Mathematical Problems in Engineering

Volume 2013 (2013), Article ID 283426, 10 pages

http://dx.doi.org/10.1155/2013/283426

## Output Feedback Control of Discrete Impulsive Switched Systems with State Delays and Missing Measurements

^{1}School of Automation, Nanjing University of Science and Technology, Nanjing 210094, China^{2}Department of Engineering, Faculty of Engineering and Science, University of Agder, 4898 Grimstad, Norway

Received 7 July 2013; Revised 31 July 2013; Accepted 19 August 2013

Academic Editor: Tao Li

Copyright © 2013 Xia Li 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.

#### Abstract

This paper is concerned with the problem of dynamic output feedback (DOF) control for a class of uncertain discrete impulsive switched systems with state delays and missing measurements. The missing measurements are modeled as a binary switch sequence specified by a conditional probability distribution. The problem addressed is to design an output feedback controller such that for all admissible uncertainties, the closed-loop system is exponentially stable in mean square sense. By using the average dwell time approach and the piecewise Lyapunov function technique, some sufficient conditions for the existence of a desired DOF controller are derived, then an explicit expression of the desired controller is given. Finally, a numerical example is given to illustrate the effectiveness of the proposed method.

#### 1. Introduction

Due to their wide applications, switched systems which are an important class of hybrid systems have drawn considerable attention in the last decade [1, 2]. During these years, there have been increasing research activities in the field of stability analysis for such systems (see [3–6], and the references cited therein). Recently, impulsive switched systems as a class of special switched systems have gained research attention. This is because impulsive switched systems can represent some practical switched systems that exhibit impulsive dynamical behavior due to sudden changes in the state of the system at certain instants of switching. Some problems on impulsive switched systems with and without delays have been successfully investigated, and a rich body of the literatures is now available [7–10].

On the other hand, control synthesis is one of the important issues in system theory. State feedback control as an effective control strategy has been widely used in various complex dynamical systems. For instance, some state feedback control problems for switched systems have been extensively studied in [11, 12]. The adaptive control for a class of nonlinear systems via backstepping method was studied in [13]. The authors in [14] considered an optimal state feedback control problem for impulsive switched systems. In [15–18], some controller design methods for impulsive switched systems were developed. In addition, output feedback control has been considered as an effective control method when the states of the system are not all measurable in practice. At present, many results on the output feedback controller design for nonlinear systems or switched systems have been obtained (see [19–24]), and less work has been done for impulsive switched systems.

In almost all the works mentioned above, the assumption of consecutive measurements has been made implicitly. Unfortunately, in many practical applications, such an assumption does not hold. For example, due to sensor temporal failure or network transmission delay/loss, at certain time points, the system measurement may contain noise only, indicating that the real signal is missing. One of the most popular ways to describe the missing measurement is to view it as a Bernoulli distributed (binary switching) white sequence specified by a conditional probability distribution in the output equation. The Bernoulli distribution description was first proposed in [25] to deal with the optimal recursive filtering problem and then has been used in [26–29] for various control and filtering problems of linear systems with probabilistic missing measurements. It is worth pointing out that the references mentioned above did not consider the effect of impulse. However, the missing measurements and impulsive jumps happening simultaneously in the systems will bring some challenges and difficulties for the analysis and synthesis. To the best of our knowledge, the issue of dynamic output feedback controller design with missing measurements for impulsive switched systems has not been fully investigated, which motivates our present study.

In this paper, we will focus our interest on the problem of dynamic output feedback (DOF) control for a class of uncertain discrete impulsive switched systems with state delays and missing measurements. The main contributions of the paper are as follows: a DOF controller is proposed for discrete impulsive switched systems, and the controller contains impulsive jumps, which is different from most of the existing ones, for example, those in [20–23]; sufficient conditions for the existence of a DOF controller are developed such that the resulting closed-loop system is exponentially stable in mean square sense.

The remainder of the paper is organized as follows. In Section 2, problem formulation and some necessary preliminaries are given. In Section 3, the main results are presented. Section 4 gives a numerical example to illustrate the effectiveness of the proposed approach. Concluding remarks are given in Section 5.

