Optimal Steady-State Control for Linear Time-Invariant Systems

Abstract

We consider the problem of designing a feedback controller that guides the input and output of a linear timeinvariant system to a minimizer of a convex optimization problem. The system is subject to an unknown disturbance that determines the feasible set defined by the system equilibrium constraints. Our proposed design enforces the Karush-Kuhn-Tucker optimality conditions in steady-state without incorporating dual variables into the controller. We prove that the input and output variables achieve optimality in equilibrium and outline two procedures for designing controllers that stabilize the closed-loop system. We explore key ideas through simple examples and simulations. I . I N T R O D U C T I O NMany engineering systems must be operated at an "optimal" steady-state that minimizes operational costs. For example, the generators that supply power to the electrical grid are scheduled according to the solution of an optimization problem which minimizes the total production cost of the generated power [1]. This same theme of guiding system variables to optimizers emerges in other areas, such as network congestion management [2], [3], chemical processing [4], [5], and wind turbine power capture [6, Section 2.7], [7].Traditionally, the optimal steady-state set-points are computed offline in advance, and then controllers are used in real time to track the set-points. However, this two-step design method is inefficient if the set-points must be updated repeatedly and often. For instance, the increased use of renewable energy sources causes rapid fluctuations in power networks, which decreases the effectiveness of the separated approach due to the rapidly-changing optimal steady-state [8]. Furthermore, the two-step method is infeasible if unmeasurable disturbances change the optimal set-points. To continuously keep operational costs to a minimum, such systems should instead employ a controller that continuously solves the optimization problem and guides the system to an optimizer despite disturbances; we will call the problem of designing such a controller the optimal steady-state (OSS) control problem.The prevalence of the OSS control problem in applications has motivated much work on its solution for general system classes. In the extremum-seeking control approach to OSS

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

“…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-Driven Synthesis of Optimization-Based Controllers for Regulation of Unknown Linear Systems

“…Online optimization problems have attracted significant attention in various disciplines, including machine learning [1], control systems [2], [3], and transportation management [4]. When considering dynamical systems, a basic online optimization setting consists in making online decisions in order to minimize a pre-specified loss function that is timevarying according to an underlying and possibly uncertain dynamical environment [5]- [7].…”

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).…”

“…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). A joint stabilization and regulation problem was considered in Lawrence et al (2021Lawrence et al ( , 2018. See also the recent survey by Hauswirth et al (2021).…”