Prediction-based adaptive control of a class of discrete-time nonlinear systems with nonlinear growth rate

www.scichina.com info.scichina.com www.springerlin.com Prediction-based adaptive control of a class of discrete-time nonlinear systems with nonlinear growth rate WEI Chen & CHEN ZongJi School of Automation Science and Electrical Engineer, Beijing University of Aeronautics and Astronautics, Beijing 100083, China Adaptive control of discrete-time parametric-strict-feedbac nonlinear systems will be studied in this paper. To the best nowledge of the authors, almost all of the existing results for such systems need Lipschitz condition. The main purpose of this paper is to tae a step towards nonlinear growth rate for second-order systems. The control law is designed based on least squares (LS) algorithms and on recursive adaptive predictors. Global stability and tracing performance are established for the closed-loop systems. adaptive nonlinear control, discrete time, global stability, prediction, nonlinear growth rate 1 Introduction In the past several decades, adaptive control of linear systems have received rather complete results. Efficient methods for both analysis and design have been well developed. The well-nown analytical methods are the so-called Key Technical Lemma 1] and linear time-variant method 2]. The former is powerful to gradient control algorithms, and the latter plays the ey role in the analysis of LS-based control system. Although most practical systems are inherently nonlinear, adaptive control of such systems was not seriously considered until recent years. The methods for linear systems can be directly applied to a class of nonlinear systems with the nonlinearities having linear growth rate. However, when dealing with the systems with essential nonlinearities, the traditional methods failed. In spite of the difficulities, nonlinear system control still attracts much attention due to its theoretical and practical importance. For continuous-time nonlinear systems, many novel techniques Received January 31, 2007; accepted June 12, 2007 doi: 10.1007/s11432-008-0012-6 Corresponding author (email: weichen@buaa.edu.cn) Supported by the National Natural Science Foundation of China (Grant Nos. 60204010 and 60474499) and the Aviation Foundation of China (Grant No. 00E51083) Sci China Ser F-Inf Sci May 2008 vol. 51 no. 5 524-534

have been proposed and can be successfully used in adaptive control design regardless of the growth rate of the nonlinearities, such as feedbac-linearization methods 3], bacstepping and nonlineardamping approaches 4], etc. Unfortunately, the existing continuous-time methods are not applicable to the discrete-time case, due to some inherent difficulties in discrete-time models, as described in refs. 5, 6]. The Lyapunov design technique, an extremely useful tool in the continuous-time case, is of little use in the discrete-time case, because the increments of the parameters do not appear linearly in the increments of Lyapunov functions. On the other hand, unlie the continuous-time control systems, the control signal in the discrete-time case is implemented discontinuously and the system could not be tuned in time. The earliest effort to remove the linear growth constraints in discrete-time case was made by ref. 5], where the stability of a first-order discrete-time deterministic adaptive system is analyzed. Later, ref. 7] studied the global stability and instability of a class of LS-based discrete-time deterministic adaptive control systems. By studying the critical stability of a class of nonlinear stochastic control systems, ref. 8] found the limatations of adaptive control for discrete-time nonlinear systems. As a consequence, ref. 8] also shows that in general it is impossible to have global stability results for stochastic adaptive control systems when the nonlinear function is a high order polynomial of its variables, which means that adaptive control design based on Weierstrass approximation may not be feasible in general. In subsequent research, ref. 9] studies the nonlinear models with multi-unnown parameters. A particular polynomial is introduced to determine the conditions under which a typical class of discrete-time nonlinear systems with uncertainties in both parameters and noises is not stabilizable by feedbac. To be worthy of being mentioned, a very interesting result is established in ref. 10] for the following affine nonlinear models: y t+1 = α τ f(φ t ) + β τ g(φ t )u t + w t+1, (1) where φ t = y t,,y t p+1,u t,,u t q+1 ]. f( ) and g( ) are called additive nonlinearity and multiplicative nonlinearity, respectively. It is shown that unlie the additive part, the multiplicative part is allowed to have arbitrarily fast nonlinear growth rate. Refs. 6, 11] studied adaptive control of discrete-time parametric-strict-feedbac nonlinear systems for deterministic case and stochastic case, respectively. Global stability and tracing performance were established under Lipschitz condition in both papers. How to relax such restriction is an interesting problem worthy of being studied. Ref. 12] studied output-feedbac tracing control for the deterministic case, global results are achieved for general nonlinearities by using a certain update law. What is the result if least-squares algorithm is used as the update law? As pointed in ref. 6], global boundedness and convergence are ensured in the special case when all the nonlinearities are globally Lipschitz if using the projection or the least-squares algorithm. On the other hand, as the results in ref. 8] show that 4 is a critical point of the nonlinear growth rate for the stability of input-output discrete-time adaptive nonlinear control. That means we could not go further when dealing with the high-order case. Although the upper bound of the nonlinear growth rate in this paper may not be critical, the methods and results proposed here may provide a start towards that end. The remainder of the paper is organized as follows. In section 2, the main results are presented. First, the problem to be considered is formulated, then the parameter estimation algorithm, WEI Chen et al. Sci China Ser F-Inf Sci May 2008 vol. 51 no. 5 524-534 525

