In this article, for Itô stochastic systems with aperiodic sampling and multiple time-delays, we provide two different types of mean square exponential stability analysis methods. One method is based on the Razumikhin-type theorems method, based on this method, the mean square exponential stability criteria of the system are proposed. Meanwhile, mean square exponential practical stability criteria are also proposed. In addition, through these theorems, the constraint of positivity on the Lyapunov function for all times can be relaxed to sampling times for Itô stochastic systems with aperiodic sampling and multiple time-delays is proved. Another method is based on the looped-functionals method. Two looped-functionals are introduced to analyze the stability of Itô stochastic systems with aperiodic sampling and multiple time-delays. One functional is one-sided, while the other functional is two-sided. Based on these two functionals and the Itô formula, two mean square exponential stability criteria are presented in the form of linear matrix inequalities. Numerical examples are used to illustrate that the proposed approach is less conservative than other methods.

In this article, a Bernstein polynomial approach is first applied to the estimation of reachable set for a class of periodic piecewise polynomial systems, whose subsystems are time-varying and can be expanded to Bernstein polynomial forms. A lemma on the negativity/positivity for a class of Bernstein polynomial matrix functions is presented, which can provide a feasible set larger than that by the existing method. Based on the integration of the presented lemma and the theory of matrix polynomials, two tractable sufficient conditions are developed. For comparison of conservatism, the reachable set estimation is achieved through optimizing the ellipsoidal bounding region. Four sets of constraints with different conservatism are derived and compared. The effectiveness and superiority of the Bernstein polynomial approach in reachable set estimation are demonstrated via an illustrative example. The results show that the proposed approach enables lower conservatism in reachable set estimation, providing an intuitive route to tackle time-varying parameter products with high powers.

This article is focused on the design of an output tracking control scheme for a class of continuous-time periodic piecewise time-varying systems (PPTVSs) with actuator saturation and nonlinear perturbations. The energy-to-peak tracking performance is studied based on an equivalent condition on the definiteness property of matrix polynomials. Considering the actuator saturation and nonlinear perturbation, matrix polynomial-based sufficient conditions are derived through the Lyapunov method using periodic matrix functions. From a perspective of subinterval segmentation aimed at PPTVSs, the proposed conditions can achieve less conservatism for tracking the output of a periodic time-varying reference system, while the controller gains can be computed using convex optimization. Moreover, a heuristic algorithm is constructed to simultaneously guarantee the closed-loop state convergence and the output tracking performance. The reduction in conservatism and the effectiveness of algorithm are demonstrated by illustrative case studies.

In this paper, the robust peak-to-peak filtering problem for periodic piecewise systems with polytopic uncertainties is investigated. Attention is focused on designing of robust peak-to-peak filters that guarantee the robust asymptotic stability of periodic piecewise filtering error system and satisfy a prescribed peak-to-peak disturbance attenuation level for all admissible uncertainties. A sufficient condition is proposed for robust stability and peak-to-peak performance by employing the constructed time-varying Lyapunov function. Two design approaches based on the Lyapunov function with parameter-dependent matrices and parameter-independent matrices are utilised to solve this problem. An algorithm is proposed to compute the periodic filter parameters governed by non-convex feasibility conditions. Two numerical examples are presented to demonstrate the effectiveness of the proposed techniques.

This article deals with the problem of fault detection observer design for continuous-time periodic piecewise linear systems. Mixed $H_{-}/H_{\mathrm{\infty }}$ performance index is introduced, which results in a trade-off between robustness to unknown input and sensitivity to a fault. Based on a class of Lyapunov functions with time-varying Lyapunov matrices, sufficient conditions are developed to investigate the existence of periodic piecewise fault detection observer with weighted $H_{-}/H_{\mathrm{\infty }}$ performance. A tractable iterative algorithm is proposed to establish the desired periodic piecewise fault detection observer. Finally, a numerical example is provided to illustrate the effectiveness of the proposed approach.