Discrete-time Lipschitz Nonlinear Systems Research Articles

A new event-triggered distributed state estimation approach for one-sided Lipschitz nonlinear discrete-time systems and its application to wireless sensor networks

This article proposes the design of a distributed state estimator for a class of one-sided Lipschitz nonlinear systems over wireless sensor networks. The suggested estimation scheme utilizes the one-sided Lipschitz constraint in conjunction with quadratic inner-boundedness, which makes it applicable to a broader class of nonlinear systems. The proposed estimator design is evaluated under a conventional event-triggered mechanism both in the absence and presence of external perturbations. Furthermore, a novel event-triggering condition is introduced that ensures error convergence to the origin in the absence of external perturbations. It is further established that the inclusion of new triggering condition reduces the estimation error upper bounds in the presence of external disturbances and noises. Sufficient conditions for boundedness of estimation errors are derived for each case, and matrix inequalities are developed for the calculation of estimator gains. Finally, a numerical example is considered to illustrate the efficacy of the proposed estimator.

Ellipsoid-Based Interval Estimation for Lipschitz Nonlinear Systems

This article proposes an intervalestimation method for discrete-time Lipschitz nonlinear systems via observer design and ellipsoidal analysis. A robust observer with novel structure is presented, which provides more degrees of design freedom and can be applied to enhance the interval estimation performance. To handle the nonlinear term in Lipschitz systems, we use the reformulated Lipschitz property to transform the nonlinear error dynamics into a linear parameter-varying form. After the transformation, the robustness performance of the proposed observer is characterized with the aid of peak-to-peak index. Furthermore, the comprehensive interval estimation results are obtained by using the peak-to-peak performance and the ellipsoidal analysis. Numerical simulation results are conducted to demonstrate the viability and validity of the presented method.

H − / L ∞ UIO‐based actuator and sensor fault diagnosis design in the finite frequency domain for discrete‐time Lipschitz nonlinear systems

This study investigates the problem of fault diagnosis observer design in the finite frequency domain for discrete-time Lipschitz nonlinear systems simultaneously subjected to actuator and sensor faults and unknown input disturbances. An augmented descriptor system is developed by constructing an augmented state consisting of system states and sensor faults. A novel H−/L∞ unknown input observer (UIO) is proposed in the finite frequency domain using the generalized Kalman–Yakubovich–Popov lemma to render the residual sensitive to actuator faults and robust against disturbances, which can achieve a better fault diagnosis result compared with existing observers in the full frequency domain. In the fault diagnosis scheme, a time-varying threshold is presented based on the L∞ performance index to detect actuator faults, and the sensor fault estimation is simultaneously achieved through an asymptotical estimation of the augmented state. Design conditions for the proposed H−/L∞ UIO are derived and converted into linear matrix inequalities to solve its design matrices together. The effectiveness of the developed H−/L∞ UIO and the fault diagnosis scheme are validated via simulations of an example of a single-link flexible joint robot.

Event-triggered H–/L∞ fault detection observer design for discrete-time Lipschitz nonlinear networked control systems in finite-frequency domain

This paper proposes an event-triggered fault detection observer (FDO) design method for discrete-time Lipschitz nonlinear networked control systems (NCSs) in finite-frequency domain. First, a discrete event-triggered transmission scheme is proposed to mitigate the utility of limited network bandwidth. Second, with the aid of a reformulated Lipschitz property, the nonlinear error dynamics are converted into a linear parameter varying (LPV) networked system model. Third, based on this model, the finite-frequency index is used to measure the worst-case fault sensitivity performance and the norm from unknown disturbance to residual is used to measure disturbance robustness performance. Next, a residual evaluation and a dynamic threshold are synthesised based on the norm. Furthermore, sufficient conditions of the FDO design are derived and transformed by a set of linear matrix inequalities (LMIs). The proposed FDO design method can significantly reduce the data transmission to relieve the communication pressure, and can also achieve better FD performance than the full frequency domain. Finally, A numerical example is provided to demonstrate the effectiveness and applicability of the proposed design approach.

Active Disturbance Rejection Control for Nonaffined Globally Lipschitz Nonlinear Discrete-Time Systems

The design and analysis of active disturbance rejection control (ADRC) are considered for a globally Lipschitz nonlinear discrete-time system, which is nonaffine to control inputs. A local dynamic linearization is proposed to transfer the original nonaffined nonlinear system into a nonlinear system affine to control input such that the open problem of selecting a control gain in the feedback control law of ADRC is solved. Furthermore, both a parameter updating law and an adaptive extended state observer are designed to estimate control gains and total uncertainty, respectively. The stability of ADRC for nonlinear nonaffine discrete-time processes is analyzed rigorously for the first time with the aid of contraction mapping principle. The proposed adaptive feedback controller considers desired reference trajectory, parameter adaption, error feedback, and the compensation of total uncertainties caused by parameter estimation error and unknown nonlinearity simultaneously to achieve better control performance. The proposed method is nearly data driven since no other model information is needed except for the globally Lipschitz condition and the nonzero property of the partial derivative of the nonlinear system with respect to the control input. The theoretical result is verified by simulations.

