Event-triggered control subject to actuator saturation

Degree project in

Event-triggered control subject to actuator saturation

GEORG A. KIENER

Stockholm, Sweden 2012

XR-EE-RT 2012:009

Automatic ControlMaster's thesis

Diploma Thesis

Event-triggered control subject to

actuator saturation

Georg A. Kiener

2011 - 2012

Examiner, KTH:

Prof. Dr. Karl H. Johansson

Supervisor, KTH:

Dr.-Ing. Daniel Lehmann

Examiner, TUM:

Prof. Dr.-Ing. Sandra Hirche

Supervisor, TUM:

Dipl.-Ing. Adam Molin

Abstract. Event-triggered control is a recent approach in control theory which aimsat reducing the communication load in networked control systems by adapting thecommunication among the components to the current needs. In more detail, theinformation exchange over the feedback link only takes place if certain event conditions,that guarantee a desired control performance, are satisfied. This thesis analyzes theconsequences of actuator saturation on the stability of the event-triggered controlloop. Based on linear matrix inequalities, stability criteria have been derived whichcan be used to determine regions in the state space that guarantee a stable behavior.Furthermore, the existence of a lower bound on the minimum inter-event time is shown.Due to integrator windup, actuator saturation may severely degrade the performanceof the event-triggered closed-loop system. In order to overcome this problem, thestability criteria have been extended by incorporating a static anti-windup structure.Finally, the effects of transmission delays in the feedback link are analyzed and aprocedure to deal with their consequences is proposed. The results are illustrated bysimulations and by experiments with a wirelessly controlled first-order tank system.

Contents

1 Introduction 1

1.1 Event-triggered control . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Deadband sampling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Actuator saturation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.4 Contribution of this thesis . . . . . . . . . . . . . . . . . . . . . . . . . 41.5 Structure of this thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2 Preliminaries 6

2.1 Notation and definitions . . . . . . . . . . . . . . . . . . . . . . . . . . 62.2 Running example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

3 Continuous-time control subject to actuator saturation 9

3.1 System representation . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.1.1 Dynamic output feedback controller . . . . . . . . . . . . . . . . 103.1.2 PI controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.1.3 Universal representation of the continuous-time control loop . . 11

3.2 Sector condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.3 Stability analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.4 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

4 Event-triggered control subject to actuator saturation 20

4.1 System representation . . . . . . . . . . . . . . . . . . . . . . . . . . . 204.1.1 Dynamic output feedback controller . . . . . . . . . . . . . . . . 204.1.2 PI controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224.1.3 Universal representation of the event-triggered control loop . . . 22

4.2 Stability analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234.3 Minimum inter-event time . . . . . . . . . . . . . . . . . . . . . . . . . 264.4 Extended stability analysis . . . . . . . . . . . . . . . . . . . . . . . . . 27

4.4.1 Alternative event generator . . . . . . . . . . . . . . . . . . . . 274.4.2 Nonzero disturbance signals . . . . . . . . . . . . . . . . . . . . 294.4.3 Nonzero reference signals . . . . . . . . . . . . . . . . . . . . . . 314.4.4 Nonzero disturbance and reference signals . . . . . . . . . . . . 32

4.5 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334.5.1 Fixed event threshold and disturbance . . . . . . . . . . . . . . 344.5.2 Variable event threshold and disturbances . . . . . . . . . . . . 37

i

5 Anti-windup compensation 39

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395.2 Modeling of a static anti-windup structure . . . . . . . . . . . . . . . . 405.3 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

6 Transmission delays 45

6.1 Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 456.2 Large delays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 486.3 Small delays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 486.4 Mutual dependencies . . . . . . . . . . . . . . . . . . . . . . . . . . . . 496.5 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

7 Optimization problems and practical implementation 54

7.1 Algorithms for stability analysis . . . . . . . . . . . . . . . . . . . . . . 547.2 Optimization functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 557.3 Implementation in MATLAB . . . . . . . . . . . . . . . . . . . . . . . 567.4 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

8 Experiments 62

9 Conclusion and outlook 67

List of Figures 69

List of Tables 71

A Mathematical tools 72

A.1 Ellipsoid contained in a symmetric polytope . . . . . . . . . . . . . . . 72A.2 The S-procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

B Implementations 74

B.1 The event-triggered control loop . . . . . . . . . . . . . . . . . . . . . . 74B.2 Algorithm 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75B.3 Algorithm 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76B.4 Plotting of ellipsoids . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

Bibliography 79

ii

1 Introduction

Due to an ongoing progress in communication engineering, a new opportunity of con-trolling complex plants has evolved: Wireless control systems. Compared to wiredinstallations, these systems provide advantages such as simple installation as well asmaintenance or a high degree of flexibility. However, it has to be considered that suffi-cient network resources are an essential requirement for wireless control systems. Forinstance, a busy communication network may evoke delays or packet losses which de-grade the control performance or can even lead to an unstable behavior of the system[1]. In order to resolve this issue, a new control scheme has gained attention, namely:Event-triggered control [2–4].

1.1 Event-triggered control

Event-triggered control is a method to reduce the communication over the feedbacklink by adapting the communication to the current needs [5, 6]. To describe its func-tionality, the event-triggered control loop as considered in this thesis is depicted inFig. 1.1. It consists of a plant with state xp, input u and output y. The plant iscontrolled by a controller with controller state xc and input y which is forwarded by awireless network. The signal d represents an unknown disturbance affecting the plantand w is a reference signal used by the controller.

Controllerxc(t)

Plantxp(t)

Event generator

Wireless network

u(t) y(t) y(tk)y(tk)

d(t)w(t)

Figure 1.1: The event-triggered control loop

1

1 Introduction

The main characteristic of an event-triggered control loop is the event generator. Theevent generator forwards information y(tk) only at certain event times tk. In Fig. 1.1,this is indicated by dashed lines. The event generator can be seen as a kind of smartsensor that forwards its information only if certain event conditions are met. Thesubstantial difference between discrete-time and event-triggered control loops is thatin discrete-time control loops, the time intervals between consecutive “events” are fixed.By contrast, in event-triggered control loops time intervals between consecutive eventsmay vary, based on the conditions of the event generator. To verify these conditions,an event generator requires a processing unit. Within this thesis, it is assumed thatsuch processing units do not pose any restrictions regarding the computational effortat the event generator. By forwarding information only at event times, network trafficis reduced.

1.2 Deadband sampling

Depending on the event conditions, different types of event generators can be defined[2]. In this thesis, deadband sampling will be considered [7].

To explain deadband sampling, a first order system and an event-generator with inputy(t) is used. The event-generator compares the current value y(t) with the last valueforwarded to the controller y(tk). As long as the absolute value of the difference iswithin the deadband

Bk = {y ∈ ℜp, |y(tk) − y(t)| < e} (1.1)

with event threshold e, where |.| denotes the absolute value of scalar, no informationis forwarded to the controller. When the absolute value of the difference meets theevent threshold at the time t, the event condition

|y(tk) − y(t)| = e (1.2)

is satisfied. Now, the current value y(tk) is forwarded to the controller and a newdeadband Bk+1 is set around the current y(t). An illustration of this behavior isshown in Fig. 1.2.

The stability and communication behavior of certain event-triggered control loops areanalyzed in various sources, for example in [2, 3, 8–12]. However, the effects of a verypractical phenomenon – actuator saturation – on event-triggered control loops stillneeds to be studied in depth.

2

1.3 Actuator saturation

t

y

tk+1 tk+2

2e

tk

y(tk) y(tk+2)

y(tk+1)

Figure 1.2: Deadband sampling

1.3 Actuator saturation

Actuator saturation is a constraint that has to be considered in virtually all practicalcontrol systems. The maximum impact an actuator can exert on a system is usuallylimited by physical or safety constraints. For instance, the output of a valve within aprocess plant is limited between fully open and fully closed. If a controller demandsmore or less flow than the valve permits by being fully open or fully closed, actua-tor saturation occurs. Further examples are deflection limits in aircrafts or ships orvoltage limits in electrical actuators. Actuator saturation does not only restrict achiev-able system performance. If it is not considered appropriately, it may even lead toinstabilities within the control loop [13]. For this, an example is given in Section 3.4.

In summary, saturating actuators represent nonlinearities within the control loop yield-ing stability problems. Especially controller with an integrator part are subject to theseproblems as additional integrator windup can occur [14]. In continuous-time controlsystems, actuator saturation and its effect is well studied [13, 14].

Controllerxc(t)

Plantxp(t)

Event generator

u(t) y(t) y(tk)

d(t)w(t)

&(t)

Figure 1.3: The event-triggered control loop subject to actuator saturation

3

1 Introduction

However, regarding actuator saturation in event-triggered control loops, further re-search is necessary. Up to now, research has mostly focused on event-triggered pro-portional controller. Event-triggered control incorporating dynamical controllers is ofrecent interest. Especially the analysis of PI controllers aims to improve the appli-cability of event-triggered control in practical applications [15]. Within this thesis,suitable methods to deal with actuator saturation within an event-triggered controlloop are developed. The respective system to be analyzed is depicted in Fig. 1.3 whichincludes the actuator saturation nonlinearity with input v(t) and output u(t).

1.4 Contribution of this thesis

The approach presented in this thesis adapts methods from continuous-time controltheory to event-triggered control. According to [13], there are two methods to deal withactuator saturation. The first is an analytical approach, in which actuator saturationis explicitly considered. In the second approach, negative effects of actuator saturationare avoided by using anti-windup techniques. Anti-windup control aims at avoidingintegrator windup within the integrator parts of the controller [13, 14].

Both of these methods are considered, leading to the following contributions of thisthesis:

1. LMI1-based stability conditions are derived which can be used to calculate regionsof stability for the event-triggered control loop subject to actuator saturation(Theorem 2, Corollaries 1, 2 and 4).

2. It is shown, that Zeno behavior can be excluded by deriving lower bounds on theminimum inter-event time (Theorem 3 and Corollary 3).

3. A static anti-windup structure is used to deal with actuator saturation andto overcome potential performance degradations caused by actuator saturation(Corollary 5).

4. The effect of transmission delays in the feedback link on the behavior of theevent-triggered control loop subject to actuator saturation is investigated (Propo-sitions 1 and 2).

5. Simulations and experiments illustrate the theoretical results by showing thedependencies of the stability region and the positive effects of anti-windup com-pensation.

Parts of this thesis were submitted for publication [17–19].

1For an introduction to linear matrix inequalities, see [16].

4

1.5 Structure of this thesis

1.5 Structure of this thesis

The thesis is organized as follows:

• Chapter 2 introduces the basic notations and definitions that are needed for thisthesis. Furthermore, a running example is introduced that is used to illustratetheoretical results of the subsequent chapters.

• Chapter 3 presents a method to deal with actuator saturation in continuous-timecontrol loops.

• Chapter 4 derives stability conditions for the event-triggered control loop subjectto actuator saturation by adapting the method presented in the previous chapter.

• Chapter 5 extends the event-triggered control loop with a static anti-windupstructure to deal with actuator saturation.

• Chapter 6 highlights the problems of transmission delays in the feedback link.

• Chapter 7 explains algorithms that are suitable to calculate regions of stabilitybased on LMI conditions.

• Chapter 8 presents an experiment with a water tank to validate the theoreticalresults.

5

2 Preliminaries

In Section 2.1, the general notation and definitions which are used throughout thisthesis are introduced. Furthermore, Section 2.2 introduces a running example that isused to illustrate the results of the subsequent chapters.

2.1 Notation and definitions

Notation. Throughout the thesis, a scalar is denoted by italic letters (x ∈ ℜ), avector by bold italic letters (x ∈ ℜn), a matrix by bold capital letters (A ∈ ℜn×n) anda signal at time t ∈ ℜ+ by x(t), where x0 describes the initial signal value at timet = 0. Identity matrices of the size n × n are denoted by In.

The i-th element of a vector x is denoted by x(i), the i-th row of a matrix A byA(i) and the transpose of a matrix or vector by ()T . Symmetric matrices of the form[

A BT

B C

]

are abbreviated by

[

A ⋆

B C

]

. Furthermore, the absolute value of a scalar is

denoted by |x|, the Euclidean vector norm by ||x|| and the induced matrix norm by||A||. ⌈x⌉ rounds x up to the next whole number, for example ⌈0.2⌉ = 1.

Regarding matrices, A > 0 (A ≥ 0) means that the matrix A is positive definite(semidefinite) and A < 0 (A ≤ 0) means that the matrix A is negative definite(semidefinite). The trace, i.e., the sum of the diagonal elements, of a matrix A isdenoted by trace(A).

E (P , η) describes elliptical sets that are defined by xT P x ≤ η−1 with P > 0.S (G, u0) describes symmetric polytopes that are defined by |G(i)x| ≤ u0(i) fori ∈ {1, . . . , m}.

Plant. The plant considered in this thesis is described by the linear continuous-timestate representation

xp(t) = Axp(t) + Bu(t) + BDd(t), xp(0) = xp0 (2.1)

y(t) = Cxp(t). (2.2)

6

2.1 Notation and definitions

xp ∈ ℜnp is the state vector, u ∈ ℜm is the input, y ∈ ℜp is the measured outputvector and d ∈ ℜq represents disturbances at the plant. A, B, BD and C are realmatrices of appropriate dimensions.

Actuator saturation. The plant input u(t) is given by

u(t) = sat(v(t)), (2.3)

where sat(.) represents the nonlinear saturation function defined by

sat(v(i)) =

umax(i) if v(i) > umax(i)

v(i) if −umin(i) ≤ v(i) ≤ umax(i)

−umin(i) if v(i) < −umin(i)

(2.4)

with i ∈ {1, ..., m}. Throughout the thesis, only symmetrical saturation functionswith

u0(i) = umax(i) = umin(i) for i ∈ {1, ..., m} (2.5)

will be considered. This is illustrated in Fig. 2.1.

Stability. A square matrix A ∈ ℜn×n is called Hurwitz if all its eigenvalues λi (i ∈{1, . . . , n}) have strictly negative real part [20].

System (2.1), (2.2) with zero inputs (u(t) = w(t) = 0 for all t ≥ 0) is asymptoticallystable, i.e., limt→∞ ‖x(t)‖ = 0, if the matrix A is Hurwitz [20].

