Reduction of the Controller Complexity
Controller complexity reduction is an issue in many applications. The key objectives of controller complexity reduction is to obtain a reduced order controller which preserves the closed-loop properties of the nominal closed-loop system (stability, robust
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Reduction of the Controller Complexity
9.1 Introduction The complexity (order of the polynomials R and S) of the controllers designed on the basis of identified models depends upon • the complexity of the identified model; • the performance specifications; and • the robustness constraints. The controller will have a minimum complexity equal to that of the plant model but as a consequence of performance specifications and robustness constraints, this complexity increases (often up to the double of the size of the model, in terms of number of parameters, and in certain cases even more). In many applications, the necessity of reducing the controller complexity results from constraints on the computational resources in real time (reduction of the number of additions and multiplications). Therefore one should ask the question: can we obtain a simpler controller with almost the same performance and robustness properties as the nominal one (design based on the plant model)? Consider the system shown in Fig. 9.1 where the plant model transfer function is given by G(z −1 ) =
z −d B(z −1 ) A(z −1 )
(9.1)
R(z −1 ) S(z −1 )
(9.2)
and the nominal controller is given by: K (z −1 ) =
© Springer International Publishing Switzerland 2017 I.D. Landau et al., Adaptive and Robust Active Vibration Control, Advances in Industrial Control, DOI 10.1007/978-3-319-41450-8_9
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9 Reduction of the Controller Complexity
Fig. 9.1 The true closed-loop system
where: R(z −1 ) = r0 + r1 z −1 + · · · + rn R z −n R S(z −1 ) = 1 + s1 z −1 + · · · + sn S z −n S = 1 + z −1 S ∗ (z −1 )
(9.3) (9.4)
Different sensitivity functions have been defined in Sect. 7.1 for the system given in Fig. 9.1. The system given in Fig. 9.1 will be denoted the “true closed-loop system”. Throughout this chapter, feedback systems which will use either an estimation of ˆ or a reduced order estimation of K (denoted Kˆ ) will be considered. G (denoted G) The corresponding sensitivity functions will be denoted as follows: • Sx y —Sensitivity function of the true closed-loop system (K , G). • Sˆ x y —Sensitivity function of the nominal simulated closed-loop system (nominal ˆ controller K + estimated plant model G). ˆ • Sˆ x y —Sensitivity function of the simulated closed-loop system using a reduced ˆ order controller (reduced order controller Kˆ + estimated plant model G). ˆˆ −1 ) when ˆ −1 ) when using K and G, ˆ P(z Similar notations are used for P(z −1 ), P(z ˆ using Kˆ and G. The specific objective will be to reduce the orders n R and n S of controller polynomials R and S. The basic rule for developing procedures for controller complexity reduction is to search for controllers of reduced orders which preserve the properties of the closed-loop as much as possible. A direct simplification of the controller transfer function by traditional techniques (cancellation of poles and zeros which are close, approximations in the frequency domain, balanced truncation, etc.) without taking into account the properties of the closed-loop leads in general to unsatisfactory
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