Rational Implementation of Distributed Delay Using Extended Bilinear Transformations

Transcription

1 Rational Implementation of Distributed Delay Using Extended Bilinear Transformations Qing-Chang Zhong School of Electronics University of Glamorgan United Kingdom

2 Outline Introduction Problem formulation Rational Implementation Stability and convergence of the implementation Numerical example Q.-C. ZHONG: RATIONAL IMPLEMENTATION OF DISTRIBUTED DELAY p. 2/29

3 What is distributed delay? A finite integral over the time: (1) v(t) = h 0 e Aζ Bu(t ζ)dζ, (h > 0). The equivalent in the s-domain: v Z u (2)Z(s) = (I e (si A)h )(si A) 1 B. It often appears in the controller of a delay system as part of finite-spectrum-assignment control law modified Smith predictor The problem is how to implement Z to guarantee that Z is stable even if A is unstable? Q.-C. ZHONG: RATIONAL IMPLEMENTATION OF DISTRIBUTED DELAY p. 3/29

4 Implementation in the z-domain v u h N ZOH Σ N 1 i=0 ei h N A Bz i S (a) Corresponding to 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) Cor. to Z f0 (s) = 1 e h N s s e h N A I h N A 1 N 1 i=0 e i h N (si A) B Q.-C. ZHONG: RATIONAL IMPLEMENTATION OF DISTRIBUTED DELAY p. 4/29

5 Implementation in the s-domain Z fǫ (s) = 1 e h N (s+ǫ) h e N A I 1 e h N ǫ s/ǫ A 1 Σ N 1 i=0 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) an extra parameter ǫ involving delay terms rational implementation? Q.-C. ZHONG: RATIONAL IMPLEMENTATION OF DISTRIBUTED DELAY p. 5/29

6 Rational implementation xn x N 1 Π x 2 Π x 1 Π u b 1 Φ B u v r Π = ( si A + Φ) 1 Φ Π = (si A + Φ) 1 Φ, Φ = ( h N 0 e Aζ dζ) 1. slowly convergent Q.-C. ZHONG: RATIONAL IMPLEMENTATION OF DISTRIBUTED DELAY p. 6/29

7 Keep points about implementations The low-pass property of distributed delay must be kept in the implementation; There is no unstable pole-zero cancellation in the implementation in order to guarantee the internal stability; The implementation itself must be stable to guarantee the stability of the closed-loop system; The implementation error should be able to be made small enough to guarantee the stability of the closed-loop system; The static gain of the distributed delay should be retained in the implementation to guarantee the steady-state performance of the system. Q.-C. ZHONG: RATIONAL IMPLEMENTATION OF DISTRIBUTED DELAY p. 7/29

8 Problem formulation Introduction Problem formulation Rational Implementation Stability and convergence of the implementation Numerical example Q.-C. ZHONG: RATIONAL IMPLEMENTATION OF DISTRIBUTED DELAY p. 8/29

9 Problem formulation How to find an approximation, which is stable fast-convergent and rational for the following finite-impulse-response block to meet the requirements on the previous slide? Z(s) = (I e (si A)h )(si A) 1 B. Q.-C. ZHONG: RATIONAL IMPLEMENTATION OF DISTRIBUTED DELAY p. 9/29

10 Rational Implementation Introduction Problem formulation Rational Implementation Stability and convergence of the implementation Numerical example Q.-C. ZHONG: RATIONAL IMPLEMENTATION OF DISTRIBUTED DELAY p. 10/29

11 The γ-operator The well-known γ-operator in digital and sampleddata control circles is defined as γ = 2 τ q 1 q + 1, where q is the shift operator and τ the sampling period. often used to digitizing a continuous-time transfer function also called the bilinear transformation, or the Tustin s transformation corresponds to the trapezoidal rule also connects to the (lower) linear fractional transformation F l and the (right) homographic transformation H r : γ = 2 τ F l( , q) = 2 τ H r( , q). Q.-C. ZHONG: RATIONAL IMPLEMENTATION OF DISTRIBUTED DELAY p. 11/29

