16
Hybrid Control of Parabolic PDEs: Handling Faults of Constrained Control Actuators Nael H. El-Farra and Panagiotis D. Christofides Department of Chemical Engineering University of California Los Angeles, CA 90095-1592 [email protected], [email protected] Abstract. This work proposes a hybrid control methodology that in- tegrates feedback and switching for fault-tolerant constrained control of linear parabolic partial differential equations (PDEs) for which the spec- trum of the spatial differential operator can be partitioned into a finite “slow” set and an infinite stable “fast” complement. Modal decomposi- tion techniques are initially used to derive a finite-dimensional system (set of ordinary differential equations (ODEs) in time) that captures the dominant dynamics of the PDE. The ODE system is then used as the basis for the integrated synthesis, via Lyapunov techniques, of a stabi- lizing nonlinear feedback controller together with a switching law that orchestrates the switching between the admissible control actuator con- figurations, based on their constrained regions of stability, in a way that respects actuator constraints and maintains closed-loop stability in the event of actuator failure. Precise conditions that guarantee stability of the constrained closed-loop PDE system under actuator switching are provided. The proposed method is applied to the problem of stabilizing an unstable, spatially unifrom steady-state of a linear parabolic PDE under constraints and actuator failures. 1 Introduction The study of distributed systems in control is motivated by the fundamentally distributed nature of the control problems arising in many chemical and phys- ical systems, such as transport-reaction processes and fluid flows. The distin- guishing feature of distributed control problems is that they involve the regula- tion of distributed variables by using spatially-distributed control actuators and measurement sensors. Several classes of spatially-distributed processes (such as transport-reaction processes) are naturally modeled by parabolic PDE systems that involve spatial differential operators whose spectrum can be partitioned into a finite slow part and an infinite stable fast complement [12]. The traditional approach to the control of parabolic PDEs involves the application of Galerkin’s method to the PDE system to derive ODE systems that describe the dynam- ics of the dominant (slow) modes of the PDE system, which are subsequently used as the basis for the synthesis of finite-dimensional controllers (e.g., [2,16, O. Maler and A. Pnueli (Eds.): HSCC 2003, LNCS 2623, pp. 172–187, 2003. c Springer-Verlag Berlin Heidelberg 2003

[Lecture Notes in Computer Science] Hybrid Systems: Computation and Control Volume 2623 || Hybrid Control of Parabolic PDEs: Handling Faults of Constrained Control Actuators

  • Upload
    amir

  • View
    212

  • Download
    0

Embed Size (px)

Citation preview

Page 1: [Lecture Notes in Computer Science] Hybrid Systems: Computation and Control Volume 2623 || Hybrid Control of Parabolic PDEs: Handling Faults of Constrained Control Actuators

Hybrid Control of Parabolic PDEs: HandlingFaults of Constrained Control Actuators

Nael H. El-Farra and Panagiotis D. Christofides

Department of Chemical EngineeringUniversity of California

Los Angeles, CA [email protected], [email protected]

Abstract. This work proposes a hybrid control methodology that in-tegrates feedback and switching for fault-tolerant constrained control oflinear parabolic partial differential equations (PDEs) for which the spec-trum of the spatial differential operator can be partitioned into a finite“slow” set and an infinite stable “fast” complement. Modal decomposi-tion techniques are initially used to derive a finite-dimensional system(set of ordinary differential equations (ODEs) in time) that captures thedominant dynamics of the PDE. The ODE system is then used as thebasis for the integrated synthesis, via Lyapunov techniques, of a stabi-lizing nonlinear feedback controller together with a switching law thatorchestrates the switching between the admissible control actuator con-figurations, based on their constrained regions of stability, in a way thatrespects actuator constraints and maintains closed-loop stability in theevent of actuator failure. Precise conditions that guarantee stability ofthe constrained closed-loop PDE system under actuator switching areprovided. The proposed method is applied to the problem of stabilizingan unstable, spatially unifrom steady-state of a linear parabolic PDEunder constraints and actuator failures.

1 Introduction

The study of distributed systems in control is motivated by the fundamentallydistributed nature of the control problems arising in many chemical and phys-ical systems, such as transport-reaction processes and fluid flows. The distin-guishing feature of distributed control problems is that they involve the regula-tion of distributed variables by using spatially-distributed control actuators andmeasurement sensors. Several classes of spatially-distributed processes (such astransport-reaction processes) are naturally modeled by parabolic PDE systemsthat involve spatial differential operators whose spectrum can be partitioned intoa finite slow part and an infinite stable fast complement [12]. The traditionalapproach to the control of parabolic PDEs involves the application of Galerkin’smethod to the PDE system to derive ODE systems that describe the dynam-ics of the dominant (slow) modes of the PDE system, which are subsequentlyused as the basis for the synthesis of finite-dimensional controllers (e.g., [2,16,

O. Maler and A. Pnueli (Eds.): HSCC 2003, LNCS 2623, pp. 172–187, 2003.c© Springer-Verlag Berlin Heidelberg 2003

Page 2: [Lecture Notes in Computer Science] Hybrid Systems: Computation and Control Volume 2623 || Hybrid Control of Parabolic PDEs: Handling Faults of Constrained Control Actuators

Hybrid Control of Parabolic PDEs 173

6]). A potential drawback of this approach is that the number of modes thatshould be retained to derive an ODE system that yields the desired degree ofapproximation may be very large, leading to complex controller design and highdimensionality of the resulting controllers.

