# A FIRST /X BASED METHOD

This section and the following one consider the general case of a MIMO nonlinearity. As a preliminary, the first subsection illustrates that the general problem of detecting a limit-cycle in the presence of parametric uncertainties can be recast into an LFT framework. A technical result is then presented in the second subsection. It is proved in the third subsection that the issue of detecting a limit-cycle can be made equivalent to the issue of detecting the singularity of a matrix, which depends on parametric uncertainties: this problem can thus be treated in the ц framework. An extension of the method is presented in the fourth subsection. The use of Ц bounds is finally discussed in the last subsection.

3.1 AN LFT FORMULATION OF THE PROBLEM

Consider the general case of a MIMO transfer matrix G(s,6), where 5 is a vector of parametric uncertainties Si. Following chapter 3, G(s,6) can be expressed as an LFT Ft(P(s), Д2),where Д 2 is a diagonal matrix:

In the generalized problem of Figure 12.3 (to be compared with Figure 12.1), the aim is twofold:

■ If no limit-cycle exists for the nominal closed loop system (<5j = 0), the minimal amount of parametric uncertainties is to be found, for which a limit-cycle is obtained in the closed loop system of Figure 12.3.

■ If a limit-cycle exists in the nominal closed loop system, it is interesting to visualize the movement of this limit-cycle (i. e. the variation

of its magnitude X and frequency ta) as a function of parametric uncertainties.

## Leave a reply