Basis-dependent Lyapunov Function Research Articles

Nonfragile H ∞ Control for Uncertain Takagi–Sugeno Fuzzy Systems Under Digital Communication Channels and Its Application

This work addresses the nonfragile <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$H_{\infty }$ </tex-math></inline-formula> controller design problem for a class of discrete-time uncertain nonlinear systems. The norm-bounded uncertainty is contained in the nonlinear plant, which is described by the well-known Takagi–Sugeno (T–S) fuzzy model. The controller gain perturbation is also considered. When the input signal is transmitted from the controller to the system through the digital communication channels, it will be quantized by a static quantizer. The main attention is to design the nonfragile <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$H_{\infty }$ </tex-math></inline-formula> state-feedback controller for the closed-loop quantized uncertain T–S fuzzy system. By introducing some auxiliary scalars and the fuzzy basis-dependent Lyapunov function approach, sufficient conditions are established in the form of linear matrix inequalities (LMIs). The construction for the nonfragile <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$H_{\infty }$ </tex-math></inline-formula> controller can be completed by solving these LMIs. In the end, the applicability and effectiveness of the proposed method have been illustrated by two simulation examples.

Nonfragile peak-to-peak filtering for discrete-time Takagi-Sugeno fuzzy systems with dynamic quantizations

This paper investigates the problem of nonfragile peak-to-peak filtering for a class of discrete-time nonlinear networked control systems with dynamic quantizations. The Takagi-Sugeno fuzzy model is applied to approximate the considered nonlinear networked control systems. The perturbations are also considered in the peak-to-peak filter to be designed. Unlike most work about the peak-to-peak filtering for dynamic systems with quantization, both the performance output signal and measurement output signal are quantized by two different dynamic quantizers in our work. The main attention is to design the nonfragile peak-to-peak filter which guarantees the filtering error system to be asymptotically stable and possess a peak-to-peak performance index. By employing the fuzzy basis-dependent Lyapunov function approach and the S-procedure, new sufficient conditions about designing the nonfragile peak-to-peak filter have been established, which are expressed in the form of linear matrix inequalities. In the end, a numerical example is offered to verify the effectiveness of the designed nonfragile peak-to-peak filter.

Finite-time synchronization for periodic T–S fuzzy master-slave neural networks with distributed delays

Finite-time (FT) synchronization for periodic T–S fuzzy master-slave neural networks (NNs) with distributed delays is addressed in this work. A fuzzy controller is designed for the salve NNs to synchronize the master NNs in FT and a synchronization error system (SES) is derived. Sufficient conditions are established to guarantee that the SES is FT bounded by using the mode and fuzzy basis dependent Lyapunov function. A new algorithm is proposed to obtain the suboptimal boundary of the SES to analyze how the periodic characteristics affect the system boundary. Finally, a numerical example is provided to demonstrate the validity of the fuzzy controller and the iterative algorithm for the boundary.

Non-fragile dissipative filtering of cyber–physical systems with random sensor delays

This paper considers the problem of non-fragile state estimation under dissipative constraint for a class of nonlinear cyber–physical systems (CPSs) with sensor delays. The dynamics of the considered CPSs is characterized by the well-known T–S fuzzy model and system measurements are valued by wireless sensors. The communication link between the filter and the plant is described by a relatively practical model and sensor delays occurred in signal transmissions are taken into consideration. A stochastic variable which yields the standard Bernoulli distribution is exploited to model sensor delays encountered by the sensor measurements. With the help of a basis-dependent Lyapunov function and predefined performance constraint, sufficient conditions are then developed to establish the stochastic stability as well as strict dissipativity for the resultant filtering error system. The existence of the corresponding filter is guaranteed and the expression of desired filter parameters are shown explicitly. In the end, the established theoretical results are validated by a tunnel diode circuit example and corresponding simulations are also provided.

Fuzzy Output Tracking Control and Filtering for Nonlinear Discrete-Time Descriptor Systems Under Unreliable Communication Links.

In this paper, the problems of output tracking control and filtering are investigated for Takagi-Sugeno fuzzy-approximation-based nonlinear descriptor systems in the discrete-time domain. Especially, the unreliability of the communication links between the sensor and actuator/filter is taken into account, and the phenomenon of packet dropouts is characterized by a binary Markov chain with uncertain transition probabilities, which may reflect the reality more accurately than the existing description processes. A novel bounded real lemma (BRL), which ensures the stochastic admissibility with H∞ performance for fuzzy discrete-time descriptor systems despite the uncertain Markov packet dropouts, is presented based on a fuzzy basis-dependent Lyapunov function. By resorting to the dual conditions of the obtained BRL, a solution for the designed fuzzy output tracking controller is given. A design method for the full-order fuzzy filter is also provided. Finally, two examples are finally adopted to show the applicability of the achieved design strategies.

