LMI formulations for designing controllers according to time response and stability margin constraints Mohamed Abbas-Turki, Gilles Duc and Benoˆ ıt Clement Abstract— Designing a controller with respect to time and frequency-domain objectives remains a difficult problem, al- though both kinds are generally present in the manufacturer specifications. In general, the temporal objectives are replaced by frequency dependent ones, which in major cases do not fit the actual expectations. In this paper, convex mathematical translations of both kinds of objectives are proposed using Linear Matrix Inequalities (LMI). The application of Youla parameterization allows to restore the linearity in the compen- sator parameters, but a huge state space representation of the system is induced. Thus the Cutting Plane Algorithm (CPA) is efficiently used to overcome the problem of having a huge number of added variables, which often occurs in Semi-Definite Programming (SDP) particulary when used in conjunction with the Youla parameterization. I. INTRODUCTION The commun way to solve a multiobjective control prob- lem is to reformulate the design specifications into more con- venient forms such as H ∞ or H 2 constraints. Unfortunately most of the manufacturer specifications cannot be exactly translated into such formulations, so that this approach leads either to more restrictive constraints or to approximate results. For instance in [1], a LMI specification is proposed to translate a template on a time response, which derives a hard constraint. The time domain specifications can be indirectly handled by H 2 constraints or frequency shaping, but the overshoot and the settling time remain difficult to be adjusted. The purpose of this work is to design a controller ac- cording to time-domain specifications together with gain and phase margins requirements. The case of H ∞ and H 2 norms constraints has been presented in [2], [3]. By using the Youla parameterization, which defines a convex set describing all stabilizing controllers [4], all these specifications are ex- pressed as matrix inequalities which are linear in the decision variables (LMI), provided a particular base is chosen for the Youla parameter. The obtained problem is therefore convex, so that it can be solved using convex optimization techniques. Furthermore, it allows to conclude on the feasibility or non- feasibility of the control problem, provided the basis chosen for the Youla parameter allows to cover appropriately the set of stable transfer functions. This work was supported by the Launch Division of the French Space Agency (CNES/DLA). M. Abbas-Turki G. Duc are with the Service Automatique, ´ Ecole Sup´ erieure d’Electricit´ e, 3 rue Joliot-Curie, 91192 Gif-sur-Yvette cedex, France; first name.last name@supelec.fr B. Clement is with the Direction des Lanceurs, CNES, Rond Point de l’Espace, 91023 Every cedex, France; benoit.clement@cnes.fr As a disadvantage, using the Youla parameterization in- duces a huge state-space representation. The most commonly used technique for solving LMI problems is the semi-definite programming (SDP): however the frequency-dependent con- traints generally require introducing a symmetric matrix of the same order as the state-space matrix. Thus this technique should be avoided when the Youla parameterization is used. In order to avoid the additional variables, Kao [5] presents an alternative based on the eigenvalues of some Hamiltonian matrix, and the application of a Cutting Plane Algorithm (CPA) instead of SDP. Although this method is more sensi- tive to numerical conditioning, it is less affected by the order of the plant. In this paper, the efficiency of using CPA in this context will be shown: the time-domain specifications will be directly expressed as LMI constraints, without any restriction nor approximation. The stability margins requirements will be considered as real uncertainties. Contrary to the approach proposed in [6], no decomposition of the Youla parameter is needed and no additional variable has to be introduced. On the other hand, the proposed condition is only sufficient but it has been verified that it is not too conservative in most practical cases. The paper is organized as follows: section 2 contains a brief presentation of the Youla parameterization; section 3 introduces the CPA. The main contributions appear in sections 4 and 5, where a time-domain template and stability margins constraints are respectively formulated on a suitable form to be used by the CPA. An illustrative example is finally presented in section 5. II. YOULA PARAMETERIZATION A. Parameterization of the set of stabilizing controllers The Youla parameterization allows describing all stabiliz- ing controllers by only one stable transfer Q, called the Youla parameter [4]. Consider a continuous or discrete-time plant G, with z the output to be controlled despite disturbance w, using control input u and measurement y. A state space realization of G can be written as: G =  G 11 G 12 G 21 G 22  : w u z y ⎛ ⎝ A B 1 B 2 C 1 D 11 D 12 C 2 D 21 D 22 ⎞ ⎠ (1) All stabilizing controllers are described by the Redheffer product K = J ∗ Q (see the interconnection structure of figure 1), where the Youla parameter Q is any stable transfer function. System J depends both on coprime factorizations Proceedings of the 44th IEEE Conference on Decision and Control, and the European Control Conference 2005 Seville, Spain, December 12-15, 2005 WeC05.2 0-7803-9568-9/05/$20.00 ©2005 IEEE 5740