Motivated by this, significant recent work has focused on the synthesis ofnonlinear low-order controllers on the basis of ODE models obtained throughcombination of Galerkin’s method with approximate inertial manifolds (see therecent book [5] for details and references). In addition to this work, other ad-vances in control of PDE systems include controller design based on the infinite-dimensional system and subsequent use of approximation theory to computelow-order finite dimensional compensators [3], results on distributed control us-ing generalized invariants [15] and concepts from passivity and thermodynamics[18], analysis and control of parabolic PDE systems with actuator saturation [7],and the stabilization of PDE systems using boundary control [14,4].

While the above efforts have led to the development of a number of system-atic methods for distributed controller design, these methods focus exclusivelyon the classical control paradigm, where a fixed control structure (single feed-back law, fixed actuator/sensor spatial arrangement) is used to achieve ceratincontrol objectives (see Fig.1a). There are many practical situations, however,where it is desirable, and sometimes even necessary, to consider a hybrid controlparadigm, such as the one depicted in Fig.1b, where the control system consistsof a family of controller configurations (e.g., a family of feedback laws and/ora family of actuator/sensor spatial arrangements) together with a higher-levelsupervisor that uses logic to orchestrate switching between these control config-urations. An example, where consideration of hybrid control is necessary, is theproblem of control actuator failure. In this case, upon the detection of a fault inthe operating actuator configuration, it is often necessary to switch to an alterna-tive, well-functioning actuator configuration, with a different spatial placementof the actuators, in order to preserve closed-loop stability. Switching betweenspatially distributed actuators in this case provides a means for fault-tolerantcontrol. Due to the use of logic-based switching and the variable structure ofthe control system, the dynamics of the control system in the hybrid paradigmare an intermix of discrete and continuous components. This is in contrast tothe purely continuous dynamics in the classical control paradigm. Furthermore,whereas a fixed control structure can at best achieve a tradeoff between mul-tiple control objectives in classical control, switching in hybrid control allowsthe accommodation and reconciliation of multiple, possibly conflicting, controlobjectives.

Motivated by these considerations, this work proposes a hybrid controlmethodology which integrates feedback and switching for the constrained stabi-lization of linear of parabolic PDEs. The central idea is the integrated synthesis,via Lyapunov techniques, of a stabilizing nonlinear feedback controller togetherwith a stabilizing switching law that orchestrates the switching between theadmissible control actuator configurations, in a way that respects actuator con-straints, accommodates inherently conflicting control objectives, and guarantees

Page 3: [Lecture Notes in Computer Science] Hybrid Systems: Computation and Control Volume 2623 || Hybrid Control of Parabolic PDEs: Handling Faults of Constrained Control Actuators

174 N.H. El-Farra and P.D. Christofides

Spatially Distributed Process

2

Feedbackcontroller

Control

objectives

Measurement sensor

Control actuator

y (t)m2y (t)m

1

u (t) u (t) u (t)31

(a)

objectives

Spatially Distributed Process

y (t)m2y (t)m

1

u (t) u (t) u (t)31 2

Supervisor

controllerA

controllerB C

controller

Switchinglogic

Control

(b)

Fig. 1. Comparison between the control structures arising in (a) the classical control,and (b) the hybrid control paradigms of spatially distributed systems.

closed-loop stability at the same time. The rest of the paper is organized as fol-lows. In section 2, we present some mathematical preliminaries to characterizethe class of parabolic PDEs considered, and formulate precisely the switchingproblem. Then in section 3, we initially use modal decomposition to derive anODE system that captures the dominant dynamics of the PDE. This ODE sys-tem is then used as the basis for the integrated synthesis of the feedback andthe switching laws. Precise conditions that guarantee stability of the constrainedclosed-loop PDE system under actuator switching are provided. Finally, the pro-posed hybrid control method is applied in section 4, through computer simula-tions, to the problem of fault-tolerant stabilization of the unstable steady-stateof a linearized model of a diffusion-reaction process. The proposed hybrid con-trol method has also been extended to the nonlinear case in [8] and applied in[11] to the problem of fault-tolerant constrained stabilization of the Kuramoto-Sivashinsky equation, a fourth-order PDE that describes wavy behavior.

2 Preliminaries

2.1 Class of Systems

In this work, we focus on control of linear parabolic PDEs of the form:

∂x

∂t= b

∂2x

∂z2 + αx + w

l∑

i=1

bi(z)ui (1)

subject to the boundary conditions:

x(0, t) = 0, x(π, t) = 0 (2)

and the initial condition:x(z, 0) = x0(z) (3)

Page 4: [Lecture Notes in Computer Science] Hybrid Systems: Computation and Control Volume 2623 || Hybrid Control of Parabolic PDEs: Handling Faults of Constrained Control Actuators

Hybrid Control of Parabolic PDEs 175

where x(z, t) ∈ IR denotes the state variable, z ∈ [0, π] ⊂ IR is the spatialcoordinate, t ∈ [0, ∞) is the time, ui ∈ [−umax

i , umaxi ] ⊂ IR denotes the i-

th constrained input, and umaxi is a positive real numbers.

∂2x

∂z2 denotes thesecond-order spatial derivative of x; α, b, and w are constant real numbers withb > 0, and x0(z) is the initial condition. The function bi(z) is a known smoothfunction of z which describes how the control action ui(t) is distributed in thefinite interval [0, π]. Whenever the control action enters the system at a singlepoint z0, with z0 ∈ [0, π] (i.e. point actuation), the function bi(z) is taken to benonzero in a finite spatial interval of the form [z0 −µ, z0 +µ], where µ is a smallpositive real number, and zero elsewhere in [0, π]. Throughout the paper, thenotation | · | will be used to denote the standard Euclidean norm in IRm, whilethe notations | · |2 and ‖ · ‖2 will be used to denote the L2 norms (as defined inEq.4 below) associated with a finite-dimensional and infinite dimensional Hilbertspaces, respectively. Finally, we recall the definition of a class KL function. Inparticular, a function β(s, t) is said to be of class KL if, for each fixed t, thefunction β(s, ·) is continuous, increasing, and zero at zero and, for each fixed s,the function β(·, t) is non-increasing and tends to zero at infinity.

