| Luise Blank | State estimation from the view point of inverse problems |
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Monitoring dynamical processes requires the estimation of the entire state,
which is only partly accessible by measurements.
Most quantities must be determined via model based state estimation.
Hence, we shortly focus in the beginning on the observability
of the system and introduce a
-to the authors knowledge- new measure of observability.
Since
in general only noisy data are given,
state estimation yields an ill-posed inverse problem.
Therefore we give also a short introduction to
ill-posed problems and its standard treatments.
For state estimation regularization techniques are commonly applied in addition to the least squares ansatz. However, to avoid undesired bias we suggest to omit the regularization of the unknown initial data. To the authors knowledge this problem formulation has not been analysed analytically yet, which is one purpose of this talk. The first order necessary conditions of the resulting minimization problem are presented and the problem is reduced by several variables. In the linear case we show that this problem formulation leads to a well-posed problem with respect to L2- and L∞- disturbances. However, we show that the condition numbers of the evolving operators can be arbitrarily large if the spectral radius of the system matrix is large, i.e. if the measure of observability is low. In this context long time horizons seem to be undesired, since they lead to larger spectral radii. This potential ill-conditioning turns out to be also an issue for the numerical calculation of the state. Small L2 measurement errors may lead to large disturbances in the states. Nevertheless, for the probably in praxis more relevant L∞-norm, perturbations yield errors in the initial data bounded independently of the system matrices. At the end we would like to discuss whether bias or possible ill-conditioning is more acceptable for state estimation, whether there is a way to avoid both and which norm is most appropriate. |
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| Georg Bock | Numerical Methods for Parameter and State Estimation |
| We review classical numerical methods for parameter and state estimation in ordinary differential (ODE) and differential algebraic equation (DAE) models. We focus on the Boundary Value Problem (BVP) approach, in particular the multiple shooting method, that treats both the simulation and the parameter estimation problem in one shot, and that has been developed to treat strongly nonlinear problems efficiently. Here, a large but structured constrained least squares problem is formulated and solved by a generalized Gauss-Newton method. We discuss important features of an efficient implementation like generation of derivatives, structure exploiting linear algebra, globalisation. We illustrate the methods at examples from engineering and biology. | |
| Moritz Diehl | A Real-Time Optimization Algorithm for Moving Horizon State Estimation |
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We present an online state and parameter estimation method for
nonlinear dynamical systems, as needed for advanced feedback control and
real-time optimization techniques, e.g. in process engineering. The
estimation method uses a moving horizon approach based on past online
measurements. The resulting state and parameter estimation problems
are solved -- online -- by a multiple shooting approach for
differential algebraic equations in conjunction with a Gauss Newton
method. For problem regularization, if necessary, we propose to employ
virtual measurements of initial state and parameters that are based on
previous measurement data (lying before the horizon) and updated by a
Kalman filter algorithm based on the current system linearization. We
show how to initialize subsequent problems by a shift, and propose a
fast "real-time iteration scheme" for online estimation, that updates
the measurement data in each Gauss Newton iteration. The numerical
performance and estimation capability of the algorithm is demonstrated
at an example from process engineering.
Joint work with H.G. Bock, L. Wirsching, P. Kühl, J. Busch, J. P. Schlöder |
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| Rolf Findeisen | State estimation and model predictive control: existing results and challenges |
| In this talk we consider the output-feedback stabilization problem for nonlinear continuous time systems employing nonlinear model predictive control in combination with suitable state observers. In the first part we review results based on so called nonlinear separation principles and the certainty equivalence principle. Even so that these results are theoretically appealing certain restrictions limit their applicability. For example the application requires that the value function is continuous, which is often difficult to guarantee. In the second part we outline an approach based on the combination of a min-max nonlinear model predictive control schemes with observers that deliver set-based state information. Provided the observer estimates are consistent and the min-max NMPC controller is designed suitably, it is show that the closed-loop can be rendered stable. | |
| Rüdiger Franke | Aspects of bringing optimization on-line |
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Modern control systems need to extend the reach of traditional automation systems
-beyond control of the process- to achieve the productivity gains necessary to succeed
in today's business markets. The Industrial IT System 800xA from ABB provides a
scalable solution that spans and integrates loop, unit, area, plant, and interplant
controls.
