Observer-Based Finite-Time <svg style="vertical-align:-4.4712pt;width:36.412498px;" id="M1" height="20.262501" version="1.1" viewBox="0 0 36.412498 20.262501" width="36.412498" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"><g transform="matrix(.022,-0,0,-.022,.062,14.638)"><path id="x1D43B" d="M865 650q-1 -4 -4 -14t-4 -14q-62 -5 -77 -19.5t-29 -82.5l-74 -394q-12 -61 -0.5 -77t75.5 -21l-6 -28h-273l8 28q64 5 82 21t29 76l36 198h-380l-37 -197q-11 -64 0.5 -78.5t79.5 -19.5l-6 -28h-268l6 28q60 6 75.5 21.5t26.5 76.5l75 394q13 66 2 81.5t-77 20.5l8 28
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q-27 49 -47 80t-50 67t-64 54t-73 18q-48 0 -81.5 -47t-33.5 -128q0 -96 37.5 -157.5t99.5 -61.5q68 0 117.5 47t94.5 128zM889 204q0 91 -35.5 151t-99.5 60q-68 0 -119 -47t-95 -127q27 -49 47 -80.5t50 -67.5t65 -54t74 -18q113 0 113 183z" /></g></svg> Control of Singular Markovian Jump Systems

Zhang, Yingqi; Cheng, Wei; Mu, Xiaowu; Guo, Xiulan

doi:https://doi.org/10.1155/2012/205727

Journal of Applied Mathematics

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Research Article | Open Access

Volume 2012 | Article ID 205727 | https://doi.org/10.1155/2012/205727

Observer-Based Finite-Time Control of Singular Markovian Jump Systems

Yingqi Zhang,¹Wei Cheng,¹Xiaowu Mu,²and Xiulan Guo¹

Academic Editor: Reinaldo Martinez Palhares

Received09 Nov 2011

Revised27 Dec 2011

Accepted04 Jan 2012

Published04 Apr 2012

Abstract

This paper addresses the problem of finite-time control via observer-based state feedback for a family of singular Markovian jump systems (SMJSs) with time-varying norm-bounded disturbance. Firstly, the concepts of singular stochastic finite-time boundedness and singular stochastic finite-time stabilization via observer-based state feedback are given. Then an observer-based state feedback controller is designed to ensure singular stochastic finite-time stabilization via observer-based state feedback of the resulting closed-loop error dynamic SMJS. Sufficient criteria are presented for the solvability of the problem, which can be reduced to a feasibility problem involving linear matrix inequalities with a fixed parameter. As an auxiliary result, we also discuss the problem of finite-time stabilization via observer-based state feedback of a class of SMJSs and give sufficient conditions of singular stochastic finite-time stabilization via observer-based state feedback for the class of SMJSs. Finally, illustrative examples are given to demonstrate the validity of the proposed techniques.

1. Introduction

In practice, there exist many concerned problems which described that system state does not exceed some bound during some time interval, for instance, large values of the state are not acceptable in the presence of saturations [1–3]. Therefore, we need to check the unacceptable values to see whether the system states remain within the prescribed bound in a fixed finite-time interval. Compared with classical Lyapunov asymptotical stability, in order to deal with these transient performances of control dynamic systems, finite-time stability or short-time stability was introduced in the literatures [4, 5]. Applying Lyapunov function approach, some appealing results were obtained to ensure finite-time stability, finite-time boundedness, and finite-time stabilization of various systems including linear systems, nonlinear systems, and stochastic systems. For instance, Amato et al. [6] investigated the output feedback finite-time stabilization for continuous linear systems. Zhang and An [7] considered finite-time control problems for linear stochastic systems. Recently, Meng and Shen [8] extended the definition of control to finite-time control for linear continuous systems, and a state feedback controller was designed to ensure finite-time boundedness of the resulting systems and the effect of the disturbance input on the controlled output satisfying a prescribed level. For more details of the literature related to finite-time stability, finite-time boundedness, and finite-time control, the reader is referred to [9–20] and the references therein.

