Research program

The research program Systems, Control and Optimization (SCO) is devoted to the analysis, control and optimization of complex dynamical systems. The focus is on fundamental mathematical research, stimulated by collaborations with colleagues from the engineering and natural sciences.

Mathematical systems and control theory deals with the modeling and control of open and interconnected systems evolving in time. The dynamical behavior is not only sought to be analyzed, but to be influenced (controlled) and optimized as well; by the addition of feedback loops, and by the interconnection to other dynamical systems (controller design), or the optimal selection of parameters. Furthermore, the systems point of view is emphasized, by viewing complex systems as networks of simpler components. From the optimization perspective, we study convergence and complexity of iterative algorithms, in connection with their underlying dynamics .

Current focal points of our research are:

• Network dynamics and control (Kanat Camlibel, Harry Trentelman, Arjan van der Schaft, Bart Besselink, Stephan Trenn, Henk van Waarde).

Network dynamics and dynamical multi-agent systems arise in many fields of engineering and natural sciences. Systems and control theory contributes to this area by providing concepts and tools for the study of structural properties such as controllability (leader-follower networks) and model reduction, and for their control including synchronization and consensus dynamics. This entails a close interplay between geometric systems and control theory on the one hand, and algebraic graph theory on the other. Applications include power and sensor networks, dynamical distribution networks, intelligent transportation systems, as well as large-scale chemical reaction networks in systems biology.

• Geometric modeling and control of multi-physics systems (Arjan van der Schaft, Bart Besselink).

Port-Hamiltonian systems constitute an extension of Hamiltonian systems where external interaction and energy-dissipating ports are taken into account, and the underlying geometry is derived from the interconnection structure of the complex system. The aim of this research is to provide a systematic geometric theory for the modeling, analysis and simulation of multi-physics, lumped- and distributed parameter systems. Current focal themes include the geometric modeling and analysis of power systems and of thermo-dynamical systems. The port-Hamiltonian formulation is employed for controller design, leading to physically inspired and robust control strategies. Applications include the analysis of neuromorphic computing systems, control of mechatronic systems, and the analysis and control of distribution networks (such as power systems and chemical reaction networks).

• Mathematical systems theory (Kanat Camlibel, Harry Trentelman, Bart Besselink, Stephan Trenn, Arjan van der Schaft, Henk van Waarde ).

Mathematical systems theory deals with the modeling and analysis of open and interconnected systems. This naturally leads to models containing differential and algebraic equations, called DAE systems. Current research themes concern equivalence and minimality notions, and model reduction. Furthermore, physical systems often do not exhibit an a priori fixed information flow direction. In the behavioral approach, all external system variables are therefore in a first instance treated on an equal footing. Hybrid systems are a mixture of interacting continuous and discrete dynamics, and arise naturally in embedded systems and physical systems modeling, including switched DAE systems and convex processes. Important research issues concern the analysis of solution trajectories, and the structural properties of controllability and stabilizability, as well as the design of controllers. The mathematical analysis of switched and piecewise-affine systems is heavily intertwined with convex optimization theory and non-smooth analysis. Another line of research concerns the compositional analysis and design of interconnected systems, developing notions and tools of assume-guarantee reasoning and contract-based design using geometric control theory.

• Modeling, control and optimization of energy systems (Stephan Trenn, Arjan van der Schaft, Kanat Camlibel).

Power networks, from high-voltage distribution networks to AC or DC micro-grids, constitute an application area of growing importance and interest. Furthermore, there is an increasing trend for integration with other energy systems such as gas distribution networks. The aim of this theme is to develop a sound mathematical framework for the modeling, optimization and control of large-scale energy systems. This includes the systematic modeling of components such as synchronous generators and converters, as well as of the transmission line network. Based on these models fundamental problems of stability and power sharing are addressed, as well as optimal demand-supply matching by dynamic pricing, coupling physical dynamics to market dynamics. Furthermore, integration of power networks with other energy systems leads to large-scale distributed optimization problems. This research is mostly carried out in a collaborative effort with colleagues from ENTEG.

• Control of distributed-parameter systems and inverse problems (Alden Waters, Stephan Trenn, Arjan van der Schaft).

This research theme is concerned with the analysis, control and estimation of systems described by partial differential equations. Current themes of interest are approximations to solutions for coupled systems of hyperbolic wave equations, and issues of short time well-posedness and parameter recovery from solution waves. Further topics are structure-preserving spatial discretization and model reduction for control as well as coupling hyperbolic PDEs with switched systems via the boundary values.

• Data-driven modeling, analysis and control (Henk van Waarde, Kanat Camlibel, Harry Trentelman, Bart Besselink).

Systems and control theory deals with the analysis and design of mathematical models of dynamical systems. However, obtaining suitable system models is an art in itself. Data-based techniques hold much potential when dynamical systems are too complex for first-principles modeling, or when their parameters are uncertain. One of our research lines focuses on the identification of interconnected networks of dynamical systems, and on the extraction of low-dimensional system models from high-dimensional time series data. Another line avoids the modeling step altogether and aims at the analysis and design of dynamical systems directly from measurements of their trajectories. Here, it is of interest to assess system-theoretic properties such as stability, controllability and dissipativity using data. In addition, we are interested in the data-driven design of control laws, with rigorous guarantees on stability and performance despite uncertainties caused by noise and imperfect data.  

• First order iterative algorithms in optimization and variational analysis (Juan Peypouquet).

We investigate several aspects of the convergence analysis of continuous and discrete-time dynamical systems, with an emphasis in the study of convergence rates and complexity, especially in the presence of uncertainty, both from a theoretical perspective and through numerical analysis and simulations.

We study dynamics given by certain ordinary differential equations and inclusions, in order to design and analyze numerical optimization algorithms, aiming at producing fast algorithms for optimization problems with large volumes of data, and establish guarantees for their convergence rates and complexity. In particular, we shall study inertial methods with damping defined by the curvature, as well as inertial iterations of fixed-point type for more general equilibrium problems, envisioning applications in mechanics and game theory. These algorithms use only  first order information, allowing them to consider problems with large volumes of data. 

Our research focuses primarily on 1) geometric properties of the functions, such as metric regularity, error bounds, Lojasiewicz inequalities and second order information; 2) dynamical aspects, such as inertia, acceleration and relaxation; 3) restarting schemes in order to improve the convergence and performance of optimization algorithms that combine acceleration and stabilization mechanisms; and 4) the effects of intrinsic (by design) and extrinsic (by uncertainty) stochasticity.