Robuste Optimierung zeitdiskreter und periodischer nichtlinearer dynamischer Systeme unter Berücksichtigung von Stabilitätsgrenzen

• Applicant: Professor Dr.-Ing. Martin Mönnigmann
• Subject Area: Automation, Mechatronics, Control Systems, Intelligent Technical Systems, Robotics
• Term: from 2007 to 2015
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 48247034

• Final Report Year: 2014

Final Report Abstract

The project deals with developing a method for an automatic, model-based, numerical optimization for a class of parametrically uncertain, nonlinear dynamical discrete time systems and periodically operated continuous time systems. The considered problem classes are large and applications exist in many engineering disciplines. In typical applications of the method, technical systems are optimized with respect to economic objectives with nonlinear programming methods, while the desired dynamical properties are ensured with the so-called normal vector constraints. The desired dynamical properties are guaranteed for all operation points in a finite neighborhood around the optimal point, where the neighborhood can be chosen to account for uncertain model parameters. The main idea of the method is to guarantee a minimal distance between the optimal point and any critical boundary during the optimization. Typical critical boundaries of interest are stability and feasibility boundaries. The former consist of bifurcation points. The latter involve points at which constraints on output or input variables are violated. Boundaries of the first type are not immediately apparent from a system model, but only defined implicitly. Essentially, the proposed method is an approach to state constraints with respect to these implicitly defined boundaries in optimization problems. These constraints are called normal vector constraints, since they can be interpreted geometrically with vectors orthogonal to critical boundaries. The normal vector method has originally been developed for robust optimization of steady states and transient modes of operation of nonlinear continuous time systems. In the project, first, we extended the method for robust optimization of fixed points of nonlinear discrete time systems. Such systems frequently arise in engineering applications, either because the model is intrinsically discrete in time, or because the model is the result of a time discretization. Attention was paid to both of these cases. The concept of the normal vector constraints was applied successfully to the optimization procedures of supply chains (which are naturally modeled as discrete time systems). Among other applications, we demonstrated the use of the proposed method for a chemical engineering example of a discrete time systems that results from sampling a continuous time process (i.e. a discretized case). Since stability properties of discrete time systems and periodically operated systems are closely related, the normal vector method was considered for the optimization of oscillating models. Note that besides processes where only oscillating states occur, there exist models that can be operated either periodically or at a steady state. Examples for such behavior are again found in the treated chemical engineering examples. Another illustration of the normal vector method application to periodically operated systems was given by optimization of a passive walking robot model.

Publications

• Robust optimization of discretized dynamical systems with stability constraints. SIAM Annual Meeting 2009, Denver, USA, 2009
  D. Kastsian and M. Mönnigmann
  
• Robust optimization of fixed points of nonlinear discrete time systems with uncertain parameters. SIAM J. Appl. Dyn. Syst., 9(2):357– 390, 2010
  D. Kastsian and M. Mönnigmann
  
• Optimal power system unit commitment with guaranteed local stability. Proc. of American Control Conference 2011, San Francisco, USA, 4526–4531, 2011
  W. Grote, D. Kastsian, and M. Mönnigmann
  
• Optimization of a vendor managed inventory supply chain with guaranteed stability and robustness. Int. J. Prod. Econ., 131(2):727–735, 2011
  D. Kastsian and M. Mönnigmann
  
• Robust stable cost optimization in a three echelon supply chain model. Proc. of 18th World Congress of IFAC, Milano, Italy, 6407–6412, 2011
  D. Kastsian and M. Mönnigmann
  
• Robust optimization of periodically operated reactors with stability constraints. Proc. of IEEE Multi-Conference on Systems and Control 2012, Dubrovnik, Croatia, 184–189, 2012
  D. Kastsian and M. Mönnigmann
  
• Guaranteed stability in optimal power generation dispatching under uncertainty. IEEE Trans. Power Syst., 28(2):1103–1112, 2013
  W. Grote, D. Kastsian, and M. Mönnigmann
  (See online at https://doi.org/10.1109/TPWRS.2012.2207970)
• Impact of delay on robust stable optimization of a CSTR with recycle stream. Proc. of the 10th Int. Symp. on Dynamics and Control of Process Systems, 433–438, 2013
  D. Kastsian and M. Mönnigmann
  
• Optimization of nonlinear dynamical systems with guaranteed stability and robustness. 5th GACM Colloquium on Computational Mechanics, Hamburg, Germany, 2013
  D. Kastsian and M. Mönnigmann
  
• Robust optimization of periodically operated nonlinear uncertain processes. Chem. Eng. Sci., 106:109–118, 2014
  D. Kastsian and M. Mönnigmann
  (See online at https://doi.org/10.1016/j.ces.2013.11.023)