# A Randomized Algorithm for Optimal PID Controllers
## 1 Introduction  
Proportional Integral Derivative (PID) controllers are the most widely-used controllers in industry despite of many new results in control theory that have been achieved in recent years. This is because in many cases PID controllers achieve close to optimal performance, because it is hard to compete with their investment-to-profit ratio and because of their well-understood influence on the closed-loop system behavior, as well as the availability of many tuning procedures for various performance objectives, at least for SISO systems or MIMO systems that can be well decoupled. Often, the performance achieved by PID controllers can be significantly improved by using more effective design techniques. Therefore the field of PID controllers is an on-going active field of research. A review of most recent techniques for tuning and designing PID-based control structures, together with methods for assessing their performance is presented in [1]. In [2] optimal decentralized MIMO PID controller design for a benchmark MIMO distillation column models is carried out using a Modified Firefly Algorithm (MFA), where the optimality is with respect to an objective function which is a weighted combination of the \(L_{1}\)-norm and the \(L_{2}\)-norm of the tracking error, with respect to a reference signal. Tuning of MIMO coupled systems PID controllers is much more challenging than tuning PID controllers for SISO system or decoupled MIMO systems, because of the number of parameters and because they should be tuned all at once, since otherwise, tuning each channel separately might end up in a very poor performance of the whole system or even destabilize the whole system. In [3] the need for effective methods for the design of PID optimal controllers for MIMO coupled systems that cannot be well approximated by second-order systems, was raised. In [4] the parametrization of all the solutions of the standard \(H_{\infty}\) control problem is used to obtain Bilinear Matrix Inequalities (BMI's) constraints on the PID gains for each frequency. Next, for each frequency, Linear Matrix Inequalities (LMI's) restricted constraints for a stabilizing PID gain are derived from the BMI constraints. Finally, the set of LMI's for a chosen finite set of frequencies are solved by the Iterated Linear Matrix Inequalities(ILMI) method. The sequence is shown to converge to a local optimum. In [5] a method for MIMO coupled systems PID controller design for stable plants (given by transfer function at an appropriate set of frequencies) is represented. The method relies on solving a sequence of SDP's, but it cannot guarantee finding a globally optimal design. In this article, we suggest a randomized algorithm for PID optimal controllers. The algorithm is based on the known reduction of [3] to the Stabilizing Static-Output-Feedback (SOF) problem, which is then solved by the Ray-Shooting Method, where the method is used both for the search for feasible starting SOF and for the optimal SOF (see [6; 7; 8] for some applications of the method). A closely related approach to ours is given in [9] where the problem is first reduced to the SOF problem, by the reduction of [3] and by using a descriptor system representation of the augmented system (in order to bypass the feasibility condition that \(I-\hat{K}_{3}C_{2}B_{2}\) should be invertible--see the following). Next, the search problem of a starting stabilizing SOF controller is solved via the ILMI approach. Finally, a Bounded-Real Lemma version for descriptor systems is used in order to get a set of Quadratic Matrix Inequalities (QMI's) that their solution guaranties a \(\gamma\) suboptimal \(H_{\infty}\) gain. The last can be solved by the ILMI method (without a proof of convergence). The method of ILMI was discussed in many articles (see [3; 10; 11; 12]). Unfortunately, the ILMI method needs a good starting point and has no proof of convergence in general.