State-dependent Switching Research Articles

Sliding mode control design for stabilization of underactuated mechanical systems

Stabilization of underactuated mechanical systems is one of the fundamental benchmark problem in the field of control theory. In this article, a hierarchical sliding mode control based on state-dependent switching gain is proposed for stabilization of underactuated mechanical systems. This controller is based on the so called first-level and second-level sliding surfaces. The asymptotic stability of these surfaces is proved by the Lyapunov stability theory. The proposed control technique is applied to two nonlinear underactuated mechanical systems and its feasibility is verified by numerical simulation. The proposed controller efficiently tackles the bounded external disturbance and shows robust performance. The convergence rate of the proposed controller is much faster as compared with the conventional decoupled sliding mode control and the integral sliding mode control.

Incremental passivity-based H∞ output tracking control for switched nonlinear systems

This paper is concerned with the problem of H∞ output tracking control for a class of switched nonlinear systems with external disturbances using incremental passivity, even if the problem of H∞ output tracking control for none of subsystems is solvable. First, an incremental passivity concept of switched nonlinear systems without external disturbances is proposed. This incremental passivity property requires each active subsystem is incrementally passive. Then, sufficient conditions to be incrementally passive are given. Second, by resorting to the established incremental passivity theory, a state-dependent switching law and a set of feedback controllers are designed to solve the problem of H∞ output tracking control for a class of switched nonlinear systems with external disturbances. This avoids solving Hamilton-Jacobi (HJ) inequality. Third, a composite state-dependent switching law and the state feedback controllers are designed to solve the H∞ output tracking problem for a class of cascaded switched nonlinear systems with external disturbances. The designed composite switching law allows the driven switched system and the driving switched system switch asynchronously. Finally, two examples are provided to verify the effectiveness of the proposed method.

Vector incremental L2-gain and incremental stability for switched nonlinear systems

This paper studies vector incremental L2-gain and incremental stability for switched nonlinear systems using individual incremental gains and multiple storage functions. Firstly, a vector incremental L2-gain concept for switched nonlinear systems is proposed. Each subsystem is not required to have incremental L2-gain, but it has its own incremental L2-gain and the related storage function, when it is active. The transformation of “energy” from the active subsystem to each inactive subsystem is characterized by cross-supply rates. Then, we show that a switched nonlinear system who has vector incremental L2-gain can be incrementally stabilized under some constraints on the energy change of inactive subsystems. Secondly, a state-dependent switching law is designed to achieve vector incremental L2-gain, even if each subsystem does not have incremental L2-gain in the classic sense. Thirdly, each switched system is not required to have vector incremental L2-gain, but the feedback interconnection of switched nonlinear systems is incrementally stabilized by the design of a composite switching law. The switching law allows the two switched systems switch asynchronously. Two examples are provided to verify the effectiveness of the proposed method.

State-dependent act-and-wait time-delayed feedback control algorithm

Time-delayed feedback control (TDFC) method is one of the most popular experimental tools used for stabilization of unstable periodic orbits in nonlinear dynamical systems. The theory of the method is difficult since systems controlled by time-delayed feedback evolve in an infinite-dimensional function space. Here we modify the TDFC by appending the feedback loop with a state-dependent switch. The switch closes and opens the feedback loop in the dependence of the phase of the output signal, which is measured by a specially designed phase detector. This modification represents a further development of the act-and-wait concept applied to time-delayed feedback controllers. The concept allows to reduce the phase space dimension of time-delayed feedback systems and to significantly simplify the theory. In previous works, the time-delayed feedback loop was tuned on and off periodically, independently of the state of the system. We discuss the advantages of our algorithm with respect to previously considered modifications. The efficacy of the proposed algorithm is numerically demonstrated with autonomous Rössler and Lorenz systems as well as a nonautonomous Duffing oscillator.

Adaptive Fault-Tolerant Control of Uncertain Switched Nonaffine Nonlinear Systems With Actuator Faults and Time Delays

This paper addresses the adaptive fault-tolerant control (FTC) problem of uncertain switched nonaffine nonlinear systems with actuator failures and time delays. Unlike the literature employing the FTC methods, the novel nonlinear fault compensate function with adjustable parameter factor, whose norm is used for approximation with fuzzy logic systems, is constructed based on the mean value theorem. Accordingly, a standard transformation scheme from the multi-input and single-output nonaffine nonlinear system to the corresponding affine nonlinear system is first to be given; then, the designed adaptation mechanism with state-dependent switching strategy can effectively eliminate the effects of multiple time delays and actuator faults, including outage and stuck modes. Furthermore, by using the Lyapunov–Krasovskii stability analysis method, it is proved that the proposed adaptive switched controller can guarantee the states of the overall closed-loop systems to asymptotically converge to zero. Finally, the simulation results are displayed to verify the feasibility of the presented approach.

