Common Lyapunov Function Research Articles

Non‐approximation‐based outputs stabilization for switched large‐scale nonlinear systems with quantized inputs and sensor uncertainties

AbstractA non‐approximation‐based output feedback control strategy for a class of switched large‐scale nonlinear systems with quantized inputs and sensor uncertainties is proposed. A dynamic gain, which is shared by the state observers and controllers of all the subsystems, is designed so that the effects of sensor uncertainties, quantized inputs, unknown parameters, and external disturbances can be compensated. By constructing some common Lyapunov functions (CLFs) shared by the switched systems, it is proved that with the proposed scheme, the closed‐loop system stability can be guaranteed under arbitrary switching, and the outputs of all the subsystems can be steered to within arbitrarily small neighborhoods of the origin.

Adaptive neural dynamic surface control with fixed-time prescribed performance for uncertain nonstrict-feedback stochastic switched systems

An adaptive neural dynamic surface control (DSC) problem with fixed-time prescribed performance (FTPP) is investigated for a class of nonstrict-feedback stochastic switched systems. Differently from the existing works for FTPP problem, the stochastic switched systems with nonstrict-feedback form and completely unknown systems are considered in this paper, and the unknown functions are approximated by some radial basis function (RBF) neural networks (NNs). The desired adaptive neural controller is designed by using common Lyapunov function method and defining fixed-time prescribed performance function (PPF). And based on the adaptive DSC scheme with the nonlinear filter, the “explosion of complexity” problem is avoided. Besides, the constructed fixed-time PPF just need to meet the requirement of second derivative exists. According to the Lyapunov stability theory, the FTPP of output tracking error is achieved, and all signals of closed-loop system remain bounded in probability. Finally, simulation results are presented to verify the availability of the designed control strategy.

Finite-time state feedback stabilization of strict-feedback switched systems with asymmetric output-constraints

Finite-time stabilization of strict-feedback switched systems with asymmetric output constraints (AOCs) via state feedback is investigated. First, an elaborately constructed fraction-type barrier Lyapunov function (BLF) is presented. Then, with a mild assumption on strict-feedback switched systems, state feedback laws are constructed by revamping the adding a power integrator approach (AAPIA) and meanwhile a common Lyapunov function (CLF) is also obtained. The resultant closed-loop systems are finite-time stable (FTS) and the asymmetric output constraint is satisfied too. The approach, proposed in this note, can make strict-feedback switched systems with/without AOCs finite-time stable in a unified frame.

Hybrid neural network control design for uncertain switched nonlinear systems with external disturbances: Application to ship maneuvering system

In this work, the problem of hybrid neural network control for a class of switched uncertain nonlinear systems in strict-feedback form is considered. To approximate unknown nonlinear functions, an improved whale optimization algorithm (IWOA)-based hybrid neural network controller is introduced. Based on hybrid neural network approximation ability, a new hybrid neural network controller is constructed via the backstepping technique and the common Lyapunov function (CLF) is used for the stability analysis of the proposed model. The suggested method ensures that all signals in the closed-loop system are semi-global uniform ultimate bounded (SGUUB), and the tracking error converges to a small neighborhood of zero. Finally, the proposed scheme is applied to a ship maneuvering system and a numerical example to verify its effectiveness.

Dynamic Event-Triggered Adaptive NN Control for Switched Uncertain Nonlinear Systems.

This article is concerned with the problem of dynamic event-triggered adaptive neural network (NN) control for a class of switched strict-feedback uncertainty nonlinear systems. A novel switched command filter-based dynamic event-triggered adaptive NN control approach is set up by exploiting the backstepping and command filter and the common Lyapunov function method. Since adaptive controllers of subsystems are event triggered, then if the switching happens between any two consecutive triggering instants, asynchronous switching will arise between candidate controllers of subsystems and subsystems. Unlike the existing literature, where maximum asynchronous time is restricted, without any strict limitations on maximum asynchronous time being needed in this article, the asynchronous switching problem is directly handled by proposing a novel switching dynamic event-triggered mechanism (DETM) and event-triggered adaptive controllers of subsystems. Moreover, a piecewise constant variable is introduced into the switching DETM, which overcomes the difficulty of switched measurement error being discontinuous. Also, a strictly positive lower bound of interevent times is obtained. Finally, a continuous stirred tank reactor system and a numerical example are presented to demonstrate the effectiveness of the developed approach.

