The gain-scheduling approach is perhaps one of the most popular nonlinear control design approaches which has been widely and successfully applied in fields ranging from aerospace to process control. Despite the wide application of gain-scheduling controllers and a diverse academic literature relating to gain-scheduling extending back nearly thirty years, there is a notable lack of a formal review of the literature. Moreover, whilst much of the classical gain-scheduling theory originates from the 1960s, there has recently been a considerable increase in interest in gain-scheduling in the literature with many new results obtained. An extended review of the gainscheduling literature therefore seems both timely and appropriate. The scope of this paper includes the main theoretical results and design procedures relating to continuous gain-scheduling (in the sense of decomposition of nonlinear design into linear sub-problems) control with the aim of providing both a critical overview and a useful entry point into the relevant literature.

A wide variety of problems in control system theory fall within the class of parameterized Linear Matrix Inequalities (LMIs), that is, LMIs whose coefficients are functions of a parameter conned to a compact set. Such problems, though convex, involve an innite set of LMI constraints, hence are inherently difficult to solve numerically. This paper investigates relaxations of parameterized LMI problems into standard LMI problems using techniques relying on directional convexity concepts. An in-depth discussion of the impacts of the proposed techniques in quadratic programming, Lyapunov-based stability and performance analysis, µ analysis and Linear Parameter Varying control is provided. Illustrative examples are given to demonstrate the usefulness and practicality of the approach.

We present new algorithms for the robust stability analysis and gain-scheduled controller synthesis for linear systems affected by time-varying parametric uncertainties. Sufficient conditions for robust stability as well as conditions for the existence of a robustly stabilizing gain-scheduled controller are given in terms of a finite number of Linear Matrix Inequalities; explicit formulae for constructing robustly stabilizing gain-scheduled controllers are given in terms of the feasible set of these LMIs. Our approach is proven to be in general less conservative than existing methods for stability analysis and gain-scheduled controller synthesis for parameter-dependent linear systems; numerical examples are presented to show that our approach offers significant improvement in practice as well. I. Introduction Our notations are standard. R m\Thetan denotes the set of real m \Theta n matrices, and C m\Thetan the set of complex m \Theta n matrices. I m is an m \Theta m identity matr...

The randomized algorithm of Polyak and Tempo (2000), which consists of random sampling and subgradient descent, is generalized in order to solve parameter-dependent linear matrix inequalities and its computational complexity is analyzed. This paper first examines an algorithm obtained by direct generalization of Polyak and Tempo's and shows that its expected time to achieve a solution is infinite. Then this paper improves this algorithm so that its expected achievement time becomes finite.

We study families of linear time-varying systems, where time-variations have to satisfy restrictions on the dwell time, that is on the minimum distance between discontinuities, as well as on the derivative in between discontinuities.

Recent results in parameter-dependent control of linear parameter-varying (LPV) systems are applied to the problem of designing gain-scheduled pitch rate controllers for the F-16 VISTA (Variable-Stability In-Flight Simulator Test Aircraft). These methods, based on parameter-dependent quadratic Lyapunov functions, take advantage of known a priori bounds on the parameters' rates of variation (the bounds may themselves be parameter-varying). The controller achieves an induced-L 2 -norm performance objective; Level 1 flying qualities are predicted. Suboptimal solutions are obtained by solving a convex optimization problem described by linear matrix inequalities (LMIs). Incorporation of D-K iteration with "constant D-scales" provides robustness to time-varying uncertainty. Parameter-varying performance weights are used to shape the desired performance at different points in the design envelope. 1 Introduction The area of analysis and control of linear parametervarying (LPV) systems has r...

: The purpose of the paper is to demonstrate the ability of LPV (Linear Parameter Varying) control techniques to handle difficult nonlinear control problems. The focus in this paper is on the wide range stabilization of an arm-driven inverted pendulum. Two different LPV control techniques are used to design nonlinear controllers that achieve stabilization of the pendulum over the maximum range of operating conditions while providing time- and frequency-domain performances. The merits of each of these techniques are investigated and the improvements over more classical LTI (Linear Time-Invariant) control schemes such as H1 or controllers are discussed. A particular emphasis is put on the real-time implementation of these controllers for the inverted pendulum experiment. It is shown that suitable multi-objective extensions of the standard characterization of LPV controllers allow to cope with sampling rate implementation constraints. Finally, a complete validation of the proposed LPV c...

Abstract — In this paper it is shown that robust stability of multiparameter affinely-dependent LTI systems is equivalent to the existence of a multi-parameter polynomially-dependent quadratic Lyapunov function of known, bounded degree in terms of the system parameters. Testing the stability of multi-parameter dependent LTI systems over a compact, connected set can be cast in terms of two finite-dimensional linear matrix inequalities (LMIs). I.

Existing control theory for linear parameter-varying systems uses a uniform-in-the-parameters upper bound (a constant scalar) on the induced-L 2 norm. In this paper, this constant is generalized to a function of the parameters; the performance criterion generalizes to an integral quadratic constraint and implies a norm bound that depends naturally on the real-time parameter trajectory. As in existing theory, the synthesis problem reduces to convex optimization involving linear matrix inequalities. A motivating example, gainscheduled flight control of the F-16 VISTA, demonstrates the flexibility gained by incorporating the new performance criterion. 1 Introduction Some recent effort has been given to solving gainscheduled control problems using linear parametervarying (LPV) systems [12]: finite-dimensional linear systems that depend on one or more time-varying, measurable parameters (e.g., families of linearized models for nonlinear aircraft dynamics). So-called LPV control design met...

Abstract—This paper proposes different parameterized linear matrix inequality (PLMI) characterizations for fuzzy control systems. These PLMI characterizations are, in turn, relaxed into pure LMI programs, which provides tractable and effective techniques for the design of suboptimal fuzzy control systems. The advantages of the proposed methods over earlier ones are then discussed and illustrated through numerical examples and simulations. Index Terms—Fuzzy systems, parameterized linear matrix inequality (PLMI). the state-space representation of the T–S model is where (4) I.