On the other hand, singular systems referred to as descriptor systems, differential-algebraic systems, generalized state-space systems, or semistate systems have attracted many researchers since the class of systems have been extensively applied to deal with mechanical systems, electric circuits, chemical process, power systems, interconnected systems, and so on; see more practical examples in [21, 22] and the references therein. A great number of results based on the theory of regular systems or state-space systems have been extensively generalized to singular systems with or without time delay, such as stability [23], stabilization [24], control [25–29], and other issues. Meanwhile, Markovian jump systems are referred to as a special family of hybrid systems and stochastic systems, which are very appropriate to model plants whose parameters are subject to random abrupt changes [30]. Thus, many attracting results and a large variety of control problems have been studied, such as stochastic Lyapunov stability [31–33], sliding mode control [34, 35], robust control [36–40], filtering [41–45], dissipative control [46], passive control [47], guaranteed cost control [48], tracking control [49], and other issues, the readers are refered to [31] and the references therein. It is pointed out that the problem of state feedback stabilization, just as was mentioned above, requires to assume the complete access to the state vector. Practically this assumption is not realistic for many reasons like the nonexistence of the appropriate sensors to measure some of the states or the limitation in the control strategies. Thus, the observer-based control and output feedback control are probably well suited in such situation for feedback control, such as stability [31], control [50–54], passive control [55], and finite-time control [6, 20]. However, to date, the problems of observer-based finite-time stabilization of singular stochastic systems have not been investigated. The problems are important and challenging in many practice applications, which motivates the main purpose of our research.

In this paper, we consider the problem of finite-time control via observer-based state feedback of singular Markovian jump systems (SMJSs) with time-varying norm-bounded disturbance. The results of this paper are totally different from those previous results, although some studies on finite-time control for singular stochastic systems have been conduced, see [18, 56, 57]. The concepts of singular stochastic finite-time boundedness (SSFTB) and singular stochastic finite-time stabilization via observer-based state feedback of singular stochastic systems are given. The main contribution of the paper is to design an observer-based state feedback controller which ensures singular stochastic finite-time stabilization via observer-based state feedback of the resulting closed-loop error dynamic SMJS. Sufficient criterions are presented for the solvability of the problem, which can be reduced to a feasibility problem in terms of linear matrix inequalities with a fixed parameter. As an auxiliary result, we also investigate the problem of observer-based finite-time stabilization via state feedback of a class of SMJSs and give sufficient conditions of singular stochastic finite-time stabilization via observer-based state feedback for the class of SMJSs.

The rest of this paper is organized as follows. In Section 2, the problem formulation and some preliminaries are introduced. The results of singular stochastic finite-time stabilization via observer-based state feedback are given for a class of SMJSs in Section 3. Section 4 presents numerical examples to show the validity of the proposed methodology. Some conclusions are drawn in Section 5.

Notations 1. Throughout the paper, and denote the sets of component real vectors and real matrices, respectively. The superscript stands for matrix transposition or vector. denotes the expectation operator respective to some probability measure . In addition, the symbol denotes the transposed elements in the symmetric positions of a matrix, and stands for a block-diagonal matrix. and denote the smallest and the largest eigenvalues of matrix , respectively. Notations sup and inf denote the supremum and infimum, respectively. Matrices, if their dimensions are not explicitly stated, are assumed to be compatible for algebraic operations.

2. Problem Formulation

Let us consider the following continuous-time singular Markovian jump system (SMJS):where is the state variable, is the controlled output, is the measured output, is the controlled input, is the controlled output, is a singular matrix with , is continuous-time Markov stochastic process taking values in a finite space with transition matrix , and the transition probabilities are described as follows: where , satisfies , and for all . Moreover, the disturbance satisfies the following constraint condition: and the matrices , and are coefficient matrices and of appropriate dimension for all .

For notational simplicity, in the sequel, for each possible , a matrix will be denoted by ; for instance, will be denoted by , by , and so on.

In this paper, we construct the following state observer and feedback controller: where and are the estimated state and output, is an estimated initial state, is to be a designed state feedback gain, and is an observer gain to be designed. Define the state estimated error and . Then the resulting closed-loop error dynamic SMJS can be written in the form as follows: where

Definition 2.1 (regular and impulse-free, see [21, 22]). The SMJS (2.1a) with is said to be regular in time interval if the characteristic polynomial is not identically zero for all . The SMJS (2.1a) with is said to be impulse-free in time interval if for all .

Definition 2.2 (singular stochastic finite-time stability (SSFTS)). The SMJS (2.1a) with is said to be SSFTS with respect to , with and , if the stochastic system is regular and impulse-free in time interval and satisfies

Definition 2.3 (singular stochastic finite-time boundedness (SSFTB)). The SMJS (2.1a) which satisfies (2.3) is said to be SSFTB with respect to , with and , if the stochastic system is regular and impulse-free in time interval , and condition (2.7) holds.

