Lyapunov Function Approach Research Articles

Robust Output Regulation for a Constrained Omnidirectional Mobile Robot

This paper aims to design a robust output control to regulate the output for an uncertain omnidirectional mobile robot with input saturation and state constraints, in the presence of external disturbances. The proposed robust control approach is composed of a linear control part, that is designed based on the attractive ellipsoid method and a barrier Lyapunov function approach, taking into account the input and state constraints, and a nonlinear control part that can compensate the effect of matched disturbances. The synthesis of the robust control is given in terms of linear matrix inequalities. Simulation results show the workability of the proposed robust control approach.

Robust fuzzy observer‐based fault‐tolerant control: A homogeneous polynomial Lyapunov function approach

In this paper, a homogeneous polynomial Lyapunov function (HPLF) is employed in robust observer-based Integrated Fault-Tolerant Control (IFTC) of nonlinear systems modelled by the polynomial fuzzy model (PFM). This makes the design problem benefit from the conservative reduction property of polynomial Lyapunov functions (PLFs) and simultaneously avoid the emergence of non-convex terms due to differentiation of the polynomial Lyapunov matrix. As a result, the less conservative sum of square (SOS) conditions are obtained in the form of polynomial matrix inequalities (PMI), which are solved via SOSTOOLS. For the first time, a non-quadratic Lyapunov function is used to design a polynomial fuzzy unknown-input observer that estimates the system's states and actuator faults in the presence of model uncertainties and external disturbances. The effectiveness of the proposed approach in providing more relaxed and less conservative results along with the better fault tolerance is illustrated through three simulating examples and compared with the quadratic Lyapunov function (QLF) approach.

Distributed optimization in predefined-time for multi-agent systems over a directed network

This paper is concerned with the distributed optimization problem (DOP) with equality constraints in predefined time for multi-agent systems (MASs) over a directed network. Firstly, a new principle with respect to the predefined-time convergence is developed for absolute continuous functions, which plays a key role in the consensus analysis later. Secondly, a novel control protocol is designed to address the solution of the considered DOP with convex objective functions. Thirdly, on the basis of Lyapunov functional approach and the proposed convergence principle, the predefined-time consensus conditions are achieved. Under the designed control protocol, the consensus state can reach feasible region in finite time, and subsequently converges to an optimal solution. Finally, the correctness of the theoretical analysis and the feasibility of the designed control strategy are verified by two illustrative simulations and an application example.

Generic Stability Criteria for Switched Nonlinear Systems With Switching-Signal-Based Lyapunov Functions Using Takagi–Sugeno Fuzzy Model

The aim of this article is to establish generic stability conditions for switched nonlinear systems with mode-dependent average dwell-time (MDADT) switching rules via Takagi–Sugeno (T–S) fuzzy modeling, which cover all unstable modes, all stable modes, and partially unstable modes as special cases. Different from traditional multiple and multiple discontinuous Lyapunov function (MDLF) methods, the proposed novel switching-signal-based multiple discontinuous Lyapunov function (SMDLF) approach divides each mode-running interval dynamically according to the actual switching of the system. The developed approach can not only yield less conservative bounds on the MDADT but can also handle all unstable subsystems, which cannot be done by means of the traditional MDLF. A class of SMDLF is constructed for continuous-time switched T–S fuzzy systems, and relaxed stability conditions are presented for the system with both stable and unstable subsystems, using a larger switching signal space. Then, novel stability conditions are deduced for all stable and unstable subsystems in terms of slow and fast MDADT switching signals, respectively. Finally, comparative simulation examples are provided to verify the advantages and effectiveness of the proposed approach.

Novel finite and fixed-time stability theorems for fractional-order impulsive discontinuous systems and their application to multi-agent systems

This article studies the finite-time (FNT) and fixed-time (FXT) stability theorems of general fractional-order impulsive discontinuous systems (FOIDSs) through an indefinite Lyapunov functional (LF) approach. Using differential inclusion theory and set-valued map concept, the general FOIDSs can be transformed into fractional-order impulsive differential inclusions (FOIDIs). Two new stability theorems on finite and FXT stability of FOIDSs with finite impulsive instants are derived using an indefinite LF and fractional derivatives of the LF on some impulsive instants, respectively. The settling time is explicitly calculated. The obtained theoretical findings are applied to fractional-order discontinuous multi-agent systems. Discrete control schemes and indefinite Lyapunov-Krasovskii functions (LKFs) are also developed for ensuring FXT synchronization in fractional-order discontinuous multiagent systems. To demonstrate the efficiency and feasibility of the proposed method, some numerical simulations are conducted.

