strongly asymptotically stable at the origin as w = 0, and hasL2-gain less than or equal to γ from w to y, where

L∗FV (x) = max

LFV (x)∈M(K[f ](x),∂CV (x)

) {LFV (x)} (2.98)

L∗QV (x) = max

LQV (x)∈M(K[gw](x),∂CV (x)

) {‖LQV (x)‖} (2.99)

Proof. According to the calculating rules of Filippov set-valued mapping Proposition A.1,the following conclusion follows

K[F ](t, x)⊂K[f ](x) + K[gww](t, x)

= K[f ](x) + w(t)K[gw](x)(2.100)


Along any Filippov solution x(t) of system (2.33), one has from (2.100) that

V (x(t))∣∣x(t)∈K[F ](t,x) ∈ ˙

V (x(t)), a.e. t ≥ 0 (2.101)

where˙V (x(t)) =

{ζ ∈ R | ∃f ∈ K[f ](x), gw ∈ K[gw](x), s.t. ζ = pT

V f + wpTV gw, pV ∈ ∂CV (x)

}For any pV ∈ ∂CV (x), it can be derived directly by (2.98) that

pTV f + wpT

V gw ≤ L∗FV (x) + wpT

V gw

In view of HJI (2.97), the above inequality satisfies that

pTV f + wpT

V gw

≤ −12‖h(x)‖2 − 1

2γ2[L∗

QV (x)]2 +1

2γ2‖pT

V gw‖2 +γ2

2‖w‖2 − γ2

2

∥∥∥∥w − 1γ2

pTV gw

∥∥∥∥2

≤ 12[γ2‖w‖2 − ‖h(x)‖2

]which indicates that

ζ ≤ 12[γ2‖w‖2 − ‖h(x)‖2

], ∀ζ ∈ ˙

V (x(t))

Furthermore due to (2.100), it is clear that

M(K[F ](t, x), ∂CV (x)) ⊆⋃

x(t) s.t. x(t)∈K[F ](t,x)

˙V (x(t))

Thus, the conclusion follows by Theorem 2.3.

It should be emphasized that it is a special case when the function V (x) for the HJI(2.98) is continuous differentiable.

2.3.2 L2-gain Controller

In this subsection the feedback design problem of L2-gain performance controller for thediscontinuous dynamical system is investigated by utilizing the HJI based approach. Weconsider that system (2.94) is represented by{

x(t) = f(x) + d(x)δ(x) + gw(x)w(t) + g(x)u,

y(t) = h(x), t ≥ 0(2.102)

which satisfies the matching condition (2.45), where f(x), d(x) and g(x) are Lipschitzcontinuous functions with appropriate dimensions.

On the basis of the stabilization design idea of Section 2.1.1, Theorem 2.4, one canprove the following by Theorem 2.11.

2.3. L2-gain Control Design 41

Theorem 2.11. Suppose that system (2.102) satisfies the following conditions:

H1: it is zero-state observable13;

H2: there exists a Lipschitz continuous, positive-definite function V : Rn → R such that

L∗FV (x) +

12γ2

[L∗QV (x)

]2 +12‖h(x)‖2 ≤ 0 (2.103)

Then a solution of the L2-gain performance design problem is

u = α(x) = −k1sgn(L∗GV (x)

) |η(x)|γ∗δ (x) (2.104)

where k1 ≥ 1 is any constant and

L∗GV (x) = max

pV ∈∂CV (x){pT

V g(x)}

Proof. Although, Lipschitz continuous function V (x) is utilized, combined with Theorem2.10, the proof is a similar procedure with Theorem 2.4.

Furthermore, it is straightforward to conclude the following for the case of f(x) beingdiscontinuous.

Corollary 2.1. Let f(x) in system (2.102) is discontinuous. Feedback control law (2.107)is a solution to the L2-gain performance design problem if system (2.102) satisfies theconditions H1 and H2.

Meanwhile, if there is the matching condition between the channels of control inputand disturbance input, i.e. there exists proper gw(x) such that

gw(x) = g(x)gw(x) (2.105)

then similar arguments with Theorem 2.5 allow one to proof the following result by The-orem 2.10.

Theorem 2.12. Suppose that system (2.102) satisfies the conditions (2.105), H1 and

H3: there exists a Lipschitz continuous, positive-definite function V : Rn → R such that

L∗FV (x) +

12η2(x) +

12‖h(x)‖2 ≤ 0 (2.106)

Then a solution of the L2-gain performance design problem is

u = α(x) = −k1

[12L∗GV (x)γ∗

δ (x) +1

2γ2L∗GV (x)‖gw‖2

](2.107)

where k1 ≥ 1/2 is any constant.13System (2.102) is said to be zero-state observable if all the Filippov solutions x(t) that satisfy h(x) = 0

converge to zero as u = 0 and w = 0.


2.4 Concluding Remarks

In this chapter, a methodology has been developed for the feedback stabilization and L2-gain synthesis problems applied to discontinuous dynamical systems. It aims at presentingfundamental theories and tools to deal with the analysis and synthesis of dynamical sys-tems in the sense of Filippov solution. The main results are established based on theLyapunov stability theory. For solving the feedback design problems, using nonsmoothfunctions to play the role of Lyapunov functions are investigated on the basis of the non-smooth analysis theory.

The basic idea of the presented design approach is to dominate the set-valued timederivative of Lyapunov function to be negative. It should be noted that the set-valued timederivative introduced in this chapter is much wider than the generalized time derivative [22,88], hence, dominating all members of the set might cause conservativeness in the feedbackdesign. However, comparing with the well-known general time derivative proposed in[22, 88], the presented design approach does not require the Lyapunov function to beregular. Furthermore, as shown in Theorem 2.7, the presented approach can be extendedto integrator-based recursive design problem for a class of discontinuous nonlinear systems.

Future developments will concern the adaptive control theory for the discontinuousdynamical systems involve uncertainties. It is believed that the presented design methodcan give feasible ways to extend the classical adaptive control methods to discontinuousdynamical systems.

Chapter 3

Analysis and Control of PiecewiseContinuous Time-delay Systems

The objective of this chapter is to establish functional differential inclusion based theo-retical tools from the view of system control and present an extended Filippov frameworkin Chapter 2 applicable to deal with analysis and synthesis problems of discontinuoustime-delay systems.

In Section 3.1, a new definition of solution for discontinuous functional differential equa-tion is given with respect to an associated functional differential inclusion formulated inthe sense of Filippov. Then, Lyapunov-Krasovskii theorem and LaSalle invariance princi-ple in the sense of Filippov solution are extended to evaluate the behavior of discontinuoustime-delay systems. Moreover, L2-gain property with related conditions are also summa-rized. On the basis of the above theoretical preparation, feedback design approaches aredeveloped to address the stabilization and L2-gain synthesis problems in Section 3.2 andSection 3.3, respectively. A summary is given in Section 3.4.

3.1 Preliminaries

Consider discontinuous time-delay systems represented by the following functional differ-ential equation

x(t) = f(xt), t ≥ t0 (3.1)

with xt0(τ) = φ(−τ), τ ∈ [0, r] (r > 0), where x ∈ Rn, xt ∈ Cr is defined by xt(τ) = x(t−τ)(xt(τ) denotes the value of xt at τ .) and φ(τ) ∈ Cr. The functional f : Cr → Rn is locallybounded and is discontinuous in xt.

43

44 Chapter 3. Analysis and Control of Piecewise Continuous Time-delay Systems

3.1.1 Extended Filippov Solution

Consider the case that the functional f of (3.1) is continuous everywhere except for onseveral surfaces defined by

Si = {φ ∈ Cr | Si(φ(−τ)) = 0, τ ∈ [0, r]} ⊂ Cr, i = 1, · · · , n (3.2)

where functional Si : Cr → R are smooth.

Let Cri , (i = 1, · · · , n) be the partitions of the space Cr, i.e. ∪i=ni=1Cri = Cr. On the

boundary of each Cri , the functionals Si = 0, i.e. Si = ∂Cri . Denote

Sf =i=n⋃i=1

Si (3.3)

which is clearly the set of points where functional f is discontinuous. In this thesis, afunctional f with the above mentioned property is said to be piecewise continuous. Asimilar representation of piecewise continuous functional can be found in [99]. Then, fora discontinuous dynamical system (3.1) with piecewise continuous functional f , we candescribe the system equivalently by

x(t) = fi(xt), xt ∈ Cri , i = 1, · · · , n, t ≥ t0 (3.4)

where fi : Cr → Rn is continuous.

Along the definition of Filippov solution and the corresponding investigations in Sec-tion 2.1.1, extensions applied to discontinuous functional differential equations are deducedand shown as follows.

Definition 3.1. A vector function x(t) defined on [−r+t0, t1] (t1 > t0) is called a Filippovsolution of functional differential equation (3.1), if it is absolutely continuous on [−r +t0, t1] and

x(t) ∈ K[f ](xt), a.e. t ∈ [t0, t1] (3.5)

where K[f ](xt) : Cr → Rn denotes the set-valued mapping defined by

K[f ](xt) = co{

limi→∞

f(xit) | xi

t → xt, xit /∈ Sf

}(3.6)

From Definition 3.1, it is clear that expression (3.6) is a set-valued mapping, andFilippov solution of functional differential equation (3.1) is represented by a solution of afunctional differential inclusion formulated by (3.6). Throughout the thesis, we call (3.6)is Filippov set-valued mapping, where K : Cr → B(Cr), (B(Cr) denotes the collection ofall possible subsets in Cr.).

Remark 3.1. It should be noted that the Filippov solution of functional differentialequation (3.1) given by Definition 3.1 is an extended solution of the equivalent Filippov


set-valued mapping (2.8) with respect to piecewise continuous vector field. The extensionof original Definition 2.1 of Filippov solution is not a straightforward work to describe thebehavior of functional differential equations since the different significance of measure zeroin Cr and Rn. In this thesis, we only focus on piecewise continuous functional differentialequations.

Similar to the description of Filippov set-valued mapping (2.4), the following facts areconsidered as natural extensions of the rules given by Proposition A.11.

1) If f : Cmr → Rn is continuous at xt ∈ Cm

r ,

K[f ](xt) = {f(xt)} (3.7)

2) If f, g : Cmr → Cr are locally bounded at xt ∈ Cm

r ,

K[f + g](xt) ⊆ K[f ](xt) + K[g](xt) (3.8)

If one of the functional is continuous at xt, then equality holds.

3) If f1 : Cmr → Cn1

r and f2 : Cmr → Cn2

r are locally bounded at xt ∈ Cmr ,

K[f1 × f2](xt) ⊆ K[f1](xt) × K[f2](xt) (3.9)

If one of the functional is continuous at xt, then equality holds.

4) If f : Cmr → Cr is locally bounded at xt ∈ Cm

r and g : Cmr → Cm×n

r continuous atx ∈ Cm

r ,K[gf ](xt) = g(xt)K[f ](xt) (3.10)

Let Si = ∂Cri = ∂Crj and N be the normal vector to the surface Si. Assume that thepositive direction of the normal vector is from ∂Crj to ∂Cri . f−(xt) and f+(xt) denotethe limiting functionals as any xt ∈ Si is approached from Crj and Cri , respectively, and isdefined by

f−(xt) = limζt(τ)→x−

t (τ)fj(ζt), f+(xt) = lim

ζt(τ)→x+t (τ)

fi(ζt), xt ∈ Si (3.11)

Let f−N (xt) and f+

N (xt) denote the projections of f−(xt) and f+(xt) at xt(τ) on the normalvector N .

Finally, for piecewise continuous functional differential equation, Filippov solution onthe surfaces Si (i = 1, · · · , n) can also be characterized theoretically. The following propo-sition gives a natural extension of Proposition 2.1.

1Throughout this thesis, for a mapping m : Cr → Cm×nr and a Filippov set-valued mapping K : Cr →

B(Cr), the calculation m(·) · K[ ](·) means the set {ξ ∈ Cmr | ξ = m(·) · γ(·), γ(·) ∈ K[ ](·)}.


Proposition 3.1. Consider system (3.4). Let a vector function x(t) be absolutely contin-uous. Suppose that for t ∈ [t1, t2] (t2 > t1), xt(τ) ∈ Si, τ ∈ [0, r], f−

N (xt) ≥ 0, f+N (xt) ≤ 0

and f−N (xt)− f+

N (xt) > 0. Then x(t) is a solution of equation (3.4), if and only if the x(t)satisfies the following functional differential equation

x(t) = f0(xt) = λ(xt)f+(xt) + (1 − λ(xt))f−(xt), a.e. t ∈ [t1, t2] (3.12)

where

λ(xt) =f−

N (xt)f−

N (xt) − f+N (xt)

Example 3.1. Consider the system given by

x(t) = f(xt) = −sgn(xt(τ)), τ ∈ [0, r] (3.13)

This system can be represented as the form (3.4) with

f1(xt) = −1, f2(xt) = 0, f3(xt) = 1

and Cr1 = {xt ∈ Cr | xt > 0}, Cr2 = {xt ∈ Cr | xt = 0}, Cr3 = {xt ∈ Cr | xt < 0}.According to (3.6), it is easy to calculate the associated Filippov set-valued mapping asfollows

K[f ](xt) =

⎧⎪⎪⎨⎪⎪⎩1, xt ∈ Cr1

[−1, 1], xt ∈ Cr2

−1, xt ∈ Cr3

Moreover, Sf = S1 = ∂Cri (i = 1, 2, 3) with

S1(xt(τ)) = xt(τ) = 0, τ ∈ [0, r]

Consider xt∗(τ) = 0, τ ∈ [0, r]. Then, x(t) = 0 ∈ S1, t ∈ [−r + t∗,∞] is a Filippov solutionof system (3.13). The demonstration is given as follows.

The normal vector to S1 is N = 1. The limiting functionals are as follows.

f+(xt) = limxt(τ)→0+

f(xt) = −1, f−(xt) = limxt(τ)→0−

f(xt) = 1

Then, we have that for all t ∈ [t∗,∞],

f+N (xt) = −1, f−

N (xt) = 1 (3.14)

It is clear that λ(xt) = 1/2 and

x(t) = f0(xt) = λ(xt)f+N (xt) + (1 − λ(xt))f−

N (xt) = 0, a.e. t ∈ [t∗,∞]

Hence, the conclusion follows by Proposition 3.1.


3.1.2 Stability Analysis

It is noted that the Filippov set-valued mapping (3.6) given in Definition 3.1 is upper semi-continuous and correspondence with non-empty convex compact values. The existencecondition of a solution of functional differential inclusion (3.5) follows by the result in [48].As is known, Lyapunov-Krasovskii theorem is the fundamental tool for stability analysisof time-delay systems [59, 41]. In this section we intend to establish generalized tools toanalyze stability with respect to functional differential inclusion proposed previously. Wewould also pay attention to the Lypunov stability of system (3.1) in the sense of strongstability.

Denote a Filippov solution of system (3.1) with an initial function φ(−τ), τ ∈ [0, r]as xt(φ). Moreover, without loss of generality, we suppose that 0 ∈ K[f ](0). The strongstability of solution x = 0 of functional differential inclusion (3.1) is defined as follows.

Definition 3.2. The system (3.1) is said to be strongly stable at the origin if for any ε > 0,there exists a δ(ε) > 0 such that for any initial condition φ satisfying ‖φ‖c ≤ δ, all theFilippov solutions satisfy ‖xt(φ)‖c ≤ ε, ∀t ≥ t0. Furthermore, it is strongly asymptoticallystable at the origin if it is strongly stable and all the Filippov solution xt(φ) → 0 as t → ∞.

With respect to functional differential inclusion, Lyapunov-Krasovskii theorem [41] canbe extended as follows.

Theorem 3.1. Consider system (3.1). If there exists a Lipschitz continuous functionalV : Cr → R such that

W1(‖φ(0)‖) ≤ V (φ) ≤ W2(‖φ‖c), ∀φ ∈ Cr (3.15)

holds for given Wi(·) ∈ K (i = 1, 2) and for any initial condition φ,

V (xt(φ)) ≤ 0, a.e. t ≥ t0 (3.16)

holds along all the Filippov solutions xt(φ), then system (3.1) is strongly stable at theorigin. Moreover, if there exists a W3(x) ∈ K such that

V (xt(φ)) ≤ −W3(‖xt(0)‖), a.e. t ≥ t0 (3.17)


Proof. The derivative of functional V calculated along a Filippov solution xt(φ) of system(3.1) is defined as the upper right-hand derivative

V (xt(φ)) = lim suph→0+

V (xt+h(φ)) − V (xt(φ))h

(3.18)

which is used to evaluate the Lyapunov-Krasovskii functional along the solution of func-tional differential equations, see [59], [41], etc. In [81], the author has proved that the


non-increasing property of a Lipschitz continuous functional V can be guaranteed if itsderivative along a solution xt(φ) of system (3.1) is nonpositive almost everywhere associ-ated with the condition that xt(φ) is absolutely continuous. Based on this argument, theproof for Theorem 3.1 is identical to the counterpart for functional differential equationwith continuous right-hand side f(xt) shown in [41] (the proof of theorem 2.1 in chapter 5)except for the corresponding relations holding almost everywhere instead of everywhere inthe sense of the definition (3.18), and therefore is omitted here. Finally, under the condi-tion of this theorem, the non-increasing property of the Lyapunov-Krasovskii functional Vis guaranteed along all Filippov solutions starting from any initial condition. This followsthe strong stability.

Furthermore, the invariance principle in the sense of LaSelle [41] can also be extendedto discontinuous time-delay system. The proof is similar to the proof of LaSalle invari-ant principle for continuous time-delay systems [41] and the argument for Theorem 3.1,therefore it is omitted here.

Definition 3.3. A set Ω ⊆ Cr is said to be a positively invariant set with respect to system(3.1), if for any initial condition φ ∈ Ω, all the Filippov solutions xt(φ) ∈ Ω, ∀t ≥ t0.

Theorem 3.2. (Invariance Principle) Consider system (3.1). Let a set Ω ⊆ Cr becompact and positively invariant with respect to system (3.1). Suppose that the Filippovsolution starting in Ω is unique. If there exists an absolutely continuous functional V :Cr → R such that V (xt) ≤ 0 in Ω, then all the Filippov solutions xt(φ) ∈ Ω of systme(3.1) converge to the invariant set M which is the largest invariant set in the set Dd ={xt(φ) ∈ Ω | V (xt(φ)) = 0}.

It should be noted that the time derivative (3.18) of a Lyapunov-Krasovskii functionalin Theorem 3.1 cannot be calculated, since the solution of the system is not given. Asshown in the literatures (see [35], [83] and references therein), for the systems with certainconditions, a candidate of Lyapunov-Krasovskii functional can be chosen with the followingstructure, i.e.

V (xt) = V1(xt(0)) + V2(xt) (3.19)

where V1 : Rn → R is a continuous differentiable, positive-definite function and V2 : Cr →R is a functional with the integral form

V2(xt) =∫ 0

−τq(xt(−s))ds

where τ ∈ [0, r] and q(x) : Rn → R is positive-definitive. Motivated by this commonapproach and on the basis of the provided nonsmooth analysis tools in Section 2.1.2, wegive the following generalizations.

