OPTIMAL CONTROL OF A CLASS OF PARABOLIC

Transcription

1 OPTIMAL CONTROL OF A CLASS OF PARABOLIC TIME-VARYING PDES James Ng a, Ilyasse Aksikas a,b, Stevan Dubljevic a a Department of Chemical & Materials Engineering, University of Alberta, Canada b Department of Mathematics, King Abdelaziz University, Jeddah, KSA 7 th Workshop on control of Distributed Parameter Systems Wuppertal, Germany July 18-22, 2011

2 INTRODUCTION MOTIVATING EXAMPLES Catalytic tubular reactor Tc T D2 ξ = v (T in T(0, t)) ξ=0 CA D1 ξ = v (CA in CA(0, t)) ξ=0 r ξ T ξ = 0 ξ=l CA ξ = 0 ξ=l Figure: Catalytic reactor Many industrial processes, e.g methanol, ammonia and other petrochemicals Diffusion-Convection-Reaction Process Tubular reactor systems with catalyst deactivation, Loss of catalyst activity Time-varying rates of reaction = Parabolic time-varying PDEs

3 INTRODUCTION MOTIVATING EXAMPLES Crystal Growth Process Important industrial process utilized for the production of semi-conductor material in the electronics and microprocessor industry. Materials produced: Silicon (Si), Germanium (Ge). Temperature dynamics: Parabolic PDE with time-varying coefficients Convective transport term is time-varying due to the motion of the domain boundary. Figure: Crystal Process diagram

4 INTRODUCTION MOTIVATING EXAMPLES Parabolic partial differential equations (PDEs) with time-varying features represent an important class of models for reaction-diffusion-convection processes. e.g. Tubular and packed bed reactor systems with catalyst deactivation, Crystal growth and annealing type processes with time-varying spatial domains These time-dependent features play an important role in the system dynamics, and therefore must be incorporated into the model based controller design. * Approach: Evolution systems representation Operator differential Riccati equation

5 INTRODUCTION RELATED WORKS Nonautonomous PDEs I. Aksikas, J.F. Forbes, and Y. Belhamadia, Optimal control design for time-varying catalytic reactors: a Riccati equation based approach, Int. J. Control., 2009 I. Aksikas and J.F. Forbes, Linear quadratic regulator for time-varying hyperbolic distributed parameter systems, IMA. J. Mathematical Control and Information, P. Acquistapace, F. Flandoli, and B. Terreni, Initial boundary value problems and optimal control for nonautonomous parabolic systems, SIAM J. Cont. & Optim., A. Smyshlyaev and M. Krstic, On control design for PDEs with space-dependent diffusivity and time-dependent reactivity, Automatica, PDEs with time-varying spatial domains A. Armaou and P. D. Christofides, Robust control of parabolic PDE systems with time-dependent spatial domains, Automatica, P.K.C.Wang, Stabilization and control of distributed systems with time-dependent spatial domains, J. Optim. Theor. & Appl., 1990.

6 GENERAL MODEL PDE DESCRIPTION Let Ω be a bounded open set of R m with smooth boundary Ω. Consider the initial and boundary value problem of the form: z(ξ, t) + A(t)z(ξ, t) = f (ξ, t) t in Ω [0, T] z(ξ, 0) = z 0 (ξ) in Ω z(ξ, t) n = 0, on Ω [0, T] The family of operators A(t) is defined as: m A(t)z := a ij (ξ, t) 2 z m + b i (t) z + c(ξ, t)z ξ i,j=1 i ξ j ξ i=1 i (i) z(ξ, t) represents, for example, temperature or concentration, with initial distribution z 0 (ξ). (ii) a ij (ξ, t) describes the heterogeneous thermal conductivity or diffusivity. (iii) b i (t) is a convective transport coefficient (e.g. time-dependent fluid superficial velocity). (iv) c(ξ, t) is a linearized reaction term (e.g. due to catalyst deactivation).

