Unstable Subsystems Research Articles

Stability of stochastic delay switched neural networks with all unstable subsystems: A multiple discretized Lyapunov-Krasovskii functionals method

This paper considers switching stochastic delay neural networks (SSDNNs) with all unstable subsystems. By using discretized Lyapunov-Krasovskii functions (DLKFs) combined with the dwell time method, exponential stability of SSDNNs with all unstable subsystems are analyzed, and several novel stability criteria in mean square are obtained. Comparing with the existing works, our results focus on all unstable subsystems rather than other combinations such as all stable or partially stable subsystems, which is of more research significance. Finally, the correctness of the conclusion is checked by the feasible solutions of two numerical examples.

Mean square exponential stability of discrete-time Markov switched stochastic neural networks with partially unstable subsystems and mixed delays

In this paper, we study the mean square exponential stability of discrete-time stochastic neural networks with partially unstable subsystems and mixed delays. The mixed delays under consideration involve discrete delay and distributed delay. Moreover, the discrete delay term satisfies the Bernoulli distribution. Different from the deterministic switching, we consider Markov switching and our system has partially unstable subsystems. By constructing a novel Lyapunov–Krasovskii functional and using the stationary distribution of Markov chain, we give sufficient conditions for the mean square exponential stability of the suggested system. Finally, two numerical examples are given to check the theory results.

Mittag-Leffler stabilization of fractional infinite dimensional systems with finite dimensional boundary controller

This paper studies the Mittag-Leffler stabilization for unstable infinite dimensional systems actuated by boundary controllers described by time fractional reaction diffusion equations. Via the Riesz basis method, the stable and the unstable part of the considered system are separated. The Kalman rank criterion, which is a classical linear algebra condition, guarantees the stabilizability of the unstable subsystem. Based on these, a controller governed by a finite dimensional system (so called the finite dimensional controller hereafter) is designed to achieve the Mittag-Leffler stability of the closed loop. From infinite to finite, this methodology is a substantial improvement for the existing control laws.

Formula omitted]th Moment Exponential Stability of Switched Discrete-Time Stochastic Systems: A Multiple Lyapunov Functions Method

This paper studies the pth moment exponential stability of switched discrete-time stochastic systems with stable and unstable subsystems. By using multiple Lyapunov functions and mode-dependent average dwell time, we provide a sufficient condition for the pth moment exponential stability with fast convergence rate for deterministic switched discrete-time stochastic systems with stable and unstable subsystems. Obviously, our result has a weaker conservatism than those previous results where the effect of random factor was ignored. In addition, the criteria for stability and instability of discrete-time stochastic systems without switching are provided. Finally, the correctness of the results is verified by two numerical examples. To the best of our knowledge, it is the first time that the stability of discrete time switching system with random disturbance has been discussed.

Stability analysis of switched stochastic delay system with unstable subsystems

This paper investigates the stability of switched stochastic delay systems (SSDSs) with unstable subsystems. By using the multiple Lyapunov–Krasovskii functionals (MLKFs) method combined with the mode-dependent average dwell time of stable subsystems (MDADTSS) and the mode-dependent average dwell time of unstable subsystems (MDADTUS) method, SSDSs with unstable subsystems are analyzed, and several novel sufficient conditions for finite-time stability and pth moment exponential stability are given. Compared with the existing related works, our work considers the influences of random disturbance and delays on the stability of SSDSs. Finally, we provide two examples to illustrate the feasibility of this method.

Stability of Switched Systems with Unstable Subsystems: A Sequence-Based Average Dwell Time Approach

Stability of Switched Systems with Unstable Subsystems: A Sequence-Based Average Dwell Time Approach

Retraction Note: Boundary behaviors for linear systems of subsolutions of the stationary Schr\xf6dinger equation

This paper investigates the boundary behaviors for linear systems of subsolutions of the stationary Schrodinger equation, which contain unstable subsystems. Our first aim is to establish a state-feedback switching law guaranteeing the continuous-time systems to be uniformly exponentially stable. And then we present sufficient and necessary for the stability of the systems with two Schrodinger subsystems. Finally, an illustrative example is given to verify the result.

Bumpless Transfer Control for Switched Linear Systems and its Application to Aero-Engines

To avoid the bumps of the control signal in the switching instants, this paper proposes a bumpless transfer control method which guarantees the continuous of control signal while keeping the asymptotic stability of the closed-loop system. Firstly, the presented bumpless transfer control method possesses a simple structure and meanwhile not have to redesign sub-controllers, which is easy to implement. Furthermore, the method is appropriate for either stable subsystems or unstable subsystems, and be suitable for either state feedback control or dynamic output feedback control. Moreover, we give the sufficient conditions of asymptotic stability for switched linear systems under bumpless transfer control for four cases. For the original switched system with stable subsystems under state or dynamic output feedback control, the stability conditions under mode-dependent average dwell time switching are derived. For the original switched system with unstable subsystems under state or dynamic output feedback control, which is stabilized by some certain switching laws, the stability conditions under the original switching law are given. Finally, a hardware-in-the-loop simulation of an aero-engine control system is employed to verify the effectiveness and superiority of the proposed method.

