Abstract
Recently, we presented an implementation of intervalbased algorithms which can be applied in realtime to control dynamical processes and to estimate internal states and disturbances. The approach is based on verified methods for sets of algebraic equations, ordinary differential equations as well as differentialalgebraic equations. Due to this fact, the same program code can be used for two different tasks. On the one hand, we can use it online to estimate nonmeasurable internal system states which are necessary for nonlinear modelbased control strategies. On the other hand, we can verify the admissibility and feasibility of these control strategies offline.
Although we use the same code for the online and offline tasks, there is an important difference between them. While the computing time is of minor importance in offline applications, we have to guarantee that the necessary online computations are completed successfully in a predefined time interval. For that reason, the role of verification is slightly different depending on the task. In offline applications, our goal is to compute tightest possible bounds for the sets of all solutions to the control problem under consideration. In contrast to that, we restrict the online mode to a search for a single solution that matches all demands on feasibility of control inputs and admissibility of the trajectories of the state variables in a reliable way.
To highlight the practical applicability of the underlying computational routines, we present the following cases for the use of verified solvers in realtime [13].
Case 1: Direct computation of feedforward control strategies with the help of differentialalgebraic equation solvers. In this application, both verified and nonverified solvers can be used to determine openloop control strategies for a dynamical system such that its output coincides with a predefined time response within given tolerances. This procedure corresponds to a numerical inversion of the dynamics of the system to be controlled. In this case, verified solvers are used to prove the existence of a control law within given physical bounds for the admissible range of the system inputs.
Case 2: If measured data and their time derivatives are available, the same procedures as in case 1 can be used to estimate nonmeasured state variables as well as nonmeasurable disturbances. Since the verified algorithms used in this context are capable of propagating bounded measurement uncertainties, the quality of the state and disturbance estimates can be expressed in terms of the resulting interval widths. Moreover, assumptions about the parameters and the structure of the underlying model can be verified.
Case 3: Routines for verified sensitivity analysis provide further information on the influence of variations of control inputs on the trajectories of the state variables. We present novel procedures implementing a sensitivitybased framework for modelpredictive control. These procedures can be integrated directly in a feedback control structure.
Sometimes it is necessary to combine verified and nonverified algorithms to solve a given control problem. In this case, it is important to certify the results of the algorithm appropriately. Based on the fourtier hierarchy presented in earlier works [4], we develop a measure for characterizing such mixed approaches.
The presentation is concluded with simulation and experimental results for the example of temperature control of a distributed heating system.
[1] Rauh, Andreas; Auer, Ekaterina: Applications of Verified DAE Solvers in Engineering, Intl. Workshop on Verified Computations and Related Topics, COE Lecture Note Vol. 15: Kyushu University, pp. 8896, Karlsruhe, Germany, 2009.
[2] Rauh, Andreas; Menn, Ingolf; Aschemann, Harald: Robust Control with State and Disturbance Estimation for Distributed Parameter Systems, Proc. of 15th Intl. Workshop on Dynamics and Control 2009, pp. 135142, Tossa de Mar, Spain, 2009.
[3] Rauh, Andreas; Auer, Ekaterina; Aschemann, Harald: RealTime Application of Interval Methods for Robust Control of Dynamical Systems, CDProc. of IEEE Intl. Conference on Methods and Models in Automation and Robotics MMAR 2009, Miedzyzdroje, Poland, 2009.
[4] Auer, Ekaterina; Luther, Wolfram: Numerical Verification Assessment in Computational Biomechanics, in A. Cuyt, W. Krämer, W. Luther, P. Markstein: Numerical Validation in Current Hardware Architectures, LNCS 5492, pp. 145160, SpringerVerlag, Berlin, Heidelberg, 2009.
BibTeX  Entry
@InProceedings{rauh_et_al:DSP:2010:2512,
author = {Andreas Rauh and Ekaterina Auer},
title = {Interval Approaches to Reliable Control of Dynamical Systems},
booktitle = {Computerassisted proofs  tools, methods and applications},
year = {2010},
editor = {B. Malcolm Brown and Erich Kaltofen and Shin'ichi Oishi and Siegfried M. Rump},
number = {09471},
series = {Dagstuhl Seminar Proceedings},
ISSN = {18624405},
publisher = {Schloss Dagstuhl  LeibnizZentrum fuer Informatik, Germany},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2010/2512},
annote = {Keywords: Robust control, Ordinary differential equations, Differentialalgebraic equations}
}
Keywords: 

Robust control, Ordinary differential equations, Differentialalgebraic equations 
Seminar: 

09471  Computerassisted proofs  tools, methods and applications 
Issue Date: 

2010 
Date of publication: 

27.04.2010 