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TECHNICAL PAPERS

# State Feedback Gain Scheduling for Linear Systems With Time-Varying Parameters

[+] Author and Article Information
Vinícius F. Montagner

School of Electrical and Computer Engineering, University of Campinas, CP 6101, 13081-970, Campinas - SP - Brazilmontagne@dt.fee.unicamp.br

Pedro L. D. Peres

School of Electrical and Computer Engineering, University of Campinas, CP 6101, 13081-970, Campinas - SP - Brazilperes@dt.fee.unicamp.br

The LMIs must be implemented with all the combinations $±$.

There always exists a similarity transformation that allows the output of a linear system to be written as in Eq. 22.

J. Dyn. Sys., Meas., Control 128(2), 365-370 (Jun 01, 2005) (6 pages) doi:10.1115/1.2194074 History: Received September 15, 2003; Revised June 01, 2005

## Abstract

This paper addresses the problem of parameter dependent state feedback control (i.e. gain scheduling) for linear systems with parameters that are assumed to be available (measured or estimated) in real time and are allowed to vary in a compact polytopic set with bounded variation rates. A new sufficient condition given in terms of linear matrix inequalities permits to determine the controller gain as an analytical function of the time-varying parameters and of a set of constant matrices. The closed-loop stability is assured by means of a parameter dependent Lyapunov function. The condition proposed encompasses the well-known quadratic stabilizability condition and allows to impose structural constraints such as decentralization to the feedback gains. Numerical examples illustrate the efficiency of the technique.

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## Figures

Figure 1

Nonlinear behavior of the entries of the scheduled state feedback control gain 32 as a function of α1(t) for system 27 with ρ1=1

Figure 2

One period of the entries of the scheduled state feedback control gain 32 for system 27 with α1(t) given by Eq. 29

Figure 3

Trajectories of the state variables of the closed-loop system for the standard gain scheduling with linear interpolation, given by Eq. 34 (top), and for the gain scheduling strategy of Theorem 1 (nonlinear function of the time-varying parameters), given by Eq. 32 (bottom), for system 27 with ρ1=1

Figure 4

Parameter dependent gain entries obtained from Theorem 1 for system 27 with ρ1=100

Figure 5

Stabilizability domains for the time-varying system 35,36,37 in terms of the bounds on the parameter time derivatives (ρ1 and ρ2)

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