Control of slowly-varying linear systems

Abstract

Aktruct-State feedback control of slowly-varying linear continuoustime and discrete-time systems with bounded coefficient matrices is studied in terms of the frozen-time approach. This note centers on pointwise stabilizable systems: that is, systems for which there exists a state feedback gain matrix placing the frozen-time closed-loop eigenvalues to the left of a line Res =y < 0 in the complex plane (or within a disk of radius p < 1 in the discrete-time case). It is shown that if the entries of a pointwise stabilizing feedback gain matrix are continuously differentiable functions of the entries of the system coefficient matrices, then the closed-loop system is uniformly asymptotically stable if the rate of time variation of the system coefficient matrices is sufficiently small.It is also shown that for pointwise stabilizable systems with a sufficiently slow rate of time variation in the system coefficients, a stabilizing feedback gain matrix can be computed from the positive definite solution of a frozen-time algebraic Riccati equation.

New stability conditions for a class of linear time-varying systems

“…Comparing with the existing results (Amato et al, 1993;Desoer, 1969;Garcia et al, 2010;Guo & Rugh, 1995;Ilchmann et al, 1987;Jetto & Orsini, 2009;Kamen et al, 1989;Krause & Kumar, 1986;Mullhaupt et al, 2007;Rosenbrock, 1963;Solo, 1994;Tan & Duan, 2009), here, the system matrix function A(t) and its derivativeȦ(t) (provided ∥Ȧ(t)∥ exists) are allowed to be unbounded at t = +∞.…”

“…Numerous important results, including but not limited to Amato, Celentano, and Garofalo (1993), Desoer (1969), Garcia, Peres, and Tarbouriech (2010), Guo and Rugh (1995), Ilchmann, Owens, and Prätzel-Wolters (1987), Jetto and Orsini (2009), Kamen, Khargonekar, and Tannembaum (1989), Krause and Kumar (1986), Mullhaupt, Buccieri, and Bonvin (2007), Solo (1994) and Tan and Duan (2009), have been obtained through the efforts…”

Model-based control of the magnetic flux profile in a Tokamak plasma

“…In particular, having all closed-loop eigenvalues with negative real parts at all times is not a sufficient condition to determine the stability of such systems without an additional condition of slow-enough variation, see for instance [1,3,6,15].…”

Identification of Systems With Regime Switching and Unmodeled Dynamics

“…It is also possible to deal with systems that experience both slowly time-varying parameter variations in normal operating conditions and sudden switching at critical moments, in other words, slowly time-varying hybrid systems. Techniques developed in this paper for switching systems and those in [5], [6], [16], [41], [1], [3] for slowly varying systems may jointly provide a promising foundation for such pursuit. Another worthwhile undertaking is to use nonlinear filtering techniques to track Markov chain states.…”

“…One typical approach of extending results from time-invariant systems to time-varying systems is to consider slowly timevarying systems in which system dynamics experience small changes over a small period of time [5], [6], [16], [41]. Similarly, stochastic algorithms are widely used to track a system whose parameters change slowly in their magnitude [1], [3], [31].…”