Remark 2.4. The definition of SSFTB is the generalization of finite-time boundedness [1]. SSFTB implies that the whole mode of the singular stochastic system is finite-time bounded since the static mode is regular and impulse-free.

Definition 2.5 (singular stochastic finite-time stabilization via observer-based state feedback). The error dynamic SMJS (2.5a) and (2.5b) is said to be singular stochastic finite-time stabilization via observer-based state feedback with respect to , with and , if there exists a state feedback control law and a state observer in the form (2.4a)–(2.4c), such that the error dynamic SMJS (2.5a) and (2.5b) is regular and impulse-free in time interval and satisfies the following constraint relation:

Definition 2.6 (see [30, 33]). Let be the stochastic function, and define its weak infinitesimal operator of stochastic process by

Definition 2.7 (singular stochastic finite-time stabilization via observer-based state feedback). The closed-loop error dynamic SMJS (2.5a) and (2.5b) is said to be singular stochastic finite-time stabilization via observer-based state feedback with respect to , with and , if there exists a state observer and feedback controller in the form (2.4a)–(2.4c), such that the error dynamic SMJS (2.5a) and (2.5b) is SSFTB with respect to , , and under the zero-initial condition, the controlled output satisfies for any nonzero which satisfies (2.3), where is a prescribed positive scalar.
The main objective of this paper being to concentrate on designing a state observer and feedback controller of the form (2.4a)–(2.4c) that ensures singular stochastic finite-time stabilization via observer-based state feedback of the error dynamic SMJS (2.5a) and (2.5b), we require the following lemmas.

Lemma 2.8 (Schur complement lemma, see [57, 58]). The linear matrix inequality is equivalent to and with and .

Lemma 2.9 (see [57]). The following items are true.
(i) Assume that , then there exist two orthogonal matrices and such that has the decomposition as where with for all . Partition , , and with and .
(ii)If satisfies then with and satisfying (2.11) if and only if with . In addition, when is nonsingular, one has and . Furthermore, satisfying (2.12) can be parameterized as where , , and is an arbitrary parameter matrix.
(iii) If is a nonsingular matrix, and are two symmetric positive definite matrices, and satisfy (2.12), is a diagonal matrix from (2.14), and the following equality holds: Then the symmetric positive definite matrix is a solution of (2.15).

3. Main Results

In this section, LMI conditions are established to design a state observer and feedback controller that guarantees the error dynamic SMJS of the class we are considering is singular stochastic finite-time stabilization via observer-based state feedback.

Theorem 3.1. The error dynamic SMJS (2.5a) and (2.5b) is singular stochastic finite-time stabilization via observer-based state feedback with respect to if there exist positive scalars , a set of mode-dependent nonsingular matrices , and sets of mode-dependent symmetric positive-definite matrices , , and for all , such that the following inequalities hold: where , , and .

Proof. Firstly, we prove that the error dynamic SMJS (2.5a) and (2.5b) is regular and impulse-free in time interval . Applying Lemma 2.8, condition (3.1b) implies Now, there exist two orthogonal matrices and such that has the decomposition as where with for all . Partition , , and with and . Denote Noting that condition (3.1a) and is a nonsingular matrix, by Lemma 2.9, we have and . Before and after multiplying (3.2) by and , respectively, this results in that the following matrix inequality holds: where the star will not be used in the following discussion. By Lemma 2.8, we have . Therefore, is nonsingular, which implies that the error dynamic SMJS (2.5a) and (2.5b) is regular and impulse-free in time interval .
For the given mode-dependent nonsingular matrix , let us consider the following quadratic function as: Computing the weak infinitesimal operator emanating from the point at time along the solution of error dynamic SMJS (2.5a) and (2.5b) and noting the condition (3.1a), we obtain where . Before and after multiplying (3.1b) by and , respectively, this results in the following matrix inequality From (3.7) and (3.8), we can obtain Further, (3.9) can be rewritten as Integrating (3.10) from 0 to , with , we obtain Noting , and condition (3.1c), we have Taking into account that we obtain Therefore, it follows that condition (3.1d) implies for all . This completes the proof of the theorem.

