control delay-like system is globally null-controllable in finite time. The feedback stabilizing controller is designed via the solution of matrix Ric...

0 downloads 0 Views 128KB Size

ARTICLE doi: 10.2306/scienceasia1513-1874.2009.35.284

ScienceAsia 35 (2009): 284–289

H∞ optimal control of linear time-varying systems with time-varying delay via a controllability approach Piyapong Niamsupa , Vu N. Phatb,∗ a b ∗

Department of Mathematics, Chiang Mai University, Chiang Mai 50200, Thailand Department of Control and Optimization, Institute of Mathematics, VAST, Hanoi 10307, Vietnam Corresponding author, e-mail: [email protected] Received 26 Mar 2009 Accepted 24 Jun 2009

ABSTRACT: This paper addresses the H∞ optimal control problem for a class of uncertain linear time-varying delay systems. The interesting features here are that the system in consideration is non-autonomous, the state delay is timevarying, and the controllers to be designed satisfy some exponential stability constraints on the closed-loop poles. Based on the Lyapunov-Krasovskii functional method, we show that the H∞ optimal control problem for the system has a solution if some appropriate linear control delay-like system is globally controllable. KEYWORDS: robust control, exponential stability, time delay, Riccati equation, Lyapunov functional

INTRODUCTION In recent decades, considerable attention has been devoted to the problem of state estimation. When a priori information on external noises is not precisely known, the celebrated Kalman filtering scheme is no longer applicable. In such cases, H∞ filtering can be used 1–4 . With H∞ filtering the input signal is assumed to be energy bounded and the main objective is to minimize the H∞ norm of the filtering error system. Other norms introduced for systems with uncertainties are L2 and L1 , which have different physical meanings when used as performance indexes. Time delays are frequently encountered in various engineering systems such as aircraft, long transmission lines in pneumatic models, and chemical or process control systems because of the time taken for transmission of measurement information. As these delays may be the source of instability and serious deterioration in the performance of closedloop systems, the H∞ control problem of systems with time delays has received considerable attention from many researchers in the last decade. A significant new development in H∞ optimal control theory has been the introduction of state-space methods. This has led to a rather transparent solution to the standard problem of H∞ control theory, which is to find a feedback controller stabilizing a given system that satisfies some normed suboptimal level on perturbations/uncertainties (see, e.g., Refs. 5, 6). In the H∞ control for time-invariant delay systems, the corresponding methods make use of the Lyapunovwww.scienceasia.org

Krasovskii functional approach and the sufficient conditions are obtained via solving either linear matrix inequalities, or algebraic Riccati-type equations 7, 8 . However, this approach may not be readily applied for systems with time-varying parameters, which are frequently encountered in process control, filtering, and mobile communication systems. The difficulty is that the solution of a Riccati-type differential equation is, in general, not uniformly positive definite as is required for use in a Lyapunov-Krasovskii functional candidate. Hence the stability analysis becomes more complicated, and in particular when the system delay and uncertainties are also time-varying. Some results on the stabilization of linear time-varying (LTV) systems have been tackled in Refs. 6, 9, but without considering time delays. To find a H∞ controller for LTV systems, the state-space approach is used in Refs. 6, 10 to derive sufficient conditions for the H∞ control problem in terms of the solution of some Riccati differential equations. Based on the assumption of uniform controllability of the nominal control systems, some sufficient conditions for H∞ of LTV systems were obtained in Refs. 7, 11. To the best of our knowledge, this paper is the first to present a unified approach that addresses the problem of the H∞ control problem for a class of LTV systems subject to time-varying state delay, system uncertainties, and an external disturbance. We consider the time-varying case of time-varying delays and norm-bounded time-varying uncertainties in the state and input matrices. By using the LyapunovKrasovskii functional method, we show that the H∞

