Robust Control of Time-delay Systems

Transcription

1 Robust Control of Time-delay Systems Qing-Chang Zhong Distinguished Lecturer, IEEE Power Electronics Society Max McGraw Endowed Chair Professor in Energy and Power Engineering Dept. of Electrical and Computer Engineering Illinois Institute of Technology Web: qzhong2/

2 Outline Notations and some preliminaries Standard H control problem Delay-type Nehari problem Controller implementation

3 Notations Given a matrix A, A T and A denote its transpose and complex conjugate transpose respectively. A stands for (A 1 ) when the inverse A 1 exists. G(s) = D +C(sI A) 1 B. = G (s) = [G( s )] = [ A B C D [ ] A C B D F l and F u are the commonly-used lower/upper linear fractional transformation (LFT). ]

4 Preliminaries Two operators Chain-scattering representation Homographic transformation

5 Completion operator A completion operator π h is defined for h 0 as: π h {G} =. [ A B Ce Ah 0 ] e sh [ A B C D ].= Ĝ(s) e sh G(s). 4 Impulse response G e sh G π h (G) shift Time (sec) h=

6 Truncation operator A truncation operator τ h is defined for h 0 as: τ h {G}. = [ A B C D ] e sh [ A e Ah B C 0 ].= G(s) e sh G(s) 4 Impulse response G τ h (G) Time (sec) h=

7 Representations of systems C l (M) (a) the left CSR Input-output rep. C r (M) (b) the right CSR

8 Chain-scattering representation [ ] M11 M For M = 12, the right and left M 21 M 22 chain-scattering representations are defined as: C r (M) =. [ M12 M 11 M21 1 M 22 M 11 M21 1 M21 1 M 22 M21 1 C l (M) =. [ M12 1 M12 1 M 11 M 22 M12 1 M 12 M 22 M12 1 M 11 provided that M 21 and M 12, respectively, are invertible. If both M 21 and M 12 are invertible, then C r (M) C l (M) = I. ] ]

9 Homographic transformations (HMT) The right HMT: H r (M,Q) = (M 11 Q +M 12 )(M 21 Q +M 22 ) 1 The left HMT: H l (N,Q) = (N 11 QN 21 ) 1 (N 12 QN 22 ) M Q N Q H r (M,Q) H l (N,Q)

10 Properties of the HMT Lemma H r satisfies the following properties: (i) H r (C r (M),S) = F l (M,S); (ii) H r (I,S) = S; (iii) H r (M 1,H r (M 2,S) = H r (M 1 M 2,S); (iv) If H r (G,S) = R and G 1 exists, then S = H r (G 1,R). Lemma Let Λ be any unimodular matrix, then the H control problem H r (G, K 0 ) < γ is solvable iff H r (GΛ, K) < γ is solvable. Furthermore, K 0 = H r (Λ, K) or K = H r (Λ 1, K 0 ).

11 Outline Notations and some preliminaries Standard H control problem Delay-type Nehari problem Controller implementation

12 Standard H problem of single-delay systems Given a γ > 0, find a proper controller K such that the closed-loop system is internally stable and Fl (P, Ke sh ) < γ. z w y P u 1 e sh I u K

13 Key steps to solve the SP h The SP h is solved via solving two simpler problems: Key steps: The one-block problem (OP h ): in the form of SP h but with P( ) = [ ] 0 I. I 0 An extended Nehari problem (ENP h ): in the form of NP h but minimising the H -norm of G β11 +e sh Q β. Step 1: reduce SP h to SP 0 + OP h Step 2: reduce OP h to ENP h Step 3: solve ENP h Step 4: recover the controllers

14 Reducing SP h to SP 0 + OP h z u 1 e sh I u C r (P) K w y z u 1 z u 1 1 e sh I u C r (P) G α C r (G β ) K w y w 1 y Delay-free problem 1-block delay problem G α is the controller generator of SP 0. G β is defined such that C r(g β ). = G 1 α. Gα and Cr(G β) are all bistable.

