Control Design of a Distributed Parameter. Fixed-Bed Reactor

Transcription

1 Workshop on Control of Distributed Parameter Systems p. 1/1 Control Design of a Distributed Parameter Fixed-Bed Reactor Ilyasse Aksikas and J. Fraser Forbes Department of Chemical and Material Engineering University of Alberta Canada

2 Workshop on Control of Distributed Parameter Systems p. 2/1 Outline Introduction Problem Statement Optimal Control Design Closed-loop Nonlinear Model Analysis Concluding Remarks

3 Workshop on Control of Distributed Parameter Systems p. 3/1 Introduction Tubular Reactors : Processes in (bio)chemical engineering. Diffusion-Convection-Reaction systems. No diffusion Plug flow reactor.

4 Workshop on Control of Distributed Parameter Systems p. 3/1 Introduction Tubular Reactors : Processes in (bio)chemical engineering. Diffusion-Convection-Reaction systems. No diffusion Plug flow reactor. Mathematical Model : Nonlinear PDE s model Nonlinear infinite dimensional system

5 Workshop on Control of Distributed Parameter Systems p. 4/1 Introduction (cont.) Plug Flow Reactor T t = v l T z + r(c,t) + h(t j T) = v l C z + r(c,t) C t

6 Workshop on Control of Distributed Parameter Systems p. 4/1 Introduction (cont.) Plug Flow Reactor T t = v l T z + r(c,t) + h(t j T) = v l C z + r(c,t) C t I. Aksikas, J.J. Winkin and D. Dochain, Optimal LQ-Feedback Regulation of a Nonisothermal Plug Flow Reactor Model by Spectral Factorization, IEEE TAC, vol. 52, 7, 2007

7 Workshop on Control of Distributed Parameter Systems p. 4/1 Introduction (cont.) Plug Flow Reactor T t = v l T z + r(c,t) + h(t j T) = v l C z + r(c,t) C t I. Aksikas, J.J. Winkin and D. Dochain, Optimal LQ-Feedback Regulation of a Nonisothermal Plug Flow Reactor Model by Spectral Factorization, IEEE TAC, vol. 52, 7, 2007 I. Aksikas, J.J. Winkin and D. Dochain, Optimal LQ-Feedback Control for a Class of First-Order Distributed Parameter Systems, submitted, under revision, 2007

8 Workshop on Control of Distributed Parameter Systems p. 5/1 Introduction (cont.) Fixed-Bed Reactor

9 Workshop on Control of Distributed Parameter Systems p. 5/1 Introduction (cont.) Fixed-Bed Reactor Assumption: [T,C] fluid phase = [T,C] solid phase

10 Workshop on Control of Distributed Parameter Systems p. 5/1 Introduction (cont.) Fixed-Bed Reactor Assumption: [T,C] fluid phase = [T,C] solid phase PDE Model ρ b c pb T t = ρ f c pf v l T z + r(c,t) + h(t j T) ǫc t = v l C z + r(c,t)

11 Workshop on Control of Distributed Parameter Systems p. 5/1 Introduction (cont.) Fixed-Bed Reactor Assumption: [T,C] fluid phase = [T,C] solid phase PDE Model ρ b c pb T t = ρ f c pf v l T z + r(c,t) + h(t j T) ǫc t = v l C z + r(c,t) P.D. Christofides, Nonlinear and Robust Control of PDE Systems, Birkhäser, 2001

12 Workshop on Control of Distributed Parameter Systems p. 6/1 Problem Statement Objective: We want to minimize the cost function 0 { Cx(s),PCx(s) + u(s),ru(s) }ds along the differential equation constraint { x t (t) = V x z (t) + Mx(t) + Nu(t) x(0) = x 0 x(t) H = L 2 (0, 1) n and u(t) L 2 (0, 1) n V,M IR n n V symmetric and N IR m n.

13 Workshop on Control of Distributed Parameter Systems p. 7/1 Stability Result Let us consider the operator A = V d dz + M I defined on D(A) = {x : x is a.c, dx dz H ;x(0) = 0} A generates an exp stable C 0 -semigroup IF

14 Workshop on Control of Distributed Parameter Systems p. 7/1 Stability Result Let us consider the operator A = V d dz + M I defined on D(A) = {x : x is a.c, dx dz H ;x(0) = 0} A generates an exp stable C 0 -semigroup IF V diagonalizable and has identical eigenvalues. OR

15 Workshop on Control of Distributed Parameter Systems p. 7/1 Stability Result Let us consider the operator A = V d dz + M I defined on D(A) = {x : x is a.c, dx dz H ;x(0) = 0} A generates an exp stable C 0 -semigroup IF V diagonalizable and has identical eigenvalues. OR The eigenvalues of V are negative.

16 Workshop on Control of Distributed Parameter Systems p. 8/1 Optimal Control Design Operator Riccati Equation [A Q o + Q o A + C PC Q o BR 1 B Q o ]x = 0, for all x D(A), where Q o (D(A)) D(A )

17 Workshop on Control of Distributed Parameter Systems p. 8/1 Optimal Control Design Operator Riccati Equation [A Q o + Q o A + C PC Q o BR 1 B Q o ]x = 0, for all x D(A), where Q o (D(A)) D(A ) If (A, B, C) exp stabilizable and exp Detectable This equation admits a unique positive self-adjoint solution. The optimal control is given by u opt (t) = R 1 B Q o x(t).

