Hybrid Control of Parabolic PDE Systems - CiteSeerX
Proceedings of the 41st IEEE Conference on Decision and Control Las Vegas, Nevada USA, December 2002
TuA07-2
Hybrid Control of Parabolic PDE Systems Nael H. El-Farra and Panagiotis D. Christofides Department of Chemical Engineering University of California, Los Angeles, CA 90095 Abstract
the number of modes that should be retained to derive an ODE system that yields the desired degree of approximaThis paper proposes a hybrid control methodology which tion may be very large, leading to complex controller design integrates feedback and switching for constrained stabiliza- and high dimensionality of the resulting controllers. tion of parabolic partial differential equation (PDE) systems for which the spectrum of the spatial differential op- Motivated by this, significant recent work has focused on erator can be partitioned into a finite slow set and an infi- the synthesis of nonlinear low-order controllers on the basis nite stable fast complement. Galerkin’s method is initially of ODE models obtained through combination of Galerkin’s used to derive a finite-dimensional system (set of ordinary method with approximate inertial manifolds (see [4] and the differential equations (ODEs) in time) that captures the recent book [3] for details and references). In addition to dominant dynamics of the PDE system. This ODE system this work, other advances in control of PDE systems have is then used as the basis for the integrated synthesis, via been made, including, for example, controller design based Lyapunov techniques, of a stabilizing nonlinear feedback on the infinite-dimensional system and subsequent use of controller together with a switching law that orchestrates approximation theory to design and compute low-order fithe switching between the admissible control actuator con- nite dimensional compensators [2], results on distributed figurations, in a way that respects input constraints, accom- control using generalized invariants [11] and concepts from modates inherently conflicting control objectives, and guar- passivity and thermodynamics [14], analysis and control of antees closed-loop stability. Precise conditions that guaran- parabolic PDE systems with actuator saturation [7], and tee stability of the constrained closed-loop PDE system un- robust H∞ control of distributed systems [9]. der switching are provided. The proposed methodology is While the above efforts have led to the development of a successfully applied to stabilize an unstable steady-state of number of systematic methods for distributed controller a diffusion-reaction process using switching between three design, these methods focus exclusively on the classical different control actuator configurations. control paradigm, where a fixed control structure (single Key words: Parabolic PDEs, Galerkin’s method, Hy- feedback law, fixed actuator/sensor spatial arrangement) is brid control, Actuator configuration switching, Input con- used to achieve ceratin control objectives (see Figure 1a). There are many practical situations, however, where it is straints. desirable, and sometimes even necessary, to consider a hybrid control paradigm, such as the one depicted in Figure 1 Introduction 1b, where the control system consists of a family of conThe study of distributed systems in control is motivated by troller configurations (e.g., a family of feedback laws and/or the fundamentally distributed nature of the control proba family of actuator/sensor spatial arrangements) together lems arising in many chemical and physical systems, such with a higher-level supervisor that uses logic to orchestrate as transport-reaction processes and fluid flows. The disswitching between these control configurations. An examtinguishing feature of distributed control problems is that ple, where consideration of hybrid control is necessary, is they involve the regulation of distributed variables by using the problem of control actuator failure. In this case, upon spatially-distributed control actuators and measurement detection of a fault in the operating actuator configurasensors. Several classes of distributed systems are natution, it is often necessary to switch to an alternative wellrally modeled by highly dissipative PDE systems, such as functioning actuator configuration, with a different spatial parabolic PDE systems (transport-reaction processes) that placement of the actuators, in order to preserve closed-loop involve spatial differential operators whose spectrum can be stability. Switching between spatially distributed actuators partitioned into a finite slow part and an infinite stable fast in this case provides a means for fault-tolerant control. complement [8]. The traditional approach to the control of parabolic PDEs involves the application of Galerkin’s Due to the use of logic-based switching and the variable method to the PDE system to derive ODE systems that structure of the control system, the dynamics of the condescribe the dynamics of the dominant (slow) modes of trol system in the hybrid paradigm are an intermix of the PDE system, which are subsequently used as the ba- discrete and continuous components. This is in contrast sis for the synthesis of finite-dimensional controllers (e.g., to the purely continuous dynamics in the classical control [1, 12, 6]). A potential drawback of this approach is that
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paradigm. Furthermore, whereas a fixed control structure 2 Preliminaries can at best achieve a tradeoff between multiple control We consider quasi-linear parabolic PDE systems of the objectives in classical control, switching in hybrid control form: allows the accommodation and reconciliation of multiple, m X ∂x ¯ ∂x ¯ ∂2x ¯ possibly conflicting, control objectives. x) = A + B 2 + w bi (z)ui + f (¯ ∂t ∂z ∂z i=1 (1) Motivated by these considerations, this work proposes Z β κ κ a hybrid control methodology which integrates feedback ym = s(z) ω¯ x(z, t)dz κ = 1, . . . , p and switching for constrained stabilization of a class of α parabolic PDE systems. The central idea is the inte- subject to the boundary conditions: grated synthesis, via Lyapunov techniques, of a stabiliz∂x ¯ ing nonlinear feedback controller together with a stabilizC1 x ¯(α, t) + D1 (α, t) = R1 ∂z ing switching law that orchestrates the switching between (2) ∂x ¯ multiple spatially-distributed control actuator configuraC2 x ¯(β, t) + D2 (β, t) = R2 tions, in a way that respects actuator constraints, accom∂z modates inherently conflicting control objectives, and guar- and the initial condition: antees closed-loop stability at the same time. The proposed x ¯(z, 0) = x ¯0 (z) methodology is applied successfully to the fault-tolerant (3) stabilization of an unstable steady state of a diffusionT n where x ¯(z, t) = [¯ x1 (z, t) · · · x ¯n (z, t)] ∈ IR denotes the reaction process. vector of state variables, z ∈ [α, β] ⊂ IR is the spatial coordinate, t ∈ [0, ∞) is the time, ui ∈ [ui,min , ui,max ] ⊂ IR deκ notes the i-th constrained manipulated input, and ym ∈ IR ∂x ¯ ∂2x ¯ denotes the κ-th measured output. , denote the 1 2 3 u (t) ∂z ∂z 2 u (t) u (t) first- and second-order spatial derivatives of x ¯, f (¯ x) is a nonlinear vector function, w, k, ω are constant row vectors, A, B, C1 , D1 , C2 , D2 are constant matrices of appropriate Spatially Distributed Process dimensions, R1 , R2 are column vectors, and x ¯0 (z) is the initial condition. The function b (z) is a known smooth i 1 2 ym (t) ym (t) function of z which describes how the control action ui (t) is distributed in the interval [α, β], and sκ (z) is a known smooth function of z which depends on the shape (point or Feedback Control controller distributing sensing) of the measurement sensors in the inobjectives terval [α, β]. Whenever the control action enters the system at a single point z0 , with z0 ∈ [α, β] (i.e. point actuation), Measurement sensor the function bi (z) is taken to be nonzero in a finite spatial Control actuator interval of the form [z0 − µ, z0 + µ], where µ is a small posi(a) u1 (t) u2(t) u3(t) tive real number, and zero elsewhere in [α, β]. Throughout the paper, the order of magnitude O(²) notation will be used. In particular, δ(²) = O(²) if there exist positive real Spatially Distributed Process numbers k1 and k2 (which are independent of ²) such that: ym1 (t) ym2 (t) |δ(²)| ≤ k1 |²| , ∀ |²| < k2 . Switching logic