Bipartite containment control of fractional multi-agent systems with input delay on switching signed directed networks

The bipartite containment control of nonlinear fractional multi-agent systems (FMASs) on fixed and switching signed directed networks is addressed in this article. Delayed control protocols are put forward to provide several simple algebraic criteria by means of using signed graph theory, fractional Razumikhin technique and common Lyapunov function approach. The presented method can well overcome the difficulty resulting from fractional calculus, time delay and switchings. Two numerical examples are presented for further clarifying the theoretical conclusions.

Leader-following consensus considering effects of agents on each other via impulsive control: A topology dependent average dwell time approach

This paper intends to investigate the consensus problem of a nonlinear multi-agent system with new nonlinear terms added to the dynamics of each agent in the leader-following framework with impulsive control. The main contribution of this paper is introducing these new terms expressing the effect of each agent on neighbor agents. The new terms called effect terms (ETs) are considered with time-varying delay. Moreover, the communication interactions among all agents are addressed by a set of consensusable and unconsensusable switching topologies. In particular, the topology-dependent average dwell time (TDADT), one of the significant practical analysis methods for switched systems, has been calculated for each topology. The globally uniformly exponentially stability (GUES) for the consensus error dynamics is analyzed by employing algebraic graph theory and a multiple discontinuous Lyapunov function approach (MDLF) regarding separate Lyapunov functions for impulse instants. Furthermore, sufficient conditions in terms of linear matrix inequalities (LMIs) are derived to ensure that consensus can be achieved. Finally, the effectiveness of the theoretical analysis is corroborated by a numerical example.

Admissible edge-dependent finite-time ℋ∞ filtering for discrete-time switched systems with average dwell time

Considering that the property of finite-time convergence is of great significance for practical filters, this article is focused on the finite-time ℋ∞ filtering for discrete-time switched systems with average dwell time. Different from the case of the general mode-dependent switching, the information of the admissible switching edge is fully exploited to derive more preferable filtering scheme. In order to further reduce the design conservatism, the multiple discontinuous Lyapunov function approach, which incorporates the concept of admissible edge-dependent average dwell time, is used to analyze the finite-time stability and ℋ∞ performance of the filtering error system. This approach assigns each subsystem of the error system with a discontinuous function rather than a continuous one, so it is less conservative compared to the conventional multiple Lyapunov function approach. First, the finite-time boundedness and ℋ∞ performance is analyzed for general switched nonlinear systems with admissible edge-dependent average dwell time. Then, a sufficient condition of the finite-time boundedness and ℋ∞ performance is derived for the filtering error system based on the obtained criterion. Subsequently, an admissible edge-dependent finite-time filtering scheme is proposed for the underlying system in terms of linear matrix inequalities. Finally, the proposed filter design scheme is applied to a DC–DC power converter circuit, and its effectiveness and superiority are verified by the simulation results.

Stability analysis of delayed fractional-order switched systems

This paper discusses stability of delayed fractional-order switched systems. First, two useful propositions are proposed based on Razumikhin approach together with properties of fractional calculus and analytical techniques, which can well solve the difficulties caused by distributed and discrete delays and fractional-order derivatives. Then, with the help of multiple Lyapunov function approach and dwell time technique, an effective way is developed to overcome the trouble caused by switching rules. A delay-dependent dwell time is derived to guarantee asymptotic stability under the hypothesis that each subsystem of delayed fractional-order switched systems is asymptotically stable. Finally, two illustrative examples further clarify the obtained conclusions.

Finite-Time Stabilization for Continuous Triangular Systems via Vector Lyapunov Function Approach

This article provides finite-time stabilizing control laws for the continuous triangular systems, in which the nominal dynamics is the chain of power integrators with the power of a positive odd rational number. For a class of continuous lower-triangular systems, the linear controllers are designed. By invoking a vector Lyapunov function theory, there is no need to use the usual back-stepping technique and a quite number of inequality computations are avoided. For a class of continuous upper-triangular systems, the nested-saturation controllers are constructed. In this case, the use of a vector Lyapunov function theory also helps to reduce computational burden, and allows us to summarize a same group of parameter conditions guaranteeing both the reduction of saturated terms and the finite-time stability of the reduced system.

