Proximal Algorithms for Large-Scale Statistical Modeling and 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 a subset of available 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. We establish linear convergence of the proximal gradient algorithm, draw contrast between the proposed proximal algorithms and alternating direction method of multipliers, and provide examples that illustrate the merits and effectiveness of our framework. Index TermsActuator selection, sensor selection, sparsity-promoting estimation and control, method of multipliers, nonsmooth convex optimization, proximal algorithms, regularization for design, semi-definite programming, structured covariances. I. INTRODUCTIONConvex optimization has had tremendous impact on many disciplines, including system identification and control design [1]- [7]. 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-principles (e.g., linearized Navier-Stokes equations), in order to accurately 2 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 [8], [9].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 architectur...

On a Relaxation of Time-Varying Actuator Placement

“…For example, fundamental limitations in terms of unavoidably large control energy have been studied in [18,17,26,11] among others. Algorithms for actuator placement, in either the fixed or time-varying regime, based on randomized sampling [3,10,19,6], convex relaxation [24,22], or greedy methods [25,4,13,20,27,21] were studied in recent works. Given the relatively large amount of work done on different versions of the problem which are not directly related to our motivating concern, we refer the reader to the above papers for a broader overview of the field.…”

Stochastic Dynamical Modeling of Turbulent Flows

“…This interpretation provides an alternative viewpoint that is closely related to a class stochastic control (68,69,70,71) and output covariance estimation (72,73) problems; see (62, Section II.C) for details. Based on this, we next describe an alternative formulation of the covariance completion problem as a state-feedback synthesis that is optimal with respect to a different design criterion (74,75).…”

“…As described in (74,75), the challenge of establishing consistency between statistical measurements and a linearized model can be alternatively cast as the problem of seeking a completion of the missing entries of a covariance matrix X along with a perturbation ∆ of the system dynamics subject to white-in-time input w,…”

“…This gives rise to convex optimization problem CC-2 described in the sidebar MINIMUM ENERGY COVARIANCE COMPLETION PROBLEM. The objective function in CC-2 provides a trade-off between the minimum-control-energy problem and the number of feedback couplings that need to be introduced to modify the dynamical generator A and achieve consistency with available data (74,75).…”

“…As demonstrated in (76,62,75), covariance completion problems CC-1 and CC-2 can be cast as semidefinite programs. For small-and medium-size problems, these can be solved efficiently using standard solvers (77,78,64).…”