obc-stochastic_15Feb10

obc-stochastic_15Feb10 - Optimization-Based Control Richard...

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Optimization-Based Control Richard M. Murray Control and Dynamical Systems California Institute of Technology DRAFT v2.1a, February 15, 2010 c ± California Institute of Technology All rights reserved. This manuscript is for review purposes only and may not be reproduced, in whole or in part, without written consent from the author.
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Chapter 4 Stochastic Systems In this chapter we present a focused overview of stochastic systems, oriented toward the material that is required in Chapters 5 and 6. After a brief review of random variables, we deFne discrete-time and continuous-time random processes, including the expectation, (co-)variance and correlation functions for a random process. These deFnitions are used to describe linear stochastic systems (in continuous time) and the stochastic response of a linear system to a random process (e.g., noise). We initially derive the relevant quantities in the state space, followed by a presentation of the equivalent frequency domain concepts. Prerequisites. Readers should be familiar with basic concepts in probability, in- cluding random variables and standard distributions. We do not assume any prior familiarity with random processes. Caveats. This chapter is written to provide a brief introduction to stochastic pro- cesses that can be used to derive the results in the following chapters. In order to keep the presentation compact, we gloss over several mathematical details that are required for rigorous presentation of the results. A more detailed (and mathemati- cally precise) derivation of this material is available in the book by ˚ Astr¨om [ ˚ Ast06]. 4.1 Brief Review of Random Variables To help Fx the notation that we will use, we brie±y review the key concepts of random variables. A more complete exposition is available in standard books on probability, such as Hoel, Port and Stone [HPS71]. Random variables and processes are deFned in terms of an underlying proba- bility space that captures the nature of the stochastic system we wish to study. A probability space has three elements: a sample space Ω that represents the set of all possible outcomes; a set of events F the captures combinations of elementary outcomes that are of interest; and a probability measure P that describes the likelihood of a given event occur- ring. Ω can be any set, either with a Fnite, countable or inFnite number of elements. The event space F consists of subsets of Ω. There are some mathematical limits on the properties of the sets in F , but these are not critical for our purposes here. The probability measure P is a mapping from P : F→ [0 , 1] that assigns a probability to each event. It must satisfy the property that given any two disjoint set A,B ⊂F ,
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4-2 CHAPTER 4. STOCHASTIC SYSTEMS P ( A B )= P ( A )+ P ( B ). The term probability distribution is also to describe a probability measure.
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obc-stochastic_15Feb10 - Optimization-Based Control Richard...

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