Feedback controls are usually designed to achieve multiple and often conflicting performance goals. These incommensurable objectives can be found in both time and frequency domains. For instance, one may want to design a control system such that the closed-loop system response to a step input has a minimum percentage overshoot , peak time, rise time, settling time, tracking error, and control effort. Another designer may want the controlled system to have a maximum crossover frequency, maximum phase margin and minimum steady-state error . However, Most of these objectives cannot be achieved concurrently. Therefore, trade-offs have to be made when the design objective space includes two or more conflicting objectives. These compromise solutions can be found by techniques called multi-objective optimization algorithms. Unlike the single optimization methods which return only a single solution, the multi-objective optimization algorithms return a set of solutions called the Pareto set and a set of the corresponding objective function values called the Pareto front.

In this thesis, we present a multi-objective optimal (MOO) design of linear and nonlinear control systems using two algorithms: the non-dominated sorting genetic algorithm (NSGA-II) and a multi-objective optimization algorithm based on the simple cell mapping. The NSGA-II is one of the most popular methods in solving multi-objective optimization problems (MOPs). The cell mapping methods were originated by Hsu in 1980s for global analysis of nonlinear dynamical systems that can have multiple steady-state responses including equilibrium states, periodic motions, and chaotic attractors. However, this method can be also used also to solve multi-objective optimization problems by using a direct search method that can steer the search into any pre-selected direction in the objective space.

Four case studies of robust multi-objective/many-objective optimal control design are introduced. In the first case, the NSGA-II is used to design the gains of a PID (proportional-integral-derivative) control and an observer simultaneously. The optimal design takes into account the stability robustness of both the control system and the estimator at the same time. Furthermore, the closed-loop system's robustness against external disturbances and measurement noises are included in the objective space.

The second case study investigates the MOO design of an active control system applied to an under-actuated bogie system of high speed trains using the NSGA-II. Three conflicting objectives are considered in the design: the controlled system relative stability, disturbance rejection and control energy consumption. The performance of the Pareto optimal controls is tested against the train speed and wheel-rail contact conicity, which have huge impact on the bogie lateral stability.

The third case addresses the MOO design of an adaptive sliding mode control for nonlinear dynamic systems. Minimizing the rise time, control energy consumption, and tracking integral absolute error and maximizing the disturbance rejection efficiency are the objectives of the design. The solution of the MOP results in a large number of trade-off solutions. Therefore, we also introduce a post-processing algorithm that can help the decision-maker to choose from the many available options in the Pareto set. Since the PID controls are the most used control algorithm in industry and usually experience time delay, a MOO design of a time-delayed PID control applied to a nonlinear system is presented as the fourth case study. The SCM is used in the solution of this problem. The peak time, overshoot and the tracking error are considered as design objectives and the design parameters are the PID controller gains.

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