Theorem 3.2. The error dynamic SMJS (2.5a) and (2.5b) is singular stochastic finite-time stabilization via observer-based state feedback with respect to if there exist positive scalars , a set of mode-dependent nonsingular matrices , and a set of mode-dependent symmetric positive-definite matrices , and for all , such that (3.1a), (3.1c), and the following inequalities: hold, where .

Proof. Note that Thus, condition (3.15a) implies that Let for all , by Theorem 3.1, conditions (3.1a), (3.1c), (3.15b), and (3.17) can guarantee that the error dynamic SMJS (2.5a) and (2.5b) is singular stochastic finite-time stabilization via observer-based state feedback with respect to . Therefore, we only need to prove that the constraint relation (2.10) holds. Let us choose the Lyapunov-Krasovskii function in the form (3.6) in Theorem 3.1 and noting (3.7) and (3.15a), we obtain Further, (3.18) can be represented as Integrating (3.19) from 0 to and noting that under-zero initial condition, we have Using the Dynkin formula, it results that Thus, for all and under-zero initial condition, we have This completes the proof of the theorem.

Let , , and , then the following theorem gives LMI conditions to ensure singular stochastic finite-time stabilization via observer-based state feedback of the error dynamic SMJS (2.5a) and (2.5b).

Theorem 3.3. There exist a state feedback controller with and a state observer such that the error dynamic SMJS (2.5a) and (2.5b) is singular stochastic finite-time stabilization via observer-based state feedback with respect to if there exist positive scalars , and sets of mode-dependent symmetric positive-definite matrices , , sets of mode-dependent matrices , , and for all , such that the following inequalities hold: where In addition, the form of is from (3.36).

Proof. We firstly prove that condition (3.23b) implies condition (3.15a). Let , , and , then conditions (3.1a), (3.1c), and (3.1d) are equivalent towhere and . By condition (3.23a), we have Thus, the inequality holds, where Noting that the inequality holds for each , thus, where . Let and , then a sufficient condition for (3.15a) to guarantee is that where . Noting the forms of , then the inequality (3.31) is equivalent to the following: where Let , we obtain Letting and applying Lemma 2.8, it follows that (3.32) is equivalent to (3.23b).
Noting that is nonsingular matrix, by Lemma 2.9, there exist two orthogonal matrices and , such that has the decomposition as where with for all . Partition , , and with and . Let , from (3.23a), is of the following form , and can be expressed as where and with a parameter matrix . If we choose as a symmetric positive definite matrix, then is a symmetric positive definite matrix. Thus, is a solution of (3.25b), and satisfies .
Let , then it is easy to check that conditions (3.23c) and (3.23d) can guarantee that conditions (3.25c) hold. This completes the proof of the theorem.

Corollary 3.4. There exist a state feedback controller with and a state observer such that the error dynamic SMJS (2.5a) is singular stochastic finite-time stabilization via observer-based state feedback with respect to if there exist positive scalars , and sets of mode-dependent symmetric positive-definite matrices , , , sets of mode-dependent matrices , , and for all , such that the following inequalities hold: where , and are the same as Theorem 3.3.

Remark 3.5. The feasibility of conditions stated in Theorem 3.3 and Corollary 3.4 can be turned into the following LMIs-based feasibility problem with a parameter , respectively: Furthermore, we can also find the parameter by an unconstrained nonlinear optimization approach, in which a locally convergent solution can be obtained by using the program fminsearch in the optimization toolbox of Matlab.

Remark 3.6. If we can find feasible solution with parameter , by the above discussion, we can obtain the designed state observer, and state feedback controller cannot only ensure SSFTB and stochastic stabilization of the error dynamic SMJS (2.5a) and (2.5b) but also the effect of the disturbance input of the disturbance input on the controlled output satisfying for the error dynamic SMJS.

4. Numerical Examples

In this section, we present numerical examples to illustrate the proposed methods.

Example 4.1. To show the results of singular stochastic finite-time stabilization via observer-based state feedback of the error dynamic SMJS (2.5a) and (2.5b), consider a two-mode SMJS (2.1a)–(2.1c) with parameters as follows:
(i) mode : mode : In addition, the transition rate matrix is described by .
Then, we choose , and Theorem 3.3 yields , , and This also gives the following gains:
Moreover, by Theorem 3.3, the optimal bound with minimum value of relies on parameter . Let , then we can find feasible solution when . Figure 1 shows the optimal value with a different value of . Furthermore, by using the program fminsearch in the optimization toolbox of Matlab starting at , the locally convergent solution can be derived as with and the optimal values and .

