Abstract
In Chap. 7, we consider the design of fixed structure controllers for uncertain systems in the H ∞ framework. Although the presented design procedures apply for any kind of structured controller, we focus mainly on the most widely used of them which is the PID. Two design approaches will be considered: the mixed sensitivity method and the H ∞ loop-shaping design procedure. Using these methods, the resulting PID design problem is formulated as an inherently non-convex optimization problem. The resulting tuning method is applicable both to stable and unstable systems, without any limitation concerning the order of the process to be controlled. Various design examples are presented to give practical insights into the methods presented.
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Notes
- 1.
The quantity 1/|S(jω)| represents the distance between the Nyquist curve of the open-loop transfer function G(s)K(s) and the critical point −1 at the frequency 2π/ω. The minimum of this distance then represents a good measure of the stability margin and is called the modulus margin. The modulus margin is given by M m =1/max ω |S(jω)|=1/∥S∥∞. This can be generalized to the MIMO case by considering \(\|\bar{\sigma}(S(j\omega))\|_{\infty}\) instead of ∥S(jω)∥∞, see Fig. 7.3 where we impose \(\|\bar{\sigma}(S(j\omega))\|_{\infty}\leq1/\underline{W}_{1}\).
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Toscano, R. (2013). H ∞ Design of Structured Controllers. In: Structured Controllers for Uncertain Systems. Advances in Industrial Control. Springer, London. https://doi.org/10.1007/978-1-4471-5188-3_7
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