15 Reducing OP h to ENP h (I) Q α(s) = H r(c r(g β ),e sh K) = F l (G β,e sh K) = G β11 + G β12 Ke sh (I G β22 Ke sh ) 1 G β21 z 1 + G β12 u 1 e sh u I K(s) G β11 G β22 2 K h (s) w 1 G β21 y - Introducing a Smith predictor 2 (s) = π h {G β22 }. = Ĝ β22 + e sh G β22

16 Reducing OP h to ENP h (II) z 1 e sh u I 2 G β12 + Q β (s) G β11 Ĝ β22 K h (s) w 1 z 2 G β21 [. Q β = Hr( The OP h is now reduced to ENP h : Q α(s) = Gβ11 +e sh Q β < γ. G β12 0 Gβ21Ĝβ22(s) 1 G 1 β21 ],K h )

17 Solution to the SP h Solvability : H 0 dom(ric) and X = Ric(H 0 ) 0; J 0 dom(ric) and Y = Ric(J 0 ) 0; ρ(xy) < γ 2 ; γ > γ h, where γ h = max{γ : detσ 22 = 0}. u y Z - V 1 Q A+B2C1 B2 Σ12Σ 1 22 C 1 Σ 22 B1 V 1 = C1 I 0 γ 2 B1 Σ 21 C2Σ 22 0 I

18 Outline Notations and some preliminaries Standard H control problem Delay-type Nehari problem Controller implementation

19 The delay-type Nehari problem Given a minimal state-space realisation G β = [ ] A B C 0, which is not necessarily stable, and h 0, characterise the optimal value γ opt = inf{ Gβ (s)+e sh K(s) L : K(s) H } and for a given γ > γ opt, parametrise the suboptimal set of proper K H such that Gβ (s)+e sh K(s) L < γ.

20 The optimal value It is well known that this problem is solvable iff γ > γ opt. = Γe sh G β, (1) where Γ denotes the Hankel operator. The symbol e sh G β is non-causal and, possibly, unstable. The problem is how to characterise it.

21 Estimation of γ opt It is easy to see that γ opt G β (s) L because at least K can be chosen as 0. It can be seen from (1) that γ opt ΓGβ. Hence, ΓGβ γopt G β (s) L. (2) When γ G β (s) L, the matrix [ H = ] A γ 2 BB C C A has at least one pair of eigenvalues on the jω-axis.

22 Two AREs [ ] A B For a minimally-realised G β = having no C 0 jω-axis zero or pole, the following two AREs [ Lc I ] [ ] [ ] I [ ] Lo H c = 0, I Lo Ho = 0 L c I (3) always have unique stabilising solutions L c 0 and L o 0, respectively, where H c = [ ] A γ 2 BB 0 A, H o = [ ] A 0 C C A.

23 The formula for γ opt Theorem For [ a given minimally-realised ] transfer A B matrix G β = having neither jω-axis zero C 0 nor jω-axis pole, the optimal value γ opt of the delay-type Nehari problem is γ opt = max{γ : det ˆΣ 22 = 0}, where ˆΣ 22 = [ L c I ] Lo Σ[ I ], Σ = [ Σ11 Σ 12 Σ 21 Σ 22 ].= e Hh.

24 Parametrisation of K For a given γ > γ opt, all K(s) H solving the problem can be parametrised as [ ] I 0 K = H r ( W Z I 1, Q), where Q(s) H < γ is a free parameter and W 1 = [ Gβ I Z = π h {F u ( I 0 ],γ 2 G β )}, A+γ 2 BB L ˆΣ c 22 (Σ 12 +L oσ 11 )C ˆΣ 22 B C I 0 γ 2 B (Σ 21 Σ 11 L c) 0 I.

25 Representation in a block diagram The controller K consists of an infinite-dimensional block Z, which is a finite-impulse-response (FIR) block (i.e. a modified Smith predictor), a finite-dimensional block W 1 and a free parameter Q. z e sh u I K G β Z W 1 Q w y -

26 Example: G β (s) = 1 s a (a > 0) ˆΣ22 aγ ah aγ aγopt ah The contour ˆΣ The surface ˆΣ 22 with respect to ah and aγ 22 = 0 on the ah-aγ plane Since I L cl o = 1 4a 2 γ 2, there is Γ Gβ = 1 2a. As a result, the optimal value γ opt satisfies 0.5 aγ opt 1.