For a precise characterization of the class of PDEs considered in this work,we formulate the PDE of Eq.1 as an infinite dimensional system in the Hilbertspace H([0, π]; IR), with H being the space of functions defined on [0, π] thatsatisfy the boundary conditions of Eq.2, with inner product and norm:

(ω1, ω2) =∫ π

0(ω1(z), ω2(z))IRdz, ||ω1||2 = (ω1, ω1)

12 (4)

where ω1, ω2 are two elements of H([0, π]; IR) and the notation (·, ·)IR denotesthe standard inner product in IR. Defining the state function x on H([0, π]; IR)as:

x(t) = x(z, t), t > 0, z ∈ [0, π], (5)

the operator A as:

Ax = b∂2x

∂z2 + αx, x ∈ D(A) =x ∈ H2(0, π) : x(0, t) = x(π, t) = 0

(6)

where H2(0, π) is a Sobolev space, and the input operator as:

Bu = wl∑

i=1

biui (7)

the system of Eqs.1-2-3 takes the form:

x = Ax + Bu, x(0) = x0 (8)

where x0 = x0(z). For A, the eigenvalue problem is defined as:

Aφj = λjφj , j = 1, . . . ,∞ (9)

Page 5: [Lecture Notes in Computer Science] Hybrid Systems: Computation and Control Volume 2623 || Hybrid Control of Parabolic PDEs: Handling Faults of Constrained Control Actuators

176 N.H. El-Farra and P.D. Christofides

where λj denotes an eigenvalue and φj denotes an eigenfunction. The eigenvalueproblem takes the form

b∂2φj

∂z2 + αφj = λjφj (10)

subject toφj(0) = φj(π) = 0 (11)

where b > 0. A direct computation of the solution of the above eigenvalue prob-lem yields

λj = α − bj2, φj(z) =√

sin(j z), j = 1, . . . ,∞ (12)

The spectrum of A, σ(A), is defined as the set of all eigenvalues of A, i.e.σ(A) = λ1, λ2, · · ·. From the expression for the eigenvalues, it is clear thatall the eigenvalues of A are real, and that, for a given α and b, only a finitenumber of unstable eigenvalues exists, and the distance between two consecutiveeigenvalues (i.e., λj and λj+1) increases as j increases. Furthermore, σ(A) canbe partitioned as σ(A) = σ1(A)

⋃σ2(A), where σ1(A) = λ1, · · · , λm contains

the first m (with m finite) “slow” (possibly unstable) eigenvalues and σ2(A) =λm+1, λm+2, · · · contains the remaining “fast” stable eigenvalues. This impliesthat the dominant dynamics of the PDE can be described by a finite-dimensionalsystem, and motivates the use of modal decomposition in section 3.1 to derivea finite-dimensional system that captures the dominant (slow) dynamics of thePDE.

It should be noted here that, while our development in this section has con-sidered only a single PDE (for the purpose of simplifying the notation and thesolution of the resulting eigenvalue problem), the results of this paper can beextended to systems with multiple parabolic PDEs.

2.2 Problem Formulation

Consider the spatially distributed system of Eq.1, where the control inputs,ui, are constrained in the interval [−umax

i , umaxi ], and assume that N (with N

finite) different spatial arrangements (configurations) of the control actuatorsare available for possible use in feedback control. The notation zk, where k ∈1, · · · , N, is used to represent the vector of spatial locations of the actuators inthe k-th configuration. Only one actuator configuration can be active for control,at any time instance, while the rest are kept dormant. To ensure controllabilityof the system, we allow only a finite number of switches over any finite intervalof time. The problem of interest is how to coordinate switching between thedifferent actuator configurations in a way that respects the actuator constraintsand guarantees closed-loop stability. To address this problem, we formulate thefollowing three objectives:

1. Initially, modal decomposition is employed to derive a finite-dimensional ap-proximation that captures the dominant dynamics of the infinite-dimensionalsystem of Eq.8.

Page 6: [Lecture Notes in Computer Science] Hybrid Systems: Computation and Control Volume 2623 || Hybrid Control of Parabolic PDEs: Handling Faults of Constrained Control Actuators

Hybrid Control of Parabolic PDEs 177

2. Next, the finite-dimensional system is used as the basis for the synthesis, viaLyapunov-based control techniques [13], of a bounded nonlinear controllerthat enforces asymptotic stability and provides an explicit characterizationof the constrained stability region, associated with each control actuatorconfiguration.

3. Finally, a switching rule is derived to orchestrate the transition betweenthe various actuator configurations. The switching law determines whichactuator configuration can be activated at a given moment, i.e. the value ofthe index k as a function of time.

3 Constrained Hybrid Control of Parabolic PDEs

3.1 Modal Decomposition

In this section, we apply standard modal decomposition to the system of Eq.8 toderive a finite-dimensional system. Let Hs, Hf be modal subspaces of A, definedas Hs = spanφ1, φ2, . . . , φm and Hf = spanφm+1, φm+2, . . . (the existenceof Hs, Hf follows from the properties of A). Defining the orthogonal projectionoperators Ps and Pf such that xs = Psx, xf = Pfx, the state x of the systemof Eq.8 can be decomposed as:

x = xs + xf = Psx + Pfx (13)

Applying Ps and Pf to the system of Eq.8 and using the above decompositionfor x, the system of Eq.8 can be written in the following equivalent form:

dxs

dt= Asxs + Bsu

∂xf

∂t= Afxf + Bfu

xs(0) = Psx(0) = Psx0, xf (0) = Pfx(0) = Pfx0

(14)

where As = PsA, Bs = PsB, Af = PfA, Bf = PfB, and the partial derivative

notation in∂xf

∂tis used to denote that the state xf belongs to an infinite-

dimensional space. In the above system, As is a diagonal matrix of dimensionm×m of the form As = diagλj, and Af is an unbounded differential operatorwhich is exponentially stable (following from the fact that λm+1 < 0 and theselection of Hs, Hf ). From the above modal decomposition procedure, we havethe following finite-dimensional system

dxs

dt= Asxs + Bsu (15)

which describes the evolution of the m slow modes of the infinite-dimensionalsystem.

