Péter Gáspár, Zoltán Szabó, József Bokor

Discrete Feedback Systems 2.

Modern Control

A.1.3 Schur complement and Schur lemma

Lemma A.1 (Schur Decomposition) Suppose A or D respectively is non non-singular. Then

, or

Lemma A.2 (Schur Lemma) Let

and

be symmetric matrices. The following are equivalent.

search		(A.16)
search	(A.17)
search	(A.18)

Lemma A.3 (Symmetric Schur Lemma) Let

and

be symmetric matrices. The following are equivalent.

search		(A.19)
search	(A.20)
search	(A.21)

We note that the equality

is redundant since

Lemma A.4 Suppose that

is nonsingular. Then

Lemma A.5 (Matrix Inversion Lemma) Let

and

be non-singular. Then

Suppose

and

are both non-singular. Then

Let us suppose that the block matrix

is non-singular. Moreover, suppose

respectively is non-singular and let

. Then

and

are all non-singular then

An easy but important observation is that for the quadratic form we have

i.e., we have the following sum of squares representations

(A.22)

exists and

(A.23)

exists, respectively. As an example, having the problem

under conditions

we have immediate the conditions

, getting

. Impozing

, we can obtain the lowest value, i.e.,

Tartalomjegyzék

Kiadó: Akadémiai Kiadó

Online megjelenés éve: 2019

ISBN: 978 963 454 373 2

The classical control theory and methods that we have been presented in the first volume are based on a simple input-output description of the plant, expressed as a transfer function, limiting the design to single-input single-output systems and allowing only limited control of the closed-loop behaviour when feedback control is used. Typically, the need to use modern linear control arises when working with models which are complex, multiple input multiple output, or when optimization of performance is a concern. Modern control theory revolves around the so-called state-space description. The state variable representation of dynamic systems is the basis of different and very direct approaches applicable to the analysis and design of a wide range of practical control problems. To complete the design workflow, finally some introduction into system identification theory is given.

Hivatkozás: https://mersz.hu/gaspar-szabo-bokor-discrete-feedback-systems-2//

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