Reduced order observer design for nonlinear systems

This work is a geometric study of reduced order observer design for nonlinear systems. Our reduced order observer design is applicable for Lyapunov stable nonlinear systems with a linear output equation and is a generalization of Luenberger’s reduced order observer design for linear systems. We establish the error convergence for the reduced order estimator for nonlinear systems using the center manifold theory for flows. We illustrate our reduced order observer construction for nonlinear systems with a physical example, namely a nonlinear pendulum without friction.     

Observer design for discrete-time nonlinear systems

This paper is a geometric study of the observer design for discrete-time nonlinear systems. First, we obtain necessary and sufficient conditions for local exponential observers for Lyaupnov stable discrete-time nonlinear systems. We also show that the definition of local exponential observers can be considerably weakened for neutrally stable discrete-time nonlinear systems. As an application of our local observer design, we consider a class of discrete-time nonlinear systems with an input generator (exosystem) and show that for this class of nonlinear systems, under some stability assumptions, the existence of local exponential observers in the presence of inputs implies and is implied by the existence of local exponential observers in the absence of inputs.     

A relation between the output regulation and the observer design for nonlinear systems

Output regulation and observer design are two important problems for nonlinear systems, and there is a vast literature addressing each problem separately in the control literature. Isidori and Byrnes [1] have solved the output regulation problem for nonlinear systems with a Poisson stable exosystem, and Sundarapandian [2] has solved the exponential observer design problem for Lyapunov stable nonlinear systems. In this paper, we demonstrate that for a special class of Lyapunov stable nonlinear systems, namely neutrally stable systems, the exponential observer design problem can be solved by converting it into an output regulation problem and then solving the new problem using the output regulation techniques of Isidori and Byrnes [1]. Finally, we present the corresponding results for the discrete-time case.     

Complements to full order observer design for linear systems with unknown inputs

This note completes the results obtained in [M. Darouach, M. Zasadzinski, S.J. Xu, Full-order observers for linear systems with unknown inputs, IEEE Trans. Automat. Control., AC-39 (3) (1994) 606–609]. It presents a full order observer for linear systems with unknown inputs in the state and in the measurement equations. It gives a more general approach for the observer’s design than that presented in the above mentioned reference and it shows that all the obtained results are independent of the choice of the generalized inverses considered in the observer’s design. Continuous- and discrete-time systems are considered.     

Observer synthesis method for Lipschitz nonlinear discrete-time systems with time-delay: An LMI approach

In this paper, we address the problem of observer design for a class of Lipschitz nonlinear discrete-time systems with time-delay. The main contribution lies in the use of a new structure of the proposed observer with a novel Lyapunov-Krasovskii functional. Thanks to these designs, new nonrestrictive synthesis conditions, expressed in terms of linear matrix inequalities (LMIs), are obtained. Indeed, the obtained LMIs contain more degree of freedom than those established by the approaches available in the literature which consider a simple Luenberger observer with a simple Lyapunov function for the stability analysis. An extension of the presented result to H_∞ performance analysis is given in order to take into account the noise (if it exists) affecting the considered system.     

基于观测器的模糊Delta算子系统的控制器设计

对于非线性模糊系统控制器和观测器的分析和设计,提出一种统一方法。利用Delta域离散T—S模糊模型对非线性系统建模,并基于李雅普诺夫稳定性理论给出模糊状态反馈控制器和观测器的设计策略,将所得结果归结为求解一组线性矩阵不等式。同时结论表明:分离性原理对Delta算子T—S模糊系统仍然成立。所得结果可将现有关于连续和离散T—S模糊系统的相关结论统一于Delta算子框架内。     

New results on general observers for discrete-time nonlinear systems

In this paper, we establish that detectability is a necessary condition for the existence of general observers (asymptotic or exponential) for discrete-time nonlinear systems. Using this necessary condition, we show that there does not exist any general observer (asymptotic or exponential) for discrete-time nonlinear systems with real parametric uncertainty, if the state equilibrium does not change with the parameter values and if the plant output function is purely a function of the state. Next, using center manifold theory for maps, we derive necessary and sufficient conditions for the existence of general exponential observers for Lyapunov stable discrete-time nonlinear systems. As an application of this result, we show that for the existence of general exponential observers for Lyapunov stable discrete-time nonlinear systems, the dimension of the state of the general exponential observer should not be less than the number of critical eigenvalues of the linearization matrix of the state dynamics of the plant.     

Minimum-order observers for discrete-time systems

We consider the synthesis of a minimum-order state or functional observer for a linear dynamical system. The synthesis problem is solved for completely certain systems of general form and for some classes of uncertain systems. Various approaches are described, which ultimately lead to the same task: finding a minimum-dimension Hurwitz solution for a system of linear equations with a Hankel matrix. For scalar and vector linear systems, prior upper and lower bounds on the observer dimension are derived, which makes it possible to switch to an iterative procedure of finding an optimal solution. The discussion is set out for discrete-time dynamical systems.     

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.     

High Gain Observers Using an Approximate Observer Normal Form

If one can transform a nonlinear system into observer normal form, it is possible to design an observer with exactly linear error dynamics. Unfortunately, many systems of practical relevance violate the existence conditions of the normal form. This drawback can be circumvented by means of an approximate observer normal form. The actual computation of the associated transformation may still be very difficult. We suggest a nonlinear observer design procedure that avoids this drawback. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

State observers for a class of multi-output nonlinear dynamic systems

This note considers the problem of observer design for a class of multi-output nonlinear systems. A new state observer design methodology for linear time-varying multi-output systems is presented. Furthermore, we show that the same methodology can be extended to a class of multi-output nonlinear systems. Some sufficient conditions for the existence of the proposed observer are obtained, which guarantee that the error of state estimation converges asymptotically to zero. An example is given to demonstrate the effectiveness of the proposed methodology.     

