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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:
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