Robust PID controller tuning based on the constrained particle swarm optimization

Abstract

This paper proposes a novel tuning strategy for robust proportional-integral-derivative (PID) controllers based on the augmented Lagrangian particle swarm optimization (ALPSO). First, the problem of PID controller tuning satisfying multiple $H_{\infty }$ performance criteria is considered, which is known to suffer from computational intractability and conservatism when any existing method is adopted. In order to give some remedy to such a design problem without using any complicated manipulations, the ALPSO based robust gain tuning scheme for PID controllers is introduced. It does not need any conservative assumption unlike the conventional methods, and often enables us to find the desired PID gains just by solving the constrained optimization problem in a straightforward way. However, it is difficult to guarantee its effectiveness in a theoretical way, because PSO is essentially a stochastic approach. Therefore, it is evaluated by several simulation examples, which demonstrate that the proposed approach works well to obtain PID controller parameters satisfying the multiple $H_{\infty }$ performance criteria.

Introduction

Tuning strategies for robust proportional-integral-derivative (PID) controllers satisfying the given $H_{\infty }$ specifications based on optimization approaches have recently received considerable attention. However, the design problem for optimal robust PID controllers based on $H_{\infty }$ techniques results in a non-convex optimization problem which suffers from computational intractability and conservatism. For such problems, Åström, Panagopoulos, and Hägglund (1998) introduced an iterative procedure using the well-known Newton–Raphson search algorithm to find the PI controller gains within the non-convex domain. However, the choice of good initial conditions is a crucial factor, since it has a considerable effect on a computational efficiency of Newton–Raphson iterations. In Hwang and Hsiao (2002), a derivative gain should be chosen in advance by the designer. Then, based on the given derivative gain, the analytical expressions for describing the boundary of an equality constraint set on proportional and integral gains are derived. The maximum allowable proportional and integral gains are obtained by tracing the boundaries of equality constraint sets using a path-following algorithm. Ho (2003) presented a synthesis of $H_{\infty }$ PID controllers based on the generalized Hermite–Biehler theorem for complex polynomials which is used to develop a linear programming based optimization algorithm for determining an admissible PID controller. However, a suitable proportional gain should be chosen in advance by the designer to determine the integral and derivative gains. Therefore, the reasonable selections of derivative gain in Hwang and Hsiao (2002) and proportional gain in Ho (2003) are important issues to improve the performance of the developed PID controllers.

From the above observations, it is required to develop a novel tuning strategy of robust PID controller, which can determine all controller gains simultaneously by solving an optimization problem subject to multiple constraints on $H_{\infty }$ specifications. Further, most of the conventional PID controller design techniques are based on simple characterization of system dynamics as first-order or second-order models, and there are very few generally accepted design methods for systems with higher order (Ho & Lin, 2003). Thus, it is one of the important research issues to develop a robust PID controller applicable to higher order systems.

On the other hand, Eberhart and Kennedy (1995) recently proposed a particle swarm optimization (PSO) algorithm which is a swarm intelligence technique and is one of the evolutionary computation algorithms. PSO has attracted a lot of attention in recent years because of the following reasons (Parsopoulos & Vrahatis, 2002): First, it requires only a few lines of computer code to realize the PSO algorithm. Second, its search technique using not the gradient information but the values of the objective function makes it an easy-to-use algorithm. Third, it is computationally inexpensive, since its memory and CPU speed requirements are very low. Fourth, it does not require a strong assumption made in conventional deterministic methods such as linearity, differentiability, convexity, separability or non-existence of constraints in order to solve the problem efficiently. Finally, its solution does hardly depend on initial states of particles, which could be a great advantage in engineering design problems based on optimization approaches. Further, Sedlaczek and Eberhard (2006) recently developed an augmented Lagrangian particle swarm optimization (ALPSO) algorithm to handle the optimization problem subject to equality and inequality constraints.

The aim of this paper is to develop a simple and computationally tractable tuning strategy for robust PID controllers satisfying multiple $H_{\infty }$ specifications. Finding such controller gains is known to be computationally intractable by the conventional techniques. Therefore, in order to solve simply and directly such a design problem without using any complicated manipulations, we first formulate the ALPSO based constrained optimization problem, and then present its distinctive features. It is important to note that a set of PID gains can be directly obtained by solving a non-convex optimization problem based on the ALPSO technique, which is the main difference from the conventional methods by Hwang and Hsiao (2002) and Ho (2003). Also, the proposed technique is applicable both to stable and to unstable systems, which is different from Kristiansson and Lennartson (2006). However, it is difficult to guarantee its effectiveness in a theoretical way, because PSO is essentially a stochastic approach. Therefore, several numerical examples are given to verify the effectiveness of our robust PID controller design technique.