Stability and L₁ × ℓ₁-to-L₁ × ℓ₁ performance analysis of uncertain impulsive linear positive systems with applications to the interval observation of impulsive and switched systems with constant delays

ABSTRACT

Solutions to the interval observation problem for delayed impulsive and switched systems with $L_1$-performance are provided. The approach is based on first obtaining stability and $L_1\times {\ell }_1$-to-$L_1\times {\ell }_1$ performance analysis conditions for uncertain linear positive impulsive systems in linear fractional form with norm-bounded uncertainties using a scaled small-gain argument involving time-varying D-scalings. Both range and minimum dwell-time conditions are formulated – the case of constant and maximum dwell-times can be directly obtained as corollaries. The conditions are stated as timer/clock-dependent conditions taking the form of infinite-dimensional linear programmes that can be relaxed into finite-dimensional ones using polynomial optimisation techniques. It is notably shown that under certain conditions, the scalings can be eliminated from the stability conditions to yield equivalent stability conditions on the so-called worst-case system, which is obtained by replacing the uncertainties by the identity matrix. These conditions are then applied to the special case of linear positive systems with delays, where the delays are considered as uncertainties, similarly to as in (Briat (2018a). Stability and performance analysis of linear positive systems with delays using input-output methods. International Journal of Control, 91(7), 1669–1692). As before, under certain conditions, the scalings can be eliminated from the conditions to obtain conditions on the worst-case system, coinciding here with the zero-delay system – a result that is consistent with all the existing ones in the literature on linear positive systems with delays. Finally, the case of switched systems with delays is considered. The approach also encompasses standard continuous-time and discrete-time systems, possibly with delays and the results are flexible enough to be extended to cope with multiple delays, time-varying delays, distributed/neutral delays and any other types of uncertain systems that can be represented as a feedback interconnection of a known system with an uncertainty.

Keywords:

• Hybrid (positive) systems
• interval observation
• uncertain and delay systems
• timer/clock-dependent Lyapunov conditions
• sum of squares programming

Disclosure statement

No potential conflict of interest was reported by the author.

Notes

1 See (Briat, 2017a) for a comparison of all these methods.

2 We add here extra conditions to constrain the values of the entries of the gains to lie within the interval $[-10,10]$. See (Briat, 2013a) for more details.