A numerical method for the optimal control of switched systems

Switched dynamical systems have shown great utility in modeling a variety of systems. Unfortunately, the determination of a numerical solution for the optimal control of such systems has proven difficult, since it demands optimal mode scheduling. Recently, we constructed an optimization algorithm to calculate a numerical solution to the problem subject to a running and final cost. In this paper, we modify our original approach in three ways to make our algorithm's application more tenable. First, we transform our algorithm to allow it to begin at an infeasible point and still converge to a lower cost feasible point. Second, we incorporate multiple objectives into our cost function, which makes the development of an optimal control in the presence of multiple goals viable. Finally, we extend our approach to penalize the number of hybrid jumps. We also detail the utility of these extensions to our original approach by considering two examples.

Two-Layered Real-Time Optimization of a Solid Oxide Fuel Cell Stack

Dominique Bonvin, Gene Bunin, Grégory François

The optimal operation of a solid oxide fuel cell stack is addressed in this paper. Real-time optimization, performed at a slow time scale via constraint adaptation, is used to account for uncertainty and degradation eﬀects, while model-predictive control is performed at a faster time scale to reject process disturbances and to safely adapt the system to the specified output constraints following changes in cell power demand. To ensure that these constraints are strictly honored, a novel adaptation algorithm that uses the built-in constraint handling of quadratic programming is implemented within the model-predictive controller. An additional feature of this algorithm - its ability to adapt the feasibility region in view of uncertainty - is shown as well. Simulation results illustrate the efficacy of this approach in the solid oxide fuel cell system.

2010

Optimal control of mass transfer in peritoneal dialysis

Diego Mastalli

The aim of this work is to set up mathematical models and numerical methods to investigate the mass transfer process occurring during the peritoneal dialysis (PD) therapy. More precisely the final goal is the set up of tools to find for each patient submitted to PD the best therapy profile to purify blood and remove water from the patients. First, we build up specific mathematical models to represent the physical phenomena we are interested in. As discussed in chapter 1, starting from the Kedem-Katchalsky equations, we end up with a systems of nonlinear ordinary differential equations describing the various aspects of the physical problem. Then, we propose a method based on nonlinear programming techniques to solve the inverse problem arising from the need to assess the peritoneal membrane characteristics which are not directly measurable on the patient. Thanks to its flexibility we are able to support the main standard tests nowadays in use to assess the kinetic properties of the peritoneum. We devise a suitable parametrization of the control function (DPD Dynamic Peritoneal Dialysis) that allows to improve the standard PD profiles with a larger set of treatments. Then we propose an optimization algorithm to improve the PD efficiency. Moreover, in the framework of control theory, we devise an algorithm based on the maximum principle of Pontryagin for switched systems to investigate deeply the PD optimal control problem. Afterwards we carry out numerical simulations, investigating the main inputs influencing the peritoneal dialysis efficiency. Specifically we show an extensive comparison between the APD (Automated Peritoneal Dialysis) and DPD (Dynamic Peritoneal Dilaysis) in order to assess the conditions under which DPD allows to improve the PD performance. Then a numerical investigation based on the algorithm devised for switched systems is carried out to assess the adequacy of DPD. Moreover we set up a procedure to reach an efficiency target and minimize the patient's exposure to glucose in order to improve the PD biocompatibility. Finally, we present the validation results of the mathematical model in order to verify its accuracy. A comparison between the APD and DPD is presented. Moreover we carry out a statistical analysis to assess the error distribution related to the most relevant quantities in order to evaluate strengths and weaknesses of this model and to identify the needs for a further improvement.

EPFL2006

A descent algorithm for the optimal control of constrained nonlinear switched dynamical systems

One of the oldest problems in the study of dynamical systems is the calculation of an optimal control. Though the determination of a numerical solution for the general non-convex optimal control problem for hybrid systems has been pursued relentlessly to date, it has proven difficult, since it demands nominal mode scheduling. In this paper, we calculate a numerical solution to the optimal control problem for a constrained switched nonlinear dynamical system with a running and final cost. The control parameter has a discrete component, the sequence of modes, and two continuous components, the duration of each mode and the continuous input while in each mode. To overcome the complexity posed by the discrete optimization problem, we propose a bi-level hierarchical optimization algorithm: at the higher level, the algorithm updates the mode sequence by using a single-mode variation technique, and at the lower level, the algorithm considers a fixed mode sequence and minimizes the cost functional over the continuous components. Numerical examples detail the potential of our proposed methodology.

ACM Press2010

Optimal control of mass transfer in peritoneal dialysis

Diego Mastalli

EPFL2006