Since information in event-triggered control loops is only exchanged at event times tk,i.e., when a certain event threshold is met, asymptotic stability cannot be ensured forthese systems. Instead a more appropriate definition for the stability of such systemsis considered in this thesis: Ultimate boundedness (practical stability). Ultimate

sat(v(i))

v(i)

u0(i)

-u0(i)

u(t)v(t)

Figure 2.1: Circuit diagram and graph of the saturation function

7

2 Preliminaries

boundedness means, that the state x(t) can be driven into a certain surroundingE (P , η) of the equilibrium point and kept there for all future times despite exogenousdisturbances.Definition 1. The solution x(t) of the system (2.1), (2.2) is ultimately bounded inE (P , η) if

x(t) ∈ E (P , η), ∀t ≥ 0

holds for every x(0) ∈ E (P , η). In this case, system (2.1), (2.2) is called ultimatelybounded or stable [2, 20].

2.2 Running example

Within this section, an exemplary system based on the system introduced in [15] ispresented. It acts as a running example and is used to illustrate the different resultsof this thesis at the end of each chapter.

The system includes a scalar plant that is given by

xp(t) = axp(t) + bu(t) + bDd(t), xp(0) = xp0 (2.6)

y(t) = xp(t). (2.7)

The plant is controlled by a PI controller which is given by

xc(t) = y(t) − w(t), xc(0) = xc0 (2.8)

v(t) = kIxc(t) + kP (y(t) − w(t)). (2.9)

The numerical values used for following simulations are a = 0.1, b = 1, bD = 0.1,kI = −1 and kP = −1.6. Furthermore, the actuator output u(t) is subject to actuatorsaturation. This is described by the saturation function (2.4) with u(t) = sat(v(t))and u0 = 0.4.

In Chapter 4, an event generator is introduced to analyze the event-triggered controlloop. In that case, the continuous-time plant output y(t) has to be replaced by theevent-triggered plant output y(tk) within the controller equations (2.8) and (2.9). Thismodification will be explained in detail within Chapter 4.

8

3 Continuous-time control subject to

actuator saturation

This chapter introduces a stability criterion for continuous-time PI control loops sub-ject to actuator saturation displayed in Fig. 3.1. First, an appropriate system repre-sentation is developed in Section 3.1. Section 3.2 introduces a general sector conditionbased on a dead zone nonlinearity. This condition is needed to derive the stabilitycriterion for the continuous-time control loop subject to actuator saturation which isdescribed in Section 3.3. Finally, an example illustrates the results in Section 3.4.

The presented stability criterion and its derivation are explained in detail in [13]. Theconcepts explained in this chapter are essential for the development of the stabilitycriterion for the event-triggered control loop subject to actuator saturation proposedin Chapter 4.

3.1 System representation

This section introduces an appropriate system representation for the continuous-timecontrol loop subject to actuator saturation [13]. In this representation, the controllerdescription is included in an extended plant description. Thereby, general methodsfor analyzing the closed-loop systems become applicable.

Plantxp(t)

u(t) y(t)

d(t)w(t)

�(t)Controllerxc(t)

Figure 3.1: The continuous-time control loop subject to actuator saturation

9

3 Continuous-time control subject to actuator saturation

3.1.1 Dynamic output feedback controller

The plant (2.1), (2.2) is controlled by a general dynamic linear output feedback con-troller, which is described by

xc(t) = Acxc(t) + Bcy(t) + BcW w(t), xc(0) = xc0 (3.1)

v(t) = Ccxc(t) + Dcy(t) + DcW w(t) (3.2)

with the controller state xc ∈ ℜnc and reference signals w ∈ ℜs. The matrices Ac, Bc,BcW , Cc, Dc and DcW are real and have appropriate dimensions.

By considering the dynamic controller equation (3.1) and the augmented state vector

x =

[

xp

xc

]

, (3.3)

the plant (2.1), (2.7) becomes

x(t) = Ax(t) + Bu(t) + BDd(t) + BW w(t), x(0) = x0, (3.4)

y(t) = Cx(t) (3.5)

with

A =

[

A 0

BcC Ac

]

, B =

[

B

0

]

, BD =

[

BD

0

]

, BW =

[

0

BcW

]

, C =[

C 0]

. (3.6)

Moreover, by using equation (2.2) and the state transformation (3.3), the controlleroutput (3.2) becomes

v(t) = Kx(t) + KW w(t) (3.7)

with

K =[

DcC Cc

]

, KW = DcW . (3.8)

This transformation is possible for every linear plant that is controlled by a dynamicoutput feedback controller and results in a general system representation where con-troller dynamics are included within the plant description (3.4). Thereby, the a pro-portional control law results as shown in equation (3.7).

10

3.2 Sector condition

3.1.2 PI controller

By setting np = nc = s and by using

xc(t) = y(t) − w(t), xc(0) = xc0

as well as

v(t) = KIxc + KP (y(t) − w(t)),

instead of equations (3.1) and (3.2), PI controllers are explicitly included in this generalsystem representation. Thereby, the respective matrices (3.6) and (3.8) become

A =

[

A 0

C 0

]

, B =

[

B

0

]

, BD =

[

BD

0

]

, BW =

[

0

−Inp

]

, C =[

C 0]

,

K =[

KP C KI

]

, KW = −KP .

(3.9)

3.1.3 Universal representation of the continuous-time control loop

Finally, by introducing the saturation function u(t) = sat(v(t)), the following repre-sentation for the continuous-time system subject to actuator saturation is obtained:

x(t) = Ax(t) + Bsat(Kx(t) + KW w(t)) + BDd(t) + BW w(t),

x(0) = x0,

y(t) = Cx(t).

(3.10)

This representation is displayed in Fig. 3.2. All considerations within this chapterwill be based on systems of the form (3.10) with x ∈ ℜn, n = np + nc, u ∈ ℜm,d ∈ ℜq, w ∈ ℜs, y ∈ ℜp and matrices A, B, BD, BW , K, KW and C of appropriatedimensions. In Section 4.1 it will be extended to cover the event-triggered case.

3.2 Sector condition

Equation (3.10) includes the saturation function as defined in equation (2.4). To applygeneralized methods for stability analysis, the system has to be transformed in a Lureproblem. Therefore, the saturation will be replaced by another nonlinearity, i.e., adecentralized dead zone nonlinearity [13]

φ(v(t)) = sat(v(t)) − v(t) (3.11)

11


Plantx(t)

u(t)

#(t)

v(t)P controller

w(t) w(t)

x(t)

y(t)

Figure 3.2: Adapted system representation of the continuous-time control loop

with elements φ(i)(v), where

φ(i)(v) = φ(v(i)) =

u0(i) − v(i) for v(i) > u0(i)

0 for −umin(i) ≤ v(i) ≤ umax(i)

−u0(i) − v(i) for v(i) < −u0(i)

(3.12)

for i ∈ {1, ..., m}. In Fig. 3.3, the dead zone nonlinearity φ(vi) is displayed.

By using the dead zone representation, system (3.10) can be rewritten as

x(t) = Ax(t) + Bφ(Kx(t) + KW w(t)) + BDd(t) + (BW + BKW )w(t),

x(0) = x0,

y(t) = Cx(t)

(3.13)

with

A = A + BK, B = B.

φ(v(i))

v(i)u0(i)

-u0(i)

sat(v(i))

v(i)

u0(i)

-u0(i)

Figure 3.3: Graph of the dead zone nonlinearity

12

3.3 Stability analysis

Defining

u0 =

u0(1)...

u0(m)

,

a generalized sector condition for symmetrical saturation functions (see equation (2.5))can be used to derive stability criteria for the continuous-time as well as for the event-triggered control loop subject to actuator saturation. The following lemma is provenin [13].

Lemma 1. If υ and ω are elements of the set

S (‖υ − ω‖, u0) = {υ ∈ ℜm, ω ∈ ℜm, |υ(i) − ω(i)| ≤ u0(i)}, i ∈ {1, . . . , m}

then the nonlinearity φ(υ) satisfies

φ(υ)T T (φ(υ) + ω) ≤ 0

for any diagonal positive definite matrix T ∈ ℜm×m.


In this section, a criterion regarding internal asymptotic stability for system (3.10) ispresented [13]. Therefore, d(t) = 0 and w(t) = 0 hold and the state equation fromrepresentation (3.10) turns into

x(t) = Ax(t) + Bsat(Kx(t)), x(0) = x0 (3.14)

or equivalently

x(t) = Ax(t) + Bφ(Kx(t)), x(0) = x0. (3.15)

By using a quadratic Lyapunov function

V (x) = xT P x, P = P T > 0 (3.16)

and applying Lemma 1, the following stability criterion can be obtained.

13


Theorem 1. If there exist a symmetric positive definite matrix W ∈ ℜn×n, apositive definite diagonal matrix S ∈ ℜm×m and a matrix Z ∈ ℜm×n satisfying

W AT

+ AW BS − W KT − ZT

SBT − KW − Z −2S

< 0 (3.17)

[

W ZT(i)

Z(i) ηu0(i)2

]

≥ 0, i ∈ {1, ..., m} (3.18)

then the ellipsoid E (P , η), with P = W −1, is a region of asymptotic stability forthe saturated system (3.14) or equivalently (3.15).

Proof. By setting υ = Kx and ω = Kx + Gx, Lemma 1 guarantees that any x

belonging to the set

S (G, u0) = {x ∈ ℜn, |G(i)x| ≤ u0(i)}, i ∈ 1, . . . , m (3.19)

satisfies the inequality

φ(Kx)T T (φ(Kx) + Kx + Gx) ≤ 0. (3.20)

Considering the quadratic Lyapunov function (3.16), the region of stability to becalculated is an ellipsoid. This ellipsoid E (P , η) is included in the set (3.19) if theinequalities

[

P GT(i)

G(i) ηu20(i)

]

≥ 0 for i ∈ {1, . . . , m}

P > 0

hold (see Appendix A.1) [13]. The second inequality is fulfilled automatically bychoosing a proper Lyapunov function (3.16). By setting W = P −1 and Z = GW ,the first inequality can be rewritten to get equation (3.18). In summary, this equationmeans that the sector condition (3.20) holds for every x ∈ E (P , η).

Furthermore, the time derivative of the Lyapunov function has to fulfill the followingcondition in order to prove asymptotic stability of system (3.14):

V (x) = xT P x + xT P x < 0.

By using sector condition (3.20) a quadratic term in φ can be added according to

V (x) ≤ V (x) − 2φT T (φ + Kx + Gx) < 0.

This estimate is valid for all x ∈ E (P , η). For clarity, the argument of φ is notdisplayed from now on.

14

3.4 Example

By using state representation (3.15), the previous inequality reads[

x

φ

]T

ATP + P A P B − KT T T − GT T T

BTP − T K − T G −2T

[

x

φ

]

< 0.

This inequality contains nonlinearities in decision variables, e.g., T G. By extendingthe states

[

x

φ

]T

=

[

xP P −1

φT T −1

]T

,

[

x

φ

]

=

[

P −1P x

T −1T φ

]

and by setting W = P −1, S = T −1 and Z = GW , this problem is solved andequation (3.17) is achieved.

Altogether, this means that asymptotic stability of the continuous time system subjectto actuator saturation (3.14) is ensured for any x ∈ E (P , η) because V (x) < 0 holdsfor all t ≥ 0.

Moreover, global stability conditions can be derived for linear plants with a matrix A

that is Hurwitz. As shown in equation (3.9), PI controllers yield a matrix A that isnot Hurwitz due to the integrator state of the controller. Therefore, these conditionsare not explained here and the reader is referred to [13].

Furthermore, in Chapter 4, exogenous signals as well as matrices BD, BW and KW

will be considered. Results of that chapter can also be used to ensure stability withinthe continuous-time control loop subject to actuator saturation with d(t) 6= 0 andw(t) 6= 0.

With Theorem 1, regions of stability for the continuous-time system subject to actuatorsaturation can be calculated which is illustrated in the next section.

3.4 Example

The running example presented in Section 2.2 is used to illustrate the methods de-scribed in this chapter. First, equations (2.6)-(2.9) are translated into the form of

(3.4) and (3.7). By introducing the augmented state vector x =[

xp xc

]T, these

equations read

x(t) =

[

a 01 0

]

x(t) +

[

b

0

]

u(t) +

[

bD

0

]

d(t) +

[

0−1

]

w(t), x(0) = x0,

y(t) =[

1 0]

x(t),

v(t) =[

kP kI

]

x(t) − kP w(t).

15


xp

xc

Region of asymptotic stability

x0 =[

1 −2]T

x0 =[

2 −2.5]T

x0 =[

2.5 −2.5]T

-1.5 -1 -0.5 0 0.5 1 1.5 2 2.5

-2

-1

0

1

2

3

4

5

6

Figure 3.4: Region of asymptotic stability and trajectories for different initial statesx0 of the continuous-time control loop

By introducing the saturation function u(t) = sat(v(t)), the state equation

x(t) =

[

a 01 0

]

x(t) +

[

b

0

]

sat([

kP kI

]

x(t) − kP w(t)) +

[

bD

0

]

d(t) +

[

0−1

]

w(t),

x(0) = x0

according to (3.10) is obtained with matrices

A =

[

a 01 0

]

, B =

[

b

0

]

, K =[

kP kI

]

, KW = −kP , BD =

[

bD

0

]

, BW =

[

0−1

]

.

In this chapter, disturbances and reference signals were not considered, i.e. d(t) =w(t) = 0. Therefore, the representation above is simplified as

x(t) =

[

a 01 0

]

x(t) +

[

b

0

]

sat([

kP kI

]

x(t)), x(0) = x0. (3.21)

16

3.4 Example

By solving the optimization problem

min{−trace(W )}subject to conditions from Theorem 1

with the YALMIP toolbox [21] for τ1 = τ2 = 0.1, Theorem 1 was used to calculate anestimate for the region of asymptotic stability of this system. Furthermore, a MAT-LAB/Simulink implementation of the control loop was used to calculate trajectories ofthe system. These methods are described in detail in Chapter 7. Necessary numericalvalues of the system parameters are included in Section 2.2.

Region of asymptotic stability. The region of asymptotic stability is presented inFig. 3.4. For every initial state x(0) = x0 within the ellipsoid, the system will convergeasymptotically towards the equilibrium point which is the origin. For instance, the

green trajectory starts at x0 =[

1 −2]T

and converges towards the origin.

However, the presented approach yields some conservatism as the presented stabil-ity conditions are sufficient conditions. For example the black trajectory starts at

x0 =[

2 −2.5]T

, i.e., outside the region of asymptotic stability but it still convergesasymptotically towards the origin. This is due to the fact that with this method onlyelliptical estimates can be calculated for the region of asymptotic stability. In general,these estimates do not fit exactly the actual region of stability. They are furthermoredependent on various optimization parameters which is described in Chapter 7. Ingeneral, it is not possible to analytically calculate the overall region of asymptoticstability [13].

