Multi-Objective Optimal Design of a PID Sliding Mode Controller With Three Different Reaching Laws

Sliding mode controllers (SMCs) are well-known nonlinear control techniques. The design of a SMC involves the selection of a sliding mode surface and reaching law. The constant, exponential, and power rate reaching laws are the most widely used. Selecting a reaching law is often based on the desired reaching time; that is how fast the state trajectory approaches the switching manifold. However, the selection of a reaching law does not only affect the reaching time (tr) but also other design specifications such as the settling time (ts), overshoot (Mp), and tracking error (JIAE). Indeed, the design of a closed-loop system usually involves multiple and often conflicting objectives. Therefore, a multi-objective optimal design approach that takes into consideration all the design requirements should be adopted. Furthermore, a systematic study is needed to evaluate and compare the performance of a SMC controller under these reaching laws in multi-objective settings. To this end, the problems of designing a PID (Proportional-Integral-Derivative) sliding mode controller applied to linear and nonlinear dynamic systems using the three reaching laws are formulated as multi-objective optimization problems (MOPs). The objective space includes tr, Mp, ts, and JIAE and the parameter space consists of the design gains of the reaching laws and the sliding mode surface. The non-dominated sorting genetic algorithm (NSGA – II) is used to solve the optimization problem. The solution of the MOP is a Pareto front of optimal design points. Therefore, comparing three Pareto fronts is not a straightforward task. As a result, sections of the Pareto fronts that satisfy some legitimate constraints on the objective space are extracted. Then, a comparison among these sections is conducted graphically. The results show that the exponential rate reaching law outperforms the other two laws in most of the objectives under investigation.