*Notations*. Throughout this paper, the superscript “” denotes the transpose, and the notation means that matrix is positive semidefinite (positive definite, resp.). denotes the Euclidean norm. stands for the mathematical expectation, and means the occurrence probability of the event “”. represents the identity matrix and denotes a diagonal matrix with the diagonal elements , . denotes the inverse of . The asterisk in a matrix is used to denote a term that is induced by symmetry. The set of all positive integers is represented by .

#### 2. Problem Formulation and Preliminaries

Consider the following uncertain discrete impulsive switched systems with state delay:where is the state vector, is the control input, is the output vector, and is a discrete vector-valued initial function on interval . is the discrete time delay. is a switching signal which takes its values in the finite set , denotes the number of subsystems. is the initial time and () denotes the th switching instant. Moreover, means that the th subsystem is activated.

The measurement output which may contain missing data is described by where is the measurement output vector and are known real constant matrices with appropriate dimensions. The stochastic variable is a Bernoulli distributed white sequence taking the values of 0 and 1 withwhere is a known positive scalar. From (3a)-(3b), we obtain that For each , and are uncertain real-valued matrices with appropriate dimensions. We assume that these uncertainties are norm-bounded and satisfy where , , , , and , , are known real constant matrices with appropriate dimensions. are unknown and are possibly time-varying matrices with Lebesgue measurable elements and satisfy

Here, we are interested in designing a DOF switched controller described bywhere is the controller state vector; , , and are constant matrices to be determined later.

*Remark 1. *Different from the existing DOF controllers proposed in [20–23], the controller proposed here contains (7b), which coincides with the structure of system (1a), (1b), (1c), and (1d).

Now, define a new state vector,
The combination of the previous DOF controller (7a), (7b), (7c), and (7d) and system (1a), (1b), (1c), and (1d) yields the following closed-loop system:where

The following definitions and lemmas will be essential for our later development.

*Definition 2 (see [30]). * For any , let denote the switching number of during the interval . If there exist and such that
then and are called the average dwell time and the chatter bound, respectively.

*Remark 3. *In this paper, the average dwell time method is used to restrict the switching number during a time interval such that the stability of system (9a), (9b), and (9c) can be guaranteed.

*Definition 4 (see [27]). *System (9a), (9b), and (9c) is said to be exponentially stable in mean square sense under the switching signal , if there exist constants and , such that the trajectory of system (9a), (9b), and (9c) satisfies

Lemma 5 (see [31]). *For a given matrix , where and are square matrices, the following conditions are equivalent:*(i)*; *(ii)*, ;*(iii)*, .*

Lemma 6 (see [32]). *Let , , , and be real matrices of appropriate dimensions with satisfying , then for all , , if and only if there exists a scalar such that .*

#### 3. Main Results

##### 3.1. Stability Analysis

The following theorem provides sufficient conditions under which the exponential stability of system (9a), (9b), and (9c) can be guaranteed in mean square sense.

Theorem 7. *Consider system (9a), (9b), and (9c), for a given scalar , if there exist positive definite symmetric matrices and () with appropriatedimensions, such that
**
where , then, under the following average dwell time scheme
**
system (9a), (9b), and (9c) is exponentially stable in mean square sense, where satisfies
*

*Proof. *Choose a piecewise Lyapunov function candidate for system (9a), (9b), and (9c) of the form
the form of is given by
where

When , we let (). Then along the trajectory of system (9a), (9b), and (9c), we have
Notice that
Thus we obtain
where .

Applying Lemma 5, it is easy to get that inequality (13) is equivalent to . Thus we can obtain from (13) that

When , we let . Along the trajectory of system (9a), (9b), and (9c), we have
It can be obtained from (15) that

It follows that
Thus, for , we have
Repeating the previous manipulation, one has that
From Definition 2, we get that , where denotes the switching number of during the interval , then it follows that
Notice that
Then, one obtains
where
Then under the average dwell time scheme (14), it is easy to get that , which implies that system (9a), (9b), and (9c) is exponentially stable in mean square sense.

This completes the proof.

