On the ship path-following control system design by using robust feedback linearization Zenon Zwierzewicz Department of Automation and Robotics Szczecin Maritime University Szczecin, Poland z.zwierzewicz@am.szczecin.pl Abstract—The paper considers the problem of ship path- following system design based on input-output feedback linearization method combined with the robust control approach. At first, the nonlinear process model is linearized by means of partial coordinate transform and a simple system nonlinearity cancellation. Since the exact values of the model parameters are not known, the ensuing inaccuracies are taken as disturbances acting on the system. Thereby we obtain a linear system with an extra term representing the uncertainty which can be treated by using robust, H ∞ optimal control techniques. The performed simulations of ship path-following process confirmed a high performance of the proposed controller despite the assumed significant errors of its parameters. I. INTRODUCTION The feedback linearization (FL) method [9,10,11,12] consists in such transformation of a given nonlinear system that results in a new, linear time-invariant one. Here, by transformation we mean the application of a proper control law combined with a possible change of the system coordinates. Once a linear system is obtained, a secondary control law (or sub-control) should be designed to ensure that the overall closed-loop system performs according to the specifications. In the simplest case of the system the FL method is reduced to the ordinary cancellation of nonlinearity by means of a properly selected control function. One of the main drawbacks of the FL method relates to inaccuracies arising during cancellation of system nonlinearities. Thus obtained transformed system is not in fact perfectly linear and, moreover, these imperfections may often prevent the use of efficient techniques of linear systems synthesis. An effective way to solve this problem is to combine feedback linearization method with the robust control techniques. In this paper H ∞ optimal control theory within the state space framework is applied, i.e. the problem is considered from the position of the differential games theory [3,4,8,14]. In this view model uncertainties are considered as an action of adversary player (or opposing nature) while our part is to invent a control strategy that is the best in terms of some given quality criterion (cost functional). In other words, we are trying to minimize the cost assuming the ‘worst-case action’ of our opponent player (disturbances). Such an approach allows to devise a controller which, taking into consideration system parametric uncertainties, guarantees at the same time a good process performance. In the paper, besides presenting a relevant portion of the above stated theory, its usefulness to ship path-following [6] problem is considered. This approach, as an alternative to adaptive control techniques [15], is less susceptible to an unacceptable state transition process. However, one should be aware that the combination of robust and adaptive control techniques in one design may lead to a better effect. The paper is divided into five sections followed by a brief conclusion. The second section presents system class definitions, basic concepts as well as the transformation of the considered system into a standard differential game form. In the third section the ship path-following process and its model are defined, and the robust controller synthesis is given. The fourth section, beside the introduction of the ship simulation model, presents a short description of simulation tests and their results . II. BASIC CONCEPTS AND DEFINITIONS Let us consider a nonlinear system in Brunovsky form (or controller canonical form for nonlinear systems) [11] ,[7] 1 2 2 3 (, ) (, ) f g n x x x x x f g u = = = +   #  x θ x θ (1a) 1 y x = (1b) where n ∈ x R is the state vector, f k ∈ R θ , g l ∈ θ R are vectors of system parameters, u ∈ R is the control input and 978-1-4673-5508-7/13/$31.00 ©2013 IEEE 836