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1.2 Overview of chapters 2 13 using barcode implement for vs .net control to generate, create data matrix image in vs .net applications. GS1 DataBar how one arrives at .net vs 2010 2d Data Matrix barcode this factorization. The existence of a doubly coprime factorization is a pure input/output property which can be stated without any reference to an underlying system.

Moreover, it is easy to construct examples of systems which are not jointly stabilizable and detectable, but whose input/output map still has doubly coprime factorizations. We address this question in Section 8.4, where we introduce the notions of coprime stabilizability and detectability.

We call a state feedback right coprime stabilizing if the closed-loop system corresponding to this feedback is stable and produces a right coprime factorization of the input/output map. Analogously, an output injection is left coprime detecting if the closed-loop system corresponding to this feedback is stable and produces a left coprime factorization of the input/output map. The last theme in this chapter is the dynamic stabilization presented in Section 8.

5. Here we show that every well-posed jointly stabilizable and detectable system can be stabilized by means of a dynamic controller, i.e.

, we show that there is another well-posed linear system (called the controller) such that the interconnection of these two systems produces a stable system. We also present the standard Youla parametrization of all stabilizing controllers. 9 By a realization of a given time-invariant causal map D we mean a (often well-posed) linear system whose input/output map is D.

In this chapter we study the basic properties of these realizations. For simplicity we stick to the L p . Reg-well-posed cas e. We begin by de ning what we mean by a minimal realization: this is a realization which is both controllable and observable. Controllability means that the range of the input map (the map denoted by B above) is dense in the state space, and observability means that the output map (the map denoted by C above) is injective.

As shown in Section 9.2, any two L p . Reg-well-posed rea datamatrix 2d barcode for .NET lizations of the same input/output map are pseudosimilar to each other. This means roughly that there is a closed linear operator whose domain is a dense subspace of one of the two state spaces, its range is a dense subspace of the other state space, it is injective, and it intertwines the corresponding operators of the two systems.

Such a pseudo-similarity is not unique, but there is one which is maximal and another which is minimal (in the sense of graph inclusions). There are many properties which are not preserved by a pseudo-similarity, such as the spectrum of the main operator, but pseudosimilarities are still quite useful in certain situations (for example, in Section 9.5 and 11).

In Section 9.3 we show how to construct a realization of a given input/output map from a factorization of its Hankel operator. The notions of controllability and observability that we have de ned above are often referred to as approximate controllability or observability.

Some other notions of controllability and observability (such as exact, or null in nite time,. Introduction and overview or exact in in nit VS .NET Data Matrix 2d barcode e time, or nal state observable) are presented in Section 9.4, and the relationships between these different notions are explained.

In particular, it is shown that every controllable L p -well-posed linear system with p < whose input map B and output map C are (globally) bounded can be turned into a system which is exactly controllable in in nite time by replacing the original state space by a subspace with a stronger norm. If it is instead observable, then it can be turned into a system which is exactly observable in in nite time by completing the original state space with respect to a norm which is weaker than the original one. Of course, if it is minimal, then both of these statements apply.

Input normalized, output normalized, and balanced realizations are presented in Section 9.5. A minimal realization is input normalized if the input map B becomes an isometry after its null space has been factored out.

It is output normalized if the output map C is an isometry. These de nitions apply to the general L p -well-posed case in a Banach space setting. In the Hilbert space setting with p = 2 a minimal system is input normalized if its controllability gramian BB is the identity operator, and it is output normalized if its observability gramian C C is the identity operator.

We construct a (Hankel) balanced realization by interpolating half-way between these two extreme cases (in the Hilbert space case with p = 2 and a bounded input/output map). This realization is characterized by the fact that its controllability and observability gramians coincide. All of these realizations (input normalized, output normalized, or balanced) are unique up to a unitary similarity transformation in the state space.

The balanced realization is always strongly stable together with its dual. A number of methods to test the controllability or observability of a system in frequency domain terms are given in Section 9.6, and some further time domain test are given in Section 9.

10. In Section 9.7 we discuss modal controllability and observability, i.

e., we investigate to what extent it is possible to control or observe different parts of the spectrum of the main operator (the semigroup generator). We say a few words about spectral minimality in Section 9.

8. This is the question about to what extent it is possible to construct a realization with a main operator whose spectrum essentially coincides with the points of singularities of the transfer function. A complete answer to this question is not known at this moment (and it may never be).

Some comments on to what extent controllability and observability are preserved under various transformations of the system (including feedback and duality) are given in Sections 9.9 and 9.10.

10 In 4 we saw that every L p . Reg-well-posed lin ear system has a control operator B mapping the input space U into the extrapolation space X 1 , and also an observation operator C mapping the domain X 1 of the.
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