The talk introduces into 800xA and the underlying Aspect Object technology. Aspect Objects relate all plant data and functions, the aspects, to specific plant assets, the objects. It is discussed, how this technology is applied to on-line optimization. This includes the relation of process signals to model variables, the estimation of the model state, the predictive optimization and the application of results. A Non-linear Modedel-based Predictive Controller (NMPC) for power plant start-up serves as example. The startup problem is challanging as it is highly non-linear in the covered large range of operation. Thermal stress occuring in thick walled components needs to be kept in given limits. However, the governing process of steam flowing through the thick walled components is not always observable during startup, in particular for low mass flow rates, parallel flow paths with lumped flow rate measurements and at the boundary between saturated and superheated steam. This is why the estimation of initial states and the use of an appropriate error model is crucial for bringing the optimization on-line. |
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| Guillaume Goffaux | Robust Kalman Filtering Techniques Applied to Train Positioning |
| PDF-file | |
| Uwe Hanebeck | Progressive Bayes: A new state estimation framework for nonlinear systems |
| PDF-file | |
| Ekaterina Kostina | Robust Parameter Estimation for Identification of Satellite Injection Orbits |
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The observations of satellites typically consist of range, range rates,
azimuth
or elevation angles. The measurement times are usually clustered,
not all states are measured and outliers may occur due to ambiguity
problems.
From such usually few measurements the orbit of a satellite has to be
recovered
in order to allow the prediction of the future trajectory.
Due to the fact that the actual injection orbit of a satellite may significantly deviate from the planned one, e.g. if the launcher exhibits underperformance or malfunction, a fast and reliable the determination of initial orbits using the available data is particularly important. Based on practical problems provided by European Space Agency (ESA) the talk describes typical difficulties. A mathematical formulation of orbit determination problems is described. An l1 objective function is choosen in order to reduce the influence of outliers. For the numerical treatment of orbit determination problems shooting strategies for estimation problems in nonlinear differential equations are described. Emphasis is put on the efficienct treatment of the resulting large-scale nonlinear constrained weighted l1 optimization problem. The performance of the resulting codes is demonstrated using practical test cases provided by ESA. The talk is based on joint work with H. G. Bock, G. Gienger, S. Pallaschke, J. P. Schlöder and G. Ziegler. |
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| Philipp Kügler | Online Parameter Estimation in Infinite Dimensional Dynamical Systems |
| Online (or real time) parameter estimation finds practical applications both by itself and as part of an adaptive control system. In the infinite dimensional case, the literature discusses online algorithms for parameter estimation in parabolic or hyperbolic partial differential equations which require spatially distributed data and their differentiation with respect to the space variables. In this talk, we introduce an online parameter estimation algorithm applicable to PDEs as well as ODEs that also allows for partial state observations and makes data differentiation unnecessary. Numerical examples are presented. | |
| Toshiyuki Ohtsuka | Continuation Method for Real-Time Computation of Nonlinear Moving Horizon State Estimation |
| In this talk, real-time algorithms for nonlinear moving horizon state estimation (MHSE) are introduced. Since a dynamic optimization problem over a finite past has to be solved at each sampling time, a fast optimization algorithm is essential for implementation of nonlinear MHSE. One of key ideas for realizing real-time optimization is exploitation of the time-dependent nature of the MHSE problem. For example, the derivative of the optimal solution with respect to time can be obtained by solving a linear two-point boundary-value problem without iterative searches, which is a kind of continuation method. This idea of the continuation method can be utilized to derive, without any discretization, an explicit differential equation for the state estimate in a similar form as the Kalman filter. It is also possible to apply the continuation method for the optimization problem discretized over the horizon. In that case, the continuation method leads to a linear algebraic equation for the derivative of the unknown variables with respect to time, and the linear algebraic equation can be solved efficiently with Krylov subspace methods. Some application examples of the continuation-based algorithms are presented, including not only MHSE but also receding horizon control of a hovercraft, ship and an automobile. | |
| Andrey Romanenko | Unscented Kalman state estimation in process systems |
|
Recently, the unscented Kalman filter (UKF) algorithm, which is a new generalization
of the Kalman filter for nonlinear systems, was proposed in the literature.
It has significant advantages over its widely used predecessor that include
better accuracy and simpler implementation.
A number of applications using the UKF have been reported since then, most of them in the area of aerospace navigation and tracking where one frequently encounters severe nonlinearity and fast dynamics. However, accounts of UKF applications in process systems engineering are relatively scarce. This talk aims to provide an introduction to the UKF technique and to outline its benefits and disadvantages in a state estimation framework of simulated chemical processes. A comparison of the UKF with the linearization used in the standard extended Kalman filter is provided from a practical point of view. |