Tarek Raïssi

Interval State Estimation for a Class of Nonlinear Systems

The goal of this technical note is to design interval observers for a class of nonlinear continuous-time systems. The first part of this work shows that it is usually possible to design an interval observer for linear systems by means of linear time-invariant changes of coordinates even if the system is not cooperative. This result is extended to a class of nonlinear systems using partial exact linearisations. The proposed observers guarantee to enclose the set of system states that is consistent with the model, the disturbances and the measurement noise. Moreover, it is only assumed that the measurement noise and the disturbances are bounded without any additional information such as stationarity, uncorrelation or type of distribution. The proposed observer is illustrated through numerical simulations.

Brief paper: Interval observer design for consistency checks of nonlinear continuous-time systems

This paper deals with fault detection for nonlinear continuous-time systems. A procedure based on interval analysis is proposed to build a guaranteed qLPV (quasi-Linear Parameter-Varying) approximation of the nonlinear model. The interval qLPV approximation makes it possible to derive two point observers which estimate respectively the lower and the upper bound of the state vector using cooperativity theory. A set guaranteed to contain the actual value of the residual is then designed. The modelling uncertainties and measurement errors are taken into account at the design stage. The proposed methodology is illustrated through numerical simulations.

Interval state observer for nonlinear time varying systems

This paper is devoted to the 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 estimation for LPV systems applying high order sliding mode techniques

In this paper, the problem of design of interval observers for Linear Parameter-Varying (LPV) systems, containing non-detectable or non-strongly-observable parts, is addressed. Firstly, a High Order Sliding Mode (HOSM) method is applied to the strongly observable subsystem, obtained by an appropriate change of coordinates, to estimate the state and its derivative. Secondly, this information is used to decrease the level of uncertainty in the rest of the system, which leads to improvement of accuracy of the set-membership estimates generated by an interval observer. Moreover, it is shown that HOSM techniques allows us to relax the applicability conditions of standard interval observer design methods. The efficiency of the proposed approach is demonstrated through simulation examples.

Control of Nonlinear and LPV Systems: Interval Observer-Based Framework

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 of interval variables to zero, 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 through two examples.

Brief paper: Actuator fault detection and compensation under feedback control

The problem of unknown input estimation and compensation is studied for actuated nonlinear systems with noisy measurements. The proposed solution is based on high-order sliding-mode differentiation and discrete-time optimization technique. Accuracy of the proposed hybrid estimation scheme is evaluated and stability conditions of the compensating mechanism are established. It is shown that the fault detection delay as well as the smallest detectable fault magnitude can be estimated. Efficiency of the proposed approach is demonstrated through oscillatory failure detection and compensation in aircraft surface servo loops.

Interval observers for continuous-time LPV systems with L 1 / L 2 performance

An approach to interval observer design for Linear Parameter-Varying (LPV) systems is proposed. It is assumed that the vector of scheduling parameters in LPV models is not available for measurement. Two different interval observers are constructed for nonnegative systems and for a generic case. Stability conditions are expressed in terms of matrix inequalities, which can be solved with respect to the observer gains using standard numerical solvers. Applying L 1 / L 2 framework the robustness and estimation accuracy with respect to model uncertainty are analyzed. The efficiency of interval estimation for LPV models is demonstrated through numerical experiments for a microfluidic system and an academic example.

Interval Observers for Time-Varying Discrete-Time Systems

This techical note deals with interval state observer design for time-varying 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.

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

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.

Robust estimation of fractional models in the frequency domain using set membership methods

In this paper, the usual definition of Grunwald-Letnikov fractional derivative is first extended to interval derivatives in order to deal with uncertainties in the differentiation orders. The Laplace transform of interval derivatives is computed and its monotonicity is studied in the frequency domain. Next, the main objectives of this paper are presented as the implementation of three methods for set membership parameters estimation of fractional differentiation models based on complex frequency data. The first one uses a rectangular inclusion function with rectangle sides corresponding to real and imaginary parts of the complex frequency response; the second one uses a polar inclusion function and the gain/phase representation; the third one uses a circular inclusion function with disk representation. Each inclusion function introduces pessimism differently. It is shown that all three approaches are complementary and that the results can be merged to obtain a smaller feasible solution set. The proposed methods can be applied to estimate parameters of certain/uncertain linear time variant/invariant systems.