predictor, and control design are given. Finally in this section, the stability results and tracing performance are shown. Section 3 is devoted to the proof of the main theorem, and some concluding remars are made in section 4. 2 Main results 2.1 Problem formulation Consider the following parametric-strict-feedbac control system x 1 ( + 1) = x 2 () + θ τ α 1 (x 1 ()), x 2 ( + 1) = u() + θ τ α 2 (x 1 (),x 2 ()), y() = x 1 (), where y(), u() denote the system output and input, respectively, x() = (x 1 (), x 2 ()) τ is measurable state vector, θ R d is unnown parameter vector, and α i ( ) (i = 1,2) are nown nonlinear real-valued functions. The control objective is to design a feedbac control u() at time, such that the system output tracs the reference signal y () 0. In order to analyze the control problem, we introduce the following condition. A1) There exist constants L 1, b i such that α i (x) α i (z) L 1 x z bi, x,z R i, i = 1,2. Remar 1. If b i 1, i = 1,2, Condition A1) will become the linear growth condition, which has been studied in ref. 11]. Therefore, we assume b i > 1, i = 1,2 in this paper. 2.2 Parameter estimation First, rewrite eq. (2) into a compact form { x( + 1) = Ax() + bu() + Φ θ, where y() = Cx(), ( ) ( ) x 1 () α τ 1 x() =, Φ = (x 1()), (4) x 2 () α τ 2(x 1 (),x 2 ()) ( ) ( ) 0 1 0 A =, b =, C = (1 0). (5) 0 0 1 Next, use the matrix version of LS algorithm to estimate θ θ +1 = θ + P +1 Φ τ (x( + 1) ˆx( + 1)), (6) P +1 = P P Φ τ Q Φ P, P 0 = αi, 0 < α 1, (7) Q = I + Φ P Φ τ ], (8) ˆx( + 1) = Ax() + bu() + Φ θ, (9) where (θ 0,P 0,Φ 0 ) are the initial conditions and P R d d. (2) (3) 526 WEI Chen et al. Sci China Ser F-Inf Sci May 2008 vol. 51 no. 5 524-534

2.3 Predictor and controller design For the purpose of control design, we first put (2) into the following input-output form where without causing any confusion y( + 2) = u() + θ τ (α 1 ( + 1) + α 2 ()), (10) α 1 () = α 1 (x 1 ()), α 2 () = α 2 (x 1 (),x 2 ()). (11) If α 1 ( + 1) were nown at time, we see from eq. (10) that the certainty equivalence adaptive tracing control would be u() = θ τ (α 1 ( + 1) + α 2 ()). However, at time, x 1 ( + 1), and hence, α 1 ( + 1) are unnown. A natural way is to use their prediction values instead. We adopt the following adaptive predictor presented in ref. 11] ˆx 1 ( + 1 ) = x 2 () + θ τ α 1(x 1 ()), (12) ˆα 1 ( + 1 ) = α 1 (ˆx 1 ( + 1 )). (13) Then, at time, the adaptive control law can be defined as u() = θ τ (ˆα 1 ( + 1 ) + α 2 ()). (14) By applying this controller to eq. (10), the closed-loop system is where y( + 2) = θ τ (α 1 ( + 1) + α 2 ()) θ τ (ˆα 1 ( + 1 ) + α 2 ()) = θ τ α 1 ( + 1) θ τ ˆα 1 ( + 1 ) + θ τ α 2 (), (15) θ = θ θ. (16) Remar 1. As mentioned in ref. 11], by transforming eq. (2) into eq. (10), the formula of the optimal controller can be written immediately, and the necessity to predict the state is obvious. 2.4 Stability and tracing performance For the case d = 1, the stability and tracing performance of the closed-loop system can be established as the following theorem. Theorem 1. Consider the adaptive control system (6) (9) and (12) (15) with d = 1. Let Condition A1) be satisfied. Further, suppose b 2 1 b 2 < 4 and b 1 b 2 < 2. Then, the closed-loop system is globally stable, and y 0. Remar 2. Although we only consider the two-order systems in this paper for simplicity, the analysis given here is also valid for the high-order case. However, as we can see later from the proof of Theorem 1, it will become much harder to remove the linear growth rate condition for high-order nonlinear systems. This coincides with the results in ref. 9]. Further research needs to be done in this area. 3 Proof of the main results Since the proof of the main result is somewhat tedious, an outline of the proofs and some other explanations will be given first for a clear understanding. WEI Chen et al. Sci China Ser F-Inf Sci May 2008 vol. 51 no. 5 524-534 527