Frequency Information-Based Observer Design for Takagi–Sugeno Fuzzy Affine Systems

This brief studies the problem of observer design for discrete-time uncertain Takagi-Sugeno (T-S) fuzzy affine systems subject to external disturbances and measurement noises with known frequency ranges. A class of novel observers with more design parameters is proposed. Based on the premise variable space partition information, piecewise Lyapunov functions are constructed together with the frequency range information of disturbances and noises, less conservative asymptotic stability with finite frequency performance analysis results are obtained. Then, based on Projection lemma, a novel observer design approach is proposed to T-S fuzzy affine systems, which tends to be less conservative than the non-frequency information and/or common Lyapunov function based observer design ones. Finally, simulation results on a circuit system are given to demonstrate the effectiveness and advantages of the proposed method.

ℋ2 reset controller design for linear systems using piecewise quadratic Lyapunov functions

SummaryThis study presents reset controller design conditions for multiple‐input and multiple‐output (MIMO) linear systems in terms of linear matrix inequalities (LMIs). The design conditions were developed using a piecewise quadratic Lyapunov function (PWQLF), leading to less conservative conditions than using a common Lyapunov function. Achieving LMI relaxation of the PWQLF‐based reset controller design conditions for MIMO linear systems is challenging because (1) dividing the flow set of the reset condition is highly complex, (2) equality constraints, which make the PWQLF continuous on the boundaries between adjacent flow sets, prevent the use of congruence transformation in matrix inequalities, and (3) indefiniteness of the matrices for regional constraints also makes it difficult to apply standard LMI relaxation techniques. To overcome these difficulties, this study introduces a two‐dimensional projection of the plant's output measurements and reset controller's states to construct a reset condition, allowing the PWQLF to be tractable in MIMO linear systems. Then, a new stability analysis condition is derived using the elimination lemma, overcoming the challenges in LMI relaxation of the controller design attributed to the equality constraints and indefinite matrices. An LMI‐based reset controller design condition is then proposed based on the proposed stability analysis. To account for transient behavior, tracking reset controller design problem is addressed. A numerical example is provided to demonstrate the effectiveness of the proposed method.

Observer-based bipartite containment control of fractional multi-agent systems with mixed delays

This paper is concerned with observer-based bipartite containment control of nonlinear fractional multi-agent systems (FMASs) with distributed and input delays. First, a follower-based observer is designed by the local output information. On the basis of the state information observed, a delayed control protocol is proposed to achieve bipartite containment control on fixed signed directed network, several feasible conditions on devising observer and delayed controller are obtained with the help of signed graph theory, fractional Razumikhin method and matrix transformation. Secondly, the observer-based bipartite containment control on switching networks is furthered addressed by means of common Lyapunov function approach. Lastly, the effectiveness of presented conclusions is clarified through two simulation examples.

Common Lyapunov Functions for Switched Linear Systems: Linear Programming-Based Approach

We study the stability of an equilibrium of arbitrarily switched, autonomous, continuous-time systems through the computation of a common Lyapunov function (CLF). The switching occurs between a finite number of individual subsystems, each of which is assumed to be linear. We present a linear programming (LP) based approach to compute a continuous and piecewise affine (CPA) CLF and compare this approach with different methods in the literature. In particular we compare it with the prevalent use of linear matrix inequalities (LMIs) and semidefinite optimization to parameterize a quadratic common Lyapunov function (QCLF) for the linear subsystems.

Zonotope-based Fault Estimation for Uncertain Discrete-Time Switched Linear Systems

This paper investigates actuator fault estimation for discrete-time switched linear systems subject to unknown but bounded disturbances and measurement noises. By taking the fault as an auxiliary state vector, the proposed method consists of two steps. First, a switched L∞-based descriptor observer attenuating the effect of uncertainties is designed to obtain simultaneous point estimate of the system state and the fault term. Second, interval estimation is achieved by integrating robust point estimation with zonotopic analysis techniques. The observer gains are calculated by solving Linear Matrix Inequality (LMI) derived using a common Lyapunov function. Finally, the effectiveness of the proposed approach is demonstrated through a numerical example.

Predefined-Time Leader-Following Formation Control of Nonlinear Multiagent Systems under Switching Topologies

The paper proposes a predefined-time leader-following formation control scheme for nonlinear multi-agent systems with external perturbations under switching topologies. Unlike previous work on predefined-time containment control, we aim to create an isosceles triangle shaped formation instead of being covered by a generic convex hull by having followers follow the leader's trajectories. To ensure convergence along the sliding surface within a predefined-time, we present a nonlinear sliding surface. A terminal sliding mode controller is then used to obtain the desired formation within a predefined-time. The convergence time is independent of both initial values and control parameters. Using, common Lyapunov function, predefined-time stability of closed-loop system is investigated. Simulation examples demonstrate that the developed control protocol is feasible and robust.

Distributed Containment Control With Prescribed Accuracy for Nonstrict-Feedback Switched Nonlinear Multiagent Systems

Distributed containment control issue with prescribed accuracy is addressed in this article for nonstrict-feedback switched nonlinear multiagent systems with aperiodically time-varying parameters. First, a novel <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">congelation of variables</i> method is adopted to deal with time-varying parameters, which are fast-varying in an unknown compact set with only their radii known a priori. Further, by means of the Gaussian basis function property of the radial basis function neural network, the backstepping recursive design method can work normally for nonstrict-feedback systems. In order to handle the switched dynamics and obtain the desired prescribed performance, a common Lyapunov function is designed by incorporating a series of continuously differentiable switching functions. Then, under the proposed containment control scheme, containment control is achieved, and meanwhile, each containment error converges with a prescribed accuracy, which is given before the containment controller is implemented. Finally, two simulations are performed to validate the theoretical results.

Fixed‐time fault‐tolerant control of uncertain MIMO switched interconnected systems subject to state delay, unknown control directions, and quantized nonlinear inputs

SummaryThis paper investigates design of an adaptive fixed‐time fault‐tolerant decentralized controller for a class of uncertain multi‐input multi‐output (MIMO) switched large‐scale non‐strict interconnected systems under arbitrary switching subject to unknown control directions, quantized nonlinear inputs, actuator failures unknown external disturbances, and unmodeled dynamics. In addition to interconnected terms, time‐varying delayed interconnected terms have been considered in the system model which makes it more general than previous works in the literature. The proposed controller can handle switched systems with unknown switching signal and different types of input nonlinearities including, saturation, backlash, and dead‐zone. The singularity problem in designing the fixed time controller has been solved. The quantizer and actuators fault parameters are assumed to be unknown. The Razumikhin lemma has been used to deal with the delayed interconnected terms. To cope with the system unknown dynamics, neural networks (NNs) have been applied and by updating the maximum norms of the networks weight vectors the computational load has been reduced. The explosion of complexity occurring in the traditional back‐stepping technique has been avoided by applying dynamic surface control (DSC). Finally, by defining an appropriate common Lyapunov function (CLF), fixed‐time convergence of system outputs and the closed‐loop system stability have been established. The effectiveness of the proposed controller has been shown via simulation study.

Hybrid neural network control of uncertain switched nonlinear systems with bounded disturbance

AbstractIn this work, we focus on the design of a hybrid neural tracking controller for a class of uncertain switched nonlinear systems with bounded disturbance. A new tracking control model is constructed using adaptive particle swarm optimization (APSO) based neural network called hybrid neural network (HNN). Hybrid neural tracking controller is developed by combining the backstepping approach and neural network approximation ability along with complexity analysis. A common Lyapunov function (CLF) is used for the stability of the proposed model, to develop a CLF for the switched system, a virtual control function is developed via adaptive law and HNN. By appropriately choosing the design parameters, it has been proven that all closed‐loop signals are semi‐globally uniformly ultimately bounded and the tracking error converges to a small bounded region around the origin. Finally, two numerical examples and a real‐life example about one‐link manipulator systems demonstrate the effectiveness of the proposed hybrid controller.

Output-Based Robust Switching Rule Design for Uncertain Switched Affine Systems: Application to DC–DC Converters

This brief presents a new output-based robust switching control rule for switched affine systems based on observer-based feedback control or dynamic output feedback control. The designed switching rule can drive the state of the switched affine system, which have various equilibrium points, to a desired one, and overcome the effects caused by the parameter uncertainty and external disturbance. The stability conditions for the closed-loop systems are established by applying a common Lyapunov function and are presented in the form of linear matrix inequalities. Finally, physical experiments in a DC-DC boost converter circuit illustrate the effectiveness of the proposed control method.

