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 in basic terms within a frequency band, in order that approximately not anything will be stated in regards to the habit of the true plant at excessive frequencies. however, if the version is derived at the foundation of actual equations, it may be parameterized as a functionality of some actual parameters, that are frequently no longer completely recognized in perform. this can be e.g. the case in aeronautical platforms: to illustrate, the ae- dynamic version of an aircraft is derived from the flight mechanics eq- tions. while synthesizing the plane keep watch over legislation, it really is then essential to consider uncertainties within the values of the steadiness derivatives, which correspond to the actual coefficients of the aerodynamic version. in addition, this plane version doesn't completely symbolize the be- vior of the true airplane. As an easy instance, the flight regulate process or the autopilot tend to be synthesized simply utilizing the aerodynamic version, therefore with no 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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Additional info for A Practical Approach to Robustness Analysis with Aeronautical Applications
Nevertheless, in the special 20 A PRACTICAL APPROACH TO ROBUSTNESS ANALYSIS case of a single full complex block (resp. v. coincides with the maximal singular value (resp. the real spectral radius When is a full complex block, The small gain theorem provides indeed a necessary and sufficient condition of stability in the context of an unstructured model perturbation. In an alternative way, the following result can be used: if A is a complex matrix, the size of the smallest unstructured complex matrix which renders the matrix singular, is In the context of the initial problem, matrix M is assumed to be invertible for the sake of simplicity.
2 CHECKING A FREQUENCY DOMAIN TEMPLATE Performance can be defined as the minimization of the weighted norm of a closed loop transfer matrix. g. e. the transfer between the reference input and the tracking error. A template is defined to reflect the design specifications. 13). 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).
It is consequently impossible to compute the exact value of for large dimension problems. A usual solution is to compute upper and lower bounds instead of the exact value. The associated algorithms can be exponential time (like the algorithms which compute the exact value of ), or more interestingly polynomial time. Even if the gap between the bounds can not be guaranteed a priori when using polynomial time algorithms, good results can be obtained in realistic examples: this will be illustrated in the following.
A Practical Approach to Robustness Analysis with Aeronautical Applications by Gilles Ferreres