7 GENERAL MODEL PDE OPERATOR PROPERTIES Assumptions: P1. For each t [0, T], the operator A(t) is strongly elliptic, i.e. m a ij (ξ, t)η i η j ε η 2, for η R m i,j=1 P2. The coefficients c(ξ, t) L 2 ([0, T], L 2 (Ω)), b i (t) C 1 ([0, T]) and a ij (ξ, t) are sufficiently Hölder continuous, i.e. a ij (ξ, t) a ij (ξ, s) L t s β for s, t [0, T], ξ Ω and constant L > 0 and β (0, 1]. P3. The function f (ξ, t) L 2 (Ω) satisfies: ( 1 f (ξ, t) f (ξ, s) dξ) 2 2 L t s β, 0 s < t T Ω {A(t)} t [0,T] forms a family of strongly elliptic operators which admit a family of eigenfunctions {φ n(t)} t [0,T] with corresponding family of eigenvalues {λ n(t)} t [0,T].

8 INFINITE-DIMENSIONAL SYSTEM REPRESENTATION NONAUTONOMOUS PARABOLIC EVOLUTION SYSTEM Under the properties of: strong ellipticity and continuity of a ij (ξ, t), b i (t) and c(ξ, t), the operator A(t) satisfies: 1. For every t [0, T], the resolvent R(λ; A(t)) exists for all λ with Reλ 0 and there exists a constant L 1 such that R(λ; A(t)) L 1 / λ ; 2. There exists constants L 2 and β (0, 1] such that (A(t) A(s))A(τ) 1 L 2 t s β for s, t, τ [0, T]. The initial and boundary value problem is represented as a non-autonomous evolution system on L 2 (Ω): dz(t) dt for 0 s < t T and z s L 2 (Ω). = A(t)z(t) + f (t), z(s) = z s

9 INFINITE-DIMENSIONAL SYSTEM REPRESENTATION TWO-PARAMETER SEMIGROUP The solution of the nonautonomous evolution system is expressed in the form of: t z(t) = U(t, s)z s + U(t, τ)f (τ)dτ s where U(t, s) is the two-parameter semigroup generated by the operator A(t). The operator A(t) is defined as: A(t) := λ n(t), ψ n(t) φ n(t) n=1 The operator A(t) : D(A(t)) L 2 (Ω) L 2 (Ω) is the infinitesimal generator of the two-parameter semigroup: U(t, s)z(s) := e µn(t) µn(s) z(s), ψ n(s) φ n(t), for 0 s t T n=1

10 INFINITE-DIMENSIONAL SYSTEM REPRESENTATION TWO-PARAMETER SEMIGROUP The operator U(t, s) satisfies the following identities: A1. U(t, t) = I, A2. U(t, s) = U(t, r)u(r, s) for 0 s r t T A3. U(t, s) is continuous on 0 s < t T, and: U(t, s) t A4. U(t, s) L 1, = A(t)U(t, s), and A5. A(t)U(t, s) L 2 (t s) 1, and U(t, s) s A6. A(t)U(t, s)a(s) 1 L 3 for constants L i > 0. = U(t, s)a(s)

11 INFINITE-DIMENSIONAL SYSTEM REPRESENTATION OPTIMAL CONTROL PROBLEM The finite-time horizon optimal control problem is given as the following: T ( min J(u) = min C(τ)z(τ) 2 + u(τ) 2) dτ + Qz(T), z(t) u u 0 where the functional J(u) is minimized over all trajectories of { ẋ(t) = A(t)x(t) + B(t)u(t) x(0) = x 0 (1) If Q 0, B( ) C([0, T; L(U; L 2 (Ω))) and C( ) C([0, T; L(Y, L 2 (Ω))), then the optimization problem has the unique minimizing solution u min (t) such that the optimal pair u min (t) C([0, T]; U) and z min (t) C 1 ([0, T]; X) C([0, T]; D(A(t)) are related by the feedback formula, u min t [0,T] (t) = B (t)π(t)z min (t) The operator Π(t) L(X) is the strongly continuous, self adjoint, nonnegative solution of the differential Riccati equation, Π(t) + A (t)π(t) + Π(t)A(t) Π(t)B(t)B (t)π(t) + C (t)c(t) = 0, Π(T) = Q