New results on observer-based robust preview tracking control for Lipschitz nonlinear systems

In this article, the observer-based robust preview tracking control problem is revisited for discrete-time Lipschitz nonlinear systems. The proposed observer-based preview control scheme is composed of the integral control action, the observer-based state feedback control action, and the preview feedforward action of the reference signal. Sufficient design condition of controller and observer gains, which are able to ensure the simultaneously convergence of both the estimation error and the tracking error toward zero, is established in terms of linear matrix inequality by applying the Lyapunov function approach and several mathematical techniques. Compared with the existing result, the system model is more general, which could describe a larger range of practical processes. The observer-based preview controller design is simplified by computing the gain matrices of both observer and tracking controller simultaneously by only one-step linear matrix inequality procedure. Robustness against external disturbance is analyzed via the H∞ performance criterion to attenuate its effect on the performance signal. Finally, two numerical examples are provided to demonstrate the effectiveness of the suggested controller.

Observer-based trajectory tracking control with preview action for a class of discrete-time Lipschitz nonlinear systems and its applications

In this article, the observer-based preview tracking control problem is investigated for a class of discrete-time Lipschitz nonlinear systems. To convert the observer-based trajectory tracking problem into a regulation problem, the classical difference technique is used to construct an augmented error system containing tracking error signal and previewable reference knowledge. Then, a state feedback controller with specific structures is taken into consideration. Sufficient design condition is established, based on the Lyapunov function approach, to guarantee the asymptotic stability of the closed-loop system. By means of some special mathematical derivations, the bilinear matrix inequality condition is successfully transformed into a tractable linear matrix inequality. Meanwhile, the gains of both observer and tracking controller can be computed simultaneously only in one step. As for the original system, the developed tracking control law is composed of an integrator, an observer-based state feedback controller, and a preview action term related to the reference signal. Finally, two numerical examples are provided to demonstrate the effectiveness of the theoretical method.

Fault Detection for Lipschitz Nonlinear Systems With Restricted Frequency-Domain Specifications

This article deals with the problem of fault detection for discrete-time Lipschitz nonlinear systems subject to a class of restricted frequency-domain specifications. We present a novel observer structure with more design parameters, which can be applied to enhance the observer performance. The performances of fault sensitivity and disturbance robustness are characterized using finite-frequency <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$H_{-}$ </tex-math></inline-formula> and <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> indices, respectively. Less restrictive design conditions are obtained based on a reformulated Lipschitz property. Moreover, to detect faults timely, a novel dynamic threshold is synthesized based on zonotopic set-membership techniques. Simulation examples are conducted to demonstrate the viability and validity of the presented method.

Formula omitted] non-fragile observer-based dynamic event-triggered sliding mode control for nonlinear networked systems with sensor saturation and dead-zone input

This paper considers the problem of robust non-fragile observer-based dynamic event-triggered sliding mode control (SMC) for a class of discrete-time Lipschitz nonlinear networked control systems subject to sensor saturation and dead-zone input nonlinearity. First, an improved dynamic event-triggered scheme (DETS) in consideration of sensor saturation is proposed to reduce the number of data transmission. Next, a non-fragile observer is designed to estimate the system state, which facilitates the construction of the discrete sliding surface. By using a reformulated Lipschitz property, the error dynamics and sliding mode dynamics are modeled as a unified linear parameter varying (LPV) networked system with time-varying delays. Then, based on this model, sufficient conditions are established to guarantee the resulting closed-loop system to be asymptotically stable with a given disturbance attenuation level. Furthermore, an observer-based event-triggered SMC law is designed to drive the trajectories of the observer system onto a region near equilibrium point in a finite time in the presence of dead-zone input nonlinearity. Finally, two practical examples are employed to demonstrate the effectiveness of the proposed method.

New LMI Conditions for Reduced-order Observer of Lipschitz Discrete-time Systems: Numerical and Experimental Results

The objective of this paper is to propose a reduced-order observer for a class of Lipschitz nonlinear discrete-time systems. The conditions that guarantee the existence of this observer are presented in the form of linear matrix inequalities (LMIs). To handle the Lipschitz nonlinearities, the Lipschitz condition and the Young′s relation are adequately operated to add more degrees of freedom to the proposed LMI. Necessary and sufficient conditions for the existence of the unbiased reduced-order observer are given. An extension to \begin{document}${H}_\infty$\end{document} performance analysis is considered in order to deal with \begin{document}${H}_\infty$\end{document} asymptotic stability of the estimation error in the presence of disturbances that affect the state of the system. To highlight the effectiveness of the proposed design methodology, three numerical examples are considered. Then, high performances are shown through real time implementation using the ARDUINO MEGA 2560 device.