Remark 4.2. To show the results of singular stochastic finite-time stabilization via observer-based state feedback of the error dynamic SMJS (2.5a), consider a two-mode SMJS (2.1a) and (2.1c) with parameters that the matrical variables and the transition rate matrix are the same as the above example. Let , by Corollary 3.4, the optimal bound with minimum value of relies on the parameter . We can find feasible solution when . Figure 2 shows the optimal value with different value of . Then using the program fminsearch in the optimization toolbox of Matlab starting at , we can obtain the locally convergent solution with .

Example 4.3. To show SSFTB and stochastic stabilization of the error dynamic SMJS (2.5a) and (2.5b) and the effect of the disturbance input of the disturbance input on the controlled output satisfying for the error dynamic SMJS, let and other matrical variables and the transition rate matrix are the same as Example 4.1.
Then, let . By Theorem 3.3, we can find the feasible solution when . Noting that when , Theorem 3.3 yields the optimal value and . Thus, the above error dynamic SMJS is stochastically stabilizable, and the effect of the disturbance input of the disturbance input on the controlled output satisfies .

5. Conclusions

This paper investigates the problem of finite-time control via observer-based state feedback for a family of SMJSs with time-varying norm-bounded disturbance. An observer-based state feedback controller is designed which ensures singular stochastic finite-time stabilization via observer-based state feedback of the resulting closed-loop error dynamic SMJS. Sufficient criterions are presented for the solvability of the problem, which can be reduced to a feasibility problem in the form of linear matrix inequalities with a fixed parameter. In addition, we also give the problem of finite-time stabilization via observer-based state feedback of a class of SMJSs and present sufficient conditions of singular stochastic finite-time stabilization via observer-based state feedback for the class of SMJSs. Numerical examples are also given to show the validity of the proposed methodology.

Acknowledgments

The authors would like to thank the reviewers and editors for their very helpful comments and suggestions which could have improved the presentation of the paper. The paper was supported by the National Natural Science Foundation of China under Grant 60874006, the Doctoral Foundation of Henan University of Technology under Grant 2009BS048, the Natural Science Foundation of Henan Province of China under Grant 102300410118, the Foundation of Henan Educational Committee under Grant 2011A120003, and the Foundation of Henan University of Technology under Grant 09XJC011.

