Problem Of Adaptive Observer Design Research Articles

Initial-Excitation-Based Robust Adaptive Observer for MIMO LTI Systems

The paper addresses the problem of adaptive observer design for multi-input multi-output (MIMO) linear time-invariant (LTI) plants, under the effects of bounded disturbances in the system dynamics and the output, without requiring the persistence of excitation (PE) condition for parameter convergence. A switched robust adaptive observer is proposed, based on initial excitation (IE), a milder condition as compared to PE in terms of excitation requirement and online verifiability. The unknown initial condition of the state is strategically included as an additional unknown parameter in the estimation framework, which, as far as the authors are aware, is a first of its kind technique in the context of adaptive observer design. The extended estimation error dynamics ensures global uniformly ultimately bounded (GUUB) stability of the estimation error dynamics. Further, estimation errors in a disturbance-free case result in uniformly global exponential stability (UGES) in a delayed sense. Simulation results are presented and estimation performance of the proposed adaptive observer is compared with existing algorithms to illustrate the effectiveness.

Adaptive observer design for uncertain nonlinear ODE‐PDE cascades

SummaryWe are revisiting the problem of adaptive observer design for systems that are constituted of an Ordinary Differential Equation (ODE), containing a globally Lipschitz function of the state, and a linear Partial Differential Equation (PDE) of a diffusion–reaction heat type. The ODE and PDE are connected in series and both are subject to parametric uncertainties. In addition to nonlinearity and uncertainty, the system complexity also lies in the fact that no sensor can be implemented at the junction point between the ODE and the PDE. In the absence of parameter uncertainty, nonadaptive state observers are available featuring exponential convergence. However, convergence is guaranteed only under the condition that either the Lipschitz coefficient is sufficiently small or the PDE domain length is sufficiently small. To get around this limitation, and also to account for parameter uncertainty, we develop a design that involves two concatenated adaptive observers, covering the two subintervals of the PDE domain. The proposed design employs one extra sensor, providing the measurement of the PDE state at an inner position close to the ODE‐PDE junction point. Both observers are shown to be exponentially convergent, under ad‐hoc persistent excitation (PE) conditions, with no limitation on the Lipschitz coefficient and domain length.

Application of Adaptive Observer to Sensorless Induction Motor via Parameter-Dependent Transformation

This brief addresses the adaptive observer design problem for sensorless induction motor (IM) drives with unknown resistances. In the literature, difficulty is found when applying several well-developed adaptive observer designs to IM dynamics. Hence, by introducing a parameter-dependent transformation, in the new coordinate, the IM dynamics manifest the attributes that facilitate the adaptive observer design. Although the classic designs that require linear-in-parameters result in overparameterization, the proposed nonlinearly parameterized adaptive observer demonstrates a good performance, as illustrated by the simulation and experimental results.

An adaptive observer for a class of nonlinear systems with a high-gain approach. Application to the twin-rotor system

In this paper, we investigate the problem of adaptive observer design for a class of nonlinear systems subject to unknown parameters and such that the classical observer matching assumption is not satisfied. We adopt the idea of generating auxiliary outputs based on a high-gain observer. The estimated auxiliary outputs are then employed by an adaptive observer to reconstruct both states and unknown parameters. The stability analysis is established based on Lyapunov theory. It is shown that the state estimation error and the adaptation error are uniformly bounded and converge to a compact set that may be reduced by an appropriate choice of the design parameters. In order to improve the robustness of our approach against disturbances, the proposed observer is appropriately modified based on sliding modes theory. Theoretical results are illustrated and validated for the twin-rotor MIMO system with numerical simulations.

A note on adaptive observer for the Lur’e differential inclusion system

This paper considers the adaptive observer design problem for the Lur’e differential inclusion system with unknown parameters. Different from the current work, the proposed observer does not hold any set-valued mapping and the uncertain parameters can also be identified. Firstly, the adaptive observer is designed for the Lur’e differential inclusion system, and it is proved to be valid by Lyapunov stability theory. Then, based on the canonical form of the system matrix, the sufficient and necessary condition for one assumption is presented in both SISO and MIMO cases. Finally, numerical examples are simulated to show the effectiveness of the proposed method.

Adaptive output-feedback stabilization of non-local hyperbolic PDEs

We address the problem of adaptive output-feedback stabilization of general first-order hyperbolic partial integro-differential equations (PIDE). Such systems are also referred to as PDEs with non-local (in space) terms. We apply control at one boundary, take measurements on the other boundary, and allow the system’s functional coefficients to be unknown. To deal with the absence of both full-state measurement and parameter knowledge, we introduce a pre-transformation (which happens to be based on backstepping) of the system into an observer canonical form. In that form, the problem of adaptive observer design becomes tractable. Both the parameter estimator and the control law employ only the input and output signals (and their histories over one unit of time). Prior to presenting the adaptive design, we present the non-adaptive/baseline controller, which is novel in its own right and facilitates the understanding of the more complex, adaptive system. The parameter estimator is of the gradient type, based on a parametric model in the form of an integral equation relating delayed values of the input and output. For the closed-loop system we establish boundedness of all signals, pointwise in space and time, and convergence of the PDE state to zero pointwise in space. We illustrate our result with a simulation.

