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Zeros and poles of linear periodic multivariable discrete-time systems. (English) Zbl 0662.93015
This paper extends the notions of transmission zero, invariant zero, structural zero and pole to linear periodic discrete-time systems. Their meaning and relations are clarified. To the p-input q-output linear \(\omega\)-periodic system \(\Sigma\) \[ (*)\quad x(k+1)=A(k)x(k)+B(k)u(k),\quad y(k)=C(k)x(k),\quad x(k)\in {\mathbb{R}}^ n \] a \(p\omega\)-input \(q\omega\)-output linear time-invariant system \(\Sigma^ a(k)\) \[ x_ k(h+1)=E_ kx_ k(h)+J_ ku_ k(h),\quad y_ k(h)=L_ kx_ k(h)+M_ ku_ k(h) \] is associated, where \(u_ k(h):=[u'(k+h\omega),\quad u'(k+1+h\omega),...,u'(k+\omega -1+h)]',\) \(x_ k(h):=x(k+h\omega),\) \(y_ k(h):=[y'(k+h\omega),...,y'(k+\omega - 1+h\omega)]'\). The system matrix \(P_ k(z)\) of \(\Sigma^ a(k)\) is defined by \[ P_ k:=\left[ \begin{matrix} -zI+E_ k\\ L_ k\end{matrix} \begin{matrix} J_ k\\ M_ k\end{matrix} \right] \] Let \(\xi^ k_ i(z)\) denote the invariant polynomial in the Smith form of \(P_ k(z)\). The zeros of the polynomial \(\zeta_ k(z):=\prod^{n+r_ k}_{i=1}\xi_ i^ k(z)\) are called the invariant polynomials, where \(r_ k\) is the rank of \(W_ k(z):=L_ k(zI-I)^{-1}J_ k+M_ k\). Let \(\epsilon_ i^ k(z)(\psi_ i^ k(z))\) denote the \(r_ k\) numerator (denominator) polynomials in the Smith-McMillan form of \(W_ k(z)\). The zeros of the polynomial \(\eta_ k(z):=\prod^{r_ k}_{i=1}\epsilon_ i^ k(z)(\chi_ k(z):=\prod^{r_ k}_{i=1}\psi_ i^ k(z))\) are called the transmission zeros of \(\Sigma\) (the poles of \(\Sigma)\) at time k, with multiplicities equal to their multiplicities in this polynomial.
Theorem 3.1. The rank \(r_ k\) of \(W_ k(z)\) is independent of k. The nonzero poles (nonzero transmission zeros) of \(\Sigma\) at time k are independent of k, together with their multiplicities.
Call \(\Sigma^ F\) the system described by (*) together with the control law \(u(k)=F(k)x(k)+v(k)\), \(F(k+\omega)=F(k).\)
Theorem 3.2. The invariant zeros of \(\Sigma^ F\) at time, together with their multiplicities, coincide with those of \(\Sigma\).
The structural zeros are defined in terms of the reachability subspace and the invariant subspace.
Theorem 5.1. For each \(k\in {\mathbb{Z}}\), the structural zeros of \(\Sigma\) at time k, with their multiplicities, coincides with the invariant zeros of \(\Sigma\).
A numerical example is given.
Reviewer: I.Ichikawa

MSC:
93B27 Geometric methods
93C05 Linear systems in control theory
93C55 Discrete-time control/observation systems
93C35 Multivariable systems, multidimensional control systems
Keywords:
time-dependent
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