Notes on Chapter 1 of Maybeck 1979,\u00a0Stochastic Models, Estimation, and Control.<\/p>\n
1.1: why stochastic models, estimation, and control?<\/p>\n
A math model isn’t perfect, and parameters are not known absolutely. \u00a0Sensors don’t provide perfect data either. \u00a0Given uncertainties, you still want to be able to estimate quantities of interest and control the system.<\/p>\n
1.2: Overview of the text<\/p>\n
skip.<\/p>\n
1.3: The Kalman filter: an introduction to concepts<\/p>\n
The Kalman filter is an “optimal linear estimator”. \u00a0Given (1) knowledge of the system and the measurement device, (2) a statistical description of noise and error, and (3) initial condition info, a Kalman filter can combine the knowledge to create an estimate. \u00a0When the system is described by a linear model, and the noise is Gaussian and is white noise, the Kalman filter is the best estimator.<\/p>\n
1.4 Basic assumptions<\/p>\n
The model is linear, the noise is white noise, and the error is Gaussian\/normally distributed.<\/p>\n
1.5 A simple example<\/p>\n
A static problem: Trying to determine your position, you have a single measurement (z1) and its precision\/standard deviation.\u00a0 This leads to a conditional probability density (conditioned on your measurement).\u00a0 The best estimate is currently z1.<\/p>\n
A friend takes a second measurement, z2, with smaller variance.<\/p>\n
The best estimate will now be a combo of these two that takes into account the precision\/variance on each.<\/p>\n
There is a predictor-corrector structure to the way the estimate is made: we can take the previous best estimate and associated standard deviation and then “correct” it with the new data and new standard deviation.<\/p>\n
A dynamic problem: Now add a motion model.\u00a0 This evolves the pdf forward in time.\u00a0 (And adds noise so will increase the standard deviation).\u00a0 This creates a new estimate and variance.\u00a0 Then take a measurement and use the corrector process from above.<\/p>\n","protected":false},"excerpt":{"rendered":"
Notes on Chapter 1 of Maybeck 1979,\u00a0Stochastic Models, Estimation, and Control. 1.1: why stochastic models, estimation, and control? A math model isn’t perfect, and parameters are not known absolutely. \u00a0Sensors don’t provide perfect data either. \u00a0Given uncertainties, you still want to be able to estimate quantities of interest and control the system. 1.2: Overview of […]<\/p>\n","protected":false},"author":8032,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_newsletter_access":"","_jetpack_dont_email_post_to_subs":false,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"_jetpack_feature_clip_id":0,"_jetpack_memberships_contains_paid_content":false,"footnotes":"","jetpack_publicize_message":"","jetpack_publicize_feature_enabled":true,"jetpack_social_post_already_shared":true,"jetpack_social_options":{"image_generator_settings":{"template":"highway","default_image_id":0,"font":"","enabled":false},"version":2},"jetpack_post_was_ever_published":false},"categories":[1],"tags":[],"class_list":["post-217","post","type-post","status-publish","format-standard","hentry","category-uncategorized"],"jetpack_publicize_connections":[],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"jetpack_shortlink":"https:\/\/wp.me\/p7E5LF-3v","jetpack-related-posts":[{"id":210,"url":"https:\/\/archive.blogs.harvard.edu\/siams\/2019\/08\/10\/notes-on-particle-filters-for-high-dimensional-geoscience-applications-a-review-van-leeuwen-et-al-2019\/","url_meta":{"origin":217,"position":0},"title":"Notes on “Particle filters for high dimensional geoscience applications: a review”. van Leeuwen et al 2019","author":"siams","date":"10 August 2019","format":false,"excerpt":"Notes on Peter Jan van Leeuwen,\u00a0Hans R. K\u00fcnsch,\u00a0Lars Nerger,\u00a0Roland Potthast,\u00a0Sebastian Reich. \u00a0Q J R Meteorol Soc. 2019;1\u201331. \u00a0Particle filters for high-dimensional geoscience applications: A review This paper is focusing on the problem of \"weight degeneracy\" for the weighting of particles in a particle filter. Introduction: \"the linear data assimilation problem\"\u2026","rel":"","context":"In "Research paper"","block_context":{"text":"Research paper","link":"https:\/\/archive.blogs.harvard.edu\/siams\/category\/research-paper\/"},"img":{"alt_text":"","src":"","width":0,"height":0},"classes":[]},{"id":220,"url":"https:\/\/archive.blogs.harvard.edu\/siams\/2019\/09\/18\/notes-on-ljung-system-identification\/","url_meta":{"origin":217,"position":1},"title":"Notes on Ljung: System Identification","author":"siams","date":"18 September 2019","format":false,"excerpt":"Reading Ljung. \u00a0System Identification: theory for the user. 1: Introduction. Goal: infer a model from observations. \u00a0\"Model\" refers to the set of relationships between variables in the system. \u00a0System identification involves