Common Lyapunov Function Approach Research Articles

Fixed-time adaptive neural tracking control for high-order nonlinear switched systems with input saturation and dead-zone

This article investigates the problem of tracking control for high-order switched nonlinear systems with input saturation and dead-zone. The uncertain nonlinear functions are estimated via applying radial basis function neural networks (RBF NNs). An improved transformation approach is designed to simplify the design complexity caused by input nonlinearities. In particular, a novel filter is presented to handle the difficulty of “explosion of complexity”. With the support of the common Lyapunov function (CLF) approach, a novel neural fixed-time dynamic surface control (DSC) scheme is proposed to assure all the signals in closed-loop systems are bounded and the output signal tracks the expected signal within fixed time. The simulation example illustrates the validity of the proposed control algorithm.

Distributed State Estimation Under Jointly Connected Switching Networks: Continuous-Time Linear Systems and Discrete-Time Linear Systems

In this paper, we address the distributed state estimation problem for both continuous-time linear time-invariant (LTI) systems and discrete-time LTI systems under switching networks. The observed system is jointly observable, i.e., each agent can only access a part of the measurement output of the observed system and cannot recover the full state by itself. The full state estimation has to be achieved by network communication of neighboring agents. In contrast to existing works, the salient feature of this work is that the developed approach can deal with the jointly connected switching network and thus is more resilient to unreliable communication. First, we propose a new observability decomposition method for linear systems in modal canonical form. Then, we design distributed observers for both the continuous-time and the discrete-time system. Based on the common Lyapunov function approach, we show that the switched estimation error system is asymptotically stable and thus, the full state estimation can be achieved under jointly connected switching networks.

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.

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.

Observer-based adaptive fuzzy finite-time tracking control of switched nonlinear systems

In this article, we investigate the finite-time output feedback control issue for switched nonlinear systems. In the design procedure, fuzzy logic systems are used for identifying the unknown functions in switched nonlinear systems. A common state observer is constructed such that the unmeasured states can be measured. Based on the framework of the backstepping algorithm, an observer-based finite-time adaptive tracking control strategy is proposed via the common Lyapunov function approach. The stability analysis process confirms in detail that all the closed-loop signals are bounded under arbitrary switchings and the tracking error can converge to a small neighbourhood of the origin after a finite period of time. In order to verify the feasibility of the proposed control scheme, a numerical simulation example is shown in this paper.

Decentralised adaptive fuzzy event-triggered control for nonlinear switched large-scale systems with unknown backlash-like hysteresis

A decentralised adaptive fuzzy event-triggered control problem of nonlinear switched large-scale systems with unknown hysteresis is investigated in this article, in which the unknown hysteresis is expressed by several nonlinear differential equations. The unknown nonlinearities in the systems are generally approximated by fuzzy logic systems due to their excellent function approximation abilities. Through employing dynamic surface control technology, the issue of complexity explosion resulting from repeated differentiation in the traditional backstepping method is overcome. A decentralised event-triggered controller is developed through adopting the common Lyapunov function approach. In the closed-loop system, the whole signals are semi-globally uniformly ultimately bounded which can be guaranteed through adopting the proposed scheme. Ultimately, the simulation results of two examples demonstrate the validity of the proposed strategy.

Quantized-State-Feedback-Based Neural Control for a Class of Switched Nonlinear Systems With Unknown Control Directions

This paper investigates the problem of unknown virtual control directions in a state-quantized adaptive recursive control design for a class of arbitrarily switched uncertain pure-feedback nonlinear systems in a band-limited network. State quantization is considered for state feedback control in a band-limited network. The primary contribution of this study is to provide a quantized state feedback adaptive control strategy to address the unknown control direction and arbitrarily switched nonaffine nonlinearities. Herein, a coupling problem between Nussbaum functions and quantization errors caused by quantized state feedback control laws is considered in the Lyapunov-based design and stability analysis. A state-quantized adaptive recursive control scheme using the function approximation is constructed without a priori knowledge of the signs of the control gain functions, where the estimated parameters and Nussbaum-type functions are adaptively updated via quantized states. Theoretical lemmas are derived to show that the adaptive parameters and quantization errors of the closed-loop signals are bounded using the proposed control scheme. The boundedness of the closed-loop signals and the convergence of tracking error to a neighborhood of the origin are proved using the common Lyapunov function approach. Two simulation examples are shown to illustrate the effectiveness of the proposed theoretical result.