Nevertheless, for most of the initial states outside the region of asymptotic stability,the system yields an instable behavior. This is illustrated by the red trajectory. After

starting in x0 =[

2.5 −2.5]T

it quickly rises to infinity.

Behavior of the continuous-time control loop. In Fig. 3.5, the trajectories ofthe states are presented. The states converge slower towards the origin for x0 =[

2 −2.5]T

than for x0 =[

1 −2]T

because the initial states are farer away from theorigin and actuator saturation keeps the control output within certain bounds. The

red trajectories of x0 =[

2.5 −2.5]T

describe a rising oscillation till infinity which iscaused by the corresponding control input depicted in Fig. 3.6. In that figure, con-troller and actuator outputs for the different initial states are portrayed. As actuatorsaturation occurs, the actuator output u lies in each case between -0.4 and 0.4. How-ever, the controller output v may extend these borders. This is the windup effect. Theintegrator of the PI controller increases the controller output even if the actuator is

17


xp

t

xc

x0 =[

1 −2.5]T

x0 =[

2 −2.5]T

x0 =[

2.5 −2.5]T

0 10 20 30 40 50 60 70 80 90

0 10 20 30 40 50 60 70 80 90

-5

0

5

10

15

-10-8-6-4-202

Figure 3.5: States of the continuous-time control loop for different initial states x0 ofthe continuous-time control loop

already saturated which affects the control performance negatively which can be seenin Fig. 3.5.

For instance, controller windup depicted in the bottom graph of Fig. 3.6 leads toan increasing integrator state xc in Fig. 3.5. Thereby, the integrator state keeps theactuator output within the saturation bounds for almost all times. Finally, this leadsto an unstable oscillation of the integrator state xc and the plant behavior. In summary,actuator saturation slows down the transient behavior of the controller or even leadsto instability if integrator windup occurs. However, there are anti-windup methodsto deal with integrator windup [14]. A static anti-windup structure is applied andanalyzed in Chapter 5.

18

3.4 Example

t

van

du

van

du

van

du

v for x0 =[

1 −2]T

u for x0 =[

1 −2]T

v for x0 =[

2 −2.5]T

u for x0 =[

2 −2.5]T

v for x0 =[

2.5 −2.5]T

u for x0 =[

2.5 −2.5]T

0 10 20 30 40 50 60 70 80 90

0 10 20 30 40 50 60 70 80 90

0 10 20 30 40 50 60 70 80 90

-1

-0.5

0

0.5

-6-5-4-3-2-10123

-40-30-20-10

01020

Figure 3.6: Controller and actuator outputs for different initial states x0 of thecontinuous-time control loop

19

4 Event-triggered control subject to actuator

saturation

Based on the derivations from the previous chapter, stability criteria for event-triggeredcontrol loops subject to actuator saturation are developed within this chapter. Theconsidered system is depicted in Fig. 1.3. Similar to Chapter 3, an appropriate sys-tem representation is introduced first. In Section 4.2, the basic stability criterion ispresented and its derivation is explained in detail. Section 4.3 presents an approxi-mation for the minimum inter-event time. Extensions to the stability conditions likedifferent definitions of the event generator or disturbances as well as reference signalsare discussed in Section 4.4. Finally, an example is used to illustrate the results inSection 4.5.


To deal with event-triggered sampling, an appropriate system representation has tobe derived first. Again, the controller dynamics will be included by using an extendedplant description in order to make methods for stability analysis introduced in Sec-tion 3.1 applicable.

4.1.1 Dynamic output feedback controller

The plant (2.1), (2.2) is controlled by a general dynamic linear output feedback con-troller. However, in difference to the continuous-time control loop, this controllerreceives new information about the plant output y(t) only at event times tk withk ∈ {0, 1, 2, ...}. Hence, the controller can be described during the time interval[tk, tk+1) by

xc(t) = Acxc(t) + Bcy(tk) + BcW w(t), xc(tk) = xck (4.1)

v(t) = Ccxc(t) + Dcy(tk) + DcW w(t), (4.2)

where xc ∈ ℜnc is the integrator state and w ∈ ℜs represents reference signals. Ac,Bc, BcW , Cc, Dc and DcW are real matrices of appropriate dimensions.

20


As this representation of the controller is only valid during the time interval [tk, tk+1),it is necessary to develop an appropriate model that holds for all times t ≥ 0 in orderto apply the presented methods for stability analysis. Therefore, the output error isintroduced:

e(t) = y(tk) − y(t). (4.3)

By using this error, the controller can be presented in a way that holds for all timest ≥ 0 according to:

xc(t) = Acxc(t) + Bcy(t) + Bce(t) + BcW w(t), xc(0) = xc0 (4.4)

v(t) = Ccxc(t) + Dcy(t) + Dce(t) + DcW w(t). (4.5)

By considering equation (4.4) and by using the augmented state vector

x =

[

xp

xc

]

, (4.6)

the plant (2.1), (2.2) can be rewritten as

x(t) = Ax(t) + Bu(t) + BDd(t) + BW w(t) + BEe(t), x(0) = x0, (4.7)

y(t) = Cx(t) (4.8)

with

A =

[

A 0

BcC Ac

]

, B =

[

B

0

]

, BD =

[

BD

0

]

, BW =

[

0

BcW

]

, BE =

[

0

Bc

]

,

C =[

C 0]

.

(4.9)

Furthermore, the controller output (4.5) can be rewritten to

v = Kx(t) + KW w(t) + KEe(t) (4.10)

with

K =[

DcC Cc

]

, KW = DcW , KE = Dc. (4.11)

Similar to Section 3.1, a general system representation with an extended plant de-scription (4.7) that includes the controller dynamics is achieved. The resulting controllaw (4.10) is again proportional. This transformation is possible for every linear plantthat is controlled by an event-triggered dynamic output feedback controller of theform (4.1), (4.2).

21

4 Event-triggered control subject to actuator saturation

4.1.2 PI controller

PI controllers are explicitly included within the general system representation by set-ting np = nc = s and by using

xc(t) = y(tk) − w(t), xc(tk) = xck

v(t) = KIxc + KP (y(tk) − w(t))

instead of equations (4.1) and (4.2). Moreover, by introducing the output error (4.3)and by using the augmented state (4.6), the general system representation (4.7), (4.10)is obtained again. Thereby, the respective matrices (4.9), (4.11) become

A =

[

A 0

C 0

]

, B =

[

B

0

]

, BD =

[

BD

0

]

, BW =

[

0

−Inp

]

, BE =

[

0

Inp

]

,

C =[

C 0]

, K =[

KP C KI

]

, KW = −KP , KE = KP .

(4.12)

4.1.3 Universal representation of the event-triggered control loop

By introducing the saturation function u(t) = sat(v(t)), a general closed-loop rep-resentation for plants that are controlled by an event-triggered controller subject toactuator saturation is developed

x(t) = Ax(t) + Bsat(Kx(t) + KW w(t) + KEe(t))+

+ BDd(t) + BW w(t) + BEe(t),

x(0) = x0,

y(t) = Cx(t).

(4.13)

This representation depicted in Fig. 4.1. All further analysis of this thesis is based onsystems that have the form of representation (4.13) with x ∈ ℜn, n = np +nc, u ∈ ℜm,d ∈ ℜq, w ∈ ℜs, e ∈ ℜp, y ∈ ℜp and matrices A, B, BD, BW , BE , K, KW , KE

and C of appropriate dimensions. Besides additional terms for the output error e(t),this representation is identical to the representation that describes continuous-timesystems subject to actuator saturation (3.10).

Finally, by adding the dead zone model (3.11), the following representation is devel-oped

x(t) = Ax(t) + Bφ(Kx(t) + KW w(t) + KEe(t))+

+ BDd(t) + (BW + BKW )w(t) + (BE + BKE)e(t),

x(0) = x0,

y(t) = Cx(t)

(4.14)

22


Plantx(t)

u(t)

x(t)

w(t)

v(t)P controller

e(t) e(t)d(t)w(t)

y(t)

Figure 4.1: Adapted system representation of the event-triggered control loop

with

A = A + BK, B = B.


In this section, a stability criterion for the event-triggered control loop subject toactuator saturation is presented. To develop such a criterion, again the quadraticLyapunov function (3.16) is used. Regarding exogenous signals, only the output errore(t) will be considered in this section, i.e., d(t) = w(t) = 0. Thereby, the stateequations from (4.13) and (4.14) become

x(t) = Ax(t) + Bsat(Kx(t) + KEe(t)) + BEe(t), x(0) = x0 (4.15)

and

x(t) = Ax(t) + Bφ(Kx(t) + KEe(t)) + (BE + BKE)e(t), x(0) = x0. (4.16)

The results of this section can be extended to cover additional disturbances or referencesignals as well. This will be explained in Section 4.4.

In [13] stability criteria for systems that are similar to equation (4.15) are described.However, only disturbances e(t) that appear either only outside or only inside thesaturation term (i.e. either KE or BE are zero) are considered there. For the event-triggered control-loop subject to actuator saturation both terms KE and BE have tobe considered. Therefore, the stability criterion from [13] has to be extended.

An important requirement is the boundedness of e(t). In this chapter, the outputerror e(t) is amplitude bounded through a quadratic norm, i.e., e(t) belongs to thefollowing set

W (R, δ) = {e ∈ ℜp, eT Re ≤ δ−1}, R = RT > 0, δ > 0. (4.17)

23


The boundedness of the output error can be achieved by a proper definition of theevent generator. An event generator which ensures that e(t) belongs to the set (4.17)for all t ≥ 0 is defined by the following event condition

eT Re = δ−1. (4.18)

Event condition (1.2) is included as a special case with R = 1 and δ−1 = e2. Methodsto handle other forms of boundedness are discussed in Section 4.4. With the set (4.17),the following stability criterion can be derived.

Theorem 2. If there exist a symmetric positive definite matrix W ∈ ℜn×n, apositive definite diagonal matrix S ∈ ℜm×m, a matrix Z ∈ ℜm×n and three positivescalars τ1, τ2 and η satisfying

W AT

+ AW + τ1W BS − W KT − ZT BE + BKE

SBT − KW − Z −2S −KE

(BE + BKE)T −KTE −τ2R

< 0 (4.19)

[

W ZT(i)

Z(i) ηu20(i)

]

≥ 0, i ∈ {1, ..., m} (4.20)

− τ1δ + τ2η < 0 (4.21)

then,1. for e = 0, the ellipsoid E (P , η), with P = W −1, is a region of asymptotic

stability of the saturated system (4.15) or equivalently (4.16).2. for any e ∈ W (R, δ) and x0 ∈ E (P , η), the trajectories of the saturated

system (4.15) or equivalently (4.16) are ultimately bounded and do not leavethe ellipsoid E (P , η).

Proof. By setting υ = Kx + KEe and ω = Kx + KEe + Gx, Lemma 1 ensuresthat the ellipsoid E (P , η) is included in S (|G|, u0) and that the sector condition

φ(Kx + KEe)T T (φ(Kx + KEe) + Kx + KEe + Gx) ≤ 0 (4.22)

holds for all x ∈ E (P , η). In the following, conditions are derived to show thatE (P , η) is a positively invariant set regarding e(t). By applying the S-procedure (seeAppendix A.2), the condition V (x) < 0 can be extended according to

V (x) + τ1(xT P x − η−1) + τ2(δ−1 − eT Re) < 0 (4.23)

with τ1, τ2 > 0 [13, 16]. The relation above ensures that V (x) < 0 holds for all x

satisfying xT P x ≥ η−1 and for any e ∈ W (R, δ).

Relation (4.23) has to be verified especially for x(t1) ∈ ∂E (P , η), i.e., at the boundaryof E (P , η). If e(t) ∈ W (R, δ) holds, it follows V (x(t1)) < 0. Thus x(t1 + ∆t) will lie in

24


the interior of the ellipsoid for any arbitrary time step ∆t. As a conclusion, inequality(4.23) ensures that the ellipsoid E (P , η) is a positive invariant set regarding e(t).

For further analysis, inequality (4.23) is split into the two inequalities

−τ1δ + τ2η < 0 (4.24)

V (x) + τ1xT P x − τ2eT Re < 0. (4.25)

Inequality (4.24) transfers directly into condition (4.21) of Theorem 2. Similar toSection 3.3, inequality (4.25) can be extended by using the sector condition (4.22) forx ∈ ∂E (P , η):

V (x) + τ1xT P x − τ2eT Re ≤≤V (x) + τ1xT P x − τ2eT Re − 2φT T (φ + Kx + KEe + Gx) < 0.

By using the state equation from (4.16) and denoting W = P −1, S = T −1 andZ = GW , finally condition (4.19) is obtained.

In summary, this procedure leads to the second statement of Theorem 2. Condition(4.20) ensures that the ellipsoid E (P , η) is included in S (G, u0) and, thereby, sectorcondition (4.22) holds for all x ∈ E (P , η). This can be used to derive the conditions(4.19) and (4.21). If these conditions are satisfied, inequality (4.23) holds yieldingV (x) < 0 for all x ∈ ∂E (P , η). Thereby, it is shown that E (P , η) is an invariant setbecause the state trajectories of system (4.15) or equivalently (4.16) will not leave theset for any e ∈ W (R, δ).

Moreover, the special case e = 0 needs to be considered. Thereby, inequality (4.25)turns into

V (x) < −τ1xT P x < 0

ensuring, that V (x) < 0 holds for all x ∈ E (P , η). Hence, it can be proven thatE (P , η) is a region of asymptotic stability of system (4.15) or equivalently (4.16) [13].The result is the first statement of Theorem 2. As this case is similar to the analysis inSection 3.3 and since event-triggered systems usually have e 6= 0, this is not explainedin detail within this thesis.

Theorem 2 can be used to calculate invariant regions for event-triggered control loopssubject to actuator saturation. Suitable algorithms are portrayed in Chapter 7. InSection 4.4, Theorem 2 will be adapted to handle further extensions.

25


4.3 Minimum inter-event time

In event-triggered control, it is important that a lower bound for the time betweentwo consecutive events can be found in order to exclude Zeno behavior. Zeno behaviormeans that an infinite amount of events could be triggered in an infinitesimal timeframe. Such a control behavior could negatively affect the performance in practicalapplications.

The lower bound for the time between two consecutive events is called minimum inter-event time and is given by

Tmin = mink

{tk+1 − tk}, k ∈ {0, 1, 2, ...}.

The following theorem shows that there exists a lower bound on the minimum inter-event time and, therefore, Zeno behavior can be excluded.

