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5 pages, 180 KiB

Open AccessEditorial

Algorithms for Reliable Estimation, Identification and Control

by Andreas Rauh, Luc Jaulin and Julien Alexandre dit Sandretto

Algorithms 2022, 15(8), 276; https://doi.org/10.3390/a15080276 - 5 Aug 2022

Viewed by 1555

Abstract

The two-part Special Issue “Algorithms for Reliable Estimation, Identification and Control” deals with the optimization of feedforward and feedback controllers with respect to predefined performance criteria as well as the state and parameter estimation for systems with uncertainty [...] Full article

(This article belongs to the Special Issue Algorithms for Reliable Estimation, Identification and Control II)

Research

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9 pages, 632 KiB

Open AccessArticle

Bounded-Error Parameter Estimation Using Integro-Differential Equations for Hindmarsh–Rose Model

by Carine Jauberthie and Nathalie Verdière

Algorithms 2022, 15(6), 179; https://doi.org/10.3390/a15060179 - 24 May 2022

Cited by 2 | Viewed by 1877

Abstract

A numerical parameter estimation method, based on input-output integro-differential polynomials in a bounded-error framework is investigated in this paper. More precisely, the measurement noise and parameters belong to connected sets (in the proposed work, intervals). First, this method, based on the Rosenfeld–Groebner elimination [...] Read more.

A numerical parameter estimation method, based on input-output integro-differential polynomials in a bounded-error framework is investigated in this paper. More precisely, the measurement noise and parameters belong to connected sets (in the proposed work, intervals). First, this method, based on the Rosenfeld–Groebner elimination algorithm, is presented. The latter provides differential equations containing derivatives, sometimes of high order. In order to improve the numerical results, a pretreatment of the differential relations is done and consists in integration. The new relations contain, essentially, integrals depending only on the outputs. In comparison with the initial relations, they are less sensitive to measurement noise. Finally, the impact of the size of the measurement noise domain on the estimated intervals is studied. Full article

(This article belongs to the Special Issue Algorithms for Reliable Estimation, Identification and Control II)

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23 pages, 6828 KiB

Open AccessArticle

Experimental Validation of Ellipsoidal Techniques for State Estimation in Marine Applications

by Andreas Rauh, Yohann Gourret, Katell Lagattu, Bernardo Hummes, Luc Jaulin, Johannes Reuter, Stefan Wirtensohn and Patrick Hoher

Algorithms 2022, 15(5), 162; https://doi.org/10.3390/a15050162 - 11 May 2022

Cited by 4 | Viewed by 2483

Abstract

A reliable quantification of the worst-case influence of model uncertainty and external disturbances is crucial for the localization of vessels in marine applications. This is especially true if uncertain GPS-based position measurements are used to update predicted vessel locations that are obtained from [...] Read more.

A reliable quantification of the worst-case influence of model uncertainty and external disturbances is crucial for the localization of vessels in marine applications. This is especially true if uncertain GPS-based position measurements are used to update predicted vessel locations that are obtained from the evaluation of a ship’s state equation. To reflect real-life working conditions, these state equations need to account for uncertainty in the system model, such as imperfect actuation and external disturbances due to effects such as wind and currents. As an application scenario, the GPS-based localization of autonomous DDboat robots is considered in this paper. Using experimental data, the efficiency of an ellipsoidal approach, which exploits a bounded-error representation of disturbances and uncertainties, is demonstrated. Full article

(This article belongs to the Special Issue Algorithms for Reliable Estimation, Identification and Control II)

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17 pages, 360 KiB

Open AccessArticle

A Truly Robust Signal Temporal Logic: Monitoring Safety Properties of Interacting Cyber-Physical Systems under Uncertain Observation

by Bernd Finkbeiner, Martin Fränzle, Florian Kohn and Paul Kröger

Algorithms 2022, 15(4), 126; https://doi.org/10.3390/a15040126 - 11 Apr 2022

Cited by 6 | Viewed by 3171

Abstract

Signal Temporal Logic is a linear-time temporal logic designed for classifying the time-dependent signals originating from continuous-state or hybrid-state dynamical systems according to formal specifications. It has been conceived as a tool for systematizing the monitoring of cyber-physical systems, supporting the automatic translation [...] Read more.

Signal Temporal Logic is a linear-time temporal logic designed for classifying the time-dependent signals originating from continuous-state or hybrid-state dynamical systems according to formal specifications. It has been conceived as a tool for systematizing the monitoring of cyber-physical systems, supporting the automatic translation of complex safety specifications into monitoring algorithms, faithfully representing their semantics. Almost all algorithms hitherto suggested do, however, assume perfect identity between the sensor readings, informing the monitor about the system state and the actual ground truth. Only recently have Visconti et al. addressed the issue of inexact measurements, taking up the simple model of interval-bounded per-sample error that is unrelated, in the sense of chosen afresh, across samples. We expand their analysis by decomposing the error into an unknown yet fixed offset and an independent per-sample error and show that in this setting, monitoring of temporal properties no longer coincides with collecting Boolean combinations of state predicates evaluated in each time instant over best-possible per-sample state estimates, but can be genuinely more informative in that it infers determinate truth values for monitoring conditions that interval-based evaluation remains inconclusive about. For the model-free as well as for the linear model-based case, we provide optimal evaluation algorithms based on affine arithmetic and SAT modulo theory, solving over linear arithmetic. The resulting algorithms provide conclusive monitoring verdicts in many cases where state estimations inherently remain inconclusive. In their model-based variants, they can simultaneously address the issues of uncertain sensing and partial observation. Full article

(This article belongs to the Special Issue Algorithms for Reliable Estimation, Identification and Control II)

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24 pages, 2603 KiB

Open AccessArticle

Kleene Algebra to Compute Invariant Sets of Dynamical Systems

by Thomas Le Mézo, Luc Jaulin, Damien Massé and Benoit Zerr

Algorithms 2022, 15(3), 90; https://doi.org/10.3390/a15030090 - 8 Mar 2022

Cited by 2 | Viewed by 2645

Abstract

In this paper, we show that a basic fixed point method used to enclose the greatest fixed point in a Kleene algebra will allow us to compute inner and outer approximations of invariant-based sets for continuous-time nonlinear dynamical systems. Our contribution is to [...] Read more.

In this paper, we show that a basic fixed point method used to enclose the greatest fixed point in a Kleene algebra will allow us to compute inner and outer approximations of invariant-based sets for continuous-time nonlinear dynamical systems. Our contribution is to provide the definitions and theorems that will allow us to make the link between the theory of invariant sets and the Kleene algebra. This link has never be done before and will allow us to compute rigorously sets that can be defined as a combination of positive invariant sets. Some illustrating examples show the nice properties of the approach. Full article

(This article belongs to the Special Issue Algorithms for Reliable Estimation, Identification and Control II)

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14 pages, 310 KiB

Open AccessArticle

An Algebraic Approach to Identifiability

by Daniel Gerbet and Klaus Röbenack

Algorithms 2021, 14(9), 255; https://doi.org/10.3390/a14090255 - 27 Aug 2021

Cited by 4 | Viewed by 2357

Abstract

This paper addresses the problem of identifiability of nonlinear polynomial state-space systems. Such systems have already been studied via the input-output equations, a description that, in general, requires differential algebra. The authors use a different algebraic approach, which is based on distinguishability and [...] Read more.

This paper addresses the problem of identifiability of nonlinear polynomial state-space systems. Such systems have already been studied via the input-output equations, a description that, in general, requires differential algebra. The authors use a different algebraic approach, which is based on distinguishability and observability. Employing techniques from algebraic geometry such as polynomial ideals and Gröbner bases, local as well as global results are derived. The methods are illustrated on some example systems. Full article

(This article belongs to the Special Issue Algorithms for Reliable Estimation, Identification and Control II)

16 pages, 1161 KiB

Open AccessArticle

Experimental Validation of a Guaranteed Nonlinear Model Predictive Control

by Mohamed Fnadi and Julien Alexandre dit Sandretto

Algorithms 2021, 14(8), 248; https://doi.org/10.3390/a14080248 - 20 Aug 2021

Cited by 7 | Viewed by 2900

Abstract

This paper combines the interval analysis tools with the nonlinear model predictive control (NMPC). The NMPC strategy is formulated based on an uncertain dynamic model expressed as nonlinear ordinary differential equations (ODEs). All the dynamic parameters are identified in a guaranteed way considering [...] Read more.

This paper combines the interval analysis tools with the nonlinear model predictive control (NMPC). The NMPC strategy is formulated based on an uncertain dynamic model expressed as nonlinear ordinary differential equations (ODEs). All the dynamic parameters are identified in a guaranteed way considering the various uncertainties on the embedded sensors and the system’s design. The NMPC problem is solved at each time step using validated simulation and interval analysis methods to compute the optimal and safe control inputs over a finite prediction horizon. This approach considers several constraints which are crucial for the system’s safety and stability, namely the state and the control limits. The proposed controller consists of two steps: filtering and branching procedures enabling to find the input intervals that fulfill the state constraints and ensure the convergence to the reference set. Then, the optimization procedure allows for computing the optimal and punctual control input that must be sent to the system’s actuators for the pendulum stabilization. The validated NMPC capabilities are illustrated through several simulations under the DynIbex library and experiments using an inverted pendulum. Full article

(This article belongs to the Special Issue Algorithms for Reliable Estimation, Identification and Control II)

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23 pages, 18886 KiB

Open AccessArticle

Iterative Solution of Linear Matrix Inequalities for the Combined Control and Observer Design of Systems with Polytopic Parameter Uncertainty and Stochastic Noise

by Andreas Rauh, Robert Dehnert, Swantje Romig, Sabine Lerch and Bernd Tibken

Algorithms 2021, 14(7), 205; https://doi.org/10.3390/a14070205 - 7 Jul 2021

Cited by 10 | Viewed by 3247

Abstract

Most research activities that utilize linear matrix inequality (LMI) techniques are based on the assumption that the separation principle of control and observer synthesis holds. This principle states that the combination of separately designed linear state feedback controllers and linear state observers, which [...] Read more.

Most research activities that utilize linear matrix inequality (LMI) techniques are based on the assumption that the separation principle of control and observer synthesis holds. This principle states that the combination of separately designed linear state feedback controllers and linear state observers, which are independently proven to be stable, results in overall stable system dynamics. However, even for linear systems, this property does not necessarily hold if polytopic parameter uncertainty and stochastic noise influence the system’s state and output equations. In this case, the control and observer design needs to be performed simultaneously to guarantee stabilization. However, the loss of the validity of the separation principle leads to nonlinear matrix inequalities instead of LMIs. For those nonlinear inequalities, the current paper proposes an iterative LMI solution procedure. If this algorithm produces a feasible solution, the resulting controller and observer gains ensure robust stability of the closed-loop control system for all possible parameter values. In addition, the proposed optimization criterion leads to a minimization of the sensitivity to stochastic noise so that the actual state trajectories converge as closely as possible to the desired operating point. The efficiency of the proposed solution approach is demonstrated by stabilizing the Zeeman catastrophe machine along the unstable branch of its bifurcation diagram. Additionally, an observer-based tracking control task is embedded into an iterative learning-type control framework. Full article

(This article belongs to the Special Issue Algorithms for Reliable Estimation, Identification and Control II)

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12 pages, 1342 KiB

Open AccessArticle

Interval Extended Kalman Filter—Application to Underwater Localization and Control

by Morgan Louédec and Luc Jaulin

Algorithms 2021, 14(5), 142; https://doi.org/10.3390/a14050142 - 29 Apr 2021

Cited by 7 | Viewed by 2906

Abstract

The extended Kalman filter has been shown to be a precise method for nonlinear state estimation and is the facto standard in navigation systems. However, if the initial estimated state is far from the true one, the filter may diverge, mainly due to [...] Read more.

