State Impulses Research Articles

Stability of multi-dimensional switched systems with an application to open multi-agent systems

A multi-dimensional switched system or multi-mode multi-dimensional (M3D) system extends the classic switched system by allowing different subsystem dimensions. The stability problem of the M3D system, whose state transitions at switching instants can be discontinuous due to the dimension-varying feature, is studied. The discontinuous state transition is formulated by an affine map that captures both the dimension variations and the state impulses, with no extra constraint imposed. In the presence of unstable subsystems, the general criteria featuring a series of Lyapunov-like conditions for the practical and asymptotic stability properties of the M3D system are provided under the proposed slow/fast transition-dependent average dwell time framework. Then, by considering linear subsystems, we propose a class of parametric multiple Lyapunov functions to verify the obtained Lyapunov-like stability conditions and explicitly reveal a connection between the practical/asymptotic stability property and the non-vanishing/vanishing property of the impulsive effects in the state transition process. Further, the obtained stability results for the M3D system are applied to the consensus problem of the open multi-agent system (MAS), whose network topology can be switching and size-varying due to the migrations of agents. It shows that through a proper transformation, the seeking of the (practical) consensus performance of the open MAS with disconnected digraphs boils down to that of the (practical) stability property of an M3D system with unstable subsystems.

Doing Boundary Work: Suburban Residents’ Associations in Johannesburg

Despite best hopes of social and urban transformation, Johannesburg’s middle-class suburbs have remained largely inaccessible to lower income and more marginalised communities. This article examines the everyday practices and repertoires of action by resident associations in Johannesburg, demonstrating their ability to moderate more progressive state impulses and other land use changes. It argues that resident associations have become the custodians of middle-class visions and aesthetics and carry out the boundary work and symbolic violence that maintains and defends suburban borders.

Nonlinear Parametric Estimation of Hamiltonian Systems: Identification as Stabilization

The paper presents two numerical procedures in the continuous time for the identification of parameters of a class of Hamiltonian systems. Employing the definition of First Integrals and their characteristics, the suggested approach permits to treat the parametric identification process as a stabilization of the derivative of the first integrals. It is realized by two proposed numerical procedures, supported by an super-twist differentiator to have online estimates of the derivatives of the generalized state coordinates and impulses. The convergence of these identification procedures and their implementation in a scalar and vector cases are presented. The numerical examples illustrate a good workability of the suggested method.

Practical output synchronization for asynchronously switched multi-agent systems with adaption to fast-switching perturbations

The asynchronously switched multi-agent systems comprising switched agents of different dynamics and switching signals are considered under arbitrarily switching communication topologies. The practical output synchronization problem is studied for such a kind of systems due to the heterogeneity brought by both the dynamics and the switchings of agents. A switching-dependent controller with an embedded virtual reference system is proposed for each agent. The original problem is then converted into tracking problems between each agent and its reference system. The analysis of resultant tracking error systems involves the analysis of switched systems with bounded but non-attenuating state impulses. By satisfying sufficient conditions featuring the average dwell time (ADT) and the newly proposed piecewise ADT, the practical output synchronization can be achieved and the ultimate bound of the output errors can also be obtained for the considered systems. Furthermore, a realistic case where the agent switching signals undergo adverse fast-switching perturbations is studied. The perturbations may potentially invalidate the “slow-switching” based method. A regulation strategy is thus developed for each agent to render it adaption to such adversity. A payload transport task is taken as the practical example to illustrate the effectiveness of the proposed method and the adaption strategy.

Periodic Orbits Analysis in a Class of Planar Liénard Systems with State-Triggered Jumps

In this paper, we investigate the existence problem of periodic orbits for a planar Liénard system, whose solution mappings are interrupted by abrupt changes of state. We first present the geometrical properties of solutions for the planar Liénard system with state impulses, then by using Bendixson theorem of impulsive differential equations and successor function method, several new criteria on the closed orbits and discontinuous periodic orbits are established in the impulsive Liénard system.

On the existence of discontinuous periodic solutions for a class of Liénard systems with impulses

In this paper, we investigate the existence of discontinuous periodic solutions for a class of Liénard systems with state impulses, whose solutions are interrupted by abrupt changes of state. By employing Poincaré mapping method, a criterion for the existence of at least one discontinuous periodic solution in the impulsive Liénard systems is established. Finally, we give an example to illustrate the effectiveness of our result, and the corresponding numerical simulation is presented to show that there is a good agreement between our theoretical result with the computer numerical analysis.

Observer design for a class of uncertain nonlinear systems with sampled outputs—Application to the estimation of kinetic rates in bioreactors

A continuous–discrete time observer is proposed for a class of uncertain nonlinear systems where the output is available only at non uniformly spaced sampling instants. The underlying correction term depends on the output observation error and is updated in a mixed continuous-discrete fashion. The proposed observer is first introduced under a set of differential equations with instantaneous state impulses corresponding to the measured samples and their estimates. Two features of the proposed observer are worth to be pointed out. The first one consists in the simplicity of its calibration while the second one lies in its comprehensive convergence analysis. More specifically, it is shown that in the case of noise-free sampled outputs, the observation error lies in a ball centered at the origin and its radius is proportional to the bounds of the uncertainties and the sampling partition diameter. Moreover, in the free uncertainties case, the exponential convergence to zero of the observation error is established under a well-defined condition on the maximum value of the sampling partition diameter. The ability of the proposed observer to perform a suitable estimation of the reactions rates in biochemical reactors is highlighted through a simulation study dealing with an ethanolic fermentation.

