José C. Geromel. Australian National University Canberra, December 7-8, 2017

Transcription

1 Differential LMI in Optimal Sampled-data Control José C. Geromel School of Electrical and Computer Engineering University of Campinas - Brazil Australian National University Canberra, December 7-8, 217 1/32

2 Contents Main notation Main motivation Bellman s principle of optimality Sampled-data control Hybrid systems Hybrid systems general performance Linearization Numerical issue Example Conclusion 2/32

3 Main notation Real (R), nonnegative real (R + ) and natural numbers (N) The trace function is tr( ) For ψ(t) defined for all t the value ψ(τ ) is the limit of ψ(t) as t goes to τ from the left Set of bounded signals: L2 equipped with w 2 2 = w(t) 2 2 dt < l2 equipped with w 2 2 = k N w[k] 2 2 < 3/32

4 Main motivation Stability: Matrix A R n n is Hurwitz (asymptotically) stable if Re{λ i (A)} <, i = 1,,n Schur (asymptotically) stable if λ i (A) < 1, i = 1,,n Simple and useful relationship: For any h > A Hurwitz stable e Ah Schur stable 4/32

5 Main motivation Consider h > and a polytopic convex set A c R n n A }{{} d = generic! Robust stability (analysis): eah : A A }{{} c polytopic A A c }{{} Hurwitz e Ah A d }{{} Schur 5/32

6 Main motivation Robust stability (synthesis): Sampled-data control Find a state feedback gain matrix L ( ) h e Ah + e Aτ Bdτ L } {{ } Schur = A A c Nonlinear and nonconvex parameter dependence. Very hard to solve. Robust performance similar difficulty! 6/32

7 Main motivation The celebrated Lyapunov Lemma states that: A R n n is Hurwitz stable if and only if A P +PA <, P > A R n n is Schur stable if and only if A SA S <, S > Simple and important fact: For any h > P > s.t. A P +PA < = e A h Pe Ah P < 7/32

8 Main motivation Quadratic stability: Continuous-time polytopic system Sampled-data polytopic system A i P +PA i <, P >, i V c { Ṗ(t)+A i P(t)+P(t)A i <, i V c,t [,h) P() = P(h) = S > e A h Se Ah S <, S >, A A c 8/32

9 Main motivation Differential Linear Matrix Inequality (DLMI): General form arising in H control design Ṗ +F P +PF PJ G γ 2 I I P(), P(h) = LMI < Importance of Two-Point Boundary Value Problem (TPBVP) Specific solutions P(t) convert DLMI into LMI. Linearity with respect to the parameters. C. Sherer (1995) and C. Briat (213), among others! 9/32

10 Bellman s principle of optimality The celebrated Bellman s principle of optimality HJB equation (continuous-time) and dynamic programming (discrete-time) are the most important theoretical devices for dynamic systems optimization Hard to solve due to generality! Consider the general problem of cost evaluation subject to min u U f(x(t),u(t))dt ẋ(t) = F(x(t),u(t)), x() = x 1/32

11 Bellman s principle of optimality Consider the sequence {t k } k N. Bellman s principle of optimality states that { tk+1 } V(x(t k ),t k ) = min f(x,u)dt +V(x(t k+1 ),t k+1 ) u U t k where: The function V(x,t) : R n R + R is called cost-to-go! Assuming (x,u U) is optimal, adding terms min u U tk+1 f(x(t),u(t))dt = k N t k = V(x,) f(x(t),u(t) )dt 11/32

12 Bellman s principle of optimality Upper bounds to the optimal cost are easily generated! If f(x,u) >, (x,u) and f(,) = then V(x,t) is positive definite V(x(tk ),t k ) > V(x(t k+1 ),t k+1 ) for all k N The discrete-time sequence {x(tk )} k N converges! The cost-to-go function satisfies the HJB inequality V t + V x F(x,u) f(x,u), u U }{{} dv dt for t [t k,t k+1 ), k N. For time invariant systems and t k+1 t k = h, k N a stationary solution (if any) is determined by imposing adequate boundary conditions on the time interval t [, h). 12/32

13 Bellman s principle of optimality Existence of a stationary (sub)optimal solution { h } v(x) min f(x,u)dt +v(x(h)), x() = x R nx u U being optimal whenever equality holds. The (sub)optimal cost is V(x,) = v(x ). The (sub)optimal control is of the form u = u(x). A (sub)optimal solution can be determined by enforcing a particular function v(x). 13/32

14 Sampled-data control Consider the LTI system ẋ(t)=ax(t)+bu(t)+e c w c (t), x() = x y[k]=c d x(t k )+E d w d [k 1] z(t)= C c x(t)+d c u(t) x( ) R n x is the state wc ( ) R rc is the exogenous continuous-time input wd [ ] R r d is the exogenous discrete-time input (w d [ 1]!) y[ ] R n y is the measured output z( ) R n z is the controlled output u( ) R n u is the control input such that u U u(t) = u(t k ), t [t k,t k+1 ), k N 14/32

15 Sampled-data control Dynamic output feedback sampled-data controller ˆx[k] =ˆx[k 1]+ ˆBy[k] u[k]=ĉˆx[k 1]+ ˆDy[k] Initial condition ˆx[ 1] = Full order controller ˆx[ ] R n c with n c = n u +n x! Controller transfer function C (ζ) = Ĉ(ζI nc Â) 1ˆB + ˆD 15/32

