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This paper proposes a new switched control design method for some classes of linear time-invariant systems with polytopic uncertainties. This method uses a quadratic Lyapunov function to design the feedback controller gains based on linear matrix inequalities (LMIs). The controller gain is chosen by a switching law that returns the smallest value of the time derivative of the Lyapunov function. The proposed methodology offers less conservative alternative than the well-known controller for uncertain systems with only one state feedback gain. The control design of a magnetic levitator illustrates the procedure.

In recent years, there has been much interest in studying switched systems, due to the considerable advance in this research field, initiating mainly with [

In general, most papers in the area of switched linear systems utilize multiple Lyapunov functions [

The design of the robust state feedback control for continuous-time systems subject to norm bounded uncertainty can be seen in [

Although outnumbered, there are papers about switched linear systems using a common Lyapunov function as in [

This paper proposes a new methodology for switched control design of a class of linear systems with polytopic uncertainties. This method uses a common Lyapunov function and quadratic stability for designing the state feedback controller gains based on linear matrix inequalities (LMIs). The proposed controller chooses a gain from a set of gains by means of a suitable switching law that returns the smallest value of the Lyapunov function time derivative. The proposed methodology allows a less conservative LMI-based design than the traditional method for uncertain plants that considers only one state feedback controller gain [

To confirm the advantages of the proposed methodology, figures comparing the regions of feasibility of the proposed methodology with the one state feedback gain classical method are presented. To compare the performance of the proposed control law with the classical control law, an application in the magnetic levitator control was simulated. The computational implementations were carried out using the modelling language YALMIP [

For convenience, in some places, the following notation is used:

Consider the linear system with polytopic uncertainties

Assuming that all state variables are available for feedback, the control law largely used in the literature is given by [

Define a feedback control law with the state vector as

This section presents some results on stability and control of linear systems with polytopic uncertainties.

The linear system with polytopic uncertainties given in (

The equilibrium point

Consider a quadratic Lyapunov candidate function

Defining

If

It is similar to the proof of Theorem

In a control design, it is important to assure stability and usually other indices of performance for the controlled system, such as the setting time, constraints on input control and output signals. The setting time is related to the decay rate of the system (

The linear system with polytopic uncertainties given in (

It is similar to that of Theorem

The equilibrium point

The proof is similar to that of Theorem

One can constraint the norm of the controller gains by imposing restrictions on

The constraint on the norm of the controller gains such that

The proof is similar to that presented in [

In this section, the design of a switched controller for the uncertain system (

Suppose that (

Note that, in (

The implementation of (

Therefore, from (

Assume that the conditions of Corollary

Consider a quadratic Lyapunov candidate function

Theorem

In this case, the linear system with polytopic uncertainties will be considered as given in (

Let

After the considerations above, note that the system (

In this case, it is assumed that the system (

Now suppose that

Assume that the gains

Within this context, the following theorem is proposed.

Suppose that the conditions from Theorem

Consider a quadratic Lyapunov candidate function

In this section, examples will be used to illustrate the three cases presented. The figures will show that the LMIs used to find the controller gains (Theorems

Stability.

Consider the uncertain linear system given by (

In this example, the idea is to show that, considering only the stability, the methodology presented in this paper (Corollary

Feasible regions using Theorem

Stability and constraint on the norm of the controller gains.

Consider the uncertain linear system given in (

Therefore, the matrix

In control problems, it is important to consider performance indices, for instance, restrictions on the norm of the controller gains. Thus, to find the regions of feasibility of the system, Theorem

Feasible regions using Theorems

Stability, decay rate, and constraint on the norm of the controller gains.

Consider the uncertain linear system given in (

As in previous examples, the region of feasibility of the system will be found, and in this case, Theorems

Feasible regions using Theorems

To illustrate, the control of a magnetic levitator is designed, whose mathematical model [

Define the state variables

Consider that during the required operation,

The goal of the simulation is to design a controller that keeps the ball in a desired position

From the second equation

Linearizing the system (

Consider the constant position

Observe that the system (

Considering the domain

Feasible regions using Theorems

Fixing the parameters related to decay rate in

As the controller gains have been found, to implement the control law (

For the numerical simulation, at

Position

Note that the control law given in (

Observe that the function

Position

This paper proposed a new switched control design method for some classes of linear systems with polytopic uncertainties. In the proposed controller, the gain is chosen by a switching law that returns the smallest time derivative value of the Lyapunov function.

The LMI used to find the gains are less conservative than that with only one state feedback gain [

The authors would like to thank the Brazilian agencies CAPES, CNPq, and FAPESP which have supported this research.