Robust <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-4.458401pt" id="M1" height="16.0827pt" version="1.1" viewBox="0 -11.6243 26.854 16.0827" width="26.854pt"><g transform="matrix(.018,0,0,-0.018,0,0)"><path id="g113-73" d="M828 650H570L564 622C646 614 650 609 634 524L604 366H253L281 524C296 608 308 615 382 622L388 650H132L126 622C211 615 214 609 198 524L121 122C106 42 102 35 23 28L17 0H276L282 28C196 35 191 40 206 122L243 324H597L559 123C544 42 537 35 452 28L446 0H710L716 28C632 35 628 43 643 122L719 524C735 610 741 615 822 622L828 650Z"></path></g><g transform="matrix(.013,0,0,-0.013,13.86,4.308)"><path id="g50-30" d="M1000 227C1000 351 905 451 780 451C658 451 590 380 530 279C465 385 388 451 261 451C168 451 38 376 38 210C38 98 121 -12 261 -12C381 -12 448 59 508 160C573 53 652 -12 780 -12C879 -12 1000 64 1000 227ZM485 199C434 109 370 25 271 25C188 25 133 117 133 241C133 355 189 414 249 414C357 414 426 302 485 199ZM903 204C903 70 852 25 790 25C681 25 611 136 552 240C603 330 669 414 768 414C855 414 903 321 903 204Z"></path></g></svg> Fuzzy Control for Nonlinear Discrete-Time Stochastic Systems with Markovian Jump and Parametric Uncertainties

Sun, Huiying; Yan, Long

doi:https://doi.org/10.1155/2014/235804

Mathematical Problems in Engineering

On this page

Abstract Introduction Preliminaries Conclusions Acknowledgments References Copyright Related Articles

Special Issue

Recent Advances on the Theory and Applications of Networked Control Systems

View this Special Issue

Research Article | Open Access

Volume 2014 | Article ID 235804 | https://doi.org/10.1155/2014/235804

Robust Fuzzy Control for Nonlinear Discrete-Time Stochastic Systems with Markovian Jump and Parametric Uncertainties

Huiying Sun¹and Long Yan¹

Academic Editor: Peng Shi

Received04 May 2014

Revised02 Aug 2014

Accepted03 Aug 2014

Published18 Aug 2014

Abstract

The paper mainly investigates the fuzzy control problem for a class of nonlinear discrete-time stochastic systems with Markovian jump and parametric uncertainties. The class of systems is modeled by a state space Takagi-Sugeno (T-S) fuzzy model that has linear nominal parts and norm-bounded parameter uncertainties in the state and output equations. An control design method is developed by using the Lyapunov function. The decoupling technique makes the Lyapunov matrices and the system matrices separated, which makes the control design feasible. Then, some strict linear matrix inequalities are derived on robust norm conditions in which both robust stability and performance are required to be achieved. Finally, a computer-simulated truck-trailer example is given to verify the feasibility and effectiveness of the proposed design method.

1. Introduction

Over the past decade, there has been a rapidly growing interest in control and filtering of nonlinear systems, and there have been many successful applications [1–4]. In [1], based on the sum of squares approach, sufficient conditions for the existence of a nonlinear state feedback controller for polynomial discrete-time systems are given in terms of solvability of polynomial matrix inequalities. Reference [3] presents a stochastic distribution control algorithm; an optimal control law is then obtained using the penalty function method. Despite the success, it has become evident that many basic issues remain to be addressed. In particular, the control technique based on the so-called Takagi-Sugeno (T-S) fuzzy model [5] has attracted a great deal of attention (see [6–12]). This is because it is regarded as a powerful solution to bridging the gap between the fruitful linear control and the fuzzy logic control targeting complex nonlinear systems. The common practice of the technique is as follows. First, the nonlinear plant is represented by the T-S fuzzy model. This fuzzy model is described by a set of fuzzy IF-THEN rules which correspond to local linear input-output relations of the system, respectively. The overall model of the system is achieved by fuzzy “blending” of these fuzzy models. Then, based on this fuzzy model, a control design is carried out based on the fuzzy models via the so-called parallel distributed compensation (PDC) scheme. The idea is that a linear feedback control is designed for each local linear model. The resulting overall controller, which is, in general, nonlinear, is again a fuzzy blending of each individual linear controller. Their works introduce the model-based analysis methods into fuzzy logic control.