Robust Preview Control for a Class of Uncertain Discrete-Time Lipschitz Nonlinear Systems

This paper considers the design of the robust preview controller for a class of uncertain discrete-time Lipschitz nonlinear systems. According to the preview control theory, an augmented error system including the tracking error and the known future information on the reference signal is constructed. To avoid static error, a discrete integrator is introduced. Using the linear matrix inequality (LMI) approach, a state feedback controller is developed to guarantee that the closed-loop system of the augmented error system is asymptotically stable with H∞ performance. Based on this, the robust preview tracking controller of the original system is obtained. Finally, two numerical examples are included to show the effectiveness of the proposed controller.

H∞ observer-based event-triggered sliding mode control for a class of discrete-time nonlinear networked systems with quantizations

This paper investigates the problem of H∞ observer-based event-triggered sliding mode control (SMC) for a class of uncertain discrete-time Lipschitz nonlinear networked systems with quantizations occurring in both input and output channels. The event-triggered strategy is used to save the limited network bandwidth. Then, based on the zero-order-hold (ZOH) measurement, a state observer is designed to reconstruct the system state, which facilitates the design of the discrete-time sliding surface. Considering the effects of quantizations, networked-induced constraints and event-triggered scheme, the nonlinear state error dynamics and sliding mode dynamics are converted into a unified linear parameter varying (LPV) time-delay system with the aid of a reformulated Lipschitz property. By using the Lyapunov-Krasovskii functional and free weighting matrix, a new sufficient condition is derived to guarantee the robust asymptotic stability of the resulting closed-loop system with prescribed H∞ performance. And then the observer gain, event-triggering parameter and sliding mode parameter are co-designed. Furthermore, a novel SMC law is synthesized to force the trajectories of the observer system onto a pre-specified sliding mode region in a finite time. Finally, a single-link flexible joint robot example is utilized to demonstrate the effectiveness of the proposed method.

Preview tracking control for a class of discrete-time Lipschitz non-linear time-delay systems

AbstractThe design of a preview tracking controller for a class of discrete-time Lipschitz non-linear time-delay systems is considered in this paper. First, based on preview control theory, we introduce a difference operator and deduce an augmented error system which contains previewable information on the reference signal. The original tracking problem is thus transformed into a regulation problem. Second, we design a memory state feedback controller for the augmented error system. Applying Lyapunov’s second method, the sufficient condition for stability of the closed-loop system is derived in the form of linear matrix inequality (LMI), and the preview controller is obtained accordingly. Finally, to show the superiority of the controller proposed, two numerical examples are presented.

Robust Fault Tolerant Control for Discrete-Time Dynamic Systems With Applications to Aero Engineering Systems

Unexpected faults in actuators and sensors may degrade the reliability and safety of aero engineering systems. Therefore, there is motivation to develop integrated fault tolerant control techniques with applications to aero engineering systems. In this paper, discrete-time dynamic systems, in the presence of simultaneous actuator/sensor faults, partially decoupled unknown input disturbances, and sensor noises, are investigated. A jointly state/fault estimator is formulated by integrating an unknown input observer, augmented system approach, and optimization algorithm. Unknown input disturbances can be either decoupled by an unknown input observer, or attenuated by a linear matrix inequality optimization, enabling the estimation error to be input-to-state stable. Estimator-based signal compensation is then implemented to mitigate adverse effects from the unanticipated actuator and sensor faults. A pre-designed controller, which maintains normal system behaviors under a fault-free scenario, is allowed to work along with the presented fault tolerant mechanism of the signal compensations. The fault-tolerant closed-loop system can be ensured to mitigate the effects from the faults, guarantee the input-to-state stability, and satisfy the required robustness performance. The proposed fault estimation and fault tolerant control methods are developed for both discrete-time linear and discrete-time Lipschitz nonlinear systems. Finally, the proposed techniques are applied to a jet engine system and a flight control system for simulation validation.

Fault detection for discrete‐time Lipschitz non‐linear systems in finite‐frequency domain

In this study, the problem of fault detection (FD) observer design for discrete-time Lipschitz non-linear systems with finite-frequency specifications is investigated. The finite-frequency H ∞ and H − indices are introduced to measure the robustness to unknown disturbances and the sensitivity to faults, respectively. Based on the use of a reformulated Lipschitz property, the non-linear error dynamics are transformed into a linear parameter varying (LPV) system. A new lemma is proposed to characterise the system performances in finite-frequency domain. To reduce the conservativeness of the multi-objective problem, slack variable techniques are used to obtain the sufficient conditions for the design of FD observer which are derived in terms of linear matrix inequalities (LMIs). The proposed design method can provide less restrictive LMI conditions and achieve a better FD performance than the existing one in full frequency domain. Two numerical examples are given to show the effectiveness and superiority of the new results.