controller A
Supervisor
controller B
Control objectives
controller C
(b)
Figure 1: Comparison between the control structures arising
For a precise characterization of the class of PDE systems considered in this work, we formulate the system of Eq.1 as an infinite dimensional system in the Hilbert space H([0, π]; IRn ), with H being the space of n-dimensional, sufficiently smooth vector functions defined on [0, π] that satisfy the boundary conditions of Eq.2, with inner product and norm: Z β 1 (ω1 , ω2 ) = (ω1 (z), ω2 (z))IRn dz, ||ω1 ||2 = (ω1 , ω1 ) 2 α
in the classical control (a) and the hybrid control (4) (b) paradigms of spatially distributed systems. where ω1 , ω2 are two elements of H([α, β]; IRn ) and the no-
tation (·, ·)IRn denotes the standard inner product in IRn . Defining the state function x on H([α, β]; IRn ) as: x(t) = x ¯(z, t),
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t > 0,
z ∈ [α, β],
(5)
the operator A in H([α, β]; IRn ) as: ∂x ¯ ∂2x ¯ +B 2, ∂z ∂z x ∈ D(A) = {x ∈ H([α, β]; IRn ) : C1 x ¯(α, t) Ax = A
∂x ¯ ∂x ¯ ¯(β, t) + D2 (β, t) = R2 +D1 (α, t) = R1 , C2 x ∂z ∂z
¾
and the input and measured output operators as: Bu = w
m X
bi ui ,
Sx = (s, ωx)
(6)
i=1
the system of Eqs.1-2-3 takes the form: x˙ = Ax + Bu + f (x),
x(0) = x0
ym = Sx
(7)
where f (x(t)) = f (¯ x(z, t)) and x0 = x ¯0 (z). We assume that the nonlinear term f (x) is locally Lipschitz with respect to its argument and satisfies f (0) = 0. For A, the eigenvalue problem is defined as: Aφj = λj φj ,
j = 1, . . . , ∞
(8)
available at our disposal is a family of N (with N finite) control actuator configurations, of which only one configuration can be used for control at any given time instance. Each of these N configurations consists of a spatiallydistinct arrangement of the control actuators, which we denote by z¯k(t) , k = 1, · · · , N . In this notation, the index k(t) denotes the actuator configuration being active at time t, while z¯k is a column vector whose components represent the corresponding spatial locations of the actuators associated with the k-th configuration. To ensure controllability of the system, we allow only a finite number of switches between configurations over finite time. The problem is how to coordinate switching between the different control actuator configurations, in the event of actuator failure, in a way that respects actuator constraints and guarantees closed-loop stability. To address this problem, we formulate the following objectives. Initially, Galerkin’s method is used to derive a nonlinear finite-dimensional ODE system that captures the dominant dynamics of the parabolic PDE system of Eq.1. Next, the ODE approximation is used as the basis for the synthesis, of bounded nonlinear controllers of the general form
where λj denotes an eigenvalue and φj denotes an eigenfunction; the eigenspectrum of A, σ(A), is defined as the set of all eigenvalues of A, i.e. σ(A) = {λ1 , λ2 , . . . , }. In this work, we consider parabolic PDE systems for which the following assumption holds.
u = p(ym , umax , z¯k )
(9)
that enforce asymptotic stability in the constrained closedloop system and provide an explicit characterization of the stability region associated with each control actuator configuration. The controller synthesis is carried out via Lyapunov-based control techniques [10]. Finally, a set of switching rules is derived to determine which of the N conAssumption 1: trol actuator configurations can be engaged at any given 1. Re {λ1 } ≥ Re {λ2 } ≥ · · · ≥ Re {λj } ≥ · · ·, where time, and an upper bound on the separation between the slow and fast eigenvalues, which guarantees stability of the Re {λj } denotes the real part of λj . infinite-dimensional switched closed-loop system, is com2. σ(A) can be partitioned as σ(A) = σ1 (A) + σ2 (A), puted. where σ1 (A) consists of the first m (with m finite) eigenvalues, i.e. σ1 (A) = {λ1 , . . . , λm }, and 3.2 Galerkin’s method |Re {λ1 }| = O(1). In this section, we apply standard Galerkin’s method |Re {λm }| to the system of Eq.1 to derive an approximate finitedimensional system. Let Hs , Hf be modal subspaces |Re {λm }| = O(²) where ² ≡ of A, defined as H = span{φ , φ , . . . , φ } and H = 3. Re {λm+1 } < 0 and s 1 2 m f |Re {λm+1 }| span{φm+1 , φm+2 , . . . , } (the existence of Hs , Hf follows |Re {λ1 }| < 1 is a small positive number. from the properties of A). Defining the orthogonal projec|Re {λm+1 }| tion operators Ps and Pf such that xs = Ps x, xf = Pf x, the state x of the system of Eq.1 can be decomposed as: Assumption 1 above states that the eigenspectrum of A can be partitioned into a finite-dimensional part consistx = xs + xf = Ps x + Pf x (10) ing of m slow eigenvalues and a stable infinite-dimensional complement containing the remaining fast eigenvalues, and Applying Ps and Pf to the system of Eq.7 and using the that the separation between the slow and fast eigenvalues above decomposition for x, the system of Eq.7 can be writof A is large. This assumption is satisfied by the major- ten in the following equivalent form: dxs ity of diffusion-convection-reaction processes (see, e.g., the = As xs + Bs u + fs (xs , xf ) models studied in [3, 13]). dt ∂xf = Af xf + Bf u + ff (xs , xf ) 3 Hybrid control of parabolic PDEs ∂t ym = Sxs + Sxf , 3.1 Problem formulation xs (0) = Ps x(0) = Ps x0 , xf (0) = Pf x(0) = Pf x0 Consider the system of Eq.1, where the control inputs ui are (11) constrained in the interval [−umax , umax ], and assume that
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where As = Ps A, Bs = Ps B, fs = Ps f , Af = Pf A, Bf = Pf B, ff = Pf f , and the partial derivative nota∂xf tion in is used to denote that the state xf belongs ∂t to an infinite-dimensional space. In the above system, As is a diagonal matrix of dimension m × m of the form As = diag{λj }, fs (xs , xf ) and ff (xs , xf ) are Lipschitz vector functions, and Af is an unbounded differential operator which is exponentially stable (following from the fact that Re{λm+1 } < 0 and the selection of Hs , Hf ). Neglecting the fast and stable infinite-dimensional xf -subsystem in Eq.11, the following m-dimensional slow system can be obtained: d¯ xs = As x ¯s + Bs u + fs (¯ xs , 0) dt y¯m = S x ¯s
where as (t) = [a1 (t) · · · am (t)]T ∈ IRm , ai (t) is the amplim X ¯ are tude of the i-th eigenmode, xs = ai (t)φi (z), F , G i=1
constant matrices and f˜(·) is a nonlinear function of its argument. The system of Eq.14 is obtained from that of Eq.12 by taking, for each mode, the inner product on both sides with respect to the corresponding eigenfunction. Finally, we define f¯s (¯ as ) = F as + f˜s (¯ as ) and denote by g¯i the ¯ i-th column of the matrix G.