Finite frequency H∞ filter design for T-S fuzzy systems: New approach

This paper studies the problem of H∞ filter analysis and design for discrete-time Takagi–Sugeno (T-S) fuzzy systems, in a finite frequency (FF) domain. Based on a fuzzy basis dependent Lyapunov function and time domain interpretations of Generalized Kalman Yakubovich Popov (GKYP) lemma, combined with generalised S-procedure, sufficient conditions are developed, guaranteeing that the filtering error system is stable and has a prescribed H∞ attenuation level, over a specified FF domain of the external disturbances. Furthermore, several additional slack variables are introduced to the derived conditions by applying Finsler’s lemma twice, which leads to performance improvement and less conservatism in the solution. Thus, new filter design conditions are proposed in the formulation of Linear Matrix Inequality (LMI) terms in three different frequency ranges, (low, middle and high frequencies). A numerical example is given to demonstrate the feasibility, the effectiveness and the less conservatism of the proposed approach in comparison with another solution that appears in the literature.

L ∞ Fuzzy filter design for nonlinear systems with missing measurements: Fuzzy basis-dependent Lyapunov function approach

In this paper, l∞ fuzzy filtering problem is dealt for nonlinear systems with both persistent bounded disturbances and missing probabilistic sensor information. The Takagi–Sugeno (T–S) fuzzy model is adopted to represent a nonlinear dynamic system. The measurement output is assumed to contain randomly missing data, which is modeled by a Bernoulli distributed with a known conditional probability. To design the l∞ fuzzy filter and guarantee tracking performance, the effect of the perturbation against persistent bounded disturbances is reduced by using the minimum l∞ performance. By using the fuzzy basis-dependent Lyapunov function approach, a sufficient condition is established that ensure the mean square exponential stability of the filtering error. The proposed sufficient condition is represented as some linear matrix inequalities (LMIs), and the filter gain is obtained by the solution to a set of LMIs. Finally, the effectiveness of the proposed design method is shown via an example.

Dissipativity-based filtering of nonlinear periodic Markovian jump systems: The discrete-time case

This paper is concerned with the problem of dissipative filtering for discrete-time periodic Markovian jump systems with bounded nonlinearity. It is assumed that the data measurements from the plant to the filter are subject to randomly occurred packet dropouts satisfying Bernoulli distribution. The purpose of this paper is to design a filter such that the filtering error system is stochastically stable and strictly (Q,S,R)-dissipative. To eliminate the cross coupling between the Lyapunov matrix and system matrices, some slack matrix variables are introduced. Based on a mode-dependent and basis-dependent Lyapunov function, a sufficient condition of the desired dissipative filter in terms of linear matrix inequalities (LMIs) for discrete-time Periodic Markovian Jump Systems with bounded nonlinearity is derived. Finally, a numerical example is exploited to demonstrate the effectiveness of the proposed filter design method.

H∞ filter design for fuzzy systems with quantized measurements

This paper addresses the problem of H∞ filtering for a class of discrete time T–S model based fuzzy systems with quantized measurements. The measurement signal is quantized by a logarithmic quantizer. By using basis dependent Lyapunov function, a sufficient condition is provided to ensure the stability of the filtering error system and to guarantee a given H∞ performance level. Based on this analysis result of the filtering error system, an explicit filter design approach is proposed and the basis dependent filtering algorithm is further improved by using a kind of relaxed technique. Two numerical examples are also presented to show that the filtering algorithm based on the basis dependent Lyapunov function has less conservative than the one based on the basis independent Lyapunov function.

Formula omitted] filtering for time-delay T–S fuzzy systems with intermittent measurements and quantization

This paper investigates the H∞ filtering problem for Takagi–Sugeno (T–S) fuzzy systems with intermittent measurements. The communication channel between the plant and the filter is supposed to be imperfect, and a Bernoulli process is employed to model the phenomenon of the missing measurements. The T–S fuzzy system is first transformed into the form of interconnection of two subsystems by employing a two-term approximation for the delayed state variables. Scaled Small Gain (SSG) Theorem is developed to analyze the stability of stochastic systems. By using a basis-dependent Lyapunov function approach, sufficient conditions for the existence of the H∞ filters are established. Based on the conditions, the corresponding H∞ filter design scheme is proposed. Finally, a numerical example is provided to demonstrate the effectiveness of the proposed approach.

