# Optimal control

This is my new test problem for the OpenAi Gym.

Nothing to see here; I don’t do optimal control. But here are some notes from when I thought I might.

Karl J. Åström and Richard M. Murray. Feedback Systems: An Introduction for Scientists and Engineers is an interesting control systems theory course from Caltech.

The online control blog post mentioned below has a summary:

Perhaps the most fundamental setting in control theory is a LDS is with quadratic costs $$c_t$$ and i.i.d Gaussian perturbations $$w_t$$. The solution known as the Linear Quadratic Regulator, derived by solving the Riccati equation, is well understood and corresponds to a linear policy (i.e. the control input is a linear function of the state).

The assumption of i.i.d perturbations has been relaxed in classical control theory, with the introduction of a min-max notion, in a subfield known as $$H_{\infty}$$ control. Informally, the idea behind $$H_{\infty}$$ control is to design a controller which performs well against all sequences of bounded perturbations.

There are some connections and dual relations to state estimation that might be worth exploring.

## Nuts and bolts

Åström et al maintain a supporting python toolkit, python-control.

OPENMODELICA is an open-source Modelica-based modeling and simulation environment intended for industrial and academic usage. Its long-term development is supported by a non-profit organization — the Open Source Modelica Consortium (OSMC).

Related:

openMDAO is an open-source high-performance computing platform for systems analysis and multidisciplinary optimization, written in Python. It enables you to decompose your models, making them easier to build and maintain, while still solving them in a tightly coupled manner with efficient parallel numerical methods.

The OpenMDAO project is primarily focused on supporting gradient-based optimization with analytic derivatives to allow you to explore large design spaces with hundreds or thousands of design variables, but the framework also has a number of parallel computing features that can work with gradient-free optimization, mixed-integer nonlinear programming, and traditional design space exploration.

## Online

New Methods in Control: The Gradient Perturbation Controller by Naman Agarwal, Karan Singh and Elad Hazan .

what is the analogue of online learning and worst-case regret in robust control? …Our starting point for more robust control is regret minimization in games. Regret minimization is a well-accepted metric in online learning, and we consider applying it to online control.

See POMDP.

## References

Agarwal, Naman, Brian Bullins, Elad Hazan, Sham M. Kakade, and Karan Singh. 2019. arXiv:1902.08721 [Cs, Math, Stat], February.
Agarwal, Naman, Elad Hazan, and Karan Singh. 2019. arXiv:1909.05062 [Cs, Math, Stat], September.
Andersson, Joel A. E., Joris Gillis, Greg Horn, James B. Rawlings, and Moritz Diehl. 2019. Mathematical Programming Computation 11 (1): 1–36.
Ariyur, Kartik B., and Miroslav Krstic. 2003. Real-Time Optimization by Extremum-Seeking Control. John Wiley & Sons.
Åström, Karl J, Richard M Murray, and EBL. 2008. Feedback systems: an introduction for scientists and engineers. Princeton: Princeton University Press.
Bardi, Martino, and Italo Capuzzo-Dolcetta. 2009. Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations. Springer Science & Business Media.
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———. 1995b. Dynamic Programming and Optimal Control Volume 2. Athena Scientific.
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Taghvaei, Amirhossein, and Prashant G. Mehta. 2019. October.

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