Impulsive observer design for a class of switched nonlinear systems with unknown inputs

This paper investigates hybrid observer design of a class of unknown input switched nonlinear systems. The distinguishing feature of the proposed method is that the stability of all subsystems of the error switched systems is not necessarily required. First, an output derivative-based method and time-varying coordinate transformation are considered to eliminate the unknown input. Then in order to maintain a satisfactory estimation performance, an impulsive full-order and switched reduced-order observer are developed with a pair of upper and lower dwell time bounds and constructing time-varying Lyapunov functions combined with convex combination technique. In addition, the time-varying Lyapunov functions method is also used to analyze the stability of a class of error switched nonlinear systems with stable subsystems. Finally, two examples are presented to demonstrate the effectiveness of the proposed method.     

Command filtered adaptive output feedback design with novel Lyapunov-based analysis for nonlinear systems with unmodeled dynamics

This paper presents a novel Lyapunov function-based backstepping controller design to tackle the tracking problems for nonlinear systems with unmodeled dynamics and unmeasurable states. The coexistence of unmodeled dynamics and unmeasurable states is the main challenge, which calls for novel techniques to take these two factors into account simultaneously. First, the classical Luenberger observer is extended with a novel transformation function to decouple the original system state and state estimation error. In this way, the effect of unmodeled dynamics on system stability can be separately considered. On this basis, a command-filtered controller is designed to simplify the backstepping design procedures. It is worthy to pointed out that, a novel Lyapunov function is developed to simplify the stability analysis with command filter, where the filter errors, the observer error, compensated tracking errors, and parameter estimation errors can be guaranteed to be semi-globally uniformly ultimate bounded. The simulation studies are investigated to validate the effectiveness of the presented design scheme.     

Observer-based sliding mode control for a class of discrete systems via delta operator approach

In this paper, an observer-based sliding mode control (SMC) problem is investigated for a class of uncertain delta operator systems with nonlinear exogenous disturbance. A novel robust stability condition is obtained for a sliding mode dynamics by using Lyapunov theory in delta domain. Based on a designed sliding mode observer, a sliding mode controller is synthesized by employing SMC theory combined with reaching law technique. The robust asymptotical stability problem is also discussed for the closed-loop system composed of the observer dynamics and the state estimation error dynamics. Furthermore, the reachability of sliding surfaces is also investigated in state-estimate space and estimation error space, respectively. Finally, a numerical example is given to illustrate the feasibility and effectiveness of the developed method.     

Observer design and stability analysis for a class of PDE chaotic systems

This paper deals with observer design and stability for a class of partial differential equation (PDE) systems governed by one-dimensional wave equations with mixed derivative terms and superlinear boundary conditions, whose dynamics exhibits chaos when the system parameters change within certain ranges. Firstly, a sufficient and necessary condition that guarantees the stability of this class of systems is obtained. Secondly, based on the method of characteristics, an observer is designed by injecting the measurement output estimation error on the boundary, and the observation error dynamics is proved to be stable with a necessary and sufficient criterion, which can identity the range of the feedback gain for the observer. Finally, two numerical examples are provided to illustrate the validity of the theoretical conclusions.     

Dissipative observers for discrete-time nonlinear systems

This paper addresses the problem of designing a state observer for a class of nonlinear discrete-time systems using the dissipativity theory. We show that the dissipative observation methodology, originally proposed by one of the authors for continuous-time nonlinear systems, can be extended to the discrete-time case. For constructing a convergent observer, the methodology is applied to the nonlinear estimation error dynamics, which is decomposed into a discrete-time Linear Time-Invariant (LTI) subsystem in the forward loop, connected to a time-varying static nonlinearity in the feedback loop. In order to assure asymptotic stability of the closed-loop, complementary dissipativity conditions are imposed on each of the subsystems: (i) the static nonlinearity is required to be dissipative with respect to a quadratic supply rate, and (ii) the observer gains are designed such that the LTI system is dissipative with respect to a complementary supply rate. As in the continuous time framework, the proposed method includes as special cases, unifies and generalizes some observer design methods proposed previously in the literature. A great advantage of the Dissipative Observer Design Method proposed here is that it leads to Matrix Inequalities for the design of the observer gains, and these can be usually converted into Linear Matrix Inequalities (LMI’s). The results are illustrated using Chua’s Chaotic system.     