Event-triggered fault detection for discrete-time Lipschitz nonlinear networked systems in finite-frequency domain

The problem of event-triggered fault detection (FD) filter design for discrete-time Lipschitz nonlinear networked systems with finite-frequency specifications is investigated. The event-triggered transmission scheme is introduced to mitigate the utility of limited network bandwidth. The developed filter combines the H∞ and H− indices. For this class of systems, the generalized Kalman–Yakubovic–Popov lemma-based finite-frequency FD filter design methods are invalid. To solve this problem, the nonlinear error dynamics are transformed into a linear parameter varying (LPV) system based on the use of a reformulated Lipschitz property and a new lemma is developed to capture the system performances in finite-frequency domain. By introducing slack variable techniques, sufficient conditions for the design of FD filter are derived in terms of linear matrix inequalities (LMIs). The proposed design method can significantly reduce the data transmission and achieve a better FD performance than that in full frequency domain. Finally, two examples are presented to validate the effectiveness and superiority of the new results.

H∞ Filtering for Discrete-Time Nonlinear Singular Systems with Quantization

This paper investigates the problem of H∞ filtering for class discrete-time Lipschitz nonlinear singular systems with measurement quantization. Assume that the system measurement output is quantized by a static, memoryless, and logarithmic quantizer before it is transmitted to the filter, while the quantizer errors can be treated as sector-bound uncertainties. The attention of this paper is focused on the design of a nonlinear quantized H∞ filter to mitigate quantization effects and ensure that the filtering error system is admissible (asymptotically stable, regular, and causal), while having a unique solution with a prescribed H∞ noise attenuation level. By introducing some slack variables and using the Lyapunov stability theory, some sufficient conditions for the existence of the nonlinear quantized H∞ filter are expressed in terms of linear matrix inequalities (LMIs). Finally, a numerical example is presented to demonstrate the effectiveness of the proposed quantized filter design method.

Krein space-based H∞ adaptive smoother design for a class of Lipschitz nonlinear discrete-time systems

In this paper, the problem of H∞ adaptive smoother design is addressed for a class of Lipschitz nonlinear discrete-time systems with l2 bounded disturbance input. By comprehensively analyzing the H∞ performance, Lipschitz conditions and unknown parameter’s bounded condition, a positive minimum problem for an indefinite quadratic form is introduced such that the H∞ adaptive smoothing problem is achieved. A Krein space stochastic system with multiple fictitious outputs is constructed by associating with the minimum problem of the introduced indefinite quadratic form. The minimum of indefinite quadratic form is derived in the form of innovations through utilizing Krein space orthogonal projection and innovation analysis approach. Via choosing the suitable fictitious outputs to guarantee the minimum of indefinite quadratic form is positive, the existence condition of the adaptive smoother and its analytical solutions are obtained in virtue of nonstandard Riccati difference equations. The quality of the estimator is checked on an example.

Fault detection for discrete-time Lipschitz nonlinear systems with signal-to-noise ratio constrained channels

In this paper, the problem of fault detection for discrete-time Lipschitz nonlinear systems with additive white Gaussian noise channels subject to signal-to-noise ratio constraints is investigated. An optimal residual generator based on the mixed H−/H∞ performance index is designed to generate the so-called residual signal, and the H−-index is used to measure the minimum effect of faults on the residual signal, while the influence of unknown disturbances and channel noise on the residual signal is maximized by the means of the H∞-index. Then, in order to detect the occurrence of faults, a norm-based residual evaluation function is provided, and a dynamic threshold including upper bounds on the modulus of the solution of Lipschitz nonlinear systems and the stochastic properties of channel noise is also constructed. Finally, a simulated example is presented to demonstrate the effectiveness of the proposed approach.

Estimation and Compensation for Lipschitz Nonlinear Discrete-Time Systems Subjected to Unknown Measurement Delays

Unknown measurement delays usually degrade system performance and even damage a system under output feedback control, which motivate us to develop an effective method to attenuate or offset the adverse effect from the measurement delays. In this paper, an augmented observer is proposed for discrete-time Lipschitz nonlinear systems subjected to unknown measurement delays, enabling a simultaneous estimation for system states and perturbed terms caused by output delays. On the basis of the estimates, a sensor compensation technique is addressed to remove the influence from the measurement delays to the system performance. Furthermore, an integrated robust estimation and compensation technique is proposed to decouple constant piecewise disturbances, attenuate other disturbances/noise, and offset the adverse effect caused by the measurement delays. The proposed methods are applied to a two-stage chemical reactor with delayed recycle and to an electromechanical servosystem, which demonstrate the effectiveness of the present techniques.