References

F. Amato, M. Ariola, and P. Dorato, “Finite-time control of linear systems subject to parametric uncertainties and disturbances,” Automatica, vol. 37, no. 9, pp. 1459–1463, 2001.
View at: Publisher Site | Google Scholar
A. El-Gohary and A. S. Al-Ruzaiza, “Optimal control of non-homogeneous prey-predator models during infinite and finite time intervals,” Applied Mathematics and Computation, vol. 146, no. 2-3, pp. 495–508, 2003.
View at: Publisher Site | Google Scholar | Zentralblatt MATH
A. El-Gohary, “Optimal control of an angular motion of a rigid body during infinite and finite time intervals,” Applied Mathematics and Computation, vol. 141, no. 2-3, pp. 541–551, 2003.
View at: Publisher Site | Google Scholar | Zentralblatt MATH
H. J. Kushner, “Finite time stochastic stability and the analysis of tracking systems,” IEEE Transactions on Automatic Control, vol. 11, pp. 219–227, 1966.
View at: Google Scholar
L. Weiss and E. F. Infante, “Finite time stability under perturbing forces and on product spaces,” IEEE Transactions on Automatic Control, vol. 12, pp. 54–59, 1967.
View at: Google Scholar | Zentralblatt MATH
F. Amato, M. Ariola, and C. Cosentino, “Finite-time stabilization via dynamic output feedback,” Automatica, vol. 42, no. 2, pp. 337–342, 2006.
View at: Publisher Site | Google Scholar | Zentralblatt MATH
W. Zhang and X. An, “Finite-time control of linear stochastic systems,” International Journal of Innovative Computing, Information and Control, vol. 4, no. 3, pp. 689–696, 2008.
View at: Google Scholar
Q. Meng and Y. Shen, “Finite-time $H_{\infty}$ control for linear continuous system with norm-bounded disturbance,” Communications in Nonlinear Science and Numerical Simulation, vol. 14, no. 4, pp. 1043–1049, 2009.
View at: Publisher Site | Google Scholar | Zentralblatt MATH
Y. Yang, J. Li, and G. Chen, “Finite-time stability and stabilization of nonlinear stochastic hybrid systems,” Journal of Mathematical Analysis and Applications, vol. 356, no. 1, pp. 338–345, 2009.
View at: Publisher Site | Google Scholar | Zentralblatt MATH
G. Garcia, S. Tarbouriech, and J. Bernussou, “Finite-time stabilization of linear time-varying continuous systems,” IEEE Transactions on Automatic Control, vol. 54, no. 2, pp. 364–369, 2009.
View at: Publisher Site | Google Scholar
R. Ambrosino, F. Calabrese, C. Cosentino, and G. De Tommasi, “Sufficient conditions for finite-time stability of impulsive dynamical systems,” IEEE Transactions on Automatic Control, vol. 54, no. 4, pp. 861–865, 2009.
View at: Publisher Site | Google Scholar
F. Amato, R. Ambrosino, M. Ariola, and C. Cosentino, “Finite-time stability of linear time-varying systems with jumps,” Automatica, vol. 45, no. 5, pp. 1354–1358, 2009.
View at: Publisher Site | Google Scholar | Zentralblatt MATH
S. Li, Z. Wang, and S. Fei, “Finite-time control of a bioreactor system using terminal sliding mode,” International Journal of Innovative Computing, Information and Control B, vol. 5, no. 10, pp. 3495–3504, 2009.
View at: Google Scholar
Z. Wang, S. Li, and S. Fei, “Finite-time tracking control of bank-to-turn missiles using terminalsliding mode,” ICIC Express Letters B, vol. 3, no. 4, pp. 1373–1380, 2009.
View at: Google Scholar
F. Amato, M. Ariola, and C. Cosentino, “Finite-time control of discrete-time linear systems: analysis and design conditions,” Automatica, vol. 46, no. 5, pp. 919–924, 2010.
View at: Publisher Site | Google Scholar
S. He and F. Liu, “Robust finite-time stabilization of uncertain fuzzy jump systems,” International Journal of Innovative Computing, Information and Control, vol. 6, no. 9, pp. 3853–3862, 2010.
View at: Google Scholar
S. He and F. Liu, “Stochastic finite-time boundedness of Markovian jumping neural network with uncertain transition probabilities,” Applied Mathematical Modelling, vol. 35, no. 6, pp. 2631–2638, 2011.
View at: Publisher Site | Google Scholar | Zentralblatt MATH
Y. Zhang, C. Liu, and X. Mu, “Stochastic finite-time guaranteed cost control of Markovian jumping singular systems,” Mathematical Problems in Engineering, vol. 2011, Article ID 431751, 20 pages, 2011.
View at: Publisher Site | Google Scholar
Y. Zhang, C. Liu, and X. Mu, “On stochastic finite-time control of discrete-time fuzzy systems with packet dropout,” Discrete Dynamics in Nature and Society, vol. 2012, Article ID 752950, 18 pages, 2012.
View at: Publisher Site | Google Scholar
