Interval state estimation for uncertain nonlinear systems
Wilfrid Perruquetti
Sign up for access to the world's latest research
checkGet notified about relevant papers
checkSave papers to use in your research
checkJoin the discussion with peers
checkTrack your impact
Abstract
The objective of this work is to develop some design methods of interval observers for a class of nonlinear continuous-time systems. It is assumed that the estimated system can be represented as a superposition of the nominal subsystem (belonged to the class of uniformly observable systems) and a Lipschitz nonlinear perturbation vanishing at the origin. Then it is shown there exists an interval observer for the system that estimates the set of admissible values for the state consistent with the output measurements. An example of the observer application is given with computer simulation results.
Related papers
Design of interval observer for a class of uncertain unobservable nonlinear systems
Automatica, 2016
This paper investigates the interval observer design for a class of nonlinear continuous systems, which can be represented as a superposition of a uniformly observable nominal subsystem with a Lipschitz nonlinear perturbation. It is shown in this case there exists an interval observer for the system that estimates the set of admissible values for the state consistent with the output measurements. An illustrative example of the observer application is given with simulation results.
Stabilization of nonlinear uncertain systems based on interval observers
2011
The problem of output stabilization of a class of nonlinear systems subject to parametric and signal uncertainties is studied. First, an interval observer is designed estimating the set of admissible values for the state. Next, it is proposed to design a control algorithm for the interval observer providing convergence to zero of the interval variables, that implies a similar convergence of the state for the original nonlinear system. An application of the proposed technique shows that a robust stabilization can be performed for linear time-varying and Linear-Parameter-Varying (LPV) systems without assumption that the vector of scheduling parameters is available for measurements. Efficiency of the proposed approach is demonstrated on two examples of computer simulation.
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.
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.
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.
Interval observer design for Linear Parameter-Varying systems subject to component faults
2016 24th Mediterranean Conference on Control and Automation (MED), 2016
In this paper an interval observer for Linear Parameter-Varying (LPV) systems is proposed. The considered systems are assumed to be subject to parameter uncertainties and component faults whose effect can be approximated by parameters deviations. Under some conditions, an interval observer with discrete-time Luenberger structure is developed to cope with uncertainties and faults ensuring guaranteed bounds on the estimated states and their stability. The interval observer design is based on assumption that the uncertainties and the faults magnitudes are considered as unknown but bounded. A numerical example shows the efficiency of the proposed technique.
From Verified Parameter Identification to the Design of Interval Observers and Cooperativity-Preserving Controllers
Acta Cybernetica, 2020
One of the most important advantages of interval observers is their capability to provide estimates for a given dynamic system model in terms of guaranteed state bounds which are compatible with measured data that are subject to bounded uncertainty. However, the inevitable requirement for being able to produce such verified bounds is the knowledge about a dynamic system model in which possible uncertainties and inaccuracies are themselves represented by guaranteed bounds. For that reason, classical point-valued parameter identification schemes are often not sufficient or should, at least, be handled with sufficient care if safety critical applications are of interest. This paper provides an application-oriented description of the major steps leading from a control-oriented system model with an associated verified parameter identification to a verified design of interval observers which provide the basis for the development and implementation of cooperativity-preserving feedback cont...
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.
Stable Interval Observers in ${\BBC}$ for Linear Systems With Time-Varying Input Bounds
IEEE Transactions on Automatic Control, 2013
This technical note deals with the design of stable interval observers and estimators for continuous-time linear dynamic systems under uncertain initial states and uncertain inputs enclosed within time-varying zonotopic bounds. No monotony assumption such as cooperativity is required in the vector field: the interval observer stability directly derives from the stability of the observer state matrix, where any poles (real or complex, single or multiple) are handled in the same way.
Simultaneous Input and State Interval Observers for Nonlinear Systems with Rank-Deficient Direct Feedthrough
2021 European Control Conference (ECC), 2021
We address the problem of designing simultaneous input and state interval observers for Lipschitz continuous nonlinear systems with rank-deficient feedthrough, unknown inputs and bounded noise signals. Benefiting from the existence of nonlinear decomposition functions and affine abstractions, our proposed observer recursively computes the maximal and minimal elements of the estimate intervals that are proven to contain the true states and unknown inputs. Moreover, we provide necessary and sufficient conditions for the existence and sufficient conditions for the stability (i.e., uniform boundedness of the sequence of estimate interval widths) of the designed observer, and show that the input interval estimates are tight, given the state intervals and decomposition functions.
