Discontinuous Lyapunov Function Research Articles

$H_\infty$ Refined Antidisturbance Control of Switched LPV Systems With Application to Aero-Engine

In this paper, a parameter-dependent multiple discontinuous Lyapunov function (PMDLF) approach is proposed to study the $H_\infty$ refined antidisturbance control problem of switched linear parameter-varying systems. The $H_\infty$ refined antidisturbance means the disturbance appearing in the control channel can be accurately compensated by means of the estimation of the disturbance and the energy bounded external disturbance can be restrained. A key point is to set up a PMDLF framework that provides an effective tool for attenuating the energy bounded disturbances and rejecting the disturbances generated by the exosystem accurately. A parameter-driven and dwell time-dependent switching law is designed, and a solvability condition ensuring the $H_\infty$ refined antidisturbance performance is developed. Then, the $H_\infty$ refined antidisturbance switched parameter-dependent disturbance observers and the disturbance observer-based refined controllers are established to achieve required disturbance attenuation and rejection. Finally, an example of an aero-engine control system is given to verify the availability of the acquired approaches.

Event-Triggered Controller Design With Varying Gains for T-S Fuzzy Systems.

This paper investigates the problem of designing event-triggered (ET) fuzzy controllers with varying gains for T-S fuzzy systems. New types of fuzzy controllers named ET fuzzy controllers with varying gains are proposed, in which the gains are automatically updated according to an online iteration algorithm at different triggering instants. By designing discontinuous Lyapunov function with time-varying Lyapunov matrix, sufficient conditions in terms of linear matrix inequalities (LMIs) are obtained to ensure the stability of the closed-loop system. It is shown that the proposed method achieves better system performances than the existing design methods with a quadratic Lyapunov function or ET fuzzy controllers with fixed gains. An example is provided to illustrate the effectiveness of the proposed method.

Dynamic event-triggered robust secondary frequency control for islanded AC microgrid

The permeability of renewable energy sources rapidly increases in the microgrid for economic and environmental concerns. However, the uncertainty of renewable energy may lead to frequency fluctuation or even instability in the microgrid. This paper proposes a dynamic event-triggered robust secondary frequency control scheme for the islanded microgrid to handle the uncertainty of the renewable energy source. In addition to the enhancement of frequency stability, the communication burden is also considered in this work. The proposed dynamic event-triggered communication scheme provides a good balance between dynamic performance and communication burden. Based on the discontinuous Lyapunov function, the stability analysis helps the optimal gain selecting. Given the designed controller, the derived theorems in this paper can help to examine whether the communication system fulfills the requirements from the controllers. To validate the proposed method, case studies are carried out based on two typical microgrids. The simulation results are compared with conventional frequency controllers. It shows that the proposed controller outperforms other types of controllers in term of the frequency response performance and communication burden.

Stability of mode-dependent linear switched singular systems with stable and unstable subsystems

This paper studies the E-exponential stability of mode-dependent linear switched singular systems with stable and unstable subsystems. First, by constructing an appropriate multiple discontinuous Lyapunov function, new sufficient conditions of E-exponential stability for linear switched singular systems are established. Considering the feature of mode-dependent average dwell time switching, we adopt the switching strategy where fast switching and slowing switching are respectively applied to unstable and stable subsystems. Compared with the existing results, our approach is more flexible and tighter bounds can be obtained. Finally, a numerical example is provided to illustrate the effectiveness of the proposed criteria.

Improved stability results for discrete-time switched systems: A multiple piecewise convex Lyapunov function approach

In this paper, the stability problem is investigated for a class of discrete-time switched systems with unstable subsystems under the mode-dependent average dwell time (MDADT) switching. A multiple convex Lyapunov function (MCLF) and a multiple piecewise convex Lyapunov function (MPCLF) are firstly proposed, which are formulated in a convex combination form of positive definite matrices with quasi-time-dependent coefficients. It is pointed out in the paper that the multiple Lyapunov function (MLF) and the multiple discontinuous Lyapunov function (MDLF) can be regarded as special cases of the proposed MCLF and MPCLF, respectively. Then, the exponential stability conditions are derived by the new Lyapunov functions. Both slow switching and fast switching are exerted on stable modes and unstable modes, respectively. Finally, two numerical examples are given to demonstrate that larger stability regions and tighter MDADT bounds can be obtained by using our developed techniques compared with some recent results.

