This paper is concerned with the stability and L 1 -gain analysis of periodic piecewise positive systems with constant time delay. λ -exponential stability, which is applied to characterize the decay rates of the considered systems, is investigated first. A co-positive Lyapunov-Krasovskii functional is used to obtain a sufficient stability condition. The stability condition characterizes the convergent speed of the state by the system matrices and the size of the time delay. One can also apply the Lyapunov-Krasovskii functional to characterize the L 1 -gain of the systems. By taking advantage of the periodic property of the system, linear inequalities are employed to characterize the L 1 -gain, and an unweighted upper bound of the L 1 -gain of the system is given.Index Terms-L 1 -gain analysis, periodic systems, positive systems, stability analysis, time-delay systems.
Summary
This article presents a descriptor observer design approach for positive Markov jump linear systems subject to interval parameter uncertainties and sensor faults. First, by taking the sensor fault term as an auxiliary state, an augmented descriptor system is constructed. A pair of positive observers with state‐bounding feature is then proposed, which enables simultaneous estimation of the system state and sensor faults. A necessary and sufficient condition on existence of the desired state‐bounding observer is derived by considering positivity and robust mean exponential stability of corresponding observer error dynamics. An iterative optimization algorithm is developed for the computation of the optimized observer matrices. Finally, a numerical example is presented to show the validity of the proposed methods.
This paper investigates the input‐output gains, including ℓ1‐ and ℓ∞‐gains and L1‐ and L∞‐gains, of discrete‐time and continuous‐time positive periodic systems. For the discrete‐time case, the input–output gain can be characterized by linear inequalities. For the continuous‐time case, the input–output gain characterization problem turns into the existence problem of a positive periodic vector function. Based on some necessary and sufficient input–output gain conditions, we find the ℓ1‐ (L1‐) gain of discrete‐time (continuous‐time) positive periodic systems is equivalent to the of ℓ∞‐ (L∞‐) gain of the associated dual systems. Finally, two numerical examples are given to illustrate the results.
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