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In this chapter alternatives to state-space descriptions were introduced and used to further study the behavior of linear time-invariant systems and to study in depth structural properties of feedback control systems.

In Part 1, the properties of systems described by Polynomial Matrix Descriptions (PMDs) were explored in Section 7.3 and background on polynomial matrices was provided in Section 7.2. The Diophantine Equation, which plays an important role in feedback systems, was studied at length in Subsection 7.2E.

An in-depth study of the theory of parameterizations of all stabilizing controllers with emphasis on PMDs was undertaken in Part 2, Subsection 7.4A, and the parameterizations of all proper stabilizing controllers in terms of proper and stable Matrix Fraction Descriptions (MFDs) were derived in Subsection 7.4C. State feedback and state estimation using PMDs were studied in Subsection 7.4B. Finally, control systems with two degrees of freedom controllers were explored in Subsection 7.4D, with an emphasis on stability, parameterizations of all stabilizing controllers, and attainable response maps.

In this chapter alternatives to state-space descriptions were introduced and used to further study the behavior of linear time-invariant systems and to study in depth structural properties of feedback control systems.

In Part 1, the properties of systems described by Polynomial Matrix Descriptions (PMDs) were explored in Section 7.3 and background on polynomial matrices was provided in Section 7.2. The Diophantine Equation, which plays an important role in feedback systems, was studied at length in Subsection 7.2E.

An in-depth study of the theory of parameterizations of all stabilizing controllers with emphasis on PMDs was undertaken in Part 2, Subsection 7.4A, and the parameterizations of all proper stabilizing controllers in terms of proper and stable Matrix Fraction Descriptions (MFDs) were derived in Subsection 7.4C. State feedback and state estimation using PMDs were studied in Subsection 7.4B. Finally, control systems with two degrees of freedom controllers were explored in Subsection 7.4D, with an emphasis on stability, parameterizations of all stabilizing controllers, and attainable response maps.

In this chapter alternatives to state-space descriptions were introduced and used to further study the behavior of linear time-invariant systems and to study in depth structural properties of feedback control systems.

In Part 1, the properties of systems described by Polynomial Matrix Descriptions (PMDs) were explored in Section 7.3 and background on polynomial matrices was provided in Section 7.2. The Diophantine Equation, which plays an important role in feedback systems, was studied at length in Subsection 7.2E.

An in-depth study of the theory of parameterizations of all stabilizing controllers with emphasis on PMDs was undertaken in Part 2, Subsection 7.4A, and the parameterizations of all proper stabilizing controllers in terms of proper and stable Matrix Fraction Descriptions (MFDs) were derived in Subsection 7.4C. State feedback and state estimation using PMDs were studied in Subsection 7.4B. Finally, control systems with two degrees of freedom controllers were explored in Subsection 7.4D, with an emphasis on stability, parameterizations of all stabilizing controllers, and attainable response maps.

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.

In this chapter alternatives to state-space descriptions were introduced and used to further study the behavior of linear time-invariant systems and to study in depth structural properties of feedback control systems.

In Part 1, the properties of systems described by Polynomial Matrix Descriptions (PMDs) were explored in Section 7.3 and background on polynomial matrices was provided in Section 7.2. The Diophantine Equation, which plays an important role in feedback systems, was studied at length in Subsection 7.2E.

An in-depth study of the theory of parameterizations of all stabilizing controllers with emphasis on PMDs was undertaken in Part 2, Subsection 7.4A, and the parameterizations of all proper stabilizing controllers in terms of proper and stable Matrix Fraction Descriptions (MFDs) were derived in Subsection 7.4C. State feedback and state estimation using PMDs were studied in Subsection 7.4B. Finally, control systems with two degrees of freedom controllers were explored in Subsection 7.4D, with an emphasis on stability, parameterizations of all stabilizing controllers, and attainable response maps.

In this chapter alternatives to state-space descriptions were introduced and used to further study the behavior of linear time-invariant systems and to study in depth structural properties of feedback control systems.

In Part 1, the properties of systems described by Polynomial Matrix Descriptions (PMDs) were explored in Section 7.3 and background on polynomial matrices was provided in Section 7.2. The Diophantine Equation, which plays an important role in feedback systems, was studied at length in Subsection 7.2E.

An in-depth study of the theory of parameterizations of all stabilizing controllers with emphasis on PMDs was undertaken in Part 2, Subsection 7.4A, and the parameterizations of all proper stabilizing controllers in terms of proper and stable Matrix Fraction Descriptions (MFDs) were derived in Subsection 7.4C. State feedback and state estimation using PMDs were studied in Subsection 7.4B. Finally, control systems with two degrees of freedom controllers were explored in Subsection 7.4D, with an emphasis on stability, parameterizations of all stabilizing controllers, and attainable response maps.