I co-organized with John Simpson-Porco an invited session on **Real-time Optimization of Power Systems** at INFORMS Annual Meeting. This session is motivated by our recent work on the topic [1, 2, 3]

[1] Z. Nelson and E. Mallada, “An integral quadratic constraint framework for steady state optimization of linear time invariant systems,” in American Control Conference, 2018.

[Bibtex] [Abstract] [Download PDF]

[Bibtex] [Abstract] [Download PDF]

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.

```
@inproceedings{nm2018acc,
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.},
author = {Nelson, Zachary and Mallada, Enrique},
booktitle = {American Control Conference},
doi = {10.23919/ACC.2018.8431231},
grants = {1544771, W911NF-17-1-0092, 1711188},
issn = {2378-5861},
keywords = {Optimization, IQCs},
month = {06},
title = {An integral quadratic constraint framework for steady state optimization of linear time invariant systems},
url = {https://mallada.ece.jhu.edu/pubs/2018-ACC-NM.pdf},
year = {2018}
}
```

[2] L. S. P. Lawrence, Z. Nelson, E. Mallada, and J. W. Simpson-Porco, “Optimal Steady-State Control for Linear Time-Invariant Systems,” in 57th IEEE Conference on Decision and Control (CDC), 2018, pp. 3251-3257.

[Bibtex] [Abstract] [Download PDF]

[Bibtex] [Abstract] [Download PDF]

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, piecewise constant in time, which shifts the feasible set defined by the system equilibrium constraints. Our proposed design combines proportional-integral control with gradient feedback, and 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 steady-state, and provide a stability criterion based on absolute stability theory. The effectiveness of our approach is illustrated on a simple example system.

```
@inproceedings{lnms2018cdc,
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,
piecewise constant in time, which shifts the feasible set defined
by the system equilibrium constraints. Our proposed design
combines proportional-integral control with gradient feedback,
and 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 steady-state, and provide a stability criterion
based on absolute stability theory. The effectiveness of our
approach is illustrated on a simple example system.},
author = {Lawrence, Liam S. P. and Nelson, Zachary and Mallada, Enrique and Simpson-Porco, John W.},
booktitle = {57th IEEE Conference on Decision and Control (CDC)},
doi = {10.1109/CDC.2018.8619812},
grants = {CPS:1544771, ARO:W911NF-17-1-0092, CAREER:1752362},
issn = {2576-2370},
month = {12},
pages = {3251-3257},
pubstate = {presented, submitted Mar. 2018.},
title = {Optimal Steady-State Control for Linear Time-Invariant Systems},
url = {https://mallada.ece.jhu.edu/pubs/2018-CDC-LNMS.pdf},
year = {2018}
}
```

[3] L. S. P. Lawrence, J. W. Simpson-Porco, and E. Mallada, Linear-Convex Optimal Steady-State Control, 2020, under review.

[Bibtex] [Abstract] [Download PDF]

[Bibtex] [Abstract] [Download PDF]

We consider the problem of designing a feedback controller for a multivariable nonlinear system that regulates an arbitrary subset of the system states and inputs to the solution of a constrained optimization problem, despite parametric modelling uncertainty and time-varying exogenous disturbances; we term this the optimal steady-state (OSS) control problem. We derive necessary and sufficient conditions for the existence of an OSS controller by formulating the OSS control problem as an output regulation problem wherein the regulation error is unmeasurable. We introduce the notion of an optimality model, and show that the existence of an optimality model is sufficient to reduce the OSS control problem to an output regulation problem with measurable error. This yields a design framework for OSS control that unifies and extends many existing designs in the literature. We present a complete and constructive solution of the OSS control problem for the case where the plant is linear time-invariant with structured parametric uncertainty, and disturbances are constant in time. We illustrate these results via an application to optimal frequency control of power networks, and show that our design procedure recovers several frequency controllers from the recent literature.

```
@unpublished{lsm2019a-preprint,
abstract = {We consider the problem of designing a feedback controller for a multivariable nonlinear system that regulates an arbitrary subset of the system states and inputs to the solution of a constrained optimization problem, despite parametric modelling uncertainty and time-varying exogenous disturbances; we term this the optimal steady-state (OSS) control problem. We derive necessary and sufficient conditions for the existence of an OSS controller by formulating the OSS control problem as an output regulation problem wherein the regulation error is unmeasurable. We introduce the notion of an optimality model, and show that the existence of an optimality model is sufficient to reduce the OSS control problem to an output regulation problem with measurable error. This yields a design framework for OSS control that unifies and extends many existing designs in the literature. We present a complete and constructive solution of the OSS control problem for the case where the plant is linear time-invariant with structured parametric uncertainty, and disturbances are constant in time. We illustrate these results via an application to optimal frequency control of power networks, and show that our design procedure recovers several frequency controllers from the recent literature.},
author = {Lawrence, Liam S. P. and Simpson-Porco, John W. and Mallada, Enrique},
month = {8},
pages = {1-8},
title = {Linear-Convex Optimal Steady-State Control},
url = {https://mallada.ece.jhu.edu/pubs/2019-Preprint-LSM.pdf},
year = {2020, under review}
}
```