On guaranteeing tracking performance and stability with LPV control for nonlinear systems

On guaranteeing tracking performance and stability with LPV control for nonlinear systems 1 Gustavo S. Mazzoccante, Roland Tóth and Siep Weiland Control Systems Group, Electrical Engineering, Eindhoven University of Technology P.O. Box 513, 5600 MB Eindhoven, For the last three decades, systematic design of Nonlinear (NL) and Time-Varying (TV) controllers capable of guaranteeing stability and performance has been in the focus of front-line research. The Linear Parameter-Varying (LPV) framework has become a popular choice, since it is based on the extension of the well-understood Linear Time-Invariant (LTI) framework and it offers computationally efficient and robust control-synthesis methodologies. LPV systems are dynamical models capable of describing a NL/TV behavior in terms of a linear structure which depends on a measurable, so-called scheduling-variable ρ. Early LPV approaches exploited Gain-Scheduling (GS) methods, which basically is the linearization of a NL system model at various operating points, described by ρ, that results in a collection of local LTI models, for which LTI controllers are designed and interpolated to give a global control solution to the entire operating region. Due to many successful applications, GS has become part of industrial practice, even though it does not guarantee overall stability of the designed LPV controller, let alone performance. Later it was realized that it is possible to embed the dynamic behavior of a nonlinear system into a LPV model in which ρ becomes a function of the output, input, or state through the so-called scheduling map. Then, through powerful convex optimization tools involving Linear Matrix Inequalities (LMI) constraints, global stability and performance analysis together with control synthesis is possible for this surrogate LPV model. Still, most of the LPV based techniques see controller synthesis confined into the system class itself without considering that the resulting closed-loop control solution is inherently nonlinear and its performance depends on the information structure hidden in ρ. Indeed, recent studies show that customary L 2 gain characterization of performance, which is a popular choice in the linear context, fail to ensure the performance specification on the resulting nonlinear performance [3, 4]. In fact, the utter motivation of the LPV control design is the synthesis of an NL/TV controller that can guarantee internal stability and a given bound of worst-case performance on the underlying system compared to the objective of only stabilizing 1 This work has received funding from the European Research Council (ERC) under the European Unions Horizon 2020 research and innovation programme (grant agreement nr ). and optimizing performance with respect to the surrogate LPV model of the plant. To ensure performance guarantees for the NL controlled system a stronger concept of incremental stability and incremental L 2 -gain performance has been proposed. A nonlinear system is said to be incrementally bounded on L 2 if there exists η 0 such that Σ(u 1 ) Σ(u 2 ) 2 η (u 1 ) (u 2 ) 2 for all u 1,u 2 L 2. Testing incremental properties is, in general, a rather difficult task, hence a more conservative but computationally more attractive notion is introduced: quadratic incremental stability and performance [3]. Also, in order to analyze robust performance the weighted incremental norm seems to be the natural framework [1]. The weighted incremental norm can be also seen as a natural extension of the well-known H linear framework concept to the nonlinear context. Under this motivation, this work is intended to carry on the studies on nonlinear controller design via the LPV framework by focusing on other signal norms such as L 1 and L, so that in the future it will serve as basis to describe the correlation between linear representations and stability and performance guarantees of the NL/TV behaviors using the L q -norms. The understanding of this problem would aid a potential breakthrough in understanding and applying LPV control on real-world systems. [1] V. Fromion, S. Monaco, and D. Normand-Cyrot. The weighted incremental norm approach: from linear to nonlinear H control. Automatica, 37(10): , [2] V. Fromion and G. Scorletti. A theoretical framework for gain scheduling. Int. J. of Robust and Nonlinear Control, 13(10): , [3] V. Fromion, G. Scorletti, and G. Ferreres. Nonlinear performance of a PI controlled missile: An explanation. Int. J. of Robust and Nonlinear Control, 9(8): , [4] G. Scorletti, V. Fromion, and S. de Hillerin. Toward nonlinear tracking and rejection using LPV control. In Proc. of the 1st IFAC Workshop on Linear Parameter Varying Systems, pp , Grenoble, France, Oct