Page 7: [Lecture Notes in Computer Science] Hybrid Systems: Computation and Control Volume 2623 || Hybrid Control of Parabolic PDEs: Handling Faults of Constrained Control Actuators

178 N.H. El-Farra and P.D. Christofides

Remark 1. We note that the above model reduction procedure which led to thefinite-dimensional system of Eq.15 can also be used, when empirical eigenfunc-tions of the system of Eq.1 computed through Karhunen-Loeve (KL) expansionare used as basis functions in Hs and Hf instead of the eigenfunctions of A. Forapproaches on how to compute empirical eigenfunctions through KL expansion,the reader is referred to [1].

3.2 Integrating Feedback and Switching

Having obtained a finite-dimensional model that describes the dominant dynam-ics of the infinite-dimensional system, we proceed in this section to describe theproposed procedure for designing the hybrid control system. To this end, andsince our objective is the stabilization of the infinite-dimensional, we assume,for simplicity, that the number of inputs is equal to the number of slow modes(i.e. l = m). Though not discussed in this paper explicitly, the results can begeneralized to address the problem of reference-input tracking. For a clear pre-sentation of our results, we consider the equivalent representation of the slowsystem of Eq.15 in terms of the evolution of the amplitudes of the eigenmodes.This finite-dimensional ODE system is given by

as(t) = Fas(t) + G(zk)u(t) (16)

where as(t) = [a1(t) · · · am(t)]T ∈ IRm, ai(t) is the amplitude of the i-th eigen-

mode, xs(t) =m∑

j=1

aj(t)φj , (xs(t), φj) = aj(t)(φj , φj), F is an m × m diagonal

matrix of the form F = diagλj, and G is an m × m matrix whose (i, k)-thelement is given by Gik = φi(zk). Using a quadratic Lyapunov function of theform V = aT

s Φas, where Φ is a positive-definite symmetric matrix that satisfiesthe Riccati inequality

FT Φ + ΦF − ΦGGT Φ < 0 (17)

we synthesize the following bounded nonlinear feedback law [13]

u = −r(as, ukmax, zk) (LGV )T (zk) (18)

where

r(as, ukmax, zk) =

L∗F V +

√(L∗

F V )2 + (ukmax|(LGV )T (zk)|)4

|(LGV )T (zk)|2[1 +

√1 + (uk

max|(LGV )T (zk)|)2] (19)

where zk = [zk1 zk2 · · · zkm ]T , k ∈ 1, · · · , N, L∗F V = aT

s (FT Φ+ΦF )as+ρ|as|2,ρ > 0, LGV (zk) = 2aT

s ΦG(zk). The notation ukmax is used to indicate the mag-

nitude of actuator constraints associated with the k-th actuator configuration.This quantity is allowed to vary from one configuration to another. The scalar

Page 8: [Lecture Notes in Computer Science] Hybrid Systems: Computation and Control Volume 2623 || Hybrid Control of Parabolic PDEs: Handling Faults of Constrained Control Actuators

Hybrid Control of Parabolic PDEs 179

function r(·) in Eqs.18-19 can be thought of as a nonlinear gain of the LGVcontroller. This Lyapunov-based gain, which depends on both the magnitudeof actuator constraints, uk

max, and the spatial arrangement of the actuators,zk, is shaped in a way that guarantees constraint satisfaction and asymptoticclosed-loop stability within a well-characterized region in the state space givenin Theorem 1 below.

We are now ready to state the main result of this work. Theorem 1 belowprovides both the state feedback control law (see the discussion in remark 5 foroutput feedback controller design and implementation) as well as the necessaryswitching law, and states precise conditions that guarantee closed-loop stabilityin the switched closed-loop infinite-dimensional system.

Theorem 1. 1. Consider the system of Eq.16 under the feedback control lawof Eqs.18-19 and let δk

s be a positive real number such that the compact setΩ(uk

max, zk) = as ∈ IRm : aTs Φas ≤ δk

s is the largest invariant set embeddedwithin the region described by the following inequality

L∗F V ≤ uk

max|(LGV )T (zk)| (20)

Without loss of generality, assume that zk(0) = z1 and as(0) ∈ Ω(u1max, z1).

If, at any given time T , the condition

as(T ) ∈ Ω(ujmax, zj) (21)

holds, for some j ∈ 1, · · · , N, then setting zk(T+) = zj guarantees that theswitched closed-loop system of Eq.15-18-19 is asymptotically stable.

2. Consider the parabolic PDE of Eq.1 under the control law of Eq.18-19and the switching law of Eq.21. Let Θk be the set of all xs(0) for whichas(0) ∈ Ω(uk

max, zk). Then given any xs(0) ∈ Θk, for any k ∈ 1, · · · , N,and given any xf (0), the infinite-dimensional switched closed-loop system isasymptotically stable.