Iterative Learning Control for Multiphase Batch Processes With Asynchronous Switching

Asynchronous switching between the controller and the active subsystems in multiphase batch processes may cause the systems to be unstable around the switching instants. In view of this, an average dwell-time method-based iterative learning control (ILC) scheme is proposed in this paper. First, the multiphase process is represented as an equivalent closed-loop two-dimensional (2-D) switched system composed of stable and unstable subsystems, based on which new relevant concepts on the stability of the switched system are given. Second, using an average dwell-time method, the ILC law is designed to guarantee the system exponentially stable. Minimum running time for the stable subsystems and maximum running time for the unstable ones are obtained. Lastly, depending on the maximum time for the unstable subsystems, the idea of putting the controller switching step forward is proposed. In this way, the asynchronous switching is removed such that the unstable subsystem can be avoided. The case study on an injection molding process demonstrates the effectiveness and superiority of the proposed method in comparison with the existing 2D-MPC and one-dimensional traditional control methods.

New stability conditions for discrete-time switched positive linear time-varying systems

In this note, we will devote to investigate the stability of discrete-time switched positive linear time-varying systems (PLTVSs). Firstly, a new asymptotic stability criterion of discrete-time PLTVSs is obtained by using time-varying copositive Lyapunov functions (TVCLFs) and this criterion is then extended to the switched case based on the multiple TVCLFs. Furthermore, the sufficient conditions are derived for stability of discrete-time switched PLTVSs with stable subsystems by means of function-dependent average dwell time and function-dependent minimum dwell time. In addition, the stability sufficient conditions are drawn for the switched PLTVSs which contain unstable subsystems. It is worth noting that the difference of TVCLFs and multiple TVCLFs are both relaxed to indefinite in our work. The theoretical results obtained are verified by two numerical examples.

Exponential stability of switched positive systems with unstable modes and distributed delays

This paper studies the exponential stability of switched positive system consisting of unstable subsystems with distributed time-varying delay. Unlike the existing results concerning delays, switching behaviors dominating the system can be either stabilizing or destabilizing. The distributed delay is supposed to be slowly varying and upper-bounded. To tackle the difficulties brought by both the switching behaviors with mixed effects and the distributed delay, a multiple discretized Lyapunov–Krasovskii functional is employed to derive sufficient conditions for the exponential stability of the system. Specifically, by adjusting the ratio of the stabilizing switching behaviors, the state divergence caused by unstable subsystems and destabilizing switching behaviors can be compensated. Simulation examples demonstrate the effectiveness of the results.

Stable Learning-Based Tracking Control of Underactuated Balance Robots

We present a Gaussian process (GP)-based tracking control of underactuated balance robots in which an actuated subsystem is required to follow a desired trajectory, while an unactuated, unstable subsystem needs to be kept balanced. The GP models are used to capture the coupling effects between the actuated/unactuated subsystems through a constructed balance equilibrium manifold (BEM). Optimization-based algorithm is used to obtain the BEM estimation. The control design takes advantage of the structural property of the robot dynamics and is built on the GP models with a data selection algorithm. Stability analysis is given to guarantee the tracking control performance. The control design and comparison with other controllers are demonstrated through experiments on a rotary pendulum.

Event-triggered [formula omitted] control for network-based uncertain Markov jump systems under DoS attacks

This paper investigates the event-triggered H∞ control problem for network-based Markov jump systems subject to denial-of-service (DoS) attacks. In order to reduce the amount of signal transmission, the event-triggering scheme (ETS) is adopted between sensor and controller. Due to DoS attacks invalidating data over networks, a new switched time-delay Markov jump model with unstable subsystems is developed based on state feedback controller. Then with the help of piecewise Lyapunov-Krasovskii functional method, a set of sufficient conditions incorporating constraints of DoS attacks are provided, which guarantees that the resulting switched time-delay Markov jump system is stochastically stable with a certain H∞ performance. Subsequently, we present criterions to obtain the parameters of state feedback gain and ETS. Finally, an example is provided to show the effectiveness of the proposed method.

Improved Stability Criteria for Discrete-Time Switched T–S Fuzzy Systems

This paper concerns with the stability analysis for a class of discrete-time switched Takagi–Sugeno (T–S) fuzzy systems. By establishing a semitime-dependent Lyapunov function and exploring the property of mode-dependent average dwell-time switching, improved stability criteria are provided for switched T–S fuzzy systems containing both stable and unstable modes. Then, slow and fast switching strategies are designed, which are adopted for stable and unstable subsystems, respectively. Especially, we also provide the stability conditions for switched systems with all modes stable and unstable. Finally, the validities and advantages of provided techniques are illustrated by some simulation examples.

