{"id":83,"date":"2016-04-20T12:35:04","date_gmt":"2016-04-20T10:35:04","guid":{"rendered":"http:\/\/eudaimonia-production.com\/?page_id=83"},"modified":"2019-01-14T04:28:40","modified_gmt":"2019-01-14T02:28:40","slug":"special-topics-in-control-theory","status":"publish","type":"post","link":"https:\/\/eudaimonia-production.com\/?p=83","title":{"rendered":"Special topics in control theory"},"content":{"rendered":"

\u0418\u0437\u0434\u0430\u0442\u0435\u043b\u0441\u043a\u0438 \u0434\u0430\u043d\u043d\u0438…\"OblojkaSpecialTopics\"<\/a><\/strong><\/p>\n

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\u0421\u0432\u0430\u043b\u0438 \u043f\u044a\u043b\u043d\u0438\u044f \u0442\u0435\u043a\u0441\u0442 \u043d\u0430 \u043a\u043d\u0438\u0433\u0430\u0442\u0430 \u0442\u0443\u043a…<\/a><\/strong><\/p>\n

\u041a\u043d\u0438\u0433\u0430\u0442\u0430 \u0432 \u041d\u0430\u0446\u0438\u043e\u043d\u0430\u043b\u043d\u0438\u044f \u0440\u0435\u0433\u0438\u0441\u0442\u044a\u0440 \u043d\u0430 \u0438\u0437\u0434\u0430\u0434\u0435\u043d\u0438\u0442\u0435 \u043a\u043d\u0438\u0433\u0438…<\/strong><\/a><\/p>\n

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<\/a>\u0418\u0437\u0434\u0430\u0442\u0435\u043b\u0441\u043a\u0438 \u0434\u0430\u043d\u043d\u0438:<\/strong><\/p>\n

“Special topics in control theory<\/span>”<\/p>\n

Controllability and observability<\/p>\n

\u041f\u044a\u0440\u0432\u043e \u0438\u0437\u0434\u0430\u043d\u0438\u0435<\/p>\n

\u0421\u043e\u0444\u0438\u044f, 2014<\/p>\n

\u0410\u0432\u0442\u043e\u0440: Daniela Gancheva Marinova<\/p>\n

\u041f\u0435\u0447\u0430\u0442\u043d\u043e \u0438\u0437\u0434\u0430\u043d\u0438\u0435: ISBN 978-619-7209-01-3<\/p>\n

\u0415\u043b\u0435\u043a\u0442\u0440\u043e\u043d\u043d\u043e \u0438\u0437\u0434\u0430\u043d\u0438\u0435: ISBN 978-619-7209-05-1<\/p>\n

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<\/a>\u0417\u0430 \u0430\u0432\u0442\u043e\u0440\u0430:<\/strong><\/p>\n

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<\/a>\u041e\u043f\u0438\u0441\u0430\u043d\u0438\u0435:<\/strong><\/p>\n

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<\/a>\u0421\u044a\u0434\u044a\u0440\u0436\u0430\u043d\u0438\u0435:<\/strong><\/p>\n

1. Kalman criterion I 005
\n1.1. Introduction 005
\n1.2. Linear systems with unconstrained control 006
\n2. Kalman criterion II 011
\n2.1. General theorem 011
\n2.2. Kalman controls 014
\n3. Kalman criterion III 017
\n3.1. Relations between attainable sets for constrained and unconstrained systems 017
\n3.2. Convex sets and support functions 020
\n3.3. Support function of attainable set 021
\n4. Kalman criterion IV 023
\n4.1. More about convex sets 023
\n4.2. More on support functions 023
\n4.3. Kalman duality 025
\n5. Brammer’s criterion 027
\n6. Stochastic approach 031
\n6.1. Least square method 031
\n6.1.1. Problem statement 031
\n6.1.2. Algorithm 032
\n6.1.3. Properties of LSM 033
\n6.1.4. Gaussian case 034
\n6.1.5. Student distribution 035
\n6.1.6. Maximum likelihood and least square 036
\n6.2. Stochastic Integrals 036
\n6.2.1. Definition of the integral 0 036
\n6.2.1.1. Integration of simple non-anticipating functions 037
\n6.2.1.2. Approximation by simple functions 037
\n6.3. Extension of Stochastic Integral 041
\n6.3.1. Doob inequality 041
\n6.3.2. Applications of Doob inequality 042
\n6.4. Ito formula 045
\n6.4.1. Statement of Ito lemma 045
\n6.4.2. Vector version of Ito formula 047
\n6.4.3. First applications of Ito formula 048
\n6.4.4. Proof of Ito formula 050
\n6.5. Stochastic Differential Equations 052
\n6.5.1. Cauchy Problem 052
\n6.5.2. Proof of Cauchy–Lipschitz theorem for SDE 054
\n6.5.3. Dependence on parameters 056
\n6.6. Diffusion Processes 058
\n7. Pole placement 063
\n7.1. Cauchy formula 063
\n4
\n7.2. Remarks on Cauchy formula 066
\n7.2.1. Remark 1 066
\n7.2.2. Remark 2 067
\n7.3. Exercises 069
\n8. Elements of Nonlinear Controllability 073
\n8.1. Lie brackets 073
\n8.2. Rashevsky \u2013 Chow theorem 075
\n8.3. H?rmander theorem 078
\n9. Black box 079
\n10. Examples 085
\n10.1. A cart under bounded force 085
\n10.2. Harmonic oscillator 086
\n11. White Noise I 089
\n11.1. Reminder on the Probability Theory 089
\n11.1.1. Measure and integral 091
\n11.1.2. Independence and conditional expectation 092
\n11.1.3. Characteristic functions and Gaussian vectors 093
\n12. White Noise II 097
\n12.1. Stochastic processes 097
\n12.2. Construction of the white noise 099
\n13. White Noise III 105
\n13.1. Why the white noise is white? 105
\n13.2. Quantum mechanics and functional integrals 107
\n13.3. Girsanov’s theorem 111
\n14. Kalman Filter I 113
\n14.1. Equations for the conditional density 113
\n14.2. Kalman filter 117
\n14.2.1. Heisenberg algebra 117
\n14.2.2. Lax type equations 119
\n14.2.3. Riccatti equation 120
\n15. Kalman Filter II 121
\n15.1. Filtering problem 121
\n15.2. Innovation process 121
\n15.3. Equations for the conditional mean 123
\n15.4. Riccati equation 126
\n15.5. Kalman-Bucy filter 127
\nReferences 131<\/p>\n