Definition 3.4. For closed sets Ωa ∈ Rn and Ωb ∈ Rn, and φ ∈ Cr, denote

Md

(Ωa, Ωb, φ(0), φ(−τ)

)=

⋃ϕ∈Ωa

{ξ ∈ R | ξ = pT ϕ + φ(0) − φ(−τ), p ∈ Ωb

}


Moreover, for a given C ∈ R

Md

(Ωa, Ωb, φ(0), φ(−τ)

) ≤ C ⇔ ξ ≤ C, ∀ξ ∈ Md

(Ωa, Ωb, φ(0), φ(−τ)

)Corollary 3.1. Consider system (3.1). If there exist a Lipschitz continuous, positive-definite function V1 : Rn ∈ R and a positive-definitive function q : Rn → R such that

Md

(K[f ](xt), ∂CV1(xt(0)), q(xt(0)), q(xt(τ))

) ≤ 0 (3.20)

then system (3.1) is strongly stable at the origin, in addition if there exists a functionW (·) ∈ K such that

Md


) ≤ −W (‖xt(0)‖) (3.21)


Proof. Choose a Lyapunov-Krasovskii functional as

V (xt) = V1 (xt(0)) +∫ 0

−τq (xt(−s)) ds (3.22)

If condition (3.20) holds, then along any Filippov solution xt(φ) of system (3.1), V (xt(φ))≤ 0, a.e. t ≥ 0. Hence, the strong stability follows by Theorem 3.1. Likewise, the condition(3.21) guarantees the strong asymptotic stability.

Moreover, we can obtain the following result by the invariance principle Theorem 3.2and Corollary 3.1.

Corollary 3.2. Let a set Ω ⊆ Cr be compact and positively invariant with respect to (3.1).Suppose that the Filippov solution of system (3.1) starting in Ω is unique. If there exist aLipschitz continuous function V1 : Rn ∈ R and a positive-definitive function q : Rn → Rsuch that

Md


) ≤ 0 (3.23)

Then all the Filippov solutions xt(φ) ∈ Ω of (3.1) converges to the invariant set M whichis the largest invariant set in Dd defined by

Dd ={xt ∈ Ω | 0 ∈ Md


)}(3.24)

3.1.3 L2-gain Analysis

The results presented above are generalized to the L2-gain analysis with respect to func-tional differential inclusion as follows.

Consider discontinuous time-delay systems described by functional differential equationof the form {

x(t) = f(xt) + gw(xt)w(t)

y(t) = h(x), t ≥ 0(3.25)


where x ∈ Rn denotes the state, w : R → Rm is the input which is continuous in t andy ∈ Rp is the output. Functional f(xt) : Cr → Rn with 0 ∈ K[f ](0) and gw(xt) : Cr →Rn×m are discontinuous in xt, and h(x) : Rn → Rp with h(0) = 0 is continuous.

We focus on the L2-gain from input w to output y. The input signal w ∈ L2[0, T ] iscalled admissible, if the corresponding output y belongs to L2[0, T ] along all the Filippovsolutions of system (3.25) with respect to any initial condition.

Let xt(φ,w) denote a Filippov solution of system (3.25) with an initial conditionx0(τ) = φ(−τ), τ ∈ [0, r].

Definition 3.5. Let γ > 0. System (3.25) is said to have L2-gain less than or equal to γif there exists a continuous functional V : Cr → R such that

W1(‖φ(0)‖) ≤ V (φ) ≤ W2(‖φ‖c), ∀φ ∈ Cr (3.26)

and for any initial condition φ,

V(xt(φ,w)

) ≤ V (φ) +∫ t

0

12[γ2‖w(s)‖2 − ‖h(xs(φ,w)

)‖2]ds (3.27)

holds along all the Filippov solutions xt(φ,w) with respect to any admissible input w, whereW1(·), W2(·) ∈ K.

It is obvious from Definition 3.5 that if we consider the special case as initial conditionφ = 0 and note that V (0) = 0, then condition (3.27) implies that for any T > 0,∫ T

0‖h(xt(0, w)

)‖2dt ≤ γ2

∫ T

0‖w(t)‖2dt (3.28)

holds for any w ∈ L2[0, T ]. Furthermore, if T is admitted to take value T = ∞, then thisinequality is the condition that restricts the L2-induced norm of the system to be less thanor equal to the given level γ [93].

Moreover, Definition 3.5 does not require the functional V to be differentiable, andit is usually taken as candidate of Lyapunov-Krasovskii functional to investigate stabilityassociated with L2-gain property. Based on Theorem 3.1 and invariance principle Theorem3.2 the following result can be established.

Definition 3.6. System (3.25) is called zero-state observable if all the Filippov solutionsxt(φ,w) that satisfy y(t) ≡ 0 converge to zero when w(t) = 0, i.e.

h(xt(φ, 0)

) ≡ 0 ⇒ xt(φ, 0) → 0, as t → ∞. (3.29)

Theorem 3.3. Consider system (3.25). Suppose that the Filippov solutions with w = 0starting in Ωh = {φ ∈ Cr | h(φ(0)) = 0} is unique. If there exists a functional V : Cr → R,which satisfies condition (3.26) and is Lipschitz continuous for all xt ∈ Cr, such that forany given initial condition φ,

V (xt(φ,w)) ≤ 12[γ2‖w(t)‖2 − ‖h(xt(φ,w))‖2

](3.30)


holds along all the Filippov solutions xt(φ,w) with respect to any admissible input w, thensystem (3.25)


(ii) is strongly asymptotically stable at the origin with w = 0 if in addition it is zero-stateobservable

Proof. It is straightforward to prove (i) by integrating (3.26). We now show (ii). First,as w ≡ 0, from (3.26), the derivative of V along all the Filippov solutions of system (3.25)satisfies

V(xt(φ, 0)

) ≤ −12‖h(xt(φ, 0)

)‖2, a.e. t ≥ 0 (3.31)

which means that V (xt) is a Lyapunov-Krasovskii functional and strong stability at theorigin follows by the Theorem 3.1. Moreover, for any xt(φ, 0) that satisfies h(x(φ, 0)) = 0and

xt(φ, 0) ∈ K[f ](xt), a.e. t ≥ 0

xt(φ, 0) → 0, since system (3.25) is zero-state observable. In other words, Ωh is positivelyinvariant set with respect to system (3.25) with zero input. Furthermore, condition (3.30)implies that the set ΩV = {xt(φ, 0) | V (xt(φ, 0)) = 0} ⊆ Ωh is invariant set. Therefore,all the trajectories xt(φ, 0) will converge to the set Ωh and finally converge to zero byTheorem 3.2. Strong asymptotic stability at the origin follows.

Denote F (t, xt) = f(xt) + gw(xt)w(t). Combining the idea of Corollary 3.2 and theproof of Theorem 3.3 obtains the following result that enables one to analyze L2-gainwithout explicit computation of the derivative of the associate functional V along thesystem trajectory.

Corollary 3.3. Consider system (3.25). Suppose that the Filippov solution with w = 0starting in Ωh = {φ ∈ Cr | h(φ(0)) = 0} is unique. If there exist a Lipschitz continuousfunction V1 : Rn → R and a function q : Rn → R such that

Md

(K[F ](t, xt), ∂CV1(xt(0)), q(xt(0)), q(xt(τ))

)≤ 1

2[γ2‖w‖2 − ‖h(xt(φ, 0))‖2

], a.e. t ≥ 0

(3.32)

then system (3.25)


(ii) is strongly asymptotically stable at the origin with w = 0 if in addition it is zero-stateobservable.


3.2 Feedback Stabilizing Control

This section addresses the stabilization problems for a class of discontinuous time-delaysystems. By using the extended Filippov framework that consists of the Filippov solution,Lypunov-Krasovskii theorem and invariance principle with respect to functional differen-tial inclusion presented in previous section, a feedback design approach is proposed, whichguarantees the strong asymptotic stability of the closed-loop systems. Moreover, an ex-tension of the approach to the robust adaptive controller is also shown for the systemswith unknown parameters.

Consider a time-delay system described by

x(t) = f(xt) + d(xt)δ(xt) + g(xt)u, t ≥ 0 (3.33)

with x0(τ) = φ(−τ), τ ∈ [0, r], where x ∈ Rn is the state, u ∈ R is the control input.f(xt) : Cr → Rn is a nonlinear functionals, δ(xt) : Cr → R denotes a discontinuousfunctional in xt, and d(xt) : Cr → Rn and g(xt) : Cr → Rn are continuous functionalssatisfying 2

d(xt) = g(xt)η(xt) (3.34)

where functional η(·) : Cr → R is continuous in xt. Moreover, the system has static trivialsolution, i.e. 0 ∈ K[F ](0), where F (xt) = f(xt) + d(xt)δ(xt).

3.2.1 Stabilization

For the system (3.33), the design problem considered in this section is to find a feedbackcontrol law

u = α(xt) (3.35)

with 0 ∈ K[α](0) such that the closed-loop system is strongly asymptotically stable at theorigin.

The Filippov solution of the closed-loop system (3.33) with (3.35) satisfies the followingfunctional differential inclusion

x(t) ∈ K[Fc](xt), a.e. t ≥ 0 (3.36)

where Fc(xt) = f(xt)+d(xt)δ(xt)+ g(xt)α(xt). Moreover, noted that the functional f(xt)can always be represented as a functional that depends on xt(0) and also xt, i.e.

f(xt) = f(x, xt) (3.37)

The main result of this section is the following theorem.2In many physical systems, discontinuities can be modeled to satisfy the condition (3.34), for instance,

the Coulomb friction in Lagrangian systems [71, 74] which can be represented as the function satisfying(3.34) when the control input is driving torque.

3.2. Feedback Stabilizing Control 53

Theorem 3.4. Consider system (3.33) that satisfies the following conditions,

H1: functional f(x, xt) is continuous and can be represented by3

f(x, xt) = f0(x) + f1(x)e(xt) (3.38)

where f0(x) : Rn → Rn is a known continuous function, e(xt) : Cr → Rq is a knowncontinuous functional and f1(x) : Rn → Rn×q is a known matrix with entries beingcontinuous functions.

H2: There exist a continuous differentiable, positive-definite function V1(x) : Rn → Rand a positive-definitive function Q(x) : Rn → R such that

Lf0V1(x) +12‖LT

f1V1(x)‖2 +

12‖e(x)‖2 ≤ −Q(x) (3.39)

Then, system (3.33) is strongly asymptotically stable at the origin under feedback controllaw given by

u = α(xt) = −ksgn(LgV1(x)) |η(xt)| γ∗δ (xt) (3.40)

with a constant k ≥ 1 and

γ∗δ (xt) = max

γδ(xt)∈K[δ](xt){|γδ(xt)|} (3.41)

Proof. The functional f(xt) is continuous, then according to Definition 3.1 and the compu-tation rules of functional differential inclusion (3.7)∼(3.10), the solution of the closed-loopsystem represented by the functional differential inclusion (3.36) satisfies the followingfunctional differential inclusion

x ∈ K[Fc](xt) ⊂ f(x, xt) + d(xt)K[δ](xt) + g(xt)K[α](xt) (3.42)

and taking condition (3.38) into account obtains

x ∈ K[Fc](xt) ⊂ f0(x) + f1(xt)e(xt) + d(xt)K[δ](xt) + g(xt)K[α](xt) (3.43)

Choose a candidate of Lyapunov-Krasovskii functional with the function V1(x) by

V (xt) = V1(xt(0)) +12

∫ 0

−τ‖e(xt(−s))‖2ds (3.44)

which is in the form consisting with the functional (3.19) with

q(x) =12‖e(x)‖2 (3.45)

3The f0(x) in (3.38) is the part f(x, 0), i.e. f(x, xt) = f(x, 0) + f1(x)e(xt).


Considering that V1(x) is continuous differentiable, then by Definition 3.4 we have that

Md

(K[Fc](xt), ∂CV1(x), q(xt(0)), q(xt(τ))

)= ∇V T

1 (x)K[Fc](xt) + q(xt(0)) − q(xt(τ))

⊂ Lf0V1(x) + Lf1V1(x)e(xt) + LdV1(x)K[δ](xt) + LgV1(x)K[α](xt)

+12‖e(x)‖2 − 1

2‖e(xt(τ))‖2

(3.46)

With the feedback control law (3.40) we obtain

LgV1(x)K[α](xt) = −k|LgV1(xt)| · |η(xt)|K[γ∗δ ](xt) (3.47)

Substituting (3.47) into (3.46) gives

Md


)⊂ Lf0V1(x) + Lf1V1(x)e(xt) + LdV1(x)K[δ](xt) − k|LgV1(x)| · |η(xt)|K[γ∗

δ ](xt)

+12‖e(x)‖2 − 1

2‖e(xt(τ))‖2

(3.48)Then, taking the conditions (3.34) and (3.39) into account yields that

∀ξVd∈ Md


),

ξVd≤ −Q(x) + |LgV1(x)| · |η(xt)|γ∗

δ (xt) − k|LgV1(xt)| · |η(xt)|γ∗δ (xt)

(3.49)

whereγ∗

δ (xt) = maxγδ∈K[γ∗

δ ](xt){γδ} ≥ 0

By the fact, γ∗δ (xt) ≤ γ∗

δ (xt), inequality (3.49) reduces to

∀ξVd∈ Md


),

ξVd≤ −Q(x) − (k − 1)|LgV1(x)| · |η(xt)| · γ∗

δ (xt)(3.50)

Hence, by Corollary 3.1, we conclude that the closed-loop system is strongly asymptoticallystable at the origin.

Corollary 3.4. Consider system (3.33). Suppose that the functional f(xt) satisfies thedecomposition of (3.38) but with piecewise continuous functions f0 : Rn → Rn and f1 :Rn → Rn×q, and the following condition

∂V1

∂xf0(x) +

12

∥∥∥∥∥(

∂V1

∂x

)T

f1(x)

∥∥∥∥∥2

+12‖e(x)‖2 ≤ −Q(x),

∀f0(x) ∈ K[f0](x), ∀f1(x) ∈ K[f1](x)

(3.51)

holds with a continuous differentiable, positive-definite function V1(x) : Rn → R and apositive-definite function Q(x) : Rn → R. Then, system (3.33) is strongly asymptoticallystable at the origin under the feedback control law (3.40).


Remark 3.2. Theorem 3.4 shows that the time-delay system (3.33) can be stabilized by adiscontinuous delay-dependent feedback control law. There is a free parameter k(≥ 1) to bechosen which determines the damping of the Lyapunov-Krasovskii functional (3.44) alongthe solution of the closed-loop system. Obviously, quick responsibility can be achieved bychoosing a large k, but as a by-product, larger k will usually cause undesired overshoot oroscillation. Hence, in practical applications, a trade-off is needed between quickness andrejection of oscillation.

Remark 3.3. It is clear that same conclusion to Theorem 3.1 is true if the involved g(xt)in system (3.33) is a discontinuous functional.

On the other hand, it should be noted that due to the assumptions H1 and H2 inTheorem 3.4, the system under consideration is a special case of (3.33). The followingshows that the proposed design approach can be extended and applied to more generalsystems.

First, we consider the case when system (3.33) can be rendered by a state feedbackcompensation to satisfy the condition (3.39), then a new control law can be obtained bythe combination of the compensation and the controller given by Theorem 3.4.

Theorem 3.5. Consider system (3.33). Suppose that condition H1 in Theorem 3.4 holds.If there exists a continuous differentiable, positive-definite function V1(xt(0)) : Rn → Rsuch that

Lf0V1(x) +12(‖LT

f1V1(x)‖2 − |LgV1(x)|2) +

12‖e(x)‖2 ≤ 0 (3.52)

then, system (3.33) is strongly asymptotically stable at the origin under the feedback controllaw given by

u = β(xt) − ksgn(LgV1(x))|η(xt)|γ∗δ (xt) (3.53)

withβ(xt) = −1

2LgV1(x) (3.54)

Proof. First, consider an associated system with (3.33),

x = f(x, xt) + g(xt)β(xt) (3.55)

Then, by taking the conditions (3.38) and (3.52) into account, we get the derivative offunction V1(xt(0)) along any solution of (3.55) satisfying

V1(x) = Lf0V1(x) + Lf1V1(x)e(xt) + LgV1(x)β(xt)

≤−12‖LT

f1V1(x)‖2 − 1

2‖e(x)‖2 + Lf1V1(x)e(xt)

with β(xt) given by (3.54). In this case, it is clear that the remainder of the proof isstraightforward from the proof of Theorem 3.4 by using the same Lyapunov-Krasovskiifunctional (3.44).


Furthermore, note that a smooth function f(x, y) depending on x and y can be alwaysdecomposed by [63]

f(x, y) = f0(x) + f1(x, y) (3.56)

and a continuous function represented by f1(x, y) can be bounded by [63]

|f1(x, y)| ≤ a(x)b(y) (3.57)

where a(x) > 1 and b(y) > 1 are smooth functions. According to the above properties, iff(x, xt) in system (3.33) is smooth, then we have

f(x, xt) = f0(x) + f1(x, xt) (3.58)

and for a given continuous differentiable function V1(xt(0)), the Lie derivative Lf1V1(x, xt)is continuous and satisfies

|Lf1V1(x, xt)| ≤ va(x)vb(xt) (3.59)

where va(x) > 1 and vb(xt) > 1 are smooth. Using this bounding condition, an argumentsimilar to Theorem 3.4 obtains the stabilizing control law with a new Lyapunov-Krasovskiifunctional.

Theorem 3.6. Consider system (3.33). Suppose that f(x, xt) is smooth. If there exists acontinuous differentiable, positive-definitive function V1(xt(0)) : Rn → R and a positive-definite function Q(x) : Rn → R such that

Lf0V1(x) +12v2a(x) +

12v2b (x) ≤ −Q(x) (3.60)

then the closed-loop system (3.33) with control law (3.40) is strongly asymptotically stableat the origin.

Proof. First, note that by using the relations (3.56) and (3.57), the derivative of functionV1(xt(0)) along the trajectory of associated system x(t) = f(x, xt) satisfies

V1(x) = Lf0V1(x) + Lf1V1(x)

≤ Lf0V1(x) + va(x)vb(xt)

≤ Lf0V1(x) +12v2a(x) +

12v2b (xt)

(3.61)

and by the condition (3.60), it yields

V1(x) ≤ −Q(x) − 12v2b (x) +

12v2b (xt) (3.62)

Then, choose a Lyapunov-Krasovskii functional in the form that consists with (3.19) asfollows

V (xt) = V1(xt(0)) +12

∫ 0

−τv2b (xt(−s))ds (3.63)


It is straightforward to get

∀ξVd∈ Md

(K[Fc](xt), ∂CV1(x)),

12v2b (xt(0)),

12v2b (xt(τ))

),

ξVd≤ −Q(x) − (k − 1)|LgV1(x)||η(xt)|γ∗

δ (xt)(3.64)

Therefore, the strong asymptotic stability follows by Corollary 3.4.

Example 3.2. Consider a second-order system given by{x1 = −2x1 + x2t(τ)

x2 = −x2 + x1t(τ)(x2 + 1)sgn(x1) + u(3.65)

where delay time τ is constant. The system can be represented by

x = f0(x) + f1(x)e(xt) + d(xt)δ(xt) + g(xt)u (3.66)

with e(xt) = x2t(τ), δ(xt) = sgn(x1),

f0(x) =

[−2x1

−x2

], f1(x) =

[x1

0

], d(xt) =

[0

x1t(τ)(x2 + 1)

], g(xt) =

[01

]

It satisfies condition (3.34) with η(xt) = x1t(τ)(x2 + 1).