In recent years, the stability issue and robust performance of T-S fuzzy control systems have been discussed in an extensive literature. Since the pioneering work on the so-called optimal control theory, there has been a dramatic progress in the control theory. Recently, the problem of nonlinear control was intensively studied (see [5, 7, 8, 13]). The design of controller for fuzzy dynamic systems was presented in the paper [5]. Reference [4] introduces a new class of discrete-time networked nonlinear systems with mixed random delays and packet dropouts, and sufficient conditions for the existence of an admissible filter are established, which ensure the asymptotical stability as well as a prescribed performance. The authors in [13] analyzed the control design problem systematically for a class of nonlinear stochastic active fault-tolerant control systems with Markovian parameters. controller design which was considered in [7] noted that most of the aforementioned research efforts focused on the use of single quadratic Lyapunov function, which tended to give more conservative conditions. This is because with the use of parameter-dependent or basis-dependent Lyapunov function less conservative control results can be obtained than with the use of single Lyapunov quadratic function. More recently, there were many results on stability analysis and control synthesis of discrete-time uncertain systems based on parameter-dependent Lyapunov functions [9] or basis-dependent Lyapunov function (see [8, 14]). It is shown that with the use of Lyapunov function well-pleasing control results can be obtained.

In practice, a lot of physical systems have variable structures subject to random changes. These changes may result from abrupt phenomena such as random failures and repairs of the components, changes in the interconnections of subsystems, and sudden environment changes. The so-called Markovian jump systems (MJSS) are special class of hybrid systems. As one of the most basic dynamics models, Markov jump nonlinear system can be used to represent these random failure processes in manufacturing and some investment portfolio models. In the MJSS, the random jumps in system parameters are governed by a Markov process which takes values in a finite set. The class of systems may represent a large variety of processes including those in the production systems, fault-tolerant systems, communication systems, and economic systems [15]. In the past two decades, many important issues of MJSS were researched extensively, such as the controllability and observability [16], stability and stabilization (see [17–22]), [23] and performance [24–27], and robustness [28, 29]. To the best of our knowledge, despite these efforts, very few results are available for the control design for nonlinear MJSS. In this study, a mode-dependent control design is proposed to achieve robust stability of uncertain discrete-time nonlinear MJSS via fuzzy control.

In the paper, robust control is investigated for nonlinear discrete-time stochastic MJSS with parametric uncertainties. Under our proposed fuzzy rules, the nonlinear discrete-time stochastic MJSS could be translated into a class of equivalent T-S fuzzy models with parametric uncertainties. The uncertainty could be translated into a linear fractional form, which includes the norm-bounded uncertainty as a special case and can describe a kind of rational nonlinearities. With the PDC scheme, the control law is designed to make the closed-loop system with norm bound stable. Besides, some matrix variables and the Lyapunov function for robust stabilization with -norm bound for the fuzzy discrete-time stochastic MJSS are given. It is shown that the solution of the control design problem can be obtained by solving a class of LMIs.

The paper is organized as follows. Section 2 discusses the T-S fuzzy models. And definitions and preliminary results are given for uncertain nonlinear discrete-time MJSS. Section 3 gives the analysis results of robust stability with performance, and the results are employed in the following to develop an control design. In Section 4, a numerical simulation example is proposed to illustrate the effectiveness of the approach. Finally, the paper is concluded in Section 5.

For convenience, the following basic notations are adopted throughout the paper. denotes the -dimensional real space, and denotes the set of all real matrices. indicates the transpose of matrix and represents a nonnegative definite (positive definite) matrix. Similarly, represents a nonpositive definite matrix (negative definite). refers to the space of square summable infinite vector sequences. stands for the usual norm. represents the mathematical expectation.

