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Stabilization of discrete-time stochastic systems via sliding mode technique
Journal of the Franklin Institute 349 (2012) 1497–1508 www.elsevier.com/locate/jfranklin
Stabilization of discrete-time stochastic systems via sliding mode technique Yugang Niua,, Daniel W.C. Hob a
Key Laboratory of Advanced Control and Optimization for Chemical Processes of Ministry of Education, East China University of Science and Technology, Shanghai 200237, PR China b Department of Mathematics, City University of Hong Kong, 83 Tat Chee Avenue, Kowloon, Hong Kong Received 13 October 2010; received in revised form 14 March 2011; accepted 4 June 2011 Available online 13 July 2011
Abstract This paper considers the problem of sliding mode control for discrete-time stochastic systems with parameter uncertainties and state-dependent noise perturbation. An integral-like sliding surface is chosen and a discrete-time sliding mode controller is designed. The key feature in this work is that both the reachability of the quasi-sliding mode and the stability of system states are simultaneously analyzed, due to the existence of state-dependent noise perturbation. By utilizing an Lyapunov function involving system states and sliding mode variables, the sufficient condition for reachability is obtained. Finally, numerical simulation results are provided. & 2011 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.
1. Introduction In the past decades, the study of stochastic systems has attracted much attention due to the extensive applications of stochastic modeling in mechanical systems, economics, and other areas. A lot of results related to robust stability analysis and stabilization for stochastic systems have been reported in, e.g., [1–4] for continuous-time cases, and [5–7] for discrete-time ones. Recently, sliding mode control (SMC) for stochastic systems has begun obtaining successful applications [8–11]. The notable advantage of SMC systems is its strong robustness against uncertainties and external disturbance, which makes it become an effective control strategy for incompletely modeled (or uncertain) systems, see [12–17] Corresponding author.
E-mail address: [email protected] (Y. Niu). 0016-0032/$32.00 & 2011 The Franklin Institute. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.jfranklin.2011.06.005
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and the references therein. To the authors’ best knowledge, little works on SMC for discrete-time case have been available in the literature so far. Due to the widespread use of computers for control purpose, the design of discrete sliding mode control (DSMC) has recently stirred considerable research interests [18–22]. In the discrete-time system, the control input is calculated once in every sampling interval and is held as constant during the sampling period. Thus, the structure of a DSMC may be changed only at discrete instants, which is in contrast to its continuous counterpart. Hence, DSMC systems cannot be obtained from their continuous counterpart by means of simple equivalence [23]. Especially, due to the structure characteristic of stochastic systems with state-dependent noise perturbation, the previous works in deterministic DSMC cannot be trivially extended to the stochastic case and some new analysis method had to be provided. This motivates the research in this work. In this work, the problem of DSMC for stochastic systems will be investigated. In the systems under consideration, there exist parameter uncertainties and state-dependent noise perturbation. An integral-like sliding surface is firstly constructed and the sufficient condition of the robustly stochastic stability for ideal sliding mode dynamics is obtained. Due to the specific structure of discrete-time stochastic systems, the analysis on the reachability of the quasi-sliding mode is made simultaneously with the stability of system states by utilizing a new Lyapunov function in Eq. (22) dependent on both the states and sliding mode variables (see Remark 2 later). These are different from some existing works on continuous-time stochastic systems [8] or discrete-time deterministic systems [24]. Moreover, this also provides an effective method for the design of DSMC systems. The paper is divided into five sections. Section 2 provides preliminaries for subsequent developments and formulates the DSMC problem to be addressed. The design of discretetime stochastic SMC is given and some sufficient conditions are derived in Section 3. The effectiveness of the proposed SMC approach will be demonstrated via numerical simulation results in Section 4, which will be followed by some concluding remarks in Section 5. Notation: for a real symmetric matrix, M40 means that M is positive definite. I is used to represent an identity matrix of appropriate dimensions. lmin ðÞ and lmax ðÞ denote the minimum and maximum eigenvalue of a matrix, respectively. ðO,F ,PÞ is a probability space with O the sample space, F the s-algebra of subsets of the sample space, and P the probability measure. Efg denotes the expectation operator with respect to probability measure P. Matrices, if not explicitly stated, are assumed to have compatible dimensions.
2. Definitions and problem formulation Consider the uncertain discrete-time stochastic systems as follows: xðk þ 1Þ ¼ AxðkÞ þ BðuðkÞ þ f ðxðkÞ,kÞÞ þ ðC þ DCðkÞÞxðkÞwðkÞ,
ð1Þ
where xðkÞ 2 Rn is the state, w(k) is a scalar Wiener process on a probability space ðO,F ,PÞ relative to an increasing family ðF k Þk2N of s-algebra F k F generated by ðwðkÞÞk2N with N the set of natural numbers, and is assumed to satisfy EfwðkÞg ¼ 0, EfwðkÞ2 g ¼ 1, and EfwðiÞwðjÞg ¼ 0 for iaj. A, B, and C are known real constant matrices. Without the loss of generality, it is assumed that the pair (A,B) is controllable and the input matrix B has full column rank. The unknown matrix DCðkÞ represents uncertainty satisfying DCðkÞ ¼ EF ðkÞH, where E and H are known real constant matrices, and F(k) is an unknown time-varying matrix
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satisfying F T ðkÞF ðkÞrI,8t. The unknown nonlinear function f ðxðkÞ,kÞ is the external disturbance satisfying Jf ðxðkÞ,kÞJrr with r known positive constant. In Eq. (1), there exists the xðkÞwðkÞ term, which is also called bilinear noise perturbation in [25]. This implies that, in the controlled system (1), the noise perturbation is not purely external, but depends on system states. Hence, it is more difficult to stabilize the system (1) than the one with only purely external noise perturbation. The objective in this work is to synthesize a discrete-time stochastic SMC system. Thus, a sliding function s(k) should firstly be chosen, and then a DSMC law u(k) will be designed such that the reachability of the specified sliding surface is attained. It is known that, in the discrete-time control system, the control input is held as a constant during the sampling period. Thus, the finite sampling rate may yield that the system state may approach the sliding surface but it is generally unable to stay on it. This means that the state trajectories would move about the surface, which yields a sliding-like mode, termed as quasi-sliding mode (QSM) in [23]. In the following, a definition on QSM is firstly made, which is a revision one in [23]. Definition 1. The motion of a discrete-time system is called a quasi-sliding mode (QSM) if this motion stays within the e-neighborhood of the specified sliding surface, i.e., jsðkÞjoe for kZkn , where e is the band of quasi-sliding domain, and kn is a positive constant. As e ¼ 0, an ideal sliding mode similar to continuous-time one will happen. Remark 1. It is worth noting that, in Definition 1, the system state is not required to cross the switching surface in each successive sampling period. Instead, the system state is only required to enter and stay within the e-neighborhood of the surface in finite time. This is different from some existing works, e.g., Ref. [23], in which the state trajectory is required to cross the switching surface in every successive sampling period, once it has crossed the surface for the first time. Actually, the zigzag motion about the surface is not necessary for the existing of an QSM. Especially, the zigzag motion about the switching may result in the undesirable chattering. Before proceeding, the following definition and lemma [26] will be given, which are useful for the development of our result. Definition 2. For the stochastic system (1), if there exist constants a40 and t 2 ð0,1Þ such that EfJZðkÞJ2 gratk EfJZð0ÞJ2 g, where ZðkÞ denotes the solution of stochastic systems with initial state Zð0Þ, then, the stochastic system (1) is said to be exponentially mean square stable. Lemma 1. Let V ðZðkÞÞ be a Lyapunov functional. If there exist real scalars lZ0, m40, n40, and 0oco1 such that mJZðkÞJ2 rV ðZðkÞÞrnJZðkÞJ2 , and EfV ðZðk þ 1ÞÞjZðkÞgV ðZðkÞÞrlcV ðZðkÞÞ,
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then the sequence ZðkÞ satisfies n l : EfJZðkÞJ2 gr EfJZð0ÞJ2 gð1cÞk þ m mc Lemma 2. For any real vectors a, b and matrix X 40 of compatible dimensions: aT b þ bT araT Xa þ bT X 1 b: Lemma 3. Let A, E, H, and F(t) be real matrices of appropriate dimensions with F(t) satisfying F T ðtÞF ðtÞrI. Then, for any real matrices Q ¼ QT , we have Q þ EF ðtÞH þ H T F T ðtÞE T o0
8F ðtÞ s:t: F ðtÞT F ðtÞrI,
if and only if there exists some scalar e40 such that Q þ eEE T þ e1 H T Ho0: 3. The synthesis of DSMC In this section, a sliding surface will be firstly chosen, and the stability of stochastic system (1) on the specified sliding surface is analyzed. Finally, a DSMC law will be designed such that the reachability of a QSM on the specified sliding surface is attained.
