Interval Observer Design Research Articles

Design of functional interval observers for non‐linear fractional‐order systems

AbstractThis paper considers the problem of designing functional interval observers for a class of non‐linear fractional‐order systems with bounded uncertainties. First, interval observers for linear functions of the state vector of the considered system are designed. Then, conditions for the existence of such interval observers are established and an effective algorithm for computing unknown observer matrices is provided in this paper. Finally, numerical examples and simulation results are given to illustrate the effectiveness of the proposed design method.

Interval observers for linear functions of state vectors of linear fractional‐order systems with delayed input and delayed output

SummaryThis paper addresses the problem of interval observer design for linear functions of state vectors of linear fractional‐order systems, which are subjected to time delays in the measured output as well as the control input. By using the information of both the delayed output and input, we design two linear functional state observers to compute two estimates, an upper one and a lower one, which bound the unmeasured linear functions of state vectors. As a particular case with output delay only, we design a linear functional state observer to estimate (asymptotically) the unmeasured linear functions of state vectors. Existence conditions of such observers are provided, and some of them are translated into a linear programming problem, in which the observers' matrices can be effectively computed. Constructive design algorithms are introduced. Numerical examples are provided to illustrate the design procedure, practicality, and effectiveness of the proposed design method.

Interval Observer Design for Discrete-Time Switched Systems by Linear Programming Method

Interval Observer Design for Discrete-Time Switched Systems by Linear Programming Method

Interval observer design for continuous-time switched systems under known switching and unknown inputs

ABSTRACTThis paper deals with unknown input estimation for switched linear systems in an unknown but bounded error (UBBE) framework. Based on a known switching signal and under the fulfilment of the relative degree property by all the subsystems, a decoupling method is used to make the state partially affected by the unknown input. Assuming that the disturbances and the measurement noises are unknown but bounded with a priori known bounds, lower and upper bounds of the unmeasured state and unknown input are then computed. A numerical example illustrates the efficiency of the proposed methodology.

Coupled boundary interval observer for LPV systems subject to uncertainties in input, output and parameters

Abstract The topic of this publication is the design of interval observers for LPV systems under consideration of parameter, input and output uncertainties that are described by bounded intervals. In the course of the design, also time-varying bounds of the parameter intervals are considered representing the case when parameters are measured with an unknown but bounded measurement error. In this case, an existing approach is extended to handle these parameter uncertainties. The effectiveness of the approach is demonstrated by two numerical examples.

Interval observer design for uncertain discrete-time linear systems

This paper proposes a novel interval observer design method for discrete-time linear systems with unknown but bounded disturbance and measurement noise. The proposed interval observer has a new structure that provides more design degrees of freedom. A direct method based on H∞ technique is used to improve the accuracy of interval estimation. The design conditions are formulated into linear matrix inequalities, which can be efficiently solved. Two numerical examples are given to illustrate the design and validate the performance of the interval observers.

Construction of hybrid interval observers for switched linear systems

This paper investigates the design of hybrid interval observers (IOs) of switched linear systems involving additive disturbances, whose subsystems need not to be cooperative. We first propose the definition of IOs for switched systems and extend the coordinate transformation for non-switched linear systems into switched ones. Based on the switched coordinate transformation, the considered switched linear system is converted into a hybrid cooperative system, whose states jump at each switching times. Then, by using the multiple Lyapunov functions method, IOs for switched linear systems are designed in two cases, that is, all of the subsystem matrices are Hurwitz and some of the subsystem matrices are not Hurwitz. Moreover, by exploiting the stabilization property of switching behaviors, IOs for switched systems, whose subsystem matrices are not all Hurwitz, are proposed based on multiple discretized Lyapunov functions. The constructed IOs for the considered switched systems are hybrid dynamical systems and the initialized states of the IOs at each switching time are coupled, since the transformation matrices for different subsystems are different. Finally, we afford an example to illustrate the validity of the derived results.

Using set invariance to design robust interval observers for discrete‐time linear systems

SummaryBased on interval and invariant set computation, an interval version of the Luenberger state observer for uncertain discrete‐time linear systems is proposed in this work. This new interval observer provides a punctual estimation of the state vector and guaranteed bounds on the estimation error. An off‐line and an on‐line approach to characterize, in a guaranteed way, the estimation error are introduced. Compared with the existing approaches, the proposed interval observer design method is not restrictive in terms of required assumptions, complexity, and on‐line computation time. Furthermore, the convergence issue of the estimation error is well established and to reduce the conservatism of the estimated state enclosure induced by the bounded additive state disturbance and noise measurement, an H∞ method to compute the optimal observer gain is proposed. The performance of the proposed state estimation approach are highlighted on different illustrative examples.

New approaches to positive observer design for discrete-time positive linear systems

This paper aims at providing new design approaches for positive observers of discrete-time positive linear systems based on a construction method of linear copositive Lyapunov function for positive systems. First, an efficient positive observer design approach is proposed by using linear programming such that the observer error system is exponentially stable. Furthermore, an interval observer design is proposed for uncertain positive systems. Then, the results are extended to positive time delay systems. In contrast with the previous design approaches, the new design method provides a general observer design with lower computational burden. Finally, three comparison examples are given to show the merit of the new design approach.

