Common Lyapunov Function Research Articles

Adaptive fault tolerant control for a class of switched nonlinear systems with unknown control directions

This paper studies the problem of adaptive fault tolerant control for a class of switched nonlinear systems with unknown control directions. Through a coordinate transformation, an equivalent system is constructed to eliminate the adverse effects of the unknown control coefficients. Then combining with fuzzy approximation and backstepping design principle, an adaptive fault tolerant control scheme is proposed. Further, by constructing common Lyapunov functions, it is proved that closed-loop system is bounded and the tracking error remains in a neighborhood of the origin. The given simulation example demonstrates the effectiveness of the proposed control approach.

Invariant set-based robust fault detection and optimal fault estimation for discrete-time LPV systems with bounded uncertainties

ABSTRACTThis paper proposes an invariant set-based robust fault detection (FD) and optimal fault estimation (FE) method for discrete-time linear parameter varying (LPV) systems with bounded uncertainties. Firstly, a novel invariant-set construction method for discrete-time LPV systems is proposed if and only if the system is poly-quadratically stable, which need not satisfy the condition that there must exist a common quadratic Lyapunov function for all vertex matrices of the system compared to the traditional invariant-set construction methods. Furthermore, by using a shrinking procedure, we provide minimal robust positively invariant (mRPI) set approximations that are always positively invariant at each step of iteration and allow a priori desired precision to obtain a high sensitivity of FD. Owing to the existence of invariant set-based FD phase, the assumption that the initial faults should be bounded by a given set can be avoided for FE. We compute an optimal parametric matrix gain by minimising the Frobenius norm-based size of the corresponding FE set to obtain the optimal FE performance. Theoretically, any trajectory in the FE tube can be chosen as a specific-value estimation for the real fault signals. Finally, a vehicle dynamics system is used to illustrate the effectiveness of the proposed method.

Neural networks-based adaptive finite-time control of switched nonlinear systems under time-varying actuator failures

This paper is dedicated to neural networks-based adaptive finite-time control design of switched nonlinear systems in the time-varying domain. More specifically, by employing the approximation ability of neural networks system, an integrated adaptive controller is constructed. The main aim is to make sure the closed-loop system in arbitrary switching signals is semi-global practical finite-time stable (SGPFS). A backstepping design with a common Lyapunov function is proposed. Unlike some existing control schemes with actuator failures, the key is dealing with the time-varying fault-tolerant job for the switched system. It is also proved that all signals in the system are bounded and the tracking error can converge in a small field of the origin in finite time. A practical example is presented to illustrate the validity of the theory.

Power control of hybrid energy systems with renewable sources (wind-photovoltaic) using switched systems strategy

Abstract This paper presents a novel strategy for output power control of a hybrid energy system, composed of renewable energy sources (wind and photovoltaic), a battery bank and a variable load. Developing an energy management strategy, the hybrid energy system is described here as a switched nonlinear system with unknown parameters. Then, an adaptive control strategy is proposed to satisfy the total power demand in different conditions, under arbitrary switching. The proposed control law only depends on angular speed and current, without requiring the wind measurements. The analytical stability proof is given, based on the common Lyapunov function method. The simulation results are also presented and discussed to demonstrate the performance of the proposed method, for a typical hybrid energy system.

A common Lyapunov function for a differential inclusion with matrices having small commutators

Let A1 and A2 be asymptotically stable matrices. We consider switched continuous time linear systems described by the differential inclusion ẋ(t)={y(t):y(t)=Ax(t),A∈{A1,A2}}. The paper demonstrates that a common quadratic Lyapunov function exists for such systems, provided the commutator [A1,A2]=A1A2−A2A1 has a sufficiently small norm. Our results generalize the ones of Narendra and Balakrishnan (1994).

Event-triggered stabilizing controllers for switched linear systems

We introduce an event-triggered algorithm for the stabilization of switched linear systems. We define a pseudo-Lyapunov function common to all the subsystems. The pseudo-Lyapunov function is compared, at every time instant, to an exponentially decreasing upper threshold. An event is generated when the two functions intersect, or when a new subsystem becomes active. The existence of a Lyapunov function common to all the subsystems is a key requirement of this method. Nevertheless, imposing this condition does not add to the complexity of the problem. Indeed, we formulate the problem in terms of Linear Matrix Inequalities, as a generalized eigenvalue problem. This formulation allows to simultaneously check for the existence of a common Lyapunov function and to obtain the optimal parameters to define the upper threshold.We prove the stability of the system under the event-triggered control and we show that successive events are separated by a minimum interval of time.