Related papers
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.
High-gain interval observer for partially linear systems with bounded disturbances
International Journal of Control, 2019
In this paper, a high-gain interval observer is proposed for a class of partially linear systems affected by unknown but bounded additive disturbances term and measurements noise. The proposed observer is based upon a classical high-gain structure from which an interval observer for the system is designed. The proposed interval observer is designed based on suitable change of coordinates which ensure the cooperativity of the system. To prove the effectiveness of the proposed approach, two numerical examples are provided and the corresponding simulation results are presented.
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 Observer Synthesis for Locally Lipschitz Nonlinear Dynamical Systems via Mixed-Monotone Decompositions
2022 American Control Conference (ACC)
This paper proposes a novel unified intervalvalued observer synthesis approach for locally Lipschitz nonlinear continuous-time (CT) and discrete-time (DT) systems with nonlinear observations. A key feature of our proposed observer, which is derived using mixed-monotone decompositions, is that it is correct by construction (i.e., the true state trajectory of the system is framed by the states of the observer) without the need for imposing additional constraints and assumptions such as global Lipschitz continuity or contraction, as is done in existing approaches in the literature. Furthermore, we derive sufficient conditions for designing stabilizing observer gains in the form of Linear Matrix Inequalities (LMIs). Finally, we compare the performance of our observer design with some benchmark CT and DT observers in the literature.
Dissipative interval observer design for discrete‐time nonlinear systems
Asian Journal of Control, 2019
This paper studies the problem of designing interval observers for a family of discrete-time nonlinear systems subject to parametric uncertainties and external disturbances. The design approach states that the interval observers are constituted by a couple of preserving order observers, one providing an upper estimation of the state while the other provides a lower one. The design aim is to apply the cooperative and dissipative properties to the discrete-time estimation error dynamics in order to guarantee that the upper and lower estimations are always above and below the true state trajectory for all times, while both estimations asymptotically converge towards a neighborhood of the true state values. The approach represents an extension to the original method proposed by the authors, which focuses on the continuous-time nonlinear systems. In some situations, the design conditions can be formulated as bilinear matrix inequalities (BMIs) and/or linear matrix inequalities (LMIs). Two simulation examples are provided to show the effectiveness of the design approach.
A New Approach to Design Interval Observers for Linear Systems
IEEE Transactions on Automatic Control, 2014
Interval observers are dynamic systems that provide upper and lower bounds of the true state trajectories of systems. In this work we introduce a technique to design interval observers for linear systems affected by state and measurement disturbances, based on the Internal Positive Representations (IPRs) of systems, that exploits the order preserving property of positive systems. The method can be applied to both continuous and discrete time systems.
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.
On Interval Observer Design for a Class of Continuous-Time LPV Systems
IFAC Proceedings Volumes, 2013
This work is devoted to interval observer design for Linear Parameter-Varying (LPV) systems under assumption that the vector of scheduling parameters is not available for measurements. Stability conditions are expressed in terms of matrix inequalities, which can be solved using standard numerical solvers. Robustness and estimation accuracy with respect to model uncertainty is analyzed using L ∞ /L 1 framework. Two solutions are proposed for nonnegative systems and for a generic case. The efficiency of the proposed approach is demonstrated through computer simulations.
Interval State and Unknown Inputs Estimation for Linear Time-Invariant Systems
Proceedings of the 19th IFAC World Congress, 2014
This paper deals with a set membership approach to design an Unknown Input Interval Observer for uncertain Linear Time-Invariant (LTI) continuous-time systems. The goal is to compute lower and upper bounds for unmeasured state as well as unknown inputs. The bounds are guaranteed under the assumption that external disturbances and noises are bounded with a priori known bounds. The proposed interval observer structure is based on decoupling the unknown input effect on the state dynamics by solving algebraic constraints on the estimation errors. Numerical simulations on a 5th-order lateral axis model of a fixed-wing aircraft are provided to demonstrate the efficiency of the proposed technique.
Loading Preview
Sorry, preview is currently unavailable. You can download the paper by clicking the button above.