Asier Ibeas

Exponential stability of hybrid switched nonlinear singular systems with time-varying delay

Abstract We address exponential stability of switched nonlinear singular systems with time-delay in which delay is time varying and presents in the states. For switched nonlinear singular time-delay systems with average dwell-time switching signals, we provide sufficient conditions, in terms of linear matrix inequalities (LMIs) to guarantee the exponential stability of such systems. By using Lyapunov–like Krasovskii approach, the relationship between the average dwell-time of the switched nonlinear singular time-delay system and the exponential decay rate of differential and algebraic states is given. A numerical example is also included to illustrate the effectiveness of the results proposed in this paper.

Stability analysis of hybrid switched nonlinear singular time-delay systems with stable and unstable subsystems

The issue of exponential stability of a class of continuous-time switched nonlinear singular systems consisting of a family of stable and unstable subsystems with time-varying delay is considered in this paper. Based on the free-weighting matrix approach, the average dwell-time approach and by constructing a Lyapunov-like Krasovskii functional, delay-dependent sufficient conditions are derived and formulated to check the exponential stability of such systems in terms of linear matrix inequalities (LMIs). By checking the corresponding LMI conditions, the average dwell-time and switching signal conditions are obtained. This paper also highlights the relationship between the average dwell-time of the switched nonlinear singular time-delay system, its stability and the exponential convergence rate of differential and algebraic states. A numerical example shows the effectiveness of the proposed method.

Exponential stability of simultaneously triangularizable switched systems with explicit calculation of a common Lyapunov function

In this note, a common quadratic Lyapunov function is explicitly calculated for a linear hybrid system described by a family of simultaneously triangularizable matrices. The explicit construction of such a function allows not only obtaining an estimate of the convergence rate of the exponential stability of the switched system under arbitrary switching but also calculating an upper bound for the output during its transient response. Furthermore, the presented result is then extended to the case where the system is affected by parametric uncertainty, providing the corresponding results in terms of the nominal matrices and uncertainty bounds.

Observer-based adaptive neural network control for a class of MIMO uncertain nonlinear time-delay non-integer-order systems with asymmetric actuator saturation

Abstract This paper focuses on the adaptive neural output feedback control of a class of uncertain multi-input–multi-output nonlinear time-delay non-integer-order systems with unmeasured states, unknown control direction, and unknown asymmetric saturation actuator. Thus, the mean value theorem and a Gaussian error function-based continuous differentiable model are used in the paper to describe the unknown asymmetric saturation actuator and to get an affine model in which the control input appears in a linear fashion, respectively. The design of the controller follows a number of steps. Firstly, based on the semigroup property of fractional-order derivative, the system is transformed into a normalized fractional-order system by means of a state transformation in order to facilitate the control design. Then, a simple linear state observer is constructed to estimate the unmeasured states of the transformed system. A neural network is incorporated to approximate the unknown nonlinear functions while a Nussbaum function is used to deal with the unknown control direction. In addition, the strictly positive real condition, the Razumikhin Lemma, the frequency-distributed model, and the Lyapunov method are utilized to derive the parameter adaptive laws and to perform the stability proof. The main advantages of this work are that: (1) it can handle systems with constant, time-varying, and distributed time-varying delays, (2) the considered class of systems is relatively large, (3) the number of adjustable parameters is reduced, (4) the tracking errors converge asymptotically to zero and all signals of the closed-loop system are bounded. Finally, some simulation examples are provided to demonstrate the validity and effectiveness of the proposed scheme.

On the stability of an SEIR epidemic model with distributed time-delay and a general class of feedback vaccination rules

This paper discusses and formulates a continuous-time SEIR -type epidemic model of pseudo-mass action type with finitely distributed delays under a very general, potentially time-varying, vaccination control rule which eventually generates feedback actions on the susceptible, infectious and recovered subpopulations. A lot of particular vaccination laws can be got from the proposed general one. The disease-free and endemic equilibrium points are characterized and their local stability properties discussed depending on the limits of the vaccination control gains provided that they converge asymptotically. Then, the global asymptotic stability to the disease-free equilibrium point is studied under an infective transmission rate below a certain maximum threshold. Later on, an extended SEIR epidemic model is discussed through simulated examples with stochastic Wiener-type perturbations around the equilibrium points.

An observer-based vaccination control law for an SEIR epidemic model based on feedback linearization techniques for nonlinear systems

This paper presents a vaccination strategy for fighting against the propagation of epidemic diseases. The disease propagation is described by an SEIR (susceptible plus infected plus infectious plus removed populations) epidemic model. The model takes into account the total population amounts as a refrain for the illness transmission since its increase makes the contacts among susceptible and infected more difficult. The vaccination strategy is based on a continuous-time nonlinear control law synthesised via an exact feedback input-output linearization approach. An observer is incorporated into the control scheme to provide online estimates for the susceptible and infected populations in the case when their values are not available from online measurement but they are necessary to implement the control law. The vaccination control is generated based on the information provided by the observer. The control objective is to asymptotically eradicate the infection from the population so that the removed-by-immunity population asymptotically tracks the whole one without precise knowledge of the partial populations. The model positivity, the eradication of the infection under feedback vaccination laws and the stability properties as well as the asymptotic convergence of the estimation errors to zero as time tends to infinity are investigated.

