Optimal nonlinear control for time delay system. Delsys 2013

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1 . Optimal. (International nonlinear control for timeworkshop delay system Delsys November 22th, 2013) 1 / 21 Optimal nonlinear control for time delay system Liliam Rodríguez-Guerrero, Omar Santos and Sabine Mondié Automatic Control Department CINVESTAV-IPN Mexico International Workshop Delsys 2013 November 22th, 2013

2 Content Inverse optimality for delay free nonlinear systems. Problem for a class of nonlinear time delay systems. Inverse optimality for a class of nonlinear time delay systems. Dehydration process.. Optimal. (CINVESTAV) nonlinear control for time delay system November 22th, / 21

3 Inverse optimality for free nonlinear systems The dificulty in delay free nonlinear systems is to solve the Hamilton-Jacobi-Bellman (HJB) equation and to propose the appropriate functional. This problem is avoided by the approach known as inverse optimality (Freeman and Kokotović, 1996) using Control Lyapunov Functions (CLF). In this approach it is possible to obtain an optimal control law without solving the HJB s equation, by defining a specific performance index that depends on the proposed CLF. For applying this approach, it is necesary to prove first that the functional is a CLF.. Optimal. (CINVESTAV) nonlinear control for time delay system November 22th, / 21

4 Control Lyapunov Function (CFL) (Freeman and Kokotović,1996). Definition A positive definite continuously diifferentiable function V (x) is a Control Lyapunov Function (CFL) of the affi ne system ẋ(t) = f 0 (x) + f 1 (x)u (1) if there exist a control law u such that the time derivative along the solutions of system (1) satisfies dv (x) dt = x V (x) T ẋ(t) = ψ 0 (x) + ψ T 1 (x)u(t) < 0 (2) (1) where: ψ 0 (x) = x V (x) T f 0 (x) and ψ T 1 (x) = x V (x) T f 1 (x). Optimal. (CINVESTAV) nonlinear control for time delay system November 22th, / 21

5 From the previous definition of CLF, it is clear that: If ψ T 1 (x) = 0 when x = 0 (the solution has not converged yet). It is only necessary to prove that ψ 0 (x) < 0 in order to satisfy inequality (2). If this condition is satisfied the asymptotic stability of the system is guaranteed.. Optimal. (CINVESTAV) nonlinear control for time delay system November 22th, / 21

6 Inverse optimality approach Freeman and Kokotović proposed to solve the inverse optimality problem applying the CLF approah for nonlinear affi ne systems (1). Suppose that V (x) is a CLF for this system and the derivative along its trajectories is dv (x) dt The positive definite functions are q(x) = [ ] Ψ T 1 (x t )Ψ 1 (x t ) + = ψ 0 (x) + ψ T 1 (x)u(t) (1) [Ψ 0 (x t )] 2 + [ Ψ T 1 (x t)ψ 1 (x t ) ] 2, r(x) = [ 1 4 Ψ T 1 (x t )Ψ 1 (x t ) ] Ψ T 1 (x t)ψ 1 (x t ) + Ψ 0 (x t ) + [Ψ 0 (x t )] 2 + [ Ψ T1 (x t)ψ 1 (x t ) ], 2. Optimal. (CINVESTAV) nonlinear control for time delay system November 22th, / 21

7 Optimal control law The performance index is J = 0 [ ] q(x) + r(x)u T u dt The HJB s equation associated to the system (1) and to the performance index is ( ) dv (x) min u dt + q(x) + r(x)u T u = 0 (1) which guarantees asymptotic stability. If we guarantee that V (x) is a CLF it is posible to synthesize the optimal control law. We compute the first derivative with respect to u and we obtain the control law of the form: u = 1 2 Ψ 1 (x t ) r(x). Optimal. (CINVESTAV) nonlinear control for time delay system November 22th, / 21

8 Inverse optimality for a class of nonlinear time delay systems Inverse optimality is an open problem for time delay systems, so the main idea in this work is to extend this approach to a class of nonlinear time delay systems, those which have a stable nominal linear part and a nonlinear part which satisfies some properties. In contrast with others approaches, our proposal is constructive. In fact, it is based in the complete type functional approach (Kharitonov and Zhabko, 2003).. Optimal. (CINVESTAV) nonlinear control for time delay system November 22th, / 21

9 We consider the class of nonlinear systems with state delay where: ẋ(t) = A 0 x(t) + A 1 x(t h) + F (x t ) + Bu(t), (3) the state x(t) R n,with inicial condition x(θ) = ϕ(θ), θ [ h, 0], ϕ PC([ h, 0], R n ), and a constant delay h > 0. the nominal system matrices are A 0, A 1 R n n. the nominal system ẋ(t) = A 0 x(t) + A 1 x(t h) (4) is stable (if not, a preliminary stabilyzing control is applied). F (x t ) R n, F (x t ) = F (x(t), x(t h)) is a nonlinear function. It is known and satisfies the Lipschitz s condition F (x) α x(t) + β x(t h). B R n m, and the control law is u(t) R m.. Optimal. (CINVESTAV) nonlinear control for time delay system November 22th, / 21

10 Inverse optimality approach apply to nonlinear affi ne time delay systems Consider the system ẋ(t) = A 0 x(t) + A 1 x(t h) + F (x t ) + }{{}}{{} B u(t). f 0 (x t ) f 1 (x t ) We first consider consider the complete type functional (Kharitonov and Zhabko, 2003) associated to the nominal system (4), 0 V (x t ) = x T (t)u(0)x(t) + 2x T (t) U( h θ)a 1 x(t + θ)dθ (5) 0 + h 0 + h h x T (t + θ) [W 1 + (h + θ) W 2 ] x(t + θ)dθ 0 h x T (t + θ 1 )A T 1 U(θ 2 θ 1 )A 1 x(t + θ 2 )dθ 1 dθ 2.. Optimal. (CINVESTAV) nonlinear control for time delay system November 22th, / 21

