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 [
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 [
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 [
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:
The remainder of the paper is organized as follows. In Section
Consider the following uncertain discrete impulsive switched systems with state delay:
The measurement output which may contain missing data is described by
Here, we are interested in designing a DOF switched controller described by
Different from the existing DOF controllers proposed in [
Now, define a new state vector,
The following definitions and lemmas will be essential for our later development.
For any
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 (
System (
For a given matrix
Let
The following theorem provides sufficient conditions under which the exponential stability of system (
Consider system (
Choose a piecewise Lyapunov function candidate for system (
When
Applying Lemma
When
It follows that
This completes the proof.
Compared with the existing results presented in [
This section will give some LMIs conditions for the controller design.
Consider system (
Then, there exists a DOF controller (
Moreover, if the previous LMI conditions are feasible, then the desired dynamic output feedback controller parameters can be designed as
Let the matrix
By
Define the following matrices:
Then we have
Use
Then, we can obtain that (
where
Using
Then combining (
By Lemma
Using Lemma
The proof is completed.
From Theorem
Based on Theorem
Given the system matrices and a constant
Applying singular value decomposition to the first equation of (
By substituting matrices
Determine the DOF controller parameters
In this section, we present an example to illustrate the effectiveness of the proposed approach. Consider system (
Let
Let
According to (
Switching signal.
State responses of the resulting closed-loop system.
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.