Composite anti-disturbance model reference adaptive control for switched systems

This paper investigates the problem of the composite anti-disturbance model reference adaptive control for switched systems with parametric uncertainties and multiple types of disturbances. It is worth emphasizing that the measurability of the system state and the disturbance generated by an exosystem is unnecessary. A key point is to design a composite anti-disturbance model reference adaptive control strategy to achieve the state tracking and the anti-disturbance performance. First, a switched adaptive state-disturbance observer is designed to estimate the system state and the disturbance generated by the exosystem simultaneously. Secondly, based on the switched adaptive state-disturbance observer, a composite switched adaptive controller and a state-dependent switching law are designed to solve the composite anti-disturbance model reference adaptive control problem for switched systems. Thirdly, a sufficient condition ensuring the solvability of the composite anti-disturbance model reference adaptive control problem for switched systems is given, even if the problem is unsolvable for individual subsystems. Finally, an example of the electrohydraulic system is used to verify the availability of the acquired approach.

Model reference adaptive control for switched linear systems using switched multiple models control strategy

In this paper, the multiple model strategy is applied to the adaptive control of switched linear systems to improve the transient performance. The solvability of the adaptive stabilization problem of each subsystem is not required. Firstly, the two-layer switching mechanism is designed. The state-dependent switching law with dwell time constraint is designed in the outer-layer switching to guarantee the stability of the switched systems. During the interval of dwell time constraint, the parameter resetting adaptive laws are designed in the inner-layer switching to improve the transient performance. Secondly, the minimum dwell time constraint providing enough time for multiple model adaptive control strategy to work fully and maintaining the stability of the switched systems is found. Finally, the proposed switched multiple model adaptive control strategy guarantees that all the closed-loop system signals remain bounded and the state tracking error converges to zero.

Output tracking control of delayed switched systems via state-dependent switching and dynamic output feedback

This paper investigates the problem of output tracking control for a class of delayed switched linear systems via state-dependent switching and dynamic output feedback control. A state-dependent switching rule and the switching regions are proposed. Then the stability and tracking performance analysis are proposed based on single Lyapunov function technique, respectively. The design problem of dynamic output feedback controllers can be solved efficiently by using linear matrix inequalities (LMIs) toolbox. Finally, a simulation example is given to illustrate the effectiveness of the proposed method.

Stabilization of Positive Switched Linear Time-Varying Systems With Delay Under State-Dependent Switching

In this paper, we focus on the stabilization of positive switched linear time-varying systems with delay. By constructing Lyapunov-Metzler inequalities, state-dependent switching signals have been designed such that the system is asymptotically stable, even if each subsystem is not asymptotically stable. Both the continuous-time and the discrete-time positive switched linear time-varying systems have been taken into consideration. Numerical examples are also given to verify the validity of our results.

A Simple Framework for Stability Analysis of State-Dependent Networks of Heterogeneous Agents

Stability and analysis of multiagent network systems with state-dependent switching typologies have been a fundamental and longstanding challenge in control, social sciences, and many other related fields. These already complex systems become further complicated once one accounts for asymmetry or heterogeneity of the underlying agents/dynamics. Despite extensive progress in analysis of conventional networked decision systems where the network evolution and state dynamics are driven by independent or weakly coupled processes, most of the existing results fail to address multiagent systems where the network and state dynamics are highly coupled and evolve based on status of heterogeneous agents. Motivated by numerous applications of such dynamics in social sciences, in this paper we provide a new direction toward analysis of dynamic networks of heterogeneous agents under complex time-varying environments. As a result we show Lyapunov stability and convergence of several challenging problems from opinion dynamics using a simple application of our framework. In particular, we introduce a new class of asymmetric opinion dynamics, namely, nearest neighbor dynamics, and show how our framework can be used to analyze their behavior. Finally, we extend our results to game-theoretic settings and provide new insights toward analysis of complex networked multiagent systems using exciting field of sequential optimization.