Design of interval observers and controls for PDEs using finite-element approximations

Synthesis of interval state estimators is investigated for the systems described by a class of parabolic Partial Differential Equations (PDEs). First, a finite-element approximation of a PDE is constructed and the design of an interval observer for the derived ordinary differential equation is given. Second, the interval inclusion of the state function of the PDE is calculated using the error estimates of the finite-element approximation. Finally, the obtained interval estimates are used to design a dynamic output stabilizing control. The results are illustrated by numerical experiments with an academic example and the Black–Scholes model of financial market.

Some recent results on the design and implementation of interval observers for uncertain systems

Abstract Based on the theory of positive systems, the goal of interval observers is to compute sets of admissible values of the state vector at each instant of time for systems subject to bounded uncertainties (noises, disturbances and parameters). The size of the estimated sets, which should be minimised, has to be proportional to the model uncertainties. An interval estimation can be seen as a conventional point estimation (the centre of the interval) with an estimation error given by the interval radius. The reliable uncertainties propagation performed in this context can be useful in several fields such as robust control, diagnosis and fault-tolerant control. This paper presents some recent results on interval observers for several dynamical systems classes such as continuous-time and switched systems.

Interval estimation for continuous-time switched linear systems

This paper deals with the design of interval observers for switched linear systems (SLS), a class of hybrid systems. Under the assumption that the disturbances and the measurement noise are bounded, upper and lower bounds for the state are calculated. New conditions of cooperativity in discrete-time instants are firstly proposed. Then, some techniques for interval estimation are developed in continuous-time. It is shown that it is possible to calculate the observer gains making the estimation error dynamics cooperative and stable via some change of coordinates under arbitrary switching sequences. The performances of the developed techniques are illustrated through numerical examples.

Stabilization of linear impulsive systems under dwell-time constraints: Interval observer-based framework

The problem of interval observer design is studied for a class of linear impulsive systems. Ranged and minimum dwell-time constraints are considered under detectability assumption. The first contribution of this paper lies in designing interval observers for linear impulsive systems under ranged and minimum dwell-time constraints, and investigating positivity of the estimation error dynamics in addition to stability. Several observers are designed oriented on different conditions of positivity and stability for estimation error dynamics. The boundedness of the estimation error (input-to-state stability property) and the observer stability conditions are stated as infinite-dimensional linear programming problems. Next, an output stabilizing feedback design problem is discussed, where the stability is checked using linear matrix inequalities (LMIs). Efficiency of the proposed approach is demonstrated by computer simulations for a commercial electric vehicle equipped with a low power range extender fuel cell, a bouncing ball, an academic linear impulsive system and for Fault Detection and Isolation (FDI) and Fault-Tolerant Control (FTC) of a power split device with clutch for heavy-duty military vehicles.

Control of non‐linear systems based on interval observer design

This study develops a Luenberger-like interval observer design methodology for a class of non-linear systems with bounded uncertainties. First, a simple procedure is given to find a positive representation that provides a method to guarantee the existence of observer gain by means of the matrix theory. Then, based on interval observer design, the gain of feedback controllers for non-linear systems is presented in terms of a linear matrix inequality formulation. Finally, a numerical example is provided to demonstrate the efficiency of the proposed approach.

Interval observers for global feedback control of nonlinear systems with robustness with respect to disturbances

This paper develops criteria for designing interval observers to guarantee robustness with respect to disturbances for feedback control of nonlinear systems. Intervals in which components of the state vector are guaranteed to stay are estimated based on the information of the range of the initial state and the disturbances. For formulating desirable properties of boundedness and convergence of estimated intervals in the presence of disturbances, the notion of integral input-to-state stability is introduced to interval observer design. Guaranteed properties of the observer-based feedback designed in the formulation are not only global, but also address nonlinearities which are broader than those covered by previously existing approaches.

An interval observer design for uncertain nonlinear systems based on the T-S fuzzy model

AbstractA 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.

On design of interval observers for parabolic PDEs

The problem of state estimation for heat non-homogeneous equations under distributed in space measurements is considered. An interval observer is designed, described by Partial Differential Equations (PDEs), for uncertain distributed parameter systems without application of finite-element approximations. Conditions of boundedness of solutions of interval observer with non-zero boundary conditions and measurement noise are proposed. The results are illustrated by numerical experiments with an academic example.

Interval observers for LPV systems and application to the guaranteed state estimation of an induction machine

The scope of this paper is the design of interval observers for linear parameter-varying systems subject to input and output uncertainties. To this, two coupled dynamical systems are designed which compute an upper bound and a lower bound of the state which enclose the real value with a bounded interval width. Besides of a full-order interval observer, a reduced-order interval observer is proposed to reduce the computational effort. In addition to sufficient conditions and the mathematical description of the interval observers, an optimization-based design procedure for their practical application is presented. The performance of the interval observers is shown by the guaranteed state estimation of an induction machine.

Unknown input interval observer with H∞ and D-stability performance

This paper investigates the design of interval observers for uncertain linear time invariant systems. The aim is to provide upper and lower bounds of the system state in a guaranteed way. The method is based on the monotone system theory. It provides a solution to the conditions of the existence of the interval observer under mixed performance criteria such as H∞ performance and D-stability. The core idea relies on the introduction of a new class of Unknown Input Interval Observers (UIIOs).

Interval observer design for discrete-time switched system

In this paper, the problems of interval observer design for discrete-time switched system are investigated. In order to make the residual system stable and non-negative, a time-varying linear transformation and a standard switched system stability analysis are used. Based on these, an interval observer for discrete-time switched system is constructed. A simulation example is given to illustrate the effectiveness and correctness of the proposed method.