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Understanding Controllability and Observability in Linear Systems Analysis

Explore the concepts of controllability and observability in linear systems analysis, using Meiling CHEN's explanations and examples. Learn how a system is completely controllable and observable, and discover the criteria for determining controllability and observability. Dive into proofs of controllability and observability matrices, understand controllable and observable canonical forms, and explore Jordan forms and blocks in linear systems. Gain insights into dealing with uncontrollable and unobservable systems through practical examples.

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Understanding Controllability and Observability in Linear Systems Analysis

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  1. Linear system 1. Analysis Lesson 7 Controllability & Observability

  2. Linear system 1. Analysis Motivation1 uncontrollable controllable Linear System by Meiling CHEN

  3. Linear system 1. Analysis Motivation2 observable unobservable Linear System by Meiling CHEN

  4. Linear system 1. Analysis Definition A linear system is said to be completelycontrollable 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 . Definition 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 . Linear System by Meiling CHEN

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

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

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

  8. Linear system 1. Analysis 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 by Meiling CHEN

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

  10. Linear system 1. Analysis Theorem III Controllable canonical form Controllable Theorem IV Observable canonical form Observable Linear System by Meiling CHEN

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

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

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

  14. Linear system 1. Analysis Linear System by Meiling CHEN

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

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

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