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Case Bibo Essay

2053 words - 9 pages

IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 39, NO. 10, OCTOBER 1994

2171

[4] P. R. Kumar and T. I. Seidman, "Dynamic instabilities and stabilization methods in distributed real time scheduling of manufacturing systems," IEEE Trans. Automat. Contr., vol. 35, pp. 289-298, 1990. [SI J. R. Perkins and P. R. Kumar, "Stable distributed real-time scheduling of flexible manufacturing/assembly/disassembly systems," I€€€ Trans. Automar. Conrr., vol. 34, pp. 139-148, 1989.

The following is a brief overview of the present work. In Section I1 we present some background material on coprime factorizations and the graph topology. In Section I11 we consider BIBO stable systems and the question of ...view middle of the document...

[ I p ) (1 5 p 5 CO) denote the usual (real) sequence spaces. A linear discrete-time system is defined as a linear convolution operator G: l p + l P . As usual the linear system G is called l p stable if

I. INTRODUCTION
This paper deals with the worst-case analysis of identification of stabilizable systems and to a smaller degree with the robustness of feedback stabilization of linear systems. Here we shall be concerned with the case when stabilization is equivalent to the closed-loop system being a bounded-input bounded-output (BIBO) stable operator, cf. I' optimal control [29], [2]. There are several ways to represent system uncertainty to deal with both stable and unstable systems: one is by considering perturbations of the graph of the system and another is by looking at perturbations of coprime factorizations of the system. The quantitative measure for the size of the perturbations depends on the particular space in which the graph, or the coprime factorization, is defined. We mention here the rich theory developed in the 1 ( L 2 )Hilbert space setup (see ' e.g., [32], [ 5 ] , [28], [8], [33], [7], [20]). Furthermore, several papers dealing with various aspects of identification of systems in the gap, graph, and/or chordal metrics have appeared recently [14], [26], [27], [16]. It appears possible to develop an equally rich theory with many applications in the I" input/output signal space setup [3], [30], [l], ~71. There is a large literature on the identification of controlled autoregressive (ARX) and controlled autoregressive moving average (ARMAX) models of systems [13], [25], [19], [18]. In the present work we are interested in a special type of ARX model: namely, in ARX models in which the AR part and the X part are coprime. It turns out that knowledge of a stabilizing controller for the unknown system is enough information to parameterize the ARX model in this way. Here it is possible to use the schemes described in [9], [22], [23]. Manuscript received March 25, 1992; revised February 20, 1993 and February 18, 1994. J. R. Partington is with the School of Mathematics, University of Leeds, Leeds LS2 9JT, United Kingdom. P. M. Makila is with the Department of Computer Science and Electrical Engineering, Lulei University of Technology, S-971 87 Lulel Sweden. IEEE Log Number 9403963.

Here llGll(p)is the induced operator norm, or the system gain, over Zp. We shall often simplify the notation somewhat and write simply IlG11. Let S p denote the Banach space of linear shift-invariant causal l p stable systems equipped with the operator norm (1). It is well known that SO= is isometrically isomorphic to I' . Thus, for a system G E S", we shall let G denote also the (unit) impulse response ( s k I k 2 0 of G. Then llGll(cr) = IlGlll Ck20 < =. lskl A convenient way of representing both l P stable and unstable ] is systems ( p E [l. x ) to consider the quotient field F ( S P )of S p . F ( S P )can be thought of as the set of all pairs (P. Q),...

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