Integral ISS for Switched Nonlinear Time-Varying Systems Using Indefinite Multiple Lyapunov Functions

This paper is concerned with studying Lyapunov characterization of integral input-to-state stability (iISS) for switched nonlinear time-varying systems. Sufficient conditions are given to verify iISS for switched nonlinear time-varying systems under a time-varying state-dependent switching law designed, which allow all subsystems to be not integral input-to-state stable (iISS) and the time derivative of Lyapunov functions of individual subsystems to be indefinite. An indefinite multiple Lyapunov functions (iMLFs) method for analyzing the dynamic behavior of switched nonlinear time-varying systems is provided. Also, an iMLFs-based small-gain theorem for switched interconnected nonlinear time-varying systems is presented, where each lower dimensional subsystem is allowed to be not iISS, which extends the small-gain technique from its original nonswitched nonlinear time-invariant version to a switched nonlinear time-varying version. Finally, an illustrative example is used to demonstrate the feasibility of the theoretical results.

Stabilization of Regime-Switching Processes by Feedback Control Based on Discrete Time Observations II: State-Dependent Case

This work investigates the almost sure stabilization of a class of regime-switching systems based on discrete-time observations of both continuous and discrete components. It develops Shao's work [SIAM J. Control Optim., 55(2017), pp. 724--740] in two aspects: first, to provide sufficient conditions for almost sure stability in lieu of moment stability; second, to investigate a class of state-dependent regime-switching processes instead of state-independent ones. To realize these developments, we establish an estimation of the exponential functional of Markov chains based on the spectral theory of linear operator. Moreover, through constructing order-preserving coupling processes based on Skorokhod's representation of jumping process, we realize the control from up and below of the evolution of state-dependent switching process by state-independent Markov chains.

Disturbance Attenuation and Rejection for Nonlinear Uncertain Switched Systems Subject to Input Saturation

This paper studies the problem of disturbance attenuation and rejection for switched systems with nonlinear uncertainty and input saturation via composite anti-disturbance control technique, in which the exosystem-generated disturbance and H 2 -norm-bounded disturbance are considered. For switched systems, the switching law and input saturation increase the difficulty for the design of the disturbance observer and the composite control scheme. A switching disturbance observer is designed to obtain the estimation of the matched disturbance, and a novel switching composite controller is further constructed based on the estimated value. By proposing a state-dependent switching law and utilizing the multiple Lyapunov function technology, the criteria are presented to ensure the local asymptotic stability with an H ∞ performance level for the closed-loop system. Furthermore, two optimal algorithms of the design of the controller are put forward to maximize the estimation of the domain of attraction of the closed-loop and the upper bound on the H 2 -norm of the disturbance, respectively. The effectiveness of the proposed technique is illustrated via the numerical examples.

Modeling left ventricular dynamics using a switched system approach based on a modified atrioventricular piston unit

The contribution of the longitudinal atrioventricular plane displacement to ventricular pumping has drawn more and more attentions. In this paper, differential equations of the left ventricle (LV) are derived via the atrioventricular piston concept. The contribution of left ventricular radial function to blood flow was converted to an equivalent coefficient. A systemic circulatory model incorporating the modified atrioventricular piston unit was developed on a switched system form by adding some state-dependent switching planes. Simulation results prove that the end-systolic pressure volume relationship of the model with a changing systemic arterial resistance is approximately linear and insensitive to perturbations in afterload. Then the LV model was validated using a data fitting method. A pressure–volume loop from a patient undergoing routine diagnostic cardiac catheterization with LV angiography was used as measurements. Model parameters and the trapezoidal profile of contraction forces were adjusted by a trial method. The root mean squared error between the measured and estimated LV pressure is 2.99 mmHg. The LV compliance is 0.34 ml/mmHg. The ratio between left ventricular and left atrial cross-section is 1.8. Therefore, parameter values used in the modified LV model match physiological data. The model can reproduce the realistic pressure-flow relationship in the LV chamber.

Adaptive Fuzzy Tracking Control for a Class of Switched Uncertain Nonlinear Systems: An Adaptive State-Dependent Switching Law Method

This paper is concerned with the problem of adaptive fuzzy tracking control for a class of switched uncertain nonlinear systems. The unknown nonlinear functions are approximated by using fuzzy logic systems, and the adaptive backstepping technique is applied to a convex combination of the subsystems. To ensure the continuousness of the Lyapunov function at switching instants, common Lyapunov function (CLF) technique is adopted and the proposed CLF permits switched parameter adaptive laws. Besides, two classes of adaptive laws are proposed, one for designing control laws and one for designing switching law. Then, the common control laws are obtained with the help of the convex combination. Based on the switched adaptive laws, common control laws, a new adaptive state-dependent switching law is proposed. It is shown that the designed state-feedback controllers and switching law can ensure that all the signals remain bounded and the tracking error converges to a small neighborhood of the origin. Moreover, a modified switching law is proposed to avoid Zeno behavior or switching too quickly. Finally, two examples are provided to show the effectiveness of the proposed approaches.