Fixed-time Synchronization of Memristive Cohen-Grossberg Neural Networks with Impulsive Effects

This paper investigates the fixed-time synchronization of memristive Cohen-Grossberg neural networks with impulsive effects. Through a nonlinear transformation and Fillipov discontinuous theory, we obtain an equivalent system from the original memristive Cohen-Grossberg neural networks. By constructing a discontinuous Lyapunov function and utilizing comparison principle, a sufficient condition is achieved to guarantee the fixed-time synchronization of drive-response system with impulsive effects. Moreover, for the purpose of reducing the cost of control, an adaptive control strategy is considered. Finally, corresponding numerical simulations are carried out to show the effectiveness of the analytic results.

Switching Stabilization for Type-2 Fuzzy Systems With Network-Induced Packet Losses.

This paper is concerned with the stabilization problem of type-2 fuzzy systems with network-induced packet losses. By regarding the packet lost process as an unstable mode of a switched system, the stability of the system is then guaranteed with the aid of the mode-dependent average dwell time approach in the sense of the slow and fast switching. The discrete-time multiple discontinuous Lyapunov function is also utilized for the analysis. Two sufficient conditions regarding the stability and the stabilization of the system are proposed. The state-feedback matrices can be then calculated from the conditions to ensure the criterion that the packet-loss rate is no larger than a specific constant. Two practical examples are given to illustrate the feasibility and effectiveness of the proposed method.

Discontinuous Lyapunov functions for a class of piecewise affine systems

In this paper, a Lyapunov-based method is provided to study the local asymptotic stability of planar piecewise affine systems with continuous vector fields. For such systems, the state space is supposed to be partitioned into several bounded convex polytopes. A piecewise affine function, not necessarily continuous on the boundaries of the polytopic partitions, is proposed as a candidate Lyapunov function. Then, sufficient conditions for the local asymptotic stability of the system, including a monotonicity condition at switching instants, are formulated as a linear programming problem. In addition, when the problem does not have a feasible solution based on the original partitions of the system, a new partition refinement algorithm is presented. In this way, more flexibility can be provided in searching for the Lyapunov function. Owing to relaxation of the continuity condition imposed on the system boundaries, the proposed method reaches to less conservative results, compared with the previous methods based on continuous piecewise affine Lyapunov functions. Simulation results illustrate the effectiveness of the proposed method.

Event-based consensus for second-order multi-agent systems with actuator saturation under fixed and Markovian switching topologies

This paper studies the event-based consensus problem of second-order multi-agent systems with actuator saturation under fixed topology and Markovian switching topologies. By a model transformation, the consensus problem is first converted into the stability problem of the error system. Using discontinuous Lyapunov functional approach, two sufficient conditions on the consensus are derived for second-order multi-agent systems with fixed topology and Markovian switching topologies, respectively. The discontinuous Lyapunov functions take full account of the characteristics of the sawtooth delay, and thus lead to a less conservative consensus criterion. It is shown that the consensus condition depends on the parameters of sampling period, Laplacian matrix, and event-triggered parameter. In addition, this paper provides an effective method to co-design both the consensus controller and the event-triggered parameter. Finally, two numerical examples are provided to illustrate the effectiveness and feasibility of the proposed algorithm.

