A Practical Approach to Robustness Analysis with by Gilles Ferreres

By Gilles Ferreres

1. MOTIVATION in lots of actual events, a plant version is frequently supplied with a qualitative or quantitative degree of linked version uncertainties. at the one hand, the validity of the version is assured purely inside of a frequency band, in order that approximately not anything might be stated in regards to the habit of the genuine plant at excessive frequencies. nonetheless, if the version is derived at the foundation of actual equations, it may be parameterized as a functionality of some actual parameters, that are often no longer completely recognized in perform. this is often e.g. the case in aeronautical structures: for instance, the ae- dynamic version of an aircraft is derived from the flight mechanics eq- tions. while synthesizing the plane keep watch over legislations, it's then essential to take note of uncertainties within the values of the soundness derivatives, which correspond to the actual coefficients of the aerodynamic version. additionally, this aircraft version doesn't completely symbolize the be- vior of the true plane. As an easy instance, the flight keep an eye on process or the autopilot tend to be synthesized simply utilizing the aerodynamic version, therefore with out accounting for the versatile mechanicalstructure: the c- responding dynamics are certainly regarded as excessive frequency missed 1 dynamics, with recognize to the dynamics of the inflexible version .

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39). Frequency is fixed. 12). 38). Using the Main Loop Theorem, it can be claimed that the two above conditions are satisfied if and only if: As a consequence, the robust performance problem reduces to an augmented robust stability problem, in which a fictitious performance block is added (Doyle, 1985). Note that this block is possibly structured: if signals and z are decomposed as and and if we are just interested in the transfer functions between scalar signals and the performance block as a complex diagonal model perturbation.

V. , so that a lower bound of the robustness margin is finally obtained as: In the context of a robust stability problem in the presence of parametric uncertainties, closed loop stability can thus be guaranteed inside the hypercube in the space of uncertain parameters. e. there exists a real model perturbation with singular (in the context of a robust stability problem, is a destabilizing model perturbation). The usefulness of a lower bound is twofold. As a first point, gives a measure of the conservatism of the upper bound by examining the tightness of the interval which contains the exact value of As a second point, an associated worst-case model perturbation is usually provided with by the computational algorithm.

In the case of a real model perturbation, a first solution is to study the robustness of the location of the closed loop poles despite parametric uncertainties. In the general context of a mixed model perturbation, a second and more classical solution consists in checking whether a frequency domain template on a closed loop transfer matrix remains satisfied despite model uncertainties. In the first case, performance is rather defined in the time domain, whereas it is defined in the frequency domain in the second one.

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