PI/PID Control for Nonlinear Systems via Singular Perturbation Technique

Abstract

The problem of output regulation for nonlinear time-varying control systems under uncertainties is one of particular interest for real-time control system design. There is a broad set of practical problems in the control of aircraft, robotics, mechatronics, chemical industry, electrical and electro-mechanical systems where control systems are designed to provide the following objectives: (i) robust zero steady-state error of the reference input realization; (ii) desired output performance specifications such as overshoot, settling time, and system type of referencemodel for desired output behavior; (iii) insensitivity of the output transient behavior with respect to unknown external disturbances and varying parameters of the system. In spite of considerable advances in the recent control theory, it is common knowledge that PI and PID controllers are most widely and successfully used in industrial applications (Morari & Zafiriou, 1999). A great attention of numerous researchers during the last few decades was devoted to turning rules (Astrom & Hagglund, 1995; O’Dwyer, 2003; Ziegel & Nichols, 1942), identification and adaptation schemes (Li et al., 2006) in order to fetch out the best PI and PID controllers in accordance with the assigned design objectives. The most recent results have concern with the problem of PI and PID controller design for linear systems. However, various design technics of integral controllers for nonlinear systems were discussed as well (Huang & Rugh, 1990; Isidori & Byrnes, 1990; Khalil, 2000; Mahmoud & Khalil, 1996). The main disadvantage of existence design procedures of PI or PID controllers is that the desired transient performances in the closed-loop system can not be guaranteed in the presence of nonlinear plant parameter variations and unknown external disturbances. The lack of clarity with regard to selection of sampling period and parameters of discrete-time counterparts for PI or PID controllers is the other disadvantage of the current state of this question. The output regulation problem under uncertainties can be successfully solved via such advanced technics as control systems with sliding motions (Utkin, 1992; Young & Ozguner, 1999), control systems with high gain in feedback (Meerov, 1965; Young et al., 1977). A set of examples can be found from mechanical applications and robotics where acceleration feedback control is successfully used (Krutko, 1988; 1991; 1995; Lun et al., 1980; Luo et al., 1985; Studenny & Belanger, 1984; 1986). The generalized approach to nonlinear control system design based on control law with output derivatives and high gain in feedback, where integral action can be incorporated in the controller, is developed as well and one is used 7