Reduced-Order Observer-Based Adaptive Backstepping Control for Fractional-Order Uncertain Nonlinear Systems

This article presents two adaptive fuzzy output-feedback dynamic surface control schemes for single-input-single-output (SISO) strict-feedback fractional-order uncertain nonlinear systems under the complete smooth nonlinearity and partial piecewise-smooth nonlinearity, respectively. For the former, complete smooth nonlinear functions are approximated by employing fuzzy logic systems, based on which a novel fractional-order reduced-order observer is constructed to estimate the unmeasurable states. For the latter, a new switched approximation and estimate strategy is proposed to deal with the problem of piecewise-smooth nonlinearity. Meanwhile, fuzzy basis function property is utilized to eliminate the assumption requirement that the nonlinear functions must satisfy the Lipschitz condition inherent in the output-feedback control, which is suitable for two cases. Under certain assumptions, the stability of two closed-loop systems are proved via fractional-order Lyapunov function stability criterion. Specifically, the stability proof of the second case is built on the foundation of the common Lyapunov function method. Finally, two diverse Chua's circuit simulation examples are provided to, respectively, validate the effectiveness of the proposed two control strategy.

Global Fixed-Time Stabilization of Switched Nonlinear Systems: A Time-Varying Scaling Transformation Approach

This brief is concerned with the problem of global fixed-time stabilization for a class of switched nonlinear systems in strict-feedback form. A novel time-varying scaling transformation is introduced to convert the original fixed-time stabilization problem into the asymptotic stabilization problem of transformed systems. Based on this, a new framework to address fixed-time state feedback for switched nonlinear systems is developed by employing the common Lyapunov function method and the adding a power integrator technique. It is proved that the proposed controller renders the states of the closed-loop system to zero in any prescribed finite time under arbitrary switchings. Finally, simulation results of a practical example are provided to confirm the efficacy of the proposed method.

Adaptive finite-time fault-tolerant controller for a class of uncertain MIMO nonlinear switched systems subject to output constraints and unknown input nonlinearities

In this work, design of an adaptive finite-time fault-tolerant controller for a class of uncertain multi-input multi-output (MIMO) nonlinear switched systems with unmodeled dynamics subject to asymmetric time-varying output constraints and unknown faulty input nonlinearities has been addressed. The number of actuator faults can be infinite. In addition, the proposed control algorithm can cope with different unknown types of input nonlinearities namely, saturation, dead zone, backlash, and hysteresis. Actuator faults and input nonlinearities can be different in different modes. To estimate the system uncertainties, neural networks (NNs) have been employed and the unmodeled dynamics has been handled utilizing a dynamical signal. Finite-time convergence of the system outputs tracking errors to the neighborhood of the origin have been guaranteed, and the time-varying outputs constraints have been satisfied. By applying the backstepping control approach and using the common Lyapunov function (CLF) method, virtual and actual controllers and adaptive law for each subsystem have been constructed. The closed-loop stability and boundedness of all signals have been established. Through the simulation study, effectiveness of the proposed control scheme has been demonstrated.

Adaptive Fuzzy Tracking Control for a Class of Uncertain Switched Nonlinear Systems with Multiple Constraints: A Small-Gain Approach

This paper deals with the problem of adaptive fuzzy tracking control for a class of uncertain switched nonlinear systems. The considered issues include arbitrary switchings, unmodeled dynamics, input saturation, unknown dead-zone output, dynamic disturbances, and unmeasurable states, which makes the results more applicable. A Nussbaum-type function is exploited in the paper to overcome the difficulty existing in tracking the dead-zone output with unknown control direction. Furthermore, fuzzy logic systems are utilized to approximate the uncertain nonlinear system functions. Also, the state observer is constructed to approximate the unmeasurable states. Then, the adaptive fuzzy tracking controller with only three adaptive laws is presented on the basis of backstepping technique, common Lyapunov function, and small-gain approach. Under the designed controller, all the signals of the switched closed-loop systems are semi-globally, uniformly and ultimately bounded, and the tracking error is driven to a small area of the origin. Finally, simulation results are provided to demonstrate the effectiveness of the proposed control scheme.

