An integral quadratic constraint framework for real-time steady-state optimization of linear time-invariant systems

Abstract

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.

Timescale Separation in Autonomous Optimization

“…Dynamical Systems: Given a vector field f : R n ×R → R n , consider the initial value probleṁ Nonlinear Optimization: Given two continuously differentiable functions ξ : R n → R s and ζ : R n → R r , let V := {v ∈ R n | ξ(v) = 0, ζ(v) ≤ 0} and let J(v) := {j | ζ j (v) = 0} denote the set of active inequality constraints at v. We call V a regular set if for all v ∈ V the matrix ∇ξ(v) T ∇ζ J(v) (v) T T has has full row rank s + |J(v)|. 1 The tangent and normal cone of V at v are respectively…”

“…Note that Assumption II.1 implies that trajectories are complete, i.e., can be extended to t → ∞. 1 The term regular alludes to the fact that these sets are in fact Clarke regular (or tangentially regular) [26]. Furthermore, the requirement that rank ∇ζ J(u) (u) = J(u) is known in the optimization literature as linear independence constraint qualification (LICQ).…”

“…By doing so, we follow a paradigm of "first stabilize, then optimize" which is in contrast to other recent works that base their designs of integral quadratic constraints [1], [2], backstepping [3], or output regulation [4]. In particular, [1], [2] pursue a holistic perspective where stabilization and tracking are considered as joint objectives. These works, however, arrive at complex and convoluted LMI conditions to certify stability that are computationally expensive at large scales and often do not In contrast, our results-although simple and potentially conservative-give immediate design prescriptions while requiring only limited model information, that can often be estimated in practice.…”

Convergence of the Expectation-Maximization Algorithm Through Discrete-Time Lyapunov Stability Theory

“…The positive definiteness follows from (12) and the definition of isolated maximizer. On the other hand, from (8), it follows that…”

“…A current trend in optimization, machine learning, and control systems, is that of leveraging on a dynamical systems interpretation of iterative optimization algorithms [6]- [8].…”