285

ScienceAsia 35 (2009)

control problem has a solution if the appropriate linear control delay-like system is globally null-controllable in finite time. The feedback stabilizing controller is designed via the solution of matrix Riccati-type equations. PRELIMINARIES The following notation will be used throughout this paper. R+ denotes the set of all non-negative real numbers, Rn denotes a n-dimensional Euclidean space with the scalar product h·, ·i, L2 ([t, ∞), Rn ) denotes the set of all strongly measurable L2 -integrable Rn -valued functions on [t, ∞), and I denotes the identity matrix. A matrix Q ∈ M n×n is called nonnegative definite (Q > 0) if hQx, xi > 0, for all x ∈ Rn . If for some c > 0 we have hQx, xi > ckxk2 for all x ∈ Rn , then Q is called positive definite (Q > 0), and A > B means A − B > 0. A matrix function Q(t) is uniformly positive definite (Q(t) >> 0) if ∃ c > 0 : hQ(t)x, xi > ckxk2 , ∀ (t, x) ∈ R+ × Rn . It is well known that if the matrix A is symmetric positive definite, then there is a matrix B such that A = B 2 and the matrix B is usually defined by B = 1 A 2 . Let BM + (0, ∞) denote the set of all symmetric non-negative definite matrix functions which are continuous and bounded on R+ , let BM U + (0, ∞) denote the set of all symmetric uniformly positive definite matrix functions which are continuous and bounded on R+ , and let C([a, b], Rn ) denote the set of all Rn -valued continuous functions on [a, b]. Consider the following uncertain LTV system with time-varying delay: x(t) ˙ = A(t)x(t) + A1 (t)x(t − h(t))

Definition 1 Linear control system (1), where w(t) = 0, is exponentially stabilizable if there exists a feedback control u(t) = K(t)x(t), such that the zero solution of the closed-loop delay system x(t) ˙ = [A(t) + B(t)K(t)]x(t) + A1 (t)x(t − h(t)), (2) is exponentially stable in the Lyapunov sense, i.e. ∃ N > 0, α > 0 : kx(t, φ)k 6 N kφke−αt , for all t > 0. In this paper, we consider the following H∞ optimal control problem with nonzero initial condition 8, 12 . Definition 2 Given γ > 0, the H∞ optimal control problem for the system (1) has a solution if there is a feedback control u(t)K(t)x(t) such that (i) the system (1), where w(t) = 0, is exponentially stabilizable, (ii) there is a number c0 > 0 such that R∞ kz(t)k2 dt 0 R∞ sup 6 γ, (3) c0 kφk2 + 0 kw(t)k2 dt where the supremum is taken over all initial states φ and non-zero admissible uncertainties w(t). In this case we say that the feedback control u(t) = K(t)x(t) exponentially stabilizes the system (1). We recall the concept of global controllability from Ref. 13 which is concerned with the possibility of steering any state to another state of the system in finite time. We will be considering the following linear time-varying control system, briefly denoted by [A(t), B(t)], x(t) ˙ = A(t)x(t) + B(t)u(t),

t ∈ R+ .

(4)

+

+ B(t)u(t) + B1 (t)w(t), t ∈ R , z(t) = C(t)x(t) + D(t)u(t), x(t) = φ(t), t ∈ [−h, 0],

(1)

where x ∈ Rn is the state, u ∈ Rm is the control, w ∈ Rp is the uncertain input, z ∈ Rq is the observation output, A(t), A1 (t), B(t), B1 (t), C(t), and D(t) are given matrix functions continuous and bounded on R+ , and φ(t) ∈ C[−h, 0] with the norm kφk = supt∈[−h,0] kφ(t)k. The time-delay function h(t) ∈ C[−h, 0] satisfies the condition 0 6 h(t) 6 h,

˙ h(t) 6 δ < 1,

∀ t ∈ R+ .

We say that the control u is admissible if u ∈ L2 ([0, s], Rm ) for every s > 0, and the uncertainty w is admissible if w ∈ L2 ([0, ∞), Rp ). Let xt be the segment of the trajectory of x(t) with the norm kxt k = sups∈[−h,0] kkx(t + s)k.

Definition 3 System (1) is globally null-controllable in finite time if for every initial state x0 , there exist a time T > 0 and an admissible control u(t) such that the solution x(t) of the system satisfies x(0) = x0 , x(T ) = 0. Proposition 1 (Klamka 14 ) Assume that the matrix functions A(t), B(t) are analytic on R+ . The system [A(t), B(t)] is globally null controllable in finite time if ∃ t0 > 0 : rank[M1 (t0 ), M2 (t0 ), . . . , Mn (t0 )] = n, (5) where M1 (t) = B(t) and Mk (t) = −A(t)Mk−1 (t) +

d Mk−1 (t), dt

for k = 2, . . . , n − 1. www.scienceasia.org

286

ScienceAsia 35 (2009)

Associated with the control system (4), we consider the following matrix Riccati equation P˙ (t)+AT (t)P (t) + P (t)A(t) T

− P (t)B(t)B (t)P (t) + Q(t) = 0.