Observer-based sliding mode control for a class of uncertain nonlinear neutral delay systems

In this paper, the observer-based sliding mode control (SMC) problem is investigated for a class of uncertain nonlinear neutral delay systems. A new robust stability condition is proposed first for the sliding mode dynamics, then a sliding mode observer is designed, based on which an observer-based controller is synthesized by using the SMC theory combined with the reaching law technique. Then, a sufficient condition of the asymptotic stability is proposed in terms of linear matrix inequality (LMI) for the overall closed-loop system composed of the observer dynamics and the state estimation error dynamics. Furthermore, the reachability problem is also discussed. It is shown that the proposed SMC scheme guarantees the reachability of the sliding surfaces defined in both the state estimate space and the state estimation error space, respectively. Finally, a numerical example is given to illustrate the feasibility of the proposed design scheme.     

Observer design for a class of nonlinear fractional-order systems with unknown input

A full order fractional-order observer is designed for a class of Lipschitz continuous-time nonlinear fractional-order systems with unknown input. Sufficient conditions of existence for the designed observer and stability of state estimation error system are developed by reconstructing state and using general quadratic Lyapunov function. By applying fractional-order extension of Lyapunov direct method, the stability of the fractional-order state estimation error system is analyzed. Due to the conditions involving a nonlinear matrix inequality, a new sufficient condition with linear matrix inequality (LMI) is reformulated, which makes the full order fractional-order observer implemented easily by using Matlab LMI toolbox. Examples are taken to show the effectiveness of the proposed approach by numerical simulations.     

Disturbance observer-based robust missile autopilot design with full-state constraints via adaptive dynamic programming

This paper aims to develop a robust optimal control method for longitudinal dynamics of missile systems with full-state constraints suffering from mismatched disturbances by using adaptive dynamic programming (ADP) technique. First, the constrained states are mapped by smooth functions, thus, the considered systems become nonlinear systems without state constraints subject to unknown approximation error. In order to estimate the unknown disturbances, a nonlinear disturbance observer (NDO) is designed. Based on the output of disturbance observer, an integral sliding mode controller (ISMC) is derived to counteract the effects of disturbances and unknown approximation error, thus ensuring the stability of nonlinear systems. Subsequently, the ADP technique is utilized to learn an adaptive optimal controller for the nominal systems, in which a critic network is constructed with a novel weight update law. By utilizing the Lyapunov's method, the stability of the closed-loop system and the convergence of the estimation weight for critic network are guaranteed. Finally, the feasibility and effectiveness of the proposed controller are demonstrated by using longitudinal dynamics of a missile.     

Actuator fault diagnosis for a class of bilinear systems with uncertainty

In this paper, the actuator fault diagnosis problem for a class of bilinear systems with uncertainty is discussed. The system is transformed into two different subsystems. One is not affected by actuator fault, so an adaptive observer can be designed such that, under certain conditions, the observer error dynamics is stable. The other whose states can be measured is affected by the faults. The observation scheme is then used for model-based fault diagnosis. Finally, an example of a semiactive suspension system is used to illustrate the applicability of the proposed method.     

Interval state estimation for positive linear systems under DoS attacks

This paper is concerned with the interval state estimation problem for continuous-time positive linear systems under intermittent denial-of-service (DoS) attacks. To solve the problem, two types of estimate strategies are proposed. One is using the interval observer at all times, the other is using the interval observer in the absence of attacks but using, instead, the interval predictor otherwise. To facilitate the analysis, the interval state estimation problem is reformulated into the positivity and stability analysis of the associated error system. Then, stability conditions and disturbance attenuation characterization of the error systems for the two strategies are established via a mode-dependent Lyapunov approach. Roughly speaking, it is shown that the interval estimation accuracy of the former strategy is higher than the latter when the open loop system is stable. Finally, several numerical examples are provided to illustrate the ascendancy of the proposed estimation strategies.     

