Impulsive Periodic System Research Articles

Distributed Filtering With Dynamic Event-Triggered Communication for Networked Periodic Systems

This article deals with the problem of distributed filtering of periodic systems over sensor networks under event-triggered communications, based on a piecewise impulsive method. The continuous-time periodic system is approximated by several linear time-invariant subsystems through a piecewise constant method. The sensor network deployed to the periodic system is taken into account for the actual needs of monitoring or estimation. A dynamic event-triggering mechanism is designed to reduce the unnecessary data transmission over the wireless network from the sensor to the filter as well as between each filter. Distributed peak-to-peak filters with periodic time-varying parameters are designed under the event-triggering mechanism to estimate the networked continuous-time periodic systems with the bounded and local differentiable disturbance. Then, a periodic piecewise filtering error system is modeled as an impulsive periodic piecewise system, for which sufficient conditions of a guaranteed peak-to-peak performance are proposed. The gains of the filters are obtained by the method of convex linearization. The validity of the proposed results is demonstrated by a pendulum tracking system.

Event‐based output feedback control of periodic systems: A piecewise impulsive method

AbstractIn this article, an event‐triggered control with periodic time‐varying parameters is designed to stabilize a class of periodic systems with exogenous disturbances in a network environment. The continuous‐time periodic system is approximated by some linear subsystems with finite‐number constant matrices based on a piecewise method. It can thus reduce the complexity of theoretical analysis. To tackle the reduction of unnecessary data transmission between the sensor (or controller) and the controller (or actuator), new event‐triggering mechanisms are designed via output feedback. Special parameters are designed to eliminate Zeno behavior, and the minimum triggering interval is obtained according to the designed condition. Then, an event‐triggered periodic piecewise system is modeled as an impulsive periodic piecewise system. The bounded stability, as well as the performance of the impulsive periodic piecewise system, are analyzed. Furthermore, the gains of the event‐triggered controller are obtained by a method of variable substitution. Finally, the validity of the proposed control is demonstrated by an application example.

Theoretical and numerical comparisons of looped functionals and clock-dependent Lyapunov functions-The case of periodic and pseudo-periodic systems with impulses

The stability of uncertain periodic and pseudo-periodic systems with impulses is analyzed in the looped-functional and clock-dependent Lyapunov function frameworks. These alternative and equivalent ways for characterizing discrete-time stability have the benefit of leading to stability conditions that are convex in the system matrices, hence suitable for robust stability analysis. These approaches, therefore, circumvent the problem of computing the monodromy matrix associated with the system, which is known to be a major difficulty when the system is uncertain. Convex stabilization conditions using a non-restrictive class of state-feedback controllers are also provided. The obtained results readily extend to uncertain impulsive periodic and pseudo-periodic systems, a generalization of periodic systems that admit changes in the ‘period’ from one pseudo-period to another. The obtained conditions are expressed as infinite-dimensional semidefinite programs, which can be solved using recent polynomial programming techniques. Several examples illustrate the approach, and comparative discussions between the different approaches are provided. A major result obtained in the paper is that despite being equivalent, the approach based on looped functional reduces to the one based on clock-dependent Lyapunov functions when a particular structure for the looped functional is considered. The conclusion is that the approach based on clock-dependent Lyapunov functions is preferable because of its lower computational complexity and its convenient structure enabling control design. Copyright © 2015 John Wiley & Sons, Ltd.

Existence results for mild solutions of impulsive periodic systems

By applying the Horn's fixed point theorem, we prove the existence of \(T_0\)-periodic \(PC\)-mild solution of impulsive periodic systems when \(PC\)-mild solutions are ultimate bounded.

A class of nonlinear integrodifferential impulsive periodic systems of mixed type and optimal controls on Banach space

A class of nonlinear integrodifferential impulsive periodic systems of mixed type on Banach space is considered. Existence of periodic PC-mild solutions is proved. Existence of periodic optimal pairs of systems governed by nonlinear impulsive integrodifferential equations of mixed type is also presented. An example is given for demonstration.

Periodic solutions of semilinear impulsive periodic system on Banach space

In this paper, a class of semilinear impulsive periodic systems on Banach space is considered. Using impulsive periodic evolution operator, the T 0 -periodic P C -mild solution is introduced and suitable Poincaré operator is constructed. Showing the compactness of the Poincaré operator and using a new generalized Gronwall inequality with mixed type integral operators, we utilize the Leray–Schauder fixed point theorem to prove the existence of the T 0 -periodic P C -mild solutions. Our method is much different from methods of other papers. An example illustrates the applicability of our results.

