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Conditions for a constrained system to have a set of impulse energy measures. (English) Zbl 0666.93017
This paper considers the existence of an $$n$$-th order single-input single-output linear time-invariant dynamical system without zeros to have a prespecified set of n impulse energy measures and gives a method to determine the parameters specifying the transfer function.
Given the transfer function $$T(s)$$ of a linear dynamical system it is possible to determine the system’s impulse energy measures, where the $$i$$-th impulse energy measure is $Y_ i = \int^\infty_0 \left(\frac{d^ i}{dt^ i} L^{-1} (T(s))\right)^2 dt, \quad i=0,1,2,\ldots$ The main result of this paper is the following theorem.
Theorem 1. Given $$Y=[Y_ 0,Y_ 1,\ldots,Y_{n-1}]$$, $$Y_ i$$ is finite and real for $$i=0,1,\ldots,n-1$$, there exists a system with a transfer function $T(s) =\frac1{s^ n+p_{n-1}s^{n-1}+p_{n-2}s^{n-2}+\cdots+p_ 0},$ $$p_ i$$ finite and real, $$i=0,1,\ldots,n-1$$, whose impulse energy measures are $$Y_0, Y_1, Y_2, \ldots, Y_{n-1}$$ if and only if $${\mathfrak M}_ n$$ is positive definite where $M_ n = \left[\begin{matrix} Y_0 & 0 & -Y_1 & 0 & Y_2 & \cdots \\ 0 & Y_1 & 0 & -Y_2 & 0 & \cdots \\ \vdots & \vdots & \vdots & \vdots & \vdots && \vdots & \vdots \\ &&&&& a+r & Y_{n-2} & 0 \\ &&&&& a+r & 0 & Y_{n-1} \end{matrix}\right].$ If $$M_ n$$ is positive definite, $$p_0, p_1, \ldots, p_{n-1}$$ can be determined from $\left[\begin{matrix} Y_{n-1} & 0 & -Y_{n-2} & 0 & \cdots \\ 0 & Y_{n-2} & 0 & -Y_{n-3} & \cdots \\ -Y_{n-2} & 0 & Y_{n-3} & 0 & \cdots \\ \vdots & \vdots & \vdots & \vdots && \vdots \\ &&&& \cdots & 0 \\ &&&& \cdots & Y_0 \end{matrix}\right] \left[\begin{matrix} p_{n-1} \\ p_{n-2} \\ \vdots \\ p_1 \\ p_0 \end{matrix}\right] = \left[\begin{matrix} 0.5 \\ Y_{n-1} \\ 0 \\ -Y_{n-2} \\ 0 \\ \vdots \end{matrix}\right].$
Reviewer: M.Kono

##### MSC:
 93B15 Realizations from input-output data 93C05 Linear systems in control theory 93B30 System identification
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##### References:
 [1] L. A. Zadeh, C. A. Desoer: Linear System Theory. McGraw-Hill, New York 1963. · Zbl 1145.93303 [2] V. F. Baklanov: Lowering the order of differential equations and transfer functions of control systems. Soviet Automatic Control 13 (1968), 1-7. [3] M. F. Hutton: Routh Approximation Method for High Order Linear Systems. Singer, Little Falls, N. J. 1973. [4] J. Lehoczky: The determination of simple quadratic integrals by Routh coefficients. Periodica Polytechnica Electrical Engineering 10 (1966), 2, 153-166. [5] C. Bruni A. Isidori, A. Ruberti: A method of realization based on the moments of the impulse response matrix. IEEE Trans. Automat. Control AC-14 (1969), 203-204. [6] R. M. Umesh: Approximate Model Matching. Unpublished Doctoral Thesis, Anna University, Madras, India 1984. [7] F. R. Gantmacher: Theory of Matrices, Volume 1, 2. Chelsea, New York 1959. · Zbl 0085.01001
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