11 The time derivative along the trajectories of system (3) is dv (x t ) dt = ω 0 (x t ) + 2 [F (x t ) + Bu(t)] T ω 1 (x t ), (3) where and ω 0 (x t ) = x T (t)w 0 x(t) + x T (t h)w 1 x(t h) + 0 h x T (t + θ)w 2 x(t + θ)dθ 0 ω 1 (x t ) = U(0)x(t) + U( h θ)a 1 x(t + θ)dθ. h. Optimal. (CINVESTAV) nonlinear control for time delay system November 22th, / 21

12 Equivalently, where: dv (x t ) dt = Ψ 0 (x t ) + Ψ T 1 (x t )u(t), (3) Ψ 0 (x t ) = ω 0 (x t ) + 2ω T 1 (x t )F (x t ) Ψ T 1 (x t ) = 2ω T 1 (x t )B.. Optimal. (CINVESTAV) nonlinear control for time delay system November 22th, / 21

13 After appropriate majorizations we get dv (x t ) dt η T E η, (3) where: [ η T = x(t) x(t h) 0 h S(θ)x(t + θ)dθ ], S(θ) = U( h θ)a 1 R n n, θ [ h, 0], E = W 0 ᾱi n W 1 βi n h s 2 W 2 (α + β) I n, where ᾱ = 2α U(0) + β U(0) + α, β = β U(0) + β, s = sup θ [ h,0] S(θ).. Optimal. (CINVESTAV) nonlinear control for time delay system November 22th, / 21

14 Suffi cient conditions Proposition: Consider the nonlinear delay system (3), suppose that the nominal linear delay system is stable and that the nonlinear function F (x t ) satisfies a Lipschitz s condition. If there exist positive definite matrices W i R n n, i = 0, 1, 2, which satisfy W = W 0 + W 1 + hw 2, and positive scalars α and β such that the matrix E is positive definite then the complete type functional V (x t ) given by (5) is a Control Lyapunov Function for system (3).. Optimal. (CINVESTAV) nonlinear control for time delay system November 22th, / 21

15 Illustrative example Consider the chemical refining process with transport lag (Ross, 1971), described by the model A 0 = A 1 = ẋ(t) = A 0 x(t) + A 1 x(t 1) + F (x t ) + Bu(t) , , B = with inicial condition x(θ) = ϕ(θ), θ [ h, 0], where ϕ(θ) = {0.1, 0.01, 0.01, 0.01}. The nonlinear function is given by F (x t ) = sin(x(t)) sin(x(t 1)),. Optimal. (CINVESTAV) nonlinear control for time delay system November 22th, / 21

16 control We compute { the optimal control law as u(t) = 1 Ψ 1 (x t ) 2 r (x ), Ψ 1(x t ) = 0 0, Ψ 1 (x t ) = x 1 (t) x 2 (t) x 3 (t) x 4 (t) time (s) 0.2 u 1 (t) u 2 (t) tim e (s). Optimal. (CINVESTAV) nonlinear control for time delay system November 22th, / 21

17 Dehydration process We want to apply our theoretical approach to dehydration process. In this process, the high consumption of energy for the temperature control, justifies the use of optimal control techniques. ẋ(t) = a 0 x(t) + a 1 x(t h) + f (x t ) + bu(t τ). Optimal. (CINVESTAV) nonlinear control for time delay system November 22th, / 21

18 In this process the distance between the heat source and the product induces a transport delay. In previous contributions, an experimental comparison betwen two control laws is reported (Santos and et al, 2012) and (Rodríguez-Guerrero and et al, 2012). A linear optimal control law with delay compensation and an industrial PID controller were applied to the dehydration process. The performance, in terms of power consumption of the optimal control law outperformed the PID controller.. Optimal. (CINVESTAV) nonlinear control for time delay system November 22th, / 21

19 Future work Our current work includes the synthesis of the optimal control law using inverse optimality approach and its application to a dehydration process. Our objective is to study the power consumption and its effects on the product quality : lycopene phenols vitamin C color in the case of tomato. We want to minimize power consumption and loss of nutrients using variational calculus techniques.. Optimal. (CINVESTAV) nonlinear control for time delay system November 22th, / 21

20 References Ross, D. W. Controller desing for time lag systems via a quadratic criterion, IEEE Transaction on Automatic Control 16 (6) (1971) Kharitonov, V.L. and A.P. Zhabko, Lyapunov-Krasovskii approach for robust stability of time delay systems, Automatica, 39:15-20, Santos, O., Rodríguez-Guerrero, L. & López-Ortega, O. Experimental results of a control time delay system using optimal control Optimal Control, Applications and Methods, Wiley Inter-Science. 2012, 33 (1): Rodríguez-Guerrero L., López-Ortega O. & Santos O. Object-oriented optimal controller for a batch dryer system, International Journal of Advanced Manufacturing Technology. 2012, 58 (1-4): Freeman, R. A., Kokotović, P. (1996). Robust Nonlinear Control Design.State-Space and Lyapunov Techniques.Birkhäuser.. Optimal. (CINVESTAV) nonlinear control for time delay system November 22th, / 21

21 Thank you. Optimal. (CINVESTAV) nonlinear control for time delay system November 22th, / 21