16 Sampled-data control Sampled-data optimal control in a general framework Hybrid Systems + 16/32

17 Sampled-data control Sampled-data optimal control in a general framework Hybrid Systems + Bellman s Principle of Optimality 17/32

18 Hybrid systems Defining ψ = [x u ˆx ] R n ψ with n ψ = n x +n u +n c, the closed-loop sampled-data system can be rewritten as, Ichicawa & Katayama (21) ψ(t) =Fψ(t)+J c w c (t) z(t)=gψ(t) ψ(t k ) =Hψ(t k )+J dw d [k 1] valid in the time interval t [t k,t k+1 ) for all k N. Necessary and sufficient condition for asymptotic stability ψ(t k ) ψ(t k+1 ) ψ(t k+1), k N Performance index calculation 18/32

19 Hybrid systems State space matrices A B F =, G = [ C c D c ] E c, J c = } {{ } data and H = I nx ˆDC d Ĉ ˆBC d Â, J d = ˆDE d ˆBE d } {{ } controller The process starts at t = with ψ() = Hψ( ) +J }{{} d w d [ 1] ψ 19/32

20 Hybrid systems general performance Given γ R +, consider the function ρ γ (ψ()) = sup z 2 2 γ 2( w c w d 2 ) 2 w c L 2, w d l 2 Optimal H sampled-data control: For zero initial condition inf { γ 2 : ρ γ () } Optimal H2 sampled-data control: For r c +r d initial conditions r c+r d inf ρ (ψ l ()) l=1 2/32

21 Hybrid systems general performance Theorem 1: Let h R +, γ R + and ψ() R n ψ be given. The hybrid linear system is asymptotically stable and ρ γ (ψ()) ψ() Sψ() holds if and only if there exists S > such that γ 2 I rd > J d SJ d satisfying the two-point boundary value problem Ṗ +F P +PF +γ 2 PJ c J c P +G G subject to the initial and final conditions S P(), P(h) H ( S 1 γ 2 J d J d) 1H }{{} controller 21/32

22 H 2 performance Corollary 1: Let h R + be given. The hybrid linear system is asymptotically stable and the H 2 performance index equals the optimal solution to the convex programming problem J 2 = inf tr(j cp(h)j c )+tr(j dp()j }{{ d ) } P( ) controller subject to the differential linear matrix inequality Ṗ +F P +PF +G G satisfying the boundary condition [ ] P(h) H P() 1 > }{{} controller 22/32

23 H 2 performance Matrix S = P() > satisfies the strict Lyapunov inequality: h e F h H SHe Fh < S e F t G Ge Ft dt } {{ } which admits a solution if and only if He Fh is Schur stable. The optimal sampled-data H 2 control follows from inf tr(j c P(h)J c)+tr(j d P()J d) }{{} CONVEX controller,p( ) controller Notice that H and J d depend on the controller matrices 23/32

24 H performance Corollary 2: Let h R + be given. The hybrid linear system is asymptotically stable and the H performance index equals the optimal solution to the convex programming problem J = inf P( ),γ γ2 subject to the differential matrix inequality Ṗ +F P +PF +γ 2 PJ c J c P +G G satisfying the boundary condition P(h) H P() 1 J d > γ 2 I rd }{{} controller 24/32

25 H performance A solution exists provided that γ > is large enough The optimal sampled-data H control follows from J = inf controller,p( ),γ γ 2 : P(h) H P() 1 J d γ 2 I rd }{{} controller CONVEX > 25/32

26 Linearization Four square blocks partitioning, for all t [,h) leads to P(t) = [ X(t) V(t) ˆX(t) ] [ Y(t) U(t), P(t) 1 = Ŷ(t) Differential LMIs Boundary constraints expressed by LMIs One-to-one change of variables yields the optimal controller Full order controller nc = n x due to pole / zero cancelations! ] 26/32

27 Numerical issue Consider the DLMI Ṗ +L(P) <, t [,h) Interval [,h] with n h segments of length η = h/n h. Denoting t p = pη, the piecewise linear function ( t tp ) P(t) = P p +(P p+1 P p ), t [t p,t p+1 ] η with P p > is feasible if and only if for all p =,,n h 1. P p+1 P p +L(P p ) < η P p+1 P p +L(P p+1 ) < η 27/32

28 Example Example borrowed from Ichicawa & Katayama (21). [ ] [ ] [ ] 1 A =, B = E 1 c =, C 1 1 d = C c = [ 1 ] [, D c = 1 ], E d = 1 with h = 1.. The optimal H sampled-data 2 nd order controller C (ζ) =.3237ζ ζ ζ ζ ζ ζ imposes the minimum cost γ = /32

29 Example For h = 1e 4 the delta-operator applied to the optimal H 2 sampled-data 2 nd order controller C (ζ) = ζ ζ ζ ζ 3 2ζ ζ provides C # (s) = s 2.746s s s which is a good approximation of the optimal analog controller at low frequency. C(s) =.754s s s /32

30 Example Magnitude and phase diagrams of C # (s) and C(s) Magnitude (db) -5-1 Bode Diagram h=.1 h=.1 h=.1 h=.1 Opt. Contr Phase (deg) Frequency (rad/s) 3/32

31 Conclusion Open problems: Dynamic output feedback control: in the context of Markov Jump Linear Systems. Nonuniform sampling: Optimal decision rule for the determination of h k T, k N, preserving stability and improving performance. 31/32

32 Conclusion Thank you for your attention!!! 32/32