Small-Gain Technique-Based Adaptive Neural Output-Feedback Fault-Tolerant Control of Switched Nonlinear Systems With Unmodeled Dynamics

In this article, the issue of adaptive neural fault-tolerant control (FTC) is addressed for a class of uncertain switched nonstrict-feedback nonlinear systems with unmodeled dynamics and unmeasurable states. In such a system, the uncertain nonlinear parts are identified by radial basis function (RBF) neural networks (NNs). Also, with the help of the structural characteristics of RBF NNs, the violation between the nontsrict-feedback form and backstepping method is tackled. Then, based on the small-gain technique, input-to-state practical stability (ISpS) theory, and common Lyapunov function (CLF) approach, an adaptive fault-tolerant tracking controller with only three adaptive laws is developed by designing an observer. It is shown that the designed controller can ensure that all the closed-loop signals are bounded under arbitrary switching, while the tracking error can converge to a small area of the origin. Finally, two simulation examples are provided to demonstrate the feasibility of the suggested control approach.

Finite-Time Stability of Hybrid Systems With Unstable Modes

In this work, we study finite-time stability of hybrid systems with unstable modes. We present sufficient conditions in terms of multiple Lyapunov functions for the origin of a class of hybrid systems to be finite-time stable. More specifically, we show that even if the value of the Lyapunov function increases during continuous flow, i.e., if the unstable modes in the system are active for some time, finite-time stability can be guaranteed if the finite-time convergent mode is active for a sufficient amount of cumulative time. This is the first work on finite-time stability of hybrid systems using multiple Lyapunov functions. Prior work uses a common Lyapunov function approach, and requires the Lyapunov function to be decreasing during the continuous flows and non-increasing at the discrete jumps, thereby, restricting the hybrid system to only have stable modes, or to only evolve along the stable modes. In contrast, we allow Lyapunov functions to increasebothduring the continuous flows and the discrete jumps. As thus, the derived stability results are less conservative compared to the earlier results in the related literature, and in effect allow the hybrid system to have unstable modes.

Adaptive control design for uncertain switched nonstrict-feedback nonlinear systems to achieve asymptotic tracking performance

In this paper, the adaptive asymptotic tracking control issue is addressed for uncertain switched nonlinear systems with nonstrict-feedback form under the dynamic surface control (DSC) framework. In contrast to most traditional control schemes that can only achieve bounded error tracking performances, the proposed control method in this paper can make the switched systems with unknown nonlinear functions achieve an asymptotic tracking performance. It is completed by introducing nonlinear filters with a compensation term to remove the boundary layer error stemming from using linear filters in the DSC process. Meanwhile, the approximation error caused by the use of radial basis function (RBF) neural networks (NNs) is compensated by an online updated parameter. Furthermore, the nonstrict-feedback form is handled by adopting the inherent properties of RBF NNs. Then, in terms of the backstepping technique and the common Lyapunov function (CLF) approach, the desired control law with only two adaptive laws is set up. Finally, the validity of the developed strategy is verified via simulation results.

Adaptive event-triggered tracking using nonlinear disturbance observer of arbitrarily switched uncertain nonlinear systems in pure-feedback form

An adaptive nonlinear disturbance observer (NDO) strategy is proposed for event-triggered tracking of switched uncertain pure-feedback nonlinear systems with mismatched external disturbances and arbitrary switching. Contrary to existing control results using NDOs for switched nonlinear systems, the main contribution of this study is the development of a common NDO-based event-triggered tracker with a triggering law to handle arbitrarily switched unknown nonaffine nonlinearities. Using the common Lyapunov function approach, a recursive NDO-based tracker design methodology is investigated to guarantee that all the closed-loop signals are uniformly ultimately bounded, and Zeno behavior is excluded in the resulting event-triggering mechanism. In each recursive step, the same adaptive estimate terms in the control laws and the NDO are derived via the function approximation technique to compensate for mismatched and switched unknown nonaffine nonlinearities. Finally, simulation examples are considered to verify the effectiveness of the proposed control technique.

Adaptive output-feedback neural tracking control for uncertain switched MIMO nonlinear systems with time delays

This paper investigates the problem of adaptive output-feedback neural tracking control for a class of uncertain switched multiple-input multiple-output (MIMO) nonstrict-feedback nonlinear systems with time delays. It should be emphasised that the design for the considered system is quite difficult due to its unknown factors caused by the unknown system coefficients and the unknown functions. In our proposed design procedure, neural networks (NNs) are introduced to identify the unknown nonlinear functions and a valid hypothesis is used to deal with the unknown system coefficients. Then, the developed switched filter can be utilised to estimate the unmeasured system states. On the basis of the backstepping technique and the common Lyapunov function (CLF) approach, an adaptive neural controller is constructed for each subsystem. It is proved that all signals existing in the switched closed-loop system are ultimately bounded under arbitrary switching and each system output can track the corresponding target trajectory within a small bounded error. Finally, simulation results are presented to illustrate the efficiency of the proposed control method.

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.

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.

