Péter Gáspár, Zoltán Szabó, József Bokor

Discrete Feedback Systems 2.

Modern Control


A.1.2 Singular value decomposition

Theorem A.1 (SVD) Suppose A is an m×n matrix whose entries come from the field F, which is either the field of real numbers or the field of complex numbers. Then there exists a factorization of the form
 
A=UΣV*,
 
where U is an m×m unitary matrix over F, the matrix Σ is m×n diagonal matrix with nonnegative real numbers on the diagonal, and V* is an n×n unitary matrix over F. Such a factorization is called a singular-value decomposition of A. A common convention is to order the diagonal entries Σi,i in non-increasing fashion. In this case, the diagonal matrix Σ is uniquely determined by A (though the matrices U and V are not). The diagonal entries of Σ are known as the singular values of A. More precisely Σ has the form
 
Σ=diag(σ1,σ2,,σp),
 
where p=min(m,n) and
 
σ1σ2σp0.
 
 
Remark A.1 (Pseudo-inverse) The singular value decomposition can be used for computing the pseudo-inverse of a matrix. Indeed, the pseudo-inverse of the matrix A with singular value decomposition A=UΣV* is
 
A=VΣU*,
 
where Σ is the pseudo-inverse of Σ with every nonzero entry replaced by its reciprocal.
 

Discrete Feedback Systems 2.

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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