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J Optim Theory Appl (2015) 165:532544 DOI 10.1007/s10957-014-0636-1
A Local Feedback Control Bringing a Linear System to Equilibrium
Alexander Ovseevich
Received: 27 January 2014 / Accepted: 29 July 2014 / Published online: 13 August 2014 Springer Science+Business Media New York 2014
Abstract We design a bounded feedback control that steers a controllable linear system to equilibrium in a nite time. The procedure amounts to solving several linear-algebraic problems, including linear matrix inequalities. We solve these problems in an efcient way. The resulting steering time and the minimum time have the same order of magnitude.
Keywords Reachable (attainable) set Common Lyapunov function Brunovsky
canonical form
Mathematics Subject Classication 39B03 37B25 93B10
1 Introduction
The problem of minimum-time steering of a controlled dynamical system to a given state is a classical and typical problem of control theory. However, the problem statement does not necessarily model a real life issue, and is subject to a justied criticism. For instance, it might turn out that the minimum-time control is too complicated even for approximate implementation. Perhaps, the performance index to be minimized should reect not only our immediate goal, like the short steering time, but should also account for complexity of the control. Therefore, it might be reasonable to relax the optimality requirement and just look for a control that solves the steering problem in a non-optimal, but stable and implementable way. In our paper, we deal with a relatively simple situation, where these relaxed goals can be achieved.
Communicated by Felix L. Chernousko.
A. Ovseevich (B)
IPMech, RAS, Moscow, Russia e-mail: [email protected]
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J Optim Theory Appl (2015) 165:532544 533
2 Korobovs Construction
Consider a linear time-invariant, completely controllable system
x = Ax + Bu, x V =
Rn, u U =
Rm. (1)
We are interested in the design of a bounded feedback control u = u(x) such that, if
a state y belongs to a sufciently small neighborhood of zero, then the phase curve of the equation x = Ax + Bu(x) subject to the initial condition x(0) = y reaches
zero in a nite time. This problem was studied, in particular, by Korobov in [1]. This paper serves as a point of departure for our notes. In particular, the construction to be presented resembles the...