Linear and Nonlinear Multivariable Feedback Control

Preface

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Chapter 1: Canonical representations and stability analysis of linear MIMO systems

¹ The terms sensitivity function and complementary sensitivity function were introduced by Bode (1945).

² The MIMO system accuracy is discussed in Chapter 2.

³ The notion of MIMO system zero as a complex number z that reduces at s = z, the local rank of the matrix W(s), is given, for example, in MacFarlane (1975).

⁴ If polynomials P(s) and Z(s) do have common roots (i.e. coincident zeros and poles), and they are cancelled in Equation (1.12), then there always exists a danger that the mentioned zeros and poles correspond to different directions of the MIMO system (see Subsection 1.2.3).

⁵ The Smith-McMillan canonical form is discussed in Remark 1.6.

⁶ The origin of the term return difference, introduced by Bode, is due to the fact that if we break the closed-loop system with unit negative feedback at an arbitrary point and inject at that point some signal y(s), then the difference between that signal and the signal –W(s)y(s) returning through the feedback loop to the break point is equal to [1 + W(s)]y(s).

⁷ That low states that the degeneracy of the product of two matrices is at least as great as the degeneracy of either matrix and, at most, as great as the sum of degeneracies of the matrices (Derusso et al. 1965).

⁸ Later on, we shall omit the case of the singular matrix [I + W (∞)], which has no practical significance.

⁹ Strictly ...