3.1. Integral sliding surface In this work, an integral-like sliding function is constructed as follows: sðkÞ ¼ GxðkÞGAxðk1Þ,
ð2Þ
where G is a design matrix to be given later so that GB is nonsingular. It can be shown from the matrix theory that the nonsingularity of GB can be ensured by choosing G ¼ BT P with P40, since B is assumed to be of full column rank. It is noted that the ideal sliding mode satisfies sðk þ 1Þ ¼ sðkÞ ¼ 0
for kZkn ,
ð3Þ
where kn is a positive constant. Thus, when the state trajectories of the system enter the ideal sliding mode, the equivalent control law may be obtained from Eqs. (1) to (3): ueq ðkÞ ¼ ðGBÞ1 GðC þ DCðkÞÞxðkÞwðkÞf ðxðkÞ,kÞ:
ð4Þ
Hence, by substituting Eq. (4) into the system (1), we obtain the sliding mode dynamics in the sliding surface sðkÞ ¼ 0 as follows: xðk þ 1Þ ¼ AxðkÞ þ ½IBðGBÞ1 GðC þ DCðkÞÞxðkÞwðkÞ:
ð5Þ
In the sequel, we shall analyze the performance of the sliding motion described by Eq. (5) and derive the condition of stochastic stability.
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3.2. Stability of sliding motion Theorem 1. Consider the stochastic system (1) and the sliding surface specified by Eq. (2). If there exist symmetric matrix P40 and scalar e40 satisfying the following LMI: 3 2 C T PB 0 P þ eH T H AT P C T P 7 6 PA P 0 0 0 7 6 7 6 6 ð6Þ PC 0 0:5P 0 PE 7 7o0, 6 7 6 BT PC 0 0 0:5BT PB BT PE 5 4 0 0 ET P E T PB eI and sliding mode matrix G ¼ BT P, then the sliding mode dynamics (5) is stochastically stable. Proof. By Schur’s complement, it is seen that LMI (6) is equivalent to the following inequality: 2 3 P AT P C T P C T PB 6 PA 7 P 0 0 6 7 ð7Þ 6 7 þ eN T N þ e1 MM T o0 4 PC 5 0 0:5P 0 BT PC
0
0
0:5BT PB
with M ¼ ½0 0 E T P E T PBT ,
N ¼ ½H 0 0 0:
On the other hand, it is noted that 2 P AT P ðC þ DCðkÞÞT P 6 6 PA P 0 6 6 PðC þ DCðkÞÞ 0 0:5P 4 BT PðC þ DCðkÞÞ 0 0 2 P AT P C T P C T PB 6 PA P 0 0 6 ¼6 4 PC 0 0:5P 0 BT PC
0
0
ðC þ DCðkÞÞT PB 0 0 0:5BT PB
3 7 7 7 7 5
3 7 7 T 7 þ MF ðkÞN þ N T F ðkÞ M T : 5
ð8Þ
0:5BT PB
Hence, by Lemma 3, it is shown from Eqs. (33) and (8) that 3 2 P AT P ðC þ DCðkÞÞT P ðC þ DCðkÞÞT PB 7 6 7 6 PA P 0 0 7o0: 6 7 6 PðC þ DCðkÞÞ 0 0:5P 0 5 4 BT PðC þ DCðkÞÞ 0 0 0:5BT PB
ð9Þ
Now, choose the following Lyapunov function candidate for stochastic system (5): V ðxðkÞ ¼ xðkÞT PxðkÞ,
ð10Þ
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then, we have EfV ðxðk þ 1ÞÞjxðkÞgV ðxðkÞÞrxðkÞAT PAxðkÞ þ2xðkÞT ðC þ DCðkÞÞT PðC þ DCðkÞÞxðkÞ þ2xðkÞT ðC þ DCðkÞÞT GT ðGBÞ1 GðC þ DCðkÞÞxðkÞxðkÞT PxðkÞ ¼ xðkÞT YxðkÞ
ð11Þ
with Y ¼ AT PA þ 2ðC þ DCðkÞÞT PðC þ DCðkÞÞ þ2ðC þ DCðkÞÞT PBðBT PBÞ1 BT PðC þ DCðkÞÞP: By Schur’s complement and the matrix inequality (9), we obtain Yo0, which implies that there exists a small scalar a40 such that YoaI. Hence, we obtain from (11) that EfV ðxðk þ 1ÞÞjxðkÞgV ðxðkÞÞoajxðkÞj2
for all xðkÞa0:
ð12Þ
Taking expectation with respect to both sides of Eq. (12), we have EfV ðxðk þ 1ÞÞgEfV ðxðkÞÞgoaEfjxðkÞj2 g:
ð13Þ
Hence, by summing up both sides of Eq. (13) from 0 to N for any integer N41, we obtain ( ) N X 2 EfV ðxðN þ 1ÞÞgEfV ðxð0ÞÞgoaE jxðkÞj , k¼0
which yields ( ) N X 1 jxðkÞj2 o ½EfV ðxð0ÞÞgEfV ðxðN þ 1ÞÞg E a k¼0 1 r EfV ðxð0ÞÞgrcEfjxð0Þj2 g ð14Þ a with c ¼ ð1=aÞlmax ðPÞ. Taking N-1, it is shown from Eq. (14) and Definition 2 that the sliding mode dynamics (5) is robustly stochastically stable. & In the above derivation, by means of the discrete-time stochastic Lyapunov method, we obtain a sufficient condition for the stability of stochastic system (1) on the specified sliding surface sðkÞ ¼ 0. In the following, a DSMC law will be further designed such that the reachability of the specified sliding surface is ensured. 3.3. The design of sliding mode controller For the stochastic system (1) and the sliding function (2), we have sðk þ 1Þ ¼ GBuðkÞ þ DðkÞ þ GðC þ DCðkÞÞxðkÞwðkÞ
ð15Þ
with DðkÞ ¼ GBf ðxðkÞ,kÞ. It follows from Jf ðxðkÞ,kÞJrr that there exist known bounds di , d i (i ¼ 1,2, . . . ,m) satisfying d i ðkÞrdi ðkÞrd i ðkÞ,
ð16Þ
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where di(k) is the ith element in D(k). Then, define d ðkÞ d ðkÞ dio ðkÞ ¼ d i ðkÞ þ i , dis ðkÞ ¼ d i ðkÞ i ð17Þ 2 2 and Do ðkÞ ¼ ½d1o ðkÞ d2o ðkÞ . . . dmo ðkÞT , Ds ðkÞ ¼ diagfd1s ðkÞ,d1s ðkÞ, . . . ,d1s ðkÞg. For notational simplicity, the variable k in the bound on D1 ðkÞ is omitted in the later design. Now, we design the SMC law as follows: uðkÞ ¼ ðGBÞ1 ½GAxðkÞ þ Do þ Ds sgnðsðkÞÞ:
ð18Þ
Remark 2. It is due to the existence of state-dependent noise perturbation in Eq. (1), the above SMC law (18) cannot ensure the reaching law similar to the one in [24]. Moreover, the structure characteristic of stochastic system (1) also makes the analysis on the reachability more complex. It will be seen from the following proof that the analysis on the reachability of the quasi-sliding mode is made simultaneously with the stability of system states, in which a Lyapunov function in Eq. (22) involving in system states and sliding mode variables will be utilized. In this following, a stochastic Lyapunov method is utilized to analyze the reachability of the sliding surface specified by Eq. (2). It will be seen from the following procedure that a Lyapunov function involving in system states and sliding mode variables will be chosen, which are different from some existing works. Theorem 2. Consider the discrete-time stochastic systems (1) and the integral sliding surface specified by Eq. (2). The sliding mode controller is constructed as Eq. (18). If there exist matrix P40 and scalar e40 satisfying the following LMI: 3 2 P þ eH T H 2AT P AT PB 2AT PB C T P C T PB 0 7 6 2PA P 0 0 0 0 0 7 6 7 6 6 BT PA 0 0:5I 0 0 0 0 7 7 6 7 6 ð19Þ 0 0 BT PB 0 0 0 7o0, 6 2BT PA 7 6 7 6 PC 0 0 0 P 0 PE 7 6 7 6 T B PC 0 0 0 0 I BT PE 5 4 0 0 0 0 E T P E T PB eI with G ¼ BT P, then the state trajectories are driven (with mean square) into a band of the specified sliding surface (2). That is, the quasi-sliding mode is attained (with mean square). Proof. It follows from Eqs. (1) and (18) that the closed-loop control system is given as xðk þ 1Þ ¼ ½ABðGBÞ1 GAxðkÞ þ ðC þ DCðkÞÞxðkÞwðkÞ þ BðGBÞ1 ½DðkÞDo Ds sgnðsðkÞÞ:
ð20Þ
Besides, substituting Eq. (18) into Eq. (15) yields sðk þ 1Þ ¼ GAxðkÞ þ GðC þ DCðkÞÞxðkÞwðkÞ þ DðkÞDo Ds sgnðsðkÞÞ:
ð21Þ
Now, choose the Lyapunov function candidate as follows: V ðZðkÞÞ ¼ xðkÞT PxðkÞ þ sðkÞT sðkÞ,
ð22Þ
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with ZðkÞ ¼ ½xðkÞT ,sðkÞT T . Thus, we have EfV ðZðk þ 1ÞÞjZðkÞgV ðZðkÞÞ ¼ Efxðk þ 1ÞT Pxðk þ 1Þ þ sðk þ 1ÞT sðk þ 1ÞjZðkÞgxðkÞT PxðkÞsðkÞT sðkÞ:
ð23Þ
Along the state trajectories of system (20), we obtain Efxðk þ 1ÞT Pxðk þ 1ÞjZðkÞg ¼ xðkÞT ½ABðGBÞ1 GAT P½ABðGBÞ1 GAxðkÞ þxðkÞT ðC þ DCðkÞÞT PðC þ DCðkÞÞxðkÞ þ2xðkÞT ½ABðGBÞ1 GAT PBðGBÞ1 ½DðkÞDo Ds sgnðsðkÞÞ þ½DðkÞDo Ds sgnðsðkÞÞT ðGBÞ1 ½DðkÞDo Ds sgnðsðkÞÞ:
ð24Þ
By Lemma 2, it is noted that ½ABðGBÞ1 GAT P½ABðGBÞ1 GAr2AT PA þ 2AT PBðBT PBÞ1 BT PA,
ð25Þ
2xðkÞT ½ABðGBÞ1 GAT PBðGBÞ1 ½DðkÞDo Ds sgnðsðkÞÞ r2xðkÞT AT PAxðkÞ þ 2xðkÞT AT PBðBT PBÞ1 BT PAxðkÞ þ ½DðkÞDo Ds sgnðsðkÞÞT ðGBÞ1 ½DðkÞDo Ds sgnðsðkÞÞ:
ð26Þ
Thus, by means of Eqs. (25) and (26), we have from Eq. (24) Efxðk þ 1ÞT Pxðk þ 1ÞjZðkÞg rxðkÞT ½4AT PA þ 4AT PBðBT PBÞ1 BT PA þ ðC þ DCðkÞÞT PðC þ DCðkÞÞxðkÞ þ 2½DðkÞDo Ds sgnðsðkÞÞT ðGBÞ1 ½DðkÞDo Ds sgnðsðkÞÞ:
ð27Þ
Similarly, along the trajectories of Eq. (20), we have Efsðk þ 1ÞT sðk þ 1ÞjZðkÞg ¼ xðkÞT AT GT GAxðkÞ 2xðkÞT AT GT ½DðkÞDo Ds sgnðsðkÞÞ þxðkÞT ðC þ DCðkÞÞT GT GðC þ DCðkÞÞxðkÞ þ½DðkÞDo Ds sgnðsðkÞÞT ½DðkÞDo Ds sgnðsðkÞÞ rxðkÞT ½2AT G T GA þ ðC þ DCðkÞÞT G T GðC þ DCðkÞÞxðkÞ þ2½DðkÞDo Ds sgnðsðkÞÞT ½DðkÞDo Ds sgnðsðkÞÞ:
ð28Þ
Note the fact that JDðkÞDo Ds sgnðsp ðkÞÞJr2JDs J
ð29Þ
It follows from Eqs. (23), (27)–(29) that EfV ðZðk þ 1ÞÞjZðkÞgV ðZðkÞÞrxðkÞT PxðkÞsðkÞT sðkÞ þ xJDs J2
ð30Þ
with x ¼ 8 þ 8JðGBÞ1 J, and P ¼ P þ 2AT G T GA þ 4AT PA þ ðC þ DCÞT PðC þ DCÞþ 4AT PBðBT PBÞ1 BT PA þ ðC þ DCÞT GT GðC þ DCÞ. It is seen from Eqs. (22) and (30) that as Po0, we have, for JsðkÞJZðx1=2 =ð1dÞ1=2 ÞJDs J, EfV ðZðk þ 1ÞÞjZðkÞgV ðZðkÞÞrxðkÞT PxðkÞdsðkÞT sðkÞ rlmin ðSÞZðkÞT ZðkÞ omZðkÞT ZðkÞ,
ð31Þ
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where 0odo1, S ¼ diagðP,dIÞ, and 0omominflmin ðSÞ,gg,
g :¼ maxflmax ðPÞ,1g:
Thus, it follows from Eq. (31) that m EfV ðZðk þ 1ÞÞjZðkÞgV ðZðkÞÞomZðkÞT ZðkÞo V ðZðkÞÞ :¼ cV ðZðkÞÞ g
ð32Þ