S. He and F. Liu, “Observer-based finite-time control of time-delayed jump systems,” Applied Mathematics and Computation, vol. 217, no. 6, pp. 2327–2338, 2010.
View at: Publisher Site | Google Scholar | Zentralblatt MATH
L. Dai, Singular Control Systems, vol. 118 of Lecture Notes in Control and Information Sciences, Springer-Verlag, New York, NY, USA, 1989.
View at: Publisher Site
F. L. Lewis, “A survey of linear singular systems,” Circuits, Systems, and Signal Processing, vol. 5, no. 1, pp. 3–36, 1986.
View at: Publisher Site | Google Scholar | Zentralblatt MATH
J. Y. Ishihara and M. H. Terra, “On the Lyapunov theorem for singular systems,” IEEE Transactions on Automatic Control, vol. 47, no. 11, pp. 1926–1930, 2002.
View at: Publisher Site | Google Scholar
Z. Wu and W. Zhou, “Delay-dependent robust stabilization of uncertain singular systems with state delay,” ICIC Express Letters, vol. 1, no. 2, pp. 169–176, 2007.
View at: Google Scholar
G. Zhang, Y. Xia, and P. Shi, “New bounded real lemma for discrete-time singular systems,” Automatica, vol. 44, no. 3, pp. 886–890, 2008.
View at: Publisher Site | Google Scholar
I. Masubuchi, Y. Kamitane, A. Ohara, and N. Suda, “ $H_{\infty}$ control for descriptor systems: a matrix inequalities approach,” Automatica, vol. 33, no. 4, pp. 669–673, 1997.
View at: Publisher Site | Google Scholar | Zentralblatt MATH
Y. Xia, P. Shi, G. Liu, and D. Rees, “Robust mixed $H_{\infty} / H_{2}$ state-feedback control for continuous-time descriptor systems with parameter uncertainties,” Circuits, Systems, and Signal Processing, vol. 24, no. 4, pp. 431–443, 2005.
View at: Publisher Site | Google Scholar | Zentralblatt MATH
L. Zhang, B. Huang, and J. Lam, “LMI synthesis of $H_{2}$ and mixed $H_{2} / H_{∞}$ controllers for singular systems,” IEEE Transactions on Circuits and Systems, vol. 50, no. 9, pp. 615–626, 2003.
View at: Publisher Site | Google Scholar | Zentralblatt MATH
S. Ma, C. Zhang, and Z. Cheng, “Delay-dependent robust $H_{\infty}$ control for uncertain discrete-time singular systems with time-delays,” Journal of Computational and Applied Mathematics, vol. 217, no. 1, pp. 194–211, 2008.
View at: Publisher Site | Google Scholar | Zentralblatt MATH
E.-K. Boukas, Communications and Control Engineering, Control of Singular Systems with Random Abrupt Changes, Springer-Verlag, Berlin, Germany, 2008.
E. K. Boukas, “Stabilization of stochastic singular nonlinear hybrid systems,” Nonlinear Analysis. Theory, Methods & Applicationss, vol. 64, no. 2, pp. 217–228, 2006.
View at: Publisher Site | Google Scholar | Zentralblatt MATH
Y. Xia, E. K. Boukas, P. Shi, and J. Zhang, “Stability and stabilization of continuous-time singular hybrid systems,” Automatica, vol. 45, no. 6, pp. 1504–1509, 2009.
View at: Publisher Site | Google Scholar
X. Mao, “Stability of stochastic differential equations with Markovian switching,” Stochastic Processes and their Applications, vol. 79, no. 1, pp. 45–67, 1999.
View at: Publisher Site | Google Scholar | Zentralblatt MATH
P. Shi, Y. Xia, G. P. Liu, and D. Rees, “On designing of sliding-mode control for stochastic jump systems,” IEEE Transactions on Automatic Control, vol. 51, no. 1, pp. 97–103, 2006.
View at: Publisher Site | Google Scholar
L. Wu, P. Shi, and H. Gao, “State estimation and sliding-mode control of Markovian jump singular systems,” IEEE Transactions on Automatic Control, vol. 55, no. 5, pp. 1213–1219, 2010.
View at: Publisher Site | Google Scholar
C. E. de Souza, “Robust stability and stabilization of uncertain discrete-time Markovian jump linear systems,” IEEE Transactions on Automatic Control, vol. 51, no. 5, pp. 836–841, 2006.
View at: Publisher Site | Google Scholar
P. Shi, E.-K. Boukas, and R. K. Agarwal, “Control of Markovian jump discrete-time systems with norm bounded uncertainty and unknown delay,” IEEE Transactions on Automatic Control, vol. 44, no. 11, pp. 2139–2144, 1999.
View at: Publisher Site | Google Scholar | Zentralblatt MATH
E. K. Boukas, “On robust stability of singular systems with random abrupt changes,” Nonlinear Analysis: Theory, Methods & Applications, vol. 63, no. 3, pp. 301–310, 2005.
View at: Publisher Site | Google Scholar | Zentralblatt MATH
J. Qiu, H. He, and P. Shi, “Robust stochastic stabilization and $H_{\infty}$ control for neutral stochastic systems with distributed delays,” Circuits, Systems, and Signal Processing, vol. 30, no. 2, pp. 287–301, 2011.
