Abstract-This paper describes how PID controllers can be designed by optimizing performance subject to robustness constraints. The optimization problem is solved using convexconcave programming. The method admits general process descriptions in terms of frequency response data and it can cope with many different constraints. Examples are presented and some pitfalls in optimization are discussed.

Summary We formulate multi‐input multi‐output proportional integral derivative controller design as an optimization problem that involves nonconvex quadratic matrix inequalities. We propose a simple method that replaces the nonconvex matrix inequalities with a linear matrix inequality restriction, and iterates to convergence. This method can be interpreted as a matrix extension of the convex–concave procedure, or as a particular majorization–minimization method. Convergence to a local minimum can be guaranteed. While we do not know that the resulting controller is globally optimal, the method works well in practice, and provides a simple automated method for tuning multi‐input multi‐output proportional integral derivative controllers. The method is readily extended in many ways, for example, to the design of more complex, structured controllers. Copyright © 2015 John Wiley & Sons, Ltd.

In this study, design of low-order feedforward controllers from both reference signal and measurable disturbance for proportional-integral-derivative (PID) controllers is considered. The feedforward controllers from reference are equivalent to the use of a PID controller with set-point weighting. The design problem is formulated as a convex optimisation problem and then solved for a batch of process models. The optimal proportional set-point weights are then used to derive tuning rules that minimise the integrated absolute error. Examples illustrate the usefulness of the proposed method and tuning rules. IntroductionTracking of reference and attenuation of disturbances are the core in the control of a system. Owing to uncertainties in the model describing the process and disturbances this is most conveniently handled using a feedback controller. The tracking and disturbance rejection can be improved without sacrificing robustness by introducing filters or controllers that have an open-loop impact on the control system. Ideally, those feedforward controllers should invert the process dynamics so that perfect tracking and disturbance rejection are obtained. This is unfortunately not possible in general due to, e.g. non-minimum-phase behaviour, time delays, and saturation in the actuators. This paper describes a method for designing feedforward controllers by solving a suitable convex optimisation problem that will handle also those cases. The same optimisation problem will be solved for a batch of processes and the results will be used to formulate tuning rules for how to choose the proportional set-point weight for proportional-integral (PI) and PI-derivative (PID) controllers.In the literature, concerning the design of low-order feedforward controllers intended for use in connection to PID controllers there seem to a bias towards feedforward action from the reference signal. This is most probable due to the fact that the reference signal, in contrast to disturbances in general, is measurable and feedforward therefore is possible.Convex optimisation has evolved into a mature branch in the tree of optimisation disciplines, see [1] and its many references. The book [2] provides a thorough analysis of linear controller design and different specifications that could be considered. It also provides a treatment of controller design with connections to convex optimisation. A variety of easy-to-use tools, for solving convex optimisation problems, are available for a number of different platforms and in several programming languages, for instance the MATLAB toolboxes CVX [3,4], YALMIP [5], and the Python software package CVXOPT [6,7].The design of feedforward compensators, assuming that the process is described by a first-order model plus dead-time (FOTD), has been addressed in a number of papers. In [8] a non-linear feedforward reference control scheme is used to obtain good tracking followed by a PID design that ensures good robustness. A tuning rule for a low-order controller that gives the optimal rejection of measurab...

Design rules for optimal feedforward controllers with lead-lag structure in the presence of measurable disturbances are presented. The design rules are based on stable firstorder models with time delays, FOTD, and are optimal in the sense of minimizing the integratedsquared error. The rules are derived for an open-loop setting, considering a step disturbance. This paper also discusses a general feedforward structure, which enables decoupling in the design of feedback and feedforward controllers, and justifies the open-loop setting.

This paper presents a convex-optimization-based technique to obtain PID parameters, used to control the infusion rate of the anesthetic drug propofol. Controller design is based on a set of identified patient models, relating propofol infusion to an EEG-based conciousness index. The main contribution lies in the method automatically taking inter-patient variability into account, i.e., it guarantees robustness (sensitivity peak) and performance (disturbance rejection) over a set of patient models, without the need for manual intervention.