*Remark 8. *Compared with the existing results presented in [16–18], we get sufficient conditions of exponential stability in mean square sense. In addition, the paper takes the missing measurement into consideration, which yields different results from those of [16–18], where the missing measurement is not considered.

##### 3.2. Design of DOF Controller

This section will give some LMIs conditions for the controller design.

Theorem 9. *Consider system (1a), (1b), (1c), and (1d) for a given positive scalar , if there exist positive-definite symmetric matrices , , , and , any matrices , , and with appropriate dimensions, and positive scalars and (), such that**where , , , , , , , .**Then, there exists a DOF controller (7a), (7b), (7c), and (7d) such that the closed-loop system (9a), (9b), and (9c) is exponentially stable in mean square sense for any switching signals with average dwell time scheme (14), where satisfies
**where
**
Moreover, if the previous LMI conditions are feasible, then the desired dynamic output feedback controller parameters can be designed as*

*Proof. *Let the matrix be partitioned as follows:
where .

By and from (34a) and (36), we have

Define the following matrices:

Then we have

Use to premultiply and to postmultiply the left-hand term of (13), and denote

Then, we can obtain that (13) is equivalent to the following inequality:

where

Using to pre- and postmultiply the left-hand term of (41) and denoting
one obtainswhere

Then combining (5) with (44), one haswhere

By Lemma 6, (46) is equivalent to
where and are positive scalars.

Using Lemma 5, we haveUsing to pre- and postmultiply the left-hand term of (49) and denoting we can obtain that (32a) is equivalent to (49), that is to say, (32a) guarantees that (13) is tenable.

The proof is completed.

*Remark 10. *From Theorem 9, it is easy to see that a larger will be favorable to the solvability of inequality (32a), (32b), and (32c) which leads to a larger value of . Considering these, we can first select a larger to guarantee the feasibility of inequality (32a), (32b), and (32c), and then decrease to obtain a smaller .

Based on Theorem 9, we present an algorithm for the design of dynamic output controller.

*Algorithm 11. *

*Step 1. *Given the system matrices and a constant ; by solving (32a), we can get the feasible solution of positive definite symmetric matrices , , , , matrices , , , and positive scalars , .

*Step 2. *Applying singular value decomposition to the first equation of (34a), we can obtain square and nonsingular matrices and . Then we can get , , and by (34a) and (34b).

*Step 3. *By substituting matrices , , , , , and into (32b)-(32c) and solving them, we can get and by (14).

*Step 4. *Determine the DOF controller parameters , , and based on (35a) and (35b).

#### 4. Numerical Example

In this section, we present an example to illustrate the effectiveness of the proposed approach. Consider system (1a), (1b), (1c), and (1d) with parameters as follows:

Let , , and , then by solving the matrix inequalities in Theorem 9, we can get the DOF controller parameters

Let

According to (32a), (32b), and (32c), we get . From (14), it can be obtained that . Choosing , simulation results are shown in Figures 1 and 2, where the initial conditions are , , , and , . Figure 1 depicts the switching signal. Under this switching signal and dynamic output feedback controller, the state responses of the resulting closed-loop system are shown in Figure 2. From Figure 1, we can see that the switching signal satisfies . Furthermore, it can be observed from Figure 2 that the resulting closed-loop system is exponentially stable in mean square sense. This indicates that the designed controller is effective although there exist missing measurements.

#### 5. Conclusions

This paper has presented a solution to the problem of dynamic output feedback controller design for a class of uncertain discrete impulsive switched systems with state delay and missing measurements. By employing the average dwell time approach, a sufficient condition for the existence of a DOF controller is presented such that the exponential stability in mean square sense of the resulting closed-loop system is ensured. An example is given to illustrate the applicability of the proposed approach. Our future work will focus on studying the problem of asynchronous control for discrete impulsive switched systems with state delay and missing measurements.

#### Acknowledgments

This work was supported by the National Natural Science Foundation of China under Grant no. 61273120 and the Alexander von Humboldt Foundation in Germany.

