Control relevant model reduction and controller synthesis for complex dynamical systems

Abstract

Research area and Problem formulation: For the control of large scale dynamical systems, complex mathematical models are used to describe the most dominant physical phenomena in sufficient detail. The controller to be designed needs to have short computation times, since it needs to be implemented in the control loop that is running in real-time, and hence needs to have a lower complexity than the mathematical model representing the dynamical system. There are two classical approaches for obtaining low order controllers. Either by first approximating the model using model reduction strategies and inferring a low order controller based on this approximation, or by directly designing a controller for the complex model and applying a reduction strategy to reduce the controller complexity. With the common model reduction strategies, both approaches have the potential disadvantage of losing relevant information for the controller in such a way that the interconnection of the controller with the original system does not have the desired performance. Similar problems occur in the design of observers for complex dynamical systems, which are used to estimate signals that are not directly measurable, and also have to be implemented in real-time. Methodology: This research aims to develop model reduction strategies that prevent these problems by providing explicit guarantees on the performance of controlled systems. The following two methodologies have been investigated: i. This methodology involves first representing the (complex) controlled system, approximating it by an efficient reduction technique and then synthesizing a controller of low complexity. ii. This methodology develops model reduction strategies that maintain control relevant information in the approximation process in such a way that a controller (or observer) designed using the approximated system, exhibits explicit guarantees on (controlled) performance. Contributions: i. We have addressed the design of models that describe the desired closed-loop system for different control and observer design problems. ii. We have provided novel results on the representation of systems with square integrable trajectories in the behavioral framework, extended this theory to rational representations, and we have provided novel computational algorithms that can be used for the synthesis of controllers using this approach. iii. We have developed model reduction strategies that keep the design of controllers and observers invariant, and ensure that disturbances on the input of the system have no influence on measurable outputs or estimated signals of the closed-loop system. Applications Part of this research has been performed at TNO, Integrated Vehicle Safety, in Helmond, where we have shown that the problem of losing control-relevant information using the two classical approaches occurs in applications in industry. For future safety systems in cars, a complex mathematical model that describes the kinematic behavior of a driver has been developed. For this model, the active muscular behavior is included by interconnection with controllers, which are based on the derived complex model. The model describing the complete active kinematic behavior, which is the interconnection of the controllers with the complex passive model, needs to be simulated in-vehicle, and therefore needs to be fast. We have shown that our proposed methodologies result in a better performance than when compared to the classical strategies.