A new approach to non-fragile H ∞ observer-based control for discrete-time fuzzy systems

In this article, a new approach regarding H ∞ and non-fragile stability analysis and observer-based controller synthesis problem for a class of discrete-time fuzzy systems is investigated. The multiplicative gain variations under consideration are contained in both the controller gain and observer gain. By defining a basis-dependent Lyapunov function and introducing some additional matrix variables, new sufficient conditions are derived to guarantee the stability of the non-fragile H ∞ observer-based control system. Seeking computational convenience, the developed results are cast in the format of linear matrix inequalities and two numerical examples are presented to illustrate the feasibility of theoretical developments.

$H_{\infty }$ Fuzzy Control of Nonlinear Systems Under Unreliable Communication Links

This paper investigates the problem of H infin fuzzy control of nonlinear systems under unreliable communication links. The nonlinear plant is represented by a Takagi--Sugeno (T-S) fuzzy model, and the control strategy takes the form of parallel distributed compensation. The communication links existing between the plant and controller are assumed to be imperfect (that is, data packet dropouts occur intermittently, which appear typically in a network environment), and stochastic variables satisfying the Bernoulli random binary distribution are utilized to model the unreliable communication links. Attention is focused on the design of H infin controllers such that the closed-loop system is stochastically stable and preserves a guaranteed H infin performance. Two approaches are developed to solve this problem, based on the quadratic Lyapunov function and the basis-dependent Lyapunov function, respectively. Several examples are provided to illustrate the usefulness and applicability of the developed theoretical results.

$H_{\bm \infty}$ Fuzzy Filtering of Nonlinear Systems With Intermittent Measurements

This paper is concerned with the problem of <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">infin</sub> fuzzy filtering of nonlinear systems with intermittent measurements. The nonlinear plant is represented by a Takagi-Sugeno (T-S) fuzzy model. The measurements transmission from the plant to the filter is assumed to be imperfect, and a stochastic variable satisfying the Bernoulli random binary distribution is utilized to model the phenomenon of the missing measurements. Attention is focused on the design of an <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">infin</sub> filter such that the filter error system is stochastically stable and preserves a guaranteed <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">infin</sub> performance. A basis-dependent Lyapunov function approach is developed to design the <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">infin</sub> filter. By introducing some slack matrix variables, the coupling between the Lyapunov matrix and the system matrices is eliminated, which greatly facilitates the filter-design procedure. The developed theoretical results are in the form of linear matrix inequalities (LMIs). Finally, an illustrative example is provided to show the effectiveness of the proposed approach.

Fault Detection for Fuzzy Systems With Intermittent Measurements

This paper investigates the problem of fault detection for Takagi-Sugeno (T-S) fuzzy systems with intermittent measurements. The communication links between the plant and the fault detection filter are assumed to be imperfect (i.e., data packet dropouts occur intermittently, which appear typically in a network environment), and a stochastic variable satisfying the Bernoulli random binary distribution is utilized to model the unreliable communication links. The aim is to design a fuzzy fault detection filter such that, for all data missing conditions, the residual system is stochastically stable and preserves a guaranteed performance. The problem is solved through a basis-dependent Lyapunov function method, which is less conservative than the quadratic approach. The results are also extended to T--S fuzzy systems with time-varying parameter uncertainties. All the results are formulated in the form of linear matrix inequalities, which can be readily solved via standard numerical software. Two examples are provided to illustrate the usefulness and applicability of the developed theoretical results.

Robust H∞ control for discrete-time fuzzy systems via basis-dependent Lyapunov functions

This paper deals with the robust H ∞ control problem for a class of discrete-time fuzzy systems with uncertainty. The uncertainty is assumed to be of structured linear fractional form. By using basis-dependent Lyapunov function, an H ∞ control design approach is developed. The control design approach is facilitated by introducing some additional instrumental matrix variables. These additional matrix variables decouple the Lyapunov and the system matrices, which makes the control design feasible. The proposed approach leads to some sufficient results in the form of strict linear matrix inequalities (LMIs). It is expected that the basis-dependent results are less conservative than the basis-independent ones due to the introduction of basis-dependent Lyapunov function. Finally, numerical examples including the discrete chaotic Lorenz system are also given to demonstrate the applicability of the proposed approach.