The ey technical lemma to yield the desired result is Lemma 1, which establishes boundedness of a certain nonlinear recursions and may be regarded as a nonlinear extension of the well-nown Bellman-Gronwall Lemma. The proof of Theorem 1 is divided into three cases as shown later. Case 1 is trivial, Case 2 is dealt with by Lemma 6, and Lemma 1 is adopted in Case 3. The purpose of Lemmas 2 5 is to get the relationship (17) in Lemma 1. It is worthy of being mentioned that Lemmas 2 4 hold for all d 1, and Lemmas 5 6 for d = 1. Lemma 1 8]. Let {S t } and {C t } be two positive nondecreasing sequences such that t Si L S t+1 C t + δ i, t 0, (17) S M i=1 i and that C t = O(S t ), where L 0,δ i 0 and δ i=1 i <. If L 2 < 4M, then { S t+1 } S t is bounded and S t = O(C t ). The following two lemmas present some properties of the parameter estimate. Lemma 2. For the parameter estimate eqs. (6) (9), we have the following properties: 1) Φ (θ +1 θ ) 2 Φ θ 2 ; 2) Φ +1 (θ +1 θ ) 2 +2 1] Φ θ 2. Proof. By eqs. (3), (6), and (9), it can be seen that Then, and θ +1 θ = P +1 Φ τ Φ θ. (18) Φ (θ +1 θ ) = (Φ P +1 Φ τ )Φ θ, (19) Φ +1 (θ +1 θ ) = Φ +1 P +1 Φ τ Φ θ = Φ +1 P 1 2 +1 P 1 2 +1 Φ τ Φ θ. (20) From eqs. (19) and (20), it can be derived respectively that and Φ (θ +1 θ ) λ max (Φ P +1 Φ τ ) Φ θ, (21) Φ +1 (θ +1 θ ) λ 1 2 max (Φ +1 P +1 Φ τ +1 ) λ 1 2 max (Φ P +1 Φ τ ) Φ θ. (22) In order to obtain the desired result, we need to estimate λ max (Φ P +1 Φ τ ) and λ max (Φ +1 P +1 Φ τ +1). First, by (7) we have Next, multiplying P +1 on both sides of (23) yields P +1 = P + Φ τ Φ. (23) P +1 P = I P +1 Φ τ Φ. Taing determinant on both sides of the above equation, we can see that i.e., = I P +1Φ τ Φ = I Φ P +1 Φ τ 1 λ max (Φ P +1 Φ τ ), λ max (Φ P +1 Φ τ ) 1 1. (24) +1 528 WEI Chen et al. Sci China Ser F-Inf Sci May 2008 vol. 51 no. 5 524-534