with 0oco1. Therefore, it is obtained from Lemma 1 and Definition 2 that for JsðkÞJZðx1=2 =ð1dÞ1=2 ÞJDs J, the closed-loop system (20) is exponentially mean square stable. This means that the quasi-sliding mode is attained (with mean square) within the band of the sliding surface specified by Eq. (2), and the system states converge to a small neighborhood of zero, i.e., JsðkÞJoðx1=2 =ð1dÞ1=2 ÞJDs J. On the other hand, by Schur’s complement, it is shown that Po0 is equivalent to the following matrix inequality: 3 2 P 2AT P AT PB 2AT PB ðC þ DCÞT P ðC þ DCÞT PB 7 6 7 6 2PA P 0 0 0 0 7 6 T 7 6 B PA 0 0:5I 0 0 0 7 6 7o0: 6 7 6 2BT PA 0 0 BT PB 0 0 7 6 7 6 PðC þ DCÞ 0 0 0 P 0 5 4 T B PðC þ DCÞ 0 0 0 0 I ð33Þ Further, by following a similar approach as in Eqs. (7)–(9), it may be shown that the matrix inequality (33) is implied by LMI (19). & Remark 3. Notice that the analysis on the reachability of quasi-sliding mode is considered simultaneously with the stability of the system states. This approach is different from the method used in the continuous-time stochastic SMC as in [8]. The proposed idea is just a typical feature of SMC for discrete-time stochastic systems. Remark 4. It is shown from Theorems 1 and 2 that if there exist matrix P40 and scalar e40 satisfying LMIs (6) and (19), the state trajectories are driven into the band of the sliding surface sðkÞ ¼ 0 and asymptotically tends to a small neighborhood of zeros. Moreover, the ideal sliding motion on sðkÞ ¼ 0 is stochastically stable. 4. Numerical simulation Consider the discrete-time stochastic system (1) with 2 3 2 2 3 0:2 0:3 0:5 0:2 0:5 1:6 6 7 6 6 7 A ¼ 4 0:1 0:2 0:2 5, B ¼ 4 2:1 1:3 5, C ¼ 4 0:1 0
0:2
E ¼ ½0:2 0:2 0:1T , F ðkÞ ¼ 0:5 sinðkÞ,
0:3
0:8
0:4
H ¼ ½0:2 0:1 0:2, " # 0:6 sinðx1 ðkÞx2 ðkÞÞ f ðxðkÞÞ ¼ : 0:3 sinðx3 ðkÞÞ
0:2
0:2 0:2 0:3
0:1
3
7 0:1 5, 0:1
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The objective is to design an SMC law such that the state trajectories can be driven onto the sliding surface and the sliding motion in the specified sliding manifold is stochastically stable. To this end, we solve LMIs (6) and (19) and obtain the solutions and the sliding mode gain matrix G as follows: 2 3 0:0245 0:0071 0:0198 6 7 P ¼ 4 0:0071 0:1407 0:0669 5, e ¼ 0:0472, 0:0198 0:0669 0:1900
T
G¼B P¼
0:0186
0:2385
0:0017
0:0564
0:1676
0:0208
:
1
x1 x2 x3
0.5 0 −0.5 −1 −1.5 −2
0
1
2
3
4
5
Fig. 1. The trajectories of state x(t).
0.15
s1 s2
0.1 0.05 0 −0.05 −0.1 −0.15 −0.2 −0.25 −0.3
0
1
2
3
4
Fig. 2. The trajectories of sliding variable s(t).
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0.6
1507
u1 u2
0.4 0.2 0 −0.2 −0.4 −0.6 −0.8
0
1
2
3
4
5
Fig. 3. The control signal u(t).
Thus, the proposed SMC law is designed as in Eq. (18), and the simulation results are shown in Figs. 1–3, in which the initial state is chosen as xð0Þ ¼ ½1 0:5 1T . It is seen that the proposed control law can quickly drive the state trajectories toward the sliding surface sðkÞ ¼ 0, and asymptotically tend to zero along the sliding surface sðkÞ ¼ 0. Hence, the simulation results show the effectiveness of the proposed method in this work. 5. Conclusions In this work, the problem of SMC for discrete-time stochastic systems has been considered. It has been shown that, for the discrete-time stochastic systems, the analysis on the reachability of the quasi-sliding mode had to be considered simultaneously with the stability of the system states. In this work, it is noted that under the quasi-sliding mode, only the boundedness of system states is ensured. Hence, it would be an interesting topic to further investigate the results along this direction. Acknowledgment The research was partially supported by a grant from Geral Research Fund (CityU 101109), NNSF from China (61074041), the Technology Innovation Key Foundation of Shanghai Municipal Education Commission (09ZZ60), and Shanghai Leading Academic Discipline Project (B504). References [1] W. Zhang, B.-S. Chen, C.-S. Tseng, Robust H1 filtering for nonlinear stochastic systems, IEEE Transactions on Signal Processing 53 (2005) 589–598. [2] S. Xu, T. Chen, H1 output feedback control for uncertain stochastic systems with time-varying delay, Automatica 40 (2004) 2091–2098. [3] M. Basin, D. Calderon-Alvarez, Optimal controller for uncertain stochastic polynomial systems with deterministic disturbances, in: American Control Conference, St. Louis, USA, 2009, pp. 778–783.