Small-gain technique-based adaptive output constrained control design of switched networked nonlinear systems via event-triggered communications

This paper investigates the problem of event-triggered tracking control for switched networked nonlinear systems with asymmetric time-varying output constraints. To handle the output constraints, an output-dependent generic constraint function is constructed to describe relationship between the output and the performance requirement. Meanwhile, an event-triggering rule is designed to reduce communication frequency between the controller and the actuator, thereby reducing the burden of the network communication. Based on the common Lyapunov function method and event-triggered control strategy, an adaptive control method is designed, which can guarantee that the closed-loop signals are bounded and avoid the Zeno behavior. Different from existing results considering constraints, the proposed scheme not only relaxes the restricted condition of constraint boundaries but also both the cases with and without output constraints can be addressed simultaneously. Furthermore, the stability of the system can be guaranteed by the small-gain technique. Finally, two simulation examples are provided to demonstrate the effectiveness of the proposed scheme.

Road-Curvature-Range-Dependent Path Following Controller Design for Autonomous Ground Vehicles Subject to Stochastic Delays

In this paper, we investigate the PID controller design problem of path following for an autonomous ground vehicle (AGV). Firstly, a bicycle model is adopted and a vehicle offset model from the target path is integrated to the bicycle model. The PID controller considers the vehicle longitudinal speed variation for adaption to the road curvature and the stochastic delay induced by the network communication. This is realized by transforming the tuning problem of proportional-integral-derivative (PID) gains for path following into a design problem of a static-output-feedback (SOF) controller for a time-delayed linear parameter varying (LPV) model form. A sufficient condition is adopted to guarantee the stability of the closed-loop system. In order to achieve better tracking performance, we propose a strategy in which the PID gains are piecewise constant and are dependent on the road-curvature ranges. The stability of the switched system is guaranteed via the common Lyapunov function method. Grey wolf optimizer (GWO) is employed to solve the optimization problem with maximum absolute tracking error as the optimization objective and stability condition and actuator dynamics as constraints. Both simulation results based on the CarSim-Simulink joint platform and hardware-in-loop experiment results are used to verify the effectiveness of the proposed control strategy.

Robust adaptive tracking control for a class of switched time‐delay systems with application of vehicle roll dynamic control

AbstractThe problems of stability and robust tracking control for a class of switched nonlinear systems with uncertain input and state delays are investigated in this study. In the presence of unknown functions and external disturbances, an adaptive fuzzy technique based on nonlinear disturbance observer is used to counteract the effects of external disturbances and approximate unknown functions. For this purpose, it is first assumed that a common Lyapunov function exists for the switched system, and then a new common Lyapunov–Krasovskii functional is built using the terms of the hypothetical common Lyapunov function. By employing this new function, the delay‐dependent input‐to‐state stability (ISS) under an arbitrary switching signal is provided. The robust tracking control problem for this system is investigated next. Finally, to assure ISS and robust tracking performance, an adaptive control law is developed that ensures conditions of the aforementioned hypothetical common Lyapunov function. In addition, to demonstrate the efficiency of the proposed strategy, the aforesaid method is applied to a vehicle roll dynamic as well as a mass‐spring system.

Secure consensus of multi‐agent systems under denial‐of‐service attacks

AbstractIn this paper, we study the secure consensus problem for multiple‐input‐multiple‐output (MIMO) linear multi‐agent systems (MASs) subject to denial‐of‐service (DoS) attacks, where an attack on an agent will block its associated communication channels until it stops. Firstly, we design an unknown input observer (UIO), based upon which we develop a resilient consensus controller, where the UIO depends only on the relative outputs. By employing a common Lyapunov function (CLF) and using the average dwell‐time (ADT) method, we show that secure consensus is achieved if the attack length rate is not greater than a positive threshold. Secondly, we design a resilient consensus controller with different control gains. By using the multiple Lyapunov functions (MLFs) technique, we show that secure consensus is achieved if the attack length rate and the attack frequency are respectively not greater than the corresponding positive thresholds. Finally, we present an example of multiple YF‐22 research UAVs to demonstrate the theoretical results.

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.