Frequency-domain L₂-stability conditions for switched linear and nonlinear SISO systems

Abstract

We consider the L ₂-stability analysis of single-input–single-output (SISO) systems with periodic and nonperiodic switching gains and described by integral equations that can be specialised to the form of standard differential equations. For the latter, stability literature is mostly based on the application of quadratic forms as Lyapunov-function candidates which lead, in general, to conservative results. Exceptions are some recent results, especially for second-order linear differential equations, obtained by trajectory control or optimisation to arrive at the worst-case switching sequence of the gain. In contrast, we employ a non-Lyapunov framework to derive L ₂-stability conditions for a class of (linear and) nonlinear SISO systems in integral form, with monotone, odd-monotone and relaxed monotone nonlinearities, and, in each case, with periodic or nonperiodic switching gains. The derived frequency-domain results are reminiscent of (i) the Nyquist criterion for linear time-invariant feedback systems and (ii) the Popov-criterion for time-invariant nonlinear feedback systems with the Lur'e-type nonlinearity. Although overlapping with some recent results of the literature for periodic gains, they have been derived independently in essentially the Popov framework, are different for certain classes of nonlinearities and address some of the questions left open, with respect to, for instance, the synthesis of the multipliers and numerical interpretation of the results. Apart from the novelty of the results as applied to the dwell-time problem, they reveal an interesting phenomenon of the switched systems: fast switching can lead to stability, thereby providing an alternative framework for vibrational stability analysis.

Keywords:

• switched systems
• absolute stability
• L ₂-stability
• time-varying systems
• feedback systems
• KPY lemma
• Lur'e problem
• periodic coefficient systems
• Nyquist's criterion
• circle criterion
• multiplier functions

Acknowledgements

The authors wish to thank the referees, the Associate Editor and the Editor-in-Chief for their valuable comments and suggestions which have led to the present, improved and updated version of this article.

Notes

Notes

1. See Section 4.

2. See Venkatesh (1974) for the earlier, preliminary results for both linear and nonlinear systems.

3. See definition in Section 2.2.

4. Applicable to a more general range , where K  ≥ 0.

5. This may be a reason for the possible anticipated difficulty in arriving at Lyapunov function candidates for the stability results corresponding to those of this article.

6. These proofs are believed to be simpler, and more transparent, than those in the literature, especially including Altshuller (2009).

7. Note here that, for the lack of suitable (additional) characters, the subscript T, which is being used for denoting a time-truncated function is retained as a subscript for the Fourier transform of the function. This is a obviously a deliberate misuse of the subscript, but there should be no cause for confusion. The subscript T in the Fourier transform does not mean a truncation in the frequency domain; it refers to the fact that the original time-function is truncated.

8. Where a system with high-frequency parametric perturbations is modelled and controlled as its average model.

9. A large switching period P does not necessarily mean slow switching. It is possible that a feedback gain k(t) switches frequently in its large period.

10. It is to be noted that no results are found for nonlinear feedback gains in Chitour and Sigalotti (2010).