Design of H_∞ Gain-Scheduled Controllers for Linear Time-Varying Systems by means of Polynomial Lyapunov Functions

Abstract

This paper proposes convex conditions to design parameter-dependent (i.e. gain-scheduled) state feedback controllers that ensure closed-loop stability with H ∞ performance for linear systems affected by time-varying parameters that belong to a polytope and have bounded time-derivatives. The conditions, based on homogeneous polynomially parameterdependent Lyapunov functions of arbitrary degree, are expressed as a set of linear matrix inequalities written in terms of the vertices of the polytope, the bounds on the rates of parametric variations and the degree of the Lyapunov function. Differently from other gain-scheduled strategies in the literature, all the system matrices here are supposed to be affected by time-varying parameters and the control gains are obtained from a finite dimension convex optimization problem, which avoids to cope with nonlinear matrix inequalities and does not require the use of gridding procedures in the space of the parameters. Numerical results illustrate the efficiency of the proposed conditions, providing progressively less conservative results as the degree of the Lyapunov function increases.

Static output feedback controller design based on LMI for linear time-varying collocated mechanical system

“…However, it is also known that the static output feedback design is not feasible since products of variables C d (t), C v (t) and P(t) appear in (13) except the case of the full states being available [7]. In order to overcome this difficulty, we specify the problem to the mechanical system (1) and propose to make the Lyapunov matrix P(t) in (13) as…”

“…which is developed for the general LTV systems based on the assumption of all states being available [7]. The optimal controller results in γ 2 = 2.34.…”

“…For the purpose, the application of the robust H ∞ output feedback controller to the linear time-invariant systems (LTI) has been extensively studied and the linear matrix inequality (LMI)-based design has become available [1], [2], [3]. Recently it has been extended to the gain scheduled optimal control for linear parameter varying (LPV) systems and linear time varying (LTV) systems [4], [5], [6], [7]. However, the obtained controller generally becomes complex and large dimensional.…”

Design and Validation of a Gain-Scheduled Controller for the Electronic Throttle Body in Ride-by-Wire Racing Motorcycles

“…In principle, one may think of avoiding the gridding procedure, as there exist approaches in the LPV literature which formulate the stability analysis problem as a feasibility LMI problem of finite dimension, see, e.g., [38]- [41]. To do this, however, the LPV system must be written either as a linear fractional representation (LFR) or as an affine LPV system.…”

FIR-type robust H 2 and H ∞ control of discrete linear time-invariant polytopic systems via memory state-feedback control laws

“…For the robust H 2 or H ∞ controller synthesis problems, the gain-scheduling control approaches have been extensively studied for LPV systems; see, e.g. references [28,30,44,45] for the robust H ∞ control and references [30,31,46] of the gain-scheduling control for LPV systems, and little progress has been made to improve the robust stabilization of LTI systems [40] in terms of conservatism until very recent years. Recently, the concept of the new kinds of controllers with finite memory structures [47][48][49][50][51] opened up a new possibility of reducing the conservatism of approaches based on the static state-feedback control.…”