12 EXAMPLE: CRYSTAL GROWTH PROCESS PROCESS MODEL Model of temperature dynamics: z(ξ, t) t = κ 2 z z v(t) ξ2 ξ + b(ξ)u(t) ξ = 0 z z (0, t) = 0, ξ z(ξ, 0) = z 0 (ξ) (l(t), t) = 0 ξ f(ξ, t) w(ξ, t) Q(ξ, t) κ thermal conductivity constant, v(t) is the boundary velocity, and b(ξ)u(t) is the distributed heat input along the length of the rod. The material domain is time-varying due to the motion of the boundary at the material-fluid interface, ξ = l(t). ξ = l(t) Crystal Melt Control objective: Optimally stabilize the temperature around some desired nominal distribution. Crucible

13 EXAMPLE: CRYSTAL GROWTH PROCESS TIME-VARYING SPATIAL DOMAIN AND FUNCTION SPACE DESCRIPTION The time-varying spatial domain at some time t [0, T] is denoted Ω = (0, l(t)) with 0 < ξ < l(t) l max. Spatial domain evolution is considered as a sequence of subdomains: Ω j Ω j+1 Ω Let φ(ξ, t) denote a family of functions defined on the subdomains Ω j with: φ(ξ, t) = { φ(ξ) for ξ Ωj 0 for ξ Ω c j R L 2 (Ω j ) forms a family of function spaces which are precompact in L 2 (Ω) for all t [0, T], (see, Adams, 1975): L 2 (Ω j ) L 2 (Ω j+1 ) L 2 (Ω) Enables the use of single inner product, on L 2 (Ω) for functions defined on an arbitrary subdomain Ω: φ, φ L 2 (Ω) = Ω φ(ξ, t)φ(ξ, t)dξ = φ(ξ)φ(ξ)dξ + 0 dξ = φ, φ L 2 Ω Ω c (Ω)

14 EXAMPLE: CRYSTAL GROWTH PROCESS PDE OPERATOR PROPERTIES The PDE operator A(t) is defined as: A(t) := κ 2 z z v(t) ξ2 ξ The operator A(t) is strongly elliptic for each t [0, T]; For each t [0, T] the family of eigenfunctions {φ n(t)} [0,T] are: ( ( ) φ n(ξ, t) = B n(t)e 2 1 κ 1 v(t)ξ nπ cos l(t) ξ 1 ( )) v(t) nπ 2 κ 1 (nπ/l(t)) sin l(t) ξ with coefficients: ( ( ) ) 2 v(t) B n(t) = 1 + l(t) 2κ (nπ/l(t)) φ n(ξ, t) are orthonormal to the eigenfunctions: ψ n(ξ, t) = e κ 1 v(t)ξ φ n(ξ, t) of the adjoint operator A (ξ, t) for each t [0, T]. The corresponding family of eigenvalues are: ( ) nπ 2 λ n(t) = κ 1 v(t)2 κ 1 l(t) 2 2

15 EXAMPLE: CRYSTAL GROWTH PROCESS INFINITE-DIMENSIONAL SYSTEM REPRESENTATION The associated family of linear operators A(t) in L 2 (Ω) is defined as: with domain D(A(t)): D(A(t)) := { φ L 2 (Ω) : φ, A(t)z = A(ξ, t)z for z D(A(t)) φ ξ are a.c., A(t)φ L2 (Ω), and φ } ξ (0, t) = 0, φ (l(t), t) = 0 ξ Initial and boundary value control problem is represented as the nonautonomous parabolic initial value problem: dz dt = A(t)z(t) + B(t)u(t), z(0) = z 0, 0 s t < T The solution of the initial value problem is expressed in terms of the two parameter semigroup U(t, s), t z(t) = U(t, 0)z 0 + U(t, τ)b(τ)u(τ)dτ 0 with z(s) L 2 (Ω).

