Yiming Lou and Panagiotis D. Christofides. Department of Chemical Engineering University of California, Los Angeles

Transcription

1 OPTIMAL ACTUATOR/SENSOR PLACEMENT FOR NONLINEAR CONTROL OF THE KURAMOTO-SIVASHINSKY EQUATION Yiming Lou and Panagiotis D. Christofides Department of Chemical Engineering University of California, Los Angeles

2 KURAMOTO-SIVASHINSKY EQUATION Kuramoto-Sivashinsky equation with distributed control: U t = ν 4 U z 4 2 U z 2 U U z + l b i u i (t) i=1 Boundary conditions: j U z j ( π, t) = j U (+π, t), j =,..., 3 zj The stability of steady-state U(z, t) = depends on ν.

3 INTRODUCTION Important systems described by the Kuramoto-Sivashinsky equation: Falling liquid films. Unstable flame fronts. Interfacial instabilities between two viscous fluids. Feedback control of the Kuramoto-Sivashinsky equation: Finite-dimensional output feedback controller design based on Galerkin s method (e.g., Armaou and Christofides, CES and Physica D, 2). Global stabilization of the zero solution of the KSE (e.g., Christofides and Armaou, SCL, 2). Boundary control (e.g., Liu and Krstic, NA, 21). Optimal actuator/sensor placement for the Kuramoto-Sivashinsky equation?

4 BACKGROUND ON ACTUATOR/SENSOR PLACEMENT Optimal actuator placement for linear controllers and PDE models. Controllability measures (e.g., Arbel, 1981). Optimal controller gain/actuator location to minimize cost on system response and control action (e.g., Rao et al, AIAA J., 1991). Optimal sensor placement for linear estimators and PDE models. Observability measures (e.g., Yu and Seinfeld, IJC. 1973; Waldraff et al, JPC, 1998). Minimum estimation error under worst measurement noise (e.g., Kumar and Seinfeld, IEEE TAC, 1978; Morari and O Dowd, Automatica 198). Review paper: Kubrusly and Malebranche, Automatica, Optimal actuators/sensors placement for nonlinear dissipative PDE systems (Antoniades and Christofides, C&CE, 2; CES, 21; C&CE, 22).

5 PRESENT WORK (Lou and Christofides, IEEE CST, 22) Optimal actuator/sensor placement for nonlinear control of the Kuramoto-Sivashinsky equation. Order reduction using linear/nonlinear Galerkin s method. Computation of optimal location of actuators and sensors through minimization of a cost that includes penalty on the closed-loop response and the control effort. Nonlinear output feedback controller design using geometric methods. Illustration of theoretical results through computer simulation of the closed-loop system using a high-order discretization of the KSE.

6 KURAMOTO-SIVASHINSKY EQUATION Kuramoto-Sivashinsky equation with distributed control: U t = ν 4 U z 4 2 U z 2 U U z + l b i u i (t) i=1 Boundary conditions: j U z j ( π, t) = j U (+π, t), j =,..., 3 zj Representation in Hilbert space: ẋ=ax + Bu + f(x), x() = x y m =kx A: linear operator. f(x): nonlinear function.

7 EIGENSPECTRUM / OPEN-LOOP DYNAMICS Eigenvalue problem: Aφ n = ν 4 φ n z 4 2 φ n z 2 = λ n φ n, n = 1,..., j φ n z j ( π) = j φ n z j (+π), j =,..., 3 Eigenvalues: λ n = νn 4 + n 2, n = 1,..., : Structure of Eigenspectrum Open-loop profile of U(z, t) Im 6 4 ν = Re 2 Uν = t z 1 2 3

8 GALERKIN S METHOD H s = span{φ 1, φ 2,..., φ m }, H f = span{φ m+1, φ m+2,..., }. x s (t) = P s x(t), x f (t) = P f x(t) P s, P f : orthogonal projection operators. Set of infinite ODEs. dx s dt x f t = A s x s + b s u + f s (x s, x f ) = A f x f + b f u + f f (x s, x f ) Finite-set of ODEs. d x s dt = A s x s + B s u + f s ( x s, ) Order reduction using Galerkin s method and approximate inertial manifolds is also possible.

9 NONLINEAR CONTROL DESIGN d x s dt = A s x s + B s u + f s ( x s, ) Assumption: l = m (i.e. number of manipulated inputs is equal to the number of slow modes) and B s is invertible. Nonlinear state feedback controller: Closed-loop ODE system: u = B 1 s ((Λ s A s ) x s f s ( x s, )) Λ s is a stable matrix. x s = Λ s x s Response depends on Λ s and x s (), but not on actuator locations: x s = e Λ st x s ()

10 OPTIMAL POINT ACTUATOR PLACEMENT Performance criterion (sum is over a set of m linearly independent x i s()): Ĵ s = 1 m m i=1 (( x s (x i s(), t), Q s x s (x i s(), t)) +u T ( x s (x i s(), t), z a )Ru( x s (x i s(), t), z a ))dt However, system s response does not depend on the actuator locations: Ĵ xs = 1 m ( x s (x i m s(), t), Q s x s (x i s(), t))dt i=1 Performance criterion reduces to: Ĵ us = 1 m u T ( x s (x i m s(), t), z a )Ru( x s (x i s(), t), z a )dt i=1 Computation of optimal actuator locations: [ ] T Ĵus Ĵus Ĵus... = [... ] T, zazaĵ us (z am ) > z a1 z a2 z al Near-optimal solution for distributed system as ɛ = λ 1 λ m+1.

