A New Approach to Design Interval Observers for Linear Systems (original) (raw)

A Positive Observer for Linear Systems

Proceedings of the 19th IFAC World Congress, 2014

We develop a positive observer for general (i.e. non necessarily positive) linear time varying systems, in both the continuous and discrete time cases. A nice feature of the approach is that no change of coordinates is needed. The observer size is twice the size of the observed system and it is stable whenever the observed system is stable. The design is based on a stable internally positive representation of linear systems that is also an original contribution of the paper. This positive observer can be used to develop interval observers and controllers for systems with several kinds of uncertainties.

Interval Observer Design for Nonlinear Systems: Stability Radii Approach

IEEE Access, 2018

This paper presents a new approach to design preserving order and interval observers for a family of nonlinear systems in absence and in presence of parametric uncertainties and exogenous disturbances. A preserving order observer provides an upper/lower estimation that is always above/below the state trajectory, depending on the partial ordering of the initial conditions, and asymptotically converges to its true values in the nominal case. An interval observer is then constituted by means of an upper and a lower preserving order observer. In the uncertain/disturbed case, the estimations preserve the partial ordering with respect to the state trajectory, and practically converge to the true values, despite of the uncertainties/perturbations. The design approach relies on the cooperativity property and the stability radii mathematical tools, both applied to the estimation error systems. The objective is to exploit the stability radii analysis for the family of linear positive systems under the time-varying nonlinear perturbations in order to guarantee the exponential convergence property of the observers, while the cooperativity condition determines the partial ordering between the trajectories of the state and the estimations. The proposed approach, defined for Lipschitz nonlinearities, depends only on two observer matrix gains. The design is reduced to the solution of linear matrix inequalities, which are given by the cooperative condition and convergence constraints. An illustrative example is presented to show the effectiveness of the theoretical results. INDEX TERMS Interval observers, preserving order observers, stability radii, positive systems.

On Interval Observer Design for Time-Invariant Discrete-Time Systems

2013

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 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 observers for time-varying discrete-time systems

2013

This paper deals with interval state observer design for timevarying discrete-time systems. The problem of a similarity transformation computation which connects a (time-varying) matrix and its nonnegative representation is studied. Three solutions are proposed: for a generic time-varying system, a system with positive state, and for a particular class of periodical systems. Numerical simulations are provided to demonstrate advantages of the developed techniques.

Interval Observers for Discrete-Time Linear Systems with Uncertainties

Symmetry

In this paper, we consider the problem involved when designing the interval observer for the system described by a linear discrete-time model under external disturbances and measurement noises. To solve this problem, we used the reduced order model of the initial system, which is insensitive or has minimal sensitivity to the disturbances. The relations involved in designing the interval observer, which has minimal dimensions and estimates the prescribed linear function of the original system state vector, were obtained. The theoretical results were illustrated by a practical example.

Observers for Interval Systems using Set and Trajectory-based Approaches

Proceedings of the 44th IEEE Conference on Decision and Control

In this paper, set and trajectory-based approaches to interval observation of uncertain systems are presented and compared. The kind of uncertain systems considered are those systems described by a discrete linear time-invariant model with parameters bounded in intervals. The aim of this paper is to study the viability of using set-based approaches coming from the interval analysis community to solve the interval observation problem. Set-based approaches are appealing because of a lower computational complexity compared to trajectory-based approaches but they suffer from the wrapping effect and do not preserve uncertain parameter time-invariance. On the other hand, trajectory-based approaches are immune to these problems but their computational complexity is higher. However, these two families of approaches are equivalent when the observer satisfies the isotonicity condition, which give criteria to tune the observer gain. Finally, these two families of interval observation philosophies will be presented, analysed and compared by using them in an example.

Interval observers design for systems with ostensible Metzler system matrices

2023

This paper attempts to resolve the problem concerning the interval observers design for linear systems with ostensible Metzler system matrices. Because system dynamics matrices are partially different from strictly Metzler structures, a solution is achieved by constructing a composed system matrix representation, which combines pre-compensated interval matrix structures fixed with a prescribed region of D-stability and the reconstructed strictly Metzler matrix structure, related to the original interval system matrix parameter definition. A novel design procedure is presented, which results in a strictly positive observer gain matrix and guarantees that the lower estimates of the positive state variables are non-negative when considering the given system structure and the non-negative system state initial values. The design is computationally simple since it is reduced to the feasibility of the set of linear matrix inequalities.

Interval state observer for nonlinear time varying systems

Automatica, 2013

This paper is devoted to design of interval observers for Linear Time Varying (LTV) systems and a class of nonlinear time-varying systems in the output canonical form. An interval observer design is feasible if it is possible to calculate the observer gains making the estimation error dynamics cooperative and stable. It is shown that under some mild conditions the cooperativity of an LTV system can be ensured by a static linear transformation of coordinates. The efficiency of the proposed approach is demonstrated through numerical simulations.

Stability radii-based interval observers for discrete-time nonlinear systems

IEEE Access, 2021

In this paper, we investigate the interval observer problem for a class of discrete-time nonlinear systems, in absence or presence of external disturbances and parametric uncertainties. The interval observers depend on the design of two preserving order observers, providing lower and upper estimations of the state. The main objective is to apply the stability radii notions and cooperativity property in the estimation error systems in order to guarantee that the lower/upper estimation is always below/above the real state trajectory at each time instant from an appropriate initialization, and the estimation errors converge asymptotically towards zero when the disturbances and/or uncertainties are vanishing. For the disturbed case, the estimation errors practically converge to a vicinity of zero, while the lower/upper estimations preserve the partial ordering with respect to the state trajectory. The design conditions, that are valid for Lipschitz nonlinearities, can be expressed as Linear Matrix Inequalities (LMIs). A numerical simulation example is provided to verify the effectiveness of the proposed method.

An effective method to interval observer design for time-varying systems

Automatica, 2014

An interval observer for Linear Time-Varying (LTV) systems is proposed in this paper. Usually, the design of such observers is based on monotone systems theory. Monotone properties are hard to satisfy in many situations. To overcome this issue, in a recent work, it has been shown that under some restrictive conditions, the cooperativity of an LTV system can be ensured by a static linear transformation of coordinates. However, a constructive method for the construction of the transformation matrix and the observer gain, making the observation error dynamics positive and stable, is still missing and remains an open problem. In this paper, a constructive approach to obtain a time-varying change of coordinates, ensuring the cooperativity of the observer error in the new coordinates, is provided. The efficiency of the proposed approach is shown through computer simulations.