18 Workshop on Control of Distributed Parameter Systems p. 9/1 Optimal Control Design (cont.) Main Result If the matrix Φ is the unique positive semi-definite solution of the MRDE V dφ dz = M Φ+ΦM+C 0P 0 C 0 ΦB 0 R 1 0 B 0Φ, Φ(1) = 0 Then Q o = Φ(z)I is the unique self-adjoint positive semi-definite solution of ORAE

19 Workshop on Control of Distributed Parameter Systems p. 10/1 Optimal Control Design (cont.) Cases Case 1: V = vi Aksikas, Winkin and Dochain, 2006

20 Workshop on Control of Distributed Parameter Systems p. 10/1 Optimal Control Design (cont.) Cases Case 1: V = vi Aksikas, Winkin and Dochain, 2006 Case 2: V = diag(v 1,...,v n )

21 Workshop on Control of Distributed Parameter Systems p. 10/1 Optimal Control Design (cont.) Cases Case 1: V = vi Aksikas, Winkin and Dochain, 2006 Case 2: V = diag(v 1,...,v n ) v i dφ i dz = 2m iiφ i +c ii b ii φ 2 i, φ i (1) = 0, i = 1,...,n 0 = m ji φ j +φ i m ij +c ij φ i b ij φ j, 1 < i < j < n,

22 Workshop on Control of Distributed Parameter Systems p. 10/1 Optimal Control Design (cont.) Cases Case 1: V = vi Aksikas, Winkin and Dochain, 2006 Case 2: V = diag(v 1,...,v n ) v i dφ i dz = 2m iiφ i +c ii b ii φ 2 i, φ i (1) = 0, i = 1,...,n 0 = m ji φ j +φ i m ij +c ij φ i b ij φ j, 1 < i < j < n, Case 3: V diagonalizable

23 Workshop on Control of Distributed Parameter Systems p. 11/1 Application to Fixed-Bed Reactor PDE Model ρ p c pb T t = ρ f c pf v l T z + k 1 Ce E RT + h(tj T) ǫc t = v l C z k 0 Ce E RT B.C and I.C T(0,t) = T in, C(0,t) = C in, T(z, 0) = T 0 (z) C(z, 0) = C 0 (z)

24 Workshop on Control of Distributed Parameter Systems p. 12/1 Linearized Model ẋ(t) = Ax(t) + Bu(t) x(0) = x 0 H := L 2 (0, 1) 2 D(A) = {x H : x is a.c, dx dz H and x(0) = 0} A = ( v 1 d. dz + α 1I α 2 I d. α 3 I v 2 dz + α 4I ) and B = ( βi 0 )

25 Workshop on Control of Distributed Parameter Systems p. 13/1 Optimal Control Design Output function y(t) = Cx(t) = ( w 1 (z)i w 2 (z)i ) x(t)

26 Workshop on Control of Distributed Parameter Systems p. 13/1 Optimal Control Design Output function y(t) = Cx(t) = ( w 1 (z)i w 2 (z)i ) x(t) Controller dφ v 1 1 dφ v 2 2 dz = 2α 1 φ 1 + pw1 2 β 2 r 1 φ 2 1, φ 1 (1) = 0, dz = 2α 4 φ 2 + pw2, 2 φ 2 (1) = 0, 0 = α 3 φ 2 + α 2 φ 1 + pw 1 w 2 = K o x = β r φ 1(z)x 1

27 Workshop on Control of Distributed Parameter Systems p. 14/1 Nonlinear closed-loop stability I. Aksikas, J.J. Winkin and D. Dochain, 2007 { ẋ(t) = A 0 x(t) + N 0 (x(t)) x(0) = x 0 D(A 0 ) F A 0 dissipative and (I λa 0 ) 1 compact. F R(I λa 0 ) lim h 0 + d(f,x + hn 0 (x)) = 0, x D(A) A 0 + N 0 strictly dissipative.

28 Workshop on Control of Distributed Parameter Systems p. 14/1 Nonlinear closed-loop stability I. Aksikas, J.J. Winkin and D. Dochain, 2007 { ẋ(t) = A 0 x(t) + N 0 (x(t)) x(0) = x 0 D(A 0 ) F A 0 dissipative and (I λa 0 ) 1 compact. F R(I λa 0 ) lim h 0 + d(f,x + hn 0 (x)) = 0, x D(A) A 0 + N 0 strictly dissipative. x 0 D(A), x(t,x 0 ) ω(x 0 )= {x}

29 Nonlinear closed-loop stability Workshop on Control of Distributed Parameter Systems p. 15/1

30 Workshop on Control of Distributed Parameter Systems p. 15/1 Nonlinear closed-loop stability Under the assumptions: lim h 0 + h 1 d(f 0,θ + hq) = 0, θ F 0 Nonlinearity Lipshitz Constant < β

31 Workshop on Control of Distributed Parameter Systems p. 15/1 Nonlinear closed-loop stability Under the assumptions: lim h 0 + h 1 d(f 0,θ + hq) = 0, θ F 0 Nonlinearity Lipshitz Constant < β = Asymptotic Stability θ 0 F 0, θ(t,θ 0 ) := Γ(t)θ 0 θ e

32 Numerical Simulations Workshop on Control of Distributed Parameter Systems p. 16/1

33 Numerical Simulations (cont.) Workshop on Control of Distributed Parameter Systems p. 17/1

34 Numerical Simulations (cont.) Workshop on Control of Distributed Parameter Systems p. 18/1

35 Workshop on Control of Distributed Parameter Systems p. 19/1 Concluding Remarks Conclusions LQ-Controller Design Application to fixed-bed reactor Perspectives : Extensions Fixed-bed reactor with diffusion Time-varying hyperbolic systems