Periodic solutions of a class of integrodifferential impulsive periodic systems with time-varying generating operators on Banach space

Periodic solutions of a class of integrodifferential impulsive periodic systems with time-varying generating operators on Banach space

A MONOTONE-ITERATIVE METHOD FOR FINDING PERIODIC SOLUTION OF AN IMPULSIVE DIFFERENTIAL EQUATIONS AND APPLICATION

In this paper, using the method of upper and lower solutions and its associated monotone iterations, we establish a new monotone-iterative scheme for finding periodic solutions of an impulsive differential equations. This method leads to the existence of maximal and minimal periodic solutions which can be computed from a linear iteration process in the same fashion as for impulsive differential equations initial value problem. This method is constructive and can be used to develop a computational algorithm for numerical solution of the periodic impulsive system. Our existence result improves a result established in [1]. The result is applied to a model of mutualism of Lotka–Volterra type which involves interactions among a mutualist-competitor, a competitor and a mutualist, the existence of positive periodic solutions of the model is obtained.

Bounded and Periodic Solutions of Semilinear Impulsive Periodic System on Banach Spaces

AbstractA class of semilinear impulsive periodic system on Banach spaces is considered. First, we introduce the "Equation missing"-periodic PC-mild solution of semilinear impulsive periodic system. By virtue of Gronwall lemma with impulse, the estimate on the PC-mild solutions is derived. The continuity and compactness of the new constructed Poincaré operator determined by impulsive evolution operator corresponding to homogenous linear impulsive periodic system are shown. This allows us to apply Horn's fixed-point theorem to prove the existence of "Equation missing"-periodic PC-mild solutions when PC-mild solutions are ultimate bounded. This extends the study on periodic solutions of periodic system without impulse to periodic system with impulse on general Banach spaces. At last, an example is given for demonstration.

The Generalized Gronwall Inequality and Its Application to Periodic Solutions of Integrodifferential Impulsive Periodic System on Banach Space

This paper deals with a class of integrodifferential impulsive periodic systems on Banach space. Using impulsive periodic evolution operator given by us, the -periodic PC-mild solution is introduced and suitable Poincaré operator is constructed. Showing the compactness of Poincaré operator and using a new generalized Gronwall's inequality with impulse, mixed type integral operators and -norm given by us, we utilize Leray-Schauder fixed point theorem to prove the existence of -periodic PC-mild solutions. Our method is much different from methods of other papers. At last, an example is given for demonstration.

Existence of Periodic Solutions for Integrodifferential Impulsive Periodic System on Banach Space

This paper deals with a class of integrodifferential impulsive periodic systems on Banach space. Using impulsive periodic evolution operator given by us, theT0-periodicPC-mild solution is introduced and suitablePoincaréoperator is constructed. By virtue of the generalized new Gronwall lemma with impulse andB-norm, the estimate on thePC-mild solutions is derived. Showing the continuity and compactness of thePoincaréoperator, we utilize Horn's fixed point theorem to prove the existence ofT0-periodicPC-mild solutions when thePC-mild solutions are bounded and ultimate bounded. This extends the study of periodic solutions of integrodifferential periodic system without impulse to integrodifferential periodic system with impulse on general Banach spaces. At last, an example is given for demonstration.

Periodic Solutions of Semilinear Impulsive Periodic System with Time‐Varying Generating Operators on Banach Space

A class of semilinear impulsive periodic systems with time‐varying generating operators on Banach space is considered. Using impulsive periodic evolution operator given by us, the T0‐periodic PC‐mild solution is introduced and suitable Poincaré operator is constructed. Showing the compactness of Poincaré operator and using a new generalized Gronwall inequality with mixed type integral operators given by us, we utilize Leray‐Schauder fixed point theorem to prove the existence of T0‐periodic PC‐mild solutions. Our method is an innovation and it is much different from methods of other papers. At last, an example is given for demonstration.

Linear Impulsive Periodic System with Time-Varying Generating Operators on Banach Space

A class of the linear impulsive periodic system with time-varying generating operators on Banach space is considered. By constructing the impulsive evolution operator, the existence of -periodic -mild solution for homogeneous linear impulsive periodic system with time-varying generating operators is reduced to the existence of fixed point for a suitable operator. Further the alternative results on -periodic -mild solution for nonhomogeneous linear impulsive periodic system with time-varying generating operators are established and the relationship between the boundness of solution and the existence of -periodic -mild solution is shown. The impulsive periodic motion controllers that are robust to parameter drift are designed for a given periodic motion. An example given for demonstration.

Robust stability of impulsive periodic hybrid systems with dependent perturbations

This paper derives sufficient conditions for the stability of impulsive periodic hybrid systems with dependent perturbations. Bounds on the size of the polytope of uncertainties which still guarantees stability are derived.