Adaptive State Estimation for a Class of Nonlinear Stochastic Systems under Generalized Lipschitz Condition

This paper is concerned with adaptive observer design problem for a class of nonlinear stochastic systems. Unknown constant parameters are assumed to be norm bounded. In order to better use the structural knowledge of the nonlinear part, a generalized Lipschitz condition is introduced to the adaptive observer design for a class of nonlinear stochastic systems for the first time. Based on a Lyapunov-Krasovskii functional approach and stochastic Lyapunov stability theory, we present a new adaptive observer design condition with ultimately exponentially bounded in sense of mean square for errors systems in terms of linear matrix inequality (LMI). A numerical example is exploited to show the validity and feasibility of the results.

Adaptive Output-Feedback Stabilization of Non-Local Hyperbolic PDEs

We address the problem of adaptive output-feedback stabilization of general first-order hyperbolic partial integro-differential equations (PIDE). Such systems are also referred to as PDEs with non-local (in space) terms. We apply control at one boundary, take measurements on the other boundary, and allow the system's functional coefficients to be unknown. To deal with the absence of both full-state measurement and parameter knowledge, we introduce a pre-transformation (which happens to be based on backstepping) of the system into an observer canonical form. In that form, the problem of adaptive observer design becomes tractable. Both the parameter estimator and the control law employ only the input and output signals (and their histories over one unit of time). Prior to presenting the adaptive design, we present the non-adaptive/baseline controller, which is novel in its own right and facilitates the understanding of the more complex, adaptive system. The parameter estimator is of the gradient type, based on a parametric model in the form of an integral equation relating delayed values of the input and output. For the closed-loop system we establish boundedness of all signals, pointwise in space and time, and convergence of the PDE state to zero pointwise in space. We illustrate our result with a simulation.

A note on improvement of adaptive observer robustness

In this paper the problem of adaptive observer design in the presence of disturbances is studied, and an augmented adaptive observer is proposed. First, the H∞ gain of a conventional adaptive observer is estimated, which characterizes the effect of disturbances on output errors. Next, it is shown that if the disturbance is “matched” in the plant equations, then it is possible to introduce additional sliding-mode feedback, dependent on the plant output, improving the accuracy of observation. Simulation results confirm the improvement.

Adaptive Observers for Nonlinearly Parameterized Systems Subjected to Parametric Constraints

We consider the problem of adaptive observer design in the settings when the system is allowed to be nonlinear in the parameters, and furthermore they are to satisfy additional feasibility constraints. A solution to the problem is proposed that is based on the idea of employing observers comprising of an exponentially converging part coupled with exploratory dynamics (cf. Tyukin et al. (2013)). The procedure is illustrated with an example.

Adaptive observer for a class of nonlinear systems with time‐varying delays

SUMMARYThe adaptive observer design problems have been extensively studied in literature for both linear and nonlinear systems. Some researches have also been carried out on adaptive observer design for linear time‐delay systems, but there is no significant work on adaptive observer design for nonlinear time‐delay systems. In this work, the adaptive observer design problem for a class of nonlinear time‐delay systems is considered. The observer is designed for the nonlinear systems whose nonlinear functions satisfy Lipschitz condition. Like conventional adaptive observers for the systems without time delays, this observer also estimates both states and unknown parameters simultaneously. For this property, it will be very much useful for many real‐time systems where time delays cannot be avoided. The sufficient conditions for existence of the observer are derived using the linear matrix inequality approach. With the help of a numerical example, effectiveness of the proposed observer is demonstrated. Copyright © 2012 John Wiley & Sons, Ltd.

Making Sense of Unstable Convergence in the Problem of Adaptive Observer Design: Case Study

AbstractWe consider the problem of adaptive observer design for systems in which the uncertainties are modelled by ill-conditioned regressors or/and unknown parameters of the model enter the equations nonlinearly. It is shown that both of these critical issues can be overcome in a unified yet simple way, provided that the usual requirement of global asymptotic stability of solutions of model-observer system is replaced with a weaker constraint of mere set-attractivity. The claim is illustrated with a particular adaptive observer design problem.

On adaptive observers for state affine systems

In this paper, the problem of adaptive observer design for the class of state affine systems is discussed. The discussion is based on recent results on adaptive observer with exponential rate of convergence obtained for multi-input–multi-output linear time varying systems on the one hand, and the well-known Kalman-like design for state affine systems on the other. In particular the relationship between both designs is emphasized, showing how they can even be equivalent. The interest of such an adaptive observer for state affine systems is illustrated by the example of state and parameter estimation for the Lorenz chaotic system. The observer performances are illustrated via simulation.

OUTPUT FEEDBACK CONTROL VIA ADAPTIVE OBSERVERS WITH PERSISTENCY OF EXCITATION

We address the problem of adaptive observer design for nonlinear time-varying systems which can be transformed in the so-called output feedback form (linear in the unmeasured variables). The observer design follows up previous work on adaptive observers for linear systems and has the form of the classical Luenberger observers for linear systems except that the observer gain is time-varying. A specific form of persistency of excitation is imposed to guarantee the convergence of the (state and parameter) estimation errors. As for the output feedback loop, we proceed using a cascade approach, i.e., we impose the appropriate conditions so that the closed loop system has a cascaded structure. Uniform global asymptotic stability may then be concluded based on cascaded systems theory. Key Words—persistency of excitation, observers, time-varying systems, output feedback control.