Robust Switched Control Design for Nonlinear Systems Using Fuzzy Models

Abstract

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.

Smoothing switched control laws for uncertain nonlinear systems subject to actuator saturation

“…In this context, this paper extends the methodology proposed in [13,14] for uncertain T-S fuzzy models, based on [31,32], taking into account actuators saturation. Additionally, an invariant region E.P; 1/ is estimated, in which for every x.0/ 2 E.P; 1/, the state vector of the controlled system x.t/, t > 0 is in a given operation region where the T-S fuzzy model exactly describes the nonlinear system and also the equilibrium point of the controlled systems is locally asymptotically stable with an adequate decay rate.…”

“…Control signal chattering is a generally undesirable phenomenon in real systems. Different control strategies try to avoid the chattering problem, for instance, in sliding mode control [17][18][19][20][21][22][23][24], to name a few, in adaptive control [25][26][27], and in control of switched linear systems [28][29][30].In this context, this paper extends the methodology proposed in [13,14] for uncertain T-S fuzzy models, based on [31,32], taking into account actuators saturation. Additionally, an invariant region E.P; 1/ is estimated, in which for every x.0/ 2 E.P; 1/, the state vector of the controlled system x.t/, t > 0 is in a given operation region where the T-S fuzzy model exactly describes the nonlinear system and also the equilibrium point of the controlled systems is locally asymptotically stable with an adequate decay rate.…”

“…A key point of the new control laws is a proposed exact representation of the minimum function using signal functions. Considering that the minimum functions are largely used in switched control laws [13,14,[28][29][30] and the signal functions are usually employed in sliding mode control laws [17-19, 21, 23, 24], the relationship between the minimum and signal functions offers a bridge between the switched control and sliding mode control strategies. Considering this fact, using a smooth approximation of the signal functions by sgn.´/ ´=.j´j C /; > 0 (sufficiently small), it is introduced the 'smooth minimum' function.…”

“…Considering this fact, using a smooth approximation of the signal functions by sgn.´/ ´=.j´j C /; > 0 (sufficiently small), it is introduced the 'smooth minimum' function. Then, the smooth minimum function is used to avoid the chattering produced by the switched control laws in [13,14], and finally, theorems that guarantee the uniform ultimate boundedness [33] of the controlled systems are proposed.The paper is organized as follows. Section 2, after a brief introduction, presents a generalization of the switched control laws presented in [13,14], taking into account the saturation of the actuators.…”

“…Then, Section 3 shows how to calculate the minimum value of a set using the signal function. After that, Section 4 uses this relationship for smoothing the switched control signal of the strategy proposed in [13,14], for nonlinear uncertain systems described by T-S fuzzy models, now subject to actuator saturation. The stability analysis of the proposed 'smooth switched' control is also presented in this section.…”