Theorem 3. Assume that the event-triggered control loop (4.15) with output (2.2)satisfies the inequalities (4.19)-(4.21), then for x0 ∈ E (P , η) the minimum inter-event time Tmin is lower bounded by

Tmin ≥ T = arg mint

{

e(t) =

√

1δ||R||

}

(4.26)

with

e(t) = maxt

∥

∥

∥C(eAt − In)∥

∥

∥ xmax +t

∫

0

∥

∥

∥CeA(t−τ)B∥

∥

∥ dτu0max (4.27)

and

xmax = maxxp∈E (P ,η)

||xp||, u0max = maxi∈{1,2,...,m}

u0(i). (4.28)

Proof. The plant is described by equations (2.1), (2.2). For d(t) = 0 the plant outputtrajectory is given by

y(t) = CeAtx0 + C

∫ t

0eA(t−τ)Bu(τ)dτ.

By using the over approximation

eT Re ≤ ||e||2||R||,

the norm of the output error

||e(t)|| = ||y(tk) − y(t)||

26

4.4 Extended stability analysis

can be upper bounded by

||e(t)|| =∥

∥

∥

∥

C(

eAt − In

)

x0 + C

∫ t

0eA(t−τ)Bu(τ)dτ

∥

∥

∥

∥

≤ e(t)

for tk = 0 and with e(t) given by (4.27).

||u|| ≤ u0max holds independently of the controller output v(t) because of the actuatorlimitations. As events are generated whenever the event condition eT Re = δ−1 issatisfied, relation (4.26) defines a lower bound on the minimum inter-event time bymeans of e(t).

This theorem shows, that the communication exchange over the feedback link is de-pendent from the event generator defined by R and δ. It can be arbitrarily adaptedby varying δ. However, as these parameters are also used in the stability conditionsof Theorem 2, the stability of the control loop may be affected as well.


In this section, the stability conditions stated in Theorem 2 are extended. The exten-sions include a different definition of the event generator and the presence of distur-bances d(t) 6= 0 and reference signals w(t) 6= 0.

4.4.1 Alternative event generator

In Section 4.2, it is stated that the output error e(t) needs to be bounded in order todevelop a stability criterion. Therefore, the event generator has to be defined in anappropriate way. In Section 4.2, the event generator triggers whenever event condition(4.18) is satisfied.

However, it might be useful to define the event generator in a different way. Espe-cially, a way to bound the amplitude of each output y(i) (i ∈ {1, ..., p}) individually isinteresting for practical applications. Such definitions

e2(1) ≤ δ−1

1 , e2(2) ≤ δ−1

2 , ..., e2(p) ≤ δ−1

p

define a p-dimensional cuboid for the set W which is illustrated in Fig. 4.2 in compar-ison to the elliptical region introduced by event condition (4.18). The event generatortriggers whenever the output error e(t) reaches the border of the cuboid (p = 2).

27


e2

e1

Figure 4.2: Elliptic and cubic event conditions

The inequalities above have to be considered individually within the derivation. There-by, inequality (4.23) turns into

V (x) + τ1(xT P x − η−1) + τ2(δ−11 − e2

1) + τ3(δ−12 − e2

2) + ... + τm+1(δ−1m − e2

m) < 0.

Furthermore, decompositions BE = [BE1, . . . , BEp] and KE = [KE1, . . . , KEp] areintroduced and inequalities (4.19), (4.21) are adapted accordingly. This leads to thefollowing corollary that enables amplitude boundedness for each individual output.

Corollary 1. If there exist a symmetric positive definite matrix W ∈ ℜn×n, apositive definite diagonal matrix S ∈ ℜm×m, a matrix Z ∈ ℜm×n and positivescalars τ1, . . . , τp+1 and η satisfying

W AT

+ AW + τ1W ⋆ ⋆ · · · · · · ⋆

SBT − KW − Z −2S ⋆ · · · · · · ⋆

(BE1 + BKE1)T −KTE1 −τ2 0 · · · 0

...... 0

. . .. . .

......

......

. . .. . . 0

(BEp + BKEp)T KTEp 0 · · · 0 −τp+1

< 0 (4.29)

[

W ZT(i)

Z(i) ηu20(i)

]

≥ 0, i ∈ {1, ..., m} (4.30)

− τ1δ1 . . . δp + τ2ηδ2 . . . δp + · · · + τm+1δ1 . . . δp−1 < 0 (4.31)


stability of the saturated system (4.15) or equivalently (4.16).2. for any e ∈ W = {e ∈ ℜp, e2

(1) ≤ δ−11 , ..., e2

(p) ≤ δ−1p } with δ1, . . . , δp > 0 and

x0 ∈ E (P , η), the trajectories of the saturated system (4.15) or equivalently(4.16) are ultimately bounded and do not leave the ellipsoid E (P , η).

28


Zeno behavior can also be excluded if cubic event conditions are used. To calculatethe minimum inter-event time Tmin, Theorem 3 can be used in combination with theadapted parameters

R =

δ1 0 · · · 0

0. . . . . .

......

. . . . . . 00 · · · 0 δp

, δ = 1.

These parameters describe an approximation of the cubic event generator that can beused to calculate the minimum inter-event time.

4.4.2 Nonzero disturbance signals

In practical applications, disturbances affecting the plant cannot be neglected. There-fore, d(t) 6= 0 needs to be considered within the stability analysis. The state equationsare

x(t) = Ax(t) + Bsat(Kx(t) + KEe(t)) + BDd(t) + BEe(t) (4.32)

or equivalently

x(t) = Ax(t) + Bφ(Kx(t) + KEe(t)) + BDd(t) + (BE + BKE)e(t) (4.33)

with x(0) = x0 in both cases. For clarity, the event generator is furthermore definedas in Section 4.2. Therefore, eT Re ≤ δ−1 holds and e belongs to the ellipsoid (4.17).

Similar to the output error signal e(t), the disturbance d(t) has to be bounded inorder to derive stability conditions. In the following, the disturbance d(t) is assumedto be bounded by a quadratic norm. In that case, d(t) belongs to the set

VD(QD, ǫD) = {d ∈ ℜq, dT QDd ≤ ǫ−1D }, QD = QT

D > 0, ǫD > 0. (4.34)

This leads to the inequality

V (x) + τ1(xT P x − η−1) + τ2(δ−1 − eT Re) + τ3(ǫ−1D − dT QDd) < 0

with τ1, τ2, τ3 > 0 which is satisfied whenever the following corollary holds.

29


Corollary 2. If there exist a symmetric positive definite matrix W ∈ ℜn×n, apositive definite diagonal matrix S ∈ ℜm×m, a matrix Z ∈ ℜm×n and four positivescalars τ1, τ2, τ3 and η satisfying

W AT

+ AW + τ1W ⋆ ⋆ ⋆

SBT − KW − Z −2S ⋆ ⋆

(BE + BKE)T −KTE −τ2R ⋆

BTD 0 0 −τ3QD

< 0 (4.35)

[

W ZT(i)

Z(i) ηu20(i)

]

≥ 0, i ∈ {1, ..., m} (4.36)

− τ1δǫD + τ2ηǫD + τ3ηδ < 0 (4.37)

then,1. for e = d = 0, the ellipsoid E (P , η), with P = W −1, is a region of asymp-

totic stability of the saturated system (4.32) or equivalently (4.33).2. for any e ∈ W (R, δ), d ∈ VD(QD, ǫD) and x0 ∈ E (P , η), the trajectories

of the saturated system (4.32) or equivalently (4.33) are ultimately boundedand do not leave the ellipsoid E (P , η).

Furthermore, Zeno behavior can still be excluded if disturbances d(t) 6= 0 occur.However, the minimum inter-event time needs to be adapted, because the output y(t)is affected by the disturbance signal d(t).

Corollary 3. Assume that the event-triggered control loop (4.32) with output (2.2)satisfies the inequalities (4.35)-(4.37), then for x0 ∈ E (P , η) the minimum inter-event time Tmin is lower bounded by

Tmin ≥ T d = arg mint

{

ed(t) =

√

1δ||R||

}

(4.38)

with

ed(t) = maxt

∥

∥

∥C(eAt − In)∥

∥

∥ xmax+

+t

∫

0

∥

∥

∥CeA(t−τ)∥

∥

∥ dτ(||B||u0max + ||BD||dmax),(4.39)

xmax and u0max as defined in (4.28) and

dmax = maxd∈VD(QD ,ǫD)

||d||. (4.40)

30


Proof. The plant is described by equations (2.1), (2.2) yielding output trajectory

y(t) = CeAtx0 + C

∫ t

0eA(t−τ)(Bu(τ) + BDd(τ))dτ.

By following the same procedure as in the proof of Theorem 3, the norm of the outputerror can be upper bounded by

||e(t)|| =∥

∥

∥

∥

C(

eAt − In

)

x0 + C

∫ t

0eA(t−τ)(Bu(τ) + BDd(τ))dτ

∥

∥

∥

∥

≤ ed(t)

for tk = 0 and with ed(t) given by (4.39). Finally, relation (4.38) is used to derive alower bound on the minimum inter-event time.

In Corollary 2 and 3, the output error e and the disturbance d at the plant belong tothe sets (4.17) and (4.34). These sets were chosen, because they allow a comfortablenotation of the stability condition for event-triggered control loops subject to actuatorsaturation and disturbances.

Nevertheless, output error signals or disturbances with individually bounded ampli-tudes can be considered straightforward as previously discussed in Section 4.4.1. Inthat case, the stability conditions and the minimum inter-event time need to beadapted accordingly.

4.4.3 Nonzero reference signals

Finally, possible reference signals w(t) 6= 0 have to be considered. For simplicityreasons, disturbances will be neglected in this section, i.e., d(t) = 0. The stateequation reads

x(t) = Ax(t) + Bsat(Kx(t) + KW w(t) + KEe(t)) + BW w(t) + BEe(t) (4.41)

or equivalently


+ (BW + BKW )w(t) + (BE + BKE)e(t)(4.42)

with x(0) = x0 in both cases. For clarity, the event generator is again defined as inSection 4.2. Hence, eT Re ≤ δ−1 holds and e belongs to the ellipsoid (4.17).

In order to derive stability conditions, the reference signal w(t) has to be bounded.In the following, the reference signal w(t) is assumed to be amplitude bounded by aquadratic norm, i.e., w(t) belongs to the following set

VW (QW , ǫW ) = {w ∈ ℜnc , wT QW w ≤ ǫ−1W }, QW = QT

W > 0, ǫW > 0. (4.43)

31


The inequality

V (x) + τ1(xT P x − η−1) + τ2(δ−1 − eT Re) + τ3(ǫ−1W − wT QW w) < 0

with τ1, τ2, τ3 > 0 which is similar to inequality (4.23) is derived. It holds, wheneverthe following corollary is satisfied.

Corollary 4. If there exist a symmetric positive definite matrix W ∈ ℜn×n, apositive definite diagonal matrix S ∈ ℜm×m, a matrix Z ∈ ℜm×n and four positivescalars τ1, τ2, τ3 and η satisfying

W AT

+ AW + τ1W ⋆ ⋆ ⋆

SBT − KW − Z −2S ⋆ ⋆

(BE + BKE)T −KTE −τ2R ⋆

(BW + BKW )T KTW 0 −τ3QW

< 0 (4.44)

[

W ZT(i)

Z(i) ηu20(i)

]

≥ 0, i ∈ {1, ..., m} (4.45)

− τ1δǫW + τ2ηǫW + τ3ηδ < 0 (4.46)

then,1. for e = w = 0, the ellipsoid E (P , η), with P = W −1, is a region of

asymptotic stability of the saturated system (4.41) or equivalently (4.42).2. for any e ∈ W (R, δ), w ∈ VW (QW , ǫW ) and x0 ∈ E (P , η), the trajectories

of the saturated system (4.32) or equivalently (4.33) are ultimately boundedand do not leave the ellipsoid E (P , η).

The minimum inter-event time Tmin is not affected by reference signals w(t) 6= 0

because these signals only affect the controller output (4.1) which is bounded by thesaturation limits and the controller state (4.2) which is not used for event generation.Therefore, the minimum inter-event time from Theorem 3 also applies for the problemformulation of this section and Zeno behavior can be excluded.

Again, output error signals or reference signals with individually bounded amplitudescan be considered straightforward as in Section 4.4.1. In that case, the stabilityconditions and the minimum inter-event time need to be adapted accordingly.

4.4.4 Nonzero disturbance and reference signals

To analyze event-triggered systems with disturbances d(t) 6= 0 as well as referencesignals w(t) 6= 0, Corollaries 2 and 4 have to be combined to extend the first and thethird LMI-conditions. For instance, if the event generator, disturbances and reference

32

4.5 Example

signals are defined as in the previous two subsections, the first and the third LMI-condition are given by

W AT

+ AW + τ1W ⋆ ⋆ ⋆ ⋆

SBT − KW − Z −2S ⋆ ⋆ ⋆

(BE + BKE)T −KTE −τ2R ⋆ ⋆

BTD 0 0 −τ3QD ⋆

(BW + BKW )T KTW 0 0 −τ4QW

< 0

− τ1δǫDǫW + τ2ηǫDǫW + τ3ηδǫW + τ4ηδǫD < 0

and the minimum inter-event time from Corollary 3 applies, excluding Zeno behavior.

4.5 Example

The running example from Section 2.2 is used to illustrate the various results presentedin this chapter. The main difference to Section 3.4 is the fact, that the event generatorwith event threshold e is introduced to the system. Information is only forwarded fromthe event generator to the controller, when the event condition |e(t)| = |y(tk)−y(t)| = e

is met. Such an event generator is modeled by R = 1 and δ = e−2.

Therefore, the controller equations (2.8) and (2.9) read

xc(t) = y(tk) − w(t), xc(k) = xck

v(t) = kIxc(t) + kP (y(tk) − w(t))

for t ∈ [tk, tk+1). By introducing the output error e(t) = y(tk) − y(t) = xp(tk) − xp(t),

the augmented state vector x =[

xp xc

]Tand the saturation function u(t) = sat(v(t)),

the state equation of the universal system representation (4.13) reads

x(t) =

[

a 01 0

]

x(t) +

[

b

0

]

sat([

kP kI

]

x(t) − kP w(t) + kP e(t))+

+

[

bD

0

]

d(t) +

[

0−1

]

w(t) +

[

01

]

e(t), x(0) = x0

which holds for all times t ≥ 0. A, B, K, KW , BD and BW are the same matricesas in Section 3.4. The matrices that apply to the output error e(t) are

KE = kP , BE =

[

01

]

.

In the examples, Theorem 2 and the corollaries proposed in Section 4.4 are usedto calculate regions of stability and trajectories. Numerical values for the systemparameters are given in Section 2.2.