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[4] R. Yang, H. Gao, P. Shi, Delay-dependent robust H1 control for uncertain stochastic time-delay systems, International Journal of Robust & Nonlinear Control 20 (2010) 1852–1865. [5] A. EL Bouhtouri, D. Hinrichsen, A.J. Pritchard, H1 -type control for discrete-time stochastic systems, International Journal of Robust & Nonlinear Control 9 (1999) 923–948. [6] S. Xu, J. Lam, T. Chen, Robust H1 control for uncertain discrete stochastic time-delay systems, Systems & Control Letters 51 (2004) 203–215. [7] W. Zhang, Y. Huang, H. Zhang, Stochastic h2 =H1 control for discrete-time systems with state and disturbance dependent noise, Automatica 43 (2007) 513–521. [8] Y. Niu, D.W.C. Ho, J. Lam, Robust integral sliding mode control for uncertain stochastic systems with timevarying delay, Automatica 41 (2005) 873–880. [9] P. Shi, Y. Xia, G.P. Liu, D. Rees, On designing of sliding-mode control for stochastic jump systems, IEEE Transactions on Automatic Control 51 (2006) 97–103. [10] Y. Niu, D.W.C. Ho, X. Wang, Robust H1 control for nonlinear stochastic systems: a sliding-mode approach, IEEE Transactions on Automatic Control 53 (2008) 1695–1701. [11] L. Huang, X. Mao, Smc design for robust H1 control of uncertain stochastic delay systems, Automatica 46 (2010) 405–412. [12] J. Zhang, Y. Xia, P. Shi, Robust adaptive sliding mode control for fuzzy systems with mismatched uncertainties, IEEE Transactions on Fuzzy Systems 18 (2010) 700–711. [13] Y. Shtessela, P. Kavehb, A. Ashrafi, Harmonic oscillator utilizing double-fold integral, traditional and second-order sliding mode control, Journal of the Franklin Institute 346 (2009) 872–888. [14] M. Basin, D. Calderon-Alvarez, Sliding mode regulator as solution to optimal control problem for non-linear polynomial systems, Journal of the Franklin Institute 347 (2010) 910–922. [15] J. Zhang, Y. Xia, Design of static output feedback sliding mode control for uncertain linear systems, IEEE Transactions on Industry Electronics 57 (2010) 2161–2170. [16] M.C. Pai, Design of adaptive sliding mode controller for robust tracking and model following, Journal of the Franklin Institute 347 (2010) 1837–1849. [17] O. Barambonesa, P. Alkorta, A robust vector control for induction motor drives with an adaptive slidingmode control law, Journal of the Franklin Institute 348 (2011) 300–314. [18] S. Janardanan, B. Bandyopadhyay, Output feedback discrete-time sliding-mode control for time delay systems, IEE Proceedings: Control Theory and Applications 153 (2006) 387–396. [19] X. Chen, Adaptive sliding mode control for discrete-time multi-input multi-output systems, Automatica 42 (2006) 427–435. [20] K. Abidi, J.-X. Xu, X. Yu, On the discrete-time integral sliding-mode control, IEEE Transactions on Automatic Control 52 (2007) 709–715. [21] N.O. Lai, C. Edwards, S.K. Spurgeon, On output tracking using dynamic output feedback discrete-time sliding-mode controllers, IEEE Transactions on Automatic Control 52 (2007) 1975–1981. [22] Y. Xia, Z. Zhu, C. Li, H. Yang, Q. Zhu, Robust adaptive sliding mode control for uncertain discrete-time systems with time delay, Journal of the Franklin Institute 347 (2010) 339–357. [23] W. Gao, Y. Wang, A. Homaifa, Discrete-time variable structure control systems, IEEE Transactions on Industry Electronics 42 (1995) 117–122. [24] Y. Niu, Daniel W.C. Ho, Z. Wang, Improved sliding mode control for discrete-time systems via reaching law, IET Control Theory Applications 4 (2010) 2245–2251. [25] E.I. Verriest, Stabilization of deterministic and stochastic systems with uncertain time delays, in: Proceedings of IEEE Conference on Decision and Control, Lake Buena Vista, FL, 1994, pp. 3829–3834. [26] F. Yang, Z. Wang, Y.S. Huang, M. Gani, H1 control for networked systems with random communication delays, IEEE Transactions on Automatic Control 51 (2006) 511–518.
Stabilization of discrete-time stochastic systems via sliding mode technique Yugang Niua,, Daniel W.C. Hob a
Key Laboratory of Advanced Control and Optimization for Chemical Processes of Ministry of Education, East China University of Science and Technology, Shanghai 200237, PR China b Department of Mathematics, City University of Hong Kong, 83 Tat Chee Avenue, Kowloon, Hong Kong Received 13 October 2010; received in revised form 14 March 2011; accepted 4 June 2011 Available online 13 July 2011
Abstract This paper considers the problem of sliding mode control for discrete-time stochastic systems with parameter uncertainties and state-dependent noise perturbation. An integral-like sliding surface is chosen and a discrete-time sliding mode controller is designed. The key feature in this work is that both the reachability of the quasi-sliding mode and the stability of system states are simultaneously analyzed, due to the existence of state-dependent noise perturbation. By utilizing an Lyapunov function involving system states and sliding mode variables, the sufficient condition for reachability is obtained. Finally, numerical simulation results are provided. & 2011 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.
1. Introduction In the past decades, the study of stochastic systems has attracted much attention due to the extensive applications of stochastic modeling in mechanical systems, economics, and other areas. A lot of results related to robust stability analysis and stabilization for stochastic systems have been reported in, e.g., [1–4] for continuous-time cases, and [5–7] for discrete-time ones. Recently, sliding mode control (SMC) for stochastic systems has begun obtaining successful applications [8–11]. The notable advantage of SMC systems is its strong robustness against uncertainties and external disturbance, which makes it become an effective control strategy for incompletely modeled (or uncertain) systems, see [12–17] Corresponding author.
E-mail address: [email protected] (Y. Niu). 0016-0032/$32.00 & 2011 The Franklin Institute. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.jfranklin.2011.06.005
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and the references therein. To the authors’ best knowledge, little works on SMC for discrete-time case have been available in the literature so far. Due to the widespread use of computers for control purpose, the design of discrete sliding mode control (DSMC) has recently stirred considerable research interests [18–22]. In the discrete-time system, the control input is calculated once in every sampling interval and is held as constant during the sampling period. Thus, the structure of a DSMC may be changed only at discrete instants, which is in contrast to its continuous counterpart. Hence, DSMC systems cannot be obtained from their continuous counterpart by means of simple equivalence [23]. Especially, due to the structure characteristic of stochastic systems with state-dependent noise perturbation, the previous works in deterministic DSMC cannot be trivially extended to the stochastic case and some new analysis method had to be provided. This motivates the research in this work. In this work, the problem of DSMC for stochastic systems will be investigated. In the systems under consideration, there exist parameter uncertainties and state-dependent noise perturbation. An integral-like sliding surface is firstly constructed and the sufficient condition of the robustly stochastic stability for ideal sliding mode dynamics is obtained. Due to the specific structure of discrete-time stochastic systems, the analysis on the reachability of the quasi-sliding mode is made simultaneously with the stability of system states by utilizing a new Lyapunov function in Eq. (22) dependent on both the states and sliding mode variables (see Remark 2 later). These are different from some existing works on continuous-time stochastic systems [8] or discrete-time deterministic systems [24]. Moreover, this also provides an effective method for the design of DSMC systems. The paper is divided into five sections. Section 2 provides preliminaries for subsequent developments and formulates the DSMC problem to be addressed. The design of discretetime stochastic SMC is given and some sufficient conditions are derived in Section 3. The effectiveness of the proposed SMC approach will be demonstrated via numerical simulation results in Section 4, which will be followed by some concluding remarks in Section 5. Notation: for a real symmetric matrix, M40 means that M is positive definite. I is used to represent an identity matrix of appropriate dimensions. lmin ðÞ and lmax ðÞ denote the minimum and maximum eigenvalue of a matrix, respectively. ðO,F ,PÞ is a probability space with O the sample space, F the s-algebra of subsets of the sample space, and P the probability measure. Efg denotes the expectation operator with respect to probability measure P. Matrices, if not explicitly stated, are assumed to have compatible dimensions.
