Robust Feedback Control for a Linear Chain of Oscillators
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Robust Feedback Control for a Linear Chain of Oscillators Alexander Ovseevich1
· Igor Ananievski2
Received: 23 May 2020 / Accepted: 3 October 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract We study the problem of bringing a linear chain of masses connected by springs to an equilibrium in finite time by means of a control force applied to the first mass. We describe explicitly the desired feedback control and establish its local equivalence to the minimum-time one. We prove the robustness of the control with respect to unknown disturbances and compute the time of transfer as well as its asymptotic estimate with respect to the length of the chain. Keywords Feedback control · Linear oscillator · Finite-time stabilization · Robustness · Orthogonal polynomials Mathematics Subject Classification 49N05 · 49N30 · 49K99 · 93D15
1 Introduction We study the feedback controllability problem for a linear finite-dimensional dynamic system. We follow the line of research initiated by Korobov [1] and elaborated by others, including the present authors [2–6]. The control used in the present paper is more smooth than the minimum-time one: its only singular point is zero, while the singular locus of optimal control is a singular hypersurface. Moreover, the time required by our control to bring a given state to the origin is not much greater than the minimal one: the ratio of these times is uniformly bounded over a neighborhood of zero. It is natural to say, that our feedback control is locally equivalent to the minimum-time one.
Communicated by Zenon Mróz.
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Alexander Ovseevich [email protected] Igor Ananievski [email protected]
1
Ishlinsky Institute for Problems in Mechanics RAS, Moscow, Russia
2
Ipmex RAS, Moscow, Russia
123
Journal of Optimization Theory and Applications Fig. 1 The linear chain of oscillators
m0
m1
mn
We consider steering of a linear chain of point masses connected via springs to the equilibrium by means of a bounded force applied to the first (leading) mass in the chain. We describe explicitly the feedback control from [5], applied to the leading mass, which brings the entire system to a prescribed equilibrium position. Then, we show its robustness with respect to unknown disturbances and study the behavior of the controlled system when the length of the chain is large. We find also the exact time of transfer of the system from one equilibrium to another. It is worth emphasizing that our control brings the system to equilibrium in a finite time. The related issues are now known as the finite-time stabilization and are a subject of an active study, e.g., in [7,8]. The study of the linear chain of oscillators goes back at least to the seminal paper [9]. Our study requires a wide spectrum of mathematical tools: from control theory to classical analysis [10–12].
2 The Chain of Oscillators We consider a linear chain of masses m i , i = 1, . . . , n, connected by springs and attached to the leading mass m 0 (Fig. 1). The entire system moves along a horizontal line unde
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