On Observer Design for a Class of Nonlinear Systems Including Unknown Time-Delay

The observer design for nonlinear systems with unknown, bounded, time-varying delays, on both input and state, is still an open problem for researchers. In this paper, a new observer design for a class of nonlinear system with unknown, bounded, time-varying delay was presented. For the proof of the observer stability, a Lyapunov–Krasovskii function was chosen. Sufficient assumptions are provided to prove the practical stability of the proposed observer. Furthermore, the exponential convergence of the observer was proved in the case of a constant time delay. Simulation results were shown to illustrate the feasibility of the proposed strategy.

     

Adaptive full-order and reduced-order observers for the Lur’e differential inclusion system

This paper deals with the observer design for the Lur’e differential inclusion system with unknown parameters. The set-valued mapping in the differential inclusion is upper semi-continuous, closed, convex, bounded and monotone. First, under some assumptions an adaptive full-order observer is designed for the system. Then, under the same assumptions, a reduced-order observer is proved to exist. An example is provided to show the validation of the designed observers.     

Sliding Mode Techniques for Robust Trajectory Tracking as well as State and Parameter Estimation

Practical applications are often affected by uncertainties—more precisely bounded and stochastic disturbances. These have to be considered in robust control procedures to prevent a system from being unstable. Common sliding mode control strategies are often not able to cope with the mentioned impacts simultaneously, because they assume that the considered system is only affected by matched uncertainty. Another problem is the offline computation of the switching amplitude. Under these assumptions, important nonlinear system properties cannot be taken into account within the mathematical model of the system. Therefore, this paper presents sliding mode techniques, that on the one hand are able to consider bounded as well as stochastic uncertainties simultaneously, and on the other hand are not limited to the matched case. Firstly, a sliding mode control procedure taking into account both classes of uncertainty is shown. Additionally, a sliding mode observer for the simultaneous estimation of non-measurable system states and uncertain but bounded parameters is described despite stochastic disturbances. This is possible by using intervals for states and parameters in the resulting stochastic differential equations. Therefore, the Itô differential operator is involved and the system’s stability can be verified despite uncertainties and disturbances for both control and observer procedures. This operator is used for the online computation of the variable structure part gain (matrix of switching amplitudes) which is advantageous in contrast to common sliding mode procedures.     

Robust output-feedback SOSM control subject to unmatched disturbances and its application: A fixed-time observer-based method

This paper investigates the output-feedback control for a general class of multi-input multi-output (MIMO) linear systems in the presence of unmatched disturbances. Firstly, a new observer composed of a Luenberger observer and a novel hierarchical high-order sliding mode (HOSM) observer is proposed to identify the system states and disturbances, simultaneously. As one of the most remarkable properties, the convergence time of the proposed observer is bounded by a positive constant which is free of the system initial error conditions. Secondly, based on the proposed observer, a new second-order sliding mode (SOSM) controller is constructed by using a novel sliding surface with unmatched disturbances compensation. The proposed control law is a simply continuous function of time and thus can certainly reduce numerical chattering. Finally, to show the effectiveness of the theoretical results, an application to inverted pendulum system is used to make simulation comparison.     

A Technique for Multiple Signal Differentiation

The state-space method is applied for noise-tolerant multiple differentiation of signals from a certain class. The construction of differentiators is reduced to the design of a state observer for a dynamical system with an unknown bounded input. Any prespecified differentiation accuracy of signals from the given class can be achieved by increasing the observer dimension, i.e., the order of the differentiator transfer function. The proposed differentiators are tolerant to high-frequency additive noise.     

Robust state estimation and fault diagnosis for uncertain hybrid systems

Robust state estimation and fault diagnosis are challenging problems in the research into hybrid systems. In this paper a novel robust hybrid observer is proposed for a class of hybrid systems with unknown inputs and faults. Model uncertainties, disturbances and faults are represented as structured unknown inputs. The robust hybrid observer consists of a mode observer for mode identification and a continuous observer for continuous state estimation and mode transition detection. It is shown that the mode can be identified correctly and the continuous state estimation error is exponentially uniformly bounded. Robustness to model uncertainties and disturbances can be guaranteed for the hybrid observer by disturbance decoupling. Furthermore, the detectability and mode identifiability conditions are rigorously analyzed. On the basis of the robust hybrid observer, a robust fault detection and isolation scheme is presented also in the paper. Simulations of a hybrid four-tank system show the proposed approach is effective.     

