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## Über dieses Buch

To describe the true behavior of most real-world systems with sufficient accuracy, engineers have to overcome difficulties arising from their lack of knowledge about certain parts of a process or from the impossibility of characterizing it with absolute certainty. Depending on the application at hand, uncertainties in modeling and measurements can be represented in different ways. For example, bounded uncertainties can be described by intervals, affine forms or general polynomial enclosures such as Taylor models, whereas stochastic uncertainties can be characterized in the form of a distribution described, for example, by the mean value, the standard deviation and higher-order moments.

The goal of this Special Volume on Modeling, Design, and Simulation of Systems with Uncertainties is to cover modern methods for dealing with the challenges presented by imprecise or unavailable information. All contributions tackle the topic from the point of view of control, state and parameter estimation, optimization and simulation.

Thematically, this volume can be divided into two parts. In the first we present works highlighting the theoretic background and current research on algorithmic approaches in the field of uncertainty handling, together with their reliable software implementation. The second part is concerned with real-life application scenarios from various areas including but not limited to mechatronics, robotics, and biomedical engineering.

## Inhaltsverzeichnis

### Chapter 1. Implementing a Rigorous ODE Solver Through Literate Programming

Interval numerical methods produce results that can have the power of a mathematical proof. Although there is a substantial amount of theoretical work on these methods, little has been done to ensure that an implementation of an interval method can be readily verified. However, when claiming rigorous numerical results, it is crucial to ensure that there are no errors in their computation. Furthermore, when such a method is used in a computer assisted proof, it would be desirable to have its implementation published in a form that is convenient for verification by human experts. We have applied Literate Programming (LP) to produce VNODE-LP, a C++ solver for computing rigorous bounds on the solution of an initial-value problem (IVP) for an ordinary differential equation (ODE).We have found LP well suited for ensuring that an implementation of a numerical algorithm is a correct translation of its underlying theory into a programming language: we can split the theory into small pieces, translate each of them, and keep mathematical expressions and the corresponding code close together in a unified document. Then it can be reviewed and checked for correctness by human experts, similarly to how a scientific work is examined in a peer-review process.
Nedialko S. Nedialkov

### Chapter 2. A New Method for Inner Estimation of Solution Sets to Interval Linear Systems

For an interval system of linear equations Ax = b, we consider the problem of inner estimation of its solution set, formed by all the solutions to point systems Ax= b with AA and bb. The so-called “center approach” to the problem is developed when the inner interval box is constructed around an a priori known center point from the solution set. Determining the size of the inner box is shown to be reduced to a maximization problem for a special quasiconcave objective function.
Sergey P. Shary

### Chapter 3. Structural Analysis for the Design of Reliable Controllers and State Estimators for Continuous-Time Dynamical Systems with Uncertainties

The task of designing feedforward control strategies for finite-dimensional systems in such a way that the output variables match predefined trajectories is a common goal in control engineering. Besides the widely used formulation of the corresponding system models as explicit sets of ordinary differential equations, differential-algebraic representations allow for a unified treatment of both system analysis and synthesis. For modeling and analysis of many real-life dynamic processes, differential-algebraic equations are a natural description to take into account interconnections between different physical components. Each component of such interconnected systems is described by a separate dynamic model, for instance the electric drive and the mechanical components in power trains.Moreover, side conditions are required to connect these component models by a description of power flow or, for example, geometric constraints imposed by links and joints. During system synthesis, control design tasks can be formulated in terms of initial value problems for sets of differential-algebraic equations. To check solvability, verified and nonverified algorithms are applicable which analyze the underlying system structures. The same holds for the reconstruction of internal variables and parameters on the basis of measured data. In this contribution, constructive approaches are discussed for solving both the control and estimator design using differential-algebraic formulations. It is demonstrated how these approaches can be used to show controllability and observability of dynamical systems. Numerical results for two applications conclude this paper.
Andreas Rauh, Harald Aschemann

### Chapter 4. Analyzing Reachability of Linear Dynamic Systems with Parametric Uncertainties

As an important approach to analyzing safety of a dynamic system, this paper considers the task of computing overapproximations of reachable sets, i.e. the set of states which is reachable from a given initial set of states. The class of systems under investigation are linear, time-invariant systems with parametric uncertainties and uncertain but bounded input. The possible set of system matrices due to uncertain parameters is represented by matrix zonotopes and interval matrices – computational techniques for both representations are presented. The reachable set is represented by zonotopes, which makes it possible to apply the approach to systems of 100 continuous state variables with computation times of a few minutes. This is demonstrated for randomized examples as well as a transmission line example.
Matthias Althoff, Bruce H. Krogh, Olaf Stursberg

