This paper concerns static output feedback design of discrete-time linear switched system using
switched Lyapunov functions (SLFs). A new characterization of stability for the switched system under
arbitrary switching is first given together with
As computers, digital networks, and embedded systems become ubiquitous and increasingly complex, one needs to understand the coupling between logic-based components and continuous physical systems. This prompted a shift in the standard control paradigm in which dynamical systems were typically described by differential or difference equations to allow the modeling, analysis, and design of systems that combine continuous dynamics with discrete logic. This paradigm is called hybrid systems [
The notations used throughout the paper are standard. The relation
Consider a linear switched system in the discrete time domain described by the following state equation:
As [ the switching rule the matrices for each mode, the pairs
We present now two useful lemmas used in the proofs later in the paper.
Given a symmetric matrix
See [
Let there exists a matrix G such that
Straightforward [
In the following sections, the problems of stability analysis with
To check stability of the switched system (
The origin equilibrium point of
The Lyapunov function (
The following statements are equivalent. There exists a Lyapunov function of the form ( There exist There exist There exist N symmetric matrices
The Lyapunov function is given by
(
(
(i) For
(ii) Since
For the following section, let the switched system be
The switched system (
The following statements are equivalent. The switched system is asymptotically stable with The switched system is asymptotically stable with
It follows much the same way as above for the first steps leading to
Of course, with
This section gives the main results of the paper. First, based on the switched quadratic Lyapunov function and Projection Lemma, new sufficient LMI conditions are deduced to obtain stabilizing SOF controller gains. Then, the method is extended to SOF controller design with
Given the measurements
If there exist symmetric matrices
(
Assume that there exist
Replacing
Letting
If there exist symmetric matrices
(
First, notice that if (
If there exist symmetric matrices
First, notice that if (
This section gives control design result that directly comes from the preceding section so that the proof will be almost omitted. Let now the linear switched system be controlled described by
We are interested by the design of a switched static output feedback control given by (
If for all
In this section, a numerical evaluation is proposed. The problem considered here is the design of a static output feedback controller stabilizing the switched system. The result obtained using the Theorem Method1 uses constant Lyapunov function (CLF)
Numerical evaluation.
Switched system | Success | |
---|---|---|
Method1 | 76 | |
Method2 | 83 | |
Method3 | 88 | |
Method4 | 95 | |
Method1 | 40 | |
Method2 | 63 | |
Method3 | 67 | |
Method4 | 87 | |
Method1 | 8 | |
Method2 | 20 | |
Method3 | 32 | |
Method4 | 52 | |
Method1 | 61 | |
Method2 | 87 | |
Method3 | 93 | |
Method4 | 97 | |
Method1 | 66 | |
Method2 | 81 | |
Method3 | 82 | |
Method4 | 100 | |
Method1 | 35 | |
Method2 | 65 | |
Method3 | 68 | |
Method4 | 92 | |
Method1 | 66 | |
Method2 | 73 | |
Method3 | 85 | |
Method4 | 90 | |
Method1 | 20 | |
Method2 | 42 | |
Method3 | 58 | |
Method4 | 73 | |
Method1 | 0 | |
Method2 | 16 | |
Method3 | 25 | |
Method4 | 30 |
This corresponds to conditions in Theorem Method2 uses the conditions given in Theorem Method3 uses the conditions given in Theorem Method4 uses the conditions given in Theorem
By using the matlab LMI Control Toolbox to check the feasibility of the LMI conditions, we introduce a counter (Success Method1, Success Method2, Success Method3, and Success method4) which is increased if the corresponding method succeeds in providing an output feedback stabilizing control.
The simple illustrative switched system is constituted of two subsystems with different dynamic, control, and output matrices. The data are (
The proposed approach improves, in terms of norm bound, the results given in [
The contribution of this paper stands in the combination of poly-quadratic stability concept [
The Tables
Numerical example.
Mode1 | Mode2 |
---|---|
Numerical results.
Approach | Controller gains | Upper bounds |
---|---|---|
Theorem | 0.96 | |
Approach [ | 0.58 | |
New approach, | 0.54 |
In this paper, the problem of stability and synthesis of discrete switched linear systems using switched Lyapunov function (SLF) is investigated. Using these Lyapunov function, the problem of stability analysis and the static output feedback design have been studied. Based on the stability of polytopic time-varying uncertain systems, condition for stability analysis of switched systems has been proposed. The existence of an SLF to ensure stability of a discrete switched system is proven to be equivalent to three-LMI-based conditions considered in [