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The theory of state-space realizations of input-output descriptions given by impulse responses, and the time-invariant case by transfer function descriptions, were studied in this chapter.

In Section 5.2 the problem of state-space realizations of input-output descriptions was defined and the existence of such realizations was addressed. In Subsection 5.3A time-varying and time-invariant continuous-time and discrete-time systems were considered. Subsequently, the focus was on time-invariant systems and transfer function matrix descriptions (). The minimality of realizations of () was studied in Subsection 5.3C, culminating in two results, Theorem 3.9 and Theorem 3.10, where it was first shown that a realization is minimal if and only if it is controllable and observable, and next, that if a realization is minimal, all other minimal realizations of a given () can be found via similarity transformations. In Subsection 5.3C it was shown how to determine the order of minimal realizations directly from (). Several realization algorithms were presented in Section 5.4, and the role of duality was emphasized in Section 5.4A.

The theory of state-space realizations of input-output descriptions given by impulse responses, and the time-invariant case by transfer function descriptions, were studied in this chapter.

In Section 5.2 the problem of state-space realizations of input-output descriptions was defined and the existence of such realizations was addressed. In Subsection 5.3A time-varying and time-invariant continuous-time and discrete-time systems were considered. Subsequently, the focus was on time-invariant systems and transfer function matrix descriptions (). The minimality of realizations of () was studied in Subsection 5.3C, culminating in two results, Theorem 3.9 and Theorem 3.10, where it was first shown that a realization is minimal if and only if it is controllable and observable, and next, that if a realization is minimal, all other minimal realizations of a given () can be found via similarity transformations. In Subsection 5.3C it was shown how to determine the order of minimal realizations directly from (). Several realization algorithms were presented in Section 5.4, and the role of duality was emphasized in Section 5.4A.

The theory of state-space realizations of input-output descriptions given by impulse responses, and the time-invariant case by transfer function descriptions, were studied in this chapter.

In Section 5.2 the problem of state-space realizations of input-output descriptions was defined and the existence of such realizations was addressed. In Subsection 5.3A time-varying and time-invariant continuous-time and discrete-time systems were considered. Subsequently, the focus was on time-invariant systems and transfer function matrix descriptions (). The minimality of realizations of () was studied in Subsection 5.3C, culminating in two results, Theorem 3.9 and Theorem 3.10, where it was first shown that a realization is minimal if and only if it is controllable and observable, and next, that if a realization is minimal, all other minimal realizations of a given () can be found via similarity transformations. In Subsection 5.3C it was shown how to determine the order of minimal realizations directly from (). Several realization algorithms were presented in Section 5.4, and the role of duality was emphasized in Section 5.4A.

The theory of state-space realizations of input-output descriptions given by impulse responses, and the time-invariant case by transfer function descriptions, were studied in this chapter.

In Section 5.2 the problem of state-space realizations of input-output descriptions was defined and the existence of such realizations was addressed. In Subsection 5.3A time-varying and time-invariant continuous-time and discrete-time systems were considered. Subsequently, the focus was on time-invariant systems and transfer function matrix descriptions (). The minimality of realizations of () was studied in Subsection 5.3C, culminating in two results, Theorem 3.9 and Theorem 3.10, where it was first shown that a realization is minimal if and only if it is controllable and observable, and next, that if a realization is minimal, all other minimal realizations of a given () can be found via similarity transformations. In Subsection 5.3C it was shown how to determine the order of minimal realizations directly from (). Several realization algorithms were presented in Section 5.4, and the role of duality was emphasized in Section 5.4A.

The theory of state-space realizations of input-output descriptions given by impulse responses, and the time-invariant case by transfer function descriptions, were studied in this chapter.

In Section 5.2 the problem of state-space realizations of input-output descriptions was defined and the existence of such realizations was addressed. In Subsection 5.3A time-varying and time-invariant continuous-time and discrete-time systems were considered. Subsequently, the focus was on time-invariant systems and transfer function matrix descriptions (). The minimality of realizations of () was studied in Subsection 5.3C, culminating in two results, Theorem 3.9 and Theorem 3.10, where it was first shown that a realization is minimal if and only if it is controllable and observable, and next, that if a realization is minimal, all other minimal realizations of a given () can be found via similarity transformations. In Subsection 5.3C it was shown how to determine the order of minimal realizations directly from (). Several realization algorithms were presented in Section 5.4, and the role of duality was emphasized in Section 5.4A.

