Optimum settings for discrete PID control of nonlinear systems

This study investigates the application of piecewise affine approximation techniques for the control of nonlinear systems, focusing on the effective linearization of systems described by the $k$th order difference equation $x[n+k]+f(x\left[n\right],x[n+1],\dots ,x[n+k-1])=u\left[n\right]$. The proposed approach employs piecewise linearization by partitioning the nonlinear function $f$ into simplices within a compact domain $D\subset R^k$. The parameter $h$, which determines the number of linear segments, governs the precision of the approximation. As $h$ increases, the linearized system’s behavior converges uniformly to that of the original nonlinear system, facilitating improved control system performance.

A key advantage of the approach is that it does not require full knowledge of the nonlinear function $f$; only values at selected nodal points are needed. Furthermore, it is sufficient that $f$ is twice continuously differentiable within each subdomain of the partition. If bounds on the gradient and Hessian of $f$ are available within each cell, the total approximation error can be rigorously estimated.

In addition, the study incorporates PID controllers and leverages the Particle Swarm Optimization (PSO) algorithm to optimize controller parameters. The optimization framework is designed to minimize key performance indices, such as the Integral Time Absolute Error (ITAE) and Integral Squared Overshoot (ISO). Numerical simulations demonstrate the efficacy of the proposed method, highlighting its ability to balance computational complexity with approximation accuracy in nonlinear control system design.