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
Workshop on Control of Distributed Parameter Systems – p. 1/19
Outline
• Introduction
• Problem Statement
• Optimal Control Design
• Closed-loop Nonlinear Model Analysis
• Concluding Remarks
Workshop on Control of Distributed Parameter Systems – p. 2/19
Introduction
• Tubular Reactors :
⊲ Processes in (bio)chemical engineering.
⊲ ”Diffusion-Convection-Reaction” systems.
⊲ No diffusion ⇒ Plug flow reactor.
Workshop on Control of Distributed Parameter Systems – p. 3/19
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
Workshop on Control of Distributed Parameter Systems – p. 3/19
Introduction (cont.)
• Plug Flow Reactor
Tt = −vlTz + r(C, T ) + h(Tj − T )
Ct = −vlCz + r(C, T )
Workshop on Control of Distributed Parameter Systems – p. 4/19
Introduction (cont.)
• Plug Flow Reactor
Tt = −vlTz + r(C, T ) + h(Tj − T )
Ct = −vlCz + r(C, T )
⊲ I. Aksikas, J.J. Winkin and D. Dochain, ”Optimal LQ-Feedback Regulation of aNonisothermal Plug Flow Reactor Model by Spectral Factoriz ation”, IEEE TAC, vol. 52, 7, 2007
Workshop on Control of Distributed Parameter Systems – p. 4/19
Introduction (cont.)
• Plug Flow Reactor
Tt = −vlTz + r(C, T ) + h(Tj − T )
Ct = −vlCz + r(C, T )
⊲ I. Aksikas, J.J. Winkin and D. Dochain, ”Optimal LQ-Feedback Regulation of aNonisothermal Plug Flow Reactor Model by Spectral Factoriz ation”, IEEE TAC, vol. 52, 7, 2007
⊲ I. Aksikas, J.J. Winkin and D. Dochain, ”Optimal LQ-Feedback Control for aClass of First-Order Distributed Parameter Systems”, subm itted, under revision, 2007
Workshop on Control of Distributed Parameter Systems – p. 4/19
Introduction (cont.)
• Fixed-Bed Reactor
Workshop on Control of Distributed Parameter Systems – p. 5/19
Introduction (cont.)
• Fixed-Bed Reactor
⊲ Assumption: [T,C]fluid phase = [T,C]solid phase
Workshop on Control of Distributed Parameter Systems – p. 5/19
Introduction (cont.)
• Fixed-Bed Reactor
⊲ Assumption: [T,C]fluid phase = [T,C]solid phase⊲ PDE Model
ρbcpbTt = −ρfcpfvlTz + r(C, T ) + h(Tj − T )
ǫCt = −vlCz + r(C, T )
Workshop on Control of Distributed Parameter Systems – p. 5/19
Introduction (cont.)
• Fixed-Bed Reactor
⊲ Assumption: [T,C]fluid phase = [T,C]solid phase⊲ PDE Model
ρbcpbTt = −ρfcpfvlTz + r(C, T ) + h(Tj − T )
ǫCt = −vlCz + r(C, T )
⊲ P.D. Christofides, ”Nonlinear and Robust Control of PDE Systems”, Birkh äser, 2001
Workshop on Control of Distributed Parameter Systems – p. 5/19
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{
xt(t) = V xz(t) + Mx(t) + Nu(t)
x(0) = x0
⋄ x(t) ∈ H = L2(0, 1)n and u(t) ∈ L2(0, 1)n
⋄ V,M ∈ IRn×n V symmetric and N ∈ IRm×n.
Workshop on Control of Distributed Parameter Systems – p. 6/19
Stability Result
Let us consider the operator
A = V ·d·
dz+ M · I
defined on D(A) = {x : x is a.c , dxdz
∈ H ;x(0) = 0}
A generates an exp stable C0-semigroup IF
Workshop on Control of Distributed Parameter Systems – p. 7/19
Stability Result
Let us consider the operator
A = V ·d·
dz+ M · I
defined on D(A) = {x : x is a.c , dxdz
∈ H ;x(0) = 0}
A generates an exp stable C0-semigroup IF
⋄ V diagonalizable and has identical eigenvalues.
