A Matlab toolbox for LMI-based analysis and synthesis of LPV/LFT self-scheduled H ∞ control systems Daniele De Vito, Aymeric Kron, Jean de Lafontaine and Marco Lovera Abstract— When controlling a Linear Parameter Varying (LPV) system, a LPV regulator is advisable, since it ensures better performance than a simple Linear Time Invariant actually does. In fact, real-time scheduling to the variations of the system allows the achievement of stability and performance requirements for a number of operating points. Within this setting, this paper discusses a Matlab toolbox achieving a self- scheduled LPV controller for an LPV model of the plant, robust in an H∞ sense in the face of uncertainties affecting the sys- tem’s dynamics, through a Linear Matrix Inequality approach. The resulting algorithm alternatively implements synthesis and analysis steps, until the desired closed-loop performance level has been reached or no improvement between two successive steps arises. I. I NTRODUCTION Many control design problems deal with systems having a strongly time-varying behavior. In these cases, Linear Parameter Varying (LPV) models are often used for the design of gain-scheduled controllers (see, e.g., [1], [2], [3] and the references therein). This happens as long as the system to be controlled is structurally linear even if with matrices depending on a set of time-varying parameters. Consequently, the controller can benefit by the knowledge of those parameters whenever they are measurable. The LPV framework was originally proposed in [4], [5], where a set of Linear Time Invariant (LTI) models for the plant to be controlled have been obtained by linearising around parameter-frozen operating points, and as many LTI controllers have been constructed for each of them in order to satisfy local performance goals. The resulting real-time task consists in adjusting (“scheduling”) the controller gains in accordance to the current operating condition through a look- up-table method or an interpolation procedure. Nevertheless, no guarantee of satisfactory performance and robustness along all possible trajectories of the scheduling parameters can be given. A similar but more accurate strategy has been firstly proposed by [6] and later also discussed by [7] and [8]. The rationale underlying such a strategy consists in defining a polytopic model of the system and in building up a controller with the same structure. In such a way, a quadratic H ∞ -like performance index can be properly defined and suitably fulfilled. Another approach to cope with This research has been partially supported by the ITAQUE Italy-Quebec cooperation project. D. De Vito and M. Lovera are with Politecnico di Milano - Dipartimento di Elettronica e Informazione, Via Ponzio 34/5, 20133, Milano, ITALY. devito,lovera@elet.polimi.it A. Kron and J. de Lafontaine are with NGC Aerospace Ltd, 1650 rue King Ouest, Office 202, Sherbrooke, Qu´ ebec J1J 2C3, CANADA. aymeric.kron,jean.delafontaine@ngcaerospace.com LPV systems stems from the Linear Fractional Transforma- tion (LFT) dependence of the plant’s model on the scheduling parameters. If this is the case, two different methods have been proposed. The first one relies on the introduction of robustly stabilizing parameter-dependent Lyapunov functions [9], while the second one resorts to the small-gain paradigm in its scaled version [10], [7], [2]. In so doing, a self- scheduled control law, robust in an H ∞ sense in the face of bounded uncertainties affecting the system’s dynamics, can be computed by means of a Linear Matrix Inequality (LMI) procedure. Moreover, the on-line elaborations required to give the controller a LPV structure readily reduce to straightforward and numerically fast matrices multiplica- tions. Finally, [11] proposed an alternative technique within a probabilistic setting, which provides a sequence of candidate solutions converging almost surely to a feasible one in a finite number of steps. Moreover, generally nonlinear dependence of the plant to be controlled on the scheduling parameters is allowed and simultaneous management of a set of LMIs is required to solve the problem. This paper deals with the design and implementation of a Matlab toolbox implementing the algorithm presented in [7] in order to achieve a robust LPV/LFT regulator given a normalized LPV/LFT model of the plant to be controlled possibly subject to bounded uncertainties. To this end, three high level functions, each of them invoking a number of low level modules, have been implemented, for the sake of mod- ularity and reusability. Each high level routine implements a precise step of the whole algorithm, which works in a loop fashion until the desired closed-loop performance has been reached or no improvement between two successive steps arises. Throughout this paper, the dependence either on the time variable t or on the Laplace variable s will be dropped whenever this does not cause any formula misunderstanding. Additionally, we denote by W ij the entry of matrix W lying in position ij . Moreover, we let the LFT state space equations of a MIMO LPV dynamic system be ˙ x = Ax + B 1 w + B 2 u z = C 1 x + D 11 w + D 12 u y = C 2 x + D 21 w + D 22 u w = diag(θ 1 I ν1 ,...,θ r I νr )z (1) where x ∈ R n is the state, w ∈ R m1 is the uncertainty input, u ∈ R m2 is the control input, z ∈ R p1 is the performance output, y ∈ R p2 is the measured output, θ =  θ 1 θ 2 ... θ r  ′ ,θ ∈P⊂ R r , is the vector gathering the time-varying scheduling parameters and I µ is the identity 2010 IEEE International Symposium on Computer-Aided Control System Design Part of 2010 IEEE Multi-Conference on Systems and Control Yokohama, Japan, September 8-10, 2010 978-1-4244-5355-9/10/$26.00 ©2010 IEEE 1397