Control and optimization are central components of many enabling techniques, and there have always been strong interactions between the two areas. The recent development of new systems with growing performance capabilities calls for an expanded and deepened understanding of the interplay of control and optimization. The small-gain theorem has played a significant role in the stability analysis and control synthesis of interconnected systems. This talk will introduce some recent results with refined small-gain techniques to handle the interaction between optimization and control algorithms for cost-critical and safety-critical systems. In particular, I will discuss how we can use the small-gain theorem to solve the stability problems arising from distributed feedback optimization and optimization-based safety control of nonlinear uncertain systems. Based on the preliminary results introduced in this talk, we expect further advancement of the interconnected systems tools and new robust algorithms for complex dynamic systems.

This talk addresses some open research questions on stability and robustness of dynamical systems controlled by Approximate Dynamic Programming (ADP)-based control laws for deterministic discrete-time nonlinear systems. This work provides a bridge between ADP literature and Lyapunov stability theory by exploiting the interplay between stability, robustness and optimality. By concentrating on stabilizing optimization-based controllers, improved near-optimality bounds and novel algorithms will be presented. These new algorithms have the potential to significantly reduce computations of ADP as well as planning algorithms, such as value iteration and OPmin, which is a new tool for the stable and near-optimal control of general nonlinear switched systems.

Exactly a century since its invention and first application, and more than two decades since the proof of its convergence, the extremum seeking control has been recognized as one of the most important model-free real-time optimization tools. However, until recently extremum seeking has been restricted to dynamic systems represented by connections of Ordinary Differential Equations (ODEs) and non-linear convex maps with unknown extremum points. This talk presents the first collection of results on the theory and design of extremum seeking strategies for systems governed by Partial Differential Equations (PDEs). The main ideas for the design of the Gradient-Newton methods and the stability analysis for infinite-dimensional systems will be discussed considering a wide class of parabolic and hyperbolic PDEs: delay equations, wave equation and reaction-advection-diffusion models. Moreover, engineering applications are presented, including problems of noncooperative games, neuromuscular electrical stimulation, biological reactors, oil-drilling systems and flow-traffic control for urban mobility.

We present a technique to design controllers from data for nonlinear systems whose model is imprecisely known. The technique is based on collecting measurements of low complexity from the systems and using them for the synthesis of controllers, which is reduced to the solution of data-dependent semidefinite programs. The method provides stability certificates in the presence of perturbations on the dataset.

Current learning-based techniques for the control of physical systems, such as reinforcement learning, require the crunching of large amounts of data for extended periods of time. In this talk we show how to obviate this hunger for data by judicious modeling. In particular, we will show how to control unknown nonlinear systems without prior data or training. Key to our approach is the re-interpretation of several results in control theory, such as Fliess and co-workers intelligent-PIDs, adaptive control, and dirty derivatives, as different examples of data-driven control. We illustrate the usefulness and applicability of the results via experimental results.

We present several approaches to estimate the forward reachable set of a dynamical system using only a finite collection of trajectory samples. Given sufficiently many samples, these data-driven methods are accompanied by probabilistic guarantees of accuracy and confidence. The first approach uses the tools of scenario optimization to construct reachable set estimates as approximate solutions to chance-constrained optimization problems. The scenario approach can be applied to a range of estimator geometries, of which we examine $p$-norm balls as a special case.The second approach uses a class of polynomials derived from empirical moment matrices called inverse Christoffel functions, whose sublevel sets act as non-convex estimates of the reachable set. We construct probabilistic guarantees for the Christoffel approach using both classical VC dimension bounds from statistical learning theory and a Bayesian variant of the Probably Approximately Correct (PAC) framework that leverages a formal connection between Christoffel functions and Gaussian process regression models.

The main challenge in designing controllers directly from data is to come up with robust control laws that guarantee stability and performance of the unknown system despite the inherent uncertainty caused by noisy data. A remedy for this challenge is to consider process noise that is described by quadratic matrix inequalities (QMIs). For such noise models, many data-driven analysis and design problems boil down to checking whether a quadratic matrix inequality implies another one. Such implications can be verified by means of the matrix S-lemma that extends the classical S-lemma to matrix-valued functions. Within this approach, a variety of problems, from feedback stabilization to Hinf control, have already been solved. The goal of this talk is to bring a fresh look to quadratic matrix inequalities appearing in the context of data-driven control. By exploiting the structure of such QMIs, we provide stronger versions of matrix S-lemma and Finsler’s lemma. In addition, we illustrate these results on several data-driven control problems.