Adaptive observers for a class of uniformly observable systems with nonlinear parametrization and sampled outputs

In this paper, we propose an adaptive observer for a class of uniformly observable nonlinear systems with nonlinear parametrization and sampled outputs. A high gain adaptive observer is first designed under the assumption that the output is continuously measured and its exponential convergence is investigated, thanks to a well defined persistent excitation condition. Then, we address the case where the output is available only at (non uniformly spaced) sampling instants. To this end, the continuous-time output observer is redesigned leading to an impulsive observer with a corrective term involving instantaneous state impulses corresponding to the measured samples and their estimates. Moreover, it is shown that the proposed impulsive observer can be put under the form of a hybrid system composed of a continuous-time observer coupled with an inter-sample output predictor. Two design features are worth to be emphasized. Firstly, the observer calibration is achieved through the tuning of a scalar design parameter. Secondly, the exponential convergence to zero of the observation and parameter estimation errors is established under a well defined condition on the maximum value of the sampling partition diameter. More specifically, the observer design is firstly carried out in the case of linear parametrization before being extended to the nonlinear one. The theoretical results are corroborated through simulation results involving a typical bioreactor.

Modeling and qualitative analysis of diabetes therapies with state feedback control

For the therapies of diabetes mellitus, a novel mathematical model with two state impulses: impulsive injection of insulin and impulsive injection of glucagon, is proposed. To avoid hypoglycemia and hyperglycemia, the injections of insulin and glucagon are determined by closely monitoring the plasma glucose level of the patients. By using differential equation geometry theory, the existence of periodic solution and the attraction region of the system have been obtained, which ensures that injections in such an automated way can keep the blood glucose concentration under control. The simulation results verify that the better insulin injection strategy in closed-loop control is a larger dose but longer interval rather than a smaller dose but shorter interval. Besides, our numerical analysis reveals that medicine studies and practice that slow down the insulin degradation are helpful for the plasma glucose control. Our findings can provide significant guidance in both design of artificial pancreas and clinical treatment.

Periodic solution of a pest management Gompertz model with impulsive state feedback control

In this paper, a new model with two state impulses is proposed for pest management. According to different thresholds, an integrated strategy of pest management is considered, that is to say if the density of the pest population reaches the lower threshold \(h_1\) at which pests cause slight damage to the forest, biological control (releasing natural enemy) will be taken to control pests; while if the density of the pest population reaches the higher threshold \(h_2\) at which pests cause serious damage to the forest, both chemical control (spraying pesticide) and biological control (releasing natural enemy) will be taken at the same time. For the model, firstly, we qualitatively analyse its singularity. Then, we investigate the existence of periodic solution by successor functions and Poincare-Bendixson theorem and the stability of periodic solution by the stability theorem for periodic solutions of impulsive differential equations. Lastly, we use numerical simulations to illustrate our theoretical results.

Continuous-Discrete Time Observers for a Class of MIMO Nonlinear Systems

This technical note addresses the observer design problem for a class of continuous-time dynamical systems with non-uniformly sampled measurements. More specifically, we propose an observer that runs in continuous-time with an output error correction term that is updated in a mixed continuous-discrete fashion. The proposed observer is actually an impulsive system since it is described by a set of differential equations with instantaneous state impulses corresponding to the measured samples and their estimates. Nevertheless, we shall show that such an impulsive system can be put under the form of a hybrid system composed of a continuous-time high gain observer coupled with an inter-sample output predictor. Two design features of the proposed observer are worth emphasizing. Firstly, the observer calibration is achieved through the tuning of a scalar design parameter. Secondly, the exponential convergence to zero of the observation error is established under a well-defined condition on the maximum value of the sampling partition diameter. Simulations results involving a flexible joint robot arm are given in order to highlight the performance of the proposed observer.

Geometric Analysis of an Integrated Pest Management Model Including Two State Impulses

According to integrated pest management strategies, we construct and investigate the dynamics of a Holling-Tanner predator-prey system with state dependent impulsive effects by releasing natural enemies and spraying pesticide at different thresholds. Applying the Dulacs criterion, the global stability of the positive equilibrium in the system without impulsive effect is discussed. By using impulsive differential equation geometry theory and the method of successor functions, we prove the existence of periodic solution of the system with state dependent impulsive effects. Furthermore, the stability conditions of periodic solutions are obtained. Some simulations are exerted to illustrate the feasibility of our main results.

Controllability Criteria for Linear Fractional Differential Systems with State Delay and Impulses

This paper is concerned with the controllability of linear fractional differential systems with delay in state and impulses. The factors of such systems including fractional derivative, impulses, and delay are taken into account synchronously. The expression of state response for such systems is derived, and the sufficient and necessary conditions of controllability criteria are established. Both the proposed criteria and illustrative examples show that the controllability property of the linear systems is dependent neither on the order of fractional derivative, on delay nor on impulses.

The geometrical analysis of a predator–prey model with two state impulses

Using successor functions and Poincaré–Bendixson theorem of impulsive differential equations, the existence of periodical solutions to a predator–prey model with two state impulses is investigated. By stability theorem of periodic solution to impulsive differential equations, the stability conditions of periodic solutions to the system are given. Some simulations are exerted to prove the results.