Interval state observer for nonlinear time varying systems

The stability and behavior of continuous-time nonlinear time-varying (NTV), nonlinear time-invariant (NTI) and linear time-varying (LTV) systems are generally very difficult to ascertain using current analytical tools. This paper presents sufficient conditions under which a NTV system with a bounded system matrix, A(t,X), is globally exponentially stable (GES). The continuous-time NTV system is modeled as an uncertain polytopic system even though A(t,X) may be reasonably known. It is shown that if all polytope-vertex matrices are Hurwitz, then the system is GES. Also, a method for determining the boundary estimate of a NTV system state-trajectory norm is developed. Presented is a unified stability analysis approach for a large class of NTV, NTI, and LTV systems. Even for large systems, application of the GES theorem presented is straight forward using readily available software and computational power. Examples are given that illustrate the application of the theorem offered.

A difference-algebraic interval observer design for perturbed discrete-time descriptor systems

A robust tracking control algorithm based mainly on the theory of variable structure systems and Lyapunov's direct method is developed in this paper, to treat the tracking control problem of a class of non-linear time-varying systems with uncertain dynamics. A nominal model is used instead of an exact one. together with a description of the bounds of plant uncertainty. The objective is to design a control law such that all the plants lying within a specified band of uncertainly will be driven to reach the goal. The modelling error, time-varying system dynamics, system uncertainties, and disturbances are all treated as general disturbances to the nominal system. This algorithm greatly eases the difficulties present in requiring a precise model of the physical system for the control purpose and for handling the non-linear time-varying system. The non-linear time-varying system is first linearized to a linear time-varying system (with time-varying coefficients completely defined) via the method of variable s...

This paper investigates the problem of interval estimation for a class of Euler-Lagrange systems with unknown disturbances. An event-triggered mechanism is applied in the design of interval observers, and unnecessary data communication burden is reduced. The sufficient conditions are derived by both the positive system theory and the Lyapunov stability theory. Moreover, the gains of observer are determined by solving a set of inequalities in the sufficient conditions. At last, the validity of the presented event-triggered interval observers is demonstrated by a numerical simulation.

The problem of interval state observer design is addressed for time-invariant discrete-time systems. Two solutions are proposed: the first one is based on a similarity transformation synthesis, which connects a constant matrix with its nonnegative representation ensuring the observation error positivity. The second contribution shows that in the discrete-time case the estimation error dynamics always can be represented in a cooperative form without a transformation of coordinates. The corresponding observer gain can be found as a solution of the formulated LMIs. The performances of the proposed observers are demonstrated through computer simulations.

Interval observer design for LPV systems with parametric uncertainty

State bounding for nonlinear time-varying systems with delay and disturbance

In this paper, an interval observer-based fault diagnosis mechanism and an interval dynamic event-triggered robust fault-tolerant control strategy are proposed for uncertain nonlinear switched systems. Firstly, the interval observer design problem is transformed into a Sylvester equation solution problem by introducing the coordinate transformation method in the absence of actuator faults, and the corresponding interval observer design method is given. Secondly, based on the interval observer output, upper and lower bound estimates of the system output are constructed, and fault diagnosis is realized by monitoring whether the system output exceeds the preset bounds. In addition, an improved interval dynamic event triggered mechanism is proposed to optimize the utilization of communication and computational resources and reduce redundant information transmission. The mechanism incorporates interval estimation information and real-time residuals, which can effectively extend the triggering time interval while improving the fault detection performance. Based on the interval fault diagnosis results, a fault-tolerant control strategy based on the interval dynamic event triggered mechanism is proposed, and the conditions for asymptotic stabilization of the closed-loop system are given. Finally, a nonlinear model of a morphing aircraft system with variable wing curvature is used to verify the validity of the designed scheme. DOI: 10.61416/ceai.v28i1.9516

An identification approach for linear and nonlinear time-variant structural systems via harmonic wavelets

A new approach to build an interval observer for nonlinear uncertain systems is presented in this paper. Nonlinear systems modeled in the Takagi-Sugeno (T-S) form are studied. A T-S proportional observer is first issued by pole-placement and LMI tools. Secondly, time-varying change of coordinates for each dynamic state estimation error is used to design an interval observer. The system state bounds are then directly deduced.

This paper deals with the problem of interval estimation. The proposed approach is based on the use of successive derivatives in continuous time (or forward measurements in discrete time) in an algebraic (or local) interval observer. The main advantage of such an observer is that it relaxes the Metzler assumption widely used in the literature of cooperative interval observers. Moreover, no prior knowledge on the initial condition is needed, and no stability nor convergence issue do appear as the estimation is local. After introducing such an observer for Linear Time Variant (LTV) systems with bounded disturbances, some extensions are proposed for Linear Parameter Varying (LPV) systems as well as for functional systems and systems affected by Unknown Input (UI). Some examples illustrate the theoretical contributions.

In previous papers we have introduced a sufficient condition for uniform attractivity of the origin of a class of nonlinear time-varying systems which is stated in terms of persistency of excitation (PE), a concept well known in the adaptive control and systems identification literature. The novelty of our condition, called uniform /spl delta/-PE, is that it is tailored for nonlinear functions of time and state and it allows us to prove uniform asymptotic stability. In this paper we present a new definition of u/spl delta/-PE which is conceptually equivalent to but technically different from its predecessors. We make connections between this property and similar properties previously used in the literature. We also show when this condition is necessary and sufficient for uniform (global) asymptotic stability for a large class of nonlinear time-varying systems.

SUMMARY Preserving Order Observers provide an estimation that is always above or below the true variable, and in the absence of uncertainties/perturbations, the estimation converges asymptotically to the true value of the variable. In this paper, we propose a novel methodology to design preserving order observers for a class of nonlinear systems in the nominal case or when perturbations/uncertainties are present. This objective is achieved by combining two important systemic properties: dissipativity and cooperativity. Dissipativity is used to guarantee the convergence of the estimation error dynamics, whereas cooperativity of the error dynamics assures the order-preserving properties of the observer. The use of dissipativity for observer design offers a big flexibility in the class of nonlinearities that can be considered while keeping the design simple: it leads in many situations to the solution of a linear matrix inequality (LMI). Cooperativity of the observer leads to an LMI. When both properties are considered simultaneously, the design of the observer can be reduced, in most cases, to the solution of both a bilinear matrix inequality and an LMI. Because a couple of preserving order observers, one above and one below, provide an interval observer, the proposed methodology unifies several interval observers design methods. The design methodology has been validated experimentally in a three-tanks system, and it has also been tested numerically and compared with an example from the literature.Copyright © 2013 John Wiley & Sons, Ltd.

Global stabilization for linear continuous time-varying systems

This paper deals with functional interval observer design for discrete-time switched systems under stealthy deception attacks. First, the boundaries of the attack are obtained by designing interval observers. Then a two-step method to design the functional observers by zonotope is presented. For the first step, an H∞ functional observer is presented and as the second step, the zonotope method is applied to obtain the boundaries of states. An illustrative example provided in the last section demonstrates the effectiveness of the proposed method.