i.e., Similarly, starting with eq. (23) yields +1 = I + Φ P Φ τ 1 + λ max (Φ P Φ τ ), (25) λ max (Φ P Φ τ ) +1 1. (26) Finally, combining eqs. (21) and (24), (22), (24) and (26), we can obtain the desired results 1) and 2), respectively. Lemma 3. For the parameter estimate eqs. (6) (9), we have where θ is defined by eq. (16) and i.e., Proof. Φ θ 2 +1 A 0 δ, δ = tr(p P +1 ). (27) By eqs. (3), (6), (9), and (23), it can be easily shown that By eq. (28), we can obtain +1 θ +1 = P θ = = P θ 0 0, P θ = P P 0 θ 0. (28) Φ θ 2 = Φ P P 0 θ 0 2 λ max (Φ P 2 Φ τ ) P 0 θ 0 2 = A 0 λ max (Φ P 2 Φ τ ), (29) where A 0 = P 0 θ 0 2. To complete the proof, it suffices to prove λ max (Φ P 2 Φ τ ) δ. (30) For this purpose, on one hand, taing trace on both sides of eq. (7) yields i.e., tr(p P +1 ) = tr(p Φ τ Q Φ P ) λ min (Q )tr(φ P 2 Φ τ ), tr(φ P 2 Φ τ ) On the other hand, by eqs. (8) and (25) we now that λ max (Q δ λ min (Q ) = δ λ max (Q ). (31) ) Finally, (30) follows directly from (31) and (32). The following corollary immediately follows from Lemmas 2 3. Corollary 1. For the parameter estimate eqs. (6) (9), we have 1) Φ (θ +1 θ ) 2 +1 A 0 σ,. (32) QED WEI Chen et al. Sci China Ser F-Inf Sci May 2008 vol. 51 no. 5 524-534 529

2) Φ +1 (θ +1 θ ) 2 1 ( 2 σ +2 ) 2 +1 + 1 ( 1 ε 2 A2 0 σ chosen arbitrarily small, and Proof. σ = max{δ,β }, First of all, we show that β = +1 +1 )2, where ε > 0 can be. (33) ε σ <. (34) =1 By the Pringsheim theorem 13], we now that β i <. Also, from eq. (27), i=0 δ i <. Hence, (34) holds, which means that σ < 1 when is sufficiently large. Without loss of generality, assume that σ < 1 in the following. Result 1) is obvious from Lemmas 2 4 and eq. (33). For result 2), we have +2 1 = +2 = +2 +2 ε +2 1 ε = β +1 i=0 +2 1 ε. Noticing ab 1 2 a2 + 1 2 b2, result 2) can be derived from Lemma 2 and the above analysis. Lemma 4. Consider the adaptive control system described by eqs. (6) (11) and (12) (13). If Condition A1) is satisfied, then x 1 ( + 1) 2 +1 K {σ ]2 ] b 1 } + σ, (35) 1 ε x 2 ( + 1) 2 +1 K {σ 1 ε where σ is defined in eq. (33), and Proof. b 1 ]2b 1 + σ ] b1b 1 } + 1, (36) = max{b 1,2}. (37) First, let us introduce the following notations for prediction errors: By eqs. (2) and (15), we have x 1 ( + 2) = y( + 2) x 1 ( + 1 ) = x 1 ( + 1) ˆx 1 ( + 1 ), (38) α 1 ( + 1 ) = α 1 ( + 1) ˆα 1 ( + 1 ). (39) = θ τ α 1 ( + 1) θ τ +1 α 1( + 1)] + θ τ +1 α 1( + 1) θ τ α 1( + 1)] + θ τ α 1 ( + 1) θ τ ˆα 1 ( + 1 )] + θ τ α 2 () = θ τ +1α 1 ( + 1) + (θ +1 θ ) τ α 1 ( + 1) + θ τ α 1( + 1 ) + θ τ α 2 (). (40) Therefore, from eqs. (4) and (40) it follows that x 1 ( + 2) 2 4 θ τα 2 () 2 + θ τ+1α ] 1 ( + 1) 2 + (θ +1 θ ) τ α 1 ( + 1) 2 + θ τ α 1( + 1 ) 2 530 WEI Chen et al. Sci China Ser F-Inf Sci May 2008 vol. 51 no. 5 524-534