2. Definitions and problem formulation Consider the uncertain discrete-time stochastic systems as follows: xðk þ 1Þ ¼ AxðkÞ þ BðuðkÞ þ f ðxðkÞ,kÞÞ þ ðC þ DCðkÞÞxðkÞwðkÞ,
ð1Þ
where xðkÞ 2 Rn is the state, w(k) is a scalar Wiener process on a probability space ðO,F ,PÞ relative to an increasing family ðF k Þk2N of s-algebra F k F generated by ðwðkÞÞk2N with N the set of natural numbers, and is assumed to satisfy EfwðkÞg ¼ 0, EfwðkÞ2 g ¼ 1, and EfwðiÞwðjÞg ¼ 0 for iaj. A, B, and C are known real constant matrices. Without the loss of generality, it is assumed that the pair (A,B) is controllable and the input matrix B has full column rank. The unknown matrix DCðkÞ represents uncertainty satisfying DCðkÞ ¼ EF ðkÞH, where E and H are known real constant matrices, and F(k) is an unknown time-varying matrix
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satisfying F T ðkÞF ðkÞrI,8t. The unknown nonlinear function f ðxðkÞ,kÞ is the external disturbance satisfying Jf ðxðkÞ,kÞJrr with r known positive constant. In Eq. (1), there exists the xðkÞwðkÞ term, which is also called bilinear noise perturbation in [25]. This implies that, in the controlled system (1), the noise perturbation is not purely external, but depends on system states. Hence, it is more difficult to stabilize the system (1) than the one with only purely external noise perturbation. The objective in this work is to synthesize a discrete-time stochastic SMC system. Thus, a sliding function s(k) should firstly be chosen, and then a DSMC law u(k) will be designed such that the reachability of the specified sliding surface is attained. It is known that, in the discrete-time control system, the control input is held as a constant during the sampling period. Thus, the finite sampling rate may yield that the system state may approach the sliding surface but it is generally unable to stay on it. This means that the state trajectories would move about the surface, which yields a sliding-like mode, termed as quasi-sliding mode (QSM) in [23]. In the following, a definition on QSM is firstly made, which is a revision one in [23]. Definition 1. The motion of a discrete-time system is called a quasi-sliding mode (QSM) if this motion stays within the e-neighborhood of the specified sliding surface, i.e., jsðkÞjoe for kZkn , where e is the band of quasi-sliding domain, and kn is a positive constant. As e ¼ 0, an ideal sliding mode similar to continuous-time one will happen. Remark 1. It is worth noting that, in Definition 1, the system state is not required to cross the switching surface in each successive sampling period. Instead, the system state is only required to enter and stay within the e-neighborhood of the surface in finite time. This is different from some existing works, e.g., Ref. [23], in which the state trajectory is required to cross the switching surface in every successive sampling period, once it has crossed the surface for the first time. Actually, the zigzag motion about the surface is not necessary for the existing of an QSM. Especially, the zigzag motion about the switching may result in the undesirable chattering. Before proceeding, the following definition and lemma [26] will be given, which are useful for the development of our result. Definition 2. For the stochastic system (1), if there exist constants a40 and t 2 ð0,1Þ such that EfJZðkÞJ2 gratk EfJZð0ÞJ2 g, where ZðkÞ denotes the solution of stochastic systems with initial state Zð0Þ, then, the stochastic system (1) is said to be exponentially mean square stable. Lemma 1. Let V ðZðkÞÞ be a Lyapunov functional. If there exist real scalars lZ0, m40, n40, and 0oco1 such that mJZðkÞJ2 rV ðZðkÞÞrnJZðkÞJ2 , and EfV ðZðk þ 1ÞÞjZðkÞgV ðZðkÞÞrlcV ðZðkÞÞ,
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then the sequence ZðkÞ satisfies n l : EfJZðkÞJ2 gr EfJZð0ÞJ2 gð1cÞk þ m mc Lemma 2. For any real vectors a, b and matrix X 40 of compatible dimensions: aT b þ bT araT Xa þ bT X 1 b: Lemma 3. Let A, E, H, and F(t) be real matrices of appropriate dimensions with F(t) satisfying F T ðtÞF ðtÞrI. Then, for any real matrices Q ¼ QT , we have Q þ EF ðtÞH þ H T F T ðtÞE T o0
8F ðtÞ s:t: F ðtÞT F ðtÞrI,
if and only if there exists some scalar e40 such that Q þ eEE T þ e1 H T Ho0: 3. The synthesis of DSMC In this section, a sliding surface will be firstly chosen, and the stability of stochastic system (1) on the specified sliding surface is analyzed. Finally, a DSMC law will be designed such that the reachability of a QSM on the specified sliding surface is attained.
3.1. Integral sliding surface In this work, an integral-like sliding function is constructed as follows: sðkÞ ¼ GxðkÞGAxðk1Þ,
ð2Þ
where G is a design matrix to be given later so that GB is nonsingular. It can be shown from the matrix theory that the nonsingularity of GB can be ensured by choosing G ¼ BT P with P40, since B is assumed to be of full column rank. It is noted that the ideal sliding mode satisfies sðk þ 1Þ ¼ sðkÞ ¼ 0
for kZkn ,
ð3Þ
where kn is a positive constant. Thus, when the state trajectories of the system enter the ideal sliding mode, the equivalent control law may be obtained from Eqs. (1) to (3): ueq ðkÞ ¼ ðGBÞ1 GðC þ DCðkÞÞxðkÞwðkÞf ðxðkÞ,kÞ:
ð4Þ
Hence, by substituting Eq. (4) into the system (1), we obtain the sliding mode dynamics in the sliding surface sðkÞ ¼ 0 as follows: xðk þ 1Þ ¼ AxðkÞ þ ½IBðGBÞ1 GðC þ DCðkÞÞxðkÞwðkÞ:
ð5Þ
In the sequel, we shall analyze the performance of the sliding motion described by Eq. (5) and derive the condition of stochastic stability.