Given γ > 0, µ = 1/(1 − δ), we set Aγ (t) = A(t) + B1 (t)B1T (t)/γ T + µA1 (t)AT 1 (t) − B(t)B (t),

(6)

Bγ (t) = [B(t)B T (t) − B1 (t)B1T (t)/γ 1

2 − µA1 (t)AT 1 (t)] .

Proposition 2 (Kalman et al 13 ) Assume that the system [A(t), B(t)] is globally null-controllable in finite time. Then for any matrix Q ∈ BM + (0, ∞), there is a solution P ∈ BM + (0, ∞) to (6). Proposition 3 (Cauchy matrix inequality) Let Q, S be symmetric matrices of appropriate dimensions and S > 0. Then 2hQy, xi − hSy, yi 6 hQS −1 QT x, xi,

∀ (x, y).

The main result is stated in the following. Theorem 1 Assume that for t > 0, B(t)B T (t) −

1 B1 (t)B1T (t) − µA1 (t)AT 1 (t) > 0, γ

and linear control system [Aγ (t), Bγ (t)] is globally null-controllable in finite time. Then the H∞ optimal control problem for the system (1) has a solution.

The proof of the above proposition is easily derived from completing the square.

The following lemma is needed for the proof of Theorem 1.

Proposition 4 For any symmetric matrix function A(t) bounded on R+ , there exists Q ∈ BM + (0, ∞) such that Q(t) − A(t) > 0.

Lemma 1 The H∞ optimal control problem for the system (1) has a solution if there exist matrix functions X, R ∈ BM U + (0, ∞) such that the following matrix inequality holds

Proof : The matrix Q(t) may be chosen as T ˙ X+A X + XA − X[BB T − 1/γB1 B1T

Q(t) = diag{q1 (t), q2 (t), . . . , qn (t)},

T − µA1 AT 1 ]X + C C + I + R 6 0,

t > 0. (8)

where qi (t) > max{|qi0 (t)|, 0} and The feedback control is

n

qi0 (t) = aii (t) +

1X 2 aij (t) + n − 1, 4

u(t) = −B T (t)X(t)x(t),

j6=i

for i = 1, . . . , n. From this it is straightforward to show that Q(t) − A(t) > 0. Proposition 5 (Lyapunov stability theorem 15 ) Consider the functional differential equation x˙ = f (t, xt ), with x(t) = φ(t) when t ∈ [−h, 0]. If there is a function V (t, xt ) and positive numbers λi , i = 1, 2, 3 such that the solution x(t) obeys λ1 kx(t)k2 6 V (t, xt ) 6 λ2 kxt k2 for t ∈ R+ , and V˙ (x(t)) 6 −λ3 kx(t)k2 , then the zero solution is asymptotically stable.

In this section we will omit the variable t of matrix functions if it does not cause any confusion. Consider the linear control system (1) where, as in Refs. 8, 12, we assume that

www.scienceasia.org

(9)

Proof : Using (9), we consider the following Lyapunov function for the closed-loop system (2), where w(t) = 0, K(t)x(t) = −B T (t)X(t)x(t): Z

t

kx(s)k2 ds.

V (t, xt ) = hX(t)x(t), x(t)i + t−h(t)

Since X >> 0, there is a positive number λ1 such that λ1 kx(t)k2 6 V (t, xt ),

t > 0.

On the other hand, since the matrix function X(t) is bounded on R+ and kx(t)k 6 kxt k, we have

MAIN RESULT

DT (t)[D(t) C(t)] = [I

t > 0.

0],

t > 0.