] 4 Φ θ 2 + Φ +1 θ+1 2 + Φ +1 (θ +1 θ ) 2 + θ τ α 1( + 1 ) 2. (41) For the last term on the RHS of eq. (41), on one hand, by eqs. (2), (16), (12), and (38), we have x 1 ( + 1 ) = x 2 () + θ τ α 1 () x 2 () θ τ α 1 () = θ τ α 1 () Φ θ. (42) Then, by eqs. (38), (39), (42), and Condition A1), it can be shown that α 1 ( + 1 ) 2 L 2 1 x 1( + 1 ) 2b1 L 2 1 Φ θ 2b1. (43) On the other hand, by eq. (28) we now that θ and then θ is bounded, i.e., there exists constant K θ > 0 such that θ K θ. (44) Therefore, combining (43) and (44) gives θ τ α 1( + 1 ) 2 (K θ L 1 ) 2 Φ θ 2b1. (45) Substituting (45) into (41) with + 2 replaced by + 1 and using Lemmas 2 3 and Corollary 1, we now there exists K 1 > 0 such that x 1 ( + 1) 2 K 1 Φ θ 2 + Φ θ 2 + Φ (θ θ ) 2 + Φ θ 2b1 ] K 1 { σ 1 ε ] 2 + σ + σ ] b1 + σ where σ is defined in eq. (33). By Condition A1), we also now that there exists L 2 0 such that x R, (46) ] 2 }, (47) α 1 (x) L 1 x b1 + L 2. (48) From (2), (4), (11), (39) and (48), we now there exists K 2 > 0 such that x 2 ( + 1) 2 = θ τ α 2 () θ τ ˆα 1 ( + 1 ) 2 = θ τ α 2 () + θ τ α 1( + 1 ) θ τ α 1 ( + 1) 2 3 Φ θ 2 + 3 θ τ α 1( + 1 ) 2 + 6 θ 2 L 2 1 x 1 ( + 1) 2b1 + 6 θ 2 L 2 2 3 Φ θ 2 + 3(K θ L 1 ) 2 Φ θ 2b1 + 6(K θ L 1 ) 2 x 1 ( + 1) 2b1 + 6(K θ L 2 ) 2 { +1 K 2 σ ]2b 1 + σ ] b1b } 1 1 ε + 1, (49) where the last inequality is derived from (37), (47), and Lemma 2. By choosing K = max{3k 1, K 2 }, (35), (36) follow from (47) and (49), respectively. Lemma 5. inequality Let Condition A1) be satisfied with d = 1. Then, t+1 satisfies the following t+1 K 3 ( =0 L 2 2 t + t =0 σ ] b1b ) 1 b2, 1 ε where b 1 is defined in eq. (37). Proof. By eqs. (4), (11) and (23), it can be shown that t+1 = t P 0 + Φ τ Φ t = α + (α 2 1 () + α2 2 ()). (50) =0 WEI Chen et al. Sci China Ser F-Inf Sci May 2008 vol. 51 no. 5 524-534 531

Therefore, by using (48) and (50), we have t t+1 2L2 1 ( x 1 () 2b1 + x() 2b2 ) + 2L 2 2 t + α. (51) =0 Finally, applying Lemma 4 to (51) and noticing (37) will give the desired result. QED Denote t r t = t+1 = α + Φ 2. (52) Then, we have the following result. Lemma 6. 0, then Proof. =0 Let the condition of Theorem 1 be satisfied. If lim t r t = and lim inf t r t t = sup t lim inf t r t t = 0 implies that r t+1 r t <. t/2] 1 lim inf t t ( Φ 2i 2] 2 + Φ 2i+1 2 ) i=0 1 lim inf t t 2] t i=0 and hence there exists a subsequence {τ n } such that By (53), we can tae n large enough such that Φ τn 2 1 and r τn For this fixed n, we proceed to prove by induction that For i = 0,1, by (54) we now that Φ i 2 1 lim inf t t t = 0, 2]r Φ τn 2 + Φ τn+1 2 0. (53) r τn+i r τn+i r τn+i r τn+i Φ τn+1 2 r τn 1. (54) 2, i 0. (55) = 1 + Φ τ n+i 2 r τn+i i.e, (55) holds for i = 0,1. Next, suppose that for some t 1, (55) holds for all i t. Since d = 1, P t+1 = r t and b1 = b 1. Since δ n 0, by Condition A1) and Lemma 4 we now that there exist positive constants L 3 and L 4 such that Φ t+1 2 = α 1 (t + 1) 2 + α 2 (t + 1) 2 rt L 3 + r ] b 1 b1b2 t + L 4. (56) r t r t 2 By the fact r t, it can be seen that for a large enough n 2, L 3 4 γ1b1b2 + L 4 r τn+t 1, 532 WEI Chen et al. Sci China Ser F-Inf Sci May 2008 vol. 51 no. 5 524-534

