An Overview of LPV Systems

The framework of Linear Parameter Varying (LPV) systems concerns linear dynamical systems whose state-space representations depend on exogenous non-stationary parameters. Since its introduction by Shamma and Athans in 1988 to model gain scheduling, the LPV paradigm has become a standard formalism in systems and controls, with many papers devoted to analysis, controller synthesis, and system identification of LPV models. This chapter reviews basic concepts and presents a representative selection of analytical approaches for LPV systems. 1 Introducing LPV Systems Origins. The framework of Linear Parameter Varying (LPV) systems concerns linear dynamical systems whose state-space representations depend on exogenous nonstationary parameters, as in ẋ = A(θ )x + B(θ )u y = C(θ )x, (1) where u is an input, y is an output, and θ is an exogenous parameter that can be time dependent. The LPV paradigm was introduced in the Ph.D. thesis of Shamma [74] for the analysis of the control design practice of “gain-scheduling” [4, Chapter 9]. In brief, gain-scheduling is a control design approach that constructs a nonlinear controller for a nonlinear plant by patching together a collection of linear controllers. These linear controllers are blended in real-time (e.g., via switching or interpolation) acJeff S. Shamma Georgia Institute of Technology, School of Electrical and Computer Engineering e-mail: [email protected]