Asymmetric prescribed performance‐barrier Lyapunov function for the adaptive dynamic surface control of unknown pure‐feedback nonlinear switched systems with output constraints

SummaryAn adaptive dynamic surface control scheme for a class of pure‐feedback nonlinear switched systems possessing completely unknown nonlinearities and output constraints is proposed in this paper. By enabling the barriers to vary with prescribed performance functions in time, an asymmetric prescribed performance‐barrier Lyapunov function (APP‐BLF) is developed to guarantee the prescribed tracking performance simultaneously with constraint satisfaction under arbitrary switching, while the initial conditional requirements are relaxed. To handle the nonaffine problem, an affine variable is constructed to transform pure‐feedback systems into a strict‐feedback form with the nonaffine and switching properties existing in time‐varying bounded parameters by utilizing the mean value theorem at each step of the backstepping design procedure. Next, by incorporating the supremum norm theory, dynamic surface control, and the Nussbaum gain technique, a novel adaptive approach is presented to overcome unknown nonlinearities, unknown control coefficients, and the explosion of complexity. At last, by using the common Lyapunov function approach in combination with the decoupled backstepping method, a common approximation‐free adaptive controller is constructed to guarantee that all the signals in the closed‐loop system are semiglobally uniformly bounded. Moreover, the proposed controller alleviates the problems of overparameterization and the computational burden. Two simulation studies are conducted to demonstrate the effectiveness, robustness, and applicability of the proposed approach.

Controller synthesis for switched T–S fuzzy positive systems described by the Fornasini–Marchesini second model

In this paper, controller synthesis problem is investigated for a class of Fornasini–Marchesini second type switched systems with time-varying delays under the Takagi–Sugeno fuzzy rules. Both the cases of state feedback controller and static output feedback controller are designed. By utilizing the common Lyapunov function approach and the multiple Lyapunov functions method, sufficient criteria are derived in the form of linear matrix inequalities to assure the resulting closed-loop 2-D systems to be positive and asymptotically stable, which can be easily computed using the existing numerical algorithms. Explicit design schemes for the state feedback/static output feedback controllers gain matrices are also presented. Finally, advantage and effectiveness of the proposed criteria are shown by a numerical example.

Local exponential stabilization for a class of uncertain nonlinear impulsive periodic switched systems with norm-bounded input

This paper investigates stabilization for a class of uncertain nonlinear impulsive periodic switched systems under a norm-bounded control input. The proposed approach studies stabilization criteria locally where the nonlinear dynamics satisfy the Lipschitz condition only on a subspace containing the origin, not on $$\mathbb {R}^{n}$$ . This makes the proposed approach applicable in most practical cases where the region of validity is limited due to physical issues. In presence of different resources of non-vanishing uncertainties, the main objective is to find a stabilizing control signal such that not only trajectories exponentially converge to a sufficient small ultimate bound, but also have the largest region of attraction. To this, for a more general model, we first propose several sufficient conditions using the common Lyapunov function approach. The proposed strategy allows the Lyapunov function to increase in some intervals, which is suitable when some of the subsystems are unstable and uncontrollable. We then apply these conditions to the targeted system, and the sufficient criteria are extracted in the forms of linear and bilinear matrix inequalities. To achieve the main goal, an optimization problem is also formulated which is solvable using augmented Lagrangian methods. Finally, some illustrative examples are presented to demonstrate the proposed approach.

Robust Fuzzy Adaptive Tracking Control for Nonaffine Stochastic Nonlinear Switching Systems.

This paper is concerned with the trajectory tracking control problem of a class of nonaffine stochastic nonlinear switched systems with the nonlower triangular form under arbitrary switching. Fuzzy systems are employed to tackle the problem from packaged unknown nonlinearities, and the backstepping and robust adaptive control techniques are applied to design the controller by adopting the structural characteristics of fuzzy systems and the common Lyapunov function approach. By using Lyapunov stability theory, the semiglobally uniformly ultimate boundness in the fourth-moment of all closed-loop signals is guaranteed, and the system output is ensured to converge to a small neighborhood of the given trajectory. The main advantages of this paper lie in the fact that both the completely nonaffine form and nonlower triangular structure are taken into account for the controlled systems, and the increasing property of whole state functions is removed by using the structural characteristics of fuzzy systems. The developed control method is verified through a numerical example.

Adaptive tracking control for a class of uncertain nonlinear systems with infinite number of actuator failures using neural networks

We consider adaptive compensation for infinite number of actuator failures in the tracking control of uncertain nonlinear systems. We construct an adaptive controller by combining the common Lyapunov function approach and the structural characteristic of neural networks. The proposed control strategy is feasible under the presupposition that the systems have a nonstrict-feedback structure. We prove that the states of the closed-loop system are bounded and the tracking error converges to a small neighborhood of the origin under the designed controllers, even though there are an infinite number of actuator failures. At last, the validity of the proposed control scheme is demonstrated by two examples.