33


xa

xc

CTET: e = 0.1, d(t) = 0ET: e = 0.1, d = 0.1

-1.5 -1 -0.5 0 0.5 1 1.5 2-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

3

Figure 4.3: Regions of stability of the even-triggered control loop (ET) in comparisonto the region of asymptotic stability of the continuous-time control loop(CT)

4.5.1 Fixed event threshold and disturbance

In this section, sets of initial states are calculated that yield stability for a givenevent generator as well as given disturbances. Since reference signals w(t) as well asoutput error signals e(t) appear both inside and outside of the saturation functionsat(.) within the general system representation (4.13), they have similar effects on theregion of stability and, therefore, reference signals are neglected, i.e., w(t) = 0.

The respective optimization problems

min{−trace(W )}subject to conditions from Theorem 2/Corollary 2

were solved with the YALMIP toolbox for τ1 = τ2 = τ3 = 0.1 to calculate estimates forthe regions of stability. To calculate trajectories of the system, a MATLAB/Simulinkimplementation of the control loop is used. For further details, the reader is referredto Chapter 7 and Appendix B.1.

Regions of stability. The general effects of event-triggered sampling and additionaldisturbances are depicted in Fig. 4.3. The blue ellipsoid describes the region ofasymptotic stability for the continuous-time control loop (3.21). It is identical to the

34

4.5 Example

xp

xc

CT

ET: e = 0.1, d(t) = 0

ET: x0 =[

0.75 0]T

ET: x0 =[

1 −2]T

ET: x0 =[

2 −2.5]T

ET: x0 =[

2.5 −2.5]T

-2 -1 0 1 2

-2

0

2

4

6

8

Figure 4.4: Trajectories of the event-triggered control loop (dashed, ET) and regionsof stability (solid, CT and ET)

region shown in the upper graph of Fig. 3.4. The other two ellipsoids describe theevent-triggered cases. In both cases, an event generator with event threshold e = 0.1is considered.

If no disturbances occur, i.e., d(t) = 0, the resulting region of stability of the event-triggered control loop is smaller then the region of asymptotic stability of the con-tinuous-time control loop. This is illustrated by the pink ellipsoid in Fig. 4.3. Theresult is reasonable, because communication within the control loop is limited due toevent generator which degrades the control performance.

An additional disturbance |d(t)| ≤ d = 0.1, which is modeled by QD = 1 and ǫD = d−2

,further decreases the region of stability which is illustrated by the brown ellipsoid inFig. 4.3.

It has to be remarked, that the ellipsoids in the event-triggered describe only regionsof stability. This means, that trajectories which begin within the respective ellipsoid,i.e. x0 ∈ E (P , η), will not leave it for all times t ≥ 0. In general asymptotic stabilitycannot be achieved in event-triggered control loops.

Fig. 4.4 illustrates trajectories of the event-triggered control loop. The event thresholdis set to e = 0.1 and no disturbances occur, i.e. d(t) = 0. As calculated before,trajectories that begin within the region of stability, i.e. the pink ellipsoid, do notleave it for any time t ≥ 0. This is illustrated by the green trajectory, starting at

x0 =[

0.75 0]T

.

35


xp

|e|

t

xc

xp(t)xp(tk)

|e(t)|Events

xc(t)

0 5 10 15 20 25 30

0 5 10 15 20 25 30

0 5 10 15 20 25 30

0

0.5

1

0

0.05

0.1

-0.5

0

0.5

1

Figure 4.5: States and output error of the event-triggered control loop for x0 =[

0.75 0]T

However, the conservatism introduced by the approach allows trajectories to start inor even outside the region of asymptotic stability of the continuous-time control loop –represented by the blue ellipsoid – to still yield a stable behavior of the event-triggered

control loop. For instance, the black and purple trajectories starting in x0 =[

1 −2]T

and x0 =[

2 −2.5]T

, still yield a stable behavior of the event-triggered control loopas illustrated in Fig. 4.4.

Nevertheless, for most initial states that are outside the region of stability, instable be-

havior can be observed. For instance the red trajectory starting at x0 =[

2.5 −2.5]T

quickly rises to infinity.

Behavior of the event-triggered control loop. The states xp(t) and xc(t), the ab-solute value of the output error |e(t)| and communication within the event-triggered

control loop for the initial state x0 =[

0.75 0]T

are illustrated in Fig. 4.5.

36

4.5 Examplev

and

u

t

v(t)u(t)

0 5 10 15 20 25 30-1.5

-1

-0.5

0

Figure 4.6: Controller and actuator outputs of the event-triggered control loop for

x0 =[

0.75 0]T

In the top graph, xp(t) and xp(tk) are drawn. According to their difference, the outputerror e(t) is defined. Its absolute value |e(t)| is depicted in the middle graph. Whenever|e(t)| = e holds, the event condition is satisfied and new information is forwarded tothe controller. This is indicated by the pink stars.

According to Theorem 3, the minimum inter-event time can be calculated. Withxmax = 0.77 obtained from Fig. 4.4,

Tmin ≤ T =1a

log(

1 +ae

axmax − u0max

)

= 0.21 < Tmin,sim = 0.29

holds for this example which is smaller than the smallest time Tmin,sim observed betweentwo consecutive events in Fig. 4.5.

Furthermore, the bottom graph shows xc(t). As mentioned before, asymptotic stabilitycan usually not be achieved with event-triggered control. Therefore, xp(t) and xc(t)are both forced into stable oscillations around zero.

Finally, the controller output v(t) and the actuator output u(t) are depicted in Fig. 4.6.Both graphs are unsteady due to the event-triggered communication between the eventgenerator and the controller. Actuator saturation only occurs during the first threeseconds.

4.5.2 Variable event threshold and disturbances

In the following, the effects of different event thresholds is analyzed. To calculate themaximal event threshold leading to a feasible problem, R = 1 has been set and the

37


xp

xc

e = 0.05

e = 0.1

e = emax = 0.148

-1.5 -1 -0.5 0 0.5 1 1.5-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

Figure 4.7: Regions of stability for different event thresholds

optimization problem

min{δ}subject to conditions from Theorem 2

was solved with the YALMIP toolbox and τ1 = τ2 = 0.1. The exact procedure isexplained in Chapter 7.

Fig. 4.7 shows various regions of stability depending on the event threshold e of theevent-triggered control loop without disturbance or reference signals d(t) = w(t) = 0.In general, it can be said that: The bigger the event threshold, the smaller the region ofstability. This is logical because a bigger event threshold yields fewer communicationbetween the controller and the event generator affecting the control performance.

In Chapter 7 it is shown, how the maximum event threshold emax can be calculated.For this example this value is emax = 0.148.

The graphs for different reference signals would look similar to Fig. 4.7. With thesame methods it would also be possible to calculate the maximal upper bounds forthe disturbance dmax and the reference signal wmax. Since the procedure is the same,this is omitted here.

38

5 Anti-windup compensation

The examples of the previous chapters showed, that integrator windup cannot beneglected within event-triggered control. Therefore, methods to deal with integratorwindup are introduced in this chapter. First, the general motivation of anti-windupcontrol is described in Section 5.1. In Section 5.2, a static anti-windup structure isintroduced. Its effects are illustrated in Section 5.3 by using the running example.

5.1 Introduction

Integrator windup is an effect, that can generally occur whenever actuators and eitherthe plant or the controller include integrator dynamics. An example for this effectis displayed in Fig. 3.6. The saturation nonlinearity creates a difference between thecontroller output v(t) and the actuator output u(t). This slows down the response ofthe feedback loop and causes the integrator state xc to wind up. In Fig. 3.5, it is shown,that a saturation nonlinearity can cause a severe windup of the controller state whicheven leads to an unstable system behavior. Furthermore, negative effects of windupmay even increase in event-triggered control because only limited communication takesplace between the controller and the plant. Thereby, old information is used by thecontroller.

A possible way to deal with these issues is to enhance the control loop with an anti-windup structure which aims at avoiding integrator windup within the integrator partof the controller. Many anti-windup structures are realized by feeding back the dif-ference u(t) − v(t) of the actuator and the controller output. When no actuatorsaturation occurs, this difference is zero and the structure has no effect on the closed-loop system. However, if actuator saturation occurs, this difference becomes nonzeroand corrective actions to reduce integrator windup are carried out.

In the following section, a static anti-windup structure is used to enhance the sys-tem performance. For further references regarding anti-windup and a wide variety ofdifferent anti-windup structures, the reader is referred to [14].

39


Controllerxc(t)

Plantxp(t)

Event generator

u(t) y(t) y(tk)

d(t)w(t)

�(t)

KAW

-

Figure 5.1: Event-triggered control loop with a static anti-windup extension

5.2 Modeling of a static anti-windup structure

In this section, the static anti-windup structure depicted in Fig. 5.1 is introduced. Theconsidered system is the same as in Section 4.1. The difference between the actuatorand the controller output corresponds to the dead zone nonlinearity (3.11) accordingto

u(t) − v(t) = sat(v(t)) − v(t) = φ(v(t)).

Using static gain KAW , this difference is fed back to the controller state. Thereby, thegeneral controller state (4.4) is extended in the following way

xc(t) = Acxc(t) + Bcy(t) + Bce(t) + BcW w(t) + KAW φ(v(t)), xc(0) = xc0.

By using the adapted controller state, the following system representation is developed


+ BDd(t) + (BW + BKW )w(t) + (BE + BKE)e(t),

x(0) = x0,

y(t) = Cx(t)

(5.1)

with

A = A + BK, B =

[

B

KAW

]

.

40

5.3 Example

The only difference between the representation of the event-triggered control loop withanti-windup compensation (5.1) and the description without anti-windup compensa-tion (4.14) is the different definition of the matrix B including new degrees of freedomfor the design of the closed-loop system through the feedback gain KAW . Theseadditional degrees of freedom can be used to improve the control performance. Forexample, the size of the region of stability can be increased by choosing an appropriateKAW which is shown in Section 5.3.

After adapting matrix B, all the tools presented in Chapter 4 can be used straightfor-ward to calculate regions of stability for control loops that are extended by a staticanti-windup structure.

For instance, the following corollary adds the presented anti-windup compensation toTheorem 2. Corollaries 1, 2 and 4 can be adapted accordingly.

Corollary 5. If there exist a symmetric positive definite matrix W ∈ ℜn×n, apositive definite diagonal matrix S ∈ ℜm×m, a matrix Z ∈ ℜm×n and three positivescalars τ1, τ2 and η satisfying

W AT

+ AW + τ1W BS − W KT − ZT BE + BKE

SBT − KW − Z −2S −KE

(BE + BKE)T −KTE −τ2R

< 0 (5.2)

[

W ZT(i)

Z(i) ηu20(i)

]

≥ 0, i ∈ {1, ..., m} (5.3)

− τ1δ + τ2η < 0 (5.4)


stability of the saturated system (5.1).2. for any e ∈ W (R, δ) and x0 ∈ E (P , η), the trajectories of the saturated

system (5.1) are ultimately bounded and do not leave the ellipsoid E(P , η).

The presented anti-windup structure does not affect the minimum inter-event timeTmin. It only affects the controller whose state xc(t) is not used for event generationand whose output v(t) is bounded by the saturation limits anyway. Therefore, theminimum inter-event time from Theorem 3 applies, excluding Zeno behavior for thesystem considered in this section.

41


xp

xc

CTET: no AWET: kAW = −0.5

ET: kAW = −1

ET: kAW = −5

-3 -2 -1 0 1 2 3 4-6

-4

-2

0

2

4

6

8

Figure 5.2: Regions of stability of the event-triggered control loop (ET, e = 0.1 andd(t) = 0) with and without anti-windup extension (AW) in comparison tothe region of asymptotic stability of the continuous-time control loop (CT)

5.3 Example

The example from the previous chapter is now extended by a static anti-windup struc-ture. The difference kAW (u(t) − v(t)) is fed back to the controller state. After intro-ducing the dead zone φ(v(t)) = sat(v(t)) − v(t), the state equation is given by

x(t) =

[

a + bkP bkI

1 0

]

x(t) +

[

b

kAW

]

φ([

kP kI

]

x(t) − kP w(t) + kP e(t))+

+

[

bD

0

]

d(t) +

[

−bkP

−1

]

w(t) +

[

bkP

1

]

e(t), x(0) = x0

with the same matrices A, B, K, KW , BD, BW , KE and BE as in Section 4.5 and

A =

[

a + bkP bkI

1 0

]

, B =

[

b

kAW

]

, BW +BKW =

[

−bkP

−1

]

, BE +BKE =

[

bkP

1

]

.

To calculate estimates for the regions of stability, the optimization problem

min{−trace(W )}subject to conditions from Theorem 2/Corollary 5

42

5.3 Example

is used again and solved with the YALMIP toolbox for τ1 = τ2 = 0.1. Furthermore,trajectories of the system are calculated again with a MATLAB/Simulink implemen-tation of the control loop (see Chapter 7 and Appendix B.1). The numerical valuesfor the system parameters are given in Section 2.2.

Regions of stability. In Fig. 5.2, regions of stability for different values kAW aredepicted. The regions are significantly bigger than the region obtained for the event-triggered control loop without anti-windup augmentation and even include the regionof asymptotic stability for the continuous-time control loop. Therefore, it can beconcluded, that anti-windup compensation is able to significantly increase the regionof stability also in event-triggered control.

Behavior of the adapted event-triggered control loop. The trajectories of the

states for x0 =[

2.5 −2.5]T

are depicted in Fig. 5.3. The red graphs describes thestates of the event-triggered control loop without anti-windup compensation whichrefers to the red trajectory in Fig. 4.4. Without anti-windup compensation the initial

value x0 =[

2.5 −2.5]T

yields an unstable system behavior.

If anti-windup compensation is applied, the initial state x0 =[

2.5 −2.5]T

lies withinthe region of stability (see Fig. 5.2). Therefore, the system behavior is stable which isillustrated by the black graphs in Fig. 5.3.

This behavior can also be observed at the controller output which is depicted in Fig. 5.4.The upper graph shows that windup rises quickly within the event-triggered controlloop without anti-windup compensation. This yields an unstable system behavior be-cause integrator windup occurs within the controller that slows its transient behavior.

In the lower graph, the anti-windup extension is applied and windup is compensatedquickly. The actuator output is driven into a stable oscillation within the saturationbounds. As a conclusion, anti-windup compensation did not only increase the size ofthe region of stability, but also improve the performance of the event-triggered controlloop significantly.