Choose a continuous differentiable, positive-definitive function V1(xt(0)) as follows

V1(x) =12x2

1 +12x2

2 (3.67)

It is easy to confirm the existence of such a proper positive-definitive Q(x) that the condi-tion (3.39) is satisfied. Hence, by Theorem 3.4, the system (3.2) is strongly asymptoticallystable at the origin by the controller (3.40), i.e.

u = −ksgn(x2)|x1t(τ)(x2 + 1)| (3.68)

where γ∗δ (xt) = 1.

The control system is simulated by Simulink with fixed step 0.001. Let r = 0.2 andchoose k = 1.2. Fig. 3.1 shows the trajectory of the system starting from (x10(τ), x20(τ)) =(1,−2), ∀τ ∈ [0, r], which indicates the asymptotic stability at the origin of the closed-loop system. Moreover, the trajectory of the above simulation result is shown in Fig. 3.2where we can see the sliding motion along the surface x2 = 0 when the trajectory achievesxt∗(τ) = (−0.23, 0), τ ∈ [0, 0.2] at a moment t∗.


-2

-1.5

-1

-0.5

0

0.5

1

0 1 2 3 4 5 6

0

0.5

1

Time [s]

x1

x2

u

0 1 2 3 4 5 6

Time [s]

Fig. 3.1: Simulation result of Example 3.2.

-0.5 0 0.5 1 1.5

-2.5

-2

-1.5

-1

-0.5

0

0.5

x1

x2

x0

x*

Fig. 3.2: A trajectory of Example 3.2.


It is clear that for the closed-loop system (3.65) with (3.68), denoted as x = Fc(xt),the set Sf = {S1,S2} is given with S1(xt(τ)) = x1t(τ) = 0 and S2(xt(τ)) = x2t(τ) = 0,τ ∈ [0, 0.2], respectively. At xt∗(τ) ∈ S2, we have

F+c (xt∗) = lim

x2t(τ)→0+Fc(xt) =

⎡⎢⎣ −2x1 + x2t(τ)

−x2 + x1t(τ)(x2 + 1)sgn(x1)−1.2|x1t(τ)(x2 + 1)|

⎤⎥⎦xt∗ (τ)

=

[ −0.46

−0.046

]

F−c (xt∗) = lim

x2t(τ)→0−Fc(xt) =

⎡⎢⎣ −2x1 + x2t(τ)

−x2 + x1t(τ)(x2 + 1)sgn(x1)+1.2|x1t(τ)(x2 + 1)|

⎤⎥⎦xt∗ (τ)

=

[−0.46

0.506

]

The normal vector to S2 is N = [0 1]T , then we have

F+cN (xt∗) = −0.046, F−

cN (xt∗) = 0.506

Hence, according to Proposition 3.1, the trajectory of the closed-loop system has slidingmotion along the surface S2 during a certain interval t ∈ [t∗, t∗ + Δt] (Δt > 0).

3.2.2 Adaptive Stabilization

Consider the dynamical systems with uncertainty represented as follows

x(t) = f(xt) + d(xt, θ)δ(xt) + g(xt)u (3.69)

which satisfies the following condition

d(xt, θ) = g(xt)η′(xt, θ) (3.70)

where θ ∈ Rl denote unknown constant parameters.

According to the bounded property (3.57) for a continuous function η′(xt, θ), we have

|η′(xt, θ)| < c(xt)Θ (3.71)

where c(xt) : Cr → R is a smooth functional and Θ : Rl → R denotes a smooth functionwith respect to unknown parameters θ.

Theorem 3.7. Consider system (3.69). If the conditions H1 and H2 given in Theorem(3.4) are satisfied, then the closed-loop system (3.69) with the following adaptive controllaw ⎧⎨⎩u = α′

(xt, Θ

)= −sgn(LgV1(x))c(xt)γ∗

δ (xt)Θ

˙Θ = �(xt) = γ−1|LgV1(x)|c(xt)γ∗δ (xt)

(3.72)

is strongly stable at x = 0 and all the Filippov solutions xt(φ) of the closed-loop systemconverge to zero as t → ∞, where γ is a given positive number.


Proof. By the computing rules (3.7)∼(3.10), we know the Filippov solution of the closed-loop system (3.69) with adaptive control law (3.72) satisfies the following functional dif-ferential inclusion,⎡⎣ x

˙Θ

⎤⎦ ∈ K[Fc](xt, Θ) ⊂[

f(xt) + d(xt, θ)K[δ](xt) + g(xt)K[α](xt, Θ)

K[�](xt)

](3.73)

Choose a Lyapunov-Krasovskii functional associated with (3.44) as follows

V(xt, Θ

)= V1(xt(0)) +

12

∫ 0

−τ‖e(xt(−s))‖2ds +

γ

2

(Θ − Θ

)2(3.74)

Denote V ′1(x, Θ) = V1(x) + 1/2(Θ − Θ)2. Along the trajectory of (3.73), we obtain that

Md

(K[Fc](xt), ∂CV ′

1(x, Θ),12‖e(x)‖2,

12‖e(xt(τ))‖2

)= ∇V T

1 (x)K[Fc](xt) +12‖e(x)‖2 − 1

2‖e(xt(τ))‖2 + γ

˙Θ(Θ − Θ

) (3.75)

With feedback control law α′(xt, Θ) of (3.72), we can obtain

LgV1(x)K[α′](xt, Θ

)= −|LgV1(xt)|c(xt)γ∗

δ (xt)Θ (3.76)

Take (3.71) and H1 and H2 in Theorem (3.4) into account. Similar to the proof of Theorem3.4, we can obtain that

∀ξVd∈ Md

(K[Fc](xt), ∂CV ′

1(x, Θ),12‖e(xt)‖2

),

ξVd≤ −Q(x) − 1

2‖LT

f1V1(x) − e(xt)‖2 − ‖e(xt)‖2

(3.77)

This implies the strong stability of the control system. Moreover, the convergency of theFilippov solutions follows by the invariance principle Corollary 3.2.

Example 3.3. Consider the system described by{x1 = −x1 + x2(t − τ)

x2 = −x2 + x1t(τ)(x2θ + 1) sgn(x1) + u(3.78)

where θ is the unknown parameter. In this case, d(xt, θ) = g(xt)η′(xt, θ) with η′(xt, θ) =x2θ + 1 which satisfies |η′(xt, θ)| ≤ (x2

2 + 1)(θ2 + 1), i.e. η′(xt, θ) is bounded with c(xt) =x2

2 + 1 and Θ = θ2 + 1 referring to (3.71).

From the illustration for system (3.65) of Example 3.2 and by Theorem 3.7, we candesign an adaptive control law for system (3.78) as follows⎧⎨⎩u = −sgn(x2)(x2

2 + 1)Θ

˙Θ = γ−1|x2|(x22 + 1)

(3.79)

3.3. L2-gain Design 61

Let r = 0.2 and choose γ = 20. Under initial condition (x10(τ), x20(τ)) = (3, 2) andΘ0(τ) = 1.6, ∀τ ∈ [0, r], a simulation result is shown in Fig. 3.3, from which we canobserve the convergence of state x(t).

-1

0

1

2

3

4

1.5

2

2.5

0 1 2 3 4 5 6-20

-15

-10

-5

0

5

Time [s]

0 1 2 3 4 5 6

Time [s]

0 1 2 3 4 5 6

Time [s]

x1

x2

��

u


3.3 L2-gain Design

Consider system (3.25) with additional input given by{x(t) = f(xt) + gw(xt)w(t) + g(xt)u

y(t) = h(x), t ≥ 0(3.80)

where w ∈ Rm denotes the disturbance input which is continuous in t, and u ∈ R is thecontrol input.


The L2-gain synthesis problem is formulated as follows: for a given constant γ > 0,find, if exists, a state feedback control law

u = α(xt) (3.81)

with 0 ∈ K[α](0) such that the closed-loop system is strongly asymptotically stable at theorigin when w = 0 and has L2-gain less than or equal to γ from w to y.

3.3.1 Hamilton-Jacobi Inequality-like Characterization

According to the theoretical results given with functional differential inclusion in Theorem3.3 and Corollary 3.3, it is obvious that a feasible way to find a solution to the synthesisproblem for system (3.80) is to find a Lyapunov-Krasovskii functional V (xt) such thatcondition (3.30), particularly, the condition (3.21), holds along all the Filippov solutionsof the closed-loop system while the undisturbed system is zero-state observable.

Definition 3.7. System (3.80) is said to be zero-state observable if all the Filippov solu-tions x(t)) that satisfy h(x) = 0 converge to zero when u = 0 and w = 0.

Theorem 3.8. Consider system (3.25). Suppose that it is zero-state observable. If fora given γ > 0, there exist a Lipschitz continuous, positive-definite function V1 : Rn → Rand a functional Q : Cr → R, Q(φ) > 0 as φ = 0, Q(0) = 0 such that

L∗FV1 (φ(0)) +

12γ2

[L∗QV1

(φ(0)

)]2 +12‖h(φ(0))‖2 + Q(φ(0)) − Q(φ(−τ)) ≤ 0, ∀φ ∈ Cr

(3.82)holds, then system (3.80) is strongly asymptotically stable at the origin as w = 0 and hasL2-gain less than or equal to γ from ω to y, where τ ∈ [0, r] and

L∗FV1 (φ(0)) = max

LFV1(φ(0))∈M(K[f ](φ),∂CV1(χ(0)))

{LFV1 (φ(0))} (3.83)

L∗QV1 (φ(0)) = max

LQV1(χ(0))∈M(K[gw ](φ),∂CV1(φ(0)))

{‖LQV1 (φ(0))‖} (3.84)

Proof. According to the calculation rules (3.7)∼(3.10), the following conclusion with re-spect to system (3.80) follows immediately

K[F ](t, xt)⊂K[f ](xt) + K[gww](xt, t)

= K[f ](xt) + w(t)K[gw](xt)(3.85)

Choose a Lyapunov-Krasovskii functional given by

V (xt) = V1 = (xt(0)) +∫ 0

−τQ(xt(−s))ds (3.86)


Along any Filippov solution xt(φ,w) of system (3.80) with respect to input w, one hasfrom (3.85) that

V (xt(φ,w))∣∣∣xt(φ,w)∈K[F ](t,xt) ∈ ˙

V (xt(φ,w)) , a.e. t ≥ 0 (3.87)

where

˙V (xt(φ,w)) =

{ζ ∈ R | ∃fd ∈ K[f ](xt), gwd ∈ K[gw](xt), s.t. ζ = pT

V1fd + wpT

V1gwd

+Q(x) − Q(xt(τ)), pV1 ∈ ∂CV1(x)}

For any pV1 ∈ ∂CV1(x), it can be derived directly by (3.83) that

pTV1

fd + wpTV1

gwd ≤ L∗FV1(x) + wpT

V1gwd

In view of the HJI-like condition (3.82), we have

pTV1

fd + wpTV1

gwd ≤−12‖h(x)‖2 − 1

2γ2[L∗

QV1(x)]2 +1

2γ2‖pT

V1gwd‖2

c

+γ2

2‖w‖2 − γ2

2

∥∥∥∥w − 1γ2

pTV1

gwd

∥∥∥∥2

c

+ Q(x) − Q(xt(τ))

which indicates that

ζ ≤ 12[γ2‖w‖2 − ‖h(x)‖2

], ∀ζ ∈ ˙

V (xt(φ,w))

Furthermore, according to Definition 3.4, we have

Md

(K[F ](t, xt), ∂CV1(x), Q(xt(0)), Q(xt(τ))

)=

⋃Fd∈K[F ](t,xt)

{ξ ∈ R | ξ = pT

V1Fd + Q(x) − Q(xt(τ)), pV1 ∈ ∂CV1(x)

}Due to (3.85), it is clear that

Md

(K[F ](t, xt), ∂CV1(x), Q(x), Q(xt(τ))

) ⊂ ⋃xt(φ,w) s.t.

xt(φ,w)∈K[F ](t,xt)

˙V (xt(φ,w))

Thus, the conclusion follows by Corollary 3.3.

3.3.2 L2-gain Controller

We now present a feedback design method with the HJI-like condition (3.82). The ap-proach is actually an extension of the result given in Section 3.2.1, which is proposed byconstructing a system-related Lyapunov-Krasovskii functional.


Consider the system (3.80) represented by{x(t) = f(xt) + d(xt)δ(xt) + gw(xt)w(t) + g(xt)u

y(t) = h(x)(3.88)

which satisfies the matching condition (3.34), where f(xt), d(xt) and g(xt) are functionalswith appropriate dimensions.

Combining the proof of Theorem 3.4 and Theorem 3.8 obtains the following conclusion.

Theorem 3.9. Consider system (3.88). If functional f(xt) satisfies the condition H1given in Theorem 3.4 and in addition the following condition holds

H3: For a given γ > 0, there exist a continuous differentiable, positive-definite functionV1(x) : Rn → R and a positive-definite function Q(x) : Rn → R such that

Lf0V1(x) +12

[1γ2

(L∗QV1(x)

)2 + ‖Lf1V1(x)‖2 + ‖e(x)‖2 + ‖h(x)‖2

]+ Q(x) ≤ 0

(3.89)

then, the system (3.88) is strongly asymptotically stable at the origin as w = 0 and hasL2-gain less than or equal to γ from input w to y with the control input u given by (3.40).

Proof. Choose (3.44) as a candidate of Lypunov-Krasovskii functional. When w = 0, inview of condition (3.89), it is easy to prove that

∀ξVd∈ Md

(K[fc](xt), ∂CV1(x), q(xt(0)), q(xt(τ))

),

ξVd≤ −(k − 1)|LgV1(x)| · |η(xt)|γ∗

δ (xt) − 12γ2

(LQV1(x))2 − 12‖h(x)‖2 − Q(x)

(3.90)

where fc(xt) = f0(x) + f1(x)e(xt) + d(xt)δ(xt) + g(xt)α(xt). The above inequality impliesthat system (3.88) with w = 0 is strongly asymptotically stable at the origin by Theorem3.4. Furthermore, it can be simultaneously obtained from the condition (3.89) that forthe closed-loop system (3.88) with (3.40),

L∗FV1(x) +

12γ2

(L∗QV1(x))2 +

12‖h(x)‖2 +

12‖e(x)‖2 − 1

2‖et(τ)‖2 ≤ 0 (3.91)

since

L∗FV1(x) +

12γ2

(L∗QV1(x))2 +

12‖h(x)‖2 +

12‖e(x)‖2 − 1

2‖et(τ)‖2

≤ Lf0V1(x) +12

[‖Lf1V1(x)‖2 +

1γ2

(L∗QV1

)2 + ‖h(x)‖2 + ‖e(x)‖2

]+ Q(x)

The proof is complicated by Theorem 3.8.


It should be noted that with the established functional differential inclusion basedframework, the presented HJI-like condition (3.89) is validated by the Lyapunov-Krasovskiitheorem with a continuous differentiable functional (3.86). Under Definition 3.5, it is asufficient condition to guarantee the L2-gain property of the considered time-delay system.Hence, using the feedback control law (3.40) will cause conservativeness in concluding thestability and the gain performance for system (3.88). To reduce the conservativeness, onecan choose, for example, the function Q(x) in (3.82) to be positive-definite with “positivitymargin” as small as possible.

Example 3.4. Consider a nonlinear time-delay system⎧⎪⎪⎨⎪⎪⎩x1 =−2x1 + x2t(τ) + sgn(x1)w1

x2 =−x2 + x1t(τ)(x2 + 1)sgn(x1) + sgn(x1)w2 + u

y = x1

(3.92)

The system is in accordance with the structure of system (3.88) and satisfies conditionsH1 in Theorem 3.4 and (3.34). By using a continuous differentiable function

V1(x) =12(x2

1 + x22)

and computing the left-hand of (3.82) yield

Lf0V1(x) +12

[1γ2

(L∗QV1(x))2 + ‖Lf1V1(x)‖2 + ‖h(x)‖2 + ‖e(x)‖2

]= −x2

1 −12x2

2 +1

2γ2(x2

1 + x22)

For a given γ, such as γ =√

2, the inequality (3.89) holds. According to Theorem 3.9, thefeedback control law from (3.40) is designed as

u = −ksgn(x2)|x1t(τ)(x2 + 1)|where γ∗

δ (xt) = 1. Let r = 0.5 and choose k = 2. The response of a simulation withinitial condition x0(τ) = (2,−3), ∀τ ∈ [0, r] and disturbance input [w1, w2]T = [0.5,−1]T

is shown in Fig. 3.4.

3.3.3 Piecewise Continuous Linear Systems

In this section, we discuss a special case of the previous proposed approach when theconsidered time-delay system has linear structure, i.e. a case of piecewise continuouslinear system.

Consider the systems represented by{x(t) = f1(x) + f2(xt) + f3(xt)w(t)

y(t) = Cx, t ≥ 0(3.93)


-3

-2

-1

0

1

2

0 5 10 15 20-2

0

2

4

6

8

Time [s]

0 5 10 15 20

Time [s]

x1

x2

u


with

f1(x) =

{A1x, S1(xt) > 0

A2x, S1(xt) < 0,

f2(xt) =

{A1xt(τ), S2(xt) > 0

A2xt(τ), S2(xt) < 0

f3(xt) =

{G1, S3(xt) > 0

G2, S3(xt) < 0

where Ai ∈ Rn×n, Ai ∈ Rn×n, Gi ∈ Rn×m (i = 1, 2) and C ∈ Rp×n are constant matrices,and Sj(xt) : Cr → R (j = 1, 2, 3) are smooth functions, which represent switchingsurfaces Sf = {xt ∈ Cr | Sj(xt(τ)) = 0, τ ∈ [0, r]}.

Let R1 > 0 and R2 > 0 be the matrices such that

xT ATi Aix ≤ xT R1x (i = 1, 2), ∀x ∈ Rn (3.94)

xT GTi Gix ≤ xT R2x (i = 1, 2), ∀x ∈ Rn (3.95)

Then, we have the following result.


Theorem 3.10. Consider system (3.93). For a given γ > 0, if there exist positive-definitematrices P and Q satisfying the following Riccati inequalities

AT1 P + PA1 + PR1Q

−1P +1γ2

PR2P + CT C + Q ≤ 0 (3.96)

AT2 P + PA2 + PR1Q

−1P+1γ2

PR2P + CT C + Q ≤ 0 (3.97)

then system (3.93) has L2-gain less than or equal to γ from w to y.