View at: Publisher Site | Google Scholar | Zentralblatt MATH
K.-C. Yao and F. Q. Cai, “Robustness and reliability of decentralized stochastic singularly- perturbedcomputer controlled systems with multiple time-varying delays,” ICIC Express Letters A, vol. 3, no. 3, pp. 379–384, 2009.
View at: Google Scholar
S. Xu, T. Chen, and J. Lam, “Robust $H_{\infty}$ filtering for uncertain Markovian jump systems with mode-dependent time delays,” IEEE Transactions on Automatic Control, vol. 48, no. 5, pp. 900–907, 2003.
View at: Publisher Site | Google Scholar
M. S. Mahmoud, P. Shi, and A. Ismail, “Robust Kalman filtering for discrete-time Markovian jump systems with parameter uncertainty,” Journal of Computational and Applied Mathematics, vol. 169, no. 1, pp. 53–69, 2004.
View at: Publisher Site | Google Scholar | Zentralblatt MATH
S. Xu and J. Lam, “Reduced-order $H_{\infty}$ filtering for singular systems,” Systems & Control Letters, vol. 56, no. 1, pp. 48–57, 2007.
View at: Publisher Site | Google Scholar | Zentralblatt MATH
S. Ma and E. K. Boukas, “Robust $H_{\infty}$ filtering for uncertain discrete Markov jump singular systems with mode-dependent time delay,” IET Control Theory & Applications, vol. 3, no. 3, pp. 351–361, 2009.
View at: Publisher Site | Google Scholar
S. Xu and J. Lam, Robust Control and Filtering of Singular Systems, Springer-Verlag, Berlin, Germany, 2006.
P. V. Pakshin, “Dissipativity of diffusion itô processes with markovain switching and problems of robust stabilization,” Automation and Remote Control, vol. 68, pp. 1502–1518, 2007.
View at: Publisher Site | Google Scholar
J. Qiu and K. Lu, “New robust passive stability criteria for uncertain singularly Markov jump systems with time delays,” ICIC Express Letters, vol. 3, no. 3, pp. 651–656, 2009.
View at: Google Scholar
E.-K. 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
Y. Yin, P. Shi, and F. Liu, “Gain-scheduled PI tracking control on stochastic nonlinear systems with partially known transition probabilities,” Journal of the Franklin Institute, vol. 348, no. 4, pp. 685–702, 2011.
View at: Publisher Site | Google Scholar | Zentralblatt MATH
Y. Q. Xia and Y. M. Jia, “ $H_{\infty}$ output feedback controller design for linear singular systems with time-delay,” Control Theory & Applications, vol. 20, no. 3, pp. 323–328, 2003.
View at: Google Scholar
E. K. Boukas, “Static output feedback control for stochastic hybrid systems: LMI approach,” Automatica, vol. 42, no. 1, pp. 183–188, 2006.
View at: Publisher Site | Google Scholar | Zentralblatt MATH
H. H. Choi and M. J. Chung, “Robust observer-based $H_{\infty}$ controller design for linear uncertain time-delay systems,” Automatica, vol. 33, no. 9, pp. 1749–1752, 1997.
View at: Publisher Site | Google Scholar
J.-D. Chen, “Robust $H_{\infty}$ output dynamic observer-based control of uncertain time-delay systems,” Chaos, Solitons and Fractals, vol. 31, no. 2, pp. 391–403, 2007.
View at: Publisher Site | Google Scholar | Zentralblatt MATH
J.-D. Chen, C.-D. Yang, C.-H. Lien, and J.-H. Horng, “New delay-dependent non-fragile $H_{\infty}$ observer-based control for continuous time-delay systems,” Information Sciences, vol. 178, no. 24, pp. 4699–4706, 2008.
View at: Publisher Site | Google Scholar | Zentralblatt MATH
Q. Li, Q. Zhang, Y. Zhang, and Y. An, “Observer-based passive control for descriptor systems with time-delay,” Journal of Systems Engineering and Electronics, vol. 20, no. 1, pp. 120–128, 2009.
View at: Google Scholar
Y. Zhang, C. Liu, and X. Mu, “Robust finite-time stabilization of uncertain singular Markovian jump systems,” Applied Mathematical Modelling. In press.
View at: Publisher Site | Google Scholar
Y. Zhang, C. Liu, and X. Mu, “Robust finite-time $H_{\infty}$ control of singular stochastic systems via static output feedback,” Applied Mathematics and Computation, vol. 218, no. 9, pp. 5629–5640, 2012.
View at: Google Scholar
S. Boyd, L. E. Ghaoui, E. Feron, and V. Balakrishnan, Linear Matrix Inequality in Systems and Control Theory. SIAM Studies in Applied Mathematics, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, Pa, USA, 1994.

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Copyright © 2012 Yingqi Zhang et al. 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.

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