which together with (56) shows Φ τn+t+1 2 r τn+t Hence, r τ n+t+1 r τn+t L 3 rτn+t r τn+t + rτn+t r τn+t 2 ] γ1b 1b 2 + L4 r τn+t L 3 4 γ1b1b2 + L 4 r τn+t 1. 2, i.e., (55) also holds for i = t + 1. Therefore, (55) holds for all i 0. Now, we are in a position to give the proof of the main results. Proof of Theorem 1. now that if then First of all, by (46), Lemma 2 2), Lemma 3, and =0 δ <, we sup t y( + 1) 2 = =0 r t+1 r t, t 0, (57) x 1 ( + 1) 2 <, =0 and Theorem 1 will follow immediately. Next, we prove (57) by considering the following three cases. i.e., Case 1. lim r t <. In this case, (57) holds trivially. t r Case 2. lim t r t = and lim inf t t = 0. By Lemma 6, (57) holds again in this case. t r Case 3. lim inf t t 0. In this case, t = O(r t t ) as t. Since d = 1, P t+1 = r t. By Lemma 5, ( t ] b1b r ) 1 b2 i r t+1 = O(t) + O. (58) i=1 σ i r 1 ε i Therefore, applying Lemma 1 to (58) we now that (57) holds if (b 1 b 1b 2 ) 2 < 4(1 ε)(b 1 b 1b 2 ), b 1 b 1b 2 < 4(1 ε). Since ε can be chosen arbitrarily small, it suffices to guarantee or equivalently, b 1 b 1 b 2 < 4, b 1 b 2 < 2 and b 2 1b 2 < 4, which is the condition of the theorem. (57) also holds in this case. This completes the proof of Theorem 1. 4 Concluding remars In this paper, we have studied adaptive control of a class of discrete-time parametric-strict-feedbac nonlinear systems with scalar unnown parameter and nonlinearities having nonlinear growth rate. The prediction/ls-based adaptive control system is shown to be globally stable and asymptotic tracing performance is achieved. As shown in Remar 2 and in section 3, the linear growth rate is WEI Chen et al. Sci China Ser F-Inf Sci May 2008 vol. 51 no. 5 524-534 533

hard to remove for the multi-unnown parameters case or high-order nonlinear systems. Can any further results be obtained? This belongs to future research. The authors would lie to than the referees for their valuable suggestions and comments, also lie to than Prof. Lei Guo for his helpful instructions. 1 Goodwin G C, Sin K S. Adaptive Filtering, Prediction and Control. Englewood Cliffs, NJ: Prentice-Hall, 1984 2 Guo L. Time-varying Stochastic Systems: Stability, Estimation and Control (in Chinese). Jilin: Jilin Science and Technology Press, 1993 3 Isidori I. Nonlinear Control Systems. Berlin: Springer-Verlag, 1989 4 Krstic M, Kanellaopoulos I, Kootovic P V. Nonlinear and Adaptive Control Design. New Yor: Wiley, 1995 5 Kanellaopoulos I. A discrete-time adaptive nonlinear system. IEEE Trans Automatic Contr, 1994, 39: 2362 2365 6 Yeh P C, Kootovic P V. Adaptive control of a class of nonlinear discrete-time systems. Int J Contr, 1995, 62(2): 303 324 7 Guo L, Wei C. Global stability/instability of LS-based discrete-time adaptive nonlinear control. In: Proc. 13th IFAC World Congr, 1996 8 Guo L. On critical stability of discrete-time adaptive nonlinear control. IEEE Trans Automatic Contr, 1997, 42(1): 1488 1499 9 Xie L L, Guo L. Fundamental limitations of discrete-time adaptive nonlinear control. IEEE Trans Automatic Contr, 1999, 44(9): 1777 1782 10 Xie L L, Guo L. Adaptive control of a class of discrete-time affine nonlinear systems. Syst Contr Lett, 1998, 35: 201 206 11 Wei C, Guo L. Prediction-based discrete-time adaptive nonlinear stochastic control. IEEE Trans Automatic Contr, 1999, 44(9): 1725 1729 12 Yeh P C, Kootovic P V. Adaptive output-feedbac design for a class of nonlinear discrete-time systems. IEEE Trans Automatic Contr, 1995, 40(9): 1663 1668 13 Knopp K. Theory and Applications of Infinite Series. London and Glasgon: Blacie & Son Ltd., 1928 534 WEI Chen et al. Sci China Ser F-Inf Sci May 2008 vol. 51 no. 5 524-534