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3.2. Stability of sliding motion Theorem 1. Consider the stochastic system (1) and the sliding surface specified by Eq. (2). If there exist symmetric matrix P40 and scalar e40 satisfying the following LMI: 3 2 C T PB 0 P þ eH T H AT P C T P 7 6 PA P 0 0 0 7 6 7 6 6 ð6Þ PC 0 0:5P 0 PE 7 7o0, 6 7 6 BT PC 0 0 0:5BT PB BT PE 5 4 0 0 ET P E T PB eI and sliding mode matrix G ¼ BT P, then the sliding mode dynamics (5) is stochastically stable. Proof. By Schur’s complement, it is seen that LMI (6) is equivalent to the following inequality: 2 3 P AT P C T P C T PB 6 PA 7 P 0 0 6 7 ð7Þ 6 7 þ eN T N þ e1 MM T o0 4 PC 5 0 0:5P 0 BT PC
0
0
0:5BT PB
with M ¼ ½0 0 E T P E T PBT ,
N ¼ ½H 0 0 0:
On the other hand, it is noted that 2 P AT P ðC þ DCðkÞÞT P 6 6 PA P 0 6 6 PðC þ DCðkÞÞ 0 0:5P 4 BT PðC þ DCðkÞÞ 0 0 2 P AT P C T P C T PB 6 PA P 0 0 6 ¼6 4 PC 0 0:5P 0 BT PC
0
0
ðC þ DCðkÞÞT PB 0 0 0:5BT PB
3 7 7 7 7 5
3 7 7 T 7 þ MF ðkÞN þ N T F ðkÞ M T : 5
ð8Þ
0:5BT PB
Hence, by Lemma 3, it is shown from Eqs. (33) and (8) that 3 2 P AT P ðC þ DCðkÞÞT P ðC þ DCðkÞÞT PB 7 6 7 6 PA P 0 0 7o0: 6 7 6 PðC þ DCðkÞÞ 0 0:5P 0 5 4 BT PðC þ DCðkÞÞ 0 0 0:5BT PB
ð9Þ
Now, choose the following Lyapunov function candidate for stochastic system (5): V ðxðkÞ ¼ xðkÞT PxðkÞ,
ð10Þ
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then, we have EfV ðxðk þ 1ÞÞjxðkÞgV ðxðkÞÞrxðkÞAT PAxðkÞ þ2xðkÞT ðC þ DCðkÞÞT PðC þ DCðkÞÞxðkÞ þ2xðkÞT ðC þ DCðkÞÞT GT ðGBÞ1 GðC þ DCðkÞÞxðkÞxðkÞT PxðkÞ ¼ xðkÞT YxðkÞ
ð11Þ
with Y ¼ AT PA þ 2ðC þ DCðkÞÞT PðC þ DCðkÞÞ þ2ðC þ DCðkÞÞT PBðBT PBÞ1 BT PðC þ DCðkÞÞP: By Schur’s complement and the matrix inequality (9), we obtain Yo0, which implies that there exists a small scalar a40 such that YoaI. Hence, we obtain from (11) that EfV ðxðk þ 1ÞÞjxðkÞgV ðxðkÞÞoajxðkÞj2
for all xðkÞa0:
ð12Þ
Taking expectation with respect to both sides of Eq. (12), we have EfV ðxðk þ 1ÞÞgEfV ðxðkÞÞgoaEfjxðkÞj2 g:
ð13Þ
Hence, by summing up both sides of Eq. (13) from 0 to N for any integer N41, we obtain ( ) N X 2 EfV ðxðN þ 1ÞÞgEfV ðxð0ÞÞgoaE jxðkÞj , k¼0
which yields ( ) N X 1 jxðkÞj2 o ½EfV ðxð0ÞÞgEfV ðxðN þ 1ÞÞg E a k¼0 1 r EfV ðxð0ÞÞgrcEfjxð0Þj2 g ð14Þ a with c ¼ ð1=aÞlmax ðPÞ. Taking N-1, it is shown from Eq. (14) and Definition 2 that the sliding mode dynamics (5) is robustly stochastically stable. & In the above derivation, by means of the discrete-time stochastic Lyapunov method, we obtain a sufficient condition for the stability of stochastic system (1) on the specified sliding surface sðkÞ ¼ 0. In the following, a DSMC law will be further designed such that the reachability of the specified sliding surface is ensured. 3.3. The design of sliding mode controller For the stochastic system (1) and the sliding function (2), we have sðk þ 1Þ ¼ GBuðkÞ þ DðkÞ þ GðC þ DCðkÞÞxðkÞwðkÞ
ð15Þ
with DðkÞ ¼ GBf ðxðkÞ,kÞ. It follows from Jf ðxðkÞ,kÞJrr that there exist known bounds di , d i (i ¼ 1,2, . . . ,m) satisfying d i ðkÞrdi ðkÞrd i ðkÞ,
ð16Þ
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where di(k) is the ith element in D(k). Then, define d ðkÞ d ðkÞ dio ðkÞ ¼ d i ðkÞ þ i , dis ðkÞ ¼ d i ðkÞ i ð17Þ 2 2 and Do ðkÞ ¼ ½d1o ðkÞ d2o ðkÞ . . . dmo ðkÞT , Ds ðkÞ ¼ diagfd1s ðkÞ,d1s ðkÞ, . . . ,d1s ðkÞg. For notational simplicity, the variable k in the bound on D1 ðkÞ is omitted in the later design. Now, we design the SMC law as follows: uðkÞ ¼ ðGBÞ1 ½GAxðkÞ þ Do þ Ds sgnðsðkÞÞ:
ð18Þ
Remark 2. It is due to the existence of state-dependent noise perturbation in Eq. (1), the above SMC law (18) cannot ensure the reaching law similar to the one in [24]. Moreover, the structure characteristic of stochastic system (1) also makes the analysis on the reachability more complex. It will be seen from the following proof that the analysis on the reachability of the quasi-sliding mode is made simultaneously with the stability of system states, in which a Lyapunov function in Eq. (22) involving in system states and sliding mode variables will be utilized. In this following, a stochastic Lyapunov method is utilized to analyze the reachability of the sliding surface specified by Eq. (2). It will be seen from the following procedure that a Lyapunov function involving in system states and sliding mode variables will be chosen, which are different from some existing works. Theorem 2. Consider the discrete-time stochastic systems (1) and the integral sliding surface specified by Eq. (2). The sliding mode controller is constructed as Eq. (18). If there exist matrix P40 and scalar e40 satisfying the following LMI: 3 2 P þ eH T H 2AT P AT PB 2AT PB C T P C T PB 0 7 6 2PA P 0 0 0 0 0 7 6 7 6 6 BT PA 0 0:5I 0 0 0 0 7 7 6 7 6 ð19Þ 0 0 BT PB 0 0 0 7o0, 6 2BT PA 7 6 7 6 PC 0 0 0 P 0 PE 7 6 7 6 T B PC 0 0 0 0 I BT PE 5 4 0 0 0 0 E T P E T PB eI with G ¼ BT P, then the state trajectories are driven (with mean square) into a band of the specified sliding surface (2). That is, the quasi-sliding mode is attained (with mean square). Proof. It follows from Eqs. (1) and (18) that the closed-loop control system is given as xðk þ 1Þ ¼ ½ABðGBÞ1 GAxðkÞ þ ðC þ DCðkÞÞxðkÞwðkÞ þ BðGBÞ1 ½DðkÞDo Ds sgnðsðkÞÞ:
ð20Þ
Besides, substituting Eq. (18) into Eq. (15) yields sðk þ 1Þ ¼ GAxðkÞ þ GðC þ DCðkÞÞxðkÞwðkÞ þ DðkÞDo Ds sgnðsðkÞÞ:
ð21Þ
Now, choose the Lyapunov function candidate as follows: V ðZðkÞÞ ¼ xðkÞT PxðkÞ þ sðkÞT sðkÞ,
ð22Þ
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with ZðkÞ ¼ ½xðkÞT ,sðkÞT T . Thus, we have EfV ðZðk þ 1ÞÞjZðkÞgV ðZðkÞÞ ¼ Efxðk þ 1ÞT Pxðk þ 1Þ þ sðk þ 1ÞT sðk þ 1ÞjZðkÞgxðkÞT PxðkÞsðkÞT sðkÞ:
ð23Þ
Along the state trajectories of system (20), we obtain Efxðk þ 1ÞT Pxðk þ 1ÞjZðkÞg ¼ xðkÞT ½ABðGBÞ1 GAT P½ABðGBÞ1 GAxðkÞ þxðkÞT ðC þ DCðkÞÞT PðC þ DCðkÞÞxðkÞ þ2xðkÞT ½ABðGBÞ1 GAT PBðGBÞ1 ½DðkÞDo Ds sgnðsðkÞÞ þ½DðkÞDo Ds sgnðsðkÞÞT ðGBÞ1 ½DðkÞDo Ds sgnðsðkÞÞ:
ð24Þ
By Lemma 2, it is noted that ½ABðGBÞ1 GAT P½ABðGBÞ1 GAr2AT PA þ 2AT PBðBT PBÞ1 BT PA,
ð25Þ
2xðkÞT ½ABðGBÞ1 GAT PBðGBÞ1 ½DðkÞDo Ds sgnðsðkÞÞ r2xðkÞT AT PAxðkÞ þ 2xðkÞT AT PBðBT PBÞ1 BT PAxðkÞ þ ½DðkÞDo Ds sgnðsðkÞÞT ðGBÞ1 ½DðkÞDo Ds sgnðsðkÞÞ:
ð26Þ
Thus, by means of Eqs. (25) and (26), we have from Eq. (24) Efxðk þ 1ÞT Pxðk þ 1ÞjZðkÞg rxðkÞT ½4AT PA þ 4AT PBðBT PBÞ1 BT PA þ ðC þ DCðkÞÞT PðC þ DCðkÞÞxðkÞ þ 2½DðkÞDo Ds sgnðsðkÞÞT ðGBÞ1 ½DðkÞDo Ds sgnðsðkÞÞ:
ð27Þ
Similarly, along the trajectories of Eq. (20), we have Efsðk þ 1ÞT sðk þ 1ÞjZðkÞg ¼ xðkÞT AT GT GAxðkÞ 2xðkÞT AT GT ½DðkÞDo Ds sgnðsðkÞÞ þxðkÞT ðC þ DCðkÞÞT GT GðC þ DCðkÞÞxðkÞ þ½DðkÞDo Ds sgnðsðkÞÞT ½DðkÞDo Ds sgnðsðkÞÞ rxðkÞT ½2AT G T GA þ ðC þ DCðkÞÞT G T GðC þ DCðkÞÞxðkÞ þ2½DðkÞDo Ds sgnðsðkÞÞT ½DðkÞDo Ds sgnðsðkÞÞ:
ð28Þ
Note the fact that JDðkÞDo Ds sgnðsp ðkÞÞJr2JDs J
ð29Þ
It follows from Eqs. (23), (27)–(29) that EfV ðZðk þ 1ÞÞjZðkÞgV ðZðkÞÞrxðkÞT PxðkÞsðkÞT sðkÞ þ xJDs J2
ð30Þ
with x ¼ 8 þ 8JðGBÞ1 J, and P ¼ P þ 2AT G T GA þ 4AT PA þ ðC þ DCÞT PðC þ DCÞþ 4AT PBðBT PBÞ1 BT PA þ ðC þ DCÞT GT GðC þ DCÞ. It is seen from Eqs. (22) and (30) that as Po0, we have, for JsðkÞJZðx1=2 =ð1dÞ1=2 ÞJDs J, EfV ðZðk þ 1ÞÞjZðkÞgV ðZðkÞÞrxðkÞT PxðkÞdsðkÞT sðkÞ rlmin ðSÞZðkÞT ZðkÞ omZðkÞT ZðkÞ,
ð31Þ
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where 0odo1, S ¼ diagðP,dIÞ, and 0omominflmin ðSÞ,gg,
g :¼ maxflmax ðPÞ,1g:
Thus, it follows from Eq. (31) that m EfV ðZðk þ 1ÞÞjZðkÞgV ðZðkÞÞomZðkÞT ZðkÞo V ðZðkÞÞ :¼ cV ðZðkÞÞ g
ð32Þ
with 0oco1. Therefore, it is obtained from Lemma 1 and Definition 2 that for JsðkÞJZðx1=2 =ð1dÞ1=2 ÞJDs J, the closed-loop system (20) is exponentially mean square stable. This means that the quasi-sliding mode is attained (with mean square) within the band of the sliding surface specified by Eq. (2), and the system states converge to a small neighborhood of zero, i.e., JsðkÞJoðx1=2 =ð1dÞ1=2 ÞJDs J. On the other hand, by Schur’s complement, it is shown that Po0 is equivalent to the following matrix inequality: 3 2 P 2AT P AT PB 2AT PB ðC þ DCÞT P ðC þ DCÞT PB 7 6 7 6 2PA P 0 0 0 0 7 6 T 7 6 B PA 0 0:5I 0 0 0 7 6 7o0: 6 7 6 2BT PA 0 0 BT PB 0 0 7 6 7 6 PðC þ DCÞ 0 0 0 P 0 5 4 T B PðC þ DCÞ 0 0 0 0 I ð33Þ Further, by following a similar approach as in Eqs. (7)–(9), it may be shown that the matrix inequality (33) is implied by LMI (19). & Remark 3. Notice that the analysis on the reachability of quasi-sliding mode is considered simultaneously with the stability of the system states. This approach is different from the method used in the continuous-time stochastic SMC as in [8]. The proposed idea is just a typical feature of SMC for discrete-time stochastic systems. Remark 4. It is shown from Theorems 1 and 2 that if there exist matrix P40 and scalar e40 satisfying LMIs (6) and (19), the state trajectories are driven into the band of the sliding surface sðkÞ ¼ 0 and asymptotically tends to a small neighborhood of zeros. Moreover, the ideal sliding motion on sðkÞ ¼ 0 is stochastically stable. 4. Numerical simulation Consider the discrete-time stochastic system (1) with 2 3 2 2 3 0:2 0:3 0:5 0:2 0:5 1:6 6 7 6 6 7 A ¼ 4 0:1 0:2 0:2 5, B ¼ 4 2:1 1:3 5, C ¼ 4 0:1 0
0:2
E ¼ ½0:2 0:2 0:1T , F ðkÞ ¼ 0:5 sinðkÞ,
0:3
0:8
0:4
H ¼ ½0:2 0:1 0:2, " # 0:6 sinðx1 ðkÞx2 ðkÞÞ f ðxðkÞÞ ¼ : 0:3 sinðx3 ðkÞÞ
0:2
0:2 0:2 0:3
0:1
3
7 0:1 5, 0:1
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The objective is to design an SMC law such that the state trajectories can be driven onto the sliding surface and the sliding motion in the specified sliding manifold is stochastically stable. To this end, we solve LMIs (6) and (19) and obtain the solutions and the sliding mode gain matrix G as follows: 2 3 0:0245 0:0071 0:0198 6 7 P ¼ 4 0:0071 0:1407 0:0669 5, e ¼ 0:0472, 0:0198 0:0669 0:1900
T
G¼B P¼
0:0186
0:2385
0:0017
0:0564
0:1676
0:0208
:
1
x1 x2 x3
0.5 0 −0.5 −1 −1.5 −2
0
1
2
3
4
5
Fig. 1. The trajectories of state x(t).
0.15
s1 s2
0.1 0.05 0 −0.05 −0.1 −0.15 −0.2 −0.25 −0.3
0
1
2
3
4
Fig. 2. The trajectories of sliding variable s(t).
5
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0.6
1507
u1 u2
0.4 0.2 0 −0.2 −0.4 −0.6 −0.8
0
1
2
3
4
5
Fig. 3. The control signal u(t).
Thus, the proposed SMC law is designed as in Eq. (18), and the simulation results are shown in Figs. 1–3, in which the initial state is chosen as xð0Þ ¼ ½1 0:5 1T . It is seen that the proposed control law can quickly drive the state trajectories toward the sliding surface sðkÞ ¼ 0, and asymptotically tend to zero along the sliding surface sðkÞ ¼ 0. Hence, the simulation results show the effectiveness of the proposed method in this work. 5. Conclusions In this work, the problem of SMC for discrete-time stochastic systems has been considered. It has been shown that, for the discrete-time stochastic systems, the analysis on the reachability of the quasi-sliding mode had to be considered simultaneously with the stability of the system states. In this work, it is noted that under the quasi-sliding mode, only the boundedness of system states is ensured. Hence, it would be an interesting topic to further investigate the results along this direction. Acknowledgment The research was partially supported by a grant from Geral Research Fund (CityU 101109), NNSF from China (61074041), the Technology Innovation Key Foundation of Shanghai Municipal Education Commission (09ZZ60), and Shanghai Leading Academic Discipline Project (B504). References [1] W. Zhang, B.-S. Chen, C.-S. Tseng, Robust H1 filtering for nonlinear stochastic systems, IEEE Transactions on Signal Processing 53 (2005) 589–598. [2] S. Xu, T. Chen, H1 output feedback control for uncertain stochastic systems with time-varying delay, Automatica 40 (2004) 2091–2098. [3] M. Basin, D. Calderon-Alvarez, Optimal controller for uncertain stochastic polynomial systems with deterministic disturbances, in: American Control Conference, St. Louis, USA, 2009, pp. 778–783.
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