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.     

Fault detection for discrete-time fuzzy systems with measurement errors using interval observers

This paper presents an interval observer-based fault detection (FD) strategy for discrete-time T–S fuzzy systems with measurement errors. The system and measurement outputs are selected as the premise variables of plant and observer respectively. The bounds of mismatch items caused by the measurement errors are established by covering matched region, mismatched left adjacent region and right adjacent region. Piecewise Lyapunov function, taking full account of possible transitions, is employed to drive observer design condition. FD is turned into optimization problem with disturbance attenuation, fault sensitivity and nonnegativity constraints. The decision is implemented by judging whether zero is excluded from the residual interval. Finally, simulation is explored to verify the scheme.     

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.     

Robust observer-based fault estimation and accommodation of discrete-time piecewise linear systems

In this paper a new integrated observer-based fault estimation and accommodation strategy for discrete-time piecewise linear (PWL) systems subject to actuator faults is proposed. A robust estimator is designed to simultaneously estimate the state of the system and the actuator fault. Then, the estimate of fault is used to compensate for the effect of the fault. By using the estimate of fault and the states, a fault tolerant controller using a PWL state feedback is designed. The observer-based fault-tolerant controller is obtained by the interconnection of the estimator and the state feedback controller. We show that separate design of the state feedback and the estimator results in the stability of the overall closed-loop system. In addition, the input-to-state stability (ISS) gain for the closed-loop system is obtained and a procedure for minimizing it is given. All of the design conditions are formulated in terms of linear matrix inequalities (LMI) which can be solved efficiently. Also, performance of the estimator and the state feedback controller are minimized by solving convex optimization problems. The efficiency of the method is demonstrated by means of a numerical example.     

Zonotopic fault detection for discrete-time systems with limited communication capability and sensor nonlinearity

In this paper, the fault detection filter (FDF) design problem based on a dynamic event-triggered mechanism (DETM) is investigated for discrete-time systems with signal quantization and sensor nonlinearity. In order to conserve the limited network resources, a newly event-triggered mechanism with dynamic threshold is adopted to reduce the number of transmitted data through network more effectively. With the consideration of DETM, signal quantization and sensor nonlinearity, a fault detection filter is constructed to achieve the robustly asymptotic stability of established model with expected fault detection objective. In addition, by influence of DETM, external interference and quantization errors, a zonotopic residual evaluation mechanism is constructed to detect the occurring fault of plant. Finally, a practical example is provided to illustrate the effectiveness of proposed design approach.     

Robust fault detection for a class of nonlinear time-delay systems

In this paper, the robust fault detection filter (RFDF) design problems are studied for nonlinear time-delay systems with unknown inputs. Firstly, a reference residual model is introduced to formulate the robust fault detection filter design problem as an H_∞ model-matching problem. Then appropriate input/output selection matrices are introduced to extend a performance index to the time-delay systems in time domain. The reference residual model designed according to the performance index is an optimal residual generator, which takes into account the robustness against disturbances and sensitivity to faults simultaneously. Applying robust H_∞ optimization control technique, the existence conditions of the robust fault detection filter for nonlinear time-delay systems with unknown inputs are presented in terms of linear matrix inequality (LMI) formulation, independently of time delay. An illustrative design example is used to demonstrate the validity and applicability of the proposed approach.     

Minimal-order observer-based distributed fault detection and isolation for stochastic multi-agent systems

In this paper, the problem of distributed fault detection and isolation (FDI) is investigated for a class of linear discrete-time stochastic multi-agent systems (MASs) with additive Gaussian white noises. By using the information received from the generalized neighbor agents, a set of residual generators are designed for one agent based on the minimal-order observers. After dividing the MAS into several first-order components, the residuals are designed to be robust to the faults in some designated components and sensitive to the faults in all the other components. Combining with FDI strategies, multiple concurrent faults in the generalized neighbor agents can be detected and isolated simultaneously. In addition, a necessary condition is established for the observer to have the minimum order. By means of the statistical method, a set of hypothesis tests are derived to detect and isolate the faults. Finally, a simulation example is presented to show the feasibility and effectiveness of the proposed methods.     