Proof. The functionals, f1, f2 and f3 of system (3.93), take set-value on the surfacesSj(xt) = 0, respectively. To prove Theorem 3.10, we first consider the case that x(t) in apartition of space Cr, i.e. R1 = {xt | S1(xt) ∈ R, S2(xt) = 0, S3(xt) = 0}, where theFilippov solution of system (3.93) satisfies the following functional differential inclusion

x(t) ∈ K[Aix] + Aixt + Giw, a.e. t ≥ 0 (3.98)

whereK[Aix] = {� | � = αA1x + (1 − α)A2x, α ∈ [0, 1]}

Take Lyapunov-Krasovskii functional (3.19) with V1(xt(0)) and V2(xt) as

V1(xt(0)) =12xT Px (3.99)

V2(xt) =12

∫ 0

−τxT

t (−s)Qxt(−s)ds (3.100)

Then, along all the Filippov solutions of system (3.93) it follows from (3.98) that

L∗FV1(x) = max

F∈K[f1+f2](xt){LFV1(x)}

= max{xT PA1x, xT PA2x} + xT PAixt(τ)(3.101)

Moreover,L∗QV1(x) = max

Q∈K[f3](xt){‖LQV1(x)‖} = ‖xT PGi‖ (3.102)

In view of (3.101) and (3.102),we obtain that

L∗FV1(x) +

12γ2

(L∗QV1(x))2 +

12‖h(x)‖2 + Q(x) − Q(xt(τ))

= max{xT PA1x, xT PA2x} + xT PAixt(τ) +1

2γ2xT PGiG

Ti Px

+12xT CT Cx +

12xT Qx − 1

2xT

t (τ)Qxt(τ)

(3.103)

Noting that

xT PAixt(τ) ≤ 12xPAiQ

−1ATi Px +

12xT

t (τ)Qxt(τ) (3.104)


and considering (3.94) and (3.95), it follows from (3.104) that

L∗FV1(x) +

12γ2

(L∗QV1(x))2 +

12‖h(x)‖2 + Q(x) − Q(xt(τ))

≤ max{xT PA1x, xT PA2x} +12xPAiQ

−1ATi Px +

12γ2

xT PGiGTi Px

+12xT CT Cx +

12xT Qx

≤ max{xT PA1x, xT PA2x} +12xPR1Q

−1Px +1

2γ2xT PR2Px

+12xT CT Cx +

12xT Qx

From Riccati inequalities (3.96) and (3.97), the above inequality implies that condition(3.82) holds. Hence the conclusion follows by Theorem 3.8 as x ∈ R1. Analysis in the aboveconsiders that discontinuity of system (3.93) is caused by Aix. With similar argument,it is easy to prove the conclusion as discontinuity is on Aixt(τ) and Giw(t), respectively,and these cases involve the following functional differential inclusions

K[Aixt(τ)] = {� | � = αA1xt(τ) + (1 − α)A2xt(τ), α ∈ [0, 1]}and

K[Gi(xt(τ))w(t)] = {G | G = αG1w(t) + (1 − α)G2w(t), α ∈ [0, 1]}respectively. In other words, the conclusion can be achieved for any x ∈ {xt ∈ Cr | S1(xt) ∈R, S2(xt) ∈ R, S3(xt) ∈ R}. This completes the proof.

Example 3.5. Consider system (3.93) given by⎧⎪⎪⎪⎨⎪⎪⎪⎩x1 = −2x1 + x2t + sgn(x1)w1

x2 = −32x2 + x1t(τ) − sgn(x2)|x1t(τ)| + sgn(x1)w2

y = x2

(3.105)

The system is with C = [0 1]T ,

A1 = A2 =

⎡⎣−2 0

0 −32

⎤⎦⎧⎪⎪⎪⎨⎪⎪⎪⎩

A1 =[0 10 0

], S2(xt) = x1t(τ)x2 > 0

A2 =[0 12 0

], S2(xt) = x1t(τ)x2 < 0⎧⎪⎪⎪⎨⎪⎪⎪⎩

G1 =[1 00 1

], S3(xt) = x1 > 0

G2 =[−1 0

0 −1

], S3(xt) = x1 < 0


For a given γ = 1, we can find a matrix P and Q, respectively as follows

P = Q =[1 00 1

]that satisfy conditions (3.96) and (3.96) of Theorem 3.10.

Let r = 0.5. The curve plotted in Fig. 3.5 is a trajectory with the initial condition(x10(τ), x20(τ)) = (1, 0), τ ∈ [0, r]. It is clear that the trajectory has a sliding motion onthe surface S2 = {xt ∈ Cr | S2(xt) = x2t(0) = 0} during the initial period, and underdisturbance input w = [0.2 − 1]T , it finally tends to a bounded set around the origin.

Moreover, the sliding motion can be checked as follows. When the surface S2 is achievedfrom S2(xt) < 0 and S2(xt) > 0, the limiting functionals are

f+(x0) = limx2t(0)→0+

f(x) =

[−2x1 + w1

w2

](1 0)

=

[−1.8−1

]

f−(x0) = limx2t(0)→0−

f(x) =

[−2x1 − w1

2x1t(τ) − w2

](1 0)

=

[−2.2

3

]

The normal vector to the surface S2 is N = [0 1]T . Then, it obtains that the projection

f−N (x0) = 3 > 0, f+

N (x0) = −1 < 0

and f−N − f+

N > 0. Therefore, the trajectory stay in the surface S2 during a certain timeinterval 0 ≤ t ≤ t1, (t1 > 0) by Proposition 3.1.

Fig. 3.6 shows two trajectories of system of Example (3.5) starting from different initialconditions under same disturbance input, in one of which sliding motion occurs alongthe switching surface and in the other, the trajectory passes through it. These behaviorsillustrate that in the sense of Filippov solution, the discontinuous system of Example 3.5has L2-gain property whenever sliding motion occurs or not.

Remark 3.4. It should be noted that the essential of Theorem 3.10 is that a commonmatrix P exists for the system in different regions where the functional differential equationof the system dynamics is continuous. Moreover, if system (3.93) is observable in the senseof zero-state observable of nonlinear system (3.25)), it is clear that strong asymptoticstability of system (3.93) with w = 0 follows by Theorem 3.3.

Remark 3.5. From the above discussion it follows that in the sense of Filippov solution,the proposed L2-gain analysis condition (3.89) is deduced to be two Riccati inequalities(3.96) and (3.96) for the piecewise continuous linear systems. Indeed, the two conditionsimply that any case of the Filippov combinations of the matrices Ai, Ai and Gi (i =1, 2) satisfies an equivalent Riccati inequality with a common positive-definite matrix P .Clearly, conservativeness is obvious in the presented approach, since the derivative of


-0.4 -0.2 0 0.2 0.4 0.6 0.8 1

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

1.2

x1

x2x0

Fig. 3.5: A trajectory of Example 3.5.

-0.4 -0.2 0 0.2 0.4 0.6 0.8 1 1.2

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

x1

x2

x01

x02

-2

Fig. 3.6: Two trajectories of Example 3.5 under different initial conditions.

the Lyapunov-Krasovskii functional along the Filippov solution of the system can not beexclusive determined at the surface where the vector field of the system is discontinuous.On the other hand, for the considered piecewise continuous linear system, the conditions(3.96) and (3.97) may be waken by using a piecewise quadratic functional instead of thecontinuous differentiable one (3.99) and (3.100). In this case, the condition may not alwaysapplicable in the sense of Filippov solution, i.e. the solutions of the system on the surfacebetween two modes in the state space can not be evaluated. Here, detailed discussion willnot be shown and we refer to [92] for a particular description on this issue.

3.4. Concluding Remarks 71


In this chapter, we propose a functional differential inclusion based framework to investi-gate the analysis and synthesis problems of discontinuous time-delay systems. The mainresults are established by means of generalized solutions satisfying a set-valued functionalformulated by extending the differential inclusion in the sense of Filippov. Under theextension, the Lyapunov-Krasovskii theorem and the invariance principle in the sense ofLaSalle for functional differential inclusions are provided to analyze stability of discontin-uous time-delay systems. Furthermore, a feedback design approach is provided such thatthe closed-loop systems are strongly stable in the sense of Filippov solution. The designapproach is also extended with adaptive control techniques to deal with systems withuncertain parameters. Moreover, a feedback design approach that concerns the L2-gainperformance combined with strong asymptotic stability is also shown achievable along thisresearch line.

In this study we focus on using a Lipschitz continuous Lyapunov-Krasovskii functionalwith special structure usually employed in the domain of time-delay system. On the basisof the nonsmooth analysis theory introduced previously, that is demonstrated useful toinvestigate discontinuous time-delay systems. Moreover, it should be noted that the pro-posed design approach is a kind of dominated-based control, which is motivated by theclassical robust design principle. It is fortunately benefited by using a smooth Lyapunov-Krasovskii functional. The approach has a key point that the Lyapunov-Krasovskii func-tional is constructed by properly using the information of the system. It turns out to beeasily applicable synthesizing the class of discontinuous time-delay systems considered.

The formulated framework provides a feasible way to deal with analysis and synthesisissues of the time-delay systems with discontinuity. Noted that in this chapter, the pro-posed control laws for stabilization and disturbance attenuation depend on the delay timeof the system. With the functional differential inclusion based framework, it is consideredto deserve attention is to develop delay-independent design approach for the discontinu-ous time-delay systems. Further effort is also being made to design stabilization controllaws for general discontinuous time-delay systems, such as integrator cascaded systems,systems with constrained inputs.

72

Chapter 4

Model-based Control of ICEngines

In this chapter, feedback speed control problem of internal combustion (IC) engines isinvestigated based on mean-value models that involve the intake-to-power delay. Twocontrol schemes that focus on dealing with transition speed control are discussed. First,by treating the system to be dual inputs single output, i.e. taking SA and throttle openingas inputs and engine speed as output, a systematic Lyapunov-based design approach ispresented without requiring the full information of model parameters; meanwhile, forimproving transient performance, an approach using switching boost control techniqueis provided. On the other hand, the case with only throttle opening as control input isconsidered; in this case, considering that the parameters in the mean-value engine modelare not constants to characterize the dynamics of engine in different operation modes,switched modeling is proposed; under this situation, a solution is given to the speedtracking control problem considering a specified operation mode; exact theoretical analysisof the main result is given by means of the proposed functional differential inclusion basedtools.

The first section includes the review for the engine model, the description of an enginetest bench and model validations served for this investigation. The control scheme for thespeed tracking control is introduced in the next section. Finally, a brief discussion of thepresented result is given.

4.1 Control-oriented Engine Model

This section gives a review of the physics based with partial parameterized mean-valueengine model. A case study of engine model validation is conducted on an engine testbench for applying the reported model to control development.

73

74 Chapter 4. Model-based Control of IC Engines

Table 4.1: Notations of physical parameters.Notation Meaning Unitκ Specific heat ratio [-]R Gas constant [J/(kg · K)]s0 Throttle area [m2]J Crank inertia moment [kg · m2]pa Atmospheric pressure [Pa]Ta Atmospheric temperature [K]ρa Atmospheric density [kg/m3]ma Air mass in intake manifold [kg]pm Intake manifold pressure [Pa]Tm Intake manifold temperature [K]Vm Intake manifold volume [m3]Vc Cylinder volume [m3]η Volumetric efficiency [-]φ Throttle opening [rad]mi Air mass flow rate through throttle [kg/s]mo Air mass flow rate into cylinders [kg/s]τe Engine torque [Nm]τf Friction torque [Nm]τl Load torque [Nm]ω Engine speed [rad/s] or [rpm]λa Air-fuel ratio (A/F) [-]uf Fuel injection command [kg]mfc Fuel film mass [kg]Sa Spark advance [rad]Mbt Minimum spark advance for best torque (MBT) [rad]

4.1.1 Mean-value Model

A schematic representation of a four-stroke SI engine is as shown in Fig. 4.1, where ECUis the electronic control unit that provides inputs and outputs signals of the engine system.In an engine with multi-cylinders, the torque to drive the crankshaft rotational motionis provided by each cylinder serially along the crank angle, and the torque generated ineach cylinder during its own expansion stroke is determined by individual air charge andfuel injection during the corresponding induction stroke. The behaviors of air charge andfuel injection are influenced by the air intake path and fuel path of the engine system,respectively. Thus, the dynamics can be basically divided into three parts: air intakedynamics, fuel injection dynamics and crankshaft rotational dynamics. Typically, thecontrol inputs are the throttle opening, the fuel injection and the spark advance (SA).

4.1. Control-oriented Engine Model 75

Air Flow Meter Intake Manifold

AirThro�le

Port Ingector

Star�ng Motor

Ou�ake Manifold

Mul�-Cylinders

Cranksha�

Spark

Wall Film

Command

Crank Angle&

Engine Speed

Direct Ingector

ECU

Fig. 4.1: Schematic representation of an IC engine system.

From the above description of an IC engine with multi-cylinders, exact descriptionsof the system dynamic behavior will generate a hybrid system that consists of continuoustime modeling and event-based discrete ones. If we focus on the mean-value characteristicsof the state variables of the dynamics, it can be represented as continuous nonlinear statespace equations, and usually the state variables can be chosen correspondingly as fuel filmmass, the intake manifold pressure and the crankshaft rotational speed, respectively, andactuation signals as throttle opening, fuel injection and SA, respectively. The followingshows a brief review of the derivation of the related dynamical models. Symbol nomencla-ture of physical parameters are shown in Table 4.1. Fig. 4.2 shows a logical block diagramof the engine dynamics where the functions fi(·) (i = 1, · · · , 5) denote the mathematicaldescriptions of engine behaviors.

f1(φ, pm) Integrator f2(pm, ω)

thro�le body

td

intake-to-power

delay

Air intake path dynamics

f3(ω, λa/f, SA)

engine torque

genera�on

Iner�al

Cranksha� rota�onal dynamics

mi

.ωmo

.pm τe

τl

τf

f4(ω)

mfcuf

+-

φ

Fuel path dynamics

SA f5(Sa)

spark �ming

+-

-

fric�on

load

wall film

Fig. 4.2: Block diagram of mean-value model.


Fuel Path Dynamics The so-called wall-film model is widely used to represent theamount of fuel mfc injected into each cylinder during the induction stroke [2], i.e.{

mf = −χmf + εuf

mfc = χmf + (1 − ε)uf

(4.1)

where mf represents the fuel mass entering the intake port per induction stroke, ε repre-sents the fraction of fuel deposited on the intake port and χ is time constant.

Air Intake Path Dynamics The air intake path dynamics relates to the engine state ofmanifold pressure is from the ideal gas law

pmVm = maRTm (4.2)

Many researches (see for example [23, 36]) point out that an adequate description of airinlet path behavior should be given by considering adiabatic intake manifold. Assumingthat the manifold temperature is constant and equals to the atmospheric temperature Ta,differentiating (4.2) obtains

pm =RTa

Vmma (4.3)

In the intake manifold, the conservation law of air mass gives the following equation

ma = mi − mo (4.4)

The air mass flow through throttle body is modeled as

mi = A(φ)ψ(pa, pm) (4.5)

where A(φ) denotes the available area of manifold modeled as

A(φ) = s0(1 − cos φ)

and ψ(pa, pm) denotes a nonlinear function used to describe air flow passing through anorifice, i.e.

ψ(pa, pm) =

⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩

pa√RTa

(pm

pa

) 1κ

√√√√ 2κ

κ − 1

[1 −

(pm

pa

)κ−1κ

],

pm

pa≥

(2

κ + 1

) κκ−1

pa√RTa

√κ

(2

κ + 1

)κ+1κ−1

, others

Consider that under the environment of manifold, the air mass into cylinders per cycleis mo with the volume Vo which associates with cylinder displacement by the parameter,volumetric efficiency [25], i.e.

Vo = ηVc (4.6)


Meanwhile, one has the following ideal gas law

pmVo = moRTm (4.7)

Combining (4.6), (4.7) and the cycle tc = 4π/ω sec. (for engines with six cylinders) gives

mo =mo

tc=

ηVc

4πRTmpmω (4.8)

Furthermore, introducing atmospheric pressure with paVm = mRTm (m is a counterpartair mass in the manifold) to the above model (4.8) obtains

mo =ρaηVc

4πpapmω (4.9)

Finally, the intake air dynamics is modeled from (4.3)∼(4.9) by

pm =RTas0

Vm(1 − cos φ)ψ(pa, pm) − RTas0

Vm

ρaηVc

4πpapmω (4.10)

Actually, the effectiveness of the model (4.9) is ensured under pseudo-static modes,and the value of η typically depends on engine speed, manifold pressure and temperatureof cylinders, etc. (see [23], [43], [75]).

Crankshaft Rotational Dynamics Crankshaft rotational dynamics is obtained fromNewton’s law as follows

Jω = τe − τf − τl (4.11)

As mentioned before, the torque of an engine with multi-cylinders is serially generated.Hence, its exact description should be represented by an event-based discontinuous math-ematical model, however, it will cause difficulties in control design. Mean-value of enginetorque is developed to character the average characteriztion of the engine torque. Con-sider that Q denotes the heat release from unit air with complete combusting at standardconditions and ηf denotes the engine efficiency per cycle. Moreover, due to the compres-sion stroke between the intake stroke and combustion stroke, engine torque generationinvolves the delay time called intake-to-power delay (denoted by td) which depends onengine speed. Under the assumption that the mechanical efficiency equals to the thermalconversion efficiency, the nominal average engine torque τe can be computed by

τe(t) =ηfQmo(t − td)

ω(t)(4.12)

On the other hand, both the A/F and SA can influence engine torque generation. Letfa(λa) ∈ [0, 1] and fs(Sa) ∈ [0, 1] denote the normalized influence of A/F and SA on theengine torque, respectively. Then, substituting the air mass flow rate expression (4.9) into(4.12) and considering that ω(t− td) equals to ω(t) approximately, the mean-value enginetorque is given as follows

τe(t) =ρaηVcηfQ

4πpapm(t − td)fa(λa)fs(Sa) (4.13)


In fact, fs(Sa) characterizes the relation between SA and MBT and in the engine litera-tures, for example [25], [69], it is modeled as

fs(Sa) = [cos(us − Mbt)]2.875 (4.14)

in which us denotes control input Sa and for a special real engine, slight modification formodel coefficient 2.875 may be needed [69]. The friction torque of engine can be simplymodeled as[25]

τf = D0ω + D1 (4.15)

where D0 and D1 are constants.

4.1.2 Model Validation

The parameters involved in the models reported above are crucial to develop engine con-trol algorithms. In automotive engineering, there is significant analysis of steady-stateoperation conditions of engines, however, very little actual dynamical engine model datafor control development is available to the control engineering. This section presents avalidation test on an engine test bench at the Laboratory of Sophia University as shownin Fig. 4.3. The engine is a 2GR-FSE IC engine supported by Toyota Motor Corporation.Basic specifications of the engine are listed in Table 4.2.

Fig. 4.4 shows a sketch of the engine test bench set up. A load torque is imposed tothe engine by a dynamometer. The sensors, including air flow meter, CAM crank angleencoder, manifold pressure sensor, sensors to measure the individual cylinder pressuresand sensors placed on individual cylinders and the gas-mixing point, etc. are equippedfor monitoring, testing and designing control algorithms. It includes the ECU controlalgorithm that serves for engine control in practical vehicles. The control software of thetest bench has been modified such that the engine can accept the control commands ofdesigners, which is realized by the dSPACE instrument. Control algorithms that are builtin Matlab/Simulink by the designers are downloaded to dSPACE, then are delivered toECU to control engine through a standard CAN bus.

Fig. 4.3: Engine test bench: instruments (left), 2GR-FSE engine (right).


Table 4.2: Basic engine specifications (2GR-FSE engine).Number of cylinders 6Bore 94.0 mmStroke 83.0 mmCompression ratio 11.8Crank radius 41.5 mmConnection rod length 147.5 mmCombustion chamber volume 53.33 mm3

Displacement 3.456 LManifold volume 6.833 LFuel injection D-4SInjection sequence 1-2-3-4-5-6

PC

ECU

dSPACE

Engine

Sensors

CAN Block, A/D, D/AMatlab/Simulink,

Control Desks

CAN Bus

Dynamometer

Fig. 4.4: Engine test bench at Sophia University.