Global synchronization for BAM delayed reaction-diffusion neural networks with fractional partial differential operator

This paper is devoted to the exploration of the global asymptotic synchronization (GAS) and global Mittag–Leffler synchronization (GMLS) issues on bidirectional associative memory reaction-diffusion neural networks (BAMRDNNs), which include a Caputo’s fractional partial differential operator, leakage and discrete delays. By means of the Green’s theorem, Jensen’s integral inequality and Lyapunov functional approach, several synchronization criteria are established by using the hybrid controllers. The presented results are described in the algebraic inequalities form, which can immensely reduce the computational complexity. Moreover, these results reveal the impact of the network system coefficient matrices on GAS and GMLS. Finally, in the three-dimensional space, the effectiveness and practicability of these synchronization criteria are confirmed by choosing different derivative orders and diffusion coefficients.

Fully distributed consensus of nonlinear fractional multi‐agent systems based on combined event‐triggered mechanism

This article focuses on the fully distributed leader‐following consensus problem of nonlinear fractional multi‐agent systems via event‐triggered control technique. The main intention of this article is to design a novel event‐triggered mechanism, which not only takes into account both the relative error and the absolute error of the samples but also ensures that the event triggering time series of each agent are independent of each other, thus effectively reducing the interaction frequency of network data. Subsequently, a fully distributed event‐triggered consensus protocol for the leader‐following fractional chaotic multi‐agent systems is proposed, and the sufficient conditions of consensus are attained via fractional Lyapunov functional approach. Finally, two simulation experiments are presented to illustrate that the proposed event‐based consensus protocol can effectively reduce the frequency of actuator data update while ensuring desired consensus performance.

Fast Finite-Time Stability and Its Application in Adaptive Control of High-Order Stochastic Nonlinear Systems

In this article, a new design method for an adaptive fast finite-time controller (FTC) is proposed for the finite-time stability (FTS) issue of a class of high-order stochastic nonlinear systems (HOSNSs) with unknown parameters. Using a power integrator technology and Lyapunov function approach, an adaptive state feedback controller is derived to ensure fast FTS of HOSNSs. The developed adaptive fast FTC is equipped with less settling time to obtain better steady-state accuracy compared with the traditional FTC. The effectiveness of the proposed adaptive control scheme is demonstrated by a numerical example.

Periodically intermittent H∞ control for discrete‐time switched interval type‐2 fuzzy systems with dwell time constraints: A switching‐path‐dependent Lyapunov function approach

AbstractThis article studies the periodically intermittent control problem of discrete‐time switched interval type‐2 fuzzy systems (DSIT2FSs) with dwell time constraints. A novel switching‐path‐dependent Lyapunov functions (SPDLFs) approach, which utilizes the switching path information over multiple controller operation periods, is proposed. First, the ‐mode DSIT2FS with periodic intermittent control inputs is regarded as a switched system with controller‐busy modes and controller‐free modes. Then, by constructing a class of non‐increasing SPDLFs, sufficient conditions on the existence of intermittent interval type‐2 (IT2) fuzzy controllers are obtained for DSIT2FS with minimum dwell time constraints. The controller gains can be gotten by solving a set of strict linear matrix inequalities. It is shown that the designed IT2 fuzzy controllers work intermittently but still can guarantee stability and performance of the close‐loop systems. Finally, two numerical examples are provided to illustrate the effectiveness and advantages of the results.

Observer-based output feedback control for networked systems with dual-channel event-triggered sampling and quantization

This paper investigates the problem of observer-based output feedback control for linear networked systems with dual-channel event-triggered mechanisms and quantization. Both continuous-time and discrete-time event detection cases are discussed. In the continuous-time case, the stability of observer error dynamics and closed-loop system are analyzed respectively, and it is proved that Zeno behavior would not occur. In order to approach engineering practice, in the discrete-time case, two types of network attacks including denial-of-service (DoS) and fault data injection (FDI) attacks are considered, whose nature property is characterized by Bernoulli variables. By combining these factors and transmission delay, a novel augmented system model is proposed, and some sufficient conditions are derived based on Lyapunov functional approach and linear matrix inequalities (LMIs). Compared with the existing results, this framework is more comprehensive and practical, and the global uniform ultimate boundedness of closed-loop systems can be guaranteed. Finally, simulation examples are given to demonstrate the effectiveness of the proposed method.