Using a quadratic Lyapunov function of the form V = a ¯Ts Φ¯ as , where Φ is a positive-definite symmetric matrix ¯G ¯T Φ < that satisfies the Riccati inequality F T Φ+ΦF −ΦG (12) 0, we synthesize the following bounded nonlinear feedback law [10] T
where the bar symbol in x ¯s and y¯m denotes that these variables are associated with a finite-dimensional system.
zk ) u = −r(as , ukmax , z¯k ) (LG¯ V ) (¯ where r(as , ukmax , z¯k ) = q 4 L∗f¯V + (L∗f¯V )2 + (ukmax |(LG¯ V )T (¯ zk )|) · ¸ q |(LG¯ V )T (¯ zk )|2 1 + 1 + (ukmax |(LG¯ V )T (¯ zk )|)2
(15)
Remark 1: Even though the fast subsystem in Eq.11 is infinite-dimensional, from a computational point view one (16) needs to consider a finite dimensional (possibly high-order) fast subsystem to work with. Therefore, in this work we consider parabolic PDE systems for which the high orzk1 z¯k2 · · · z¯km ]T , k = 1, · · · , N , L∗f¯V = der discrteization obtained through Galerkin’s method con- where z¯k = [¯ verges to the solution of the infinite dimensional system in Lf¯V + ρ|as |2 , ρ > 0, LG¯ V is a row vector of the form the sense that, for any initial condition: [Lg¯1 V · · · Lg¯m V ]. The notation ukmax is used to indicate the magnitude of actuator constraints associated with the k-th |xs | ≤ |ˆ xs | + d1 actuator configuration. This quantity is allowed to vary (13) from one configuration to another. The scalar function r(·) ||xf ||2 ≤ ||ˆ xf ||2 + d2 in Eqs.15-16 can be thought of as a nonlinear gain of the where the offsets d1 > 0, d2 > 0 can be made arbitrar- LG¯ V controller. This Lyapunov-based gain, which depends k ily small by sufficiently increasing the order of the fast on both the magnitude of actuator constraints, umax , and subsystem, and x ˆs and x ˆf are the solutions of the slow the spatial arrangement of the actuators, z¯k , is shaped in a and fast subsystems, respectively, obtained for a high (but way that guarantees constraint satisfaction and asymptotic finite)-dimensional fast subsystem. The property of Eq.13 closed-loop stability within a well-characterized region in holds for most parabolic PDEs arising in the modeling of the state space given in Theorem 1 below. diffusion-convection-reaction processes (see also the examWe are now ready to state the main result of this work. ple in section 4). Theorem 1 below provides both the state feedback control law (see the discussion in remark 2 for output feedback con3.3 Integrating feedback and switching troller design and implementation) as well as the necessary Having obtained a finite-dimensional model that describes switching law and states precise conditions that guarantee the dominant dynamics of the PDE system, we proceed closed-loop stability in the switched closed-loop system. in this section to describe the proposed procedure for designing the hybrid control system, on the basis of the finite- Theorem 1: dimensional approximation in Eq.12. To this end, and since our objective is the stabilization of the PDE system, we 1. Consider the system of Eq.14 under the feedback contake as controlled outputs the slow modes of the PDE systrol law of Eq.15-16 and let δsk be a positive real numtem, and assume, for simplicity, that the number of inputs ber such that the compact set Ω(umax , z¯k ) = {as ∈ is equal to the number of slow modes. Though not discussed IRm : V (as ) ≤ δsk } is the largest invariant set emin this paper explicitly, the results can be generalized to adbedded within the unbounded region described by the dress the problem of reference-input tracking. For a clear following inequality presentation of our results, we consider the representation of the slow system of Eq.12 in terms of the evolution of the L∗f¯V ≤ ukmax |(Lg¯ V )T (¯ zk )| (17) amplitudes of the eigenmodes. This system is given by Without loss of generality, assume that z¯k(0) = z¯1 ¯ zk )u(t) + f˜s (¯ and as (0) ∈ Ω(u1max , z¯1 ). If, at any given time T , the (14) a ¯˙ s (t) = F a ¯s (t) + G(¯ as (t))
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condition
Part 2: Substituting the controller of Eq.15-16 into Eq.11 as (T ) ∈ (18) and using the fact that ² = |Re{λ1 }| < 1, the closed|Re{λm+1 }| holds, for some j ∈ {1, · · · , N }, then setting z¯k(T + ) = loop system can be written in the following form: z¯j guarantees that the closed-loop system of Eq.12-15dxs 16 is asymptotically stable. = As xs − Bs k(xs , ukmax, z¯k )(LG¯ V )T + fs (xs , 0) dt + [fs (xs , xf ) − fs (xs , 0)] 2. Consider the parabolic PDE system of Eq.1 under ∂xf the control law of Eq.15-16 and the switching law ² = Af ² xf + ²f¯f (xs , xf ) ∂t of Eq.18. Let δhk be a positive real number such (22) that |xs | ≤ δhk whenever as ∈ Ω(ukmax , z¯k ). Then where Af ² is an unbounded differential operator defined as given any pair of positive real numbers (d, δb ) such Af ² = ²Af , and f¯f (xs , xf ) = −Bs k(xs , ukmax, z¯k )(LG¯ V )T + that δb + d ≤ δh1 , and given any positive real numf (x , x ). The system of Eq.22 is in the standard singuber δf , there exists ²∗ > 0 such that if ² ∈ (0, ²∗ ], f s f larly perturbed form, with xs being the slow states and xf |xs (0)| ≤ δh1 , ||xf (0)||2 ≤ δf , the closed-loop infinitebeing the fast states. Applying two-time scale decomposidimensional switched system is asymptotically (and tion to the above system and analyzing the resulting slow locally exponentially) stable. and fast subsystems, one can show that the slow system obtained is exactly the switched system analyzed in part 1 and satisfies the bound in Eq.21, while the fast subsystem Proof of Theorem 1: satisfies the following exponential bound Part 1: Consider the finite-dimensional closed-loop system ||¯ xf (τ )||2 ≤ k3 ||¯ xf (0)||2 e−a3 τ ∀ t ≥ 0 (23) of Eq.14-15-16. Using a Lyapunov argument, one can show that, whenever Eq.17 is satisfied, V˙ satisfies for some k3 ≥ 1, a3 > 0. Exploiting the stability properties of the fast and slow systems, it can be shown then, with 2 −ρ|¯ as | ¸ the aid of calculations similar to those performed in [5] for V˙ ≤ · ≡ −α(¯ z ) < 0 q k the analysis of singularly perturbed systems, that given any 1 + 1 + (umax |(Lg¯ V (¯ zk ))T |)2 pair of positive real numbers (δb , d), where δb +d ≤ δh , and (19) given any δ , there exists ²(1) > 0 such that if ² ∈ (0,1 ²(1) ], f ∀ a ¯s 6= 0, k = 1, · · · , N . Since the set Ω(ukmax , z¯k ) is the |x (0)| ≤ δ , ||x (0)|| ≤ δ , then, for all t ≥ 0, the states s b f 2 f largest invariant set within the region described by Eq.17, of the closed-loop singularly perturbed system satisfy k then starting from any a ¯s (0) ∈ Ω(umax , z¯k ), the closedloop state satisfies Eq.17 for all time, and consequently, |xs (t)| ≤ ϕ|xs (0)|e−βt + d ˙ V decreases monotonically according to V ≤ −α(¯ zk ) < 0, (24) t −a3 which implies that, for each actuator configuration z¯k , the ² +d ||xf (t)||2 ≤ k3 ||xf (0)||2 e closed-loop slow subsystem is asymptotically stable. Furthermore, as long as the k-th configuration is switched in The above inequalities imply that the trajectories of the at time T when a ¯s (T ) ∈ Ω(ukmax , z¯k ), V˙ will always satisfy switched closed-loop system will be ultimately bounded the following worst-case dissipation inequality for all t ≥ 0 with a bound that depends on d. Since d is arbitrary, we can Ω(ujmax , z¯j )
choose it small enough such that after a sufficiently large (20) time, say t˜, the trajectories of the closed-loop system are k=1,···,N confined within a small compact neighborhood of the origin which implies that the finite-dimensional switched closed- of the closed-loop system. Let d = b/2 and t˜ be the smallest ˜ t loop system of Eq.14-15-16-18 is asymptotically stable and time such that max{ϕ|¯ xs (0)|e−β t˜, k3 ||xf (0)||2 e−a3 ² } ≤ d. its state is bounded (and therefore the denominator term in Then it can be easily verified that Eq.19 is bounded). Therefore, given δs1 > 0, there exists |xs (t)| ≤ b, ||xf (t)||2 ≤ b ∀ t ≥ t˜ (25) a positive real number γ > 0, such that if |¯ as (0)| ≤ δs1 , 2 V˙ satisfies V˙ ≤ −γ|¯ as | and the switched closed-loop Recall from Eq.23 and Eq.21 that both the fast and slow system is exponentially stable. From this and the fact that subsystems are exponentially stable within the ball of m X (2) (2) x ¯s = ai (t)φi (z), we deduce that the closed-loop system Eq.25. Therefore there exists ² > 0 such that if ² ≤ ² , the singularly perturbed closed-loop system of Eq.22 is loi=1 of Eq.12 is asymptotically stable. Also, given the positive cally exponentially stable and, therefore, once inside the real number δh1 , there exists b > 0 such that if |¯ xs (0)| ≤ ball of Eq.25, the closed-loop trajectories converge to the δh1 , the following bound holds origin as t → ∞. This completes the proof of the theorem. V˙ ≤
max {−α(¯ zk )} < 0
|¯ xs (t)| ≤ ϕ|¯ xs (0)|e−βt
∀ t≥0
for all |¯ xs (t)| ≤ b, for some ϕ ≥ 1, β > 0.
