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Constrained Optimal Control for Complex Systems - Analysis and Applications

Author(s):

K. Margellos
Conference/Journal:

ETH Zurich, Switzerland
Abstract:

Many applications in control engineering, involve solving a constrained optimal control problem. Especially for the case of complex systems, the presence of constraints gives rise both to theoretical and computational challenges. Developing a general theoretical framework is crucial to capture all features and inter-dependencies among the components of the underlying system, but numerical computations for such detailed models might not be tractable. Therefore, a trade-off between the generality of the developed theory and the scale of applications that can be addressed numerically needs to be reached. This dissertation deals with three of these problems, all of which resulting in a constrained optimal control formulation. The first problem concentrates on the development of an optimal control framework to deal with state constrained reachability problems for continuous and hybrid systems. The main objective is to design suitable controllers to steer or keep the state of the system in a “safe” part of the state space. The synthesis of such safe controllers relies on the ability to solve target problems for the case where state constraints are also present, giving rise to reach-avoid calculations. We first focus on continuous time systems and show how such reach-avoid problems can be related to the viscosity solutions of certain optimal control problems. We then extend our framework to reachability/viability problems for hybrid systems and provide a complete characterization based entirely on optimal control, logic theorems and the definition of executions of hybrid automata. Hybrid automata with nonlinear continuous dynamics and bounded competing inputs can be captured by the proposed framework. The theoretical results are illustrated on a 4D trajectory management problem in air traffic control and various benchmark examples. The second problem involves applying model predictive control for nonlinear, but feedback linearizable systems, with input constraints. The challenge is that after feedback linearization, even though the systems dynamics are rendered linear, the initial input constraints are mapped to a set of state dependent, and in general nonlinear and possibly non convex bounds. To circumvent this difficulty, an iterative scheme is proposed, where a set of initially arbitrarily chosen input constraints is continuously refined by solving a sequence of quadratic or linear problems. Despite the fact that there are still no rigorous guarantees regarding the convergence properties of the developed algorithm and the optimality of the resulting solution, we evaluate the effectiveness of the scheme by means of several single input single-output and multi-input multi-output examples. Systems with continuous or discrete time, input affine dynamics, that can be at least partially feedback linearized, can be solved using the proposed methodology. The last problem investigated in this dissertation deals with chance constrained optimization. A new method which lies between robust (or worst-case) optimization and scenario based methods is proposed. It does not require prior knowledge of the underlying probability distribution as in standard robust optimization methods, nor is it based entirely on randomization as in scenario approaches. It instead involves solving a robust optimization problem with bounded uncertainty, where the uncertainty bounds are randomized and are computed using the scenario approach. The proposed approach is applied to the problem of reserve scheduling for power networks with renewable generation, whereas generic robust optimization problems with discrete time, linear dynamics can be casted in the developed framework.

Year:

2012
Type of Publication:

(03)Ph.D. Thesis
Supervisor:

J. Lygeros

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% Autogenerated BibTeX entry
@PhDThesis { Xxx:2012:IFA_4287,
    author={K. Margellos},
    title={{Constrained Optimal Control for Complex Systems - Analysis
	  and Applications}},
    school={},
    year={2012},
    address={ETH Zurich, Switzerland},
    month=nov,
    url={http://control.ee.ethz.ch/index.cgi?page=publications;action=details;id=4287}
}
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