Operators and Feedback Control Theory: Linear Switched Systems

Switching is a common feature in models for systems comprised of interacting software and physical processes, and in this talk we will focus on a special type of hybrid model called a linear switched system. In discrete time, these systems are represented by difference equations in which the defining system matrices are functions of a parameter taking values in a finite set; further, this discrete parameter evolves, or indeed switches, according to a transition system which in the simplest case is an automaton. The talk will focus on such linear switched systems in a feedback control context—both centralized and decentralized—and how they can be systematically analyzed using a combination of state space methods, operator theory and semidefinite programming. As a start, we will investigate the property of stability, and also the more involved attribute stabilizability—whether a feedback policy exists to stabilize an inherently unstable system. In each case we show that the property can be checked exactly from a nested chain of semidefinite programs: feasibility of any program in the chain provides a mathematical certificate that the property holds; using the concept of a multi-norm, we further show that infeasibility provides information about the degree to which the property may be attainable. More generally, we consider performance metrics for switched systems, and present results on performance verification, and automated synthesis of feedback policies. We will also discuss connections to the joint spectral radius of a set of matrices, and Markovian jump linear systems

On design methods for sampled-data systems

In this paper we compare, via example, the standard approaches to sampled-data design with recently developed direct design methods for these hybrid systems. Simple intuitive examples are used to show that traditional design heuristics provide no performance guarantees whatsoever. Even when the sampling rate is a design parameter that can be chosen as fast as desired, using design heuristics can lead to either severe performance degradation or extreme over-design. These effects are apparently well-known to practitioners, but may not be widely appreciated by the control community at large. The paper contains no new theoretical results and is intended to be of a tutorial nature

Decomposition of Nonlinear Dynamical Systems Using Koopman Gramians

In this paper we propose a new Koopman operator approach to the decomposition of nonlinear dynamical systems using Koopman Gramians. We introduce the notion of an input-Koopman operator, and show how input-Koopman operators can be used to cast a nonlinear system into the classical state-space form, and identify conditions under which input and state observable functions are well separated. We then extend an existing method of dynamic mode decomposition for learning Koopman operators from data known as deep dynamic mode decomposition to systems with controls or disturbances. We illustrate the accuracy of the method in learning an input-state separable Koopman operator for an example system, even when the underlying system exhibits mixed state-input terms. We next introduce a nonlinear decomposition algorithm, based on Koopman Gramians, that maximizes internal subsystem observability and disturbance rejection from unwanted noise from other subsystems. We derive a relaxation based on Koopman Gramians and multi-way partitioning for the resulting NP-hard decomposition problem. We lastly illustrate the proposed algorithm with the swing dynamics for an IEEE 39-bus system.Comment: 8 pages, submitted to IEEE 2018 AC

Modeling of Transitional Channel Flow Using Balanced Proper Orthogonal Decomposition

We study reduced-order models of three-dimensional perturbations in linearized channel flow using balanced proper orthogonal decomposition (BPOD). The models are obtained from three-dimensional simulations in physical space as opposed to the traditional single-wavenumber approach, and are therefore better able to capture the effects of localized disturbances or localized actuators. In order to assess the performance of the models, we consider the impulse response and frequency response, and variation of the Reynolds number as a model parameter. We show that the BPOD procedure yields models that capture the transient growth well at a low order, whereas standard POD does not capture the growth unless a considerably larger number of modes is included, and even then can be inaccurate. In the case of a localized actuator, we show that POD modes which are not energetically significant can be very important for capturing the energy growth. In addition, a comparison of the subspaces resulting from the two methods suggests that the use of a non-orthogonal projection with adjoint modes is most likely the main reason for the superior performance of BPOD. We also demonstrate that for single-wavenumber perturbations, low-order BPOD models reproduce the dominant eigenvalues of the full system better than POD models of the same order. These features indicate that the simple, yet accurate BPOD models are a good candidate for developing model-based controllers for channel flow.Comment: 35 pages, 20 figure