Discrete-Time Static Output-Feedback H ∞ Controller Design for Vehicle Suspensions Francisco Palacios-Qui˜ nonero Josep Rubi´ o-Masseg´ u and Josep M. Rossell Department of Applied Mathematics III Universitat Polit` ecnica de Catalunya (UPC) 08242-Manresa, Barcelona, Spain Emails: francisco.palacios@upc.edu, josep.rubio@upc.edu josep.maria.rossell@upc.edu Hamid Reza Karimi Department of Engineering Faculty of Engineering and Science University of Agder (UiA) N-4898 Grimstad, Norway Email: hamid.r.karimi@uia.no Abstract—This paper provides a direct and practical presenta- tion of a novel methodology for static output-feedback controller design. The proposed design strategy has been successfully ap- plied in the fields of control systems for seismic protection of large buildings and multi-building structures, control of offshore wind turbines, and active control of vehicle suspensions. The positive results obtained in these initial applications clearly indicate that this approach could be an effective tool in a large variety of control problems, for which an LMI formulation of the state- feedback version of the problem is available. The main objective of the paper is to facilitate a brief and friendly presentation of the main ideas involved in the new design methodology. To this end, a discrete-time static output-feedback H∞ controller is designed for a simplified quarter-car suspension system. Numerical simula- tions indicate that the proposed controller exhibits a remarkably good behavior when compared with the corresponding state- feedback H∞ controller. I. INTRODUCTION When designing a feedback control system, the amount of information available for feedback purposes is an element of particular relevance. In the ideal (and uncommon) case that the entire state vector is available, many advanced state- feedback controller designs can be formulated as Linear Matrix Inequality (LMI) optimization problems, and efficiently com- puted using standard computational tools as those provided by the MATLAB Robust Control Toolbox [1]. In a more realistic scenario, however, the complete state vector is rarely accessible and the available feedback information consists only in a reduced set of linear combinations of the states. In this context, static output-feedback control strategies are an excellent option that can facilitate a simpler implementation in practice. The main difficulty of this alternative approach is the numerical complexity of the non-convex problems associated to static output-feedback controller designs [2], [3]. To deal with these challenging problems, a number of multi-step algorithms have been proposed [4]–[7]. Typically, these multi- step methods require solving complex matrix equations or LMI optimization problems at each step, which can be a critical issue in large-scale designs. Some single-step methods, based on a proper transformation of the state variables, are also available [8]–[10]. In these single-step methods, the static output-feedback controller design is formulated in terms of a single LMI optimization problem. The main drawback of this second kind of methods is that they are highly problem- dependent, in the sense that a complete derivation of the LMI optimization problem needs to be carried out for most controller designs. A new design strategy has been recently proposed [11], [12], which can be applied to control problems that admit an LMI-based state-feedback controller design. In this case, an LMI formulation to compute the output-feedback control gain matrix can be easily derived by introducing a simple change of variables in the LMI state-feedback formulation. This de- sign methodology is computationally effective, conceptually simple and easy to implement. Moreover, it makes possible to obtain static output-feedback controllers for a wide variety of problems by taking advantage of the rich literature on LMI formulations for state-feedback controller design. The new approach was initially motivated by large-scale control problems associated to vibration control of large struc- tures, and has been successfully applied in designing static output-feedback controllers for seismic protection of large buildings [13], [14] and multi-building systems [15]. Other successful applications include optimal design of passive- damping systems for large structures [16], control of offshore wind turbines [17], [18], and active control of vehicle suspen- sions [19]. The main objective of the paper is to provide a sum- marized, direct and practical presentation of this new design methodology, which we believe can be of general interest for researchers and control engineers in different fields. For clarity and simplicity, a small-scale control problem and the H ∞ control approach have been selected to introduce and illus- trate the fundamental ideas. Specifically, a discrete-time static output-feedback H ∞ controller is designed for a simplified quarter-car suspension system. A discrete-time state-feedback H ∞ controller is also designed to be used as a reference in the performance assessment. Moreover, the LMI formulation of the state-feedback controller serves as a natural starting point to derive the LMI formulation for the static output-feedback controller design. The paper is organized as follows: In Section II, a mathe- matical model for a simplified quarter-car suspension system is Presented in 2014 International Conference on Mechatronics and Control (ICMC)