Optimized state-dependent switching law design for a class of switched nonlinear systems with two unstable subsystems

This paper proposes a new method for designing a state-dependent switching law to stabilize a class of switched nonlinear system with two unstable subsystems. The main idea is to convert each subsystem to a second-order mechanical system by introducing a reversible transformation, thus the summation of its kinetic and potential energies is calculated as an energy function. Then, by defining a performance index with energy function and using the variational principle, two optimal switching curves are derived from the Euler equation. By adopting a switching law designed by such switching curves, the state of the switched system can approach to the origin at the fastest speed in a sense. In addition, for the case of switched systems with linear vibration subsystems, a critical stability condition related to the stiffness and negative damping coefficients of the subsystems is obtained to make the switched system periodic. Finally, simulation results show that the proposed method can effectively solve the stability problem of switched systems, in which each subsystem does not have any stability factors.

Stabilizing Control for a Ball and Beam System Considering Restricted Beam Angle

This paper develops for a ball and beam system a design method of control system transferring the ball to a target position considering restricted beam angle. First, we compensate the beam drive system as a second-order delay system to reduce the beam angle constraint to an input one. Using modal decomposition methods, we decompose the plant into the stable and unstable subsystems, and derive a nested saturating control law globally asymptotically stabilizing the unstable one. The control system using this control law has the property that when the ball is transferred, its speed is mostly limited less than a specified value. To reduce the steady-state error of the ball position as well as to alleviate the reset windup, a new type of IMC controller is introduced where a saturated output error between the actual plant stabilized by the saturating control law and the model is fed back. The effectiveness of the control law is demonstrated numerically and experimentally.

Impedance analysis considering unstable subsystem poles for MMC-HVdc based wind farm integration system

Harmonic resonance issues of the wind farm connected MMC-HVdc system have been well investigated by ac/dc impedance analysis in past 5 years. These methods generally divided the studied system into two subsystems at ac or dc section. Then, the impedance of these subsystems can be used to identify the system stability by applying Nyquist criterion or generalized Nyquist criterion. However, the existing impedance analyses always assume subsystems are stable and overlook some special cases where subsystems may are unstable. In these special cases, the unstable poles of subsystems must be evaluated before applying Nyquist criterion. Based on this consideration, a simplified method for identifying unstable subsystem poles is proposed in this paper. With consideration of unstable subsystem poles, the gap of existing impedance analysis is filled. The proposed method and stability analyses are validated by electromagnetic transient simulation in PSCAD/EMTDC.

Mixed Incremental H ∞ and Incremental Passivity Analysis for Markov Switched Stochastic Nonlinear Systems

The current study addresses the mixed incremental H∞ and incremental passivity analysis for Markov switched stochastic (MSS) nonlinear systems. The multiple Lyapunov functions approach and the structure of Markov framework are utilized to establish some sufficient conditions for the MSS nonlinear systems, which will be used for the incrementally globally asymptotically stable in the mean(IGASiM) and performance index analysis. Then, the mixed incremental H∞ and incremental passivity performance issues are solved for two instances: in the first case, all subsystems are not IGASiM, while in the second one, both of IGASiM and unstable subsystems exist. Hence, it is shown that when none of the subsystems is IGASiM, the MSS nonlinear systems are IGASiM and possess the mixed incremental H∞ and incremental passivity performance metric in the presence of specified conditions. The mathematical induction is selected to guarantee the robust incremental stability of MSS systems with IGASiM and non-IGASiM subsystems and the performance index can be exhibited a prescribed decay rate. The effectiveness of the proposed results is demonstrated by two simulation examples.

Stability Analysis of Multi-Agent Tracking Systems with Quasi-Cyclic Switching Topologies

In this paper, the stability problem of a class of multi-agent tracking systems with quasi-cyclic switching topologies is investigated. The existing results of systems with switching topologies are usually achieved based on the assumption that the piecewise constant communication topologies are connected and the switchings are cyclic. The communication topologies are possible to be unconnected and it is difficult to guarantee the topologies switch circularly. The piecewise unconnected topology makes the interactive multi-agent tracking system to be an unstable subsystem over this time interval. In order to relax the assumption constraint, a quasi-cyclic method is proposed, which allows the topologies of multi-agent systems to switch in a less conservative way. Moreover, the stability of the tracking system with the existence of unstable subsystems is analyzed based on switched control theory. It is obtained that the convergence rate is affected by the maximum dwell time of unstable subsystems. Finally, a numerical example is provided to demonstrate the effectiveness of the theoretical results.

Consensus stability in multi-agent systems with periodically switched communication topology using Floquet theory

The stability of consensus in linear and nonlinear multi-agent systems with periodically switched communication topology is studied using Floquet theory. The proposed strategy is illustrated for the cases of consensus in linear single-integrator, higher-order integrator, and leader-follower consensus. In addition, the application of Floquet theory in analyzing special cases such as switched systems with joint connectivity, unstable subsystems, and nonlinear systems, including the use of feedback linearization and local linearization in the Kuramoto model, is also studied. By utilizing Floquet theory for multi-agent systems with periodically switched communication topologies, one can simultaneously evaluate the effects of each subsystem’s convergence rate and dwell time on overall behavior. Numerical simulation results are presented to demonstrate the efficacy of the proposed approach in stability analysis of all these cases.