Under the situation that the A/F and SA are controlled at the optimized value byother control loops, i.e. fa(λa) = 1 and fs(Sa) = 1, engine models related to the air intakepath dynamics (4.10) and crankshaft rotational dynamics (4.11) presented in the previoussection can be summarized as follows

[ω(t) pm(t)]T = fe(ω(t), pm(t − td))

=

[a1pm(t − td) − D0ω(t) − TD

auuth − a2pm(t)ω(t)

] (4.16)

where the parameters are defined by

a1 =ρaηVcηfQ

4Jπpa, D0 =

D0

J, TD =

D1 + τl

J,

au =RTas0ψ(pa, pm)

Vm, a2 =

RTa

Vm· ρaηVc

4πpa

(4.17)


and the throttle opening related input uth is defined by

uth = 1 − cos φ (4.18)

Under the case that the structure of mean-value engine model is determined, the aim isto identify the parameters a1, D0, TD, au and a2 with dynamic experimental data.

To perform identification, we take the approximate discrete-time models of enginedynamics (4.16) with sampling interval T = 0.5 × 10−3 sec.. First, for simplicity, weconsider the operation of engine is around an equilibrium (ω0, pm0) and rewrite the crankrotational dynamics as

˙ω = a1pm(t − td) − D0ω

where ω = ω − ω0 and pm = pm − pm0. In this case, we have that TD = a1pm0 − D0ω0.Then, the discrete-time models to be identified are as follows,

pm(tk+1) − pm(tk) = auTuth(tk) − a2Tpm(tk)ω(tk) (4.19)

ω(tk+1) = (1 − D0T )ω(tk) + a1T ω (4.20)

where tk = kT , k = 1, · · · , n denote the sampling instants. Based on the above discretemodels, it is now possible to use parameter identification techniques to find the modelvalues.

After the engine is full warm-up (water temperature is 80 ◦C), by sending step com-mands to throttle opening, we recall experimental data of engine speed, intake manifoldpressure and throttle opening from ECU. With these off-line data, recursive least squarealgorithm is applied to the dynamical models (4.19), (4.20), respectively. Table 4.3 liststhe identified parameters under two operation conditions of engine with respect to throttleopening. Fig. 4.5 and Fig. 4.6 show the commands signals and outputs of model versusthe corresponding experimental data.

From the identification results, it can be noted that engine models (4.16) with constantparameters are basically effective to characterize the dynamics under certain operation con-ditions with respect to a nominal engine speed. In fact, it is clear that from the definition(4.17), physical parameters depend on engine variables under different operations, such asthe volumetric efficiency that depends on the engine states (ω, pm) extremely [75].

Table 4.3: Model parameter of mean-value engine modelMode Model parameter

No. ω/rpm a1 D0 TD au a2

×10−3 ×109

1 1500 7.8 0.208 118.97 3.38 1.942 2000 6.8 0.107 128.237 3.88 2.61


0 10 20 30 40 50 60 70 803

3.2

3.4

3.6

Time [s]

Th

ro�

le o

pe

nin

g [

de

g.]

0 10 20 30 40 50 60 70 801.8

1.9

2

2.1

2.2

Time [s]

Ma

nif

old

pre

ssu

re [

Pa

]

0 10 20 30 40 50 60 70 801450

1500

1550

1600

1650

Time [s]

En

gin

e s

pe

ed

[rp

m]

x 104

Exp. dataModel output

CommandExp. data


Fig. 4.5: Experimental data vs. model output of (4.16): ω = 1500 rpm.

0 10 20 30 40 50 60 70 804

4.2

4.4

4.6

Time [s]

Th

ro�

le o

pe

nin

g [

de

g.]

0 10 20 30 40 50 60 70 801.8

1.9

2

2.1

2.2

Time [s]

Ma

nif

old

pre

ssu

re [

Pa

]

0 10 20 30 40 50 60 70 801950

2000

2050

2100

Time [s]

En

gin

e s

pe

ed

[rp

m]

x 104


CommandExp. data


Fig. 4.6: Experimental data vs. model output of (4.16): ω = 2000 rpm.


4.2 Speed Tracking Control

Based on the mean-value engine model (4.16), speed tracking control problem is inves-tigated. For the speed control of SI engines, it is well-known that the intake-to-powerdelay is one of the main difficulties in the phases of control design and evaluating. In thissection, nonlinear feedback control algorithms are developed for speed tracking controlproblems, i.e. for a given desired speed trajectory ωd(t), design a feedback controller

uth = α(ω, pm) (4.21)

such that ω(t) → ωd(t) as t → ∞. Taking the intake-to-power delay into account, two casesare investigated. First, the control problem is solved under the situation that a feedbackcompensation of SA is introduced to deal with the effort of time delay. In this case, themain control algorithm for throttle is developed based on a delay-free dynamical system,while in the second case, a feedback design approach for throttle control is provided onthe basis of the mean-value engine mode involving the time delay exactly.

4.2.1 Feedback Control with Delay Compensation

Considering the mean-value engine torque (4.13) and model (4.14), we choose us to satisfythe following equations, which is a slight modification of the control law in [95].

us = Mbt + arccos

[(pm(t)[cos(−Mbt)]2.875

pm(t − td)

) 12.875

](4.22)

Then, the engine dynamics including the air intake path dynamics and crankshaft rota-tional dynamics can be rewritten as

[ω(t) pm(t)]T = fe(ω(t), pm(t))

=

[a1pm(t) − D0ω(t) − TD


] (4.23)

In the following we show a Lyapunov-based design approach to the speed control prob-lem. It can be noted that system (4.23) is exactly feedback linearizable if the parametersof the system and external load torque are known and measurable, respectively. In thiscase, the problem is easy by a routine work with linear control techniques. However, whenthe parameters are unknown or/and the external load torque is not exactly measurable,the problem becomes much complicated. In the following, we will show that by puttingphysical characteristics into Lyapunov-based design, a simple feedback controller can beobtained although the parameters are partially unknown and the disturbance is not mea-surable. Furthermore, we will show later that the proposed controller can be extendedusing boost action for further improvement of the performance.

4.2. Speed Tracking Control 83

Consider system (4.23). For a given desired trajectory ωd(t), ωd(t), ωd(t), when thetracking performance is achieved, i.e. the speed error eω(t) = ω(t) − ωd(t) = 0 andeω(t) = 0, the manifold pressure pm must satisfy

pm(t) = pmd(t) :=1a1

(D0ωd(t) + ωd(t) + TD(t)

)(4.24)

Define ep(t) = pm(t) − pmd(t). Let x = [eω ep]T . Then, the tracking dynamics thatdetermines the error between the desired trajectory and system 4.23 can be coordinatedby (eω ep) as follows:

x(t) = fe(ω(t), pm(t)) − fe(ωd(t), pmd(t))

=

⎡⎣ −D0eω + a1ep(t)

auuth − a2pmω − 1a1


)⎤⎦ (4.25)

It should be noted that if all parameters and signals of the tracking dynamics (4.25)are exactly known and measurable, then the problem is trivial. Since in this case, byintroducing the following nonlinear compensation

uth =1au

[a2pmω +

1a1


)]+ v (4.26)

the poles of the second order linear system can be assigned to any desired location by thenew control signal v. However, if the parameters are unknown or the signals TD, ep areunmeasurable, this idea is not realizable.

In the following, we consider three cases where it can be seen that by putting thephysics to the design, it is possible to obtain a desired controllers with simple structure.

Case 1. Suppose that the parameters a1 and D0 are unknown and the external inputsignal TD is not measurable but constant. In this case, it is obvious that the pressure errorep is also unmeasurable. However, from the physical consideration, we have a1 > 0 andD0 > 0. Making better use of these physical characteristics, we have the following result.

Proposition 4.1. Consider regulation problem, i.e. ωd = 0, ωd = 0. Let the feedbackcontrol law (4.21) be given with

α(ω, pm) =1au

(−keω + a2pmω) (4.27)

where k is any positive constant. Then, the closed-loop system (4.23) with (4.27) is globallyLyapunov stable and for any given initial condition (eω(0), ep(0)) and any constant externalinput signal TD, the tracking error of engine speed eω asymptotically converges to zero ast → ∞.


Proof. Notice that a1 > 0. Consider a candidate of Lyapunov function as

V (eω, ep) =12e2ω +

a1

2ke2p (4.28)

Along any trajectory of the closed-loop system, it can be shown that

V (eω, ep) = −D0e2ω, ∀t ≥ 0 (4.29)

holds for any constant TD. The physical characteristic D0 > 0 means that V (eω, ep) ≤0, ∀t ≥ 0, which concludes the global stability by Lyapunov stability theory. Moreover,(4.29) implies the set Ω = {(eω, ep) | V (eω, ep) = 0} = {(eω, ep) | eω = 0}. Hence, theconvergency of eω follows by LaSalle invariance principle.

Case 2. Suppose that ωd = 0 and ωd = 0. In this case, it is necessary to know thevalues of parameters a1, D0, au and a2 exactly in order to compensate the effort of ωd

and ωd in the tracking dynamics. In the following, we will discuss this issue by a slightmodification of Proposition 4.1. A desired tracking controller can also be obtained withoutmeasurement of signal ep and external input signal TD.

Proposition 4.2. Consider system (4.23). For any given ωd, ωd, ωd, let the feedbackcontrol law (4.21) be given by

α(ω, pm) =1au

[−keω + a2pmω +

1a1

(D0ωd + ωd

)](4.30)

Then the closed-loop system is globally Lyapunov stable, and for any initial condition(eω(0), ep(0)) and any constant external input signal TD, the tracking error of enginespeed eω asymptotically converges to zero as t → ∞.

Proof. Notice that the closed-loop system is represented by{eω(t) = −D0eω + a1ep

ep(t) = −keω

(4.31)

Thus, using the candidate of Lyapunov function (4.28), the conclusion can be deduced bythe same argument with LaSalle invariance principle.

Remark 4.1. As shown in Lyapunov function (4.29), the damping coefficient with respectto the speed error eω is D0 which is determined by the physical parameters. The controllergain k does not contribute directly to this damping. In other words, it will cause thelimitation in improving the transient performance. As we will shown in the following, it ispossible to improve the transient performance by additional actions in the control inputduring the transient mode, but it will lose the smoothness as extra cost.


A possible approach to improve the transient performance is via introducing additionaldriving force for the control law proposed above to drive the manifold pressure error ep

to zero more quickly such that the speed error can be forced by the desired pressure assoon as possible. The approach has the proposed controller-based structure, i.e. let thecontroller be given with the following structure

uth = α(ω, pm) +1au

β(t) (4.32)

where

α(ω, pm) =1au

[−keω + a2pmω +

1a1

(D0ωd + ωd

)]and β(t) is the additional control action to be designed. For simplicity, we suppose thatall parameters a1, D0, au and a2 are known, and the external input signal TD is unknownbut constant.

Case 3. Consider the acceleration operation mode, i.e. ωd(0) ≥ ω(0). In order toobtain rapid acceleration rate, the throttle opening, from physical consideration, shouldbe enough such that the generated torque can provide the required acceleration. However,the feedback control laws proposed in Proposition 4.1 and 4.2, which guarantee the sta-bility only, can not provide enough throttle opening during the acceleration period. Thefollowing proposition shows that by introducing a boost action with β(t), we are able toimprove the acceleration performance with guaranteed stability.

Proposition 4.3. Consider system (4.23) with control law (4.32). Let

β(t) =

{β0, 0 ≤ t ≤ tb

0, tb < t(4.33)

where β0 and tb are given positive numbers. Then, for any given ω(0) and ωd (ω(0) <ωd(0)), there exist β0 and tb such that the tracking dynamics is stable and eω converges tozero asymptotically for any external input TD. Furthermore, t′e0

< te0, where t′e0denotes

the first timing where the trajectory eω(t) reaches zero from the initial condition, and te0

is the corresponding timing when β0 = 0, respectively.

Proof. The closed-loop system (4.23) with (4.32) and (4.33) can be rewritten as

x = Ax + Bβ(t) (4.34)

where

A =[−D0 a1

−k 0

], B =

[01

]Denote the eigenvalues of matrix A as λ1,2 = −m ± jn, (m,n > 0). Then, we can obtainthe response when β(t) = 0 as follows

eω(t) = γ1(t)eω(0) + γ2(t)ep(0) (4.35)


where[γ1(t) γ2(t)] = L−1

{[1 0](SI − A)−1

},

γ1(t) = a1e−mt sin(b1t + d1), γ2(t) = a2e

−mt sin(b2t + d2)

and ai, bi, di (i = 1, 2) are coefficients determined by the eigenvalues. When the initialcondition eω < 0, ep < 0, an image of (4.34) is sketched by the dashed line in Fig. 4.7.For simplicity, we now consider the case ep(0) = 0. Under the boosting action β(t) = 0,the response of the tracking error during t ≤ tb can be given by

e′ω(t) = γ1(t)eω(0) + [1 0]∫ t

0eA(t−τ)Bβ0dτ (4.36)

Note that in the acceleration period, γ1(t)eω(0) < 0, ∀ 0 ≤ t ≤ te0 and β0 > 0. It is clearthat for any given t′e0

(< te0), a sufficient large β0 > 0 can ensure e′ω(t) reaches zero at t′e0

as shown in Fig. 4.7.

tb te0 t

eω

0

eω(0)

eω(t)

e′ω(t)

β(t)

t′e0

Fig. 4.7: An image of speed error response.

Remark 4.2. Proposition 4.3 implies that with the sufficient boost action, the responseability of acceleration can be improved in the sense of shortening the rising time te0 .However, as a by-product, undesired overshoot will be caused by larger value of β0. Indeed,if we want to shorten te0 to a given t′e0

for the same initial condition, the chosen of tb shouldbe close to the given t′e0

, and β0 can be determined by

e′ω(t) = γ1(t)eω(0) + γ2(t)β0, ∀ t < tb (4.37)

i.e.

β0 = −γ1(t′e0)

γ2(t′e0)eω(0)

We simulate the system (4.23) with the presented control laws (4.27), (4.30) and (4.32)with (4.33), respectively for comparing the design methods. The model parameters in thesimulation are a1 = 1.748 × 10−3, D0 = 0.339, au = 1.485 × 108 and a2 = 5.66 × 10−2. In


the simulations, command signals are given at t = 5 sec.. Case 1 is simulated under thefollowing external input signal

TD =

{0, t < 50 sec.20, t ≥ 50 sec.

By giving a step reference signal, Fig. 4.8 shows the response curves under two feedbackgains, i.e. k = 200, and k = 500, respectively. It can be observed that in both cases,the speed tends to the desired trajectory even in the presence of the constant externalinput. To simulate case 2, a smooth time-varying reference signal is given. From theresult shown in Fig. 4.9, we can observe that the tracking performance is same as case1. For comparison, the results under the switching boost control in case 3 are shown inFigure 4.10. The feedback gain is set at k = 300, and two different boost parameters aredemonstrated with

(I) : β0 = 0, tb = 0; (II) : β0 = 0.005×10−8, tb = 0.3; (III) : β0 = 0.0055×10−8, tb = 0.5

The results show that the rising time is shortened under the boost action, and faster speedresponse can be achieved if β0 is larger, however, it causes undesired overshoot.

Finally, validation experiments are conducted on the engine test bench (shown inFig. 4.3) with a 3GR-FES (V6-3L-4D) engine. The identified model parameters area1 = 3× 10−3, D0 = 0.3, au = 1.78× 107 and a2 = 5.7× 10−3. Fig. 4.11 shows the resultsunder control law 4.27, in which the feedback gain is set at k = 400, and external inputload provided by the dynamometer is as follows

τl =

⎧⎪⎪⎨⎪⎪⎩0, t < t1

20, t1 ≤ t < t2

0, t ≥ t2

The tracking performance under control law (4.30) is shown in Fig. 4.12. The performanceunder the boost action β(t) given by control law (4.33) is tested with the control parametersin Table 4.4, and the results are shown in Fig. 4.13. It can be seen that by choosing enoughcontrol parameters tb and β0, the acceleration performance can be improved.

Table 4.4: Parameters of the boost control law (4.33).I II

β0(×10−8) 0 0.45 0.45 0.75 1.2tb 0 0.3 0.5 0.3 0.3


14

00

16

00

18

00

2.53

3.54

4.5 3

3.54

4.5

01

02

03

04

05

06

07

08

09

01

00

Tim

e [

s]

x 1

04

ωωd

τ lk

=2

00

k=

50

0

01

02

03

04

05

06

07

08

09

01

00

Tim

e [

s]

01

02

03

04

05

06

07

08

09

01

00

Tim

e [

s]

Engine speed [rpm] Maniford pressure [Pa]Thro�le opening [deg.]

Fig

.4.8

:Si

mul

atio

nre

sult

ofca

se1.

14

00

16

00

18

00 3

3.54

3.4

3.6

3.84

4.2

01

02

03

04

05

0T

ime

[s]

01

02

03

04

05

0T

ime

[s]

01

02

03

04

05

0T

ime

[s]

x 1

04


Fig

.4.9

:Si

mul

atio

nre

sult

ofca

se2.

14

00

16

00

18

00 3

3.54

4.5 3

3.54

4.5

05

10

15

20

25

30

35

40

45

50

05

10

15

20

25

30

35

40

45

50

05

10

15

20

25

30

35

40

45

50


ωωd

x 1

04

Tim

e [

s]

Tim

e [

s]

Tim

e [

s]

II

I

III

Fig

.4.

10:

Sim

ulat

ion

resu

ltof

case

3.


3

3.5

4

1500

1600

1700

1800

1.8

2

2.2

2.4

0 5 10 15 20 25 30 35 40 45 50

0 5 10 15 20 25 30 35 40 45 50

0 5 10 15 20 25 30 35 40 45 50

En

gin

e s

pe

ed

[rp

m]

Ma

nif

ord

pre

ssu

re [

Pa

]T

hro

�le

op

en

ing

[d

eg

.]

ω

ωd

x 104

τl

τl

Time [s]

Time [s]

Time [s]

t1 t2

Fig. 4.11: Experimental result of case 1.

1550

1600

1650

1700

1750

1.95

2

2.05

2.1

2.15

2.6

2.8

3

3.2

3.4

0 5 10 15 20 25 30 35 40 45 50

Time [s]

0 5 10 15 20 25 30 35 40 45 50

Time [s]

0 5 10 15 20 25 30 35 40 45 50

Time [s]

3.6

x 104

En

gin

e s

pe

ed

[rp

m]

Ma

nif

ord

pre

ssu

re [

Pa

]T

hro

�le

op

en

ing

[d

eg

.]

ω

ωd


1550

1600

1650

1700

1750

1.95

2

2.05

2.1

2.15

2.2

0 5 10 15 20 25 30 35 402.8

3

3.2

3.4

0 5 10 15 20 25 30 35 40

0 5 10 15 20 25 30 35 40

En

gin

e s

pe

ed

[rp

m]

Ma

nif

ord

pre

ssu

re [

Pa

]T

hro

�le

op

en

ing

[d

eg

.]

x 104

Time [s]

Time [s]

Time [s]

I1

I2

I3

(a)

1550

1600

1650

1700

1750

2

2.1

2.2

2.3

3

3.5

4

0 5 10 15 20 25 30 35 40

Time [s]

0 5 10 15 20 25 30 35 40

Time [s]

0 5 10 15 20 25 30 35 40

Time [s]

En

gin

e s

pe

ed

[rp

m]

Ma

nif

ord

pre

ssu

re [

Pa

]T

hro

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op

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[d

eg

.]