43


t

xp

xc

t

ET: no AWET: kAW = −1

0 10 20 30 40 50 60 70 80 90

0 10 20 30 40 50 60 70 80 90

0

5

10

15

-20246

Figure 5.3: Behavior of the event-triggered control loop with and without anti-windup

extension for x0 =[

2.5 −2.5]T

t

van

du

van

du

t

ET: no AW, v

ET: no AW, u

ET: kAW = −1, v

ET: kAW = −1, u

0 10 20 30 40 50 60 70 80 90

0 10 20 30 40 50 60 70 80 90

-2.5-2

-1.5-1

-0.50

0.5

-150

-100

-50

0

Figure 5.4: Controller and actuator outputs with and without anti-windup compensa-

tion for x0 =[

2.5 −2.5]T

44

6 Transmission delays

This chapter will focus on communication imperfections that can occur if a wirelessnetwork is used to realize the feedback link. The focus of this chapter are transmissiondelays, which are modeled by a constant dead time T . This is portrayed in Fig. 6.1.Thereby, messages y(tk) get delayed causing the controller to rely on old informa-tion longer than intended by the event-triggered control scheme which may affect thecontrol performance.

In order to investigate the problems of dead times, an appropriate modeling is neces-sary which is presented in Section 6.1. In the subsequent Sections 6.2 and 6.3, two caseswill be analyzed: Dead times T bigger or smaller than the minimum inter-event timeTmin . The problem of mutual dependencies between different variables is highlightedin Section 6.4. Finally the effects of delays in the feedback link will be illustrated inSection 6.5.

6.1 Modeling

Unlike in the rest of the thesis, this chapter will rely on a scalar problem formulation.The plant is given by

xp(t) = axp(t) + bu(t) + bDd(t), xp(0) = xp0 (6.1)

y(t) = cxp(t). (6.2)

Controllerx!(t)

Plantxp(t)

Event generator

"(t) y(t) y(tk)

$(t)w(t)

v(t)

Dead time

%

Figure 6.1: The event-triggered control loop subject to transmission delays

45


An event is generated whenever the event condition

|y(tk) − y(t)| = e (6.3)

is satisfied. In that case, y(tk) is forwarded to the controller. The controller, withoutconsidering the dead time T , is given by

xc(t) = acxc(t) + bcy(tk) + bcW w(t), xc(tk) = xck (6.4)

v(t) = ccxc(t) + dcy(tk) + dcW w(t) (6.5)

for t ∈ [tk, tk+1). This representation of the controller with y(tk) is rather inadequateto model dead time delays because it is not possible to include delays T properly. Nev-ertheless, since the controller uses values y(tk) until a new one y(tk+1) is transmitted,it is possible to introduce a piecewise defined signal

ytk(t) = y(tk), t ∈ [tk, tk+1). (6.6)

By using this signal, it is possible to rewrite the controller equations in a way thatincludes the constant dead time T and holds for all t ≥ T :

xc(t) = acxc(t) + bcytk(t − T ) + bcW w(t), xc(T ) = xcT (6.7)

v(t) = ccxc(t) + dcytk(t − T ) + dcW w(t). (6.8)

To continue the modeling process, it makes sense to study the impact of the dead timeT on the signal that the controller receives. Therefore, exemplary trajectories aredrawn in Fig. 6.2. The signal y(t), which is drawn in the upper graph, is the regularplant output (6.2) which is subject to event-triggered sampling. The event generatorforwards messages y(tk) whenever the event condition (6.3) is met, i.e., when y(t)reaches the horizontal dashed borders around its trajectory.

In the middle graph, ytk(t) is displayed. This signal describes the values the controllerwould use if no delays would occur. The difference between ytk(t) and y(t) is theoutput error

e(t) = y(tk) − y(t)

which is known from the previous chapters (see relation (4.3)).

However, delays occur. Therefore, the signal which the controller receives is ytk(t−T ).This signal may vary from ytk(t). To model the overall difference between ytk(t − T )and y(t), the extended output error

ee(t) = ytk(t − T ) − y(t) = e(t) + eT (t) (6.9)

is introduced. This signal consists of the regular output error signal e(t) caused bythe event generation and an additional error signal eT (t) caused by the time delay.

46

6.1 Modeling

t

y(t)

tk tk+1

2e

ytk(t)

ytk(t-T)

t

t

tk+2

T

Figure 6.2: Comparison of y(t), ytk(t) and ytk(t − T )

Similar to the previous chapters, plant and controller equations can be combinedby using the extended output error (6.9) as well as an augmented state vector x =[

xp xc

]T. The universal system representation reads

x(t) = Ax(t) + Bsat(Kx(t) + KW w(t) + KEee(t))+

+ BDd(t) + BW w(t) + BEee(t),

x(0) = x0,

y(t) = Cx(t).

(6.10)

This system representation is distinguished from (4.13) by the error signal ee(t) definedin equation (6.9). If the extended error signal ee(t) can be bounded, the stability anal-yses as presented in the previous chapters can be also used in the case of transmissiondelays.

47


6.2 Large delays

Dead times T are considered as large, if they are bigger as the minimum inter-eventtime Tmin. In that case, several events can be generated during the dead time T . Dueto the event condition (6.3), the output error e(t) is upper bounded by

|e(t)| ≤ e

and the additional error caused by the delay eT (t) can be upper bounded by

|eT (t)| ≤⌈

T

Tmin

⌉

e.

Altogether, these results can be used to bound the extended error signal according tothe following proposition.Proposition 1. For transmission delays T > Tmin, the norm of the extended outputerror ee(t) is upper bounded by

|ee(t)| ≤ ee,max =(⌈

T

Tmin

⌉

+ 1)

e. (6.11)

The problem with this bound is, that it includes the minimum inter-event time Tmin

which is defined in Corollary 3. This definition includes xmax from (4.28) which dependson the region of stability E (P , η). Thereby, ee,max depends on E (P , η) which dependson ee,max itself. This problem is discussed in Section 6.4.

6.3 Small delays

For dead times T that are smaller than the minimum inter-event time Tmin, rela-tion (6.11) can be used to immediately receive an upper bound for the extended errorsignal

|ee(t)| ≤ 2e.

However, it is possible to receive an even more precise bound, since the output errorcan only increase to a certain level

|ee(t)| = |e(t) + eT (t)| ≤{

edmax + e for t ∈ [tk, tk + T )e + 0 for t ∈ [tk + T, tk+1)

(6.12)

depending on the dead time T . During the first interval between two consecutiveevents, the controller still uses the old value y(tk−1). Therefore, |eT (t)| = e holds.

48

6.4 Mutual dependencies

However, the output error |e(t)| can only rise to a certain level edmax during that time.Equation (4.39) can be used to calculate this value

edmax =∣

∣

∣c(eaT − 1)∣

∣

∣ xmax +T

∫

0

∣

∣

∣cea(T −τ)∣

∣

∣ dτ(|b|u0max + |bD|dmax) (6.13)

with u0max and dmax as defined in equations (4.28) and (4.40). During the second timeinterval, the controller already received the updated value y(tk+1). Therefore, eT (t)becomes zero. The output error is bounded by |e(t)| ≤ e during that time intervalsince a new event is about to occur. The results above are summarized in the followingproposition.Proposition 2. For transmission delays T < Tmin, the norm of the extended outputerror ee(t) is upper bounded by

|ee(t)| ≤ ee,max = edmax + e (6.14)

with edmax as defined in equation (6.13).

However, relation (6.13) uses xmax again which depends on the region of stabilityE (P , η). Thereby, ee,max depends on E (P , η) which relies on ee,max itself. This problemis discusses in the following section.

6.4 Mutual dependencies

The last two sections revealed a problem for calculating bounds on the extended outputerror ee(t): The bounds rely on xmax from relation (4.28). However, xmax can only beobtained if the region of stability E (P , η) is known and for that, the extended outputerror ee(t) has to be bounded. Furthermore, the definition of the minimum inter-eventtime in Corollary 3 also relies on xmax which makes a clear a priori distinction betweenlarge and small delays impossible. The mutual dependencies of the different variables,considered that u0max and dmax are known, are portrayed in Fig. 6.3.

To overcome these issues further investigations are necessary. However, the followingprocedure can be used to show stability for practical systems subject to time delays.

1. Fix e.

2. Calculate the region of stability by using either Theorem 2 or Corollary 2, 4 or 5without considering the dead time T .

3. Calculate xmax and Tmin.

4. Calculate ee,max according to Proposition 1 or 2.

49


Event generator

e

Tmi

emax

~

Bounds for e(t)

Bounds for eT(t)

Bounds for ee(t)

xmax

T

Figure 6.3: Mutual dependencies of different variables

5. Calculate the region of stability for ee(t) by using either Theorem 2 or Corol-lary 2, 4 or 5.

6. Validate if Proposition 1 or 2 still holds for the new xmax and Tmin. If not, iterateadapting E (P , η) by fixing xmax.

6.5 Example

In this section, transmission delays within the feedback link of the system from Sec-tion 4.5 with e = 0.05 are considered.

Behavior of the event-triggered control loop. The effects of different transmission

delays are highlighted in Fig. 6.4. The trajectories of x0 =[

0.75 0]T

for T = 0.1 andT = 0.8 yield a stable system behavior. However, compared to the trajectory withouttransmission delays T = 0, the control performance degrades. If the delays are toolarge, the system behavior becomes unstable which can be seen by the trajectory forT = 1.

Different trajectories of the event-triggered control loop subject to time delays for

x0 =[

0.75 0]T

are depicted in Fig. 6.5. The upper plot depicts xp(t) with eventsindicated by stars and circles. Furthermore, the middle plot depicts xc(t) and thelower plot depicts u(t) (solid) and v(t) (dashed). It is clearly demonstrated, thattransmission delays deteriorate the control performance in terms of the amount ofevents, the stationary behavior and occurring actuator saturation.

50

6.5 Example

xp

xc ROS for T = 0

T = 0T = 0.1T = 0.8T = 1

-1.5 -1 -0.5 0 0.5 1 1.5

0.6

-2.5-2

-1.5-1

-0.50

0.51

1.52

2.5

Figure 6.4: Trajectories of the event-triggered control loop subject to transmissiondelays and region of stability (ROS) for e = 0.05

Regions of stability. The minimum inter-event time of the control loop withouttransmission delays T = 0 can be calculated according to Theorem 3. With xmax = 0.81obtained from Fig. 6.4, the minimum inter-event time

Tmin ≤ T =1a

log(

1 +ae

axmax − u0max

)

= 0.1034 < Tmin,sim = 0.14

is obtained for this example, which is smaller than the smallest time Tmin,sim observedbetween two consecutive events in Fig. 6.5.

Furthermore, for small delays T < Tmin, edmax is defined by

edmax =(

xmax +u0max

a

)

(

eaT − 1)

,

where xmax is defined by the region of stability that belongs to the control loop withthe respective transmission delay T . For xmax = xmax = 0.81, edmax = 0.0483 isobtained. In Fig. 6.6, the region of stability for |ee(t)| ≤ edmax +e = 0.0983 is depicted.Despite the fact that xmax = 0.81 does not belong to the new region anymore, it is stilla region of stability for the event-control loop with e = 0.05 subject to transmissiondelays T = 0.1, because the xmax obtained from that region leads to a Tmin = 0.1043that still fulfills T < Tmin. Furthermore, bigger xmax lead to a more conservativeestimate for edmax and, hence, the region of stability.

Exemplary trajectories in Fig. 6.6 for e = 0.05, T = 0.1 as well as x0 =[

0.75 0]T

,

x0 =[

−0.7 1.81]T

or x0 =[

0.4 −1.7]T

illustrate that the trajectories do not leavethe respective region of stability for e = 0.0983.

51


xc

t

uan

dv

xp

T = 0T = 0.8

0 1 2 3 4 5 6 7 8 9 10

0 1 2 3 4 5 6 7 8 9 10

0 1 2 3 4 5 6 7 8 9 10

-101

-2

0

2

-2

0

2

Figure 6.5: Behavior of the event-triggered control loop subject to transmission delays

for x0 =[

0.75 0]T

xp

xc

for e =

ROS for e = 0.05

ROS for e = 0.0983

-1.5 -1 -0.5 0 0.5 1 1.5-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

Figure 6.6: Region of stability for e = 0.05 and e = 0.0983. The second region alsorefers to a control loop with e = 0.05 and T = 0.1.

52

6.5 Example

Large delays T > Tmin yield bounds according to equation (6.11) that are at least3e = 0.15. Since the maximum event threshold for that a region of stability can becalculated is emax = 0.148 (see Section 4.5.2), stability cannot be shown for largedelays T > Tmin with the presented methods.

53

7 Optimization problems and practical

implementation

Theorems 1 and 2 as well as Corollaries 1, 2, 4 and 5 can be used to test if givensets of initial conditions E (P , η) yield a stable behavior of the respective systems. Todo so, P and η are put to verify the stability conditions of the respective theorem orcorollary. However, testing random sets of initial conditions is quite inefficient andyields only few practical value. Therefore, the following two optimization problemsare considered in this thesis:

• Maximize the “size” of the set of initial conditions E (P , η) that yield a stablesystem behavior for a given event generator and/or exogenous signals.

• Maximize the “size” of the set of the output error W (R, δ) and/or the set(s) ofexogenous signals VD(QD, ǫD)/VW (QW , ǫW ) that yield a stable system behavior.

Suitable algorithms for both problems are presented in Section 7.1.

In order to execute these algorithms, it is necessary to determine a suitable “size”criterion of the sets that should be maximized. There are several possibilities thattranslate into different objective functions f(E (P , η)). Some examples are given inSection 7.2.

The algorithms can be implemented with various tools. In Section 7.3, the approachof this thesis is described, i.e., the usage of specific toolboxes for solving linear matrixinequalities in MATLAB.

Finally, examples are given in Section 7.4 to illustrate the results of different objectivefunctions and parameters.

7.1 Algorithms for stability analysis

The stability conditions presented in this thesis include linear decision variables W ,S, Z and η if the parameters τi are fixed. The influence of the τi is discussed inSection 7.4. Furthermore, parameters that describe the sets of the output error (R,δ), the disturbance signals (QD, ǫD) and the reference signals (QW , ǫW ) are either apriori given or additional decision variables.

54

7.2 Optimization functions

For clarity and as this thesis focuses on event-triggered control, the sets of exogenoussignals VD(QD, ǫD), VW (QW , ǫW ) are fixed within the following algorithms. How-ever, the sets of admissible disturbance signals VD(QD, ǫD) and/or reference signalsVW (QW , ǫW ) could also be considered or maximized accordingly after adapting thealgorithms with the respective decision variables (QD, ǫD and/or QW , ǫW ).