Remark 2. Owing to the dependence of the input operator Bs in Eq.15 on thespatial locations of the control actuators (through the actuator distribution func-tions bi(z)), the inequality of Eq.20 is parameterized by the actuator locationsand can therefore be used to explicitly identify the admissible control actuatorconfigurations (zk) for which stability of the constrained closed-loop system isguaranteed. For a given actuator configuration (fixed zk), the inequality of Eq.20describes the state space region where the control action satisfies the constraintsand enforces the negative-definiteness of the time-derivative of the Lyapunovfunction along the trajectories of the finite-dimensional closed-loop system. Bycomputing the largest invariant set, Ω(uk

max, zk), within this region, we obtainan estimate of the stability region associated with each actuator configuration.The requirement that the set Ω(uk

max, zk) be invariant is needed to ensure thatat no time do the closed-loop trajectories violate Eq.20, under any actuatorconfiguration.

Page 9: [Lecture Notes in Computer Science] Hybrid Systems: Computation and Control Volume 2623 || Hybrid Control of Parabolic PDEs: Handling Faults of Constrained Control Actuators

180 N.H. El-Farra and P.D. Christofides

Remark 3. Due to the linear structure of the system of Eq.8, and the absenceof any nonlinear terms, the only source of coupling between the slow and fastsubsystems of Eq.14 (obtained through the application of modal decomposi-tion to the system of Eq.8) is the control input, u(·), which is shared by bothsubsystems. This fact, together with the fact that the control input is, by de-sign, a function of the slow states only, implies that the evolution of the xs

states, in the closed-loop system, is independent of xf (note however that theevolution of xf depends on xs). An important consequence of this outcome isthe fact that the stability region obtained (for each actuator configuration) onthe basis of the finite-dimensional system of Eq.15 is exactly the same as thestability region for the slow subsystem in the infinite-dimensional closed-loopsystem. In other words, the set of initial states that stabilize the constrainedswitched finite-dimensional closed-loop system of Eq.15 (for a given actuatorconfiguration) is the same as the set of initial slow states needed to stabilize theinfinite-dimensional switched closed-loop system of Eq.14 (for the same actua-tor configuration). It should be noted here that this result is different from itscounterpart in the nonlinear case [8], where the dependence of the closed-loopxs-subsystem on xf introduces a discrepancy (which can nonetheless be madearbitrarily small) between the stability regions for the finite-dimensional closed-loop slow system (obtained through Galerkin’s method) and the slow subsystemof the infinite-dimensional closed-loop system.

Remark 4. The switching law of Eq.21 orchestrates the transition between thevarious actuator configurations in a way that respects the constraints and guardsagainst any potential instability that may arise due to switching. The basic prob-lem here owes to the limitations imposed by constraints on the set of feasibleinitial conditions that can be used, for a given actuator configuration, to stabilizethe closed-loop system. Different actuator configurations possess different stabil-ity regions and, depending on where the state trajectory is at a given momentin time, a switch from one configuration to another may land the state outsidethe stability region associated with the target configuration, thus leading to in-stability. To guard against this possibility, the switching law of Eq.21 tracks thetemporal evolution of the slow state and allows switching to another actuatorconfiguration only when the state is within the corresponding stability region forthat configuration. This determines, implicitly, the earliest safe switching time.This idea of connecting the switching logic with the stability regions was firstproposed in [9] for constrained control of switched nonlinear systems. Extensionsof this work to switched systems with uncertainty can be found in [10].

4 Simulation Example

In this section, we demonstrate through computer simulations how the concept ofcoupling feedback and actuator-switching can be used to deal with the problem ofconstrained stabilization of parabolic PDEs in the presence of actuator failures.

Page 10: [Lecture Notes in Computer Science] Hybrid Systems: Computation and Control Volume 2623 || Hybrid Control of Parabolic PDEs: Handling Faults of Constrained Control Actuators

Hybrid Control of Parabolic PDEs 181

To this end, consider the following parabolic PDE

∂x

∂t=

∂2x

∂z2 +(βT e−γγ − βU

)x + βUb(z)u(t)

x(0, t) = 0, x(π, t) = 0, x(z, 0) = x0(z)(22)

which represents a linearized model of a typical diffusion-reaction process, aroundthe zero steady-state (see [7] for some results on constrained control of the non-linear model), where x denotes the dimensionless temperature in the reactor, βT

denotes a dimensionless heat of reaction, γ denotes a dimensionless activationenergy, βU denotes a dimensionless heat transfer coefficient, u(t) denotes themanipulated input (the temperature of the cooling medium) and b(z) is the cor-responding actuator distribution function. The following typical values are givento the process parameters: βT = 50.0, βU = 2.0, γ = 4.0. For these values,the operating steady state x(z, t) = 0 is an unstable one, as can be seen fromFig.2. The control objective is to stabilize the reactor temperature profile at this

01

230

12

34

5

0

5

10

15

20

25

zt

x(t,z)

0 0.5 1 1.5 2 2.5 3 −4

−3

−2

−1

0

1

2

3

4

a1

zAz

C

Actuator location

umax

=1.0 u

max=0.5

umax

=1.5

zB

Fig. 2. (Left) Open-loop temperature profile showing the instability of the x(t, z) = 0steady-state. (Right) Stability regions as a function of actuator location for umax = 1.5(dotted), umax = 1.0 (dashed), and umax = 0.5 (solid).

unstable steady state by manipulating the temperature of the cooling medium,subject to hard constraints. The eigenvalue problem for the spatial differentialoperator of the process can be solved analytically and its solution is

λj = 1.66 − j2, φj(z) =√

sin(j z), j = 1, . . . ,∞ (23)

For this system, we consider the first eigenvalue as the dominant one and usestandard modal decomposition to derive an ODE that describes the tempo-ral evolution of the amplitude, a1(t), of the first eigenmode, where xs(t) =a1(t)φ1(z). This ODE is used for the synthesis of the controller, using Eq.18-19, which is then implemented on a 30-th order Galerkin discretization of theparabolic PDE of Eq.22 (higher order discretizations led to identical results).