Multistability of Switched Neural Networks With Piecewise Linear Activation Functions Under State-Dependent Switching.

This paper is concerned with the multistability of switched neural networks with piecewise linear activation functions under state-dependent switching. Under some reasonable assumptions on the switching threshold and activation functions, by using the state-space decomposition method, contraction mapping theorem, and strictly diagonally dominant matrix theory, we can characterize the number of equilibria as well as analyze the stability/instability of the equilibria. More interesting, we can find that the switching threshold plays an important role for stable equilibria in the unsaturation regions of activation functions, and the number of stable equilibria of an n -neuron switched neural network with state-dependent parameters increases to 3n from 2n in the conventional one. Furthermore, for two-neuron switched neural networks, the precise attraction basin of each stable equilibrium point can be figured out, and its boundary is composed of the stable manifolds of unstable equilibrium points and the switching lines. Two simulation examples are discussed in detail to substantiate the effectiveness of the theoretical analysis.

On the importance of the coexistence of time and state-dependent switching

Switching systems have many applications in various fields of physics and engineering. Almost all the switching systems contain either the state-dependent switching or the time-dependent switching. However systems with the coexistence of time and state-dependent switching are very rare in the literature, as we are not aware of the importance and applications of such systems. In this short communication, we focus on the importance of such switched systems, where state and time-dependent switching coexist. In this article, we answer an important open question related to the bifurcation theory of dynamical systems. We show that all kind of periodic transitions are possible due to the border collision bifurcation in this type of switching systems. We have also derived the explicit mathematical expressions of the periodic attractors, existing before and after the border collision bifurcation. Besides, we present some shreds of evidence to show that the complexity of the human behaviour and human intelligence can sometimes be modelled nicely by combination of the state-dependent and the time-dependent switching.

Quadratic Performance Analysis of Switched Affine Time–Varying Systems

Abstract We analyze quadratic performance for switched systems which are composed of a finite set of affine time-varying subsystems, where both subsystem matrices and affine vectors are switched, and no single subsystem has desired quadratic performance. The quadratic performance indexes we deal with include stability, tracking and L2 gain. We show that if a linear convex combination of subsystem matrices is uniformly Hurwitz and another convex combination of affine vectors is zero, then we can design a state-dependent switching law (state feedback) and an output-dependent switching law (output feedback) such that the entire switched affine system is quadratically stable at the origin. In the case where the convex combination of affine vectors is nonzero, we show that the tracking control problem can be posed and solved using a similar switching strategy. Finally, we consider the L2gain analysis problem for the switched affine time-varying systems under state feedback.

Incremental H∞ performance for a class of stochastic switched nonlinear systems

In this paper, we investigate the incremental H∞ performance problem for a class of stochastic switched nonlinear systems by using a state-dependent switching law and the maximum and minimum dwell time approach. By resorting to the state-dependent switching law, some sufficient conditions are provided to cope with the incremental H∞ performance problem, which can be applied even if all subsystems are unstable. Then, based on the maximum and minimum dwell time scheme, the incremental H∞ performance problem to be solvable is derived for two cases: one is all subsystems are incrementally globally asymptotically stable in the mean(IGASiM), another is both IGASiM subsystems and unstable subsystems coexist. When all subsystems are IGASiM, the stochastic switched nonlinear system is IGASiM and possesses a incremental L2-gain under given conditions. When both IGASiM subsystems and unstable subsystems coexist, if the activation time ratio between IGASiM subsystems and unstable ones is not less than a specified constant, the sufficient conditions for the incremental H∞ performance of the stochastic switched nonlinear system are given. Two numerical examples are given to illustrate the validity of methods proposed.

Exponentially incremental dissipativity for nonlinear stochastic switched systems

ABSTRACTIn this paper, we investigate the exponentially incremental dissipativity for nonlinear stochastic switched systems by using the designed state-dependent switching law and multiple Lyapunov functions approach. Specifically, using incremental supply rate as well as a state dissipation inequality in expectation, a stochastic version of exponentially incremental dissipativity is presented. The sufficient conditions for nonlinear stochastic switched systems to be exponentially incrementally dissipative are given by the designed state-dependent switching law. Furthermore, the extended Kalman–Yakubovich–Popov conditions are derived by using two times continuously differentiable storage functions. Moreover, the incremental stability conditions in probability for nonlinear stochastic switched systems are derived based on exponentially incremental dissipativity. The exponentially incremental dissipativity is preserved for the feedback-interconnected nonlinear stochastic switched systems with the composite state-dependent switching law; meanwhile, the incremental stability in probability is preserved under some certain conditions. A numerical example is given to illustrate the validity of our results.