(21) Remark 2: When only output measurements are available, an output feedback controller can be constructed under the assumption that the number of measurements is
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equal to the number of slow modes and that the inverse of the operator S exists, so that x ˆs = S −1 ym (which can be ensured by appropriate choice of the location of the measurement sensors). The synthesis of the output feedback controller can be carried out by combining the state feedback controller of Eq.15-16 with a procedure proposed in [3] for obtaining estimates for the states of the approximate ODE model of Eq.12 from the measurements. While the estimation error leads to some loss in the size of the stability region obtained under state feedback, this loss can be made small by increasing the order of the ODE approximation and including more measurements. This approach therefore allows us to asymptotically (as ² → 0) recover the stability region associated with each control actuator configuration. 4 Application to a diffusion-reaction process In this section, we illustrate through computer simulations how the concept of coupling feedback and switching, proposed in Theorem 1, can be used to deal with the problem of control actuator failure in control of a diffusion-reaction process. To this end, consider a long, thin rod in a reactor. The reactor is fed with pure species A and a zeroth order exothermic catalytic reaction of the form A → B takes place on the rod. Since the reaction is exothermic, a cooling medium in contact with the rod is used for cooling. Under standard assumptions, the spatiotemporal evolution of the dimensionless rod temperature is described by the following parabolic PDE γ − ∂x ¯ ∂2x ¯ 1 + x ¯ + βU (b(z)u(t) − x ¯) − βT e−γ = + βT e ∂t ∂z 2 x ¯(0, t) = 0, x ¯(π, t) = 0, x ¯(z, 0) = x ¯0 (z) (26) where x ¯ denotes the dimensionless temperature in the reactor, βT denotes a dimensionless heat of reaction, γ denotes a dimensionless activation energy, βU denotes a dimensionless heat transfer coefficient, u(t) denotes the vector of manipulated inputs and b(z) the vector of the corresponding actuator distribution functions. The following typical values are given to the process parameters: βT = 50.0, βU = 2.0, γ = 4.0. For the above values, it was verified that the operating steady state x ¯(z, t) = 0 is an unstable one. The control objective therefore is to stabilize the rod temperature profile at this unstable steady state by manipulating the temperature of the cooling medium, subject to hard constraints. To achieve this objective, the controlled output is defined as Z πr 2 (27) sin(z)x(z, t)dz yc (t) = π 0
For this system, we consider the first eigenvalue as the dominant one and use standard Galerkin’s method to derive an ODE that describes the temporal 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.15-16, which is then implemented on a 30th order Galerkin discretization of the parabolic PDE of Eq.26 (higher order discretizations led to identical results, implying that the property of Eq.13 holds for Eq.26). In order to demonstrate the utility of the switching scheme proposed in Theorem 1 for dealing with actuator failure, we consider the following problem where three point control actuators, A, B, and C, located at zA = 0.5π, zB = 0.23π, and zC = 0.36π, respectively, are available for stabilization. The three actuators have different constraints of umaxA = umaxB = 2.5 and umaxC = 0.5. 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 that the operating actuator fails. The state feedback results will be presented first. Using Eq.17 with ρ = 0.02, the stability regions for umaxA = umaxB = 2.5 (solid line) and for umaxC = 0.5 (dashed line) are computed as functions of actuator location and shown in Figure 2a. To simplify the presentation of our results, we plot in Figure 2a the variation of the set of admissible initial conditions for the amplitude, a1 (0), of the first eigenmode, with actuator location (note that xs (0) = a1 (0)φ1 (z)). From this figure, it is easy to see that for an initial condition a1 (0) = 1.3, only actuator A can be used initially since the initial condition is outside the stability regions for actuators B and C. Now, suppose that sometime after startup, say at t = 1.4, a fault is detected in actuator A and it becomes necessary to switch to a backup actuator. Without using the switching logic of Theorem 1, it is not clear whether actuator B or C should be activated at this time. Figures 2b and 2d (solid lines) depict, respectively the resulting closed-loop temperature and manipulated input profiles when actuator C is switched in. We see in this case that the controller is unable to stabilize the system at the desired steady-state. In contrast, when the switching law of Theorem 1 is implemented, we track the evolution of a1 (t) in time and find that, at the time of actuator A’s failure, we have a1 (1.4) = 0.62, i.e. the state is inside the stability region for actuator B and outside the stability region for actuator C. Based on this, we decide to switch to actuator B. Figures 2c and 2d (dashed line) depict the results for this case which show that the controller successfully stabilizes the closed-loop system at the desired steady-state.
For the case of output feedback, we used a single point senThe eigenvalue problem for the spatial differential operator sor located at z = 0.33π, to obtain estimates of the first of the process can be solved analytically and its solution is eigenmode, which was subsequently used for implementing the output feedback controller. Owing to the small O(²) r 2 2 λj = −j , φj (z) = sin(j z), j = 1, . . . , ∞ (28) discrepancy between the stability regions obtained under π
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state and output feedback (due to the estimation error), the switching law of Theorem 1 was used as an approximate guide for switching between the different actuators in the case of output feedback (see remark 2). The simulation results for this case were found to be consistent with the state feedback results and are omitted due to space limitations.
1.5
1
a1
0.5
0
−0.5
ZC
ZB
−1 0
0.5
Z
A
1
1.5 Z
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3
References [1] M. J. Balas. Feedback control of linear diffusion processes. Int. J. Contr., 29:523–533, 1979.
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[2] J. A. Burns and B. B. King. A reduced basis approach to the design of low-order feedback controllers for nonlinear continuous systems. Journal of Vibration and Control, 4:297–323, 1998.
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[3] P. D. Christofides. Nonlinear and Robust Control of PDE Systems: Methods and Applications to TransportReaction Processes. Birkh¨ auser, Boston, 2001.
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[4] P. D. Christofides and P. Daoutidis. Finitedimensional control of parabolic PDE systems using approximate inertial manifolds. J. Math. Anal. Appl., 216:398–420, 1997.
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[5] P. D. Christofides and A. R. Teel. Singular perturbations and input-to-state stability. IEEE Trans. Autom. Contr., 41:1645–1650, 1996.
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[6] R. F. Curtain. Finite-dimensional compensator design for parabolic distributed systems with point sensors and boundary input. IEEE Trans. Automat. Contr., 27:98– 104, 1982.
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[7] N. H. El-Farra, A. Armaou, and P. D. Christofides. Analysis and control of palabolic PDE systems with input constraints. Automatica, to appear, 2002.
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[8] A. Friedman. Partial Differential Equations. Holt, Rinehart & Winston, New York, 1976.
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[9] B. Keulen. H∞ -Control for Distributed Parameter Systems: A State-Space Approach. Birkhauser, Boston, 1993.
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[10] Y. Lin and E. D. Sontag. A universal formula for stabilization with bounded controls. Systems & Control Letters, 16:393–397, 1991.