Control of a small helicopter with linear matrix inequality-based design assuring stability and performance

This article focuses on linear matrix inequality-based controller designs that can achieve stabilization and reference tracking for a small unmanned helicopter at various flight conditions. A nonlinear mathematical model of a small-scale helicopter is constructed. Then trim conditions are found and linearized around different equilibrium points. Local [Formula: see text] controllers are designed at trim conditions based on the local linear models. The pointwise controllers achieve local stability and performance, but fail at stabilization and tracking over the full envelope. A scheduling controller is built by blending the local controller outputs. In addition, grid-based [Formula: see text] controllers are designed at each operating point with common Lyapunov function. This allows controller scheduling between the adjacent design points with guaranteed stability and performance across the design envelope. Based on the family of linear systems which are obtained from the nonlinear model, an affine parameter-dependent model is built to exploit the approximate linear parameter dependency. Then, a parameter-dependent linear parameter varying controller is synthesized for the affine parameter-dependent model. Although local performance is satisfactory for all given design methods, local [Formula: see text] controllers and affine parameter-dependent controller cannot yield satisfactory performance over the full flight envelope apart from the grid-based controller with common Lyapunov function approach.

Adaptive fuzzy output‐feedback tracking control for switched stochastic pure‐feedback nonlinear systems

SummaryThis paper considers the problem of adaptive fuzzy output‐feedback tracking control for a class of switched stochastic nonlinear systems in pure‐feedback form. Unknown nonlinear functions and unmeasurable states are taken into account. Fuzzy logic systems are used to approximate the unknown nonlinear functions, and a fuzzy observer is designed to estimate the immeasurable states. Based on these methods, an adaptive fuzzy output‐feedback control scheme is developed by combining the backstepping recursive design technique and the common Lyapunov function approach. It is shown that all the signals in the closed‐loop system are semiglobally uniformly ultimately bounded in mean square in the sense of probability, and the observer errors and tracking errors can be regulated to a small neighborhood of the origin by choosing appropriate parameters. Finally, a simulation result is provided to show the effectiveness of the proposed control method.

Circular formation flight control for unmanned aerial vehicles with directed network and external disturbance

This paper proposes a new distributed formation flight protocol for unmanned aerial vehicles ( UAVs ) to perform coordinated circular tracking around a set of circles on a target sphere. Different from the previous results limited in bidirectional networks and disturbance-free motions, this paper handles the circular formation flight control problem with both directed network and spatiotemporal disturbance with the knowledge of its upper bound. Distinguishing from the design of a common Lyapunov function for bidirectional cases, we separately design the control for the circular tracking subsystem and the formation keeping subsystem with the circular tracking error as input. Then the whole control system is regarded as a cascade connection of these two subsystems, which is proved to be stable by input-to-state stability ( ISS ) theory. For the purpose of encountering the external disturbance, the backstepping technology is introduced to design the control inputs of each UAV pointing to North and Down along the special sphere ( say, the circular tracking control algorithm ) with the help of the switching function. Meanwhile, the distributed linear consensus protocol integrated with anther switching anti-interference item is developed to construct the control input of each UAV pointing to east along the special sphere ( say, the formation keeping control law ) for formation keeping. The validity of the proposed control law is proved both in the rigorous theory and through numerical simulations.

Adaptive fuzzy finite-time fault-tolerant control for switched nonlinear large-scale systems with actuator and sensor faults

This paper investigates the problem of decentralized adaptive fuzzy finite-time control for switched nonlinear large-scale systems with actuator and sensor faults. To estimate unmeasured system states, a state observer is constructed based on the actual measurement value. Fuzzy logic systems are employed to approximate unknown nonlinear functions. By combining the common Lyapunov function approach and the recursive design method, a decentralized fault-tolerant controller is designed. It can be proven that all states of the closed-loop system are bounded and the tracking error can converge to a small neighborhood of the origin in finite time under arbitrary switchings. Finally, simulation results are given to verify the effectiveness of the proposed scheme.