(7)

V (t, xt ) 6 ( sup kX(t)k + h)kxt k2 , t∈R+

t > 0,

and hence λ1 kx(t)k2 6 V (t, x) 6 λ2 kxt k2 ,

∀ t > 0, (10)

287

ScienceAsia 35 (2009)

where λ2 = supt∈R+ kX(t)k + h. Taking the derivative of V (t, xt ) along the solution x(t) of the closed-loop system, we have ˙ V˙ (t, xt ) = hX(t)x(t), x(t)i + 2hX(t)x(t), ˙ x(t)i 2 ˙ + kx(t)k − (1 − h(t)kx(t − h(t))k2 = h(X˙ + AT X + XA + I)x(t), x(t)i + 2hXBu(t), x(t)i + 2hXA1 x(t − h(t)), x(t)i ˙ − (1 − h(t)kx(t − h(t))k2 6 h(X˙ + AT X + XA + I)x(t), x(t)i − 2hXBB T Xx(t), x(t)i + 2hXA1 x(t − h(t)), x(t)i − (1 − δ)kx(t − h(t))k2 Using Proposition 1 we have

Applying the inequality (10) again gives kx(t, φ)k 6

p

V (0, x0 )/λ1 e−(λ3 /2λ2 )t 6 N kφke−(λ3 /2λ2 )t

p where N = λmax (X(0)) + h)/λ1 . The last inequality implies that the closed-loop system is exponentially stable, i.e., the system is exponentially stabilizable. To complete the proof of the lemma, it remains to show the γ-suboptimal condition (3). For this we consider the relation Z t [kz(s)k2 − γkw(s)k2 ] ds 0 Z th = kz(s)k2 − γkw(s)k2 0 Z t i + V˙ (s, xs ) ds − V˙ (s, xs ) ds, 0

2hXA1 x(t − h(t)), x(t)i − (1 − δ)kx(t − h(t))k2 6 µhXA1 AT 1 x(t), x(t)i.

where V˙ (t, xt ) is estimated as V˙ (t, xt ) 6 −λ3 kx(t)k2 − hC T Cx(t), x(t)i − hXBB T Xx(t), x(t)i

Therefore,

− 1/γhXB1 B1T Xx(t), x(t)i + 2hXB1 w(t), x(t)i.

V˙ (t, xt ) 6 h(X˙ + AT X + XA + I)x(t), x(t)i

(12)

− 2hXBB T Xx(t), x(t)i + µhXA1 AT 1 x(t), x(t)i. Taking the matrix inequality (8) into account, we have

Since V (t, xt ) > 0, we have Z t V˙ (s, xs ) ds = V (t, x(t)) − V (0, x(0)) 0

V˙ (t, xt ) 6 −hXB1 B1T Xx(t), x(t)i/γ − hXBB T Xx(t), x(t)i − hC T Cx(t), x(t)i − hRx(t), x(t)i. (11) Since hC T Cx(t), x(t)i > 0, hXBB T Xx(t), x(t)i > 0, hXB1 B1T Xx(t), x(t)i > 0, and R(t) >> 0, from (11) it follows that there is a number λ3 > 0 such that V˙ (t, x(t)) 6 −λ3 kx(t)k2 ,

∀ t ∈ R+ .

Therefore, by Proposition 5, the system is asymptotically stable. To determine the exponential factors, from (10) we have V˙ (t, xt ) 6 −(λ3 /λ2 )V (t, xt ),

Therefore, Z t Z th [kz(s)k2 − γkw(s)k2 ] ds 6 kz(s)k2 0 0 i − γkw(s)k2 + V˙ (s, xs ) ds + hX(0)x0 , x0 i. (13) Taking the estimation of V˙ (s, xs ) from (12) and using (7) for kz(t)k2 = h[C T C + XBB T X]x, xi,

∀ t > 0.

we obtain Z t Z th [kz(s)k2 − γkw(s)k2 ] ds 6 [−λ3 kx(s)k2

t > 0.

−hXB1 B1T Xx(s), x(s)i/γ + 2hXB1 w(s), x(s)i i −γhw(s), w(s)i ds + hX(0)x0 , x0 i.

Therefore, V (t, xt ) 6 V (0, x0 )e−(λ3 /λ2 )t ,

> −V (0, x(0)) := −hX(0)x0 , x0 i.