2. Problem Formulation and Preliminaries

Consider the following nonlinear MJSS: Assume that , , , , , , and are Borel measurable on . And is the system state, and represent the system control inputs and disturbance signal, and is system output. is a sequence of real random variables defined on a complete probability space , which is wide sense stationary, second-order processes with and , where refers to a Kronecker function; that is, if and if . is a measurable Markov chain taking values in a finite set , with transition probability matrix , where The paper considers the nonlinear discrete-time MJSS which can be described by the following T-S fuzzy model with uncertainties.

Rule i. If is , is is , then The are fuzzy sets; is the number of IF-THEN rules. Moreover, , , , , , , and are system matrices with parametric uncertainties. Besides, are the premise variables of the fuzzy modes and they are the functions of state variables.

Following the PDC scheme, we consider a state feedback fuzzy controller which shares the same structure of the above T-S fuzzy system, as follows.

Rule j. IF is , is is , then where are the local feedback gain matrices.

And the fuzzy basis functions are given by where is the grade of membership of in . By definition, the fuzzy basis functions satisfy A more compact presentation of the discrete-time T-S fuzzy model is given by where And , , , , , , and are assumed to be deterministic matrices with appropriate dimension; , and are unknown matrices which represent the time-varying parameter uncertainties and are assumed to be of the form where , and are known real constant matrices with appropriate dimension and unknown nonlinear time-varying matrix satisfying . To guarantee that the matrix is invertible for all admissible , it is necessary that .

The following definition and lemmas will be used later.

Definition 1. The discrete-time unforced uncertain fuzzy system is said to be robust stable with norm bound if it is stable with norm bound for all uncertainly admissible . For a given control law (4) and a prescribed level of disturbance attenuation to be achieved, the discrete-time fuzzy system (3) is said to be stabilizable with norm bound if for all , , the closed-loop system (7)-(8) is asymptotically stable and the response of the system under the zero initial condition satisfies

Lemma 2. It is supposed that , , and , so the discrete-time MJSS becomes The system (12) is globally asymptotically stable if there exists a symmetric piecewise matrix such that where .

Lemma 3. Given a matrix , , suppose , , . If then

Proof. It is noted that which gives Therefore, the result of Lemma 3 can be easily proven.

Lemma 4 (see [14]). Suppose is given by (10). With matrices , , and with appropriate dimension, the inequality holds for all such that , if and only if, for some ,

3. Robust Stability and Performance Analysis

In this section, the stability and performance for the nominal fuzzy system will be analyzed. Under control law (4), the closed-loop fuzzy system becomes When , the nominal closed-loop system becomes where

Theorem 5. The nominal closed-loop fuzzy system (22) is stable with norm bound ; that is, for all nonzero under the zero initial condition, if there exists matrices for all such that where

Proof. Obviously, inequality (25) implies the following inequality: It can be checked from the result of Lemma 3 that for all , When , the inequality (28) becomes Let When , the system (22) becomes In what follows, we will drop the argument of for clarity. By some algebraic manipulation, with , the difference of Lyapunov function given by along the solution of system (31) is where It follows from (13), (28), and (29) that which proves the stability of system (31).
It follows from (25) that for all exists Let For zero initial condition , one has Therefore, where defines the difference of along system (22). It is noted that, with defined by we have Then, it is obtained that which can be rewritten as By following similar line as in the proof of (34) it can be checked that, for any , which gives for any nonzero , , and .

Theorem 6. For the nominal fuzzy system (20), there exists a state feedback fuzzy control law (4) such that the closed-loop system is stable with norm bound , if there exist matrices , , and , satisfying the following LMIs: where represents the transposed matrices that are readily inferred by symmetry for all and except the pairs such that , for all . A robust stabilizing controller gain can be given by

Proof. By using (24) and (44), (43) becomes which gives . implies that yielding Note that is invertible. Premultiplying and postmultiplying to (46) give Set and ; there is It follows from Schur complement equivalence that The result then follows from Theorem 5.