OR
Workshop on Control of Distributed Parameter Systems – p. 7/19
Stability Result
Let us consider the operator
A = V ·d·
dz+ M · I
defined on D(A) = {x : x is a.c , dxdz
∈ H ;x(0) = 0}
A generates an exp stable C0-semigroup IF
⋄ V diagonalizable and has identical eigenvalues.
OR
⋄ The eigenvalues of V are negative.
Workshop on Control of Distributed Parameter Systems – p. 7/19
Optimal Control Design
• Operator Riccati Equation
[A∗Qo + QoA + C∗PC − QoBR
−1B∗Qo]x = 0,
for all x ∈ D(A), where Qo(D(A)) ⊂ D(A∗)
Workshop on Control of Distributed Parameter Systems – p. 8/19
Optimal Control Design
• Operator Riccati Equation
[A∗Qo + QoA + C∗PC − QoBR
−1B∗Qo]x = 0,
for all x ∈ D(A), where Qo(D(A)) ⊂ D(A∗)
• If (A,B,C) exp stabilizable and exp Detectable⋄ This equation admits a unique positiveself-adjoint solution.⋄ The optimal control is given by
uopt(t) = −R−1B∗Qox(t).
Workshop on Control of Distributed Parameter Systems – p. 8/19
Optimal Control Design (cont.)
Main Result
If the matrix Φ is the unique positive semi-definitesolution of the MRDE
VdΦ
dz= M ∗Φ+ΦM+C∗
0P0C0−ΦB0R
−10
B∗0Φ, Φ(1) = 0
Then
Qo = Φ(z)I
is the unique self-adjoint positive semi-definite solutionof ORAE
Workshop on Control of Distributed Parameter Systems – p. 9/19
Optimal Control Design (cont.)
Cases
• Case 1: V = vIAksikas, Winkin and Dochain, 2006
Workshop on Control of Distributed Parameter Systems – p. 10/19
Optimal Control Design (cont.)
Cases
• Case 1: V = vIAksikas, Winkin and Dochain, 2006
• Case 2: V = diag(v1, . . . , vn)
Workshop on Control of Distributed Parameter Systems – p. 10/19
Optimal Control Design (cont.)
Cases
• Case 1: V = vIAksikas, Winkin and Dochain, 2006
• Case 2: V = diag(v1, . . . , vn)
vidφi
dz= 2miiφi+cii−biiφ
2
i , φi(1) = 0,∀i = 1, . . . , n
0 = mjiφj +φimij + cij −φibijφj, 1 < i < j < n,
Workshop on Control of Distributed Parameter Systems – p. 10/19
Optimal Control Design (cont.)
Cases
• Case 1: V = vIAksikas, Winkin and Dochain, 2006
• Case 2: V = diag(v1, . . . , vn)
vidφi
dz= 2miiφi+cii−biiφ
2
i , φi(1) = 0,∀i = 1, . . . , n
0 = mjiφj +φimij + cij −φibijφj, 1 < i < j < n,
• Case 3: V diagonalizable
Workshop on Control of Distributed Parameter Systems – p. 10/19
Application to Fixed-Bed Reactor
• PDE Model
ρpcpbTt = −ρfcpfvlTz + k1Ce− E
RT + h(Tj − T )
ǫCt = −vl Cz − k0Ce− E
RT
• B.C and I.C
T (0, t) = Tin, T (z, 0) = T0(z)
C(0, t) = Cin, C(z, 0) = C0(z)
Workshop on Control of Distributed Parameter Systems – p. 11/19
Linearized Model
ẋ(t) = Ax(t) + Bu(t)
x(0) = x0 ∈ H := L2(0, 1)2
D(A) = {x ∈ H : x is a.c,dx
dz∈ H and x(0) = 0}
A =
(
v1d.dz
+ α1I α2I
α3I v2d.dz
+ α4I
)
and B =
(
βI
0
)
Workshop on Control of Distributed Parameter Systems – p. 12/19
Optimal Control Design
Output function
y(t) = Cx(t) =(
w1(z)I w2(z)I)
x(t)
Workshop on Control of Distributed Parameter Systems – p. 13/19
Optimal Control Design
Output function
y(t) = Cx(t) =(
w1(z)I w2(z)I)
x(t)
Controller
v1dφ1dz
= 2α1φ1 + pw2
1− β2r−1φ2
1, φ1(1) = 0,
v2dφ2dz
= 2α4φ2 + pw2
2, φ2(1) = 0,
0 = α3φ2 + α2φ1 + pw1w2
=⇒ Kox = −β
rφ1(z)x1
Workshop on Control of Distributed Parameter Systems – p. 13/19
Nonlinear closed-loop stability
I. Aksikas, J.J. Winkin and D. Dochain, 2007
{
ẋ(t) = A0x(t) + N0(x(t))
x(0) = x0 ∈ D(A0) ∩ F
• A0 dissipative and (I − λA0)−1 compact .