11 OPTIMAL LOCATION OF POINT SENSORS d x s dt = A s x s + B s u + f s ( x s, ) y m = S x s Assumption: p = m(i.e. the number of sensors is equal to the number of slow modes) and S is invertible. Computation of estimate of x s from the measurements: ˆx s = S 1 y m y m : sensor measurements, ˆx s estimate of x s. Compute point sensor locations to minimize the estimation error in the closed-loop system: Ĵ(e) = 1 m ( x s (x i m s(), t) ˆx s (x i s(), t) 2 )dt i=1 x s : slow state of infinite set of ODEs, e(t) = x s ˆx s 2 Near-optimal solution for distributed system as ɛ.

12 KURAMOTO-SIVASHINSKY EQUATION Kuramoto-Sivashinsky equation with distributed control: U t = ν 4 U z 4 2 U z 2 U U z + l b i u i (t) i=1 Boundary conditions: j U z j ( π, t) = j U (+π, t), j =,..., 3 zj The stability of steady-state U(z, t) = depends on ν.

13 CONTROLLER SYNTHESIS Kuramoto-Sivashinsky equation, ν =.2 ν =.2 Two unstable eigenmodes. Order of ODE model used for controller design:2 Number of control actuators:2; Number of measurement sensors:2 Approximate two-dimensional ODE system used for controller design. [ ] [ ] [ ] ( x s1, φ 1 ) ( x s2, φ 2 ) = λ 1 λ 2 ( x s1, φ 1 ) ( x s2, φ 2 ) + [ φ 1 (z a1 ) φ 1 (z a2 ) φ 2 (z a1 ) φ 2 (z a2 ) ] [ u 1 u 2 ] + [ (f 1 ( x s, ), φ 1 ) (f 2 ( x s, ), φ 2 ) ] z a1, z a2 : location of point actuators.

14 CONTROLLER SYNTHESIS Nonlinear state feedback controller: [ ] [ u 1 u 2 = φ 1 (z a1 ) φ 1 (z a2 ) φ 2 (z a1 ) φ 2 (z a2 ) ([ [ α λ 1 ] 1 β λ 2 (f 1 ( x s, ), φ 1 ) (f 2 ( x s, ), φ 2 ) ]) ] [ ( x s1, φ 1 ) ( x s2, φ 2 ) ] α and β are positive real numbers. The structure of the closed-loop finite-dimensional system under the feedback control : [ ] [ ] [ ] ( x s1, φ 1 ) ( x s2, φ 2 ) = α β which is independent of the actuator locations. ( x s1, φ 1 ) ( x s2, φ 2 )

15 OPTIMAL LOCATION OF ACTUATORS Cost for optimal location of control actuators: Ĵ s = 1 2 Q s = R = [ 2 i=1 1 1 ] (( x T s (x i s(), t), Q s x s (x i s(), t)) +u T ( x s (x i s(), t), z a )Ru( x s (x i s(), t), z a ))dt, x 1 s() = [ φ 1 ], x 2 s() = [ φ 2 ]. Optimal actuator locations: z a1 =.31π and z a2 =.69π.

16 OPTIMAL LOCATION OF SENSORS Cost for optimal location of measurement sensors: Ĵ(e) = i=1 ( x s (x i s(), t) ˆx s (x i s(), t) 2 )dt where x s can be obtained from the simulation of the high order system, and ˆx s is calculated from the output: [ ˆx s1 ˆx s2 ] = [ φ 1 φ 2 ] [ φ 1 (z s1 ) φ 1 (z s2 ) φ 2 (z s1 ) φ 2 (z s2 ) ] 1 [ y m1 (z s1, t) y m2 (z s2, t) ] Optimal sensor locations: z s1 =.35π and z s2 =.64π.

17 SIMULATION RESULTS State feedback control with two actuators. Case Actuator locations Ĵ s Optimal.31π,.69π π,.8π π,.6π π,.8π Closed-loop norm of u for the optimal case (solid line), case 2 (long-dashed line), case 3 (short-dashed line), and case 4 (dotted line). Left plot: x s () = [ φ 1 ]. Right plot: x s () = [ φ 2 ] u u t t

18 SIMULATION RESULTS Output feedback control with two sensors (optimal actuator locations). Case Sensor locations Ĵ(e) Ĵ s Optimal.35π,.64π 2.86e π,.8π 1.841e π,.5π 5.2e Closed-loop norm of the closed-loop estimation error e versus time, for the optimal actuator/sensor locations. Dotted line: x s () = [ φ 1 ]. Solid line: x s () = [ φ 2 ]. 7 x e t

19 SIMULATION RESULTS Profiles of the evolution of U under output feedback control, for optimal actuator/sensor locations, for x s () = [ φ 1 ] (left figure), and x s () = [ φ 2 ] (right figure)..6 U U t z t z 1 2 3

20 SIMULATION RESULTS Profiles of KSE under output feedback control, for the optimal actuator/sensor locations, for x s () = [ φ 1 ] and uncertainty in ν. Manipulated input profiles - Solid line u 1 and dashed line u U.2 u1, u t z t

21 SIMULATION RESULTS Profiles of KSE under output feedback control, for the optimal actuator/sensor locations, for x s () = [ φ 2 ] and uncertainty in ν. Manipulated input profiles - Solid line u 1 and dashed line u U.2 u1, u t z t

22 CONCLUSIONS Optimal actuator/sensor placement for nonlinear control of the KSE. Order reduction using Galerkin s method Nonlinear output feedback controller design using geometric methods. Computation of optimal actuator/sensor location through minimizing a cost that includes penalty on the close-loop response and control effort. ACKNOWLEDGMENT Financial support from the National Science Foundation, CTS-2626, and the Air Force Office of Scientific Research is gratefully acknowledged.

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