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[11] A. Palazoglu and A. Karakas. Control of nonlinear distributed parameter systems using generalized invariants. Automatica, 36:697–703, 2000.
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Figure 2: (a) Stability region as a function of actuator location [12] W. H. Ray. Advanced Process Control. McGraw-Hill, for umax = 2.5 (solid) and umax = 0.5 (dashed),(b) New York, 1981. Closed-loop temperature profile (state feedback) for a1 (0) = 1.3 when actuator A (umax = 2.5, zA = 0.5π) fails at t = 1.4 and actuator C (umax = 0.5, zA = 0.36π) is activated, (c) Closed-loop temperature profile (state feedback) for a1 (0) = 1.3 when actuator A fails at t = 1.4 and actuator B (umax = 2.5,zA = 0.23π) is activated, (d) Corresponding manipulated input profiles for case b (solid) and case c (dashed).
[13] J. J. Winkin, D. Dochain, and P. Ligarius. Dynamical analysis of distributed parameter tubular reactors. Automatica, 36:349–361, 2000. [14] E. B. Ydstie and A. A. Alonso. Process systems and passivity via the Clausius-Planck inequality. Syst. & Contr. Lett., 30:253–264, 1997.
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TuA07-2
Hybrid Control of Parabolic PDE Systems Nael H. El-Farra and Panagiotis D. Christofides Department of Chemical Engineering University of California, Los Angeles, CA 90095 Abstract
the number of modes that should be retained to derive an ODE system that yields the desired degree of approximaThis paper proposes a hybrid control methodology which tion may be very large, leading to complex controller design integrates feedback and switching for constrained stabiliza- and high dimensionality of the resulting controllers. tion of parabolic partial differential equation (PDE) systems for which the spectrum of the spatial differential op- Motivated by this, significant recent work has focused on erator can be partitioned into a finite slow set and an infi- the synthesis of nonlinear low-order controllers on the basis nite stable fast complement. Galerkin’s method is initially of ODE models obtained through combination of Galerkin’s used to derive a finite-dimensional system (set of ordinary method with approximate inertial manifolds (see [4] and the differential equations (ODEs) in time) that captures the recent book [3] for details and references). In addition to dominant dynamics of the PDE system. This ODE system this work, other advances in control of PDE systems have is then used as the basis for the integrated synthesis, via been made, including, for example, controller design based Lyapunov techniques, of a stabilizing nonlinear feedback on the infinite-dimensional system and subsequent use of controller together with a switching law that orchestrates approximation theory to design and compute low-order fithe switching between the admissible control actuator con- nite dimensional compensators [2], results on distributed figurations, in a way that respects input constraints, accom- control using generalized invariants [11] and concepts from modates inherently conflicting control objectives, and guar- passivity and thermodynamics [14], analysis and control of antees closed-loop stability. Precise conditions that guaran- parabolic PDE systems with actuator saturation [7], and tee stability of the constrained closed-loop PDE system un- robust H∞ control of distributed systems [9]. der switching are provided. The proposed methodology is While the above efforts have led to the development of a successfully applied to stabilize an unstable steady-state of number of systematic methods for distributed controller a diffusion-reaction process using switching between three design, these methods focus exclusively on the classical different control actuator configurations. control paradigm, where a fixed control structure (single Key words: Parabolic PDEs, Galerkin’s method, Hy- feedback law, fixed actuator/sensor spatial arrangement) is brid control, Actuator configuration switching, Input con- used to achieve ceratin control objectives (see Figure 1a). There are many practical situations, however, where it is straints. desirable, and sometimes even necessary, to consider a hybrid control paradigm, such as the one depicted in Figure 1 Introduction 1b, where the control system consists of a family of conThe study of distributed systems in control is motivated by troller configurations (e.g., a family of feedback laws and/or the fundamentally distributed nature of the control proba family of actuator/sensor spatial arrangements) together lems arising in many chemical and physical systems, such with a higher-level supervisor that uses logic to orchestrate as transport-reaction processes and fluid flows. The disswitching between these control configurations. An examtinguishing feature of distributed control problems is that ple, where consideration of hybrid control is necessary, is they involve the regulation of distributed variables by using the problem of control actuator failure. In this case, upon spatially-distributed control actuators and measurement detection of a fault in the operating actuator configurasensors. Several classes of distributed systems are natution, it is often necessary to switch to an alternative wellrally modeled by highly dissipative PDE systems, such as functioning actuator configuration, with a different spatial parabolic PDE systems (transport-reaction processes) that placement of the actuators, in order to preserve closed-loop involve spatial differential operators whose spectrum can be stability. Switching between spatially distributed actuators partitioned into a finite slow part and an infinite stable fast in this case provides a means for fault-tolerant control. complement [8]. The traditional approach to the control of parabolic PDEs involves the application of Galerkin’s Due to the use of logic-based switching and the variable method to the PDE system to derive ODE systems that structure of the control system, the dynamics of the condescribe the dynamics of the dominant (slow) modes of trol system in the hybrid paradigm are an intermix of the PDE system, which are subsequently used as the ba- discrete and continuous components. This is in contrast sis for the synthesis of finite-dimensional controllers (e.g., to the purely continuous dynamics in the classical control [1, 12, 6]). A potential drawback of this approach is that
0-7803-7516-5/02/$17.00 ©2002 IEEE
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paradigm. Furthermore, whereas a fixed control structure 2 Preliminaries can at best achieve a tradeoff between multiple control We consider quasi-linear parabolic PDE systems of the objectives in classical control, switching in hybrid control form: allows the accommodation and reconciliation of multiple, m X ∂x ¯ ∂x ¯ ∂2x ¯ possibly conflicting, control objectives. x) = A + B 2 + w bi (z)ui + f (¯ ∂t ∂z ∂z i=1 (1) Motivated by these considerations, this work proposes Z β κ κ a hybrid control methodology which integrates feedback ym = s(z) ω¯ x(z, t)dz κ = 1, . . . , p and switching for constrained stabilization of a class of α parabolic PDE systems. The central idea is the inte- subject to the boundary conditions: grated synthesis, via Lyapunov techniques, of a stabiliz∂x ¯ ing nonlinear feedback controller together with a stabilizC1 x ¯(α, t) + D1 (α, t) = R1 ∂z ing switching law that orchestrates the switching between (2) ∂x ¯ multiple spatially-distributed control actuator configuraC2 x ¯(β, t) + D2 (β, t) = R2 tions, in a way that respects actuator constraints, accom∂z modates inherently conflicting control objectives, and guar- and the initial condition: antees closed-loop stability at the same time. The proposed x ¯(z, 0) = x ¯0 (z) methodology is applied successfully to the fault-tolerant (3) stabilization of an unstable steady state of a diffusionT n where x ¯(z, t) = [¯ x1 (z, t) · · · x ¯n (z, t)] ∈ IR denotes the reaction process. vector of state variables, z ∈ [α, β] ⊂ IR is the spatial coordinate, t ∈ [0, ∞) is the time, ui ∈ [ui,min , ui,max ] ⊂ IR deκ notes the i-th constrained manipulated input, and ym ∈ IR ∂x ¯ ∂2x ¯ denotes the κ-th measured output. , denote the 1 2 3 u (t) ∂z ∂z 2 u (t) u (t) first- and second-order spatial derivatives of x ¯, f (¯ x) is a nonlinear vector function, w, k, ω are constant row vectors, A, B, C1 , D1 , C2 , D2 are constant matrices of appropriate Spatially Distributed Process dimensions, R1 , R2 are column vectors, and x ¯0 (z) is the initial condition. The function b (z) is a known smooth i 1 2 ym (t) ym (t) function of z which describes how the control action ui (t) is distributed in the interval [α, β], and sκ (z) is a known smooth function of z which depends on the shape (point or Feedback Control controller distributing sensing) of the measurement sensors in the inobjectives terval [α, β]. Whenever the control action enters the system at a single point z0 , with z0 ∈ [α, β] (i.e. point actuation), Measurement sensor the function bi (z) is taken to be nonzero in a finite spatial Control actuator interval of the form [z0 − µ, z0 + µ], where µ is a small posi(a) u1 (t) u2(t) u3(t) tive real number, and zero elsewhere in [α, β]. Throughout the paper, the order of magnitude O(²) notation will be used. In particular, δ(²) = O(²) if there exist positive real Spatially Distributed Process numbers k1 and k2 (which are independent of ²) such that: ym1 (t) ym2 (t) |δ(²)| ≤ k1 |²| , ∀ |²| < k2 . Switching logic
controller A
Supervisor
controller B
Control objectives
controller C
(b)
Figure 1: Comparison between the control structures arising
For a precise characterization of the class of PDE systems considered in this work, we formulate the system of Eq.1 as an infinite dimensional system in the Hilbert space H([0, π]; IRn ), with H being the space of n-dimensional, sufficiently smooth vector functions defined on [0, π] that satisfy the boundary conditions of Eq.2, with inner product and norm: Z β 1 (ω1 , ω2 ) = (ω1 (z), ω2 (z))IRn dz, ||ω1 ||2 = (ω1 , ω1 ) 2 α
in the classical control (a) and the hybrid control (4) (b) paradigms of spatially distributed systems. where ω1 , ω2 are two elements of H([α, β]; IRn ) and the no-
tation (·, ·)IRn denotes the standard inner product in IRn . Defining the state function x on H([α, β]; IRn ) as: x(t) = x ¯(z, t),
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t > 0,
z ∈ [α, β],
(5)
the operator A in H([α, β]; IRn ) as: ∂x ¯ ∂2x ¯ +B 2, ∂z ∂z x ∈ D(A) = {x ∈ H([α, β]; IRn ) : C1 x ¯(α, t) Ax = A
∂x ¯ ∂x ¯ ¯(β, t) + D2 (β, t) = R2 +D1 (α, t) = R1 , C2 x ∂z ∂z
¾
and the input and measured output operators as: Bu = w
m X
bi ui ,
Sx = (s, ωx)
(6)
i=1
the system of Eqs.1-2-3 takes the form: x˙ = Ax + Bu + f (x),
x(0) = x0
ym = Sx
(7)
where f (x(t)) = f (¯ x(z, t)) and x0 = x ¯0 (z). We assume that the nonlinear term f (x) is locally Lipschitz with respect to its argument and satisfies f (0) = 0. For A, the eigenvalue problem is defined as: Aφj = λj φj ,
j = 1, . . . , ∞
(8)
available at our disposal is a family of N (with N finite) control actuator configurations, of which only one configuration can be used for control at any given time instance. Each of these N configurations consists of a spatiallydistinct arrangement of the control actuators, which we denote by z¯k(t) , k = 1, · · · , N . In this notation, the index k(t) denotes the actuator configuration being active at time t, while z¯k is a column vector whose components represent the corresponding spatial locations of the actuators associated with the k-th configuration. To ensure controllability of the system, we allow only a finite number of switches between configurations over finite time. The problem is how to coordinate switching between the different control actuator configurations, in the event of actuator failure, in a way that respects actuator constraints and guarantees closed-loop stability. To address this problem, we formulate the following objectives. Initially, Galerkin’s method is used to derive a nonlinear finite-dimensional ODE system that captures the dominant dynamics of the parabolic PDE system of Eq.1. Next, the ODE approximation is used as the basis for the synthesis, of bounded nonlinear controllers of the general form
where λj denotes an eigenvalue and φj denotes an eigenfunction; the eigenspectrum of A, σ(A), is defined as the set of all eigenvalues of A, i.e. σ(A) = {λ1 , λ2 , . . . , }. In this work, we consider parabolic PDE systems for which the following assumption holds.
u = p(ym , umax , z¯k )
(9)
that enforce asymptotic stability in the constrained closedloop system and provide an explicit characterization of the stability region associated with each control actuator configuration. The controller synthesis is carried out via Lyapunov-based control techniques [10]. Finally, a set of switching rules is derived to determine which of the N conAssumption 1: trol actuator configurations can be engaged at any given 1. Re {λ1 } ≥ Re {λ2 } ≥ · · · ≥ Re {λj } ≥ · · ·, where time, and an upper bound on the separation between the slow and fast eigenvalues, which guarantees stability of the Re {λj } denotes the real part of λj . infinite-dimensional switched closed-loop system, is com2. σ(A) can be partitioned as σ(A) = σ1 (A) + σ2 (A), puted. where σ1 (A) consists of the first m (with m finite) eigenvalues, i.e. σ1 (A) = {λ1 , . . . , λm }, and 3.2 Galerkin’s method |Re {λ1 }| = O(1). In this section, we apply standard Galerkin’s method |Re {λm }| to the system of Eq.1 to derive an approximate finitedimensional system. Let Hs , Hf be modal subspaces |Re {λm }| = O(²) where ² ≡ of A, defined as H = span{φ , φ , . . . , φ } and H = 3. Re {λm+1 } < 0 and s 1 2 m f |Re {λm+1 }| span{φm+1 , φm+2 , . . . , } (the existence of Hs , Hf follows |Re {λ1 }| < 1 is a small positive number. from the properties of A). Defining the orthogonal projec|Re {λm+1 }| tion operators Ps and Pf such that xs = Ps x, xf = Pf x, the state x of the system of Eq.1 can be decomposed as: Assumption 1 above states that the eigenspectrum of A can be partitioned into a finite-dimensional part consistx = xs + xf = Ps x + Pf x (10) ing of m slow eigenvalues and a stable infinite-dimensional complement containing the remaining fast eigenvalues, and Applying Ps and Pf to the system of Eq.7 and using the that the separation between the slow and fast eigenvalues above decomposition for x, the system of Eq.7 can be writof A is large. This assumption is satisfied by the major- ten in the following equivalent form: dxs ity of diffusion-convection-reaction processes (see, e.g., the = As xs + Bs u + fs (xs , xf ) models studied in [3, 13]). dt ∂xf = Af xf + Bf u + ff (xs , xf ) 3 Hybrid control of parabolic PDEs ∂t ym = Sxs + Sxf , 3.1 Problem formulation xs (0) = Ps x(0) = Ps x0 , xf (0) = Pf x(0) = Pf x0 Consider the system of Eq.1, where the control inputs ui are (11) constrained in the interval [−umax , umax ], and assume that
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where As = Ps A, Bs = Ps B, fs = Ps f , Af = Pf A, Bf = Pf B, ff = Pf f , and the partial derivative nota∂xf tion in is used to denote that the state xf belongs ∂t to an infinite-dimensional space. In the above system, As is a diagonal matrix of dimension m × m of the form As = diag{λj }, fs (xs , xf ) and ff (xs , xf ) are Lipschitz vector functions, and Af is an unbounded differential operator which is exponentially stable (following from the fact that Re{λm+1 } < 0 and the selection of Hs , Hf ). Neglecting the fast and stable infinite-dimensional xf -subsystem in Eq.11, the following m-dimensional slow system can be obtained: d¯ xs = As x ¯s + Bs u + fs (¯ xs , 0) dt y¯m = S x ¯s
where as (t) = [a1 (t) · · · am (t)]T ∈ IRm , ai (t) is the amplim X ¯ are tude of the i-th eigenmode, xs = ai (t)φi (z), F , G i=1
constant matrices and f˜(·) is a nonlinear function of its argument. The system of Eq.14 is obtained from that of Eq.12 by taking, for each mode, the inner product on both sides with respect to the corresponding eigenfunction. Finally, we define f¯s (¯ as ) = F as + f˜s (¯ as ) and denote by g¯i the ¯ i-th column of the matrix G.