#### References

- D. Liberzon,
*Switching in Systems and Control*, Systems & Control: Foundations & Applications, Birkhäuser, Boston, Mass, USA, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - D. Liberzon and A. S. Morse, “Basic problems in stability and design of switched systems,”
*IEEE Control Systems Magazine*, vol. 19, no. 5, pp. 59–70, 1999. View at Publisher · View at Google Scholar · View at Scopus - J. P. Hespanha and A. S. Morse, “Stability of switched systems with average dwell-time,” in
*Proceedings of the 38th IEEE Conference on Decision and Control (CDC '99)*, pp. 2655–2660, December 1999. View at Scopus - W.-A. Zhang and L. Yu, “Stability analysis for discrete-time switched time-delay systems,”
*Automatica*, vol. 45, no. 10, pp. 2265–2271, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - L. Zhang and B. Jiang, “Stability of a class of switched linear systems with uncertainties and average dwell time switching,”
*International Journal of Innovative Computing, Information and Control*, vol. 6, no. 2, pp. 667–676, 2010. View at Google Scholar · View at Scopus - P. Yan and H. Özbay, “Stability analysis of switched time delay systems,”
*SIAM Journal on Control and Optimization*, vol. 47, no. 2, pp. 936–949, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Z.-H. Guan, D. J. Hill, and X. Shen, “On hybrid impulsive and switching systems and application to nonlinear control,”
*IEEE Transactions on Automatic Control*, vol. 50, no. 7, pp. 1058–1062, 2005. View at Publisher · View at Google Scholar · View at MathSciNet - B. Liu and H. J. Marquez, “Controllability and observability for a class of controlled switching impulsive systems,”
*IEEE Transactions on Automatic Control*, vol. 53, no. 10, pp. 2360–2366, 2008. View at Publisher · View at Google Scholar · View at Scopus - H. Ye, A. N. Michel, and L. Hou, “Stability analysis of systems with impulse effects,”
*IEEE Transactions on Automatic Control*, vol. 43, no. 12, pp. 1719–1723, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S.-H. Lee and J.-T. Lim, “Stability analysis of switched systems with impulse effects,” in
*Proceedings of the IEEE International Symposium on Intelligent Control, Intelligent Systems and Semiotics*, pp. 79–83, Semiotics, Cambridge, UK, September 1999. View at Scopus - J. C. Geromel and P. Colaneri, “${H}_{\infty}$ and dwell time specifications of switched linear systems,” in
*Proceedings of the 47th IEEE Conference on Decision and Control (CDC '08)*, pp. 5318–5323, Cancun, Mexico, December 2008. View at Publisher · View at Google Scholar · View at Scopus - J. C. Geromel and G. S. Deaecto, “Switched state feedback control for continuous-time uncertain systems,”
*Automatica*, vol. 45, no. 2, pp. 593–597, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. Tong and Y. Li, “Observer-based fuzzy adaptive control for strict-feedback nonlinear systems,”
*Fuzzy Sets and Systems*, vol. 160, no. 12, pp. 1749–1764, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - R. Li, Z. G. Feng, K. L. Teo, and G. R. Duan, “Optimal piecewise state feedback control for impulsive switched systems,”
*Mathematical and Computer Modelling*, vol. 48, no. 3-4, pp. 468–479, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - F. Long and S. Fei, “Neural networks stabilization and disturbance attenuation for nonlinear switched impulsive systems,”
*Neurocomputing*, vol. 71, no. 7–9, pp. 1741–1747, 2008. View at Publisher · View at Google Scholar · View at Scopus - W. Xiang and J. Xiao, “Stability analysis and control synthesis of switched impulsive systems,”
*International Journal of Robust and Nonlinear Control*, vol. 22, no. 13, pp. 1440–1459, 2012. View at Publisher · View at Google Scholar · View at MathSciNet - X. Li, H. R. Karimi, and Z. Xiang, “Robust reliable control of uncertain discrete impulsive switched systems with state delays,”
