By Gilles Ferreres
1. MOTIVATION in lots of actual events, a plant version is usually supplied with a qualitative or quantitative degree of linked version uncertainties. at the one hand, the validity of the version is assured merely within a frequency band, in order that approximately not anything might be acknowledged concerning 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 now not completely recognized in perform. this can be e.g. the case in aeronautical platforms: for example, the ae- dynamic version of an plane is derived from the flight mechanics eq- tions. whilst synthesizing the airplane regulate 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 characterize the be- vior of the true airplane. As an easy instance, the flight regulate process or the autopilot are typically 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 appreciate to the dynamics of the inflexible version .
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Extra resources for A Practical Approach to Robustness Analysis with Aeronautical Applications
To this aim, pre- and post-compensators and are added at the plant inputs and outputs. These transfer matrices are chosen so that the singular values reflect the design objectives. 4). A controller is obtained, which stabilizes the augmented plant. The final controller is K(s) = Note that the above robust stabilization procedure presents special properties, since it tries as much as possible to keep the desired open loop shaping while enforcing a stability closed loop constraint. 2 APPLICATION The choice of the compensators is not detailed, the interested reader is referred to (Ferreres and M’Saad, 1996) for a related work.
Indeed, can be interpreted as the derivative of with respect to (with the restrictions above). 16 illustrates this point. Let where the are real scalars. 16 represents the space of the Each point of the curve C corresponds to a value of the for which the matrix is singular. Conversely, all points strictly inside the domain, whose border is C, correspond to values of the for which the matrix is nonsingular. In an obvious way, the zero point (which corresponds to and belongs to this domain of nonsingularity.
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. The singularity of is then equivalent to the singularity of As a consequence, the size of the smallest unstructured complex matrix which renders the matrix singular, is: Consider now the case of a single real repeated scalar Then where the real spectral radius is the magnitude of the largest real eigenvalue of M: is zero if M has no real eigenvalue.