Using a quadratic Lyapunov function of the form V = a ¯Ts Φ¯ as , where Φ is a positive-definite symmetric matrix ¯G ¯T Φ < that satisfies the Riccati inequality F T Φ+ΦF −ΦG (12) 0, we synthesize the following bounded nonlinear feedback law [10] T
where the bar symbol in x ¯s and y¯m denotes that these variables are associated with a finite-dimensional system.
zk ) u = −r(as , ukmax , z¯k ) (LG¯ V ) (¯ where r(as , ukmax , z¯k ) = q 4 L∗f¯V + (L∗f¯V )2 + (ukmax |(LG¯ V )T (¯ zk )|) · ¸ q |(LG¯ V )T (¯ zk )|2 1 + 1 + (ukmax |(LG¯ V )T (¯ zk )|)2
(15)
Remark 1: Even though the fast subsystem in Eq.11 is infinite-dimensional, from a computational point view one (16) needs to consider a finite dimensional (possibly high-order) fast subsystem to work with. Therefore, in this work we consider parabolic PDE systems for which the high orzk1 z¯k2 · · · z¯km ]T , k = 1, · · · , N , L∗f¯V = der discrteization obtained through Galerkin’s method con- where z¯k = [¯ verges to the solution of the infinite dimensional system in Lf¯V + ρ|as |2 , ρ > 0, LG¯ V is a row vector of the form the sense that, for any initial condition: [Lg¯1 V · · · Lg¯m V ]. The notation ukmax is used to indicate the magnitude of actuator constraints associated with the k-th |xs | ≤ |ˆ xs | + d1 actuator configuration. This quantity is allowed to vary (13) from one configuration to another. The scalar function r(·) ||xf ||2 ≤ ||ˆ xf ||2 + d2 in Eqs.15-16 can be thought of as a nonlinear gain of the where the offsets d1 > 0, d2 > 0 can be made arbitrar- LG¯ V controller. This Lyapunov-based gain, which depends k ily small by sufficiently increasing the order of the fast on both the magnitude of actuator constraints, umax , and subsystem, and x ˆs and x ˆf are the solutions of the slow the spatial arrangement of the actuators, z¯k , is shaped in a and fast subsystems, respectively, obtained for a high (but way that guarantees constraint satisfaction and asymptotic finite)-dimensional fast subsystem. The property of Eq.13 closed-loop stability within a well-characterized region in holds for most parabolic PDEs arising in the modeling of the state space given in Theorem 1 below. diffusion-convection-reaction processes (see also the examWe are now ready to state the main result of this work. ple in section 4). Theorem 1 below provides both the state feedback control law (see the discussion in remark 2 for output feedback con3.3 Integrating feedback and switching troller design and implementation) as well as the necessary Having obtained a finite-dimensional model that describes switching law and states precise conditions that guarantee the dominant dynamics of the PDE system, we proceed closed-loop stability in the switched closed-loop system. in this section to describe the proposed procedure for designing the hybrid control system, on the basis of the finite- Theorem 1: dimensional approximation in Eq.12. To this end, and since our objective is the stabilization of the PDE system, we 1. Consider the system of Eq.14 under the feedback contake as controlled outputs the slow modes of the PDE systrol law of Eq.15-16 and let δsk be a positive real numtem, and assume, for simplicity, that the number of inputs ber such that the compact set Ω(umax , z¯k ) = {as ∈ is equal to the number of slow modes. Though not discussed IRm : V (as ) ≤ δsk } is the largest invariant set emin this paper explicitly, the results can be generalized to adbedded within the unbounded region described by the dress the problem of reference-input tracking. For a clear following inequality presentation of our results, we consider the representation of the slow system of Eq.12 in terms of the evolution of the L∗f¯V ≤ ukmax |(Lg¯ V )T (¯ zk )| (17) amplitudes of the eigenmodes. This system is given by Without loss of generality, assume that z¯k(0) = z¯1 ¯ zk )u(t) + f˜s (¯ and as (0) ∈ Ω(u1max , z¯1 ). If, at any given time T , the (14) a ¯˙ s (t) = F a ¯s (t) + G(¯ as (t))
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condition
Part 2: Substituting the controller of Eq.15-16 into Eq.11 as (T ) ∈ (18) and using the fact that ² = |Re{λ1 }| < 1, the closed|Re{λm+1 }| holds, for some j ∈ {1, · · · , N }, then setting z¯k(T + ) = loop system can be written in the following form: z¯j guarantees that the closed-loop system of Eq.12-15dxs 16 is asymptotically stable. = As xs − Bs k(xs , ukmax, z¯k )(LG¯ V )T + fs (xs , 0) dt + [fs (xs , xf ) − fs (xs , 0)] 2. Consider the parabolic PDE system of Eq.1 under ∂xf the control law of Eq.15-16 and the switching law ² = Af ² xf + ²f¯f (xs , xf ) ∂t of Eq.18. Let δhk be a positive real number such (22) that |xs | ≤ δhk whenever as ∈ Ω(ukmax , z¯k ). Then where Af ² is an unbounded differential operator defined as given any pair of positive real numbers (d, δb ) such Af ² = ²Af , and f¯f (xs , xf ) = −Bs k(xs , ukmax, z¯k )(LG¯ V )T + that δb + d ≤ δh1 , and given any positive real numf (x , x ). The system of Eq.22 is in the standard singuber δf , there exists ²∗ > 0 such that if ² ∈ (0, ²∗ ], f s f larly perturbed form, with xs being the slow states and xf |xs (0)| ≤ δh1 , ||xf (0)||2 ≤ δf , the closed-loop infinitebeing the fast states. Applying two-time scale decomposidimensional switched system is asymptotically (and tion to the above system and analyzing the resulting slow locally exponentially) stable. and fast subsystems, one can show that the slow system obtained is exactly the switched system analyzed in part 1 and satisfies the bound in Eq.21, while the fast subsystem Proof of Theorem 1: satisfies the following exponential bound Part 1: Consider the finite-dimensional closed-loop system ||¯ xf (τ )||2 ≤ k3 ||¯ xf (0)||2 e−a3 τ ∀ t ≥ 0 (23) of Eq.14-15-16. Using a Lyapunov argument, one can show that, whenever Eq.17 is satisfied, V˙ satisfies for some k3 ≥ 1, a3 > 0. Exploiting the stability properties of the fast and slow systems, it can be shown then, with 2 −ρ|¯ as | ¸ the aid of calculations similar to those performed in [5] for V˙ ≤ · ≡ −α(¯ z ) < 0 q k the analysis of singularly perturbed systems, that given any 1 + 1 + (umax |(Lg¯ V (¯ zk ))T |)2 pair of positive real numbers (δb , d), where δb +d ≤ δh , and (19) given any δ , there exists ²(1) > 0 such that if ² ∈ (0,1 ²(1) ], f ∀ a ¯s 6= 0, k = 1, · · · , N . Since the set Ω(ukmax , z¯k ) is the |x (0)| ≤ δ , ||x (0)|| ≤ δ , then, for all t ≥ 0, the states s b f 2 f largest invariant set within the region described by Eq.17, of the closed-loop singularly perturbed system satisfy k then starting from any a ¯s (0) ∈ Ω(umax , z¯k ), the closedloop state satisfies Eq.17 for all time, and consequently, |xs (t)| ≤ ϕ|xs (0)|e−βt + d ˙ V decreases monotonically according to V ≤ −α(¯ zk ) < 0, (24) t −a3 which implies that, for each actuator configuration z¯k , the ² +d ||xf (t)||2 ≤ k3 ||xf (0)||2 e closed-loop slow subsystem is asymptotically stable. Furthermore, as long as the k-th configuration is switched in The above inequalities imply that the trajectories of the at time T when a ¯s (T ) ∈ Ω(ukmax , z¯k ), V˙ will always satisfy switched closed-loop system will be ultimately bounded the following worst-case dissipation inequality for all t ≥ 0 with a bound that depends on d. Since d is arbitrary, we can Ω(ujmax , z¯j )
choose it small enough such that after a sufficiently large (20) time, say t˜, the trajectories of the closed-loop system are k=1,···,N confined within a small compact neighborhood of the origin which implies that the finite-dimensional switched closed- of the closed-loop system. Let d = b/2 and t˜ be the smallest ˜ t loop system of Eq.14-15-16-18 is asymptotically stable and time such that max{ϕ|¯ xs (0)|e−β t˜, k3 ||xf (0)||2 e−a3 ² } ≤ d. its state is bounded (and therefore the denominator term in Then it can be easily verified that Eq.19 is bounded). Therefore, given δs1 > 0, there exists |xs (t)| ≤ b, ||xf (t)||2 ≤ b ∀ t ≥ t˜ (25) a positive real number γ > 0, such that if |¯ as (0)| ≤ δs1 , 2 V˙ satisfies V˙ ≤ −γ|¯ as | and the switched closed-loop Recall from Eq.23 and Eq.21 that both the fast and slow system is exponentially stable. From this and the fact that subsystems are exponentially stable within the ball of m X (2) (2) x ¯s = ai (t)φi (z), we deduce that the closed-loop system Eq.25. Therefore there exists ² > 0 such that if ² ≤ ² , the singularly perturbed closed-loop system of Eq.22 is loi=1 of Eq.12 is asymptotically stable. Also, given the positive cally exponentially stable and, therefore, once inside the real number δh1 , there exists b > 0 such that if |¯ xs (0)| ≤ ball of Eq.25, the closed-loop trajectories converge to the δh1 , the following bound holds origin as t → ∞. This completes the proof of the theorem. V˙ ≤
max {−α(¯ zk )} < 0
|¯ xs (t)| ≤ ϕ|¯ xs (0)|e−βt
∀ t≥0
for all |¯ xs (t)| ≤ b, for some ϕ ≥ 1, β > 0.