Theorem 7. For the uncertain discrete-time MJSS (7), there exists a state feedback fuzzy control law (4) such that the closed-loop system is stable with norm bound , if there exist matrices , , and , satisfying the following LMIs: where represents the transposed matrices that are readily inferred by symmetry for all and except the pairs such that , for all . A robust stabilizing controller gain can be given by

Proof. By using Lemma 4, it can be checked that the feasibility of inequality (50) is equivalent to where It follows from (51)–(53) that which implies that the closed-loop system (20) and (21) is robustly stable with norm bound by Theorem 6.

Corollary 8. Consider the nominal unforced system The nominal unforced fuzzy system is stable with norm bound ; that is, for all nonzero under the zero initial condition, if there exist matrices , for all such that where .

Corollary 9. The nominal unforced fuzzy system (55) is stable with norm bound , if there exist matrices , , satisfying the following LMIs:

Corollary 10. When , the unforced system (7) and (8) is robustly stable with norm bound , if there exist matrices and , satisfying the following LMIs:

4. Numerical Simulation Example

In this section, to illustrate the proposed new fuzzy control method, the backing-up control of a computer-simulated truck-trailer is considered [30]. For example, the truck-trailer model is given by where is the angle difference between truck and trailer, is the angle of trailer, and is the vertical position of rear end of trailer. The parameter is used to describe the actuator failure, where implies no failure, implies a total failure, and implies a partial failure. It is assumed that there is no actuator failure. Equations (59) and (60) are linear, but (61) is nonlinear. The model parameters are given as , , , , and .

The transition probability matrix that relates the three operation modes is given as follows: As in [30], we set and the nonlinear term as where , and . By solving the equations, it can be seen that the membership functions , have the following relations. When is about 0 rad, , , and when is about rad, , . Then the following fuzzy models can be used to design the fuzzy controller for the uncertain nonlinear MJSS.

Rule 1. If is about 0 rad, then

Rule 2. If is about rad, then where In the above uncertain fuzzy models, the uncertainty to describe the modeling error is assumed to be in the following form: where We choose the following matrices for the uncertain discrete-time MJSS: We set , , , and , , , , respectively. Based on Theorem 7 and using LMI control toolbox in Matlab to solve LMIs (50), we can obtain the feasible set of solutions as follows: By (51), there are the local state feedback gains given by Then, we can get It can be found that the LMIs of Theorem 7 have some feasible solution for the uncertain discrete-time MJSS. Setting , , the aforementioned simulation results are obtained. Employing the feedback gains , the fuzzy controller can be obtained by (4). By using Theorem 7, with the fuzzy control applied, if the LMIs in (50) and (25) have a positive-definite solution for and , respectively, then the system (7) and (8) driven by the designed fuzzy controller is stable with satisfying the performance constraint.

5. Conclusions

In the paper, the robust control has been discussed for a class of nonlinear discrete-time stochastic MJSS. First, a new LMI characterization of stability with norm bound for uncertain discrete-time stochastic MJSS has been given. Moreover, sufficient conditions on robust stabilization and performance analysis and control have been presented on LMIs. Furthermore, there are some corollaries of the stability, and the nominal unforced system has been given. Finally, a numerical simulation example has been presented to show the effectiveness.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The work is supported by the National Natural Science Foundation of China (no. 61304080 and no. 61174078), a Project of Shandong Province Higher Educational Science and Technology Program (no. J12LN14), the Research Fund for the Taishan Scholar Project of Shandong Province of China, and the State Key Laboratory of Alternate Electrical Power System with Renewable Energy Sources (no. LAPS13018).