The theory of state-space realizations of input-output descriptions given by impulse responses, and the time-invariant case by transfer function descriptions, were studied in this chapter.

In Section 5.2 the problem of state-space realizations of input-output descriptions was defined and the existence of such realizations was addressed. In Subsection 5.3A time-varying and time-invariant continuous-time and discrete-time systems were considered. Subsequently, the focus was on time-invariant systems and transfer function matrix descriptions (). The minimality of realizations of () was studied in Subsection 5.3C, culminating in two results, Theorem 3.9 and Theorem 3.10, where it was first shown that a realization is minimal if and only if it is controllable and observable, and next, that if a realization is minimal, all other minimal realizations of a given () can be found via similarity transformations. In Subsection 5.3C it was shown how to determine the order of minimal realizations directly from (). Several realization algorithms were presented in Section 5.4, and the role of duality was emphasized in Section 5.4A.

The theory of state-space realizations of input-output descriptions given by impulse responses, and the time-invariant case by transfer function descriptions, were studied in this chapter.

In Section 5.2 the problem of state-space realizations of input-output descriptions was defined and the existence of such realizations was addressed. In Subsection 5.3A time-varying and time-invariant continuous-time and discrete-time systems were considered. Subsequently, the focus was on time-invariant systems and transfer function matrix descriptions (). The minimality of realizations of () was studied in Subsection 5.3C, culminating in two results, Theorem 3.9 and Theorem 3.10, where it was first shown that a realization is minimal if and only if it is controllable and observable, and next, that if a realization is minimal, all other minimal realizations of a given () can be found via similarity transformations. In Subsection 5.3C it was shown how to determine the order of minimal realizations directly from (). Several realization algorithms were presented in Section 5.4, and the role of duality was emphasized in Section 5.4A.

The theory of state-space realizations of input-output descriptions given by impulse responses, and the time-invariant case by transfer function descriptions, were studied in this chapter.

In Section 5.2 the problem of state-space realizations of input-output descriptions was defined and the existence of such realizations was addressed. In Subsection 5.3A time-varying and time-invariant continuous-time and discrete-time systems were considered. Subsequently, the focus was on time-invariant systems and transfer function matrix descriptions (). The minimality of realizations of () was studied in Subsection 5.3C, culminating in two results, Theorem 3.9 and Theorem 3.10, where it was first shown that a realization is minimal if and only if it is controllable and observable, and next, that if a realization is minimal, all other minimal realizations of a given () can be found via similarity transformations. In Subsection 5.3C it was shown how to determine the order of minimal realizations directly from (). Several realization algorithms were presented in Section 5.4, and the role of duality was emphasized in Section 5.4A.

The theory of state-space realizations of input-output descriptions given by impulse responses, and the time-invariant case by transfer function descriptions, were studied in this chapter.

In Section 5.2 the problem of state-space realizations of input-output descriptions was defined and the existence of such realizations was addressed. In Subsection 5.3A time-varying and time-invariant continuous-time and discrete-time systems were considered. Subsequently, the focus was on time-invariant systems and transfer function matrix descriptions (). The minimality of realizations of () was studied in Subsection 5.3C, culminating in two results, Theorem 3.9 and Theorem 3.10, where it was first shown that a realization is minimal if and only if it is controllable and observable, and next, that if a realization is minimal, all other minimal realizations of a given () can be found via similarity transformations. In Subsection 5.3C it was shown how to determine the order of minimal realizations directly from (). Several realization algorithms were presented in Section 5.4, and the role of duality was emphasized in Section 5.4A.

The theory of state-space realizations of input-output descriptions given by impulse responses, and the time-invariant case by transfer function descriptions, were studied in this chapter.

In Section 5.2 the problem of state-space realizations of input-output descriptions was defined and the existence of such realizations was addressed. In Subsection 5.3A time-varying and time-invariant continuous-time and discrete-time systems were considered. Subsequently, the focus was on time-invariant systems and transfer function matrix descriptions (). The minimality of realizations of () was studied in Subsection 5.3C, culminating in two results, Theorem 3.9 and Theorem 3.10, where it was first shown that a realization is minimal if and only if it is controllable and observable, and next, that if a realization is minimal, all other minimal realizations of a given () can be found via similarity transformations. In Subsection 5.3C it was shown how to determine the order of minimal realizations directly from (). Several realization algorithms were presented in Section 5.4, and the role of duality was emphasized in Section 5.4A.