*Mathematical Problems in Engineering*, vol. 2013, Article ID 197819, 8 pages, 2013. View at Publisher · View at Google Scholar · View at MathSciNet - X. Li, Z. Xiang, and H. R. Karimi, “Asynchronously switched control of discrete impulsive switched systems with time delays,”
*Information Sciences*, vol. 249, pp. 132–142, 2013. View at Google Scholar - S.-C. Tong, X.-L. He, and H.-G. Zhang, “A combined backstepping and small-gain approach to robust adaptive fuzzy output feedback control,”
*IEEE Transactions on Fuzzy Systems*, vol. 17, no. 5, pp. 1059–1069, 2009. View at Publisher · View at Google Scholar · View at Scopus - D.-W. Ding and G.-H. Yang, “${H}_{\infty}$ static output feedback control for discrete-time switched linear systems with average dwell time,”
*IET Control Theory & Applications*, vol. 4, no. 3, pp. 381–390, 2010. View at Publisher · View at Google Scholar · View at MathSciNet - L. Wu, T. Qi, and Z. Feng, “Average dwell time approach to ${L}_{2}-{L}_{\infty}$ control of switched delay systems via dynamic output feedback,”
*IET Control Theory & Applications*, vol. 3, no. 10, pp. 1425–1436, 2009. View at Publisher · View at Google Scholar · View at MathSciNet - J. C. Geromel, P. Colaneri, and P. Bolzern, “Dynamic output feedback control of switched linear systems,”
*IEEE Transactions on Automatic Control*, vol. 53, no. 3, pp. 720–733, 2008. View at Publisher · View at Google Scholar · View at MathSciNet - L. Galbusera, P. Bolzern, G. S. Deaecto, and J. C. Geromel, “State and output feedback ${H}_{\infty}$ control of time-delay switched linear systems,”
*International Journal of Robust and Nonlinear Control*, vol. 22, no. 15, pp. 1674–1690, 2012. View at Publisher · View at Google Scholar · View at MathSciNet - S. Cong, Y. Zou, and Y. Zhang, “Stability and output feedback stabilization for systems with Markovian switching and impulse effects,”
*Journal of Control Theory and Applications*, vol. 8, no. 4, pp. 453–456, 2010. View at Publisher · View at Google Scholar · View at MathSciNet - N. E. Nahi, “Optimal recursive estimation with uncertain observation,”
*IEEE Transactions on Information Theory*, vol. 15, no. 4, pp. 457–462, 1969. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Z. Wang, D. W. C. Ho, Y. Liu, and X. Liu, “Robust ${H}_{\infty}$ control for a class of nonlinear discrete time-delay stochastic systems with missing measurements,”
*Automatica*, vol. 45, no. 3, pp. 684–691, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Z. Wang, F. Yang, D. W. C. Ho, and X. Liu, “Robust ${H}_{\infty}$ filtering for stochastic time-delay systems with missing measurements,”
*IEEE Transactions on Signal Processing*, vol. 54, no. 7, pp. 2579–2587, 2006. View at Publisher · View at Google Scholar · View at Scopus - X. He and D. Zhou, “Robust $H$-infinity filtering for time-delay systems with missing measurements: a parameter-dependent approach,”
*Journal of Control Theory and Applications*, vol. 5, no. 4, pp. 336–344, 2007. View at Publisher · View at Google Scholar · View at MathSciNet - H. Zhang, Q. Chen, H. Yan, and J. Liu, “Robust ${H}_{\infty}$ filtering for switched stochastic system with missing measurements,”
*IEEE Transactions on Signal Processing*, vol. 57, no. 9, pp. 3466–3474, 2009. View at Publisher · View at Google Scholar · View at MathSciNet - A. Mahmoudi, A. Momeni, A. G. Aghdam, and P. Gohari, “On observer design for a class of impulsive switched systems,” in
*Proceedings of the American Control Conference (ACC '08)*, pp. 4633–4639, Seattle, Wash, USA, June 2008. View at Publisher · View at Google Scholar · View at Scopus - S. Boyd, L. El Ghaoui, E. Feron, and V. Balakrishnan,
*Linear Matrix Inequalities in System and Control Theory*, vol. 15 of*SIAM Studies in Applied Mathematics*, SIAM, Philadelphia, Pa, USA, 1994. View at Publisher · View at Google Scholar · View at MathSciNet - L. Xie, “Output feedback ${H}_{\infty}$ control of systems with parameter uncertainty,”
*International Journal of Control*, vol. 63, no. 4, pp. 741–750, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet