#
Proximal algorithms for large-scale statistical modeling

and optimal sensor/actuator selection

###### Abstract

Several problems in modeling and control of stochastically-driven dynamical systems can be cast as regularized semi-definite programs. We examine two such representative problems and show that they can be formulated in a similar manner. The first, in statistical modeling, seeks to reconcile observed statistics by suitably and minimally perturbing prior dynamics. The second, seeks to optimally select sensors and actuators for control purposes. To address modeling and control of large-scale systems we develop a unified algorithmic framework using proximal methods. Our customized algorithms exploit problem structure and allow handling statistical modeling, as well as sensor and actuator selection, for substantially larger scales than what is amenable to current general-purpose solvers.

1.3

Actuator selection, sensor selection, sparsity-promoting estimation and control, method of multipliers, nonsmooth convex optimization, proximal methods, regularization for design, semi-definite programming, structured covariances.

## I Introduction

Convex optimization has had tremendous impact on many disciplines, including system identification and control design [boyelgferbal94, dulpag00, fazhinboy01, boyvan04, fazhinboy04, liuvan09, jovdhiEJC16]. The forefront of research points to broadening the range of applications as well as sharpening the effectiveness of algorithms in terms of speed and scalability. The present paper focuses on two representative control problems, statistical control-oriented modeling and sensor/actuator selection, that are cast as convex programs. A range of modern applications require addressing these over increasingly large parameter spaces, placing them outside the reach of standard solvers. A contribution of the paper is to formulate such problems as regularized semi-definite programs (SDPs) and to develop customized optimization algorithms that scale favorably with size.

Modeling is often seen as an inverse problem where a search in parameter space aims to find a parsimonious representation of data. For example, in the control-oriented modeling of fluid flows, it is of interest to improve upon dynamical equations arising from first-principle physics (e.g., linearized Navier-Stokes equations), in order to accurately replicate observed statistical features that are estimated from data. To this end, a perturbation of the prior model can be seen as a feedback gain that results in dynamical coupling between a suitable subset of parameters [zarchejovgeoTAC17, zarjovgeoJFM17]. On the flip side, active control of large-scale and distributed systems requires judicious placement of sensors and actuators which again can be viewed as the selection of a suitable feedback or Kalman gain. In either modeling or control, the selection of such gain matrices must be guided by optimality criteria as well as simplicity (low rank or sparse architecture). We cast both types of problems as optimization problems that utilize suitable convex surrogates to handle complexity. The use of such surrogates is necessitated by the fact that searching over all possible architectures is combinatorially prohibitive.

Applications that motivate our study require scalable algorithms that can handle large-scale problems. While the optimization problems that we formulate are SDP representable, e.g., for actuator selection, worst-case complexity of generic solvers scales as the sixth power of the sum of the state dimension and the number of actuators. Thus, solvers that do not exploit the problem structure cannot cope with the demands of such large-scale applications. This necessitates the development of customized algorithms that are pursued herein.

Our presentation is organized as follows. In Section II, we describe the modeling and control problems that we consider, provide an overview of literature and the state-of-the-art, and highlight the technical contribution of the paper. In Section LABEL:sec.prelim, we formulate the minimum energy covariance completion (control-oriented modeling) and sensor/actuator selection (control) problems as nonsmooth SDPs. In Section LABEL:sec.algorithms, we present a customized Method of Multipliers (MM) algorithm for covariance completion. An essential ingredient of MM is the proximal gradient method which we also use for sensor/actuator selection. In Section LABEL:sec.example, we offer two motivating examples for actuator selection and covariance completion and discuss computational experiments. We conclude with a brief summary of the results and future directions in Section LABEL:sec.remarks.

## Ii Motivating applications and contribution

We consider dynamical systems with additive stochastic disturbances. In the first instance, we are concerned with a modeling problem where the statistics are not consistent with a prior model that is available to us. In that case, we seek to modify our model in a parsimonious manner (a sparse and structured perturbation of the state matrix) so as to account for the partially observed statistics. In the second, we are concerned with the control of such stochastic dynamics via a collection of judiciously placed sensors and actuators. Once again, the architecture of the (now) control problem calls for the selection of sparse matrix gains that effect control and estimation. These problems are explained next.

### Ii-a Statistical modeling and covariance completion

It is well-established that the linearized Navier-Stokes equations driven by stochastic excitation can account for qualitative [farioa93, bamdah01, jovbamJFM05, moajovJFM12] and quantitative [zarjovgeoJFM17] features of shear fluid flows. The goal has been to provide insights into the underlying physics as well as to guide control design. A significant advance was achieved recently by recognizing the value of nontrivial stochastic excitation; in [zarjovgeoJFM17], it was shown that colored-in-time noise can account for features of the flow field that white noise (used in earlier literature) cannot [jovbamCDC01]. Furthermore, it has been pointed out that the effect of colored-in-time excitation is equivalent to a structural perturbation of the system dynamics subject to white-in-time excitation [zarchejovgeoTAC17, zarjovgeoJFM17]. Such structural perturbations may reveal important dynamical couplings between states and, thereby, enhance understanding of basic physics [zarjovgeoJFM17, Section 6.1].