16 EXAMPLE: CRYSTAL GROWTH PROCESS INFINITE-DIMENSIONAL SYSTEM REPRESENTATION The operator A(t) : D(A(t)) L 2 (Ω) L 2 (Ω) generates the two-parameter semigroup U(t, s), 0 s t T expressed as: U(t, s)z(s) = n=1 { ( exp κ 0 (nπ) 2 t l(t) 2 s ) l(s) 2 1 4κ 0 ( tv(t) 2 sv(s) 2)} z(s), ψ n(s) φ n(t) One can note that in the case where the boundary motion ceases and Ω is constant, i.e. l(t) = l, v(t) = 0, the expression for U(t, s) becomes: { ( ) nπ 2 U(t, s)z(s) = exp κ 0 (t s)} z(s), φ φ = T(t s)z(s) l n=1 where T(t), t 0 is the C 0 -semigroup of operators on L 2 (Ω) which is generated by the standard heat equation.

17 EXAMPLE: CRYSTAL GROWTH PROCESS CONTROLLER FORMULATION Let consider the weighted inner product on L 2 (Ω), for z 1, z 2 D(A(t)), z 1, z 2 r = r(ξ, t)z 1 (ξ, t)z 2 (ξ, t)dξ Ω with weight function r(ξ, t) := exp( (v(t)/κ 0 )ξ). Note that φ n, φ m r = δ nm where δ nm = 1 if n = m and 0 otherwise. The differential Riccati equation can be written under the form: φ n(t), Π(t)φ m(t) r + A(t)φ n(t), Π(t)φ m(t) r + φ n(t), Π(t)A(t)φ m(t) r Π(t)B(t)B (t)π(t)φ n(t), φ m(t) r + C(t)φ n(t), C(t)φ m(t) r = 0 with φ n(t), Π(T)φ m(t) r = φ n(t), Qφ m(t) r.

18 EXAMPLE: CRYSTAL GROWTH PROCESS CONTROLLER FORMULATION In the case where B(t) = I and C(t) = I the differential Riccati equation becomes the system of infinitely many ordinary differential equations: Π nn(t) + 2λ n(t)π nn(t) + 1 Π 2 nn (t) = 0, Πnn(T) = Q The input u min is determined from the solution of the system of ODEs, and the t [0,T] optimal state trajectory is the mild solution of the state feedback system and is expressed as: ż = (A(t) B(t)B (t)π(t))z(t), z(0) = z 0 t z(t) = e µn(t) z 0, ψ n(0) φ n(t) U(t, τ) n=1 0 n Π nn(t) z 0 (τ), ψ n(τ) φ n(τ)dτ

19 NUMERICAL RESULTS HEAT INPUT AND TEMPERATURE DISTRIBUTION l(t) v(t) t Figure: Slab domain length l(t) and boundary velocity v(t). t u(t) z(ξ, t) t t closed-loop open-loop Figure: (Top) Optimal input profile u min (t) applied to the slab at input location ξ c = (Bottom) Total open and closed loop system energy.

20 NUMERICAL RESULTS CLOSED LOOP SYSTEM z(ξ, t) t ξ 3.0 Figure: Slab temperature evolution in the time-dependent spatial domain with diffusivity κ 0 = Input applied at ξ c =

21 SUMMARY CONCLUDING REMARKS The general two-parameter semigroup representation of a class of nonautonomous parabolic PDE has been presented. The optimal control formulation for the infinite-dimensional system representation is considered. A practical example of an crystal growth process is considered with PDE model defined on time-varying spatial domain. The infinite-dimensional system representation of the PDE is determined, and the explicit two-parameter semigroup expression is provided. The corresponding optimal control problem is considered and numerical results demonstrate the stabilization of the temperature distribution in the time-dependent region.