(21) Remark 2: When only output measurements are available, an output feedback controller can be constructed under the assumption that the number of measurements is
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equal to the number of slow modes and that the inverse of the operator S exists, so that x ˆs = S −1 ym (which can be ensured by appropriate choice of the location of the measurement sensors). The synthesis of the output feedback controller can be carried out by combining the state feedback controller of Eq.15-16 with a procedure proposed in [3] for obtaining estimates for the states of the approximate ODE model of Eq.12 from the measurements. While the estimation error leads to some loss in the size of the stability region obtained under state feedback, this loss can be made small by increasing the order of the ODE approximation and including more measurements. This approach therefore allows us to asymptotically (as ² → 0) recover the stability region associated with each control actuator configuration. 4 Application to a diffusion-reaction process In this section, we illustrate through computer simulations how the concept of coupling feedback and switching, proposed in Theorem 1, can be used to deal with the problem of control actuator failure in control of a diffusion-reaction process. To this end, consider a long, thin rod in a reactor. The reactor is fed with pure species A and a zeroth order exothermic catalytic reaction of the form A → B takes place on the rod. Since the reaction is exothermic, a cooling medium in contact with the rod is used for cooling. Under standard assumptions, the spatiotemporal evolution of the dimensionless rod temperature is described by the following parabolic PDE γ − ∂x ¯ ∂2x ¯ 1 + x ¯ + βU (b(z)u(t) − x ¯) − βT e−γ = + βT e ∂t ∂z 2 x ¯(0, t) = 0, x ¯(π, t) = 0, x ¯(z, 0) = x ¯0 (z) (26) where x ¯ denotes the dimensionless temperature in the reactor, βT denotes a dimensionless heat of reaction, γ denotes a dimensionless activation energy, βU denotes a dimensionless heat transfer coefficient, u(t) denotes the vector of manipulated inputs and b(z) the vector of the corresponding actuator distribution functions. The following typical values are given to the process parameters: βT = 50.0, βU = 2.0, γ = 4.0. For the above values, it was verified that the operating steady state x ¯(z, t) = 0 is an unstable one. The control objective therefore is to stabilize the rod temperature profile at this unstable steady state by manipulating the temperature of the cooling medium, subject to hard constraints. To achieve this objective, the controlled output is defined as Z πr 2 (27) sin(z)x(z, t)dz yc (t) = π 0
For this system, we consider the first eigenvalue as the dominant one and use standard Galerkin’s method to derive an ODE that describes the temporal 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.15-16, which is then implemented on a 30th order Galerkin discretization of the parabolic PDE of Eq.26 (higher order discretizations led to identical results, implying that the property of Eq.13 holds for Eq.26). In order to demonstrate the utility of the switching scheme proposed in Theorem 1 for dealing with actuator failure, we consider the following problem where three point control actuators, A, B, and C, located at zA = 0.5π, zB = 0.23π, and zC = 0.36π, respectively, are available for stabilization. The three actuators have different constraints of umaxA = umaxB = 2.5 and umaxC = 0.5. 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 that the operating actuator fails. The state feedback results will be presented first. Using Eq.17 with ρ = 0.02, the stability regions for umaxA = umaxB = 2.5 (solid line) and for umaxC = 0.5 (dashed line) are computed as functions of actuator location and shown in Figure 2a. To simplify the presentation of our results, we plot in Figure 2a the variation of the set of admissible initial conditions for the amplitude, a1 (0), of the first eigenmode, with actuator location (note that xs (0) = a1 (0)φ1 (z)). From this figure, it is easy to see that for an initial condition a1 (0) = 1.3, only actuator A can be used initially since the initial condition is outside the stability regions for actuators B and C. Now, suppose that sometime after startup, say at t = 1.4, a fault is detected in actuator A and it becomes necessary to switch to a backup actuator. Without using the switching logic of Theorem 1, it is not clear whether actuator B or C should be activated at this time. Figures 2b and 2d (solid lines) depict, respectively the resulting closed-loop temperature and manipulated input profiles when actuator C is switched in. We see in this case that the controller is unable to stabilize the system at the desired steady-state. In contrast, when the switching law of Theorem 1 is implemented, we track the evolution of a1 (t) in time and find that, at the time of actuator A’s failure, we have a1 (1.4) = 0.62, i.e. the state is inside the stability region for actuator B and outside the stability region for actuator C. Based on this, we decide to switch to actuator B. Figures 2c and 2d (dashed line) depict the results for this case which show that the controller successfully stabilizes the closed-loop system at the desired steady-state.
For the case of output feedback, we used a single point senThe eigenvalue problem for the spatial differential operator sor located at z = 0.33π, to obtain estimates of the first of the process can be solved analytically and its solution is eigenmode, which was subsequently used for implementing the output feedback controller. Owing to the small O(²) r 2 2 λj = −j , φj (z) = sin(j z), j = 1, . . . , ∞ (28) discrepancy between the stability regions obtained under π
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state and output feedback (due to the estimation error), the switching law of Theorem 1 was used as an approximate guide for switching between the different actuators in the case of output feedback (see remark 2). The simulation results for this case were found to be consistent with the state feedback results and are omitted due to space limitations.
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Figure 2: (a) Stability region as a function of actuator location [12] W. H. Ray. Advanced Process Control. McGraw-Hill, for umax = 2.5 (solid) and umax = 0.5 (dashed),(b) New York, 1981. Closed-loop temperature profile (state feedback) for a1 (0) = 1.3 when actuator A (umax = 2.5, zA = 0.5π) fails at t = 1.4 and actuator C (umax = 0.5, zA = 0.36π) is activated, (c) Closed-loop temperature profile (state feedback) for a1 (0) = 1.3 when actuator A fails at t = 1.4 and actuator B (umax = 2.5,zA = 0.23π) is activated, (d) Corresponding manipulated input profiles for case b (solid) and case c (dashed).
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