Lipschitz stability analysis of fractional-order impulsive delayed reaction-diffusion neural network models

In this paper, the concept of Lipschitz stability is introduced to impulsive delayed reaction-diffusion neural network models of fractional order. Such networks are an appropriate modeling tool for studying various problems in engineering, biology, neuroscience and medicine. Fractional derivatives of Caputo type are considered in the model. The effects of impulsive perturbations and delays are also under consideration. Lipschitz stability analysis is performed and sufficient conditions for global uniform Lipschitz stability of the model are established. The Lyapunov function approach combined with the comparison principle are employed in the development of the main results. The proposed criteria extend some existing stability results for such models to the Lipschitz stability case. The introduced concept is also very useful in numerous inverse problems.

Finite/fixed-time bipartite consensus for networks of diffusion PDEs via event-triggered control

This paper investigates the finite/ fixed-time bipartite consensus for networks of diffusion partial differential equations (PDEs) via the event-triggered control strategy. A global convergence principle in fixed time, which plays an important role in the theoretical analysis later, is developed for nonlinear systems. The C0-semigroup method is adopted to explain the well-posedness of the considered network systems. Two new event-triggered control protocols are designed to realize the finite/ fixed-time bipartite consensus goal. By applying the proposed convergence principle, Lyapunov functional approach and inequality analysis technique, the finite/ fixed-time bipartite consensus conditions are addressed under the designed control mechanism. Moreover, the settling time is estimated accurately. Finally, two numerical examples are provided to illustrate the validity of the theoretical results.

Prescribed-time synchronization in networks of piecewise smooth systems via a nonlinear dynamic event-triggered control strategy

This paper discusses the prescribed-time synchronization (PTs) issue in networks of piecewise smooth (PWS) systems. Firstly, two new lemmas with respect to the prescribed-time stability are developed for nonlinear PWS systems, where the stable time can be preset arbitrarily, and is irrelevant to any control parameters and initial values. Secondly, a feedback controller and a novel event-triggered controller are designed to realize PTs objective, respectively. Different from the existing linear dynamic event-triggered condition, the adjustable internal dynamic variable is governed by a nonlinear system, which is related to the prescribed time. Thirdly, by adopting set-valued analysis technique and generalized Lyapunov functional approach, the conditions of PTs are achieved in terms of linear matrix inequalities (LMIs). Finally, two application examples to chaotic Sprott circuit are provided to certificate the correctness of the obtained results and the effectiveness of the designed control schemes.

Prescribed performance spacecraft attitude tracking with disturbance observer: A performance-adjustable policy

In this paper, a prescribed performance control (PPC) scheme for spacecraft attitude tracking with anti-disturbance and performance-adjustable properties is proposed. The proposed controller is based on the barrier Lyapunov function (BLF) approach, which can arbitrarily adjust the state convergence performance and possess high steady-state performance. Compared with previous works, this strategy is more robust to external disturbances and thus has higher steady-state performance with low computing resource occupancy. What’s more, it has the property of freely adjusting performance to meet the control objectives and performance requirements in different scenarios. Simulation results in this paper verify the effectiveness and superiority of the proposed method.

A delayed diffusive HBV model with nonlinear incidence and CTL immune response

In this paper, we develop a diffusive and delayed hepatitis B virus (HBV) infection model, which incorporates both virus‐to‐cell and cell‐to‐cell transmissions along with cytotoxic T lymphocyte (CTL) immune response. Two threshold parameters (basic reproduction number) and (immune response reproduction number) are defined, which determine the dynamic behavior of the model. Employing the approach of Lyapunov functionals, we show that the infection‐free steady state is globally asymptotically stable when ; the immune‐free infected steady state is globally asymptotically stable when ; and the immune‐present steady state is globally asymptotically stable when , and . We perform the sensitivity and uncertainly analysis on with respect to parameters and find that the most important parameter that drives the HBV infection is the mortality of HBV DNA capsids. Moreover, the theoretical results are supported with numerical simulations.