Nonlinear analysis in the presence of parametric uncertainties

Abstract

The aim of this paper is to study the existence of limit-cycles in a closed loop, which simultaneously contains nonlinearities and parametric uncertainties. Three methods are presented: we ® rst consider the problem of detecting a limit-cycle using a necessary condition of oscillation. A graphical method and a ® rst ¹ based method are proposed. We then consider the problem of checking the absence of limit-cycles despite parametric uncertainties. A second ¹ based method is proposed, which uses a su cient condition of non-oscillation. An example is ® nally presented: the idea is to use the necessary condition of oscillation, so as to synthesize a controller which modi® es the characteristics (magnitude and frequency) of the limit-cycle.

Robust prevention of limit cycle for nonlinear control systems with parametric uncertainties both in the linear plant and nonlinearity

“…Usually, the overall picture turns out to be the proper tuning of the adjustable parameters to effectively suppress or to remove the limit cycle. To achieve this, the extensions of the describing function method-based limit cycle analysis to nonlinear systems with parametric uncertainties have recently been proposed [14][15][16][17][18][19][20]. In one of such works, Barmish and Khargonekar attempted to address the former point in [14].…”

“…Ferreres and Fromion employed the structured singular value µ-based method to analyze limit cycles of control systems containing nonlinearity in [16], but the method assumes uncertainties only in the LTI element. Extensions of [16] to the generalized interval polynomial framework whereby some of the parameters are assumed to vary in prescribed ranges are recast in [17] and [18]. In addition, a new approach to the stability analysis and limit cycle occurrence of nonlinear systems with uncertainty in the linear part is proposed in [19].…”

“…Later, the unknown coefficients of the system characteristic equation have been overbounded and the Kharitonov theorem in [21] has been used to perform the overall stability test [15]. Ferreres and Fromion employed the structured singular value µ-based method to analyze limit cycles of control systems containing nonlinearity in [16], but the method assumes uncertainties only in the LTI element. Extensions of [16] to the generalized interval polynomial framework whereby some of the parameters are assumed to vary in prescribed ranges are recast in [17] and [18].…”

Limit cycle computation for describing function approximable nonlinear systems with box-constrained parametric uncertainties

“…His method requires a good rational approximation to the describing function of the nonlinear element. Ferreres and Promion [9] propose a m-based method for limit cycle analysis, but the method assumes uncertainties only in the LTI element. Impram and Munro [10] pose the describing function analysis problem in a generalized interval polynomial framework, addressing both structured and unstructured uncertainties in Reference [11].…”

Exact Nonlinear Modelling Using Symbolic Linear Fractional Transformations

“…and Zhang, Y., 1995;Ferreres, G. and Fromion, V., 1998), it is shown how the DF approach can be used to transform a linear system with a nonlinear input into an LFT model by using the approximation error of the harmonic linearization and the corresponding SIDF for the nonlinearity. Furthermore, these two references present methods for performing nonlinear analysis, i.e.…”