In order to demonstrate the utility of the switching scheme proposed in Theo-rem 1 for dealing with actuator failures, we consider the following problem where

Page 11: [Lecture Notes in Computer Science] Hybrid Systems: Computation and Control Volume 2623 || Hybrid Control of Parabolic PDEs: Handling Faults of Constrained Control Actuators

182 N.H. El-Farra and P.D. Christofides

three point control actuators, A, B, and C, located at zA = 0.5π, zB = 0.36π,and zC = 0.22π, respectively, are available for stabilization. In practice, this canbe realized through multiple cooling systems, distributed along the length of thereactor (similar to the idea of having multiple heating zones discussed in [16]).The three actuators have different constraints of umaxA

= 1.5 umaxB= 0.5 and

umaxC= 1.0. Only one actuator can be active at any given moment while the

other two remain dormant. The question is how to choose the alternative or“backup” actuator, in order to maintain closed-loop stability, in the event thatthe operating actuator fails.

Using Eq.20 with ρ = 0.02, the stability region for umax = 1.5 (dottedline), the stability region for umax = 0.5 (solid line) and the stability regionfor umax = 1.0 (dashed line) are computed as functions of actuator locationand shown in Fig.2 (right plot). To simplify the presentation of our results,we plot in Fig.2 (right) the variation of the set of admissible initial conditionsfor the amplitude of the first eigenmode, a1(0), with actuator location (notethat xs(0) = a1(0)φ1(z)). For example, for a given actuator location, the set ofamplitudes between the two solid curves represents the set of stabilizing initialconditions for a1 when an actuator with umax = 0.5 is placed at the givenlocation. Similarly, the set of amplitudes between the two dashed curves, at agiven location, represents the set of stabilizing initial conditions for a1 whenan actuator with umax = 1.0 is placed at that location. From Fig.2 (right

0

1

2

3

0

5

10

0

1

2

3

4

5

6

zt

x(t,z)

0 2 4 6 8 10−1.5

−1

−0.5

0

t

u

Fig. 3. Closed-loop temperature profile (top) and manipulated input profile (bottom)when actuator A (z = 0.5π, umax = 1.5) is used, starting at a1(0) = 3.0.

plot), it is easy to see that for an initial condition a1(0) = 3.0, only actuatorA (zA = 0.5π, umax = 1.5) can be used initially since the initial condition liesoutside the stability regions for both actuator B and actuator C. Fig.3 depicts thecorresponding closed-loop state (top) and manipulated input (bottom) profileswhen actuator A is used (with no actuator failures). As expected the boundedcontroller stabilizes the closed-loop system at the zero steady state. Now, supposethat sometime after startup, say at t = 2.4, a fault occurs in actuator A andit becomes necessary to switch to an alternative actuator to maintain closed-loop stability. Without using the switching logic of Theorem 1, it is not clear

Page 12: [Lecture Notes in Computer Science] Hybrid Systems: Computation and Control Volume 2623 || Hybrid Control of Parabolic PDEs: Handling Faults of Constrained Control Actuators

Hybrid Control of Parabolic PDEs 183

whether actuator B or C should be activated at this time. Fig.4 depicts theresulting closed-loop state (top) and manipulated input (bottom) profiles whenactuator B is switched in at the time of the failure of actuator A. It is clear inthis case that the controller is unable to stabilize the closed-loop system at thedesired steady-state. Note that the control action stays saturated at the lower

01

230

5

10

0

5

10

15

20

25

30

35

40

zt

x(t,z)

0 2 4 6 8 10−1.5

−1

−0.5

0

t

u

Fig. 4. Closed-loop temperature profile (top) and manipulated input profile (bottom)when actuator A (umax = 1.5, zA = 0.5π) fails at t = 2.4 and actuator B (umax = 0.5,zB = 0.36π) is activated in its place.

constraint for actuator C. In contrast, when the switching law of Theorem 1 isimplemented, we track the evolution of a1(t) over time and find that, at thetime of the failure of actuator A, we have a1(2.4) = 1.4, i.e. the state is insidethe stability region of actuator C and outside the stability region of actuatorB. Based on this, we decide to switch to actuator B. Fig.5 depicts the resultingclosed-loop state (top) and manipulated input (bottom) profiles when actuatorC is switched in at t = 2.4. As expected, activation of actuator C preservesclosed-loop stability at the desired steady-state.

01

230

10

20

30

0

1

2

3

4

5

6

zt

x(t,z)

0 5 10 15 20 25 30−1.5

−1

−0.5

0

t

u

Fig. 5. Closed-loop temperature profile (left) and manipulated input profile (right)when actuator A fails at t = 2.4 and actuator C (umax = 1.0,zC = 0.22π) is activatedin its place.

Page 13: [Lecture Notes in Computer Science] Hybrid Systems: Computation and Control Volume 2623 || Hybrid Control of Parabolic PDEs: Handling Faults of Constrained Control Actuators

184 N.H. El-Farra and P.D. Christofides

Acknowledgements. Financial support from the Air Force Office of ScientificResearch and the National Science Foundation is gratefully acknowledged.

References

1. J. A. Atwell and B. B. King. Proper orthogonal decomposition for reduced basisfeedback controllers for parabolic equations. Mathematical and Computer Model-ing, 33:1–19, 2001.

2. M. J. Balas. Feedback control of linear diffusion processes. Int. J. Contr., 29:523–533, 1979.

3. J. A. Burns and B. B. King. A reduced basis approach to the design of low-orderfeedback controllers for nonlinear continuous systems. Journal of Vibration andControl, 4:297–323, 1998.