Cascade Observer Design Method for Systems with Uncertainty

The asymptotic observer design problem is considered for a system with uncertainty (i.e., with an unknown bounded input) and with arbitrary relative order. If the zero dynamics is stable, then part of the state vector can be reconstructed asymptotically exactly, but so far there has not been an exhausting asymptotic observer design method for the part of the state vector formed by the derivatives of the measured output. We propose such a method for a system of second relative order and generalize the result to systems of arbitrary relative order.     

Robust state estimation and fault diagnosis for uncertain hybrid nonlinear systems

Robust state estimation and fault diagnosis are challenging problems in the research of hybrid systems. In this paper, a novel robust hybrid observer is proposed for a class of uncertain hybrid nonlinear systems with unknown mode transition functions, model uncertainties and unknown disturbances. The observer consists of a mode observer for discrete mode estimation and a continuous observer for continuous state estimation. It is shown that the mode can be identified correctly and the continuous state estimation error is exponentially uniformly bounded. Robustness to unknown transition functions, model uncertainties and disturbances can be guaranteed by disturbance decoupling and selecting proper thresholds. The transition detectability and mode identifiability conditions are rigorously analyzed. Based on the robust hybrid observer, a robust fault diagnosis scheme is presented for faults modeled as discrete modes with unknown transition functions, and the analytical properties are investigated. Simulations of a hybrid three-tank system demonstrate that the proposed approach is effective.     

Least Square Control Problems in Nonreflexive Spaces

We consider a control problem in a Banach space with a bounded observer, but an unbounded controller which takes values in the extrapolated Favard class. A least square regulator problem can be formulated if the observer and the admissible controls take values in Hilbert spaces. We prove that for this type of LQR-problem the value function is given by a Riccati operator, and that a bounded state feedback based on the Riccati operator yields the optimal control. April 30, 1999     

On synchronization of chaotic systems based on the Thau observer design

In this paper we revisit the Thau observer design and concern its application to the synchronization problem of two Lorenz name related systems in the master-slave formalism. The first one is the Lorenz-Stenflo system possessing a positively invariant ellipsoid while another one is the hyperchaotic Lorenz system possessing a positively invariant cylinder. Information about loci of these invariant domains is applied for the observer design. Further, we present one assertion related to one spectral inequality arisen in the process of assigning stable spectrum to the observer matrix and show its use in the observer design. We demonstrate the efficiency of synchronization schemes for the both of systems with help of numerical simulation.     

Lyapunov reducibility and stabilization of nonstationary systems with an observer

For a linear nonstationary control system with an observer, we assume that the coefficients are locally Lebesgue integrable and integrally bounded on ℝ and construct a linear feedback such that the closed-loop plant-controller system is Lyapunov reducible to the special triangular form corresponding to an independent shift of the diagonal coefficients in the original system and in the system of asymptotic estimation of the state by an arbitrary pregiven quantity. For a periodic system, we prove that the constructed controls and Lyapunov transformation are periodic. We obtain corollaries on the uniform stabilization and global controllability of the central and singular exponents of the system.     

A semi-global finite-time convergent observer for a class of nonlinear systems with bounded trajectories

This paper considers the problem of designing finite-time convergent observers for a class of lower-triangular nonlinear systems with bounded solution trajectories. Using the homogeneous domination approach, we construct an observer with homogeneous structure and saturation design, whose states will converge to the real states in a finite time by adjusting the observer gain. Several application examples of this finite-time convergent observer are discussed in this paper.     

Design of observers for nonlinear systems with H ∞ performance analysis

This paper investigates the problem of observer design for nonlinear systems. By using differential mean value theorem, which allows transforming a nonlinear error dynamics into a linear parameter varying system, and based on Lyapunov stability theory, an approach of observer design for a class of nonlinear systems with time‐delay is proposed. The sufficient conditions, which guarantee the estimation error to asymptotically converge to zero, are given. Furthermore, an adaptive observer design for a class of nonlinear system with unknown parameter is considered. A method of H _∞ adaptive observer design is presented for this class of nonlinear systems; the sufficient conditions that guarantee the convergence of estimation error and the computing method for observer gain matrix are given. Finally, an example is given to show the effectiveness of our proposed approaches. Copyright © 2013 John Wiley & Sons, Ltd.     

基于输入输出线性化方法的一类非线性系统观测器设计

研究一类非线性系统的观测器设计问题.基于输入输出线性化方法提出了一类非线性系统的观测器设计.并且此非线性系统具有多输入多输出的特点,证明了在适当条件下,提出的观测器保证了观测误差渐近趋于零.仿真例表明了所得结果的有效性.     

