Acta Cybernetica

Preface

2020-04-14T17:13:07+02:00

The Summer Workshop on Interval Methods (SWIM) is an annual meeting initiated in 2008 by the French MEA working group on Set Computation and Interval Techniques of the French research group on Automatic Control. A special focus of the MEA group is on promoting interval analysis techniques and applications to a broader community of researchers, facilitated by such multidisciplinary workshops. Since 2008, SWIM has become a keystone event for researchers dealing with various aspects of interval and set-based methods.

In 2018, the 11^th edition in this workshop series was held at the University of Rostock, Germany, with a focus on research topics in the fields of engineering, computer science, and mathematics. A total of 31 talks were given during this workshop, covering the following areas:

verified solution of initial value problems for ordinary differential equations, differential-algebraic system models, and partial differential equations,
scientific computing with guaranteed error bounds,
design of robust and fault-tolerant control systems,
modeling and quantification of errors in engineering tasks,
implementation of software libraries, and
usage of the aforementioned approaches for system models in control engineering, data analysis, signal and image processing.

After a peer-review process, 15 high-quality articles were selected for publication in this special issue. They are roughly divided into two thematic groups: Uncertainty Modeling, Software, Verified Computing and Optimization as well as Interval Methods in Control and Robotics.

The first part, Uncertainty Modeling, Software, Verified Computing and Optimization, contains methodological aspects concerning reliable modeling of dynamic systems as well as visualization and quantification of uncertainty in the fields of measurement and simulation. Moreover, existence proofs for solutions of partial differential equations and their reliable optimal control synthesis are considered. A paper making use of quantifier elimination for robust linear output feedback control by means of eigenvalue placement concludes this section.

The second part of this special issue, Interval Methods in Control and Robotics, is focused on the design as well as numerical and experimental validation of robust state observation and control procedures along with reliable parameter and state estimation approaches in the fields of control for thermal systems, robotics, localization of drones and global positioning systems.

Sound Over-Approximation of Probabilities

2020-04-14T16:01:42+02:00

Safety analysis of high confidence systems requires guaranteed bounds on the probability of events of interest. Establishing the correctness of algorithms that compute such bounds is challenging. We address this problem in three steps. First, we use monadic transition systems (MTS) in the category of sets as a general framework for modeling discrete time systems. MTS can capture different types of system behaviors, but here we focus on a combination of non-deterministic and probabilistic behaviors that arises often when modeling complex systems. Second, we use the category of posets and monotonic maps as general setting to define and compare approximations. In particular, for the MTS of interest, we consider approximations of their configurations based on complete lattices of interval probabilities. Third, we obtain an algorithm that computes over-approximations of system configurations after a finite number of steps, by restricting to finite lattices.

Reliable Visual Analytics, a Prerequisite for Outcome Assessment of Engineering Systems

2020-04-14T16:01:43+02:00

Various evaluation approaches exist for multi-purpose visual analytics (VA) frameworks. They are based on empirical studies in information visualization or on community activities, for example, VA Science and Technology Challenge (2006-2014) created as a community evaluation resource to 'decide upon the right metrics to use, and the appropriate implementation of those metrics including datasets and evaluators'. In this paper, we propose to use evaluated VA environments for computer-based processes or systems with the main goal of aligning user plans, system models and software results. For this purpose, trust in VA outcome should be established, which can be done by following the (meta-)design principles of a human-centered verification and validation assessment and also in dependence on users' task models and interaction styles, since the possibility to work with the visualization interactively is an integral part of VA. To define reliable VA, we point out various dimensions of reliability along with their quality criteria, requirements, attributes and metrics. Several software packages are used to illustrate the concepts.

Another Multibody Dynamics in Natural Coordinates through Automatic Differentiation and High-Index DAE Solving

2020-04-14T16:01:45+02:00

The Natural Coordinates (NCs) method for Lagrangian modelling and simulation of multi-body systems is valued for giving simple, sparse models. We describe our version of it (NPNCs) and compare with the classical ap- proach of Jalón and Bayo (JBNCs). NPNCs use the high-index differential- algebraic equation solver DAETS. Algorithmic differentiation, not symbolic algebra, forms the equations of motion from the Lagrangian. NPNCs give significantly smaller equation systems than JBNCs, at the cost of a non- constant mass matrix for fully 3D models—a minor downside in the DAETS context. A 2D and a 3D example are presented, with numerical results.

Towards Analyzing the Influence of Measurement Errors in Magnetic Resonance Imaging of Fluid Flows

2020-04-14T16:01:46+02:00

Magnet resonance imaging does not only have a large number of applications in the field of medical examinations. In addition, several promising applications were also reported for the measurement of technical fluid flows and for the measurement of temperature fields in technical devices which do not allow for a classical access by either arrays of flow meters on the one hand or by arrays of temperature sensors such as thermocouples on the other hand. Due to the fact that magnet resonance imaging can be performed in a non-invasive manner, it has the advantage to provide relevant data without disturbing the velocity and temperature fields by external sensor devices. Moreover, measurement information can also be obtained for scenarios in which a direct access to the media under investigation is hardly possible due to constructive limitations. To make this kind of measurement applicable also for dynamic scenarios, not only the spatial resolution but also the temporal one needs to be sufficiently accurate. If the temporal resolution is of interest, an acceleration of the measurement process becomes possible by compressed sensing techniques which make use of an undersampling of the so-called $k$-space. However, such compressed sensing approaches require a reconstruction of the original fields of the physical variables to be measured. In this paper, it is shown how interval arithmetic approaches can be employed to solve the necessary optimality criteria for the fluid velocity reconstruction under the assumption of bounded measurement errors.

Computer-assisted Existence Proofs for One-dimensional Schrödinger-Poisson Systems

2020-04-14T16:01:47+02:00

Motivated by the three-dimensional time-dependent Schrödinger-Poisson system we prove the existence of non-trivial solutions of the one-dimensional stationary Schrödinger-Poisson system using computer-assisted methods.

Starting from a numerical approximate solution, we compute a bound for its defect, and a norm bound for the inverse of the linearization at the approximate solution. For the latter, eigenvalue bounds play a crucial role, especially for the eigenvalues "close to" zero. Therefor, we use the Rayleigh-Ritz method and a corollary of the Temple-Lehmann Theorem to get enclosures of the crucial eigenvalues of the linearization below the essential spectrum.

With these data in hand, we can use a fixed-point argument to obtain the desired existence of a non-trivial solution "nearby" the approximate one. In addition to the pure existence result, the used methods also provide an enclosure of the exact solution.

Verified Solution to Optimal Control Problems of Elastic Rod Motion Based on the Ritz Method

2020-04-14T16:01:48+02:00

To model vibrations in flexible structures, a generalized variational formulation of PDE control problems is considered in the frame of the method of integro-differential relations. This approach allows us to estimate a posteriori the quality of finite-dimensional approximations and, as a result, either to refine or coarsen them if necessary. Such estimates also make it possible to correct the related control signals. The procedures for solving optimization problems in dynamics of linear elasticity have been proposed based on the Ritz method and FEM. An original FEM solver for mechanical systems with
varying distributed parameters is described. The resulting control law is regularized via a quadratic cost functional including the discrepancy of the constitutive equations. The verification of optimized control for elastic rod motion involves local and integral error estimates proposed.

Eigenvalue Placement by Quantifier Elimination - the Static Output Feedback Problem

2020-04-14T16:01:49+02:00

This contribution deals with the static output feedback problem of linear time-invariant systems. This is still an area of active research, in contrast to the observer-based state feedback problem, which has been solved decades ago. We consider the formulation and solution of static output feedback design problems using quantifier elimination techniques. Stabilization as well as more specified eigenvalue placement scenarios are the focus of the paper.

Characterizing Sliding Surfaces of Cyber-Physical Systems

2020-04-14T16:01:49+02:00

When implementing a non-continuous controller for a cyber-physical system, it may happen that the evolution function of the closed-loop system is not anymore piecewise continuous along the trajectory, mainly due to if statements inside the control algorithm. As a consequence, an unwanted chattering effect may occur. This behavior is often difficult to observe even in simulation. We propose here a set-membership method based on interval analysis to detect different types of discontinuities. One of them is the sliding surface where the state trajectory jumps indefinitely between two distinct behaviors. As an application, we consider the validation of a sailboat controller. We show that our approach is able to detect and explain some unwanted sliding effects that may be observed in rare and specific situations on our actual sailboat robots.

Optimal Switching Instants for the Control of Hybrid Systems

2020-04-14T16:01:50+02:00

The problem of determining optimal switching instants for the control of hybrid systems under reachability constraints is considered. This optimization problem is cast into an interval global optimization problem with differential constraints, where validated simulation techniques and dynamic time meshing are used for its solution. The approach is applied on two examples, one being the well-known example of the Goddard's problem where a rocket has to reach a given altitude while consuming the smallest amount of fuel.

Verified Interval Enclosure Techniques for Robust Gain Scheduling Controllers

2020-04-14T20:50:11+02:00

In real-life applications, dynamic systems are often subject to uncertainty due to model simplications, measurement inaccuracy or approximation errors which can be mapped to specific parameters. Uncertainty in dynamic systems can come either in stochastic forms or as interval representations, when they are considered as bounded as it will be done in this paper. The main idea, here, is to find a joint approach for an interval-based gain scheduling controller while simultaneously reducing overestimation by enclosing state intervals with the least amount of conservativity. The robust and/ or optimal control design is realized using linear matrix inequalities (LMIs) to find an efficient solution and aims at a guaranteed stabilization of the system dynamics over a predefined time horizon. Since the resulting system is assumed to be asymptotically stable, a temporal reduction of the widths of intervals representing worst-case bounds of the system states at a specific point of time should occur. However, for commonly used approaches in the computation of interval enclosures those interval widths seemingly blow up due to the wrapping effect in many cases. To avoid this, we provide two interval enclosure techniques --- an exploitation of cooperativity and an exponential approach --- and discuss their applicability taking into account two real-life applications, a high-bay rack feeder and an inverse pendulum.

Interval Predictors for a Class of Uncertain Discrete-Time Systems

2020-04-14T16:01:52+02:00

This work presents set-valued algorithms to compute tight interval predictions of the state trajectories for a certain class of uncertain dynamical systems. Based on interval analysis and the analytic expression of the state response of discrete-time linear systems, non-conservative numerical schemes are proposed. Moreover, under some stability conditions, the convergence of the width of the predicted state enclosures is proved. The performance of the proposed set-valued algorithms are illustrated through two numerical examples and the results are compared to that obtained with an other method selected from the literature.

From Verified Parameter Identification to the Design of Interval Observers and Cooperativity-Preserving Controllers

2020-04-14T16:01:53+02:00

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 controllers. The corresponding computational steps are described and visualized for the temperature control of a laboratory-scale test rig available at the Chair of Mechatronics at the University of Rostock.

On Interval Observer Design for Continuous-Time LPV Switched Systems

2020-04-14T16:01:53+02:00

State estimation for switched systems with time-varying parameters has received a great attention during the past decades. In this paper, a new approach to design an interval observer for this class of systems is proposed. The scheduling vector is described by a convex combination so that the parametric uncertainties belong into polytopes. The considered system is also subject to measurement noise and state disturbances which are supposed to be unknown but bounded.
The proposed method guarantees both cooperativity and Input to State Stability (ISS) of the upper and lower observation errors. Sufficient conditions are given in terms of Linear Matrices Inequalities (LMIs) using a common quadratic Lyapunov function. Finally, a numerical example is provided to show the effectiveness of the designed observer.

Cooperative Localization of Drones by using Interval Methods

2020-04-14T16:01:55+02:00

In this article we address the problem of cooperative pose estimation in a group of unmanned aerial vehicles (UAV) in a bounded error context. The UAVs are equipped with cameras to track landmarks positions, and a communication and ranging system to cooperate with their neighbours. Measurements are represented by intervals, and constraints are expressed on the robots poses (positions and orientations). Each robot of the group first computes a pose domain using only its sensors measurements using set inversion via interval analysis. Then, through position boxes exchange, positions are cooperatively refined by constraint propagation in the group. Results with real robot data are presented, and show that the position accuracy is improved
thanks to cooperation.

Reliable Bounding Zones and Inconsistency Measures for GPS Positioning using Geometrical Constraints

2020-04-14T16:01:55+02:00

Reliable confidence domains for Global Navigation Satellite System (GNSS) positioning and inconsistency measures of the observations are of great importance for any navigation system, especially for safety critical applications. In this work, deterministic error bounds are introduced in form of intervals to assess remaining observation errors. The intervals can be computed based on expert knowledge or based on a sensitivity analysis of the measurement correction process. Using convex optimization, bounding zones are computed for GPS positioning using the geometrical constraints imposed by the observation intervals. The bounding zone is a convex polytope, where exploiting only the navigation geometry, confidence domain is computed in form of zonotope. We show that the relative volume between the polytopes and the zonotope is an inconsistency measures. Small polytope volume indicates bad consistency of the observations. In extreme cases empty sets are obtained which indicates large outliers. We determine the observation intervals via sensitivity analysis of the Klobuchar ionospheric model and Saastamoinen tropospheric model. The remaining errors are treated as white noise. We explain how the shape and the volume of the polytope are related to the positioning geometry. We show that this assignment has to be interpreted with care. Furthermore, we propose a new concept of Minimum Detectable Biases (MDB). Taking GPS data from simulations and real experiments, a comparison analysis between the proposed deterministic bounding method and the classical least-squares adjustment has been conduct in terms of accuracy and reliability. This helps validating that our proposed deterministic bound methods shows high internal and external reliability compared to the probabilistic approaches and that it provides rigorous inconsistency measures.