### Chapter 5. Robustness Comparison of Tracking Controllers Using Verified Integration

In this contribution we discuss a method for investigating the robustness properties of tracking controllers using verified simulation. This method allows to compare the controllers with respect to robustness against uncertainty in the parameters of the plant and in the initial conditions of measured and unmeasured states. A robustness criterion is formulated, which can be evaluated using interval methods. To illustrate the approach, we compare the robustness properties of three conceptually different flatness based tracking controllers with dynamic output feedback, which are applied to a simple example.
Marco Kletting, Felix Antritter

### Chapter 6. Probabilistic Set-Membership State Estimator

Interval constraint propagation methods have been shown to be efficient, robust and reliable to solve difficult nonlinear bounded-error state estimation problems. However they are considered as unsuitable in a probabilistic context, where the approximation of a probability density function by a set cannot be accepted as reliable. This paper proposes a new probabilistic approach which makes it possible to use classical set-membership observers which are robust with respect to outliers. The approach is illustrated on a localization of robots in situations where there exist a large number of outliers.
Luc Jaulin

### Chapter 7. Verified Global Optimization for Estimating the Parameters of Nonlinear Models

Nonlinear parameter estimation is usually achieved via the minimization of some possibly non-convex cost function. Interval analysis allows one to derive algorithms for the guaranteed characterization of the set of all global minimizers of such a cost function when an explicit expression for the output of the model is available or when this output is obtained via the numerical solution of a set of ordinary differential equations. However, cost functions involved in parameter estimation are usually challenging for interval techniques, if only because of multi-occurrences of the parameters in the formal expression of the cost. This paper addresses parameter estimation via the verified global optimization of quadratic cost functions. It introduces tools for the minimization of generic cost functions. When an explicit expression of the output of the parametric model is available, significant improvements may be obtained by a new box exclusion test and by careful manipulations of the quadratic cost function. When the model is described by ODEs, some of the techniques available in the previous case may still be employed, provided that sensitivity functions of the model output with respect to the parameters are available.
Michel Kieffer, Mihály Csaba Markót, Hermann Schichl, Eric Walter

### Chapter 8. Optimal Control of Induction Heating: Theory and Application

The theoretic background of an optimal control task for a precision induction heating problem is studied in this work. The basics of electro-magnetic and heat transfer theory are used to describe the dynamics of induction heating processes of rectangle workpieces. The main result of this work, presented as the first-order necessary conditions for the optimal solution of the considered control task, allows one to employ interval representations of the mathematical model’s main parameters in order to study the influence of environment uncertainties which have dominant effects on induction heating processes.
Darya Filatova, Marek Grzywaczewski

### Chapter 9. Coherent Upper and Lower Conditional Previsions Defined by Hausdorff Outer and Inner Measures

A new model of coherent upper conditional previsions is proposed to represent uncertainty and to make previsions in complex systems. It is defined by the Choquet integral with respect to Hausdorff outer measure if the conditioning event has positive and finite Hausdorff outer measure in its Hausdorff dimension. Otherwise, when the conditioning event has Hausdorff outer measure equal to zero or infinity in its Hausdorff dimension, it is defined by a 0-1 valued finitely, but not countably, additive probability. If the conditioning event has positive and finite Hausdorff outer measure in its Hausdorff dimension, it is proven that a coherent upper conditional prevision is uniquely represented by the Choquet integral with respect to the upper conditional probability defined by Hausdorff outer measure if and only if it is monotone, comonotonically additive, submodular and continuous from below.Moreover sufficient conditions are given such that the upper conditional previsions satisfy the disintegration property and the conglomerability principle.
Serena Doria

### Chapter 10. Two Approaches for Guaranteed State Estimation of Nonlinear Continuous-Time Models

This paper deals with the estimation of the state vector of a nonlinear continuous-time state-space model, such as those frequently encountered in the context of knowledge-based modeling. Unknown and possibly time-varying parameters may be included in an extended state vector to deal with the simultaneous estimation of state and parameters. Observations depending on the (possibly extended) state are assumed to take place at discrete measurement times. Given bounds on the size of the additive measurement errors, guaranteed estimation should then provide bounds on the possible values of the state at any given time. Two recently developed approaches are presented and their performance is compared on a simple test case.
Marco Kletting, Michel Kieffer, Eric Walter

### Chapter 11. Quantifying Spacecraft Failure in an Uncertain Environment: the Case of Jupiter Europa Orbiter

Study of the Outer Planets is considered as a high priority activity by the Planetary Science community. One candidate for the next Outer Planets Flagship Mission (OPFM –missions in the $2B–$4B range) is the Jupiter Europa Orbiter (JEO) concept. In this work, we address the interplay of various types of uncertainties to probe the possibility of characterizing the reliability of a proposed mission concept. By combining the aleatory characterization of spacecraft subsystems and the epistemic uncertainties of the Jovian environment we describe an approach for quantifying possible ranges of mission durations for a potential JEO concept. The work here illustrates the potential for probabilistic representations of epistemic uncertainties by introducing temporal correlations. In addition the effects of failure correlations among similar components in a spacecraft are incorporated to assess their impact on the failure likelihood.