Neural adaptive quantized output-feedback control-based synchronization of uncertain time-delay incommensurate fractional-order chaotic systems with input nonlinearities

This research is concerned with the problem of generalized function projective synchronization of nonlinear uncertain time-delay incommensurate fractional-order chaotic systems with input nonlinearities. The considered problem is challenging owing to the presence of unmeasured master-slave system states, external dynamical disturbances, unknown nonlinear system functions, unknown time-varying delays, quantized outputs, unknown control directions unknown actuator nonlinearities (backlash-like hysteresis, dead-zone and asymmetric saturation actuators) and distinct fractional-orders. Under some mild assumptions and using Caputos definitions for fractional-order integrals and derivatives, the design procedure of the proposed neural adaptive controller consists of a number of steps to solve the generalized function projective synchronization problem. First, smooth functions and the mean value theorem are utilized to overcome the difficulties from actuator nonlinearities and distributed time-varying delays, respectively. Then, a simple linear observer is established to estimate the unknown synchronization error variables. In addition, a Nussbaum function is incorporated to cope with the unknown control direction and a neural network is adopted to tackle the unknown nonlinear functions. The combination of the frequency distributed model, the Razumikhin Lemma, the neural network parameterization, the Lyapunov method and the Barbalats lemma is employed to perform the stability proof of the closed-loop system and to derive the adaption laws. The major advantages of this research are that: (1) the Strictly Positive Real (SPR) condition on the estimation error dynamics is not required, (2) the considered class of master-slave systems is relatively large, (3) all signals in the resulting closed-loop systems are semi-globally uniformly ultimately bounded and the synchronization errors semi-globally converge to zero. Finally, numerical examples are presented to illustrate the performance of the proposed synchronization scheme.

Robust Sliding Control of Robotic Manipulators Based on a Heuristic Modification of the Sliding Gain

A task space robust trajectory tracking control is developed for robotic manipulators. A second order linear model, which defines the desired impedance for the robot, is used to generate the reference position, velocity and acceleration trajectories under the influence of an external force. The control objective is to make the robotic manipulator’s end effector track the reference trajectories in the task space. A sliding mode based robust control is used to deal with system uncertainties and external perturbations. Thus, a sliding manifold is defined by a linear combination of the tracking errors of the system in the task space built from the difference between the real and the desired position, velocity and acceleration trajectories in comparison with previous works where the sliding manifold was defined by the desired impedance and the external force. Moreover, the ideal relay has been substituted by a relay with a dead-zone in order to fit in with the actual way in which a real computational device implements the typical sign function in sliding mode control. Furthermore, a higher level supervision algorithm is proposed in order to reduce the amplitude of the high frequency components of the output associated to an overestimation of the system uncertainty bounds. Then, the robust control law is applied to the case of a robot with parametric uncertainty and unmodeled dynamics. The closed-loop system is proved to be robustly stable with all signals bounded for all time while the control objective is fulfilled in practice. Finally, a simulation example which shows the usefulness of the proposed scheme is presented.

A Tunnel-diode Trigger Circuit Using a Regulation Multimodel Scheme

A multimodel scheme is designed for a tunnel-diode circuit with the aim of improving the transient response in a transition or switching from a system equilibrium point to another one. Each model of the circuit is obtained by a linearization of the circuit near an equilibrium point. Moreover, different discretizations for each linear model are considered. A discrete-time controller is parameterized from each of such linearized models in order to match a reference model. The scheme selects online the linear model of the circuit with the best tracking performance to generate the control law. The set of possible linear models of the circuit is updated online during the switching process according to the state of the circuit at some prefixed instants so that a tracking performance index is supervised. A Butterworth second order filter is used as reference transfer function in the simulation

Results on proximal and generalized weak proximal contractions including the case of iteration-dependent range sets

This paper presents some further results on proximal and asymptotic proximal contractions and on a class of generalized weak proximal contractions in metric spaces. The generalizations are stated for non-self-mappings of the forms Tn:An→Bn for n∈Z0+ and T:⋃j∈Z0+A0j→⋃j∈Z0+B0j, or T:A→(⋃Bn), subject to T(A0n)⊆B0n and Tn(An)⊆Bn, such that Tn converges uniformly to T, and the distances Dn=d(An,Bn) are iteration-dependent, where A0n, An, B0n and Bn are non-empty subsets of X, for n∈Z0+, where (X,d) is a metric space, provided that the set-theoretic limit of the sequences of closed sets {An} and {Bn} exist as n→∞ and that the countable infinite unions of the closed sets are closed. The convergence of the sequences in the domain and the image sets of the non-self-mapping, as well as the existence and uniqueness of the best proximity points, are also investigated if the metric space is complete. Two application examples are also given, being concerned, respectively, with the solutions through pseudo-inverses of both compatible and incompatible linear algebraic systems and with the parametrical identification of dynamic systems.