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Output Feedback Control of Discrete Impulsive Switched Systems with State Delays and Missing Measurements
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.
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 . The authors in  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  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 ). 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.
Definition 4 (see ). 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 ). For a given matrix , where and are square matrices, the following conditions are equivalent:(i); (ii), ;(iii), .
Lemma 6 (see ). 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
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
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 thatwhere , , , , , , , .
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:
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:
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.
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 , .
4. Numerical Example
Let , , and , then by solving the matrix inequalities in Theorem 9, we can get the DOF controller parameters
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.
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.
This work was supported by the National Natural Science Foundation of China under Grant no. 61273120 and the Alexander von Humboldt Foundation in Germany.
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