Fuzzy Gain Scheduling of Coupled PID Controllers for Stabilization of the Inverted Pendulum Mariagrazia Dotoli, Biagio Turchiano Politecnico di Bari, Dipartimento di Elettrotecnica ed Elettronica Via Re David, 200, I-70125 Bari, Italy Phone: +39-080-5963312, Fax +39-080-5963410 email:{dotoli, turchiano}@poliba.it ABSTRACT: Changes in operating conditions can alter performances of nonlinear processes controlled with a fixed PID algorithm. One way of preventing degradation is to develop a supervised control system. In this paper we present some results about fuzzy gain scheduling of a double PID controller for stabilization of an inverted pendulum on a cart. First, on the basis of the responses obtained with an unsupervised and locally optimized double PID, a simplified fuzzy supervisor is designed, including the cart set-point as the scheduling variable. Subsequently, the rule base is modified with two additional inputs: the pole and cart errors. Finally, a peak observer is introduced to moderate the steady-state error through integral action in the cart PID. The resulting fuzzy supervisor adapts the coupled PIDs to the changing operating conditions. The proposed control strategy is implemented in the MatLab software environment. Results are discussed in comparison with an unsupervised PID algorithm. KEYWORDS: PID control, gain scheduling, adaptive control, fuzzy supervisor, non linear system. INTRODUCTION In most control applications the most popular control algorithm is by far the PID controller with a fixed configuration and structural parameters determining the amount of proportional, derivative and integral action in the overall control law. This regulator is so popular that many rules of thumbs exist in the literature for tuning the parameters [1]. When the controlled process is nonlinear, however, a fixed gain PID controller cannot produce satisfactory control performance in all process operating regions. Hence, using the linear PID control technique for controlling a nonlinear system results in a tuning configuration that must be adjusted when a change in the operating conditions occurs. The resulting gain scheduling of PID controllers technique has over the years become one of the most popular methodologies to solve non linear control problems [1], [9]. The main advantage of conventional gain scheduling (CGS) is that controller parameters can be changed very quickly in response to changes in the plant dynamics, since no parameter estimation is required. One drawback of CGS is that the parameter change may be rather abrupt across boundaries of the process operating regions, which may result in unsatisfactory or even unstable control performance. A successful way of solving nonlinear control problems using PID controllers consists in employing fuzzy gain scheduling (FGS) to adjust the controller parameters [11]. The resulting control strategy is hierarchical, consisting of a fuzzy supervisor and of a PID control algorithm. The former is designed on the basis of operator or expert knowledge, while the latter is optimally tuned for each operating condition [1]. In addition, a fuzzy inference mechanism interpolates the controller parameters in the transition regions. Not mentioning adaptivity and the ability to cope with incomplete knowledge of the process, the main advantage in using fuzzy logic to adapt the PID parameters is that switching between operating conditions is smoothly operated, since the controller parameters are adjusted via a bumpless transfer algorithm. For further insights on FGS of a PID controller and its theoretical aspects see [2], [4], [6], [10], [11]. Recently, there has been increasing interest in multivariable PID controllers [5], [8] and FGS of MIMO PID controllers [7]. In this paper a double PID controller is proposed for controlling a well-known MIMO control benchmark: the inverted pendulum on a cart. The control task is to stabilize the pole in its upwards equilibrium point, while simultaneously controlling the cart to a varying reference position on a track. In the sequel, FGS of the coupled PID controllers is investigated: the structure and design issues of the proposed technique are discussed and different implementations of the methodology are illustrated with simulation examples conducted on the case study. It is shown that the technique guarantees a satisfactory performance under changing operating conditions, characterized by bumpless transfers across the transition regions.