II1

II2

(b)



4.2.2 Feedback Control without Delay Compensation

In this case, we provide a solution to the speed control problem based on the time-delayengine model (4.16). Moreover, it is noted from Section 4.1 that, the normalized parame-ters of the mean-value engine model (4.16) are not always constants in different operationmodes with respect to engine states. In this situation, the variation of the parameters isone of the important issues to apply model-based speed tracking control schemes in a wideoperation range. We consider a case that the engine dynamics is captured with differentparameters by the mean-value engine model (4.16) corresponding to two operation modes.The purpose is to provide a feedback control law based on the Lyapunov-Krasovskii theo-rem in terms of functional differential inclusion in Chapter 3 to guarantee the asymptoticconvergency of engine speed.

Consider that the switching of engine operation modes depends on engine speed ω∗.Let the engine dynamics (4.16) be characterized with parameters (a∗1, D∗

0, T ∗D, a∗u, a∗2) in

the operation range where locates the equilibrium (ωd, pmd). Otherwise, the dynamics ischaracterized with parameters (a′1, D′

0, T ′D, a′u, a′2). Denote

a1 = a′1 − a∗1, D0 = D′0 − D∗

0, TD = T ′D − T ∗

D, au = a′u − a∗u, a2 = a′2 − a∗2

Then, engine dynamics (4.16) can be represented with

fe (ω(t), pm(t − td)) =[a∗1pm(t − td) − D∗

0ω(t) − T ∗D

a∗uuth − a∗2pm(t)ω(t)

]+

[a1pm(t − td) − D0ω(t) − TD


]δ(ω, ω∗)

(4.38)

where

δ(ω, ω∗) =

{0, ω > ω∗

1, ω < ω∗(4.39)

Consider regulation problem, i.e. ωd(t) = 0, ωd(t) = 0. We first discuss the case whenωd > ω∗. The error dynamics of system (4.38) can be obtained as follows:

x(t) = fe(ω(t), pm(t − td)) − fe(ωd, pmd)

=

⎡⎢⎢⎢⎢⎢⎢⎢⎣

−D∗0eω + a∗1ep(t − td) + (a1ep(t − td) − D0eω)δ(eω)

+(a1p∗m − D0ωr − TD)δ(eω)

(a∗u + auδ(eω)) uth − a∗2p∗meω − a∗2ωep

+(auu∗th − a2pmω)δ(eω)

⎤⎥⎥⎥⎥⎥⎥⎥⎦(4.40)

where uth = uth − u∗th, u∗

th denotes corresponding control input of throttle opening at theequilibrium (ωd, pmd) and satisfies the following condition

a∗uu∗th − a∗2pmdωd = 0 (4.41)


and

δ(eω) =

{0, eω > ω∗ − ωd

1, eω < ω∗ − ωd

It is clear that the speed control problem is actually to find a feedback control law

uth = α(eω, ep) (4.42)

with K[α](0) = 0 such that eω → 0 as t → ∞. For the sake of simplicity, we treat thedelay time td in the model (4.38) as constant, i.e., it takes the nominal value at the desiredspeed ωd, td = π/ωd. Taking the physics that the model parameters a1 > 0 and D0 > 0,we propose the following result.

Proposition 4.4. Consider system (4.40). If the following condition

a1pmd − D0ωd − TD > 0 (4.43)

holds, then for any initial condition x0(td), the state x(t) converges to zero as t → ∞under feedback controller (4.42) with

α(ω, pm) =1

a∗u + auδ(eω)[a∗2p

∗meω + a∗2ωep − k1ep − k2eω − (auu∗

th − a2pmω) δ(eω)]

(4.44)where k1 and k2 are any given constants satisfying the following conditions, respectively,

k1 >12, 0 < k2 <

D∗0 − |D0|

a∗1 + |a1|√

2k1 (4.45)

In other words, for a given ωd, the feedback controller (4.21) with

α(ω, pm) = u∗th + uth (4.46)

given by (4.41) and (4.44) guarantees that ω → ωd, as t → ∞.

Proof. The closed-loop system (4.40) with (4.44) is represented by

x(t) = fc(xt) =

⎡⎢⎢⎢⎣−(D∗

0 + D0δ(eω))eω + (a∗1 + a1δ(eω)) ep(t − td)

+(a1p∗m − D0ωr − TD)δ(eω)

−k2eω − k1ep

⎤⎥⎥⎥⎦ (4.47)

It is noted that function fc(x) is piecewise continuous associated with the following Filip-pov set-valued mapping,

K[fc](xt) ⊂

⎡⎢⎢⎢⎣−(D∗

0 + D0K[δ](eω))

eω + (a∗1 + a1K[δ](eω)) ep(t − td)

+(a1p∗m − D0ωr − TD)K[δ](eω)

−k1ep − k2eω

⎤⎥⎥⎥⎦


where

K[δ](eω) =

⎧⎪⎪⎪⎨⎪⎪⎪⎩0, eω > ω∗ − ωd

[0, 1], eω = ω∗ − ωd

1, eω < ω∗ − ωd

Choose a continuous differentiable, positive-definite function V1 as follows

V1(xt(0)) =γ

2e2ω +

12e2p (4.48)

where γ is given by

γ =D∗

0 − |D0| +√(

D∗0 − |D0|

)2 − 2 (a∗1 + |a1|)2 ε

(a∗1 + |a1|)2(4.49)

with a given ε satisfying

0 < ε <

(D∗

0 − |D0|)2

2 (a∗1 + |a1|)2(4.50)

Furthermore, take (3.19) with (4.48) associated with a q(x) chosen by

q(x) =12‖ep‖2 (4.51)

as candidate of Lyapunov-Krasovskii functional. According to Definition 3.4, we have

Md

(K[fc](xt), ∂CV1(xt(0)),

12‖ep(t)‖2,

12‖ep(t − td)‖2

)=

⋃fcd

∈K[fc](xt)

{ξVd

∈ R | ξVd= pT

V1fcd

+12‖ep‖2 − 1

2‖ep(t − td)‖2, pV1 ∈ ∂CV1(x)

}

Then along the solution of system (4.47) we can deduce that

∀ξVd∈ Md

(K[fc](xt), ∂CV1(xt(0)),

12‖ep(t)‖2,

12‖ep(t − td)‖2

),

∃δd ∈ K[δ](eω),

s.t. ξVd≤ −1

2(2γD0 − a2

1γ2)e2ω − k2eωep − k1e

2p

+γ(a1p∗m − D0ωr − TD)δd

In view of (4.49), the above inequality reduces to be

ξVd≤ −εe2

ω − k2eωep − k1e2p + γ(a1p

∗m − D0ωr − TD)δd


By the physics (4.43), we have

γ(a1p

∗m − D0ωr − TD

)δdeω ≤ 0, ∀δd ∈ K[δ](eω)

This results inξVd

≤ −xT Qx (4.52)

where

Q =

⎡⎢⎣ εk2

2k2

2k1

⎤⎥⎦It is easy to confirm from the condition (4.50) that a positive constant � can be foundsuch that

xT Qx ≥ �‖x‖2

with the given k1 and k2 satisfy the condition (4.45). Hence, inequality (4.52) means that

Md

(K[fc](xt), ∂CV1(xt(0)),

12‖ep(t)‖2,

12‖ep(t − td)‖2

)≤ −�‖x‖2 (4.53)

This completes the proof by Corollary 3.1.

Remark 4.3. It should be noted that benefiting from the condition (4.43), Proposition 4.4provides a sufficient condition for selecting the feedback gains such that the tracking systemis asymptotically stable in the practical sense. Observing the proof of the proposition, itcan be seen that the parameters D∗

0, D0, a∗1 and a1 affect the damping rate � of theLyapunov functional. A larger � is recommended if a quick response is required.

For the closed-loop system (4.47), the switching surface Sfc = S1 with S(xt(td)) =eωt(0) = ω∗ − ωr. The limiting functional at e∗ω(∈ Sfc) are given by

f+c (e∗ω) = lim

eω→(ω∗−ωr)+=

[−D∗0(ω∗ − ωr) + a∗1ep(t − td)

−keep − k2eω

]

f−c (e∗ω) = lim

eω→(ω∗−ωr)−=

[−(D∗0 + D0)(ω∗ − ωr) + (a∗1 + a1)ep(t − td)

−keep − k2eω

]The projections on the normal vector N = [1 0]T are{

f+cN = −D∗

0(ω∗ − ωr) + a∗1ep(t − td)

f−cN = −(D∗

0 + D0)(ω∗ − ωr) + (a∗1 + a1)ep(t − td)

respectively. Clearly, by Proposition 3.1, it can be confirmed that the solution of system(4.47) is the following equation⎧⎪⎨⎪⎩

eω(t) = ω∗ − ωr

ep(t) = e−k1t − k2

k1(ω∗ − ωr)


if D∗0/a∗1 > (D∗

0+D0)/(a∗1+a1), then during t ∈ [t∗, t∗+Δt] and ep(t) satisfies the followingcondition

D∗0 + D0

a∗1 + a1≤ ep(t − td) ≤ D∗

0

a∗1(4.54)

In other words, there is sliding motion on the surface Sfc (i.e. ω = ω∗), if condition (4.54)is satisfied. From practical point of view, it is clear that sliding motion is not desiredoperation.

Furthermore, as mentioned before, the delay time td is determined by engine speedas π/ω. If we take this time-variability into account, the time derivative of the chosenLyapunov-Krasovskii functional (3.19) depends on dtd/dt. However, as we can see inthe following if the delay time td does not vary much quickly, the stability can also beguaranteed by a slightly modified Lyapunov-Kraosvskii functional.

From the demonstration of Proposition 4.4, the condition (4.50) implies that thereexists a σ (0 < σ < 1) such that

2 (a∗1 + |a1|)2 ε(D∗

0 − |D0|)2 = 1 − σ (4.55)

Let λd be a constant satisfying λd < σ. Then, there exists a constant γ′(> 0) such that

γ′(D∗

0 + D0

)− (a∗1 + |a1|)2

2(1 − λd)γ′2 = ε

We now chose a candidate of Lyapunov-Krasovskii functional

V (xt) = V ′1(xt(0)) +

12

∫ 0

−τ‖ept(−s)‖2ds

with

V ′1(xt(0)) =

γ′

2e2ω +

12e2p (4.56)

By same demonstration for Proposition 4.4, we obtain that there exists �′ > 0 such that

Md

(K[fc](xt), ∂CV ′

1(xt(0)),12‖ep(t)‖2,

12‖ep(t − td)‖2

)≤ −�′‖x‖2 (4.57)

From the above discussion, we have the following result which is guaranteed by Corollary3.1.

Corollary 4.1. Consider system (4.16) with feedback controller (4.43) where the param-eters k1, k2 satisfy conditions (4.45), respectively. For any given ωr, ω → ωr as t → ∞,if the delay time td satisfies

dtddt

≤ λd (4.58)


Remark 4.4. It is of interest to note that condition (4.58) is sufficient to cover the realdtd/dt(= −πω/ω2) in practical situation. For instance, if the engine operates in 1500 rpm∼ 2000 rpm, the acceleration should be larger than −7854 rpm/s2.

Finally, under the case ωd < ω∗, the discontinuous function in the system (4.38) isdefined by

δ(ω∗, ω) =

{0, ω < ω∗

1, ω > ω∗(4.59)

and the following condition holds,

a1pmd − D0ωd − TD < 0 (4.60)

From these facts, it is straightforward to demonstrate the conclusion of Proposition 4.4when ωd < ω∗.

The effectiveness of the proposed engine speed controller is validated by simulationwith the identification results shown in Section 4.1. Under this situation, the feedbackgain in the feedback controller (4.46) can be selected as follows:⎧⎪⎪⎨⎪⎪⎩

k1 >12, k2 ∈ (0, 1.037

√k1), ωd > ω∗

k1 >12, k2 ∈ (0, 17.095

√k1), ωd < ω∗

Set ω∗ = 1900 rpm. Simulations are conducted under the situation that the dynamicalsystem is initially stable at an equilibrium, then, acceleration and deceleration commandsignals are given at t = 5 sec., respectively. Fig. 4.14 shows the results of two caseoperations, acceleration (Fig. 4.14(a)) and deceleration (Fig. 4.14(b)), where the controlparameters are set as k1 = 20, k2 = 4.6 and k1 = 20, k2 = 75.7, respectively. The responseresults demonstrate the convergence of engine speed.

It can be observed that by choosing a larger free coefficient k1, manifold pressureconverges to the steady-state value more quickly than the speed, especially after ω > ω∗(acceleration case) and ω < ω∗ (deceleration case), since the freedom to choose a feedbackgain k2 is limited due to the dependence on model parameters, moreover, when the speedreaching the ω∗, the force of switching control law (4.44) is reduced comparing to beforethe switching. Motivated by the control algorithm of Proposition 4.3, boost action isemployed to improve the transient performance of speed response. For the accelerationoperation, an extended control law of (4.44) is introduced as follows:

α′(eω, ep) = α(eω, ep) +δ′(eω)

a∗u + auδ(eω)β(t) (4.61)

where β(t) is given by (4.33), and

δ′(eω) =

{1, eω > ωr

0, eω > ωr


Under the control law (4.61) with feedback gain k1 = 20, k2 = 4.6 and boost parametersβ0 = 1.7 × 105 and tb = 0.2 sec., simulation result in Fig. 4.15 shows that the rising timeis shortened and the transient performance of speed response is improved clearly.

Fig. 4.16 shows a simulation result under the control law (4.44) with constant δeω, i.e.δeω = 1. This result indicates that in the case of the engine system dynamics is differentin a wide operation mode, we cannot obtain the convergence of engine speed by means ofa model-based feedback control law with constant parameters.


In this chapter, nonlinear speed control schemes for IC engines are discussed based on thecontrol oriented engine models.

In the first control scheme, a feedback compensation of spark advance is introduced tohandle the effort of the intake-to-power delay. With this compensation, a Lyapunov-basedfeedback design approach of throttle control algorithm is presented. Moreover, switch-

0 10 20 30 40 501500

1600

1700

1800

1900

2000

2100

Time [s]

En

gin

e s

pe

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[rp

m]

0 10 20 30 40 501.95

2

2.05

2.1

2.15

2.2

2.25x 104

Time [s]

Ma

nif

ord

pre

ssu

re [

Pa

]

0 10 20 30 40 50

3.5

4

4.5

Time [s]

Th

ro�

le o

pe

nin

g [

de

g.]

ω

ωr

(a)

0 10 20 301700

1800

1900

2000

2100

2200

Time [s]

En

gin

e s

pe

ed

[rp

m]

0 10 20 302

2.1

2.2

2.3

x 104

Time [s]

Ma

nif

ord

pre

ssu

re [

Pa

]

0 10 20 30

4

4.5

5

Time [s]

Th

ro�

le o

pe

nin

g [

de

g.]

ω

ωr

(b)

Fig. 4.14: Simulation result of engine speed control with different model parameters.


0 5 10 15 201500

1600

1700

1800

1900

2000

2100

2200

Time [s]

En

gin

e s

pe

ed

[rp

m]

0 5 10 15 201.5

2

2.5

3

3.5

Time [s]

Ma

nif

ord

pre

ssu

re [

Pa

]

0 5 10 15 203

4

5

6

Time [s]

Th

ro�

le o

pe

nin

g [

de

g.]

x 104

ω

ωr

Fig. 4.15: Simulation result withcontrol law (4.61).

0 5 10 151500

1600

1700

1800

1900

2000

2100

2200

Time [s]

En

gin

e s

pe

ed

[rp

m]

0 5 10 150

0.5

1

1.5

2

2.5

3

Time [s]

Ma

nif

ord

pre

ssu

re [

Pa

]

0 5 10 152

3

4

5

Time [s]

Th

ro�

le o

pe

nin

g [

de

g.]

x 104

ω

ωr

Fig. 4.16: Simulation result withnon-switching control law.

ing boost control technique is further introduced to the design approach to improve thetransient performance, and qualitative theoretical analysis is given to show the feasibil-ity. The key feature of the control scheme is using physical characteristics of the system,which makes the control system structure simple. Simulation results are presented to ver-ify the theoretical analysis first. With the identified model parameters on an engine testbench, experimental results demonstrate the availability for the stability and the transientperformance requirements.

In the second control scheme, mean-value models that involve intake-to-power delaywith different parameters are used to characterize engine dynamics in a wide operationmode. A nonlinear switching control approach is presented on the basis of the switchedmodel with explicit consideration of the intake-to-power delay. The asymptotic conver-gence of speed is achieved by selecting proper feedback gains. Gain conditions are obtainedbased on the functional differential inclusion based stability condition. It is also shownthat under the proposed control scheme, the stability is guaranteed even though the delaytime is varied according to the engine speed. Moreover, the proposed boost technique inthe previous control scheme is applied to improve the transient performance. Simulationresults verify the control methods.

98

Chapter 5

Model-based Starting Control ofSI Engines

5.1 Background

The staring behavior of SI engine depends on the initial states referring to the physicalcondition and the thermal environment and the starting control strategy. During thestarting stage, there are significant transient processes of engine states, such as speed,A/F, that deteriorate the engine performance in terms of fuel economy, emission, etc. (see[24], [94]). The transient performance of starting speed in terms of fast idling stable andsmoothness becomes an important issue in the recent advanced powertrains such as inhybrid electrical vehicles and in idling stop vehicles [14]. Rich literatures on idle speedcontrol are available where many classical and advanced control approaches are provided asintroduced in Chapter 1; nevertheless precise speed control strategy for SI engines duringtransient operation modes is still a challenging issue [77, 78, 101] in particular duringstarting transient mode [77, 78].

The mentioned situation indicates that the starting speed control problem should bethe interest of both industry and academic. To reinforce the collaboration between them,the SICE Research Committee on Advanced Powertrain Control Theory proposed a bench-mark problem on starting speed control in 2006 [77, 78]. Moreover, an industrial enginemodel is provided such that the challengers apply the proposed control schemes to runit as to start a real engine. A number of academic groups and research institutes haveparticipated in this benchmark problem. Five challenging results have been reported in[62], [79], [84], [100] and [115], respectively, and brief descriptions of these results are listedas follows.

A. The challengers give a coordinate control scheme in [62]. It includes a model-basedfuel injection control law and a SA regulation method. Moreover, extremum seeking

99

100 Chapter 5. Model-based Starting Control of SI Engines

approach by minimizing an unknown performance index is employed on the throttlecontrol (Keio University).

B. A large scale database-based online modeling method is introduced in [79]; themethod is an extension of the ”just-in-time” modeling technique and is operatedbased on accumulating the measurable data to a database (Waseda University).

C. A particle swarm optimization based feedforward control combining with generalizedpredictive feedback control is applied in [84] (Tokyo Institute of Technology).

D. In [100], a new 24-order state space model is proposed to represent the dynamicsof the considered SI engine; the engine model is time-invariant which is obtainedby using a new concept, role state variables; finally based on the proposed enginemodel, optimal design approach combining with feedback and feedforward control isapplied to solve the starting speed control problem (Toyota Center R&D Labs).