12 Properties of the γ-operator [ ] q = H r (, τ γ) = 1+ τ 2 γ 1 τ γ. 2 Since q e τs when τ 0, e τs q 1 = 1 τ 2 γ 1 + τ 2 γ. Furthermore, γ holds the following limiting property: lim γ = lim τ 0 τ 0 2 τ e τs 1 e τs + 1 = s. γ can be regarded as an approximation of the differential operator p = d dt. Q.-C. ZHONG: RATIONAL IMPLEMENTATION OF DISTRIBUTED DELAY p. 12/29

13 Extended bilinear transformation Γ The γ-operator: Define γ = 2 τ q 1 q + 1 (3) Γ = (e τ(si A) I)(e τ(si A) + I) 1 Φ, with τ = h N and h N Φ = ( 0 e Aζ dζ) 1 (e A h N + I). Q.-C. ZHONG: RATIONAL IMPLEMENTATION OF DISTRIBUTED DELAY p. 13/29

14 Properties of Γ (i) the limiting property (4) si A = lim τ 0 Γ, (ii) the static property si A s=0 = Γ s=0 = A, (iii) the cancellation property Γ si A Γ si A=0 = 0. Due to the above properties, this Γ is able to bring a rational implementation to guarantee the stability of the closed-loop system and the steady-state performance. Q.-C. ZHONG: RATIONAL IMPLEMENTATION OF DISTRIBUTED DELAY p. 14/29

15 Approximation of Z From (3), we have e (si A) h N = (Φ Γ)(Φ + Γ) 1. Substitute this into Z, then (5) Z(s) = (I (Φ Γ) N (Φ + Γ) N )(si A) 1 B. Since Γ si A, Z can be approximated as (6) (7) Z r (s) = (I (Φ si + A) N (si A + Φ) N )(si A) 1 B = (I (Φ si + A)(sI A + Φ) 1 ) Σ N 1 k=0 (Φ si + A)k (si A + Φ) k (si A) 1 B = 2(sI A + Φ) 1 Σ N 1 k=0 (Φ si + A)k (si A + Φ) k B = Σ N 1 k=0 Πk ΞB, with Π = (Φ si + A)(sI A + Φ) 1, Ξ = 2(sI A + Φ) 1. Q.-C. ZHONG: RATIONAL IMPLEMENTATION OF DISTRIBUTED DELAY p. 15/29

16 Structure of Z r 1 xn x N 1 Π x 2 Π x Ξ B u v r Π = ( si A + Φ) 1 ( A si + Φ) Ξ = 2( si A + Φ) 1 Z r = Σ N 1 k=0 Πk ΞB bi-proper nodes Π plus a strictly proper node Ξ no zero-pole cancellation if all the nodes are stable then Z r is stable Z r converges to Z when N + Q.-C. ZHONG: RATIONAL IMPLEMENTATION OF DISTRIBUTED DELAY p. 16/29

17 Stability and convergence Introduction Problem formulation Rational Implementation Stability and convergence of the implementation Numerical example Q.-C. ZHONG: RATIONAL IMPLEMENTATION OF DISTRIBUTED DELAY p. 17/29

18 Stability of the nodes Denote an eigenvalue of A as σ + j ω. Then the corresponding eigenvalue of h N A is σ + jω with σ = h N σ and ω = h N ω. Theorem 1 The following conditions are equivalent: Z r, Π, Ξ or A Φ is stable; h N 0 e Aζ dζ is antistable; σ cosω + ω sinω σe σ > 0, ignoring the case when σ = 0 and ω = 0. Note: If all the eigenvalues of A are real, then each node Π or Ξ is stable for any natural number N. Q.-C. ZHONG: RATIONAL IMPLEMENTATION OF DISTRIBUTED DELAY p. 18/29

19 f(σ, ω) = σ cos ω+ω sin ω σe σ ω σ Q.-C. ZHONG: RATIONAL IMPLEMENTATION OF DISTRIBUTED DELAY p. 19/29