New Results on Stability of Slowly Switched Systems: A Multiple Discontinuous Lyapunov Function Approach

In this technical note, the problem of stability for a class of slowly switched systems is investigated. By developing a novel multiple discontinuous Lyapunov function approach and exploring the feature of mode-dependent dwell time switching, new stability conditions are established for systems with a designed switching strategy where fast switching and slow switching are respectively applied to unstable and stable subsystems. In particular, stability conditions for linear switched systems are also given via choosing multiple discontinuous Lyapunov functions in the quadratic form. Moreover, stability criteria for the systems consisting of stable subsystems are also derived. It is shown that our proposed results cover some existing ones in literature as special cases, and provide tighter bounds on the dwell time. Finally, some simulation results are provided to show the advantages of the theoretic results obtained.

Stability Analysis for Discrete-Time Switched Nonlinear System Under MDADT Switching

The problem of stability analysis for discrete-time switched nonlinear system is investigated with mode-dependent average dwell time (MDADT) method in this paper. A slow switching strategy is adopted in the discrete-time nonlinear stable subsystems and unstable subsystems are handled by a fast switching strategy. Takagi-Sugeno (T-S) fuzzy model is utilized to approximate the switched nonlinear system. By constructing a multiple discontinuous Lyapunov function approach, the stability condition of switched T-S fuzzy system is built to get tighter bound on MDADT, which shows that our proposed method outperforms the classical one. Finally, through a numerical example, the effectiveness of the presented control approach is illustrated by comparison with result from classical one.

Stability, stabilization and [formula omitted]-gain analysis of periodic piecewise linear systems

In this paper, the stability, stabilization and L2-gain problems are investigated for periodic piecewise linear systems, in which not all subsystems are Hurwitz. First, some sufficient and necessary conditions for the exponential stability are established. By employing a discontinuous Lyapunov function with time-varying Lyapunov matrix, stabilization and L2-gain conditions of periodic piecewise linear systems are proposed by allowing the corresponding Lyapunov function to be possibly non-monotonically decreasing over a period. A state-feedback periodic piecewise controller is developed to stabilize the system, and the corresponding algorithm is proposed to compute the controller gain. The L2-gain criteria with continuous time-varying Lyapunov matrix and piecewise constant Lyapunov matrices are studied as well. Numerical examples are given to show the validity of the proposed techniques.

Global exponential synchronization of nonlinear time-delay Lur’e systems via delayed impulsive control

This paper is concerned with the global exponential synchronization problem of two identical nonlinear time-delay Lur’e systems via delayed impulsive control. Some novel impulsive synchronization criteria are obtained by introducing a discontinuous Lyapunov function and by using the Lyapunov–Razumikhin technique, which are expressed in forms of linear matrix inequalities. The derived criteria reveal the effects of impulsive input delays and impulsive intervals on the stability of synchronization error systems. Then, sufficient conditions on the existence of a delayed impulsive controller are derived by employing these newly-obtained synchronization criteria. Additionally, some synchronization criteria for two identical time-delay Lur’e systems with impulsive effects are presented by using delayed continuous feedback control. The synchronization criteria via delayed continuous feedback control can deal with the case when the impulsive control strategy fails to synchronize two identical impulsive time-delay Lur’e systems. Three numerical examples are provided to illustrate the efficiency of the obtained results.

Discontinuous Lyapunov Functions for Nonasymptotic Stability Analysis

The paper presents mathematical tools required for finite-time stability analysis of discontinuous control systems using discontinuous Lyapunov functions. Elements of Filippov theory of differential equations with discontinuous right-hand sides and stability notions are briefly observed. Concepts of generalized derivatives and non-smooth Lyapunov functions are considered. The generalized Lyapunov theorems for stability analysis and convergence time estimation are presented and supported by examples.

