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L10-StochOptControl - Elements of Stochastic Optimal...

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Elements of Stochastic Optimal Control: ICDNS MSci/MSc Nick Jones [email protected]
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Aspects of Stochastic Optimal Control In this lecture we will particularly briskly investigate selected topics in Stochastic Optimal Control. Our objective will be obtain enough background so that we can consider a rejection sampling approach to solving a class of Stochastic Optimal Control problems in Ref. [3]. We’ll try and keep notation broadly consistent with Refs. [2, 3]. We will then investigate this last in our practical lecture. Note: we will have one more lecture on topics in Control in which we will investigate the bridge between topics in inference and control. This lecture will be the first of 5 - the remaining three will be on classic theory to study the driving of systems and the final lecture will draw together some of the strands of the course. Nick Jones Elements of Stochastic Optimal Control: ICDNS
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Tasks Beyond asking you to check the more obvious statements in the handouts I have made the following requests. The objective here is to ensure you’ve actually followed up on the material and understand it. Beautiful expositions are not required - just demonstrations of understanding. 1 I Research how rejection sampling performs for higher dimensional problems and how the number of rejections depends on Q , P * and c . You’ll find answers to this in MacKay. Output: This can be summarized in a page or less. 2 I It also helps understand the following two papers: Libby et al [4] and Kobayashi et al [5] which you are expected to read (but you are by no means expected to understand these fully). Output: read them. 3 I These reviews are reasonably easy read and also combine to give an introduction to Bayesian cognitive science more generally: a) Probabilistic brains: knowns and unknowns. Alexandre Pouget, Jeffrey Beck, Wei Ji Ma and Peter Latham, Nature Neuroscience, 2013. b) Statistically optimal perception and learning: from behavior to neural representations. J´ozsef Fiser, Pietro Berkes, Gerg˝o Orb´ an and M´ at´ e Lengyel, Trends in Cognitive Sciences, 2010. Please read them (they are pretty interesting). Output: read them. 4 I Find and understand a brief proof of Landauer’s principle (if you can’t find one by the time we hit control theory ask me). Output: half a page or less. 5 I Gibbs sampling is called Glauber dynamics in the physics literature (go and have a very brief read about Glauber dynamics). Output: Just a few sentences. 6 C Read the introduction to Sontag [11] and, in particular , convince yourself of the role of PID control in stabilizing an inverted pendulum. Output: A page or less of explanation of this system. 7 C “For the linear system described to be controllable we require the matrix with columns ˜ A i b where i = 0 , .., n - 1 (and is an exponent not an index) with x R n to be invertible. Look at the treatment of controllability by Zabczyk [9] (or in the other sources provided) and prove that this holds.” Output: A precis of the proof in a page.
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