References

S. Saat, D. Huang, and S. K. Nguang, “Robust state feedback control of uncertain polynomial discrete-time systems: an integral action approach,” International Journal of Innovative Computing, Information and Control, vol. 9, no. 3, pp. 1233–1244, 2013.
View at: Google Scholar
R. N. Yang, P. Shi, G. P. Liu, and H. J. Gao, “Network-based feedback control for systems with mixed delays based on quantization and dropout compensation,” Automatica, vol. 47, no. 12, pp. 2805–2809, 2011.
View at: Publisher Site | Google Scholar | MathSciNet
J. H. Zhang, M. F. Ren, Y. Tian, G. L. Hou, and F. Fang, “Constrained stochastic distribution control for nonlinear stochastic systems with non-Gaussian noises,” International Journal of Innovative Computing, Information and Control, vol. 9, no. 4, pp. 1759–1767, 2013.
View at: Google Scholar
R. Yang, P. Shi, and G. Liu, “Filtering for discrete-time networked nonlinear systems with mixed random delays and packet dropouts,” IEEE Transactions on Automatic Control, vol. 56, no. 11, pp. 2655–2660, 2011.
View at: Publisher Site | Google Scholar | MathSciNet
S. G. Cao, N. W. Rees, and G. Feng, “ $H_{\infty}$ control of uncertain dynamical fuzzy discrete-time systems,” IEEE Transactions on Systems, Man, and Cybernetics, vol. 31, no. 5, pp. 802–812, 2001.
View at: Publisher Site | Google Scholar
X. J. Su, P. Shi, L. G. Wu, and S. K. Nguang, “Induced l₂ filtering of fuzzy stochastic systems with time-varying delays,” IEEE Transactions on Cybernetics, vol. 43, no. 4, pp. 1251–1264, 2013.
View at: Google Scholar
Y. Y. Cao and P. M. Frank, “Robust $H_{\infty}$ disturbance attenuation for a class of uncertain discrete-time fuzzy systems,” IEEE Transactions on Fuzzy Systems, vol. 8, no. 4, pp. 406–415, 2000.
View at: Publisher Site | Google Scholar
D. J. Choi and P. Park, “H∞ state-feedback controller design for discrete-time fuzzy systems using fuzzy weighting-dependent lyapunov functions,” IEEE Transactions on Fuzzy Systems, vol. 11, no. 2, pp. 271–278, 2003.
View at: Publisher Site | Google Scholar
G. Feng, “Stability analysis of piecewise discrete-time linear systems,” IEEE Transactions on Automatic Control, vol. 47, no. 7, pp. 1108–1112, 2002.
View at: Publisher Site | Google Scholar | MathSciNet
G. Feng and J. Ma, “Quadratic stabilization of uncertain discrete-time fuzzy dynamic systems,” IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, vol. 48, no. 11, pp. 1337–1343, 2001.
View at: Publisher Site | Google Scholar
R. Qi, L. Zhu, and B. Jiang, “Fault-tolerant reconfigurable control for MIMO systems using online fuzzy identification,” International Journal of Innovative Computing, Information and Control, vol. 9, no. 10, pp. 3915–3928, 2013.
View at: Google Scholar
X. J. Su, L. G. Wu, and P. Shi, “Sensor networks with random link failures: distributed filtering for T-S fuzzy systems,” IEEE Transactions on Industrial Informatics, vol. 9, no. 3, pp. 1739–1750, 2013.
View at: Publisher Site | Google Scholar
Z. Lin, Y. Lin, and W. H. Zhang, “ $H_{\infty}$ stabilisation of non-linear stochastic active fault-tolerant control systems: fuzzy-interpolation approach,” IET Control Theory &; Applications, vol. 4, no. 10, pp. 2003–2017, 2010.
View at: Publisher Site | Google Scholar | MathSciNet
S. Zhou, G. Feng, J. Lam, and S. Xu, “Robust $H_{\infty}$ control for discrete-time fuzzy systems via basis-dependent Lyapunov functions,” Information Sciences, vol. 174, no. 3-4, pp. 197–217, 2005.
View at: Publisher Site | Google Scholar | MathSciNet
M. Mariton, Jump Linear Systems in Automatic Control, Marcel Dekker, New York, NY, USA, 1990.
Y. D. Ji and H. J. Chizeck, “Controllability, observability and discrete-time Markovian jump linear quadratic control,” International Journal of Control, vol. 48, no. 2, pp. 481–498, 1988.