### Chapter 12. Robust State and Parameter Estimation for Nonlinear Continuous-Time Systems in a Set-Membership Context

This chapter deals with joint state and parameter estimation for nonlinear continuous-time systems. Based on an appropriate LPV approximation, the problem is formulated in terms of a set adaptive observer design problem which can be efficiently solved. The resolution methodology avoids the exponential complexity obstruction often met in set-membership parameter estimation. The efficacy of the proposed set adaptive observers is demonstrated on several examples.
Denis Efimov, Tarek Raässi, Ali Zolghadri

### Chapter 13. Nonlinear Adaptive Control of a Bioprocess Model with Unknown Kinetics

In this paper we consider a nonlinear model of an anaerobic wastewater treatment process, in which biodegradable organic is decomposed to produce methane. The model, described by a four-dimensional dynamic system, is known to be practically validated and reliable.We propose a feedback control law for asymptotic stabilization of the closed-loop system towards a fixed operating point. Moreover, a model-based numerical extremum seeking algorithm is applied to stabilize the control system towards an equilibrium point with maximal methane flow rate. The robustness of the feedback control is demonstrated by assuming uncertainties in the growth rate functions. Computer simulations are reported to illustrate the theoretical results.
Neli S. Dimitrova, Mikhail I. Krastanov

### Chapter 14. Verified Analysis of a Model for Stance Stabilization

The stabilization of stance is a subject of continuing research in biology, biomechanics and robotics. It plays an important role in many clinical applications as well as in forward dynamical gait simulation. In this paper, we propose a new model relying on a two cylinder foot contact scheme. This contact model has the advantage of simple and smooth dynamic behavior which in turn results in better efficiency in comparison with other contact models. However, a number of parameters in this model, such as position or mass of the pelvis, are known only with some uncertainty. To deal with the situation, we analyze the model using verifiedmethods, which includes propagating the uncertainty through the system and computing the sensitivities of the equations of motion in the first time interval. To perform verified simulations of the whole model, a verified initial value problem solver for a hybrid system is required, which can switch from one system of the equations of motion to the other depending on a certain switching function. While research in this direction remains a topic of high complexity, a simplified kinetostatic version of the model allows one to analyze the sensitivity of the model to parameter variations, as presented in this paper.
Ekaterina Auer, Haider Albassam, Andrés Kecskeméthy, Wolfram Luther

### Chapter 15. Adaptive Control Strategies in Heat Transfer Problems with Parameter Uncertainties Based on a Projective Approach

Control problems for distributed heating systems described by parabolic partial differential equations are considered in this paper. This type of mathematical model is also a common description for other distributed parameter systems involving diffusion as well as heat and mass transfer. The goal of the paper is to develop an adaptive strategy including online parameter identification for efficient control of heat transfer systems. The developed strategy is based on the method of integrodifferential relations, a projective approach, and a suitable finite element technique. An adaptive control algorithm with predictive estimates of the desired output trajectories is proposed and its specific features are discussed. We use the parameters, geometry, and actuation principles of a real test setup available at the University of Rostock for the numerical simulation and verification. The test setup consists of a metallic rod equipped with a finite number of Peltier elements which are used as distributed control inputs allowing for active cooling and heating. A validation of the control laws derived in this contribution is performed taking into account the explicit local and integral error estimates resulting directly from the method of integrodifferential relations.
Vasily V. Saurin, Georgy V. Kostin, Andreas Rauh, Harald Aschemann

### Chapter 16. State and Disturbance Estimation for Robust Control of Fast Flexible Rack Feeders

Rack feeders as automated conveying systems for high bay rackings are of high practical importance. To shorten the transport times by using trajectories with increased kinematic values accompanying control measures for a reduction of the excited structural vibrations are necessary. In this contribution, the controloriented modeling for an experimental set-up of such a high bay rack feeder and the model-based design of a gain-scheduled feedforward and feedback control structure is presented. The rack feeder is modeled as an elastic multibody system. For the mathematical description of the bending deflections a Ritz ansatz is introduced for the first two bending modes. The tracking control design is performed separately for both axes using decentralized state space representations. Unmeasurable states as well as remaining uncertainties are estimated by a combined state and disturbance observer. Both the achievable performance and the resulting tracking accuracy of the proposed control concept are shown by measurement results from the experimental set-up.
Harald Aschemann, Dominik Schindele, Jèran Ritzke

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