Online Optimization of Switched LTI Systems Using Continuous-Time and Hybrid Accelerated Gradient Flows

Abstract

This paper studies the design of feedback controllers that steer the output of a switched linear time-invariant system to the solution of a possibly time-varying optimization problem. The design of the feedback controllers is based on an online gradient descent method, and an online hybrid controller that can be seen as a regularized Nesterov's accelerated gradient method. Both of the proposed approaches accommodate output measurements of the plant, and are implemented in closedloop with the switched dynamical system. By design, the controllers continuously steer the system output to an optimal trajectory implicitly defined by the time-varying optimization problem without requiring knowledge of exogenous inputs and disturbances. For cost functions that are smooth and satisfy the Polyak-Lojasiewicz inequality, we demonstrate that the online gradient descent controller ensures uniform global exponential stability when the time-scales of the plant and the controller are sufficiently separated and the switching signal of the plant is slow on the average. Under a strong convexity assumption, we also show that the online hybrid Nesterov's method guarantees tracking of optimal trajectories, and outperforms online controllers based on gradient descent. Interestingly, the proposed hybrid accelerated controller resolves the potential lack of robustness suffered by standard continuous-time accelerated gradient methods when coupled with a dynamical system. When the function is not strongly convex, we establish global practical asymptotic stability results for the accelerated method. Our results suggest the existence of a trade-off between rate of convergence and accuracy of the solutions in online optimization problems where controllers use a dynamic momentum. Our theoretical results are illustrated via different numerical examples.

Online Stochastic Optimization for Unknown Linear Systems: Data-Driven Synthesis and Controller Analysis

“…Most of the recent literature on online optimization for dynamical systems critically relies on the assumption that the system dynamics are known [2]- [4], [6], [7], [9], [12]. Unfortunately, perfect system knowledge is rarely available in practice -especially when exogenous disturbances are not observable and/or inputs are not persistently exciting -because maintaining and refining full system model often requires ad-hoc system-identification phases.…”

“…Several works applied online convex optimization to control plants modeled as algebraic maps [25]- [27] (corresponding to cases where the dynamics are infinitely fast). When the dynamics are non-negligible, LTI systems are considered in [4], [5], [7], [10], stable nonlinear systems in [6], [28], switching systems in [12], and distributed multi-agent systems in [3], [29]. All these works consider continuoustime dynamics and deterministic optimization problems, and derive results in terms of asymptotic or exponential stability.…”

Data-Enabled Gradient Flow as Feedback Controller: Regulation of Linear Dynamical Systems to Minimizers of Unknown Functions

“…In this paper, we consider the problem of designing feedback controllers to steer the output of a linear time-invariant (LTI) dynamical system towards the solution of a convex optimization problem with unknown costs. The design of controllers inspired by optimization algorithms has received attention recently; see, e.g., Jokic et al (2009); Brunner et al (2012); Lawrence et al (2018); Hauswirth et al (2020); Colombino et al (2020); Zheng et al (2020); Bianchin et al (2020) and the recent survey by Hauswirth et al (2021). These methods have been utilized to solve control problems in, e.g., power systems in Hirata et al (2014); Menta et al (2018), transportation systems in Bianchin et al (2021a), robotics in Zheng et al (2020), and epidemics in Bianchin et al (2021b).…”

“…We note that a key differentiating aspect relative to extremum seeking methods (see, e.g., Krstic and Wang (2000); Ariyur and Krstić (2003); Teel and Popovic (2001) and many others), the Q-learning of Devraj and Meyn (2017), and methods based on concurrent learning Chowdhary and Johnson (2010); Chowdhary et al (2013); Poveda et al (2021) is that we consider a setting where only sporadic functional evaluations are available (i.e., we do not have continuous access to functional evaluations). Regarding the problem of regulating LTI systems towards solutions of optimization problems, existing approaches leveraged gradient flows in Menta et al (2018); Bianchin et al (2020), proximal-methods in Colombino et al (2020), saddle-flows in Brunner et al (2012), prediction-correction methods in Zheng et al (2020), and the hybrid accelerated methods proposed in Bianchin et al (2020). Plants with (smooth) nonlinear dynamics were considered in Brunner et al (2012); Hauswirth et al (2020), and switched LTI systems in Bianchin et al (2021a).…”

“…(C2) We leverage singularperturbation arguments (as in (Khalil, 2002, Ch. 11) and in, e.g., Hauswirth et al (2020); Bianchin et al (2020)) and theory of perturbed systems (Khalil, 2002, Ch. 9) to derive sufficient conditions on the learning error and the controller gain to ensure that the error between the optimizer of the problem and the state of the closed-loop system is ultimately bounded.…”