4. C. A. Byrnes, D. S. Gilliam, and V. I. Shubov. Global lyapunov stabilization ofa nonlinear distributed parameter system. In Proc. of 33rd IEEE Conference onDecision and Control, pages 1769–1774, Orlando, FL, 1994.

5. P. D. Christofides. Nonlinear and Robust Control of PDE Systems: Methods andApplications to Transport-Reaction Processes. Birkhauser, Boston, 2001.

6. R. F. Curtain. Finite-dimensional compensator design for parabolic distributedsystems with point sensors and boundary input. IEEE Trans. Automat. Contr.,27:98–104, 1982.

7. N. H. El-Farra, A. Armaou, and P. D. Christofides. Analysis and control of pal-abolic PDE systems with input constraints. Automatica, to appear, 2003.

8. N. H. El-Farra and P. D. Christofides. Hybrid control of parabolic PDE systems.In Proceedings of 41st IEEE Conference on Decision and Control, pages 216–222,Las Vegas, NV, 2002. An extended version of this work has been submitted toComp. & Chem. Eng.

9. N. H. El-Farra and P. D. Christofides. Switching and feedback laws for control ofconstrained switched nonlinear systems. In Lecture Notes in Computer Science,Proc. of 5th Intr. Workshop on Hybrid Systems: Computation and Control, volume2289, pages 164–178, Tomlin, C. J. and M. R. Greenstreet Eds., Berlin, Germany:Springer-Verlag, 2002.

10. N. H. El-Farra and P. D. Christofides. Coordinating feedback and switching forrobust control of hybrid nonlinear processes. AIChE J., accepted, 2003.

11. N. H. El-Farra, Y. Lou, and P. D. Christofides. Coordinated feedback and switchingfor wave suppression. In Proceedings of 15th International Federation of AutomaticControl World Congress, 2002. An extended version of this work has been submit-ted to Comp. & Chem. Eng.

12. A. Friedman. Partial Differential Equations. Holt, Rinehart & Winston, New York,1976.

13. Y. Lin and E. D. Sontag. A universal formula for stabilization with boundedcontrols. Systems & Control Letters, 16:393–397, 1991.

14. W. J. Liu and M. Krstic. Stability enhancement by boundary control in thekuramoto-sivashinsky equation. Nonlinear Analysis: Theory, Methods & Appli-cations, 43:485–507, 2001.

15. A. Palazoglu and A. Karakas. Control of nonlinear distributed parameter systemsusing generalized invariants. Automatica, 36:697–703, 2000.

16. W. H. Ray. Advanced Process Control. McGraw-Hill, New York, 1981.

Page 14: [Lecture Notes in Computer Science] Hybrid Systems: Computation and Control Volume 2623 || Hybrid Control of Parabolic PDEs: Handling Faults of Constrained Control Actuators

Hybrid Control of Parabolic PDEs 185

17. P. K. C. Wang. Asymptotic stability of distributed parameter systems with feed-back controls. IEEE Trans. Automat. Contr., 11:46–54, 1966.

18. E. B. Ydstie and A. A. Alonso. Process systems and passivity via the Clausius-Planck inequality. Syst. & Contr. Lett., 30:253–264, 1997.

5 Appendix

Proof (of Theorem 1).Part 1: Consider the finite-dimensional closed-loop system of Eqs.16-18-19. Usinga standard Lyapunov argument, one can show that, for each actuator configu-ration, the time-derivative of the Lyapunov function satisfies

V ≤ −ρ|as|2[1 +

√1 + (uk

max|(LGV (zk))T |)2] ≡ −α(zk) < 0, ∀ as = 0

(24)

k = 1, · · · , N , whenever Eq.20 is satisfied. Since the set Ω(ukmax, zk) is, by defini-

tion, the largest invariant set within the region described by Eq.20, then startingfrom any initial condition as(0) ∈ Ω(uk

max, zk), the closed-loop state satisfiesEq.20 for all time, and consequently, V decreases monotonically according toV ≤ −α(zk) < 0, which implies that, for each actuator configuration, zk, theorigin of the closed-loop system of Eqs.16-18-19 (without switching) is asymp-totically stable.

Without loss of generality, assume that as(0) ∈ Ω(u1max, z1). Since the set

Ω(u1max, z1) is invariant, then for all times that the configuration zk = z1 is

active, the closed-loop state satisfies Eq.20 with zk = z1 and V decreases mono-tonically according to the dissipation inequality V ≤ −α(z1) < 0. Now if, at anytime T where as(T ) ∈ Ω(uj

max, zj) for some j ∈ 1, · · · , N, configuration zj isswitched in (and z1 switched out), then it is clear from the invariance of the setΩ(uj

max, zj) that for all times that the configuration zk = zj remains active, theclosed-loop state will satisfy Eq.20 with zk = zj , and, consequently (from Eq.24)V will continue to decrease monotonically, albeit with a (possibly) different dis-sipation rate V ≤ −α(zj) < 0. From this analysis, we conclude that as long asthe k-th configuration is switched in at a time T , where as(T ) ∈ Ω(uk

max, zk), thetime-derivative of the Lyapunov function (for the switched closed-loop system)will always satisfy the following worst-case dissipation inequality for all t ≥ 0

V ≤ maxk=1,···,N

−α(zk) < 0 (25)

which implies that the origin of the finite-dimensional switched closed-loop sys-tem of Eq.16-18-19 is asymptotically stable. Hence, there exists a function β ofclass KL such that

|as(t)| ≤ β(|as(0)|, t) ∀ t ≥ 0 (26)

which implies that as(t) → 0 as t → ∞.