27 Outline Notations and some preliminaries Standard H control problem Delay-type Nehari problem Controller implementation

28 Controller implementation All the above control laws associated with delay systems include a distributed delay like ˆ h v(t) = e Aζ Bu(t ζ)dζ, 0 or in the s-domain, Z(s) = (I e (si A)h ) (si A) 1. The implementation of Z is not trivial because A may be unstable. This problem had confused the delay community for several years and was proposed as an open problem in Automatica. It was reported that the quadrature implementation might cause instability however accurate the implementation is. My investigation shows that: The quadrature approximation error converges to 0 in the sense of H -norm. Approximation error N=1 N=5 N= Frequency (rad/sec)

29 Numerical integration v(t) = It is well known that ˆ h 0 e Aζ Bu(t ζ)dζ v(t) v w (t) = h N ΣN 1 i=0 eia h N Bu(t i h N ), but what will happen if transferred into the s-domain using the Laplace transform? Z w (s) = h N ΣN 1 i=0 e i h N (si A) B Not strictly proper = No guarantee of stability!

30 A trivial result y(τ) y(t) p(t) = 1 t h/n t τ 0 t 0 h/n t ˆ h N 0 y(t τ)dτ = ˆ t t h N y(τ)dτ = y(t) p(t),

31 Approximation of Z v(t) = ˆ h 0 N 1 = i=0 N 1 i=0 N 1 = i=0 N 1 = i=0 e Aζ Bu(t ζ)dζ ˆ (i+1) h N i h N e ia h N Bˆ (i+1) h N e Aζ Bu(t ζ)dζ i h N e ia h N Bˆ h N 0 u(t ζ)dζ u(t i h N τ i)dτ i e ia h N Bu(t i h N ) p(t) = v f(t)

32 Hidden sampling-hold effect N 1 v w (t) = i=0 N 1 v f (t) = i=0 e ia h N Bu(t i h N ) h N e ia h N Bu(t i h N ) p(t) This is equivalent to the sampling-hold effect.

33 Implementation in the z-domain v h N ZOH Σ N 1 u i=0 ei h N A Bz i S (a) Z f (s) = 1 e s h N s N 1 i=0 e i h N (si A) B v (e A h N I)A 1 ZOH Σ N 1 i=0 ei h N A Bz i S u (b) Z f 0 (s) = 1 e h N s s e N h A I h A 1 N 1 i=0 e i h N (si A) B N

34 Implementation in the s-domain Z fǫ (s) = 1 e h N (s+ǫ) h e N A I N 1 A 1 Σ 1 e h N ǫ i=0 s/ǫ+1 e i h N (si A) B. lim N + Z fǫ (s) Z(s) = 0, (ǫ 0) Approximation error N=1 N=5 N= Frequency (rad/sec)

35 Rational implementation The δ-operator is defined as δ = (q 1)/τ, where q is the shift operator and τ is the sampling period. We have δ = eτs 1 τ because q e τs when τ 0. Define Φ = 1 τ I and then = (e τ(si A) I)Φ, si A = lim τ 0, e (si A)τ = (Φ 1 +I) 1

36 Z(s) = (I e (si A)h ) (si A) 1 = (I e (si A)τN )(si A) 1 B = (I (Φ 1 +I) N )(si A) 1 B. Since si A, Z can be approximated as Z r (s) = (I (Φ 1 (si A)+I) N )(si A) 1 B = (I (Φ 1 (si A)+I) 1 ) Σ N 1 k=0 (Φ 1 (si A)+I) k (si A) 1 B = Φ 1 Σ N k=1 (Φ 1 (si A)+I) k B = Σ N k=1 Πk Φ 1 B, where Π = (si A+Φ) 1 Φ.

37 xn x N 1 Π x 2 Π x 1 Π u b 1 Φ B u v r 1 Π = ( si A + Φ) Φ Π = (si A+Φ) 1 Φ In order to guarantee zero static error, Φ = ( h N 0 e Aζ dζ) 1.

38 Fast-converging rational implementation Define ˆ h N Φ = ( 0 e Aζ dζ) 1 (e A h N +I) Γ = (e τ(si A) I)(e τ(si A) +I) 1 Φ, τ = h/n Then, si A = lim τ 0 Γ, si A s=0 = Γ s=0 = A, Γ si A=0 = 0.

39 xn x N 1 Π x 2 Π x 1 Ξ B u v r 1 Π = ( si A + Φ) ( A si + Φ) 1 Ξ = 2( si A + Φ) Π = (Φ si +A)(sI A+Φ) 1, Ξ = 2(sI A+Φ) Implementation error δ operator discrete delay bilinear trans Frequency (rad/sec)

40 Summary Standard H control problem Delay-type Nehari problem Controller implementation

41