20 A sufficient condition for the stability of nodes The conditions in Theorem 1 are all satisfied, i.e., the nodes are stable, for any number N > N with h (8) N = 2.8 max λ i (A), i where is the ceiling function ω Q.-C. ZHONG: RATIONAL IMPLEMENTATION σ OF DISTRIBUTED DELAY p. 20/29

21 Convergence of the implementation Φ h N = h N ( h N 0 e Aζ dζ) 1 (e A h N + I) = A h N (I e A h N ) 1 (e A h N + I) 6 is antistable if N > N ω 0 2 I IV II III φ = c 1+e c 1 e c = 0.5 Im I IV II III σ Re Q.-C. ZHONG: RATIONAL IMPLEMENTATION OF DISTRIBUTED DELAY p. 21/29

22 Theorem 2 Denote the approximation error of Z r as E r = Z Z r. Then lim E r(s) = 0. N + This theorem indicates that there alway exists a number N such that the implementation is stable and, furthermore, the H -norm of the implementation error is less than a given positive value. According to the wellknown small-gain theorem, the stability of the closedloop system can always be guaranteed. Q.-C. ZHONG: RATIONAL IMPLEMENTATION OF DISTRIBUTED DELAY p. 22/29

23 Numerical example Introduction Problem formulation Rational Implementation Stability and convergence of the implementation Numerical example Q.-C. ZHONG: RATIONAL IMPLEMENTATION OF DISTRIBUTED DELAY p. 23/29

24 Numerical example Consider the simple plant ẋ(t) = x(t) + u(t 1) with the control law u(t) = (1 + λ d ) ( e 1 x(t) e ζ u(t ζ)dζ ) + r(t), where r(t) is the reference. The distributed delay is v(t) = 1 and the s-domain equivalent is 0 e ζ u(t ζ)dζ, Z(s) = 1 e1 s s 1. Q.-C. ZHONG: RATIONAL IMPLEMENTATION OF DISTRIBUTED DELAY p. 24/29

25 The implementation Z r Z r (s) = Σ N 1 k=0 ( 2 ǫs ) k 2ǫ 2 2ǫ+ǫs 2 2ǫ+ǫs with ǫ = 1 e 1 N. Since A = 1 has no non-real eigenvalues, Z r is always stable, even for N = Implementation error N=1 N=5 N= Frequency (rad/sec) Q.-C. ZHONG: RATIONAL IMPLEMENTATION OF DISTRIBUTED DELAY p. 25/29

26 Comparative studies 10 1 Implementation error δ operator discrete delay proposed Frequency (rad/sec) N = 5 Q.-C. ZHONG: RATIONAL IMPLEMENTATION OF DISTRIBUTED DELAY p. 26/29

27 System response r(t) = 1(t) x(t) the ideal response N=2 N= Time (sec) N = 1: unstable N = 2: stable but slightly oscillatory N = 5: very close to the ideal response λ d = 1 u = (1+λ d ) ( e 1 x + v ) +r v = Z r u All the stable responses guarantee the steady-state performance. Q.-C. ZHONG: RATIONAL IMPLEMENTATION OF DISTRIBUTED DELAY p. 27/29

28 Summary A rational implementation has been proposed for distributed delay based on the bilinear transformation. The implementation consists of a series of bi-proper nodes cascaded with a low-pass node. The H -norm of the implementation error approaches 0 when the number N of nodes goes to and the stability of the closed-loop system can always be guaranteed. The steady-state performance of the system is guaranteed. It does not involve any extra parameter to choose apart from the number N of the nodes. In particular, no parameter for a low-pass filter is needed to choose. Simulation examples and comparisons are given. Q.-C. ZHONG: RATIONAL IMPLEMENTATION OF DISTRIBUTED DELAY p. 28/29

29 Acknowledgment This work was supported by EPSRC under grant No. EP/C005953/1. Q.-C. ZHONG: RATIONAL IMPLEMENTATION OF DISTRIBUTED DELAY p. 29/29