Page 15: [Lecture Notes in Computer Science] Hybrid Systems: Computation and Control Volume 2623 || Hybrid Control of Parabolic PDEs: Handling Faults of Constrained Control Actuators

186 N.H. El-Farra and P.D. Christofides

Using the fact that xs =m∑

j=1

aj(t)φj(z), and the definition of the L2 norm

defined for the Hilbert space Hs, we have

|xs|2 =∫ π

0

(√a21(t)φ

21(z) + · · · + a2

m(t)φ2m(z)

)dz (27)

From Eq.12, we have φ2j (z) =

sin2(j z) ≤ 2π

which yields

|xs|2 ≤∫ π

0

(√2π

√a21(t) + · · · + a2

m(t)

)dz

=∫ π

0

(√2π

|as(t)|)

dz ≤√

2π β(|as(0)|, t)(28)

which implies that the state xs is bounded and |xs|2 → 0 as t → ∞. There-fore the origin of the switched closed-loop slow subsystem of Eq.15-18-19-21 isasymptotically stable.Part 2: Substituting the controller of Eq.18 into Eq.15, the infinite-dimensionalclosed-loop system takes the form

xs = Asxs + Bsu(as, ukmax, zk)

xf = Afxf + Bfu(as, ukmax, zk)

(29)

We have already established, in part 1 above, asymptotic stability of the xs-closed-loop subsystem under the switching law of Eq.21. In this part, we showthat the fast, xf -subsystem, under the control law of Eq.18-19 and the switchinglaw of 21, is also asymptotically stable, for any xf (0) (provided that as(0) ∈Ω(uk

max, zk) or, equivalently, xs(0) ∈ Θk, for any k), and that, therefore, theinfinite-dimensional closed-loop system of Eq.29 (and hence the PDE of Eq.1),under the switching rule of Eq.21, is asymptotically stable. To this end, we firstnote that, from exponential stability of the operator Af and boundedness of thecontrol input (recall that |u(as, u

kmax, zk)| ≤ uk

max ∀ as ∈ Ω(ukmax, zk)), the

xf -subsystem can be treated as an exponentially stable system with a bounded(and decaying) input. Specifically, from the exponential stability of A (whichfollows from the fact that λm+1 < 0 and the selection of Hs, Hf ), and using theconverse Lyapunov theorem for infinite-dimensional systems [17], we have thatthere exists a Lyapunov functional W : Hf → IR≥0 and a set of positive realnumbers (b1, b2, b3, b4) such that for all xf ∈ Hf , the following conditions hold

b1‖xf‖22 ≤ W (xf ) ≤ b2‖xf‖2

2

∂W

∂xfAfxf ≤ −b3‖xf‖2

2

∥∥∥∥∂W

∂xf

∥∥∥∥2

≤ b4‖xf‖2

(30)

Page 16: [Lecture Notes in Computer Science] Hybrid Systems: Computation and Control Volume 2623 || Hybrid Control of Parabolic PDEs: Handling Faults of Constrained Control Actuators

Hybrid Control of Parabolic PDEs 187

Evaluating the time derivative of the Lyapunov functional, W , along the trajec-tories of the closed-loop fast subsystem in Eq.29, we obtain

W (xf ) =∂W

∂xf

[Afxf + Bfu(as, ukmax, zk)

]

≤ −b3‖xf‖22 + σ

∥∥∥∥∂W

∂xf

∥∥∥∥2|u(as, u

kmax, zk)|

(31)

for some σ > 0 which bounds the bounded input operator Bf . Note thatthe control input u(as, u

kmax, zk) is a continuous bounded function, for all

as ∈ Ωk, k ∈ 1, · · · , N, that converges to zero as as → 0. Therefore, giventhat we have only a finite number of actuator configurations to switch between,the following bounds hold irrespective of which configuration is active, wheneverswitching is carried according to the rule of Eq.21

W (xf ) ≤ −b3‖xf‖22 + σb4‖xf‖2 max

k=1,···,N(|u(as, u

kmax, zk)|)

≤ −b3

2‖xf‖2

2 ∀ ‖xf‖2 ≥ 2σb4

b3max

k=1,···,N(|u(as, u

kmax, zk)|) ≡ µ(|as|)

(32)Note that µ(|as|) → 0 as |as| → 0. The above inequality implies that Wf isnegative outside some residual set whose size depends on as. This, in turn,implies that, for any xf (0), there exists a finite time, t1, such that the state ofthe closed-loop switched fast subsystem satisfies

‖xf (t)‖2 ≤ ϕ‖xf (0)‖2e−αt ∀ 0 ≤ t < t1

‖xf (t)‖2 ≤ γ(µ(|as|)) ∀ t ≥ t1(33)

for some ϕ > 1, α > 0, where γ(·) is a class K function of its argument (γ(µ(·)) =√b2

b1µ(·)). This implies that ‖xf (t)‖2 is ultimately bounded with an ultimate

bound that depends on as. We have already shown in part 1 that, starting fromany as(0) ∈ Ωk, for any k ∈ 1, · · · , N, the state as (under the control law ofEq.18-19 and the switching law of Eq.21) is bounded and converges to zero ast → ∞. Therefore, by taking the limit of both sides of the second inequality inEq.33, we have

limt→∞‖xf (t)‖2 ≤ lim

t→∞γ(µ(|as(t)|)) = 0 (34)

which implies that ‖xf‖2 → 0 as t → ∞. To summarize, we have that,starting from any xs(0) ∈ Θk (or equivalently as(0) ∈ Ω(uk

max, zk) for anyk ∈ 1, · · · , N), and for any xf (0), the states of the slow and fast switchedclosed-loop subsystems are bounded and converge to the origin as t → ∞ whichimplies that origin of the switched infinite dimensional closed-loop system ofEq.29 is asymptotically stable. This completes the proof of the theorem.