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\u0418\u0437\u0434\u0430\u0442\u0435\u043b\u0441\u043a\u0438 \u0434\u0430\u043d\u043d\u0438… \u0417\u0430 \u0430\u0432\u0442\u043e\u0440\u0430… \u041e\u043f\u0438\u0441\u0430\u043d\u0438\u0435… \u0421\u044a\u0434\u044a\u0440\u0436\u0430\u043d\u0438\u0435… \u0421\u0432\u0430\u043b\u0438 \u043f\u044a\u043b\u043d\u0438\u044f \u0442\u0435\u043a\u0441\u0442 \u043d\u0430 \u043a\u043d\u0438\u0433\u0430\u0442\u0430 \u0442\u0443\u043a… \u041a\u043d\u0438\u0433\u0430\u0442\u0430 \u0432 \u041d\u0430\u0446\u0438\u043e\u043d\u0430\u043b\u043d\u0438\u044f \u0440\u0435\u0433\u0438\u0441\u0442\u044a\u0440 \u043d\u0430 \u0438\u0437\u0434\u0430\u0434\u0435\u043d\u0438\u0442\u0435 \u043a\u043d\u0438\u0433\u0438… \u0418\u0437\u0434\u0430\u0442\u0435\u043b\u0441\u043a\u0438 \u0434\u0430\u043d\u043d\u0438: “Special topics in control theory” Controllability and observability \u041f\u044a\u0440\u0432\u043e \u0438\u0437\u0434\u0430\u043d\u0438\u0435 \u0421\u043e\u0444\u0438\u044f, 2014 \u0410\u0432\u0442\u043e\u0440: Daniela Gancheva Marinova \u041f\u0435\u0447\u0430\u0442\u043d\u043e \u0438\u0437\u0434\u0430\u043d\u0438\u0435: ISBN 978-619-7209-01-3 \u0415\u043b\u0435\u043a\u0442\u0440\u043e\u043d\u043d\u043e \u0438\u0437\u0434\u0430\u043d\u0438\u0435: ISBN 978-619-7209-05-1 \u0417\u0430 \u0430\u0432\u0442\u043e\u0440\u0430:   \u041e\u043f\u0438\u0441\u0430\u043d\u0438\u0435:  <\/p>\n","protected":false},"author":2,"featured_media":179,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[11],"tags":[],"class_list":["post-83","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-book"],"_links":{"self":[{"href":"https:\/\/eudaimonia-production.com\/index.php?rest_route=\/wp\/v2\/posts\/83","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/eudaimonia-production.com\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/eudaimonia-production.com\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/eudaimonia-production.com\/index.php?rest_route=\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/eudaimonia-production.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=83"}],"version-history":[{"count":2,"href":"https:\/\/eudaimonia-production.com\/index.php?rest_route=\/wp\/v2\/posts\/83\/revisions"}],"predecessor-version":[{"id":189,"href":"https:\/\/eudaimonia-production.com\/index.php?rest_route=\/wp\/v2\/posts\/83\/revisions\/189"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/eudaimonia-production.com\/index.php?rest_route=\/wp\/v2\/media\/179"}],"wp:attachment":[{"href":"https:\/\/eudaimonia-production.com\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=83"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/eudaimonia-production.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=83"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/eudaimonia-production.com\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=83"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}