View at: Publisher Site | Google Scholar | MathSciNet
O. L. V. Costa and M. D. Fragoso, “Stability results for discrete-time linear systems with Markovian jumping parameters,” Journal of Mathematical Analysis and Applications, vol. 179, no. 1, pp. 154–178, 1993.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
W. H. Zhang and B. S. Chen, “On stabilizability and exact observability of stochastic systems with their applications,” Automatica, vol. 40, no. 1, pp. 87–94, 2004.
View at: Publisher Site | Google Scholar | MathSciNet
Y. G. Fang and K. A. Loparo, “Stabilization of continuous-time jump linear systems,” IEEE Transactions on Automatic Control, vol. 47, no. 10, pp. 1590–1603, 2002.
View at: Publisher Site | Google Scholar | MathSciNet
X. Feng, K. A. Loparo, Y. Ji, and H. J. Chizeck, “Stochastic stability properties of jump linear systems,” IEEE Transactions on Automatic Control, vol. 37, no. 1, pp. 38–53, 1992.
View at: Publisher Site | Google Scholar | MathSciNet
L. Arnold, W. Kliemann, and E. Oelijeklaus, “Lyapunov exponents of linear stochastic systems,” in Lyapunov Exponents, vol. 1186 of Lecture Notes in Mathematics, pp. 85–125, Springer, Berlin, Germany, 1986.
View at: Google Scholar
O. L. V. Costa, J. B. R. do Val, and J. C. Geromel, “A convex programming approach to H₂-control of discrete-time Markovian jump linear systems,” International Journal of Control, vol. 66, no. 4, pp. 557–580, 1997.
View at: Publisher Site | Google Scholar | MathSciNet
H. Sun, L. Jiang, and W. Zhang, “Feedback control on nash equilibrium for discrete-time stochastic systems with markovian jumps: finite-horizon case,” International Journal of Control, Automation and Systems, vol. 10, no. 5, pp. 940–946, 2012.
View at: Publisher Site | Google Scholar
O. L. V. Costa and R. P. Marques, “Mixed $H_{2}$ / $H_{\infty}$ control of discrete-time Markovian jump linear systems,” IEEE Transactions on Automatic Control, vol. 43, no. 1, pp. 95–100, 1998.
View at: Publisher Site | Google Scholar | MathSciNet
D. Hinrichsen and A. J. Pritchard, “Stochastic $H_{\infty}$ ,” SIAM Journal on Control and Optimization, vol. 36, no. 5, pp. 1504–1538, 1998.
View at: Publisher Site | Google Scholar | MathSciNet
W. Zhang and B. Chen, “State feedback $H_{\infty}$ control for a class of nonlinear stochastic systems,” SIAM Journal on Control and Optimization, vol. 44, no. 6, pp. 1973–1991, 2006.
View at: Publisher Site | Google Scholar | MathSciNet
E. Boukas and P. Shi, “Stochastic stability and guaranteed cost control of discrete-time uncertain systems with Markovian jumping parameters,” International Journal of Robust and Nonlinear Control, vol. 8, no. 13, pp. 1155–1167, 1998.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
T. Takagi and M. Sugeno, “Fuzzy identification of systems and its applications to modeling and control,” IEEE Transactions on Systems, Man and Cybernetics, vol. 15, no. 1, pp. 116–132, 1985.
View at: Google Scholar | Zentralblatt MATH
H. J. Lee, J. B. Park, and G. Chen, “Robust fuzzy control of nonlinear systems with parametric uncertainties,” IEEE Transactions on Fuzzy Systems, vol. 9, no. 2, pp. 369–379, 2001.
View at: Publisher Site | Google Scholar
K. Tanaka and M. Sano, “Robust stabilization problem of fuzzy control systems and its application to backing up control of a truck-trailer,” IEEE Transactions on Fuzzy Systems, vol. 2, no. 2, pp. 119–134, 1994.
View at: Publisher Site | Google Scholar

Copyright

Copyright © 2014 Huiying Sun and Long Yan. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

PDF Download Citation

Download other formats

Order printed copies

Views

840

Downloads

831

Citations