Stabilization of a Class of Hybrid Systems by Switching Controllers with Input Constraints

In this paper, a hybrid control strategy using appropriate switching between a set of linear controllers is presented for stabilization of a class of hybrid systems with input constraints. The input constraints under consideration are novel reverse polytopic constraints avoiding a set of specified input values. The design procedure of the switching controllers is performed by solving a set of linear matrix inequalities for the full state feedback case and by solving a set of bilinear matrix inequalities for the output feedback case. Under the designed controllers, the closed-loop system is shown to be globally uniformly pre-asymptotically stable if a specific set of matrix inequalities ensuring the existence of a common Lyapunov function are satisfied. To avoid the Zeno behavior, a tunable parameter related to the controller gains is introduced and assessed. It is shown that the proposed switching control is superior to continuous state feedback control with input constraints. In addition, besides continuous linear systems, the proposed control strategy is applicable to hybrid or impulsive systems. Numerical examples are included to illustrate the proposed results.

Adaptive neural tracking control of nonlinear stochastic switched non-lower triangular systems with input saturation

This paper focuses on the problem of a class of nonlinear stochastic switched non-lower triangular systems with input saturation. A novel adaptive neural tracking controller is developed by constructing the appropriately common Lyapunov function and applying backstepping technique. The difficulties in the design process are how to deal with the non-lower triangular structure and input saturation. In response to these questions, the variable separation technique is used to address the problem of non-lower triangular structure and the input saturation function is approximated by the efficient dynamical system. Anything else, neural networks, as universal function approximators, are employed to estimate the unknown continuous functions. Finally, it is shown that all signals in the resulting closed-loop system are uniformly bounded and the tracking error converges to a small neighbourhood around zero. In order to highlight the effectiveness of the presented control strategy, two vivid simulation examples are presented at the end.

Adaptive learning-based finite-time performance of nonlinear switched systems with quantization behaviors and unmodeled dynamics

This paper proposes a novel finite-time adaptive learning strategy for a class of nonlinear switched systems with quantization behaviors and unmodeled dynamics under unpredictable switchings. The neural network learning framework is introduced to manifest the characteristics of nonlinear systems to obtain better performances. The system nonlinearities include more complicated non-strict feedback structure and full-state dependent unmodeled dynamics. In virtue of the nonlinear decomposition technique and the finite-time criterion, an adaptive learning strategy is developed step by step. Under the proposed strategy, a common Lyapunov function for all the subsystems is constructed to guarantee that in finite time all the signals of the switched system converge to a small domain near the origin under arbitrary switchings. An illustrative example is finally given to verify the effectiveness of the proposed main results.

H∞ control of periodic piecewise polynomial time-varying systems with polynomial Lyapunov function

In this paper, the H∞ control problem of periodic piecewise systems with polynomial time-varying subsystems is addressed. Based on a periodic Lyapunov function with a continuous time-dependent Lyapunov matrix polynomial, the H∞ performance is studied. The result can be easily reduced to the conditions for periodic piecewise systems with constant subsystems or linear time-varying systems based on a common Lyapunov function or a linear time-varying Lyapunov matrix. Moreover, an H∞ controller with time-varying polynomial controller gain is proposed as well, which could be directly solved with the linear matrix inequalities. A numerical example is presented to demonstrate the effectiveness of the proposed method.

Simultaneous assessment of cortisol levels in saliva, hair and fingernail specimens in depressive and healthy people

This paper concentrates on the adaptive tracking control problem for a class of switched nonlinear systems in nonstrict-feedback form with uncertain actuator backlash. The switching signal considered in our model is arbitrary. The adaptive tracking controller design procedure is divided into three parts. Firstly, a novel fuzzy backlash model is presented to describe the uncertain backlash. Then, the fuzzy backlash is defuzzified and approximated by the linear combination of a linear term and a disturbance-like term without constructing a backlash inverse. Finally, an adaptive backstepping-based tracking controller via common Lyapunov function (CLF) method is presented to guarantee the tracking error asymptotically converges to an adjustable region of zero. The stability analysis and simulation examples are presented at the end of this paper to further verify effectiveness of our control scheme.

Robust exponential stability of interconnected switched systems with mixed delays and impulsive effect

A class of nonlinear switched dynamical systems in the presence of impulsive effect, uncertainties and both variable and continuously distributed delays are investigated. By resorting to M-matrix property, inequality technique and the idea of vector Lyapunov function, several sufficient conditions are derived to check the robust exponential stability of the interconnected switched systems under arbitrary switching. The proposed method does not require that each subsystem shares a common Lyapunov function, nor that the value of suitable Lyapunov function at every switching time is known. Moreover, average dwell time approach is employed to obtain criteria for ensuring the global exponential stability of the system under constrained switching. Finally, two numerical examples are presented to demonstrate the correctness and effectiveness of the main results.