Integrated event-triggered fault estimation and fault-tolerant control for discrete-time fuzzy systems with input quantization and incomplete measurements

This paper addresses the problem of robust integrated fault estimation (FE) and fault-tolerant control (FTC) for a class of discrete-time networked Takagi–Sugeno (T–S) fuzzy systems with two-channel event-triggered schemes, input quantization and incomplete measurements. The incomplete information under consideration includes randomly occurring sensor saturation and randomly occurring quantization. In order to save the limited networked resources, this paper firstly proposed a novel dynamic event-triggered scheme on the sensor side and a static one on the controller side. Secondly, an event-triggered FE observer for the T–S fuzzy model is designed to estimate actuator faults and system states, simultaneously. Then, a specified discrete sliding surface in the state-estimation space is constructed. By using time-delay analysis technique and considering the effects of event-triggered scheme, quantization, networked conditions, actuator fault and external disturbance, the sliding mode dynamics and error dynamics are unified into a new networked time-delay model. Based on this model, sufficient conditions are established such that the resulting augmented fuzzy system is stochastically stable with a prescribed H_∞ performance level with a single-step linear matrix inequality (LMI) formulation. Furthermore, an observer-based sliding mode controller for reaching motion is synthesized to guarantee the reachability of the sliding surface. Finally, a single-link flexible manipulator example is present to illustrate the effectiveness of the proposed method.     

Parameter fault detection and estimation of a class of nonlinear systems using observers

The focus of this paper is on the detection and estimation of parameter faults in nonlinear systems with nonlinear fault distribution functions. The novelty of this contribution is that it handles the nonlinear fault distribution function; since such a fault distribution function depends not only on the inputs and outputs of the system but also on unmeasured states, it causes additional complexity in fault estimation. The proposed detection and estimation tool is based on the adaptive observer technique. Under the Lipschitz condition, a fault detection observer and adaptive diagnosis observer are proposed. Then, relaxation of the Lipschitz requirement is proposed and the necessary modification to the diagnostic tool is presented. Finally, the example of a one-wheel model with lumped friction is presented to illustrate the applicability of the proposed diagnosis method.     

Unknown input observers for singular systems designed by eigenstructure assignment

In this paper, any unknown input observers with orders between minimum and full orders can be established for singular systems by eigenstructure assignment method. The complete and parametric solutions for the observer matrices and for the generalized eigenvectors are obtained. The completeness and parametric forms of the solutions and the flexibility in selecting the observer order allow the designer to choose a suitable observer according to the control purposes; hence, the solutions are quite suitable for advanced applications.     

On fault detection of discrete-time switched systems via designing time-varying residual generators

This paper investigates the problem of fault detection (FD) for discrete-time switched systems. Under a dwell time constraint, a switching rule that depends on the measured output is constructed for the system. Time-varying residual generators are designed such that the switched system is asymptotically stable and also with the detection performance under this switching rule. The advantages of the proposed technique are threefold: 1) It has the advantages of both slow switching and fast switching. 2) It can extend the classic design of time-invariant residual generator. 3) It can guarantee the switched system still has the desired fault detection performance even if all subsystems are without it. This feature reduces the performance requirements for each subsystem. A numerical example illustrates the effectiveness of the proposed method.     

Observer-based distributed fault detection and isolation for second-order multi-agent systems using relative information

This paper considers fault detection and isolation problem for second-order leader-follower multi-agent systems in a distributed setting. With the existing tracking protocol, an observer is constructed at each agent based on the concept of unknown input observer using relative information between the agent and its neighbors. Fault in position, velocity, both position and velocity dynamics are considered separately. Based on the residual signal generated from the designed observer, two algorithms are proposed so that the faulty agent in network can be detected and isolated simply. Effectiveness of the presented strategy is validated by some application examples.     

On functional observers for linear systems with unknown inputs and HOSM differentiators

The problem of constructing functional observers for linear systems with unknown inputs is considered. Necessary and sufficient conditions for the existence of a proper observer (without differentiations) are revisited. A simple and explicit form of a functional observer is presented. It is shown that when such observer is not proper, it is still possible to use the High-Order Sliding Mode differentiator to implement it. Nevertheless, in such case, additional conditions on the system and the unknown input are required.