Robust sensor fault estimation scheme for satellite attitude control systems

The present paper proposes two new schemes of sensor fault estimation for a class of nonlinear systems and investigates their performances by applying these to satellite control systems. Both of the schemes essentially transform the original system into two subsystems (subsystems 1 and 2), where subsystem-1 includes the effects of system uncertainties, but is free from sensor faults and subsystem-2 has sensor faults but without any uncertainties. Sensor faults in subsystem-2 are treated as actuator faults by using integral observer based approach. The effects of system uncertainties in subsystem-1 can be completely eliminated by a sliding mode observer (SMO). In the first scheme, the sensor faults present in subsystem-2 are estimated with arbitrary accuracy using a SMO. In the second scheme, the sensor faults are estimated by designing an adaptive observer (AO). The sufficient condition of stability of the proposed schemes has been derived and expressed as a linear matrix inequality (LMI) optimization problem and the design parameters of the observers are determined by using LMI techniques. The effectiveness of the schemes in estimating sensor faults is illustrated by considering an example of a satellite control system. The results of the simulation demonstrate that the proposed schemes can successfully estimate sensor faults even in the presence of system uncertainties.     

A variable gain impulsive observer for Lipschitz nonlinear systems with measurement noises

This paper investigates the variable gain impulsive observer design problem for Lipschitz nonlinear systems. It is assumed that the measurements are contaminated by noise and received by observer at aperiodic instants. To establish a tractable design condition for impulsive observers, the piecewise linear interpolation method is used to construct the variable gain function. To quantify the impact of the measurement noises and exogenous disturbance on the estimation error, a Lyapunov-based condition for establishing exponential input-to-state stability (EISS) property of the observation error dynamics is presented. Then it is shown that the EISS condition can be expressed as a set of linear matrix inequalities (LMIs) by introducing a piecewise quadratic Lyapunov function. A convex optimization problem is proposed in which the EISS gain is minimized. Comparisons with the existing methods show the effectiveness of the proposed design technique.     

Dynamical robust H∞ filtering for nonlinear uncertain systems: An LMI approach

In this paper, a new approach to robust H_∞ filtering for a class of nonlinear systems with time-varying uncertainties is proposed in the LMI framework based on a general dynamical observer structure. The nonlinearities under consideration are assumed to satisfy local Lipschitz conditions and appear in both state and measured output equations. The admissible Lipschitz constants of the nonlinear functions are maximized through LMI optimization. The resulting H_∞ observer guarantees asymptotic stability of the estimation error dynamics with prespecified disturbance attenuation level and is robust against time-varying parametric uncertainties as well as Lipschitz nonlinear additive uncertainty.     

Dissipative state observer design for nonlinear time-delay systems

This paper is concerned with the design of dissipative state observers for a family of time-delay nonlinear systems. The Dissipativity method, proposed by one of the authors for delay-free nonlinear systems, is extended here to a class of time-delay nonlinear systems. The design method consists in decomposing the time-delay estimation error dynamics into a time-delay linear subsystem and a time-varying memoryless nonlinearity, connected in a negative feedback loop. By using some storage functionals, both delay-independent and delay-dependent dissipativity criteria are derived in order to guarantee the exponential convergence property of the observer. The exponential stability of the estimation error is then ensured, assuming that the nonlinearity is dissipative with respect to a quadratic supply rate and the linear part is designed, through the observer gains, to be dissipative with respect to a complementary supply rate. The design conditions are formulated in terms of tractable bilinear (BMI’s) or linear matrix inequalities (LMI’s). An interesting advantage is that the proposed dissipative design extends and generalizes under a unified framework several methods available in the literature, since a wide diversity of nonlinearities can be considered. Numerical examples are provided to demonstrate the effectiveness of the theoretical results.     

Event-triggered fault estimation and sliding mode fault-tolerant control for a class of nonlinear networked control systems

This paper is concerned with integrated event-triggered fault estimation (FE) and sliding mode fault-tolerant control (FTC) for a class of discrete-time Lipschtiz nonlinear networked control systems (NCSs) subject to actuator fault and disturbance. First, an event-triggered fault/state observer is designed to estimate the system state and actuator fault simultaneously. And then, a discrete-time sliding surface is constructed in state-estimation space. By the use of a reformulated Lipschitz property and delay system analysis method, the sliding mode dynamics and state/fault error dynamics are converted into a unified linear parameter varying (LPV) networked system model by taking into account the event-triggered scheme, actuator fault, external disturbance and network-induced delay. Based on this model and with the aid of Lyapunov–Krasovskii functional method, a delay-dependent sufficient condition is derived to guarantee the stability of the resulting closed-loop system with prescribed H_∞ performance. Furthermore, an observed-based sliding mode FTC law is synthesized to make sure the reachability of the sliding surface. Finally, simulation results are conducted to verify the effectiveness of the proposed method.     

