The aim of this chapter is to study the existence of limit-cycles in a closed loop, which simultaneously contains nonlinearities and parametric uncertainties (Ferreres and Fromion, 1998). Three methods are presen­ted. First, the issue of detecting a limit-cycle with a necessary condition of oscillation is considered: a graphical method and a /r based method are proposed. A second ц based method is then proposed, which uses a sufficient condition of non oscillation, i. e. the issue is now to check the absence of limit-cycles despite parametric uncertainties. An example is finally presented: the necessary condition of oscillation is used to syn­thesize a controller which modifies the characteristics (magnitude and frequency) of the limit-cycle.


The analysis of nonlinear control systems remains a challenging prob­lem, despite numerous years of extensive research. As the starting point of this chapter, a classical problem is considered, namely a closed loop system which simultaneously contains an LTI transfer function and a separable autonomous nonlinear element:

■ If no a priori knowledge of the nonlinearity is available, this one is simply assumed to belong to a sector. Closed loop stability is checked with the circle or Popov criteria (Desoer and Vidyasagar, 1975).

■ Conversely, if the characteristics of the nonlinear element are a priori known, a solution is to replace this element by its Sinusoidal Input Describing Function (SIDF): see e. g. (Gray and Nakhla, 1981; Katebi and Zhang, 1995; Khalil, 1992) and included references. The har­monic linearization method is then applied, either for detecting the

presence of limit-cycles in the closed loop (see e. g. (Chin and Fu, 1994; Anderson and Page, 1995) for realistic applications of this clas­sical nonlinear analysis method), or for checking their absence.

It would be interesting to extend the above classical problems to the case of a closed loop, which is also subject to LTI parametric uncertain­ties. This was partially done for the case of a sector-type non linearity: see e. g. (Chapellat et al., 1991; Gahinet et al., 1995). The aim of this chapter is to extend the classical harmonic linearization method to the case of parametric uncertainties. Existing results in robustness analysis (i. e. analysis of the robustness properties of an LTI closed loop system, subject to LTI model uncertainties) are used as the basis

Two different problems are considered. The first one is to detect a limit-cycle with a necessary condition of oscillation: a graphical method is first proposed, which can be considered as an extension of the clas­sical method (section 2.). A g, based method is then proposed in section 3.. As a second problem, the problem of checking the absence of limit – cycles despite parametric uncertainties is studied. An alternative ц based method is proposed, which uses a sufficient condition of non oscillation (section 4.). An example is presented in section 5.. Concluding remarks end the chapter.

Note that the starting point of this chapter is to remark that the necessary condition of oscillation leads to a problem of detecting the singularity of a matrix, which depends on the parametric uncertainties, whereas the sufficient condition of non oscillation leads to a problem of checking a small gain condition despite parametric uncertainties. On the other hand, the problem of checking the presence or the absence of limit-cycles in the face of parametric uncertainties has primarily an en­gineering interest. The main purpose of this chapter is consequently to show that this difficult nonlinear problem can be solved efficiently with the s. s.v. )JL. Because many д tools are already available, this chapter, in the same spirit as (Katebi and Zhang, 1995), has primarily a practical interest, beyond the necessary theoretical justifications.