# A PRACTICAL. APPROACH TO. ROBUSTNESS. ANALYSIS

In many physical situations, a plant model is often provided with a qualitative or quantitative measure of associated model uncertainties. On the one hand, the validity of the model is guaranteed only inside a frequency band, so that nearly nothing can be said about the behavior of the real plant at high frequencies. On the other hand, if the model is derived on the basis of physical equations, it can be parameterized as a function of a few physical parameters, which are usually not perfectly known in practice.

This is e. g. the case in aeronautical systems: as an example, the aero­dynamic model of an airplane is derived from the flight mechanics equa­tions. When synthesizing the aircraft control law, it is then necessary to take into account uncertainties in the values of the stability derivatives, which correspond to the physical coefficients of the aerodynamic model.

Moreover, this airplane model does not perfectly represent the beha­vior of the real aircraft. As a simple example, the flight control system or the autopilot are usually synthesized just using the aerodynamic model, thus without accounting for the flexible mechanical structure: the cor­responding dynamics are indeed considered as high frequency neglected dynamics, with respect to the dynamics of the rigid model[1].

Summarizing, a model never perfectly represents the real plant to be controlled, and it is necessary to deal with associated model uncertain­ties. These correspond, either to uncertainties in the physical parameters of the plant (and more generally model perturbations inside the control bandwidth), or to high frequency unmodeled or neglected dynamics (un­
certainties beyond the control bandwidth).

However, many control design procedures only use the nominal model of the plant, and treat uncertainties in an incomplete or heuristic way. H synthesis schemes exist, which account for the available information on the nature and structure of these uncertainties: even if a great deal of work has been devoted to this subject, these design methods remain difficult to use, and it is not easy to control the order and the structure of the resulting controller. Note moreover that a properly designed control law can be robust in the face of uncertainties, even if these ones were not explicitly taken into account during the design process. Engineers often use their physical knowledge of the plant to design in an heuristic way control laws, which appear a posteriori sufficiently robust.

In this context, the issue is rather to validate a control law by analyz­ing its robust stability and performance properties. Various methods are available for solving this problem, depending on the nature and struc­ture of the uncertainties. We focus here on the structured singular value (s. s.v.) approach. The first reason is that the s. s.v. /xprovides a general framework to robustness analysis problems. As a second justification of this choice, the ц approach has been successfully applied to industrial problems.

Two general issues arise when applying this method, which have mo­tivated a great deal of work since the beginning of the Eighties. The first one is to put a specific control problem into a standard form, which is called an LFT (Linear Fractional Transformation). When applying the Ц tools to this standard LFT form, the second problem concerns the computational requirement, which must remain reasonable even for large dimension problems.

More precisely, the s. s.v. ц(ш) is to be computed at each point w of a frequency gridding, and /x lower and upper bounds are computed instead of the exact value of /x 2. Methods for computing these /x bounds can be divided into two large categories, namely the exponential and polynomial time ones: the computational requirement of exponential (resp. polyno­mial) time methods increases exponentially (resp. only polynomially) with the size of the problem.