Convex conditions for robust stability analysis and stabilization of linear aperiodic impulsive and sampled-data systems under dwell-time constraints

Stability analysis and control of linear impulsive systems is addressed in a hybrid framework, through the use of continuous-time time-varying discontinuous Lyapunov functions. Necessary and sufficient conditions for stability of impulsive systems with periodic impulses are first provided in order to set up the main ideas. Extensions to the stability of aperiodic systems under minimum, maximum and ranged dwell-times are then derived. By exploiting further the particular structure of the stability conditions, the results are non-conservatively extended to quadratic stability analysis of linear uncertain impulsive systems. These stability criteria are, in turn, losslessly extended to stabilization using a particular, yet broad enough, class of state-feedback controllers, providing then a convex solution to the open problem of robust dwell-time stabilization of impulsive systems using hybrid stability criteria. Relying finally on the representability of sampled-data systems as impulsive systems, the problems of robust stability analysis and robust stabilization of periodic and aperiodic uncertain sampled-data systems are straightforwardly solved using the same ideas. Several examples are discussed in order to show the effectiveness and reduced complexity of the proposed approach.

Stability and Invariance Analysis of Uncertain Discrete-Time Piecewise Affine Systems

This note proposes a method to analyze uniform asymptotic stability and uniform ultimate boundedness of uncertain piecewise affine systems whose dynamics are only defined in a bounded and possibly non-invariant set X of states. The approach relies on introducing fake dynamics outside X and on synthesizing a piecewise affine and possibly discontinuous Lyapunov function via linear programming. The existence of such a function proves stability properties of the original system and allows the determination of a region of attraction contained in X. The procedure is particularly useful in practical applications for analyzing the stability of piecewise affine control systems that are only defined over a bounded subset X of the state space, and to determine whether for a given set of initial conditions the trajectories of the state vector remain within the domain X.

Sampled-Data Control of Spacecraft Rendezvous with Discontinuous Lyapunov Approach

This paper investigates the sampled-data stabilization problem of spacecraft relative positional holding with improved Lyapunov function approach. The classical Clohessy-Wiltshire equation is adopted to describe the relative dynamic model. The relative position holding problem is converted into an output tracking control problem using sampling signals. A time-dependent discontinuous Lyapunov functionals approach is developed, which will lead to essentially less conservative results for the stability analysis and controller design of the corresponding closed-loop system. Sufficient conditions for the exponential stability analysis and the existence of the proposed controller are provided, respectively. Finally, a simulation result is established to illustrate the effectiveness of the proposed control scheme.

Discontinuous piecewise quadratic Lyapunov functions for planar piecewise affine systems

For planar piecewise affine systems, this paper proposes sufficient stability conditions based on discontinuous Lyapunov functions. The monotonicity condition for discontinuous functions at switching instants is presented based on the behavior of state trajectories on the switching surfaces. First, the stability conditions are derived for a typical multiple Lyapunov function and then these conditions are formulated as a set of linear matrix inequalities for piecewise quadratic Lyapunov functions. The implementation of the proposed method is illustrated by an example.

Synchronization of chaotic systems under sampled-data control

The problem of global asymptotical synchronization of chaotic Lur’e systems using sampled-data controller is considered in this paper. Sufficient conditions are obtained in terms of effective synchronization linear matrix inequality using a piecewise sawtooth structure of the sampling in time by constructing the new discontinuous Lyapunov functionals. The sampled-data feedback control gain is obtained from the derived condition. The Chua system and horizontal platform system are taken for numerical demonstration to show the effectiveness of the proposed condition.

Network-based H ∞ tracking control with event-triggering sampling scheme

In this study, H ∞ tracking control design for a class of networked control systems (NCSs) with event-triggering sampling scheme is investigated. First, a novel event-triggering sampling scheme is proposed. Next, considering the network-induced delays and using such a triggering scheme together with delay system approach, a new tracking model of NCSs is established. Attention is focused on the design of H ∞ controller that guarantees the prescribed H ∞ tracking performance of the closed-loop-networked tracking system. Two approaches are developed to solve this problem, based on continuous Lyapunov function and discontinuous Lyapunov function, respectively. Moreover, the obtained results are further extended to the case that the system matrices of the plant contain parameter uncertainties, represented in either norm-bounded or polytopic forms. Finally, an illustrative example is presented to show the usefulness and applicability of the propose method.