State and unknown input estimations for discrete-time switched linear systems with average dwell time

In this paper, the problem of state and unknown input estimations for a class of discrete-time switched linear systems with average dwell time switching is investigated. First, a proportional integral observer with an exponential H_∞ performance is constructed to estimate the system state and unknown input simultaneously. Second, both of the observability and the stability of the estimation error system are analyzed, then the derivation of the observer gain matrices is transformed into the calculation of linear matrix inequalities. Third, the obtained results are extended to the systems with output disturbances. Finally, two simulation examples are provided to show the validity and effectiveness of the proposed methods.     

Approximate decoupling and output tracking for MIMO nonlinear systems with mismatched uncertainties via ADRC approach

In this paper, we consider output tracking for a class of MIMO nonlinear systems which are composed of coupled subsystems with vast mismatched uncertainties. First, all uncertainties influencing the performance of controlled outputs, which include internal unmodelled dynamics, external disturbances, and uncertain nonlinear interactions between subsystems, are refined into the total disturbance in the control channels of subsystems. The total disturbance is shown to be sufficiently reflected in the measured output of each subsystem so that it can be estimated in real time by an extended state observer (ESO) in terms of the measured outputs. Second, we decouple approximately the MIMO systems by cancelling the total disturbance based on ESO estimation so that each subsystem becomes approximately independent linear time invariant one without uncertainty and interaction with other subsystems. Finally, we design an ESO based output feedback for each subsystem separately to ensure that the closed-loop state is bounded, and the closed-loop output of each subsystem tracks practically a given reference signal. This is completely in comply with the spirit of active disturbance rejection control (ADRC). Some numerical simulations are presented to demonstrate the effectiveness of the proposed output feedback control scheme.     

Event-triggered adaptive fault-tolerant control of uncertain non-affine systems with predefined performance

This paper is concerned with an event-triggered adaptive fault-tolerant problem for an uncertain non-affine system. The implicit function theorem and mean value theorem are utilized to transform a non-affine system into an affine one, and an extended state observer and a tracking differentiator are used to estimate unknown dynamics and the derivative of virtual control laws, respectively. Adaptive laws are designed for unknown faults, and an event-triggered control scheme with a time-varying threshold, based on a tracking error and adaptive parameters, is developed. The tracking error is steered to converge to a bounded set with the help of a predefined performance function, and its transient performance is improved despite of faults. The stability of the closed-loop system is analyzed by the theorem of the input-to-state practically stability, and the Zeno behavior is excluded. Finally, two examples are given to illustrate the effectiveness of the proposed scheme.     

Active fault tolerant control of wind turbine systems based on DFIG with actuator fault and disturbance using Takagi–Sugeno fuzzy model

This paper presents an active fault tolerant control (FTC) for doubly fed induction generator (DFIG) with actuator fault and disturbance using Takagi–Sugeno (TS) fuzzy model. The control structure has two parts: fault and disturbance estimation part and FTC part. First, a TS fuzzy model is used to describe the DFIG system. Using a special linear transformation, the original system is decoupled into three independent subsystems: state subsystem without fault and disturbance, disturbance subsystem without fault, and fault subsystem without disturbance. By solving linear matrix inequalities (LMIs), a TS fuzzy observer is designed for the state subsystem without fault and disturbance. Second, estimations of faults and disturbance are obtained using the other subsystem models. Third, an active FTC scheme is developed to reduce the effect of disturbance and actuator faults. Finally, the performance of the proposed FTC is tested for a wind turbine system based on DFIG with actuator faults and disturbance. The simulation results demonstrate that the new FTC scheme makes possible to obtain an efficient fault and disturbance estimation and to reduce the peak current in the transient process.     

Observer based fault-tolerant control for a class of nonlinear networked control systems

In this paper, we present a fault-tolerant control (FTC) framework for a class of nonlinear networked control systems (NCSs). Firstly, the plant is transformed into two subsystems with one of them decoupled from the system fault. Then, the nonlinear observer is designed to provide the estimation of unmeasurable state and modelling uncertainty, which are used to construct fault estimation algorithm. Considering the sampling intervals occurred by net, a fault-tolerant control method is proposed for such nonlinear NCSs using the impulsive system techniques. The controller gain and the maximum sampling interval, which make the faulty system stable are given. An example is included to show the efficiency of the proposed method.