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Non-linear estimation is easy

Non-linear state estimation and some related topics like parametric estimation, fault diagnosis and perturbation attenuation are tackled here via a new methodology in numerical differentiation. The corresponding basic system theoretic definitions and properties are presented within the framework of differential algebra, which permits to handle system variables and their derivatives of any order. Several academic examples and their computer simulations, with online estimations, illustrate our viewpoint.

Numerical differentiation with annihilators in noisy environment

Numerical differentiation in noisy environment is revised through an algebraic approach. For each given order, an explicit formula yielding a pointwise derivative estimation is derived, using elementary differential algebraic operations. These expressions are composed of iterated integrals of the noisy observation signal. We show in particular that the introduction of delayed estimates affords significant improvement. An implementation in terms of a classical finite impulse response (FIR) digital filter is given. Several simulation results are presented.

Model-free control

‘Model-free control’and the corresponding ‘intelligent’ PID controllers (iPIDs), which already had many successful concrete applications, are presented here for the first time in an unified manner, where the new advances are taken into account. The basics of model-free control is now employing some old functional analysis and some elementary differential algebra. The estimation techniques become quite straightforward via a recent online parameter identification approach. The importance of iPIs and especially of iPs is deduced from the presence of friction. The strange industrial ubiquity of classic PIDs and the great difficulty for tuning them in complex situations is deduced, via an elementary sampling, from their connections with iPIDs. Several numerical simulations are presented which include some infinite-dimensional systems. They demonstrate not only the power of our intelligent controllers but also the great simplicity for tuning them.

A revised look at numerical differentiation with an application to nonlinear feedback control

We are presenting new and efficient methods for numerical differentiation, i.e., for estimating derivatives of a noisy time signal. They are illustrated, via convincing numerical simulations, by the analysis of an academic signal and by the feedback control of a nonlinear system.

Intelligent PID controllers

Intelligent PID controllers, or i-PID controllers, are PID controllers where the unknown parts of the plant, which might be highly nonlinear and/or time-varying, are taken into account without any modeling procedure. Our main tool is an online numerical differentiator, which is based on easily implementable fast estimation and identification techniques. Several numerical experiments demonstrate the efficiency of our method when compared to more classic PID regulators.

Robust residual generation for linear fault diagnosis: an algebraic setting with examples

We are generating residuals for linear fault diagnosis and isolation which are 1. robust with respect to a large variety of additive disturbances,  A reader who is already familiar with this algebraic setting may directly proceed to § 3. The numerous examples on various aspects of diagnosis, like the previous one, are written on the other hand in such a way that they may be understood by anyone who is not mastering those mathematical tools. We hope therefore that our results might be understood by a large community. 2. working in closed-loop, even with uncertain parameters. Several examples with their computer simulations, including a concrete case-study of a two mass system, are enlightening our viewpoint. Our methods, which are mainly of algebraic flavour (module theory, differential algebra, and operational calculus), are borrowed from recent works on control and identification.

Flatness-Based Trajectory Planning/Replanning for a Quadrotor Unmanned Aerial Vehicle

A flatness-based flight trajectory planning/replanning strategy is proposed for a quadrotor unmanned aerial vehicle (UAV). In the nominal situation (fault-free case), the objective is to drive the system from an initial position to a final one without hitting the actuator constraints while minimizing the total time of the mission or minimizing the total energy spent. When actuator faults occur, fault-tolerant control (FTC) is combined with trajectory replanning to change the reference trajectory in function of the remaining resources in the system. The approach employs differential flatness to express the control inputs to be applied in the function of the desired trajectories and formulates the trajectory planning/replanning problem as a constrained optimization problem.

Real-time estimation for switched linear systems

We extend previous works on real-time estimation, via algebraic techniques, to the recovering of the switching signal and of the state for switching linear systems. We characterize also singular inputs for which the switched systems become undistinguishable. Several convincing numerical experiments are illustrating our techniques which are easily implementable.

CONTROL OF AN UNCERTAIN THREE-TANK SYSTEM VIA ON-LINE PARAMETER IDENTIFICATION AND FAULT DETECTION

Abstract We provide fast and on-line methods for failure detection and isolation of uncertain nonlinear systems which are operating in closed-loop. They are based on accurate values of the derivatives of a time-signal, which are obtained via new algebraic estimation techniques. The applicability and efficiency of our approach are illustrated by numerical simulations for a most popular case-study namely the three-tank system.

COMPLEX CONTINUOUS NONLINEAR SYSTEMS: THEIR BLACK BOX IDENTIFICATION AND THEIR CONTROL

Recent advances in estimation theory permit a new approach to nonlinear black box identification, where a phenomenological model is replacing a precise mathematical description. Convincing simulations are provided for two examples: 1) the classic ball and beam system, 2) a large scale linear system, where our setting may be regarded as a powerful alternative to model reduction.