Lesson 7 Controllability & Observability Linear system 1. Analysis.

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Presentation transcript:

Lesson 7 Controllability & Observability Linear system 1. Analysis

Linear System by Meiling CHEN2 Motivation1 controllable uncontrollable Linear system 1. Analysis

Linear System by Meiling CHEN3 Motivation2 observable unobservable Linear system 1. Analysis

Linear System by Meiling CHEN4 Definition A linear system is said to be completely controllable if, for all initial times and all initial states, there exists some input function (or sequence for discrete systems) that drives the state vector to any final state at some finite time. A linear system is said to be completely observable if, for all initial times, the state vector can be determined from the output function (or sequence), defined over a finite time. Definition Linear system 1. Analysis

Linear System by Meiling CHEN5 Proof of controllability matrix Initial condition Linear system 1. Analysis

Linear System by Meiling CHEN6 Proof of observability matrix Inputs & outputs Linear system 1. Analysis

Linear System by Meiling CHEN7 Controllability matrix Observability matrix Then: (1) controllable (2) observable Linear system 1. Analysis

Linear System by Meiling CHEN8 Example 1 uncontrollable observable The rank of a matrix is defined by the number of linearly independent rows and/or the number of linearly independent columns Linear system 1. Analysis

Linear System by Meiling CHEN9 Example 2 controllable unobservable Linear system 1. Analysis

Linear System by Meiling CHEN10 Theorem III Controllable canonical formControllable Theorem IV Observable canonical formObservable Linear system 1. Analysis

Linear System by Meiling CHEN11 example Controllable canonical form Observable canonical form Linear system 1. Analysis

Linear System by Meiling CHEN12 Theorem V Jordan form Jordan block Least row has no zero row First column has no zero column Linear system 1. Analysis

Linear System by Meiling CHEN13 Example If uncontrollable If unobservable Linear system 1. Analysis

Linear System by Meiling CHEN14 Linear system 1. Analysis

Linear System by Meiling CHEN15 controllable observable In the previous example controllable unobservable Linear system 1. Analysis

Linear System by Meiling CHEN16 L.I. L.D. Example Linear system 1. Analysis