Common Lyapunov Function Research Articles

Fuzzy Adaptive Switching Control for Stochastic Systems With Finite-Time Prescribed Performance.

The issue of fuzzy adaptive switching control for stochastic systems with arbitrary switching signal and finite-time prescribed performance is investigated in this article. A piecewise function is adopted to characterize finite-time prescribed performance, and the error signal is converted to a new state variable via the tangent function. Unknown functions are approximated via fuzzy-logic systems (FLSs). Based on the stochastic stability theory and common Lyapunov function, a fuzzy adaptive switching control scheme is presented. The control law is proposed for the stochastic switched closed-loop system so that not only all the signals are ensured to be semiglobally uniformly ultimately bounded (SGUUB) in probability but also a residual error related to the finite-time prescribed performance bound is guaranteed. Eventually, simulation studies for a practical system are given to show the effectiveness of the presented fuzzy adaptive switching control scheme.

On one inner point algorithm for common Lyapunov functions

In this paper, we consider the problem of existence and evaluation of common quadratic Lyapunov functions for a finite number of matrices. For this purpose, an inner point algorithm is proposed. The technique is based on Khalil (1982), where an algorithm is given which determines the diagonal stability of a matrix A . We obtain a necessary and sufficient condition for the existence and give an efficient algorithm which gives common solution in the case of existence and negative answer after a few iterations in the case of non existence. A number of examples are presented verifying the fast speed and efficiency of the obtained results.

Adaptive Fuzzy Output-Feedback Control for Switched Uncertain Nonlinear Systems With Full-State Constraints.

This article investigates an adaptive fuzzy tracking control approach via output feedback for a class of switched uncertain nonlinear systems with full-state constraints under arbitrary switchings. The adaptive observer and controller are designed based on fuzzy approximation. The main characteristic of discussed systems is that the state variables are not available for measurement and need to be kept within the constraint set. In order to estimate the unmeasured states, the adaptive fuzzy state observer is constructed. To guarantee that all the states do not violate the time-varying bounds, the tangent barrier Lyapunov functions (BLF-Tans) are selected in the design procedure. Based on the common Lyapunov function method, the stability of considered systems is analyzed. It is demonstrated that all the signals in the resulting system are bounded, and all the states are limited in their constrained sets. Furthermore, the simulation example is used to validate the effectiveness of the presented control strategy.

Stability analysis of switched ARX models and application to learning with guarantees

The main subject of this work is the stability analysis of Switched Auto-Regressive models with eXogenous inputs (SARX), which constitute a reference class for switched and hybrid system identification. The work introduces novel conditions for the arbitrary switching stability of multiple-input multiple-output SARX models which exploit the peculiar structure of their state-space realization. The analysis relies on the properties of block companion matrices, and partly leverages results from the theory of non-negative matrices, without nevertheless asking for an input–output positive behavior of the model. The novel stability conditions have a simple formulation in terms of linear co-positive common Lyapunov functions, and come at a remarkably low computational cost, being solvable by Linear Programming. The low computational burden is particularly attractive in an identification context, as it allows to efficiently constrain learning procedures in order to obtain SARX models with stability guarantees. The latter is itself a contribution of the work, as it fills a gap in the literature on the estimation of SARX models. The results are validated on a particular learning technique based on Regression Trees – a well known machine learning algorithm – which has shown remarkable accuracy in experimental environments.

Smooth Compositions Made Stabilization of Fuzzy Systems: Easy and More Robust.

Smooth fuzzy systems are the new structures of the fuzzy system which have recently taken attention for their capacity in system modeling. Hence, this article studies the stability of smooth fuzzy control systems and develops the sufficient conditions of the parameters for the stable closed-loop performance of the system. A major advantage of the presented conditions is that they do not call for a common Lyapunov function and therefore, no LMI is required to be solved to guarantee the stability of the fuzzy model. Besides, although they are the type-1 fuzzy model in nature, however, they show the high level of robustness to the noises and parametric uncertainties, which is comparable to the type-2 fuzzy models. Several comparative simulations demonstrate the capacity of the fuzzy models with the smooth compositions rather than the classical fuzzy models with the min-max or product-sum compositions.

Decentralized Event-Triggered Adaptive Fuzzy Control for Nonlinear Switched Large-Scale Systems With Input Delay Via Command-Filtered Backstepping

This article presents a decentralized fuzzy adaptive event-triggered command-filtered control scheme for switched large-scale nonlinear systems with input delay. Fuzzy logic systems are employed to approximate uncertain nonlinearities and a novel observer is constructed to estimate unmeasured states. The “explosion of complexity” defect inherent in the backstepping approach is overcome by employing command filter technology. An auxiliary system is designed to compensate for the effect of input delay, and a novel event-triggered decentralized controller is derived based on the common Lyapunov function method. This article shows that the presented strategy guarantees that all closed-loop variables are semiglobally uniformly ultimately bounded. Finally, simulation results are shown to further confirm the presented strategy’s validity.

Tracking control for switched nonlinear systems subject to output hysteresis via adaptive multi-dimensional Taylor network approach

For switched nonlinear systems with output hysteresis, by working on multi-dimensional Taylor network (MTN), an adaptive control strategy is put forward, which can effectively solve the tracking control problem. Firstly, the modified Bouc–Wen hysteresis model is employed to cope with the nonlinear problem caused by output hysteresis. Secondly, the Nussbaum function is introduced into the control design process, and MTN is used to approach nonlinear structures. Thirdly, a novel network adaptive control strategy is constructed with the help of a common Lyapunov function, which can realize the tracking control and stability of the system simultaneously. It is worth noting that as a new type of neural network, MTN is first used for control of switched systems with output hysteresis. Finally, two simulation examples show that the proposed control strategy is effective.