0

0

www.scienceasia.org

288

ScienceAsia 35 (2009)

Putting X(t) = P (t) + I and

Applying Proposition 3 gives 2hXB1 w, xi − γhw, wi 6 hXB1 B1T Xx, xi/γ. Then, Z t

[kzk2 − γkwk2 ] ds 6 −λ3

0

Z

t

kx(s)k2 ds

R(t) = [B(t)B T (t)−1/γB1 (t)B1T (t)−µA1 AT 1 ]+I, we see that the matrices X(t) >> 0 and R(t) >> 0 satisfy the matrix Riccati inequality (8) and hence the proof is completed by using Lemma 1.

0

+hX(0)x0 , x0 i 6 hX(0)x0 , x0 i, ∀ t ∈ R+ . Letting t → ∞ we finally obtain that Z ∞ kz(t)k2 − γkw(t)k2 ] dt 6 hX(0)x0 , x0 i, 0

and hence Z ∞ [kz(t)k2 dt 6 0 Z ∞ γ kw(t)k2 dt + (kX(0)k/γ)kφk2 . 0

Setting c0 = kX(0)k/γ, we note that kX(0)k 6= 0 because X >> 0, and hence c0 > 0. From the last inequality we have R∞ kz(t)k2 dt 0 R∞ 6 γ, c0 kφk2 + 0 kw(t)k2 dt for all φ and non-zero w(t) ∈ L2 ([0, ∞), W ). This completes the proof of the lemma. We are now in position to prove the main result. Proof of Theorem 1: Assume that the system [Aγ (t), Bγ (t)] is globally null controllable in finite time. Using Proposition 4, we find a matrix function Q ∈ BM + (0, ∞) such that Q(t) > A(t) + AT (t) + C T (t)C(t) + 2I.

(14)

By Proposition 2, the matrix Riccati equation T P˙ + AT γ P + P Aγ − P Bγ Bγ P + Q = 0,

(15)

has a solution P ∈ BM + (0, ∞). We can reformulate (15) as P˙ + AT (P + I) + (P + I)A − (P + I)[BB T

Remark 1 From Theorem 1, to verify that a solution of the H∞ control problem for system (1) exists, it suffices to check the global null-controllability of the linear control delay-like system [Aγ (t), Bγ (t)]. The stabilizing feedback control is defined by u(t) = −B T (t)[P (t) + I]x(t),

t ∈ R+ ,

where P (t) is a solution of the matrix Riccati equation (15). The problem of finding solutions of matrix Riccati equations is in general still complicated. However, some efficient approaches to solving this problem can be found, for instance, in Refs. 16, 17 and the references therein. Example 1 Consider (1) with h(t) = 0.25 sin2 t, and −1.5 − 0.5e−2 sin t 1 A(t) = , −1 −7/4 − 0.5e−2 sin t 0.5 0 1 A1 (t) = , B(t) = B1 (t) = , 0 0 0 0.5e− sin t −0.5e− sin t C(t) = , −0.5e− sin t 0.5e− sin t √ 1/√2 . D= 1/ 2 The assumption (7) holds as DT (t)D(t) = I and DT (t)C(t) = 0. Since δ = 0.5, for γ = 4, we have 0.25 0 BB T − 0.25B1 B1T − 2A1 AT = > 0, 1 0 0 1/2 Bγ = [BB T − 0.25B1 B1T − 2A1 AT 1] 0.5 = 0

0 , 0

T − B1 B1T /γ − µA1 AT 1 ](P + I) + Q − (A + A )

+ BB T − B1 B1T /γ − µA1 AT 1 = 0. Therefore, by taking (14) into account, we obtain P˙ + AT (P + I) + (P + I)A − (P + I)[BB T T − B1 B1T /γ − µA1 AT 1 ](P + I)C C

+ [BB T − B1 B1T /γ − µA1 AT 1 ] + 2I 6 0. www.scienceasia.org

Aγ = A − BB T + 0.25B1 B1T + 2A1 AT 1 −2 sin t −1 − 0.5e 1 , = −1 −0.25 − 0.5e−2 sin t and it is clear that both matrix functions Aγ and Bγ are analytic. Moreover, M1 (t) = Bγ (t), M2 (t) = −Aγ (t)Bγ (t)

