Abstract
A robust fuzzy digital PID control methodology based on gain and phase margins specifications is proposed. A mathematical formulation based on gain and phase margins specifications, the Takagi-Sugeno fuzzy model of the process to be controlled, the structure of the digital PID controller, and the time delay uncertain system are developed. A multiobjective genetic strategy is defined to tune the fuzzy digital PID controller parameters, so the gain and phase margins specified to the fuzzy control system are found. An analysis of necessary and sufficient conditions for fuzzy digital PID controller design with robust stability, with the proposal of the two theorems, is presented. Experimental results show the efficiency of the proposed methodology in this chapter, applying a platform control in real time of a thermic process through tracking the reference and the gain and phase margins keeping closed the specified ones.
TopBackground
New methodologies have been developed to ensure high performance in control systems. Reaching this objective in front of uncertainties is not a trivial task, and still constitutes a challenge in the control theory. Especially, the robust control theory has received great interest from scientific community to develop controllers to deal with complex dynamics such as nonlinearities, uncertainties, time delay, among others, and guarantee robust control design (Mueller, 2011).
Nowadays, the interest for strategies to model and control of complex process has been motivated by the following factors (Serra, 2012):
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Development of efficient identification methods and greater applicability of computational resources;
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Improvement of softwares and hardwares technologies, making it possible to incorporate complex models in the control systems design.
The robust control theory has grown remarkably in recent years, gaining ground even in the industrial environment where it is a valuable tool for analysis and dynamic systems design (Wu et al., 2010). Due to its practical applicability, the control theory is used in solving engineering problems and it has been applied in several areas where practical requirements of performance are relevant (Zhong, 2006; Serra, 2012).
Key Terms in this Chapter
Phase Margin: Is the difference between phase of the system and -180° at the gain crossover frequency, (i. e., the frequency where the module of the open loop transfer function is unitary). It is an important measure of robustness, because it is related to the stability of a closed loop system.
Mutation Rate: It is the probability of the mutation process in GA. The mutation inserts new genetic material in the population. Low mutation rate prevents that a value remains as the best solution for generations. High mutation rate becomes excessively random the research of solutions in the algorithm, resulting in loss of optimal solutions.
Multiobjective Genetic Algorithm: Genetic Algorithm which contains several objectives to optimize.
Fuzzy Control: Control based on fuzzy rules. The control action is performed from membership degrees of each controller of rule base.
Gain Margin: Corresponds to reciprocal of the magnitude of the system, at the phase crossover frequency, (i. e., the frequency where the phase angle is -180°). It is an important measure of robustness, because it is related to the stability of a closed loop system.
Uncertain Dynamic Systems: System in which uncertainties and dynamics are observed. Most real systems are dynamic and have uncertainties (time delay, nonlinearity, parametric variations). The dynamics and uncertainties can make a system unstable, so they should be considered in a control project.
Robust Control: A control action designed to track a reference trajectory and keep stability, despite of uncertainties of the process to be controlled.
Digital PID Control: Control action by controller type PID (with proportional, integral and derivative gains), in the discrete time domain.
Crossover Rate: Parameter of AG that determines the percentage of individuals to be selected for the crossover process. When the crossover rate is too high, many individuals of the population can be replaced, so it results in a loss of potentials individuals with good characteristics, but if the crossover rate is too low, may increase the time convergence of the algorithm.