Adaptive tracking control of time‐varying delay switched nonstrict‐feedback nonlinear systems with unknown backlash‐like hysteresis

SummaryIn this article, the adaptive tracking control issue is addressed for uncertain switched nonstrict‐feedback nonlinear systems with unknown time‐varying delay and backlash‐like hysteresis under neural networks (NNs) estimation framework. In contrast to most existing results of nonlinear systems with time‐delay, the time‐varying delay we consider depends entirely on the number of switching subsystems. Based on the inherent properties of radial basis function NNs, the nonstrict‐feedback structure is effectively compensated. Meanwhile, with the help of the combination of Razumikhin function method, adaptive backstepping scheme, and dynamic surface control approach, the quadratic common Lyapunov function is established for all subsystems, and then an adaptive common control strategy is developed. For arbitrary switching signals, it is shown that all the signals in the closed‐loop system can be remained bounded and the system output can follow a desired signal under the designed controller. Finally, the validity of the presented strategy is verified via two simulation examples.

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.

Research on Microgrid Connected and Islanded Switching Control Based on Switched System

This paper focuses on the switching stability control of microgrid in the process of grid-connected and island operation. Firstly, the grid-connected operation model and island operation model are established, and they are regarded as two subsystems of switched system; then the cross backstepping method is used to design the switching controllers of the two subsystems, and the common Lyapunov function is constructed at the same time. Finally, matlab simulation verifies the effectiveness of the designed controller and switching law.

Model reference safety‐critical adaptive control for nonlinear and switched systems

SummaryThis paper studies the problem of model reference safety‐critical adaptive control for a class of nonlinear and switched systems with nonlinear reference models. An adaptive law is designed to estimate unknown parameters, while an energy function is proposed to guarantee the boundedness and safety of nonlinear systems. Also, a model reference safety‐critical adaptive control framework is established to ensure that the state trajectory of nonlinear systems is always within the safe set in the presence of unknown parameters. Moreover, the results obtained are extended to switched uncertain nonlinear systems under arbitrary switchings. A switched adaptive controller is designed to reduce the conservativeness caused by adoption of a common update law for all subsystems. Also, the safety of switched systems under arbitrary switchings is guaranteed as well as the boundedness of the error states between the switched nonlinear systems and the switched reference models by utilizing the common Lyapunov function method. Finally, two illustrative numerical examples are given to demonstrate the effectiveness of the proposed control architecture.

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.

Controller design for switched systems with nonlinear sub-systems using common Lyapunov functions

Controller design for switched systems with nonlinear sub-systems using common Lyapunov functions

Indirect Adaptive Control of Piecewise Affine Systems Without Common Lyapunov Functions

Indirect Adaptive Control of Piecewise Affine Systems Without Common Lyapunov Functions

Simultaneous interval state and fault estimation for continuous-time switched systems

This paper deals with simultaneous interval state and fault estimation for continuous-time switched systems subject to unknown but bounded state disturbances. By taking the actuator fault vector as a part of an extended state vector, the original system is transformed into an augmented descriptor system. Under this augmented descriptor form, and by using a new structure of interval observers, the conservatism of observer gain matrices can be significantly reduced. The effect of disturbances is attenuated based on an L∞ approach. Then, the interval observer gains are computed by solving Linear Matrix Inequalities (LMIs) derived from a common Lyapunov function. Finally, a simulation example is given to illustrate the effectiveness of the proposed method.

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.

Adaptive robust design for nonlinear switched systems with unmodeled dynamics and input quantization

AbstractIn this paper, we focus on the adaptive robust input quantized control problem for a class of uncertain nonlinear switched systems under a state‐dependent switching law. The considered systems admit the non‐ISS unmodeled dynamics, the unknown virtual control coefficients, and an unknown parameter. In order to reduce the conservativeness of the common Lyapunov function, the multiple Lyapunov functions method is used to design a state‐dependent switching law. Besides, to overcome the design difficulty arising from the quantization behavior, a nonlinear decomposition method is used to establish the relationship between the control signal and the quantized signal. We put forward the adaptive robust controllers for the individual subsystems, based on which the resulted closed‐loop system is uniformly ultimately bounded. Finally, to verify the effectiveness of the proposed control scheme, two examples are given to construct the adaptive robust controllers.

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.

Adaptive Consensus of Non-Strict Feedback Switched Multi-Agent Systems With Input Saturations

This paper considers the leader-following consensus for a class of nonlinear switched multi-agent systems (MASs) with non-strict feedback forms and input saturations under unknown switching mechanisms. First, in virtue of Gaussian error functions, the saturation nonlinearities are represented by asymmetric saturation models. Second, neural networks are utilized to approximate some unknown packaged functions, and the structural property of Gaussian basis functions is introduced to handle the non-strict feedback terms. Third, by using the backstepping process, a common Lyapunov function is constructed for all the subsystems of the followers. At last, we propose an adaptive consensus protocol, under which the tracking error under arbitrary switching converges to a small neighborhood of the origin. The effectiveness of the proposed protocol is illustrated by a simulation example.

Adaptive fuzzy asymptotic control for switched nonlinear systems with state constraints

This article presents an adaptive fuzzy asymptotic tracking control method for switched nonlinear systems with state constraints. The fuzzy logic systems are employed to dispose the lumped unknown dynamics. The state constraints are addressed with the aid of the barrier Lyapunov function. Moreover, drawing support from some well-defined smooth functions, the asymptotic tracking controller is recursively constructed based on backstepping technique. Through Lyapunov analysis and the common Lyapunov function method, it is shown that the system output tracking error asymptotically converges to zero and all system signals are bounded under the switching signal. Finally, the validity of the developed scheme is elucidated by simulation example.