The extended Kalman filter has been shown to be a precise method for nonlinear state estimation and is the facto standard in navigation systems. However, if the initial estimated state is far from the true one, the filter may diverge, mainly due to an inconsistent linearization. Moreover, interval filters guarantee a robust and reliable, yet unprecise and discontinuous localization. This paper proposes to choose a point estimated by an interval method, as a linearization point of the extended Kalman filter. We will show that this combination allows us to get a higher level of integrity of the extended Kalman filter. Full article

(This article belongs to the Special Issue Algorithms for Reliable Estimation, Identification and Control II)

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30 pages, 9016 KiB

Open AccessArticle

Union and Intersection Operators for Thick Ellipsoid State Enclosures: Application to Bounded-Error Discrete-Time State Observer Design

by Andreas Rauh, Auguste Bourgois and Luc Jaulin

Algorithms 2021, 14(3), 88; https://doi.org/10.3390/a14030088 - 14 Mar 2021

Cited by 12 | Viewed by 3689

Abstract

Thick ellipsoids were recently introduced by the authors to represent uncertainty in state variables of dynamic systems, not only in terms of guaranteed outer bounds but also in terms of an inner enclosure that belongs to the true solution set with certainty. Because [...] Read more.

Thick ellipsoids were recently introduced by the authors to represent uncertainty in state variables of dynamic systems, not only in terms of guaranteed outer bounds but also in terms of an inner enclosure that belongs to the true solution set with certainty. Because previous work has focused on the definition and computationally efficient implementation of arithmetic operations and extensions of nonlinear standard functions, where all arguments are replaced by thick ellipsoids, this paper introduces novel operators for specifically evaluating quasi-linear system models with bounded parameters as well as for the union and intersection of thick ellipsoids. These techniques are combined in such a way that a discrete-time state observer can be designed in a predictor-corrector framework. Estimation results are presented for a combined observer-based estimation of state variables as well as disturbance forces and torques in the sense of an unknown input estimator for a hovercraft. Full article

(This article belongs to the Special Issue Algorithms for Reliable Estimation, Identification and Control II)

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18 pages, 9372 KiB

Open AccessArticle

Transformation of Uncertain Linear Systems with Real Eigenvalues into Cooperative Form: The Case of Constant and Time-Varying Bounded Parameters

by Andreas Rauh and Julia Kersten

Algorithms 2021, 14(3), 85; https://doi.org/10.3390/a14030085 - 8 Mar 2021

Cited by 6 | Viewed by 2489

Abstract

Continuous-time linear systems with uncertain parameters are widely used for modeling real-life processes. The uncertain parameters, contained in the system and input matrices, can be constant or time-varying. In the latter case, they may represent state dependencies of these matrices. Assuming bounded uncertainties, [...] Read more.

Continuous-time linear systems with uncertain parameters are widely used for modeling real-life processes. The uncertain parameters, contained in the system and input matrices, can be constant or time-varying. In the latter case, they may represent state dependencies of these matrices. Assuming bounded uncertainties, interval methods become applicable for a verified reachability analysis, for feasibility analysis of feedback controllers, or for the design of robust set-valued state estimators. The evaluation of these system models becomes computationally efficient after a transformation into a cooperative state-space representation, where the dynamics satisfy certain monotonicity properties with respect to the initial conditions. To obtain such representations, similarity transformations are required which are not trivial to find for sufficiently wide a-priori bounds of the uncertain parameters. This paper deals with the derivation and algorithmic comparison of two different transformation techniques for which their applicability to processes with constant and time-varying parameters has to be distinguished. An interval-based reachability analysis of the states of a simple electric step-down converter concludes this paper. Full article

(This article belongs to the Special Issue Algorithms for Reliable Estimation, Identification and Control II)

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