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

Achieving optimal steady-state performance in real-time is an increasingly necessary requirement of many critical infrastructure systems. In pursuit of this goal, this paper builds a systematic design framework of feedback controllers for Linear Time-Invariant (LTI) systems that continuously track the optimal solution of some predefined optimization problem. The proposed solution can be logically divided into three components. The first component estimates the system state from the output measurements. The second component uses the estimated state and computes a drift direction based on an optimization algorithm. The third component computes an input to the LTI system that aims to drive the system toward the optimal steady-state.We analyze the equilibrium characteristics of the closed-loop system and provide conditions for optimality and stability. Our analysis shows that the proposed solution guarantees optimal steady-state performance, even in the presence of constant disturbances. Furthermore, by leveraging recent results on the analysis of optimization algorithms using integral quadratic constraints (IQCs), the proposed framework is able to translate input-output properties of our optimization component into sufficient conditions, based on linear matrix inequalities (LMIs), for global exponential asymptotic stability of the closed loop system. We illustrate the versatility of our framework using several examples.

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