E. The authors present a mean-value engine model based solution for this benchmarkproblem [115]. Similar to [62], the proposed control scheme takes into account thedifferent roles of engine actuators on the engine performance. This control schemegives a fuel injection control system with dual sampling rate: the cycle-based fuelinjection command is individually adjusted for each cylinder by using a TDC-basedair charge estimation; a coordinated control system by SA and throttle operation forthe starting speed regulation. Moreover, the speed error convergence of the closed-loop system is proved for a simplified, second-order model with a time-delay (SophiaUniversity).

Under the background of SICE benchmark problem, the investigation of starting speedcontrol problem is discussed in this chapter. The control problem is described first. Themodel-based control scheme E mentioned above is introduced. Then, validations con-ducted by simulation and the engine test bench are shown, respectively. Finally, moti-vated by the results, we give some directions for future research on this challenging controlproblem of SI engines.

5.2 Statement of the Problem

The starting process is a typical transient operation mode of SI engines. The primaryphysical mechanism can be described briefly as follows. The crankshaft is initially drivento have a low constant rotational speed by the starting motor (see the image shown inFig. 4.1). The engine strokes of each cylinder follow along the crank angle. After the firstignition occurs in a cylinder, the driven torque on the crankshaft will be shifted from thestarting motor to the engine. This will cause an acceleration of crank rotational velocityto achieve stable idle speed. The starting process can be roughly divided into four phases:cranking, unstable combustion, combustion stabilization and steady idling [30, 94]. Thebasic requirement of engine starting is fast acceleration followed by quick convergence to a

5.2. Statement of the Problem 101

target idle speed. However, the significant characteristic is that many engine parameterschange during the starting evolution. For instance, following the transition of enginespeed, manifold pressure will be down from the atmospheric pressure to a static valuewhich is extremely far from its initial value. The constraint of unstable combustion makesthe regulation of the speed response during the second and the third phases and theircoordination a challenging task.

Consider the engines with six cylinders. The starting speed control problem is athirteen input one output system, where the inputs are the throttle opening and the port-fuel injection commands and SA of each cylinder and the output is the engine speed. Threesignals of the engine system are measurable for control design, i.e. the engine speed, airmass flow rate through the throttle and the crank angle. The structure of the controlsystem is as shown in Fig. 5.1. On the basis of the mean-value engine models introducedin Section 4.1, the purpose is to provide a starting speed control scheme which satisfies apre-specified engine speed performance; the design specification, as indicated in Fig. 5.2,is given by the SICE benchmark problem [77], i.e.,

• the engine speed should converge to the target idle speed, e.g. 650 rpm;

• engine speed should reach the range 650 ± 50 rpm in 1.5 sec.;

• the control system should be stable;

• the overshoot should be reduced as much as possible;

• the A/F of individual cylinder should be proper for ignition, i.e., the air-fuel mixturein each cylinder can be ignited only when λa ∈ [λmin, λmax], in addition, the A/Fshould be possibly stoichiometric for the requirements of optimal combustion andemission.

Clearly, before addressing the starting speed control, successful combustion should beguaranteed first which depends on the A/F in each cylinder. The uncertain air mass flowrate (4.9) and fuel path dynamics (4.1) will be an unavoidable obstacle to achieve this goal.Furthermore, even if the A/F is under control to ensure successful combustion, open-loop

ControllerEngine speedEngine

system

Idle speedcommand Fuel inj. ×6

SA×6

Thro�le opening

Air mass flow rate through thro�le

Crank angle

Fig. 5.1: Structure diagram of the control system.


En

gin

e s

pe

ed

[rp

m]

0

250

650

0 Time [s]

Motoring

50 rpm±Speed stable

Reduce overshootTarget idle speed

Ini�al speed

1.5

Fig. 5.2: Image of starting speed with required specification.

En

gin

e s

pe

ed

[rp

m]

0

1000

2000

pm

×105

ω

Inta

ke m

an

ifo

ld p

ress

ure

[P

a]

0

0.5

1

1.5

0 1 2 3 4 5 6 7 8Time [s]

Fig. 5.3: Response of engine speed and manifold pressure during startingwith constant control inputs.

control, which keeps throttle and SA to be equal to the constant values determined bynominal model parameters according to static operation mode, can not provide satisfactorytransient speed performance. Fig. 5.3 shows a response of the engine speed and themanifold pressure during starting obtained on a full-scale engine simulator [77, 78], wherethe desired stable speed is set at 650 rpm and control signals are throttle opening at φ = 5.2deg. and Sa = 20 deg.. In this case, the starting speed shows an extremely large overshootwhich is more than twice the required one, and therefore a long settling time which is alsomore than twice the required value. This is due to the over torque generation during thefirst few cycles. To suppress the undesired torque generation, delicate torque managementis required during the starting stage: fine air charge estimation and fuel injection arerecommended for individual cylinders so that the A/F remains in the interval [λmin, λmax]for successful combustion. Moreover, it should be essential to utilize feedback control forengine starting control problem.

5.3. Control Scheme 103

5.3 Control Scheme

The control scheme provides a TDC-scaled air charge estimation, a fuel injection controlmethod for each individual cylinder and a scheduled control strategy for throttle openingand SA adjusting law. The main structure of the control scheme is summarized as shownby Fig. 5.4. The details of the control scheme are given as follows.

.

Engine

Individualfuel inj. commands

SA commands

Thro�le openingcommand

Idle speedcommand

Engine speed

Cranking torque

Scheduling

ConstantSwitch 1

Observer+

Inverse dynamics

P-Gain

MBT

Switch 2

Star�ngmotorSupervisor

+-

Air mass flow

Constant

Model-based control

Switch 3

.

.

.

.

Fig. 5.4: Control scheme of starting speed control.

Fuel Path Control It is known that the A/F control in SI engines is usually based onthe estimation of air charge into each cylinder. With the estimated air mass, the neededamount of fuel to be injected into the corresponding cylinder is determined on the basisof the inverse dynamics of (4.1) with a desired A/F (λd). In steady state, air charge canbe calculated approximately according to the air mass flow rate mo and engine speed ω.But, this method can not work well during starting due to the dramatic speed change. Inthe following, a simple approximate calculation method is used to provide an estimationof the air charge into individual cylinders per cycle.

Since the air mass flow rate mo in the considered case is unmeasurable, an open-loopobserver is introduced which is based on the air intake path dynamics. The models (4.3),(4.4) and (4.8) are used to construct the following observer⎧⎪⎪⎨⎪⎪⎩

˙pm =RTa

Vm(mi − ˆmo)

ˆmo =ρaVcη

4πpaωpm

(5.1)


where the initial condition pm(0) is the atmospheric pressure pa.

Furthermore, it is known that the air and fuel are inducted into each cylinder syn-chronously during the induction stroke. Therefore, the air charge estimation should beachieved at each TDC timing so that the fuel injection command is able to meet the A/Fconstraint condition. However, the real air charge of each cylinder is obtained after thewhole induction stroke. The following estimation method for the air charge into eachcylinder denoted by mcyl is proposed with ˆmo given by the observer (5.1)

mcyl(l) = ˆmo(lTc) · tTdc(l) (5.2)

where l denotes the index of TDC-based sampling rate Tc(= 2π/(3ω) sec.) , and tTdc(l) =

2π/(3ω(lTc)) is a predicted time of the induction stroke with ω measured at each TDCtiming.

With the estimated air charge (5.2), the injected fuel for the i-th cylinder is calculatedas follows

mfci(k) =mcyl

(6(k − 1) + i

)λd

(5.3)

where k is the cycle-based sampling index. The relation between k and l is k = fix((l −1)/6) + 1.

Finally, the fuel dynamics (4.1) is discretized with cycle-based sampling rate Ts(= 2π/ωsec.) as follows, {

mfi(k + 1) = (1 − χ)mfi(k) + εufi(k)

mfci(k) = χmfi(k) + (1 − ε)ufi(k)(5.4)

where i = 1, · · · , 6. Then, the individual fuel injection commands are given by its inversedynamics, i.e.

ufi(k + 1) = Aufi(k) + Bmfci(k) + Cmfci(k + 1) (5.5)

whereA = −χε − (1 − χ)(1 − ε)

1 − ε, B = −1 − χ

1 − ε, C =

11 − ε

To compensate the influence of engine temperature on the fuel dynamics (5.4), the coeffi-cients ε and χ are suggested to be scheduled during the first few cycles.

Remark 5.1. It should be noted that in (5.1), constant volumetric efficiency, that iseffective in the steady-state idling operation mode is used for simplicity. Therefore, theair mass flow rate ˆmo given by observer (5.1) is an approximate estimation during thetransient starting operation mode, and a parameter scheduling is suggested to improvethe accuracy of air mass flow rate estimation ˆmo.

The developed fuel path control algorithm is summarized in Fig. 5.5.

Coordinate Control Law between Throttle and SA On the basis of the mean-valueengine modes in Section 4.1, starting speed regulation is obtained by a coordinated control

5.3. Control Scheme 105

Intake manifoldobserver

Discrete inverse dynamics

of fuel path

Scheduling

Constant

Switching signal from supervisor

Injec�on commands

for each cylinder

Sampling rate=Tc

Con�nuous �me Discrete �me domain

Engine speed

Air massflow rate

×

Air chargemass

Fig. 5.5: Control scheme for fuel injection path.

between throttle operation and SA adjusting: the control system is a multi-input single-output with us and uth as inputs and ω as output, respectively. SA feedback control isused to reject the over torque generation during the acceleration stage of engine starting,and then the feedback control will be definitely switched to the throttle opening action.The presented approach is motivated by the typical transient response of starting speedgiven in Fig. 5.3, where it can be seen that there are two phases: the first phase is therapid acceleration stage while the second phase is the idle speed regulation.

A reference model-based control approach is used for the starting speed regulation. Itis introduced to generate a desired speed trajectory ωd(t) during engine starting. Drivenby the idle speed ωr, the model which is chosen as a first-order system, i.e.

ωd(t) = −σ(ωd − ωr), t ≥ t0 (5.6)

with ωd(t0) = ω(t0), where σ is a positive constant and t0 denotes the first spark timingof the selected cylinder which is ignited first.

Feedback control action of SA is the conventional proportional control. The model-based control of throttle is a combined feedforward and feedback controller by using thereference trajectory, where a nonlinear feedback gain is used in the feedback control partand the nominal engine dynamic model is used to obtain the feedforward controller. Theswitching logic of the coordinated control algorithm is given based on the speed error asfollows, condition

0 ≤ eω(t) ≤ 50 rpm & eω(t) < 0 (5.7)

According to the above switching control logic, the coordinated control law is given by

us(t) =

{kseω(t) + us0, t0 ≤ t ≤ t1

Mbt, t ≥ t1(5.8)

uth(t) =

{0, t0 ≤ t ≤ t1

u∗th + α(ω, ωd), t ≥ t1

(5.9)


where t1 is the first moment when the condition (5.7) is satisfied, us0 is any given constantof SA, u∗

th is the feedforward compensation obtained from (4.41), and α(ω, ωd) denotesthe feedback control law designed as

α(ω, ωd) =1au

[a2p∗meω + kt(ω)(ω − ωd)] (5.10)

where kt(ω) is the feedback gain function.

As mentioned above, the control authority is definitely switched to throttle at t = t1.Therefore, the speed error convergence must be guaranteed for this control loop. In thissituation, we consider the engine dynamics is characterized by the mean-value engine model(4.16) with constant parameters under idling operations. Then, by using the feedbackcontroller (5.10), the aim is to choose a feedback gain kt(ω) such that the starting speedcan converge to the desired idle speed. To show the convergence, we prove the asymptoticstability of the error dynamic system consisting of the engine dynamics (4.16) with thereference model (5.6) and the control law (5.9).⎧⎪⎪⎨⎪⎪⎩

eω = −D0eω + a1ep(t − td)

ep = kt(ω)eω − a2ωep − kt(ω)er

er = −σer

(5.11)

where er = ωd − ωr.

Denote x = [eω ep er]T . To analyze the stability of system (5.11), we choose a candidateof Lyapunov-Krasovskii functional as follows

V(xt) =

γ1

2e2ω +

12e2p +

γ2

2e2r +

12

∫ 0

−td

e2pt(−s)ds (5.12)

where γ1 and γ2 are given by

γ1 =D0 +

√D2

0 − 2a21ε

a21

, γ2 =ε

σ(5.13)

with a given ε satisfying

0 < ε <D2

2a21

(5.14)

For the sake of simplicity, we treat the delay time td in the model (4.16) as constant,i.e. it takes the nominal value at the desired speed ωr, td = π/ωr. In addition, it is noticedthat 2a2ω > 1 under the allowable operation condition of the engine.

Proposition 5.1. For any given σ > 0, if the feedback gain kt(ω) is given by

kt(ω) = ρ(t)√

ε (2a2ω − 1) (5.15)

5.4. Validation Results 107

with a given function |ρ(t)| < 1, ∀t ≥ 0, then the time derivative of Lyapunov-Krasovskiifunctional (5.12) along the solution of system (5.11) satisfies

V (xt) ≤ −�‖x‖2 (5.16)

for a sufficiently small � > 0. In other words, for any initial condition x0(td) ∈ Cr, xt

asymptotically converges to zero as t → ∞

Proof. Calculating the derivative of (5.12) along the solution of system (5.11) obtains

V (xt) =−γ1D0e2ω + γ1a1ep(t − td)eω + kt(ω)eωep − a2ωe2

p − kt(ω)eper − γ2σe2r

+12e2p −

12e2p(t − td)

≤−12(2γ1D0 − a2

1γ21

)e2ω − γ2σe2

r + kt(ω)eωep − 12

(2a2ω − 1) e2p − kω(ω)eper

In view of (5.13), it yieldsV (xt) ≤ −xT Q(ω)x

where

Q(ω) =

⎡⎢⎢⎢⎢⎣ε −1

2kt(ω) 0

−12kt(ω) a2ω − 1

212kt(ω)

012kt(ω) ε

⎤⎥⎥⎥⎥⎦Taking the condition (5.15) into account, it is easy to verify that a sufficiently small � > 0can be found such that

xT Q(ω)x ≥ �‖x‖2

Namely, condition (5.16) holds. Moreover, it is clear that for the Lyapunov-Krasovskiifunctional (5.12), there exist functions Wi ∈ K (i = 1, 2) such that

W1(‖x‖) ≤ V (xt) ≤ W2(‖xt‖c) (5.17)

Hence, the asymptotic stability of system (5.11) at the origin follows by the Lyapunov-Krasovskii functional stability theorem 2.1 in [41].

Supervisor Block It is clear that the proposed control system is a multi-input, multi-sampling rate system, and the speed regulation loop is with dual actuators. All necessaryswitching signals for the multi-sampling rates will be delivered by this supervisor block.

5.4 Validation Results

The presented starting speed control scheme for SI engines undergos two sequences ofvalidation test: first, in the provided engine simulator; then, in a real engine test bench.


5.4.1 Simulation Results

Simulations are performed using the engine simulator provided by the SICE benchmarkproblem as shown in Fig. 5.6 . It is an industrial engine model that consists of the engineblock, the control block and the starting motor block. The basic engine specifications arelisted in Table 5.1. The engine system in the simulator has six cylinders, thirteen inputs(throttle angle, fuel injection and SA of each cylinder). The thermal and fluid dynamicsof each cylinder, of the inlet pipes and of the intake manifold are individually modeled;the stochastic features in the combustion phase are also taken into account; the modelparameters are adjusted so that the experimental dynamic behavior is reproduced. It isa hybrid system which combines continuous and discrete dynamical systems including a46th-order model. The simulator is used by automotive companies to perform hardwarein the loop tests for engine ECU. It is also used to validate the starting control strategieswhich are proposed to solve the SICE benchmark problem. Detail descriptions for thesimulator can be found in [77, 78]. The simulation condition of the engine model isdefined as follows:

• water temperature Tw0 = 25◦C;

• initial crankshaft angle, CA(0)=0 deg.;

• initial speed from starting motor is 250±50 rpm; the most motoring time is 1.5 sec..

• ignitions in each cylinder can occur only as λa ∈ [λmin, λmax].

The model parameters are listed in Table 5.2; the nominal parameters obtained byusing model identification techniques based on the simulator data under steady-state op-eration mode are as follows.

a1 = 2.5 × 10−3, D0 = 3.0 × 10−3, TD = 35.89, au = 1.072 × 107,

a2 = 4.79 × 10−2, ε = 0.1, χ = 0.01

In order to compensate the fuel requirement, a scheduling for the fuel path parametersε and χ is used in the first three cycles as shown in Table 5.3. In the speed regulationloop, the time coefficient of the reference model (5.6) is set at σ = 15. In the accelerationstage, the proportional feedback gain in (5.8) is chosen as ks = 0.07 and us0 is simply setat us0 = 10 deg..

Table 5.1: Basic specification of the engine simulator.

Number of cylinder 6 [-] Bore 86 [mm]Stroke 86 [Pa] Compression ratio 9.8 [-]Crank radius 43 [mm] Connection rod length 146.65 [mm]


Fig. 5.6: Engine simulator.

Table 5.2: The physical parameters of engine.

R 287.1 [J/(kg· K)] Vm 6.0 × 10−6 [m3]pa 1.01 × 105 [Pa] Vc 3.0 × 10−3 [m3]ρa 1.1837 [kg/m3] η 1 [-]Ta 298.15 [K] λd 14.5 [-]s0 3.5 × 10−3 [m2] Mbt 20 [deg.]

Table 5.3: Parameter compensation for fuel path.

k = 1 k = 2 k = 3 k = 4 · · ·χ 0.08 0.6 0.2 0.01 · · ·ε 0.5 0.1 0.1 0.1 · · ·

When the feedback control shifts from SA to throttle at t = t1, according to condition(5.14), the advisable ε in the feedback gain kt(ω) in (5.15) for the throttle control (5.9)belongs to the interval (0, 0.7632). We choose ε = 0.458 and the design parameter ρ(t) =0.9. The responses recorded from the simulator are reported in Fig. 5.7∼Fig. 5.9.

Fig. 5.7 shows the response of the open-loop observer (5.1) for the estimation of airmass flow rate leaving manifold into the cylinders. It can be seen that the estimated ˆmo

follows the simulated value closely during the initial period. Based on this estimation, thefuel injection commands for each cylinder are obtained. The A/F output curves shown inFig. 5.8 indicate that the proposed fuel injection algorithm for each individual cylinderguarantees the combustion condition. Moreover, from the curves of the fuel injectioncommands, it can be seen that in the first few cycles, the required fuel quantity is almost


five times the steady state fuel injection. After the engine being started successfully, thestarting speed response under the coordinated control between throttle and SA is shownin Fig. 5.9. We can see that the controller with the chosen feedback gain guarantees thespeed convergence. Moreover, these results shown in Fig. 5.8 and Fig. 5.9 indicate thatthe specifications of the SICE benchmark problem are met.

Considering that in the control scheme for cold starting, both the fuel path controland speed regulation loop are constructed based on the mean-value models, simulation isconducted to test the tolerance of the control scheme with respect to parameter fluctu-ations. Take the volumetric efficiency coefficient η used in the open-loop observer (5.1)as an example. The simulations are performed with three different values for η in theobserver (5.1). The comparison results shown in Fig. 5.10 indicates that the coefficientη in the observer affects the estimation of air mass flow rate, the A/F control and enginespeed clearly, but, in a certain range for η (roughly speaking for η ∈ [0.8, 1] according tosimulation testing), the control scheme can work effectively.