The first algorithm can be used to calculate the maximal region of stability E (P , η)for a given event generator (fixed R, δ).Algorithm 1. Maximization of the set of initial conditions E (P , η) [13]:

1. Given R and δ, choose a suitable objective function f(E (P , η)).

2. Fix τ1, τ2, τ3 and τ4.

3. Solve for W , S, Z and η the optimization problem

min{f(E (P , η))}subject to inequalities of either Theorem 1, 2 or Corollary 1, 2, 4 or 5.

The second algorithm can be used to calculate the maximum set of the output errorW (R, δ). The parameters R and δ of this set can be used to determine event conditionsthat yield a stable system behavior.Algorithm 2. Maximization of the set of the output error W (R, δ) [13]:

1. Choose a suitable objective function f(W (R, δ)).

2. Fix τ1, τ2, τ3 and τ4.

3. Solve for W , S, Z, R, η and δ the optimization problem

min{f(W (R, δ))}subject to inequalities of either Theorem 1, 2 or Corollary 1, 2, 4 or 5.

7.2 Optimization functions

An appropriate size criterion has to be determined in order to maximize the respectiveset by using the algorithms introduced in the previous section. Several possibilitiesthat translate into different objective functions f(E (P , η)) are given in [13]. Twoexamples are:

• f(E (P , η)) = n log(η) + log(det(P )) referring to volume maximization.

• f(E (P , η)) = β0η +β1trace(P ) with weighting parameters β0 and β1 which leadsto ellipsoids that are homogeneous in all directions.

55

7 Optimization problems and practical implementation

However, these objective functions become nonlinear due to the required transforma-tion P = W −1. To avoid this, the objective functions have to be adapted accord-ingly [13]:

• f(E (W −1, η)) = n log(η) − log(det(W ))

• f(E (W −1, η)) = β0η − β1trace(W )

The results by using different objective functions are discussed in Section 7.4.

7.3 Implementation in MATLAB

MATLAB/Simulink offers convenient tools to simulate the behavior of control loops.In fact, all the trajectories in the examples of the previous chapters were calculatedby using MATLAB/Simulink models of the respective control loops. An exemplarymodel can be found in Appendix B.1.

However, the tools MATLAB offers for solving optimization problems based on LMIconditions are rather inconvenient. Additional toolboxes like YALMIP [21] or CVX[22] overcome these issues and allow for a simple implementation and execution of thealgorithms described in this chapter.

For this thesis, the YALMIP toolbox was used to calculate regions of stability. Exem-plary implementations of the algorithms can be found in Appendix B.2 and B.3.

7.4 Example

This section discusses the results obtained by using different objective functions anddifferent τi for the system considered in Section 4.5.

Algorithm 1. Fig. 7.1 shows the results of Algorithm 1 for different objective func-tions, e(t) ∈ W (R, δ), d(t) = w(t) = 0 and τ1 = τ2 = 0.1. It can be seen that theobjective functions that include either the term −trace(W ) or − log(det(W )) yieldsimilar results. The term η yields rather conservative results. The objective functionβ0η −β1trace(W ) yields the same result as −trace(W ) for β0 = β1 = 1. However, theresults become more conservative if β0 increases. Unfortunately, objective functionswith the term n log(η) could not be solved with the used tools. As the simple objectivefunction −trace(W ) yields good results, it was used for all the simulations within theexamples of the previous chapters.

56

7.4 Example

xp

xc

−trace(W )η

6η − trace(W )η − trace(W )− log(det(W ))

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

Figure 7.1: Results of Algorithm 1 for different objective functions

xp

xc

τ1 = 0.1τ1 = 0.3τ1 = 0.5τ1 = 0.7τ1 = 0.9τ1 = 0.08τ1 = 0.06τ1 = 0.04

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

Figure 7.2: Results of Algorithm 1 for different τ1

57


xp

xc

τ2 = 0.1

τ2 = 1000

τ2 = 0.001

-1.5 -1 -0.5 0 0.5 1 1.5-2

-1.5

-1

-0.5

0

0.5

1

1.5

2


The results of Fig. 7.2, 7.3 and 7.4 were calculated by using the objective function−trace(W ). However, the general statements of the following paragraphs also holdfor other objective functions.

The influence of τ1 is portrayed in Fig. 7.2. The algorithm is feasible for τ1 ∈ [0.04, 1.4].The size of the region of stability varies significantly with different τ1. For this example,there is a maximum around τ1 ≈ 0.08.

The obtained region of stability remains the same for a wide range of τ2 which isshown in Fig. 7.3. Furthermore, despite of changing τ2, the domain of τ1 ∈ [0.04, 1.4]for which the algorithm is feasible remains unchanged with similar results as describedin the previous paragraph. The reason for this is, that a change in τ2 only leads to adifferent scaling between P and η within the inequality xT P x ≤ η−1 that describesthe ellipsoid E (P , η). However, if for certain reasons η is fixed, the influence of τ2 onthe region of stability would have to be considered, too.

Furthermore, the influence of τ3 for d(t) 6= 0, w(t) = 0 and τ1 = τ2 = 0.1 is portrayedin Fig. 7.4. The algorithm is feasible for τ3 ∈ [0.001, 0.3]. The size of the region ofstability varies for different τ3. A maximum is obtained for τ3 ≈ 0.01.

As a summary, τ1 and τ3 have to be chosen carefully, while different values for τ2 donot affect the results if P and η are both decision variables. To retrieve the largestregion of stability, it is recommended to iterate Algorithm 1 on a grid of different τi

values [13].

58

7.4 Example

xc

xp

τ3 = 0.3

τ3 = 0.1

τ3 = 0.05

τ3 = 0.01

τ3 = 0.005

τ3 = 0.003

τ3 = 0.001

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-2

-1.5

-1

-0.5

0

0.5

1

1.5

2


Algorithm 2. Algorithm 2 is used to maximize the set W (R, δ) and thereby toretrieve parameters R and δ that can be used to define the event generator. Thesystem of Section 4.5 uses a scalar event generator. Therefore, R = 1 is set and thetarget function δ is used. For the event generator, the relation |e(t)| ≤ e holds withe =

√δ−1.

The influence of τ1 on the region of stability is shown in Fig. 7.5. The algorithm isfeasible for τ1 ∈ [10−5, 1.4]. The resulting e are displayed in Table 7.1. There seemsto be a maximum for e around τ1 ≈ 0.5. However, in some practical applications, acompromise between a large region of stability and a big event threshold might beneeded. Therefore, it is recommended to try different τ1 until satisfying sets E (P , η)and W (R, δ) are retrieved.

Table 7.1: Calculated event thresholds for different τ1

τ1 0.005 0.01 0.05 0.1 0.5 0.9 1.4e 0.0377 0.0529 0.1121 0.1485 0.2146 0.2007 0.1195

Fig. 7.6 shows, that the results remain unchanged for a wide range of τ2 (τ1 = 0.1).Again, a change in τ2 only leads to a different scaling between P and η. The event-threshold e is neither affected by changing τ2 which is documented in Table 7.2.

59


Table 7.2: Calculated event thresholds for different τ2

τ1 0.005 0.005 0.005 0.1 0.1 0.1 0.9 0.9 0.9τ2 0.001 0.1 1000 0.001 0.1 1000 0.001 0.1 1000e 0.0377 0.0377 0.0377 0.1485 0.1485 0.1485 0.1121 0.1121 0.1121

xp

xc

τ1 = 0.1

τ1 = 0.5

τ1 = 0.9

τ1 = 1.4

τ1 = 0.05

τ1 = 0.01

τ1 = 0.005

-1 -0.5 0 0.5 1-1.5

-1

-0.5

0

0.5

1

1.5


Controller design. Eventually, the obtained region of stability is also greatly depen-dent of the controller design. This is illustrated in Fig. 7.7 where regions of stabilityare depicted for different kP and kI values. It can be seen, that the size of the regionof stability is bigger for bigger ratios kP

kI(provided that the absolute values of |kP |

and |kI | are sufficiently small) due to the lower influence of kI and potential integratorwindup. These results show, that appropriate controller design can also be used toincrease the region of stability.

60

7.4 Example

xp

xc

τ2 = 0.1

τ2 = 1000

τ2 = 0.001

-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8-1.5

-1

-0.5

0

0.5

1

1.5


xp

xc

0

kP = −1.6, kI = −1

kP = −1.6, kI = −0.5

kP = −1, kI = −1

kP = −1, kI = −0.5

-1.5 -1 -0.5 0 0.5 1 1.5-6

-4

-2

0

2

4

6

Figure 7.7: Regions of stability for different controller parameters

61

8 Experiments

The theoretical results of the previous chapters are used in a practical experimentconsisting of a first-order tank system controlled by an event-triggered PI controller.The overall setup of the experiment is depicted in Fig. 8.1. The main component isa cylindric tank from Quanser Inc. [23] which can be filled with water. It is depictedin Fig. 8.2. Its inflow can be controlled by a pump with input u ∈ [0, 12] V. Theoutflow results from a hole at the bottom of the tank and depends on the water level.The plant state xp(t) describes the water level within the tank with unit cm. It iscontinuously measured by a pressure sensor at the bottom of the tank.

The controller, the actuator and the sensor are connected via a wireless communica-tion network consisting of four Tmote Sky sensor nodes [24]. The event generator isimplemented within the sensor node and new information is only forwarded to thecontroller if the event condition |∂xp(tk) − ∂xp(t)| = e is met, whereas the controllersends its information v(t) to the actuator with a high frequency to meet the structuralrequirements used throughout this thesis as presented in Fig. 1.3. All the software isrealized with TinyOS [25].

Tank systems usually have nonlinear dynamics [26]. Therefore, the system is linearized

Wireless network

v(t)

PI controller

��(tk)

Event generator

��(tk) L

�(t)

v(t)

Figure 8.1: Setup of the tank experiment

62

Figure 8.2: Water tank from Quanser Inc.

around the operating point

xOP = 10 cm, uOP = 5.43 V,

leading to the following stable linear state-space model

∂xp(t) = − 115

∂xp(t)1s

+415

∂u(t)cmVs

, ∂xp(0) = ∂xp0

with ∂xp(t) = xp(t) − xOP and ∂u(t) = u(t) − uOP .

The plant is furthermore controlled by an event-triggered PI controller which is givenby

xc(t) = ∂xp(tk)1s, xc(tk) = xck

∂v(t) = −0.1xc(t)Vcm

− ∂xp(tk)Vcm

for t ∈ [tk, tk+1). New information is forwarded to the controller by the sensor wheneverthe event condition |∂xp(tk) − ∂xp(t)| = e is satisfied, where the event threshold e hasbeen varied in the experiments.

The overall non-symmetric saturation bounds u ∈ [0, 12] V are artificially adaptedaround the operating point uOP = 5.43V according to

umax = uOP + u0

umin = uOP − u0

in order to obtain symmetric actuator saturation u(t) = sat(v(t)) which meets therequirements of this thesis. Similar to the event threshold e, u0 has also been variedin the experiments.

Regions of stability. Regions of stability for different event thresholds e and differ-ent saturation limits u0 are depicted in Fig. 8.3. As expected, increasing the eventthreshold e and decreasing the saturation bound u0 both lead to smaller regions ofstability.

63

8 Experiments

xp in cm

xc

incm

e = 1 cm, u0 = 1 Ve = 2 cm, u0 = 1 Ve = 1 cm, u0 = 0.5 V

-8 -6 -4 -2 0 2 4 6 8-100

-80

-60

-40

-20

0

20

40

60

80

100

Figure 8.3: Regions of stability for the tank experiment

xc

incm

t in s

uin

Vx

pin

cm

60 70 80 90 100 110 120 130 140 150 160

60 70 80 90 100 110 120 130 140 150 160

60 70 80 90 100 110 120 130 140 150 160

68

1012

0

5

10

05

1015

Figure 8.4: Behavior of the tank experiment for u0 = 1 V, e = 1 cm (solid lines) andu0 = 0.1 V, e = 2 cm (dashed lines)

64

Behavior of the adapted event-triggered control loop. Fig. 8.4 depicts the behav-ior of the event-triggered control loop for xp(0) = 5 cm and xc(0) = 0. In the upperplot, the plant state xp(t) is depicted with event times indicated as stars and circles.Furthermore, the controller state xc(t) and the plant input u(t) are depicted in themiddle/lower plot. The plots all start at t = 60 s. The first sixty seconds t ∈ [0, 60) sconsist of an initial phase to drive the tank system into the desired initial state x0.

For u0 = 1 V and e = 1 cm (solid lines), the behavior of the control loop remainsstable. In fact, since the plant is stable and the actuator output is limited due to theartificial actuator saturation, the plant state xp(t) cannot become unstable.

Nevertheless, by reducing the saturation limit to u0 = 0.1 V and increasing the eventthreshold to e = 2 cm (dashed lines), the controller state xc(t) becomes unstable.Heavy actuator saturation in combination with a lose event threshold lead to a situa-tion in which no further events are created (after the second star in the upper plot).This phenomenon is called sticking effect [11]. Since the old state information usedby the controller deviates from the operating point xOP = 10 cm, the controller statexc(t) continues to increase over time.

Anti-windup compensation. To avoid instabilities caused by the sticking effect, thecontrol loop is extended by a static anti windup-structure as described in Chapter 5.Therefore, the first controller equation is adapted according to

xc(t) = ∂xp(tk)1s

+ kAW (u(t) − v(t))cmVs

, xc(tk) = xck.

In Fig. 8.5 and 8.6 the positive effects of anti-windup compensation with kAW = −1are depicted. Fig. 8.5 shows that anti-windup compensation increases the region ofstability drastically. Moreover, the behavior of the control loop is improved which isshown in Fig. 8.6. The transient behavior is better (solid lines) and the integratorstate xc remains stable (dashed lines).

65

8 Experiments

xp in cm

xc

incm

e = 1 cm, u0 = 1 V, no AW

e = 1 cm, u0 = 1 V, kAW = −1

-150 -100 -50 0 50 100 150-600

-400

-200

0

200

400

600

Figure 8.5: Regions of stability for the tank experiment with and without anti-windup(AW) extension

xp

incm

xc

incm

t in s

uin

V

60 70 80 90 100 110 120 130 140 150 160

60 70 80 90 100 110 120 130 140 150 160

60 70 80 90 100 110 120 130 140 150 160

4.55

5.56

6.5

-4-20

68

1012

Figure 8.6: Behavior of the tank experiment with anti-windup extension kAW = −1for u0 = 1 V, e = 1 cm (solid lines) and u0 = 0.1 V, e = 2 cm (dashedlines)

66

9 Conclusion and outlook

This thesis has dealt with the influence of actuator saturation on the stability of event-triggered control loops. Its main result consists of LMI conditions that allow for calcu-lating regions of stability for the event-triggered control loop. Moreover, to overcomea possible performance degradation due to integrator windup, the stability conditionshave been extended by incorporating a static anti-windup structure. Thereby, thecontrol behavior of the closed-loop system can be improved and the respective size ofregion of stability can be increased.