• F ⊂ R(I − λA0)
• limh→0+ d(F, x + hN0(x)) = 0, ∀x ∈ D(A)
• A0 + N0 strictly dissipative .
Workshop on Control of Distributed Parameter Systems – p. 14/19
Nonlinear closed-loop stability
I. Aksikas, J.J. Winkin and D. Dochain, 2007
{
ẋ(t) = A0x(t) + N0(x(t))
x(0) = x0 ∈ D(A0) ∩ F
• A0 dissipative and (I − λA0)−1 compact .
• F ⊂ R(I − λA0)
• limh→0+ d(F, x + hN0(x)) = 0, ∀x ∈ D(A)
• A0 + N0 strictly dissipative .
∀x0 ∈ D(A), x(t, x0) → ω(x0)= {x}
Workshop on Control of Distributed Parameter Systems – p. 14/19
Nonlinear closed-loop stability
Workshop on Control of Distributed Parameter Systems – p. 15/19
Nonlinear closed-loop stability
Under the assumptions:⋄
limh→0+
h−1d(F0, θ + hq) = 0, ∀θ ∈ F0
⋄Nonlinearity Lipshitz Constant < β
Workshop on Control of Distributed Parameter Systems – p. 15/19
Nonlinear closed-loop stability
Under the assumptions:⋄
limh→0+
h−1d(F0, θ + hq) = 0, ∀θ ∈ F0
⋄Nonlinearity Lipshitz Constant < β
=⇒ Asymptotic Stability
∀θ0 ∈ F0, θ(t, θ0) := Γ(t)θ0 −→ θe
Workshop on Control of Distributed Parameter Systems – p. 15/19
Numerical Simulations
Workshop on Control of Distributed Parameter Systems – p. 16/19
Numerical Simulations (cont.)
Workshop on Control of Distributed Parameter Systems – p. 17/19
Numerical Simulations (cont.)
Workshop on Control of Distributed Parameter Systems – p. 18/19
Concluding Remarks
• Conclusions⋄ LQ-Controller Design⋄ Application to fixed-bed reactor
• Perspectives : Extensions⋄ Fixed-bed reactor with diffusion⋄ Time-varying hyperbolic systems
Workshop on Control of Distributed Parameter Systems – p. 19/19
�f yellow Outline �f yellow Introduction�f yellow Introduction
�f yellow Introduction (cont.)�f yellow Introduction (cont.)�f yellow Introduction (cont.)
�f yellow Introduction (cont.)�f yellow Introduction (cont.)�f yellow Introduction (cont.)�f yellow Introduction (cont.)
�f yellow Problem Statement�f yellow Stability Result�f yellow Stability Result�f yellow Stability Result
�f yellow Optimal Control Design�f yellow Optimal Control Design
�f yellow Optimal Control Design (cont.)�f yellow Optimal Control Design (cont.)�f yellow Optimal Control Design (cont.)�f yellow Optimal Control Design (cont.)�f yellow Optimal Control Design (cont.)
�f yellow Application to Fixed-Bed Reactor{yellow Linearized Model}�f {yellow Optimal Control Design }�f {yellow Optimal Control Design }
yellow Nonlinear closed-loop stabilityyellow Nonlinear closed-loop stability
yellow Nonlinear closed-loop stabilityyellow Nonlinear closed-loop stabilityyellow Nonlinear closed-loop stability
yellow Numerical Simulationsyellow Numerical Simulations (cont.)yellow Numerical Simulations (cont.)�f yellow Concluding Remarks
