Um novo método para sintonia de controladores PID não lineares sob restrições usando modelo Fuzzy Takagi-Sugeno

Abstract

Most of the existing controllers in industries are from the PID family (Proportional-Integral-Derivative) developed for linear systems, although the processes are non-linear. Therefore, they present a significant loss of performance when operating outside the operating point for which they were designed. An alternative controller is the so-called PDC (Distributed Parallel Compensator) which, based on a Fuzzy Takagi-Sugeno model, results in a non-linear controller. The Fuzzy Takagi-Sugeno model is intrinsically non-linear, due to the membership functions that make up its structure. Thus, this model can represent nonlinear systems in such a way that, from the physical model of the system, it is possible to obtain a Fuzzy Takagi-Sugeno model that represents it exactly, which contributes to obtaining the PDC. The Fuzzy Takagi-Sugeno model can be represented by the composition of local models. In the PDC controller methodology, local controllers are designed, which can be a PID, for each local model, thus presenting the same rules structure as the Fuzzy Takagi-Sugeno model, that is, the controller shares the same membership functions. Normally, the PDC stability guarantee considers a quadratic Lyapunov function as a candidate, thus allowing it to be tested for linear matrix inequalities (LMIs) that are solvable using efficient convex optimization algorithms. However, polyhedral Lyapunov functions have shown advantages over the quadratic function, especially with regard to satisfying constraints on state, input and output variables, showing their potential to be used to guarantee stability and performance of a PDC-controlled system. In view of this, this work proposes an optimization-based method for tuning Proportional-Integral (PI) controllers for nonlinear systems subject to constraints on the controlled variable and actuator saturation. Conditions are presented for a polyhedron contained in the set of constraints to be invariant with respect to a closed-loop system with a controller subject to saturation. These conditions are used in the formulation of a bilinear programming problem whose solution provides the controller parameters that satisfy the constraints and an associated invariant set. Throughout the development of the work, several variations of the main technique were explored and their advantages and disadvantages were shown through the results. The results are presented in numerical examples. Starting with linear systems and testing I-P controllers and a traditional PI controller with a filter for the reference, we explore case studies with: saturation allowance, state constraints, time-delay, high order systems, servo and regulator problem. Then we explore nonlinear systems testing systems with and without the affine term and controllers with and without the feedfoward term, exploring case studies with: saturation permission, state constraints and servo problem. The results show that the developed method obtains a tuning for the controller that does not violate any state constraint and allows saturation, thus illustrating its efficiency