In addition to that, it has been shown that the communication over the feedbacklink is bounded for all considered schemes by deriving lower bounds on the minimuminter-event time between two consecutive events.

Since the control behavior depends on the performance of the wireless network usedfor realizing the feedback link of the closed-loop system, the effects of transmissiondelays in the feedback link have been additionally considered. The difficulties offinding analytical results have been outlined and a practical procedure to treat event-triggered control loops subject to actuator saturation and transmission delays has beenproposed.

The size of all the determined regions of stability highly depends on the event condi-tions but also on the controller parameters, additional external signals, transmissiondelays and optimization parameters which has been analyzed in this thesis. The find-ings have been illustrated by simulations.

Finally, experiments with a wirelessly controlled first-order tank system have validatedthe theoretical results. Especially the positive effects of anti-windup compensation onthe performance of the event-triggered control loop have been shown.

Eventually, possible directions for future research are the following:

• Static anti-windup compensation has improved the control performance and hasincreased the size of the region of stability of the event-triggered control loop.Despite these results, more elaborate anti-windup mechanisms including event-triggered anti-windup structures should be considered in future work. Thereby,it has to be evaluated if it is possible and reasonable to aim at a further improve-ment of the control performance by putting additional effort in more elaborateanti-windup mechanisms.

67

9 Conclusion and outlook

• Furthermore, it is necessary to intensively study the effects of communicationimperfections within wireless networks. Transmission delays within the feed-back link are common and it would be good to have an analytical tool to dealwith them. Additionally, the effects of further communication imperfections, likepacket losses or quantization, have to be analyzed.

• Finally, alternative methods for obtaining regions of stability should be consid-ered. It would be desirable to have a method tailored to event-triggered control,because the method presented in this thesis yields a certain conservatism. As astarting point, it could be evaluated if other methods presented in [13] can alsobe applied to event-triggered control leading to less conservative results.

68

List of Figures

1.1 The event-triggered control loop . . . . . . . . . . . . . . . . . . . . . . 11.2 Deadband sampling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3 The event-triggered control loop subject to actuator saturation . . . . . 3

2.1 Circuit diagram and graph of the saturation function . . . . . . . . . . 7

3.1 The continuous-time control loop subject to actuator saturation . . . . 93.2 Adapted system representation of the continuous-time control loop . . . 123.3 Graph of the dead zone nonlinearity . . . . . . . . . . . . . . . . . . . . 123.4 Region of asymptotic stability of the continuous-time control loop . . . 163.5 States of the continuous-time control loop . . . . . . . . . . . . . . . . 183.6 Controller and actuator outputs of the continuous-time control loop . . 19

4.1 Adapted system representation of the event-triggered control loop . . . 234.2 Elliptic and cubic event conditions . . . . . . . . . . . . . . . . . . . . 284.3 Regions of stability of the even-triggered control loop . . . . . . . . . . 344.4 Trajectories of the event-triggered control loop . . . . . . . . . . . . . . 354.5 States and output error of the event-triggered control loop . . . . . . . 364.6 Controller and actuator outputs of the event-triggered control loop . . . 374.7 Regions of stability for different event thresholds . . . . . . . . . . . . . 38

5.1 Event-triggered control loop with a static anti-windup extension . . . . 405.2 Regions of stability with anti-windup extension . . . . . . . . . . . . . 425.3 Behavior of the event-triggered control loop with anti-windup extension 445.4 Controller and actuator outputs with anti-windup compensation . . . . 44

6.1 The event-triggered control loop subject to transmission delays . . . . . 456.2 Comparison of y(t), ytk(t) and ytk(t − T ) . . . . . . . . . . . . . . . . . 476.3 Mutual dependencies of different variables . . . . . . . . . . . . . . . . 506.4 Trajectories of the control loop subject to transmission delays . . . . . 516.5 Behavior of the control loop subject to transmission delays . . . . . . . 526.6 Region of stability for transmission delays . . . . . . . . . . . . . . . . 52

7.1 Results of Algorithm 1 for different objective functions . . . . . . . . . 577.2 Results of Algorithm 1 for different τ1 . . . . . . . . . . . . . . . . . . . 577.3 Results of Algorithm 1 for different τ2 . . . . . . . . . . . . . . . . . . . 58

69

List of Figures

7.4 Results of Algorithm 1 for different τ3 . . . . . . . . . . . . . . . . . . . 597.5 Results of Algorithm 2 for different τ1 . . . . . . . . . . . . . . . . . . . 607.6 Results of Algorithm 2 for different τ2 . . . . . . . . . . . . . . . . . . . 617.7 Regions of stability for different controller parameters . . . . . . . . . . 61

8.1 Setup of the tank experiment . . . . . . . . . . . . . . . . . . . . . . . 628.2 Water tank from Quanser Inc. . . . . . . . . . . . . . . . . . . . . . . . 638.3 Regions of stability for the tank experiment . . . . . . . . . . . . . . . 648.4 Behavior of the tank experiment . . . . . . . . . . . . . . . . . . . . . . 648.5 Regions of stability for the tank experiment with anti-windup extension 668.6 Behavior of the tank experiment with anti-windup extension . . . . . . 66

B.1 Implementation of the event-triggered control loop in Simulink . . . . . 74

70

List of Tables

7.1 Calculated event thresholds for different τ1 . . . . . . . . . . . . . . . . 597.2 Calculated event thresholds for different τ2 . . . . . . . . . . . . . . . . 60

71

A Mathematical tools

A.1 Ellipsoid contained in a symmetric polytope

In the following, conditions are presented according to [13] that ensure E (P , η) ⊂S (G, u0).

By definition of the ellipsoid and the symmetric polytope in Section 2.1, the conditions

x ∈ E (P , η) if ηxT P x ≤ 1

x ∈ S (G, u0) if xTGT

(i)G(i)

u20(i)

x ≤ 1, i ∈ {1, . . . , m}

are obtained. To ensure E (P , η) ⊂ S (G, u0),

1 ≥ ηxT P x ≥ xTGT

(i)G(i)

u20(i)

x, i ∈ {1, . . . , m}

has to hold. If G and u0 are given, ellipsoids E (P , η) that are contained in S (G, u0)are characterized by parameters P and η that fulfill

GT(i)G(i)

ηu20(i)

− P ≤ 0, i ∈ {1, . . . , m}, P > 0.

By using Schur’s complement, the following lemma is obtained.Lemma 2. The ellipsoid E (P , η) is contained in the symmetric polytope S (G, u0),i.e., E (P , η) ⊂ S (G, u0), if

[

P GT(i)

G(i) ηu20(i)

]

≥ 0 for i ∈ {1, . . . , m}

P ≥ 0

hold.

72

A.2 The S-procedure

A.2 The S-procedure

There exist problems where a quadratic function has to be negative whenever otherquadratic functions are all negative. By applying the S-procedure, this problem can bereplaced by one single inequality, containing additional positive scalars, to be satisfied.Lemma 3. [13, 16] Let F0, . . . , Fp be quadratic functions of the variable x ∈ ℜn

defined by

Fi(x) = xT Tix + 2aTi x + bi, i ∈ {0, . . . , p}, Ti = T T

i .

If there exist τ1 ≥ 0, . . . , τp ≥ 0, such that for all x

F0(x) −p

∑

i=1

τiFi(x) ≥ 0

holds, then

F0(x) ≥ 0

holds for all x such that Fi(x) ≥ 0 with i ∈ {0, . . . , p}.

If p = 1, the converse holds if there exists x0 such that F1(x0) > 0.

73

B Implementations

B.1 The event-triggered control loop

The event-triggered control loop is implemented in Simulink. The model is depictedin Fig. B.1.

plantstate x_pplant

input u

k_P

−1.6

k_I

−1

k_AW

−1

int_p

1s

int_c

1s

event−based

plantstate

ykcontrollerstate x_c controller

output v

b

1

a

0.1

SaturationEvent generator

y ykfcn

Dead time

uv

x_c

x_p y(t_k)

Figure B.1: Implementation of the event-triggered control loop in Simulink withplant (cyan), controller (blue), event generator (grey), actuator satura-tion (pink), anti-windup structure (green), transmission delay (red) andworkspace outputs (yellow)

The event generator of the Simulink model is realized by the following code:

1 function yk = fcn (y )2 p e r s i s t e n t ybu f f e r ;3

4 i f isempty ( ybu f f e r )5 ybu f f e r = i n f ;

74

B.2 Algorithm 1

6 end

7

8 i f ( abs (y−ybu f f e r ) > 0 . 1 )9 ybu f f e r = y ;

10 yk = ybu f f e r ;11 else

12 yk = ybu f f e r ;13 end

B.2 Algorithm 1

The following code is an exemplary realization of Algorithm 1 using MATLAB andthe YALMIP toolbox for an event-triggered control loop with d(t) = w(t) = 0.

1 % −−− Parameters −−−2 u0 = 0 . 4 ; % Actuator s a tu ra t i o n3

4 R = 1 ; % Event genera tor5 e = 0 . 1 ;6 de l t a = 1/( e ^2) ;7

8 A = [ 0 . 1 0 ; % Plant9 1 0 ] ;

10 B = [ 1 ; 0 ] ;11 Be = [ 0 ; 1 ] ;12

13 K = [ −1.6 −1]; % C o n t ro l l e r14 Ke = −1.6;15

16 Bs = B; % no anti −windup compensation17 %Bs = [ 1 ; −1] % anti −windup compensation18

19 t1 = 0 . 1 ; % Optimizat ion parameters20 t2 = 0 . 1 ;21 sd = 1e −10;22

23 % −−− Optimizat ion v a r i a b l e s −−−24 W = sdpvar (2 , 2 )25 S = sdpvar (1 , 1 , ’ d iagona l ’ )26 Z = sdpvar (1 , 2 , ’ f u l l ’ )

75

B Implementations

27 eta = sdpvar (1 , 1 )28

29 % −−− LMI Cons tra in ts −−−30 C = [ eta >= sd ,31 W >= sd∗eye ( 2 ) ,32 S >= sd ,33

34 [W∗(A+B∗K) ’+(A+B∗K) ∗W+t1 ∗W, Bs∗S−W∗K’−Z ’ , Be+B∗Ke ;35 S∗Bs’−K∗W−Z , −2∗S , −Ke ;36 (Be+B∗Ke) ’ , −Ke’ , − t2 ∗R] <= −sd∗eye ( 4 ) ,37

38 [W , Z ( 1 , : ) ’ ;39 Z ( 1 , : ) , eta ∗u0 (1 ) ∗u0 (1 ) ] >= 0 ,40

41 −t1 ∗ de l t a + t2 ∗ eta <= −sd ]42

43 % −−− O b j e c t i v e func t i on −−−44 O = −trace (W)45

46 % −−− So lve op t im i za t i on −−−47 so lvesdp (C,O)48

49 % −−− Return r e s u l t s −−−50 P = inv ( double (W) ) ;51 eta = double ( eta ) ;

B.3 Algorithm 2

Algorithm 2 can for instance be realized with the YALMIP toolbox and MATLAB byusing the following code. For this example no exogenous signals are considered, i.e.,d(t) = w(t) = 0.

1 % −−− Parameters −−−2 u0 = 0 . 4 ; % Actuator s a tu ra t i o n3

4 R = 1 ; % Event genera tor5 e = 0 . 1 ;6 de l t a = 1/( e ^2) ;7

8 A = [ 0 . 1 0 ; % Plant

76

B.3 Algorithm 2

9 1 0 ] ;10 B = [ 1 ; 0 ] ;11 Be = [ 0 ; 1 ] ;12

13 K = [ −1.6 −1]; % C o n t ro l l e r14 Ke = −1.6;15

16 Bs = B; % no anti −windup compensation17 %Bs = [ 1 ; −1] % anti −windup compensation18

19 t1 = 0 . 1 ; % Optimizat ion parameters20 t2 = 0 . 1 ;21 sd = 1e −10;22

23 % −−− Optimizat ion v a r i a b l e s −−−24 W = sdpvar (2 , 2 )25 S = sdpvar (1 , 1 , ’ d iagona l ’ )26 Z = sdpvar (1 , 2 , ’ f u l l ’ )27 eta = sdpvar (1 , 1 )28 R = 129 de l t a = sdpvar (1 , 1 )30

31 % −−− LMI Cons tra in ts −−−32 C = [ eta >= sd ,33 W >= sd∗eye ( 2 ) ,34 S >= sd ,35 de l t a >= sd ,36

37 [W∗(A+B∗K) ’+(A+B∗K) ∗W+t1 ∗W, Bs∗S−W∗K’−Z ’ , Be+B∗Ke ;38 S∗Bs’−K∗W−Z , −2∗S , −Ke ;39 (Be+B∗Ke) ’ , −Ke’ , − t2 ∗R] <= −sd∗eye ( 4 ) ,40

41 [W , Z ( 1 , : ) ’ ;42 Z ( 1 , : ) , eta ∗u0 (1 ) ∗u0 (1 ) ] >= 0 ,43

44 −t1 ∗ de l t a + t2 ∗ eta <= −sd ]45

46 % −−− O b j e c t i v e func t i on −−−47 O = de l t a48

49 % −−− So lve op t im i za t i on −−−50 so lvesdp (C,O)

77

B Implementations

51

52 % −−− Return r e s u l t s −−−53 P = inv ( double (W) ) ;54 eta = double ( eta ) ;55 e = sqrt (1/ double ( d e l t a ) ) ;

B.4 Plotting of ellipsoids

Ellipsoids (P , η) can be plotted by running the following function in MATLAB.

1 function ep l o t (P, eta , s t y l e )2 alpha = 0:2∗ pi /720:2∗ pi ;3 x = [ cos ( alpha ) ; sin ( alpha ) ] ;4

5 for i = 1 :7216 p( i ) = sqrt (x ( : , i ) ’∗P∗x ( : , i ) ) ;7 end

8

9 plot ( x ( 1 , : ) . / ( p∗sqrt ( eta ) ) , x ( 2 , : ) . / ( p∗sqrt ( eta ) ) ,s t y l e )

10 end

78

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http://cvxr.com/cvx/

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http://www.tinyos.net

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Event-triggered control subject to actuator saturation