Equivalent linear observer-based tracker for stochastic chaotic system with delays and disturbances

Based on the state-space self-tuning control methodology, aunified method for determining the equivalent linear observer-basedtracker for a stochastic chaotic system with input and statedelays and deterministic disturbances is developed. In thisapproach, an equivalent linear observer is obtained for theconcerned complex system, which may have unknown system parameters,system and measurement noises, and inaccessible system states.More precisely, an equivalent delay-free system is obtainedin the estimating process of the self-tuning control loop, andthen a linear controller and an observer are designed. The proposedmethod significantly simplifies the design and the implementationprocedures of the tracker.     

Lower-order state-space self-tuning control for a stochastic chaotic hybrid system

^** Email: shtsai{at}mail.ncku.edu.tw In this paper, subject to acceptable closed-loop performance,an effective lower-order tuner for a stochastic chaotic hybridsystem is designed using the observer/Kalman filter identification(OKID) method, in which the system state in a general coordinateform is transformed to one in an observer form. The OKID methodis a time-domain technique that identifies a discrete input–outputmap by using known input–output sampled data in the generalcoordinate form, through an extension of the eigensystem realizationalgorithm. Moreover, it provides a lower-order realization ofthe tracker, with computationally effective initialization,for on-line "auto-regressive moving average process with exogenousmodel" -based identification and a lower-order state-space self-tuningcontrol technique. Finally, the chaotic Chen's system is usedas an illustrative example to demonstrate the effectivenessof the proposed methodology.     

Low-order self-tuner for fault-tolerant control of a class of unknown nonlinear stochastic sampled-data systems

Based on the modified state-space self-tuning control (STC) via the observer/Kalman filter identification (OKID) method, an effective low-order tuner for fault-tolerant control of a class of unknown nonlinear stochastic sampled-data systems is proposed in this paper. The OKID method is a time-domain technique that identifies a discrete input–output map by using known input–output sampled data in the general coordinate form, through an extension of the eigensystem realization algorithm (ERA). Then, the above identified model in a general coordinate form is transformed to an observer form to provide a computationally effective initialization for a low-order on-line “auto-regressive moving average process with exogenous (ARMAX) model”-based identification. Furthermore, the proposed approach uses a modified Kalman filter estimate algorithm and the current-output-based observer to repair the drawback of the system multiple failures. Thus, the fault-tolerant control (FTC) performance can be significantly improved. As a result, a low-order state-space self-tuning control (STC) is constructed. Finally, the method is applied for a three-tank system with various faults to demonstrate the effectiveness of the proposed methodology.     

Adaptive output feedback asymptotic stabilization of nonholonomic systems with uncertainties

A global adaptive output feedback control strategy is presented for a class of nonholonomic systems in generalized chained form with drift nonlinearity and unknown virtual control parameters. The purpose is to design a nonlinear output feedback switching controller such that the closed-loop system is globally asymptotically stable. By using the input-state scaling technique and an integrator back-stepping approach, an output feedback controller is given. A filter of observer gain is introduced for state and parameter estimates. Meanwhile, in order to avoid the over-parameters, a tuning function technique is utilized. A novel switching control strategy based on the output measurement of the first subsystem rather than time is used to overcome the uncontrollability of the x₀-subsystem in the origin. The proposed controller can guarantee that all the system states globally converge to the origin, while other signals maintain bounded. The numerical simulation testifies the effectiveness.     

A new adaptive observer design for a class of nonautonomous complex chaotic systems

This article is concerned with designing of a robust adaptive observer for a class of nonautonomous chaotic system with unknown parameters having unknown bounds. The proposed observer is established from the offered output measurement and robust against model uncertainties and external disturbances. Convergence analysis of the observation error dynamics is realized and proved by Lyapunov stabilization theory. Finally, for verification and demonstration, the proposed method is applied to the Chen as an autonomous chaotic system and the electrostatic transducer as a nonautonomous chaotic system. The numerical simulations illustrate the excellent performance of the proposed scheme. © 2014 Wiley Periodicals, Inc. Complexity 21: 145–153, 2015     

Pullback exponential attractors for the viscous Cahn-Hilliard equation in bounded domains

This work is devoted to the construction of a pullback exponential attractor for a viscous Cahn-Hilliard system in bounded domains. Our con-struction is based on the results obtained by Langa, Miranville and Real in [7].