Abstract
AbstractRobust stability results for nominally linear hybrid systems are obtained from total stability theorems for purely continuous-time and discrete-time systems by using the powerful tool of fixed point theory. The class of hybrid systems dealt consists, in general, of coupled continuous-time and digital systems subject to state perturbations whose nominal (i.e., unperturbed) parts are linear and, in general, time-varying. The obtained sufficient conditions on robust stability under a wide class of harmless perturbations are dependent on the values of the parameters defining the over-bounding functions of those perturbations. The weakness of the coupling dynamics in terms of norm among the analog and digital substates of the whole dynamic system guarantees the total stability provided that the corresponding uncoupled nominal subsystems are both exponentially stable. Fixed point stability theory is used for the proofs of stability. A generalization of that result is given for the case that sampling is not uniform. The boundedness of the state-trajectory solution at sampling instants guarantees the global boundedness of the solutions for all time. The existence of a fixed point for the sampled state-trajectory solution at sampling instants guarantees the existence of a fixed point of an extended auxiliary discrete system and the existence of a global asymptotic attractor of the solutions which is either a fixed point or a limit "Equation missing" globally stable asymptotic oscillation.
Highlights
Stability of both continuous-time and discrete-time singularly perturbed dynamic systems has received much attention 1–5
An assumption used in previous work to carry out the stability analysis of singularly perturbed systems is relaxed in 1 where an upper-bound on the singular perturbation parameters is included to derive such an analysis
A usual example, very common in practice, is the case when a digital controller operates over a continuous-time plant to stabilize it or to improve its performance
Summary
Stability of both continuous-time and discrete-time singularly perturbed dynamic systems has received much attention 1–5. Fcc t, xc1 − fcc t, xc2 ≤ βcfc xc1 − xc2 ; 2.6 fcd t, xc1 − fcd t, xc2 ≤ βcfd xc1 − xc[2 ], fdc k, xd1 − fdc k, xd2 ≤ βdfc xd1 − xd2 ; 2.7 fdd k, xd1 − fdd k, xd2 ≤ βdfd xd1 − xd[2 ]; C4 gcc t, xc1 ≤ βcgcr; 2.8 gcd t, xd1 ≤ βcgdr, gdc t, xci ≤ βdgcr; 2.9 gdd t, xdi ≤ βdgdr; for all xci ≤ r, xdi ≤ r and all integer k ≥ 0 and all t ≥ 0, with h being any of the vector real functions or sequences of 2.3 and 2.4 and βh· h f or h g being known nonnegative real constants It turns out from Picard-Lindeloff theorem that the hybrid dynamic system. The state-trajectory of Σ at sampling instants is calculated in the following subsection
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