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26) immediately imply 2 ∞ ω [φ(ω) − φ(ω0 )] dω , π 0 ω2 − ω20 2 ω0 ∞ m(ω) − m(ω0 ) dω . 34) Note that the assumption that H has no zeros or poles in the CRHP guarantees the validity of the integrals above, since log H is analytic in the finite CRHP. The singularity at ∞ arising from a strictly proper H is ruled out in the chosen contour of integration (see footnote 12 on page 40). The fact that | log H(s)|/|s| → 0 when |s| → ∞ eliminates the integral along the large semicircle in the ORHP. 34).

2 is taken from Åström (1996), where additional limitations on the achievable phase margin are also discussed. 3 SISO Control In this chapter we present results on performance limitations in linear single-input single-output control systems. These results are the cornerstones of classical design. We focus on the sensitivity and complementary sensitivity functions, S and T . Although this chapter is based on contributions by many authors, mainly during the 80’s, they may be seen with justice as direct descendants of the many ideas contained in Bode (1945).

3, consisting of the imaginary axis and a semicircle of infinite radius into the ORHP. If L has poles on the imaginary axis, then C must have small indentations to avoid them. Such a contour is called Nyquist contour and its image through L(s) is called the Nyquist plot of L. A necessary and sufficient condition for closed-loop stability is furnished by the following theorem. 3. Nyquist contour. 2 (Nyquist Stability Criterion). 1 is internally stable if and only if L(∞) = −1 and the number of counter-clockwise revolutions made by the Nyquist plot of the open-loop transfer function L = GK around the point s = −1, is equal to the number of unstable poles of G plus the number of unstable poles of K.