Moreover, simulation tests are conducted at other water temperatures, which meanthat the engine is started under certain warm conditions if we refer to the room temper-ature for the above tests as cold condition. Using the proposed control scheme with thesame design parameters, the speed responses shown in Fig. 5.11 indicate that startingspeed performance depends on the temperature condition significantly, especially at thecold condition and at an almost warm-up condition.

x 103

x 103

Air

ma

ss fl

ow

ra

te [

g/s

]A

ir m

ass

flo

w r

ate

[g/s

]

-20

0

20

0 0.5 1 1.50

5

10

15

Time [s]

0 0.5 1 1.5Time [s]

mo �

mo ��

mo ��

�mi

Fig. 5.7: Estimated air mass flow rate by observer (5.1) vs. measured airmass flow rate.


0

2

4

6

8x 10-5

Fu

el

inj.

cm

d.

[kg

]

0 1 2 3 4 50

5

10

15

20

Ind

ivid

ua

l A

/F

Time [s]

0

2

4

6

8

Fu

el

inj.

cm

d.

[kg

]

0

5

10

15

20

25

Ind

ivid

ua

l A

/F

x 10-5

#2#4#6

#1#3#5

#2#4#6

#1#3#5

0 1 2 3 4 5Time [s]

0 1 2 3 4 5Time [s]

0 1 2 3 4 5Time [s]

Fig. 5.8: Fuel injection and individual A/F.

0

200

400

600

800

En

gin

e s

pe

ed

[rp

m]

-10

0

10

20

30

SA

cm

d.

[de

g.]

0

1

2

3

4

5

6

Th

ro�

le c

md

. [d

eg

.]

ω

ωd

ωr

0 1 2 3 4 5Time [s]

0 1 2 3 4 5Time [s]

Fig. 5.9: Starting speed control result.

0 1 2 3 4 50

5

10

15

Time [s]

Air

ma

ss fl

ow

ra

te [

g/s

]

0 1 2 3 4 50

200

400

600

800

1000

Time [s]

En

gin

e s

pe

ed

[rp

m]

0 1 2 3 4 50

20

40

60

80

100

Time [s]

#1

cy

l. A

/F

0 1 2 3 4 50

50

100

150

Time [s]

#2

cy

l. A

/F

x 103

η=0.75η=1 η=0.8

Fig. 5.10: Simulation result with different η in observer (5.1).


0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

200

400

600

800

1000

Time [s]

En

gin

e s

pe

ed

[rp

m]

Tw=25ºC Tw=45ºC Tw=65ºC

Fig. 5.11: Simulation result with different water temperature.

5.4.2 Experimental Results

In this section, we show experimental validations of the proposed starting speed controlscheme. The experiments are conducted by the engine test bench by the Runit-Dtypeprocessing system. First, we introduce the conditions how the proposed control law isimplemented. The starting experiments are defined as follows.

• start the tests from water temperature almost 60◦C;

• the angle of engine stop position, CA(0), is random;

• from practical point of view, the SA should be limited for safety, combustion suf-ficiency, etc. Hence, the SA control input of the control scheme is restricted whenconducts experiments, where the low constraint is −10 deg. and the up constraintis 25 deg.

The physical parameters of the engine system in the control laws are same to onesin Table 5.2 but here Vm = 6.0 × 10−6 [m3] and Vc = 3.456 × 10−6 [m3]. The nominalparameters of the fuel path model (5.4) are taken χ = 0.01 and ε = 0.1. To guarantee thecombustion condition during the first a few cycles, an adjusting coefficient is applied o thefuel injection control laws. In fact, a scheduling is utilized in the air charge estimationduring the first 5 cycles, i.e. the equation (5.3) in the control scheme is modified by

mfci(k) = μkmcyl(6(k − 1) + i)

λd(5.18)

and the scheduling for parameter μk are show in the Table 5.4.

Table 5.4: Scheduling parameters in fuel path control.

k = 1 k = 2 k = 3 k = 4 k = 5 k = 6 · · ·μk 0.6 0.5 0.5 0.8 0.8 1.4 · · ·


The nominal parameters of mean-value engine model (4.16) obtained by using modelidentification technique are

a1 = 2.6 × 10−3, D = 6 × 10−3, D0 = 56.9677, au = 2.5623 × 107, a2 = 3.6 × 10−3

The switching time t0 and t1 in the control scheme are decided as follows: t0 is the timimgwhen ω(t0) = 300 rpm; t1 is the timing when ω(t1) = 700 rpm and eω < 0. In thereference model (5.6), σ = 5. The parameters in the control law (5.9) and 5.8 are selectedas ρ(t) = 0.1, ε = 0.01, ks = 0.06, us0 = 20 × π/180 and Mbt = 20, respectively.

Finally, the presented thirteen control inputs are delivered to ECU of engine in thefollowing form, respectively: the fuel injection command of each cylinder is given by Ufi

with unite [mm3/str.], i.e.Ufi =

ufi

ρf× 106 (5.19)

where ρf = 0.735 [L/kg] denotes the density of the fuel; the throttle command φ [deg.] isgiven from (4.18) by

φ =180π

arccos(1 − uth) (5.20)

the SA command isSA =

180π

us (5.21)

First, Fig. 5.12 shows an experimental result, the output of engine speed and individualA/F during starting under the original control law of ECU. We can observe that theresponse of starting speed is with large overshoot, and takes long time to achieve steadystate value. A/F also cannot reach the desired value until engine speed is stable. Moreover,it seems more difficult to achieve a lower target idle speed.

Decide the 1st fuel injection quantity of each cylinder experientially for successfulcombustion. Validation experiments of the proposed control laws are conducted by down-loading the proposed control laws to ECU. Two results are shown in Fig. 5.13 and Fig.5.14 under different CA(0)s, respectively. We present engine speed (ω, ωd, ωr), the mea-sured individual A/F of #2, #4, #6 cylinders and the corresponding mix point (mixR)A/F, the commands of throttle, SA and individual fuel injection , and the measured airmass flow rate through throttle (mi) and the estimated air mass flow rate into cylinders( ˆmo). Similar to the simulation results, we can see that the overshoot of the startingspeed response is reduced obviously; the engine is idling stable at the target speed in nomore than 1.5 sec.; individual A/F guarantees the combustion condition and converges tosteady value in almost 4 sec. Moreover, comparing the two results, the engine stop posi-tion, CA(0) seems to influence the smoothness of the speed response. It is considered dueto the influence of CA(0) to the torque generation before the switching time t1 involvedin the control scheme.

Finally, it is well-known that spark timing control provides fast influence on the enginetorque generation and is usually a pre-control during engine torque increase as well as


0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 160

200

400

600

800

1000

1200

1400

1600

1800

2000

Sp

ee

d [

rpm

]

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

11

12

13

14

15

16

17

18

19

20

A/F

Time [s]

10

Time [s]

ω

CA

#2#4#6

mixR

Fig. 5.12: Experimental result by original control law of ECU: Tw = 60◦C,CA(0)=540 deg..

decrease; however, excessive can cause bad effective to engine operations, such as com-bustion instability [46]. In the proposed control scheme, we use SA feedback control onlyduring acceleration stage. In order to improve the transient performance of starting speedas much as possible, the SA feedback control is also considered during the idling stablestage. A discussion of this method is associated with experimental results and is shownlater. The control law (5.8) is updated as

us(t) =

{ks1eω(t) + us0, t0 ≤ t ≤ t1

Mbt + ks2eω, t ≥ t1(5.22)

For simplicity, the above SA feedback control law is reduced as follows,

us(t) = us0 + kseω, t ≥ t0 (5.23)

By using the modified control law with same feedback gains, ks, us0, ε and ρ, and sameScheduling for fuel path, Table 5.4, from the result shown in Fig. 5.15 and Fig. 5.16, wecan see that SA feedback control can reduce the influence caused by different CA(0) suchthat the starting speed response is more smooth.


01

23

45

67

80

10

02

00

3

00

40

0

50

06

00

7

00

80

0

90

01

00

0

Tim

e [

s]

Speed [rpm]

01

23

45

67

8-2

0

-100

10

20

30

SA cmd. [deg.]

Tim

e [

s]

00.4

0.8

1.2

1.6

Thro�le cmd. [deg.]

01

23

45

67

805

10

Air mass flow rate [g/s]

Tim

e [

s]

01

23

45

67

80

10

20

30

40

50

Fuel inj. [mm3/str]

Tim

e [

s]

01

23

45

67

80

10

20

30

40

50

Tim

e [

s]

Fuel inj. [mm3/str]

01

23

45

67

81

01

11

21

31

41

51

61

71

81

92

0

A/F

Tim

e [

s]

ωωd

ωr

CA

#1

#3

#5 #2

#4

#6

#2

#4

#6

mix

Rm

o

� � � mi

Fig

.5.

13:

Exp

erim

enta

lre

sult

:T

w0

=56

◦ C,C

A(0

)=54

0de

g..


01

23

45

67

80

10

02

00

3

00

40

0

50

06

00

7

00

80

0

90

01

00

0

Tim

e [

s]

Speed [rpm]

01

23

45

67

8-2

0

-10010

20

30

SA cmd. [deg.]

Tim

e [

s]

00.4

0.8

1.2

1.6


01

23

45

67

80

2

4

6

8

10


Tim

e [

s]

01

23

45

67

8010

20

30

40

50

Fuel inj. [mm3/str]

Tim

e [

s]

01

23

45

67

8010

20

30

40

50

Tim

e [

s]

Fuel inj. [mm3/str]

01

23

45

67

81

01

11

21

31

41

51

61

71

81

92

0

A/F

Tim

e [

s]

ωωd

ωr

CA

#1

#3

#5 #2

#4

#6

#2

#4

#6

mix

Rm

o

� � � mi

Fig

.5.

14:

Exp

erim

enta

lre

sult

:T

w0

=58

.5◦ C

,C

A(0

)=18

0de

g..


01

23

45

67

80

10

02

00

3

00

40

0

50

06

00

7

00

80

0

90

01

00

0

Tim

e [

s]

Speed [rpm]

01

23

45

67

8-2

0

-100

10

20

30

SA cmd. [deg.]

Tim

e [

s]

01

23

45

67

80

2

4

6

8

10

12


Tim

e [

s]

01

23

45

67

80

10

20

30

40

50

Fuel inj. [mm3/str]

Tim

e [

s]

01

23

45

67

80

10

20

30

40

50

Tim

e [

s]

Fuel inj. [mm3/str]

01

23

45

67

81

01

11

21

31

41

51

61

71

81

92

0

A/F

Tim

e [

s]

00.4

0.8

1.2

1.6


ωωd

ωr

CA

#1

#3

#5 #2

#4

#6

#2

#4

#6

mix

Rm

o

� � � mi

Fig

.5.

15:

Exp

erim

enta

lre

sult

:T

w0

=62

.5◦ C

,C

A(0

)=18

0de

g..


02

46

80

10

02

00

3

00

40

0

50

06

00

7

00

80

0

90

01

00

0

Tim

e [

s]

Speed [rpm]

02

46

8-2

0

-10010

2030

SA cmd. [deg.]

Tim

e [

s]

0

0.4

0.8

1.2

1.6


02

46

805

10


Tim

e [

s]

02

46

80

10

20

30

40

50

Fuel inj. [mm3/str]

Tim

e [

s]

02

46

80

10

20

30

40

50

Tim

e [

s]

Fuel inj. [mm3/str]

02

46

81

01

11

21

31

41

51

61

71

81

92

0

A/F

Tim

e [

s]

ωωd

ωr

CA

#1

#3

#5 #2

#4

#6

#2

#4

#6

mix

R

mo

� � � m

i

Fig

.5.

16:

Exp

erim

enta

lre

sult

:T

w0

=66

◦ C,C

A(0

)=54

0de

g..



For the SI engines during starting operation mode, a control scheme that focuses on im-proving the transient performance of the starting speed is proposed under the backgroundof SICE benchmark problem. In contrast to the speed control problems discussed in Chap-ter 4, starting speed control involves challenging issues due to the dramatic variations ofthe engine dynamics. The focus of this research is on designing accurate and robust con-trollers for engine inputs: air intake, fuel injection and spark timing and developing controlschemes that can compensate the uncertainties caused by the involved dynamic systemsduring the starting and/or restarting of engine. Validation testings for the presented solu-tion are conducted by simulation and experiment, respectively. The results show that thecontrol scheme satisfies the speed control performance specified by the SICE benchmarkproblem.

For the starting speed control problem, there are several suggested ideas for furtherresearch on theoretical and experimental developments to improve the engine performanceinclude the following.

First, inspired by the experimental investigation of the modified control scheme withSA, theoretical analysis should be carried out. By utilizing the feedback control laws (5.9)and (5.23) in the control loops of throttle and SA, respectively, the system is discontinuouswith respect the state of engine system, hence providing quantitative analysis from the viewof developed differential inclusion and functional differential inclusion based framework ishelpful in applying the control scheme to get improved starting performance of SI engines.

From the application point of view, the optimal fuel injection of the 1st cycle in termsof fuel economy, emission, etc. is not very clearly, and should be calibrated carefully.Meanwhile, effective methods to improve individual A/F response performance should besignificant. Seeking a good scheduling of the air charge estimation during the startingstage is anticipated; in particular, accurate modeling of fuel path during engine starting ismore better for the engine stating speed control. Furthermore, experimental results onlyshow that the presented control laws with same design parameters are basically effectivewhen the water temperature in a certain range. Hence, identify model parameters underdifferent starting temperature (from 20◦C to 80◦C), then, validate the case that if watertemperature is measurable, using gain-schedule technique to the design parameters toobtain more globally robust controller.

120

Appendix A

Basic Mathematical Tools

Definition A.1. (Absolutely Continuous Function) A function f : [a, b] → R isabsolutely continuous if for all ε ∈ (0,∞), there exists δ ∈ (0,∞) such that any finitecollection (a1, b1), · · · , (an, bn) of disjoint open intervals contained in [a, b] with Σn

i=1(bi −ai) < δ satisfies

Σni=1|f(bi) − f(ai)| < ε

• Absolutely continuous functions are differentiable almost everywhere.

• Lipschitz continuous functions are absolutely continuous. An example of a functionthat is absolutely continuous but not Lipschitz continuous at 0 is

Example A.1. f : R → R defined by

f(x) =√|x|

• Absolutely continuous functions are continuous. An example of a function that iscontinuous but not absolutely continuous is


f(x) =

⎧⎨⎩x sin(

1x

), x = 0

0, x = 0

Fig. A.1 shows the graph of the function f(x) in Example A.2.

Definition A.2. (Set-valued Mapping)[3, 22]

A set-valued mapping is a mapping thathas a set as image. More formally, Let B(D) be the collection of all possible subsets ofD ∈ Rn. Consider (autonomous) set-valued mappings F : Rn → B(Rn). The mapping Fassigns each point x ∈ Rn to the set F (x) ∈ Rn.

121

122 Appendix A. Basic Mathematical Tools

0.05

-0.05

0.05

0-0.05 x

f(x)

Fig. A.1: Grough of the function in Example A.2

Definition A.3. (Regular Function) A function f : Rn → R is said to be regular atx ∈ Rn if for all v ∈ Rn,

(i) the right directional derivative f ′(x, v) exists;

(ii) f ′(x; v) = f◦(x; v)

where

f ′(x; v) = limh→0+

f(x + hv)h

f◦(x; v) = lim supy→xh→0

f(y + hv) − f(y)h

• A continuous differentiable function at x is regular at x.

• A convex function at x is regular at x (and is also Lipschitz continuous at x).

• An example of non-regular function at 0 is


f(x) = −|x|

Proposition A.1. (Calculus for Filippov Differential Inclusion)[22, 80]

The set-valued mapping K : Rn → B(Rn) has the following properties.

1) If f : Rm → Rn is continuous at x ∈ Rm,

K[f ](x) = {f(x)}

123

2) If f, g : Rm → Rn are locally bounded at x ∈ Rm,

K[f + g](x) ⊆ K[f ](x) + K[g](x)

If one of the vector field is continuous at x, then equality holds.

3) If f1 : Rm → Rn1 and f2 : Rm → Rn2 are locally bounded at x ∈ Rm,

K[f1 × f2](x) ⊆ K[f1](x) × K[f2](x)

If one of the vector field is continuous at x, then equality holds.

4) If g : Rm → Rn is continuous differentiable at x ∈ Rm with rank n, and f : Rn → Rd

is locally bounded at g(x) ∈ Rn,

K[f ◦ g](x) = K[f ](g(x))

5) If f : Rm → Rn is locally bounded at x ∈ Rm and g : Rm → Rm×n is continuous atx ∈ Rm,

K[gf ](x) = g(x)K[f ](x)

124

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Acknowledgements

First and foremost, I would like to express deepest gratitude to my supervisor, ProfessorTielong Shen, for his enthusiastic and wise guidance through these years. His good advice,valuable discussion and great patience lead me to apperceive the significance of researchand through a full graduate student’s life. I’m also indebted to him for generous support,kindness help and constant encouragement during my life of studying aborad.

I would like to greatly acknowledge Professor Riccardo Marino of University of Rome“Tor Vergata”. His friendly guidance and valuable discussions to my research were in-strumental in complementing this work. I owe special thanks to Professor Xiaohong Jiaoof Yanshan University for giving me the opportunity to come to Sophia University. Herinstruction during the early stage of my studies was invaluable.

My thesis committee has provided insightful discussion and comments. Special thanksgo to Professor Yasuhiko Mutoh, Professor Tetsushi Sasagawa and Professor TakashiSuzuki of Sophia University.

I would like to thank Mr. Yasufami Oguri of Sophia University for many support andhelpful discussions on experiment tests in the research.

Sincere thanks go to Mr. Akira Ohata, Mr. Junichi Kako, Mr. Kota Sata, Mr. KenjiSuzuki and Mr. Shozo Yashida of Toyota Motor Corporation for their generosity andinvaluable instructions on many problems in automotive control engineering.

I thank Professor Xiaowu Mu of Zhengzhou University for helpful comments to mythesis. I’m also thankful to Professor Linda Grove of Sophia University for kind help toimprove the English presentation in the thesis.

When at Sophia I greatly profited from my interaction with graduate students andour colleagues. I’m especially grateful to Dr. Kai Zheng, Dr. Po Li, Dr. Munan Hongand Dr. Xiangpeng Yu for valuable communications and aspiring assistance on experimentdesigns. Further, I wish to thank sincerely all members in Control Engineering Laboratoryfor creative discussions at the seminars and friendly atmosphere over the past years. I’malso thankful to the Secretary Hayakawa and other staffs as well as many friends, whoalways offered me their generous help.

Finally, I’m forever grateful to my parents, J.W. Zhang and R.G. An, and family

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members for their encouragement, confidence and support for making this work possible.Without their love and care, I would not have been able to complete this work.

The financial support of Toyota Motor Corporation and the Research Fellowship forYoung Scientists of Japan Society for the Promotion of Science (JSPS) is greatly acknowl-edged.

Jiangyan Zhang

in Sophia, January 2011

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