ScienceAsia 35 (2009)

and it is easy to see that there exists t0 > 0 so that the condition (5), namely rank [M1 (t0 ), M2 (t0 )] = 2 holds. Thus, by Proposition 1, the linear control system [Aγ (t) , Bγ (t)] is globally null-controllable in finite time and by Theorem 1 the H∞ optimal control problem for the system has a solution. To find the feedback stabilizing control, we take 15/4 + e−2 sin t −0.5 Q= . −0.5 7/4 + e−2 sin t It is straightforward to show that Q ∈ BM + (0, ∞) and Q > A(t) + AT (t) + C T (t)C(t) + 2I. We can find the solution P (t) of Riccati equation (15) as 1 0 > 0. P = 0 0.5 Therefore, the feedback stabilizing control is u (t) = −B T (t)[P (t) + I]x(t) = − 2 0 x(t) = −2x1 (t) T where x(t) = x1 (t) x2 (t) . CONCLUSIONS In this paper, we have shown that the H∞ optimal control problem for LTV systems with time-varying delay has a solution if the appropriate linear control delay-like system is globally null-controllable in finite time. The feedback stabilizing controller is designed via the solution of matrix Riccati equations. Acknowledgements: This work was supported by the Thailand Research Fund, the Commission on Higher Education, and the Centre of Excellence in Mathematics, Thailand, and the National Foundation for Science and Technology Development, Vietnam.

289 4. van Keulen B (1993) H∞ Control for Distributed Parameter Systems: A State-Space Approach, Birkhauser, Boston. 5. Glover K, Doyle JC (1989) A state-space approach to H∞ optimal control. In: Nijmeijer H, Schumacher JM (eds) Three Decades of Mathematical Systems Theory, Springer, Berlin, pp 179–218. 6. Ravi R, Nagpal KM, Khargonekar PP (1991) H∞ control of linear time-varying systems: A state-space approach. SIAM J Contr Optim 29, 1394–413. 7. Niamsup P, VN Phat (2006) Stability of linear timevarying delay systems and applications to control problems. J Comput Appl Math 194, 343–56. 8. de Souza CE, Xi L (1999) Delay-dependent robust H∞ control of uncertain linear state-delayed systems. Automatica 35, 1313–21. 9. Phat VN, Niamsup P (2006) Stabilization of linear nonautonomous systems with norm-bounded controls. J Optim Theor Appl 131, 135–49. 10. Ichikawa A (2000) Product of nonnegative operators and infinite-dimensional H∞ Riccati equations. Syst Contr Lett 41, 183–8. 11. Phat VN, Vinh DQ (2007) Controllability and H∞ control for linear continuous time-varying uncertain systems. In: Differential Equations and Applications, vol 4, Nova Science, New York, pp 105–11. 12. Niculescu S-I (1998) H∞ memoryless control with an α-stability constraint for time-delay systems: an LMI approach. IEEE Trans Automat Contr 43, 739–43. 13. Kalman RE, Ho YC, Narenda KS (1963) Controllability of linear dynamical systems. In: LaSall LP (ed) Contributions to Differential Equations, vol 1, Wiley, New York, pp 716–29. 14. Klamka J (1991) Controllability of Dynamical Systems, Kluwer Academic, Dordrecht. 15. Hale JK, Verduyn Lunel SM (1993) Introduction to Functional Differential Equations, Springer-Verlag, New York. 16. Abou-Kandil H, Freiling G, Ionescu V, Jank G (2003) Matrix Riccati Equations in Control and Systems Theory, Birkhauser, Basel. 17. Laub AJ (1982) Schur techniques for solving Riccati equations. In: Hinrichsen D, Isidori A (eds) Feedback Control of Linear and Nonlinear Systems, Springer, Berlin, pp 165–74.

REFERENCES 1. Elsayed A, Grimble MJ (1989) A new approach to H∞ design of optimal digital linear filters. IMA J Math Contr Inform 6, 233–51. 2. Zames G (1981) Feedback and optimal sensitivity: Model reference transformations, multiplicative seminorms, and approximate inverses. IEEE Trans Automat Contr 26, 301–20. 3. Francis BA (1987) A Course in H∞ Control Theory, Springer-Verlag, Berlin. www.scienceasia.org

Our partners will collect data and use cookies for ad personalization and measurement. Learn how we and our ad partner Google, collect and use data. Agree & close