analyzing input and output signals from the system. Example: assume a linear difference equation relates inputs to outputs. \u00a0Use\u2026","rel":"","context":"Similar post","block_context":{"text":"Similar post","link":""},"img":{"alt_text":"","src":"","width":0,"height":0},"classes":[]},{"id":132,"url":"https:\/\/archive.blogs.harvard.edu\/siams\/2019\/06\/12\/dynamical-systems-strogatz-chapter-5\/","url_meta":{"origin":217,"position":2},"title":"Dynamical Systems: Strogatz Chapter 5","author":"siams","date":"12 June 2019","format":false,"excerpt":"This chapter is mainly review of topics from prerequisite courses. \u00a0Steve does introduce the (Delta, tau)-plane for classifying fixed points of linear systems. \u00a0This chapter is a return to linear systems. There isn't a \"summary\" section in between Chapter 4 and Chapter 5. \u00a0That is probably a worthwhile spot to\u2026","rel":"","context":"In "Dynamical Systems"","block_context":{"text":"Dynamical Systems","link":"https:\/\/archive.blogs.harvard.edu\/siams\/category\/dynamical-systems\/"},"img":{"alt_text":"","src":"","width":0,"height":0},"classes":[]},{"id":108,"url":"https:\/\/archive.blogs.harvard.edu\/siams\/2019\/06\/10\/dynamical-systems-math-1b-differential-equations-background\/","url_meta":{"origin":217,"position":3},"title":"Dynamical systems: Math 1b differential equations background.","author":"siams","date":"10 June 2019","format":false,"excerpt":"I have been using the Strogatz textbook for teaching dynamical systems. \u00a0The course has multivariable calculus and linear algebra prerequisites. \u00a0Students might take the prerequisite courses different places. \u00a0For students who have taken Math 1b, AM\/Math 21a, Math 21b, there was 6-7 week of differential equations background (11 classes in\u2026","rel":"","context":"In "Dynamical Systems"","block_context":{"text":"Dynamical Systems","link":"https:\/\/archive.blogs.harvard.edu\/siams\/category\/dynamical-systems\/"},"img":{"alt_text":"","src":"","width":0,"height":0},"classes":[]},{"id":118,"url":"https:\/\/archive.blogs.harvard.edu\/siams\/2019\/06\/10\/dynamical-systems-strogatz-chapter-2\/","url_meta":{"origin":217,"position":4},"title":"Dynamical Systems: Strogatz Chapter 2","author":"siams","date":"10 June 2019","format":false,"excerpt":"Following this text, students study 1d, then 2d, then 3d flows. \u00a0In 1d, we find stability, construct phase portraits, and in chapter 3, make bifurcation diagrams. \u00a0We loop back to these topics with more complexity in 2d. \u00a0This creates natural \"spacing\". A few notes on spacing: Spacing improves induction\/generalization from\u2026","rel":"","context":"In "Dynamical Systems"","block_context":{"text":"Dynamical Systems","link":"https:\/\/archive.blogs.harvard.edu\/siams\/category\/dynamical-systems\/"},"img":{"alt_text":"","src":"","width":0,"height":0},"classes":[]},{"id":112,"url":"https:\/\/archive.blogs.harvard.edu\/siams\/2019\/06\/10\/dynamical-systems-math-21b-differential-equations-background\/","url_meta":{"origin":217,"position":5},"title":"Dynamical Systems: Math 21b differential equations background","author":"siams","date":"10 June 2019","format":false,"excerpt":"For students who have taken Math 1b, AM\/Math 21a, Math 21b, there was 6-7 week of differential equations background (11 classes in Math 1b + 9 classes in 21b). \u00a0See my prior post for the Math 1b diff eq content that is relevant to Dynamical Systems. Student diff eq background\u2026","rel":"","context":"In "Dynamical Systems"","block_context":{"text":"Dynamical Systems","link":"https:\/\/archive.blogs.harvard.edu\/siams\/category\/dynamical-systems\/"},"img":{"alt_text":"","src":"","width":0,"height":0},"classes":[]}],"_links":{"self":[{"href":"https:\/\/archive.blogs.harvard.edu\/siams\/wp-json\/wp\/v2\/posts\/217","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/archive.blogs.harvard.edu\/siams\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/archive.blogs.harvard.edu\/siams\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/archive.blogs.harvard.edu\/siams\/wp-json\/wp\/v2\/users\/8032"}],"replies":[{"embeddable":true,"href":"https:\/\/archive.blogs.harvard.edu\/siams\/wp-json\/wp\/v2\/comments?post=217"}],"version-history":[{"count":2,"href":"https:\/\/archive.blogs.harvard.edu\/siams\/wp-json\/wp\/v2\/posts\/217\/revisions"}],"predecessor-version":[{"id":219,"href":"https:\/\/archive.blogs.harvard.edu\/siams\/wp-json\/wp\/v2\/posts\/217\/revisions\/219"}],"wp:attachment":[{"href":"https:\/\/archive.blogs.harvard.edu\/siams\/wp-json\/wp\/v2\/media?parent=217"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/archive.blogs.harvard.edu\/siams\/wp-json\/wp\/v2\/categories?post=217"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/archive.blogs.harvard.edu\/siams\/wp-json\/wp\/v2\/tags?post=217"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}