^{1}

^{2}

^{3}

^{2}

^{2}

^{1}

^{2}

^{3}

The paper proposes a new switched control design method for some classes of uncertain nonlinear plants described by Takagi-Sugeno fuzzy models. 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 eliminates the need to find the membership function expressions to implement the control laws. The control designs of a ball-and-beam system and of a magnetic levitator illustrate the procedure.

There has been much interest in recent years to study switched systems, mainly linear systems, as can be seen in [

Results on switching laws based on the premise variable can be seen in [

Switching laws based on the plant state vector were proposed, for instance, in [

This paper proposes a new method of switched control for some classes of uncertain nonlinear systems described by Takagi-Sugeno fuzzy models. This new control law, which also depends on the state variables, generalizes the results given in [

The main advantage of this new procedure is its practical application because it eliminates the need to find the explicit expressions of the membership functions, which can often have long and/or complex expressions or may not be known due to the uncertainties. Furthermore, for certain classes of nonlinear systems, the switched controller can operate even with an uncertain reference control signal. Additionally, with the proposed methodology the closed-loop systems usually present a settling time that is smaller than those obtained with fuzzy controllers, without using switching, that are widely studied in the literature. Moreover, performance indices such as decay rate and constraints on the plant's input and output can be added in the control design procedure.

Simulation results of the control of a ball-and-beam system and of a magnetic levitator are presented to compare the performance of the proposed control law with the traditional PDC fuzzy control law [

The paper is organized as follows. Section

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

Consider the Takagi-Sugeno fuzzy model as described in [

From [

Assuming that the state vector

Similar to (

From (

The following theorem, whose proof can be seen in [

The equilibrium point

In this paper, for simplicity, the new design method of the controller gains was based on Theorem

In this section the design of a switched controller for the Takagi-Sugeno fuzzy system (

Suppose that (

Therefore, from (

Assume that the conditions of Theorem

Consider a quadratic Lyapunov candidate function

Theorem

In this case a fuzzy system similar to (

Let

After the aforementioned considerations, note that the system (

In this case, it is assumed that the plant given by

Now consider 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

Observe that the function

To illustrate this case, presented is the control design of a ball-and-beam system, in Figure

Ball-and-beam system.

Define the state variables

Note that, for implementing the switched controller (

After the calculations the following maximum and minimum values of the functions

Thus, the nonlinear function

Therefore, from (

Similarly, from (

Recall that

Now, define

Thus, using the LMIs (

The goal of the simulation is to keep the ball at the origin

State variables of the ball-and-beam system (

Control signal of the ball-and-beam system (

Note that the controller gains have been found using the generalized form proposed in [

To illustrate this case, consider the control system design of a magnetic levitator presented in Figure

Magnetic levitator.

Define the state variables

The objective of the paper is to design a controller that keeps the ball in a desired position

From the second equation in (

Note that the equilibrium point is not in the origin

Therefore,

Hence, the system (

Now, define

After this adjustment it is seen that the problem falls into Case 1. Thus, the procedure stated in Case 1 can be used for designing a switched control law

Thus, to find the local models, the maximum and minimum values of functions

As expected, after the calculations, considering (

Therefore, from (

Using the LMIs (

For numerical simulation, at ^{2} (assuming that

State variables of the magnetic levitator (

Switched control signal

Consider the magnetic levitator from Section

Thus, as described in Section

Observe that the system (

After the calculations, the maximum and minimum values of the functions

Thus, using the LMIs (

Setting

Consider

For the simulation illustrated in Figure

Position

Note that in this case it is not possible to obtain the membership functions, since the mass is uncertain, but the proposed method overcomes this problem, because it does not depend on such functions. Observe also that even with uncertainty in the reference control signal (because

In a control design it is important to assure stability and usually other indices of performance for the controlled system, such as the settling time (related to the decay rate), constraints on input control and output signals. The proposed methodology allows specifying these performance indices, without changing the LMIs given in [

This paper proposed a new switched control design method for some classes of uncertain nonlinear plants described by Takagi-Sugeno fuzzy models. The proposed controller is based on LMIs and the gain is chosen by a switching law that returns the smallest time derivative value of the Lyapunov function. An advantage of the proposed methodology is that it does not change the LMIs given in the control design methods commonly used for plants described by Takagi-Sugeno fuzzy models as proposed, for instance, in [

The authors declare that there is no conflict of interests regarding the publication of this paper.

The authors gratefully acknowledge the financial support by FAPESP (Grant 2011/17610-0), CNPq, and CAPES from Brazil.

_{∞}controller design of fuzzy dynamic systems based on piecewise lyapunov functions

_{∞}control for nonlinear systems

_{∞}linear parameter-varying state feedback control

_{∞}control of Takagi-Sugeno fuzzy systems