# Week 16 controllability and observability june 1 final

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Information about Week 16 controllability and observability june 1 final

Published on June 3, 2016

Author: CharltonInao

Source: slideshare.net

1. Week 16 Controllability and Observability Prof Charlton S. Inao Defence University College of Engineering 16/3/2016

2. Instructional Objectives 6/3/2016 2

3. CONTROLLABILITY 6/3/2016 3

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8. State Controllability • Controllability Matrix CM • System is said to be state controllable if  BABAABBCM n 12    )( nCMrank 

9. State Controllability (Example) • Consider the system given below • State diagram of the system is  xy uxx 21 0 1 30 01                 1 1 )(sU )(sY 1 -1 s 3 -1 s 2 1x 2x

10. State Controllability (Example)  ABBCM          00 11 CM        0 1 B        0 1 AB System order(state variable) is 2 but rank is 1, therefore not controllable

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14. Workout Exercise 146/3/2016

15. OBSERVABILITY 156/3/2016

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19. State Observability • Observable Matrix (OM) • The system is said to be completely state observable if                  1 2 MMatrixityObservabil n CA CA CA C O  nOMrank )( n= system order ,based on the number of state variable

20. State Observability (Example) • Consider the system given below • OM is obtained as • Where  xy uxx 40 1 0 20 10                       CA C OM  40C    120 20 10 40        CA

21. State Observability (Example)         120 40 MO 1)(sU -1 s -1 s 1x2x 2 4 )(sY Rank =1 n=system order =2

22. Output Controllability • Output controllability describes the ability of an external input to move the output from any initial condition to any final condition in a finite time interval. • Output controllability matrix (OCM) is given as  BCABCACABCBCM n 12 O   

23. Work Out Exercise • Check the state controllability, state observability and output controllability of the following system  10, 1 0 , 01 10              CBA

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26. Reference/Basis 6/3/2016 27

27. 6/3/2016 28 Reference/Basis

28. 296/3/2016 If determinant is zero , i.e singular… the system is non observable N=rank=3 System order=full rank, there fore it is observable

29. Finding the determinant 6/3/2016 30 Down (+) UP (-)

30. Unobservability via Observability Mtrix 316/3/2016 If determinant of the observability matrix is zero , the system is unobservable

31. Calculation of Determinant 6/3/2016 32 If determinant of the observability matrix is zero , the system is unobservable

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40. 416/3/2016 All zero column System order =2 Rank=1 Not equal , therfore UNOBSERVABLE

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42. 436/3/2016 Identical but negated(opposite sign) Identical but negated(opposite sign)

43. Controllability and Observability Using Matlab Prof Charlton S. Inao

44. • % State Space Representation % x' = Ax + Bu % y = Cx + Du % % Problem 1 --------------------------------------------------------------- % • Check Controllability and Observability of a 2nd order System % • Given ------------------------------------------------------------------- MatrixA = [0 1;-2 -3]; MatrixB = [0;1]; MatrixC = [1 -1]; MatrixD = 0; % • Objective --------------------------------------------------------------- % • 1) To Find Controllable Matrix Qc, its rank and check controllability • % 2) To Find Observable Matrix Qb, its rank and check observability %------ • --- % Controllable Matrix ----------------------------------------------------- Qc = ctrb(MatrixA,MatrixB); rankQc = rank(Qc); disp('Controllable Matrix is Qc = '); disp(Qc); if(rankQc == rank(MatrixA)) disp('Given System is Controllable.'); else disp('Given System is Uncontrollable'); end % Observable Matrix ------------------------------------------------------- Qb = obsv(MatrixA, MatrixC); rankQb = rank(Qb); disp('Observable Matrix is Qb = '); disp(Qb); if(rankQb == rank(MatrixA)) disp('Given System is Observable.'); else disp('Given System is Unobservable'); end % End of Program ----------------