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