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1 The time-domain backward shift operator used in discrete-time systems is often represented by q −1 in the literature, while the notation z is generally used for the corresponding frequency-domain Z-transform variable. Here, for the sake of simplicity and although mathematical rigour would require such distinction, we shall use the same notation for both the operator and the variable. Similarly, the time differentiation operator used in continuoustime systems is often represented by p, but we shall make no distinction between it and the frequency-domain Laplace-transform variable s.

It should be clear that it is easier to design a stabilising controller for a system with a large supK bG0 ,K than for a system with a small one. We refer the reader to (Zhou and Doyle, 1998) and references therein for more detail about the links between the generalised stability margin and controller performance. The following proposition will be used several times throughout this book. 1. 17b) or, equivalently, for some ε > 0. 18) It tells us that the stability margin achieved by a controller K connected to a system G will be close to that achieved by the same K with G0 if the closed-loop transfer matrices Ti (G, K) and Ti (G0 , K) are close in the H∞ norm.

1 Definition The ν-gap metric between two continuous-time transfer matrices G1 (s) and G2 (s) is a measure of distance between these two systems. It was first introduced by G. Vinnicombe in (Vinnicombe, 1993a, 1993b). 5. 65) ⎪ ⎪ ⎩ 1 otherwise where • Ξ(s) N2 (s)N1 (s) + M2 (s)M1 (s); ˜2 (jω)M1 (jω)+ M ˜ 2 (jω)N1 (jω) is called the chordal • κ(G1 (jω), G2 (jω)) −N distance between G1 and G2 at frequency ω; ˜ −1 (s)N ˜2 (s) are nor• G1 (s) = N1 (s)M1−1 (s) and G2 (s) = N2 (s)M2−1 (s) = M 2 malised coprime factorisations of G1 and G2 ; • wno(P (s)) = η(P −1 (s)) − η(P (s)) is called the winding number of the transfer function P (s) and is defined as the number of counterclockwise encirclements (a clockwise encirclement counts as a negative encirclement) around the origin of the Nyquist contour of P (s) indented around the right of any imaginary axis pole of P (s); • η(P (s)) denotes the number of poles of P (s) in C+ 0.

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