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Dynamic stabilization in .NET Encode Data Matrix 2d barcode in .NET Dynamic stabilization




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8.5 Dynamic stabilization use .net datamatrix 2d barcode printer todraw data matrix barcodes for .net ISO Specification 8.5 Dynamic stabilization It is now time to take a clos er look at the the notion of dynamic feedback stabilization which plays an important role in optimal control theory. For example, H control theory deals extensively with measurement feedback stabilization, which is a special case of dynamic feedback stabilization. De nition 8.

5.1 Let =. A B C D be a well-posed linear system on (U, X, Y ).. 1 1 (i) The well-posed linear data matrix barcodes for .NET system 1 = C1 D1 on (Y, X 1 , U ) is called a stabilizing dynamic feedback system for if the dynamic feedback connection in Figure 8.5 (cf.

Example 7.2.5) is (admissible and) stable.

The system is stabilizable by dynamic feedback if there exists a stabilizing dynamic feedback system 1 for . (ii) To this de nition we add one of the words weakly , strongly , or exponentially whenever the closed-loop system is stable in the corresponding sense (see De nition 8.1.

1). (iii) If to these de nitions we add one or several of the quali ers state/state , input/state , state/output , input/output , input , or output , then we mean that only the corresponding part of the closed-loop system has to be bounded or stable in the appropriate sense..

Observe that and 1 can be int erchanged with each other: 1 stabilizes if and only if stabilizes 1 . Therefore we shall also say that the two systems (dynamically) stabilize each other. In the sequel we shall primarily be concerned with dynamic input/output stabilization.

This is partially motivated by the following facts: Theorem 8.5.2 Let and let.

A1 B1 C1 D1 A B C D be a well-posed linear system gs1 datamatrix barcode for .NET on (U, X, Y ),. be a well-posed linear system on (Y, X 1 , U ).. + x0 1 A1 B1t C1 D1 y1 p+w + Figure 8.5 Dynamic stabilization Bt D x0 x1 u1 Stabilization and detection (i) Suppose that at least one of the following conditions hold: (a) both and 1 are stabilizable and detectable (not necessarily jointly); (b) both and 1 are right coprime stabilizable; (c) both and 1 are left coprime detectable. Then and 1 stabilize each other if and only if they input/output stabilize each other. (ii) Suppose both and 1 are L p -well-posed with p < , and that at least one of the following conditions hold: (a) both and 1 are strongly right coprime stabilizable; (b) both and 1 are stabilizable and strongly detectable; (c) both and 1 are strongly left coprime detectable.

Then, and 1 stabilize each other strongly if and only if they input/output stabilize each other. (iii) Suppose both and 1 are Reg-well-posed, detectable and strongly stabilizable. Then they stabilize each other strongly if and only if they input/output stabilize and strongly input/state stabilize each other.

(iv) Suppose both and 1 are Reg-well-posed, and that at least one of the following conditions hold: (a) both and 1 are stabilizable and strongly detectable; (b) both and 1 are strongly left coprime detectable. Then, and 1 stabilize each other strongly if and only if they input/output stabilize and strongly state/output stabilize each other. (v) Suppose that at least one of the following conditions hold: (a) both and 1 are Reg-well-posed or L p -well-posed with p > 1, detectable and exponentially stabilizable; (b) both and 1 are exponentially right coprime stabilizable; (c) both and 1 are L p -well-posed with p < , stabilizable and exponentially detectable; (d) both and 1 are exponentially left coprime detectable; Then and 1 stabilize each other exponentially if and only if they input/output stabilize each other.

Proof First use Propositions 8.2.10(ii)(c) and 8.

4.12 to show that the dynamic feedback connection of and 1 inherits the stabilizability and detectability properties of the two subsystems, and then use Theorems 8.2.

11 and 8.4.8 to conclude that the dynamic feedback connection is stable in the appropriate sense.

As we mentioned above, we shall in the sequel concentrate on the problem of dynamic input/output stabilization. This is motivated by Theorem 8.5.

2, which.
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