Fuzzy Partial State Feedback Control of Discrete Nonlinear Systems with Unknown Time-Delay

A new discrete-time fuzzy partial state feedback control method for the nonlinear systems with unknown time-delay is proposed. Ma et al. proposed the design method of the fuzzy controller based on the fuzzy observer and Cao and Frank extend this result to be applicable to the case of the nonlinear systems with the time-delay. However, the time-delay is likely to be unknown in practical. In this paper, the sufficient condition for the asymptotic stability is derived with the assumption that the time-delay is unknown by applying Lyapunov–Krasovskii theorem and this condition is converted into the LMI problem.     

Arbitrary observer scaling of all chaotic drive system states via a scalar synchronizing signal

This paper introduces a straightforward method to arbitrarily scale a drive system attractor using a synchronized linear observer. This scaling is controlled by a single observer parameter. Theoretical and simulation results for both continuous- and discrete-time systems demonstrate that linear observers can duplicate all chaotic system states in any desired scale using only a scalar synchronizing signal.     

Design of a nonlinear observer using automatic differentiation

Observer design for a class of observable nonlinear systems is considered. By a coordinate transformation and linearization around a desired trajectory the problem is reduced to the design for a linear time varying system. Calculation of the observer gain, depending on the desired trajectory, requires several derivatives. These are calculated using automatic differentiation by a combination of forward and reverse mode. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Local observer design for nonlinear systems

This paper is a geometric study of the local observer design for nonlinear systems. First, we obtain necessary and sufficient conditions for local exponential observers for Lyaupnov stable nonlinear systems. We also show that the definition of local exponential observers can be considerably weakened for neutrally stable nonlinear systems. As an application of our local observer design, we consider a class of nonlinear systems with an input generator (exosystem) and show that for this class of nonlinear systems, under some stability assumptions, the existence of local exponential observers in the presence of inputs implies and is implied by the existence of local exponential observers in the absence of inputs.     

Some Remarks on High Gain Observer Design

High gain observer design can be carried out for nonlinear systems, whose dynamics can be decomposed into a linear and a Lipschitz continuous nonlinear part. The observer uses a linear output injection, which is specified by a constant gain matrix. Due to this simple structure, high gain observers are frequently used in practical applications. The observer gain is often chosen via eigenvalue placement. However, the formal existence conditions are more complicated. In many cases, there is a finite bound on the maximum feasible Lipschitz constant of the nonlinear part for which the error dynamics can be stabilized. The existing results can be improved significantly, if the structure of the linear part is taken into account. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Dynamic output feedback fault tolerant controller design for discrete-time switched systems with actuator fault

This paper investigates the problem of dynamic output feedback fault tolerant controller design for discrete-time switched systems with actuator fault. By using reduced-order observer method and switched Lyapunov function technique, a fault estimation algorithm is achieved for the discrete-time switched system with actuator fault. Then based on the obtained online fault estimation information, a switched dynamic output feedback fault tolerant controller is employed to compensate for the effect of faults by stabilizing the closed-loop systems. Finally, an example is proposed to illustrate the obtained results.     

A numerical approach to design control invariant sets for constrained nonlinear discrete-time systems with guaranteed optimality

A numerical approach to design control invariant sets for constrained nonlinear discrete-time systems with guaranteed optimality is proposed in this paper. The addressed approach is based on the fact that zonotopes are more flexible for representing sets than boxes in interval analysis. Then the solver of set inversion via interval analysis is extended to set inversion via zonotope geometry by introducing the novel idea of bisecting zonotopes. The main feature of the extended solver of set inversion is the bisection and the evolution of a zonotope rather than a box. Thus the shape of admissible domains for set inversion can be broadened from boxes to zonotopes and the wrapping effect can be reduced as well by using the zonotope evolution instead of the interval evolution. Combined with global optimization via interval analysis, the extended solver of set inversion via zonotope geometry is further applied to design control invariant sets for constrained nonlinear discrete-time systems in a numerical way. Finally, the numerical design of a control invariant set and its application to the terminal control of the dual-mode model predictive control are fulfilled on a benchmark Continuous-Stirred Tank Reactor example.     

Observer-based optimal fault detection using PDFs for time-delay stochastic systems

This paper concerns the problem of fault detection (FD) for general stochastic systems with time delays. Different from the classical FD problem, the available information for the FD is the input and the measured output probability density functions (PDFs) of the system. Based on this, firstly, square root B-spline expansions technique is applied to model the PDFs so that the concerned FD problem is transformed into a nonlinear FD problem subject to the weight dynamical systems. Secondly, by using the FD observer as residual generator and guaranteed cost performance index as objective function, design of the FD observer is formulated as an optimization problem. A delay-dependent condition to solve this problem is established in terms of the feasibility of linear matrix inequality (LMI). Moreover, in order to improve FD performance, the guaranteed cost algorithm is applied to optimal observer design. Finally, two examples are given to demonstrate the applicability of the proposed approach.     

Non-Exponential Stabilization of Nonlinear Discrete-Time Systems

In this paper, we investigate the stabilization problem for a class of nonlinear discrete-time control systems. We consider semilinear systems with a nonlinear perturbation satisfying a Hölder-type condition. Based on a new discrete inequality of Gronwall type, we establish sufficient conditions for the global and local weak stabilizability of semilinear and nonlinear discrete-time systems, respectively.