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![• % 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 ----------------](https://image.slidesharecdn.com/week16controllabilityandobservabilityjune1final-160603092314/75/Week-16-controllability-and-observability-june-1-final-49-2048.jpg)
Uploaded byCharlton Inao
Week 16 controllability and observability june 1 final
This document discusses controllability and observability of systems using Matlab. It provides an example of checking the controllability and observability of a 2nd order system. The controllability and observability matrices are calculated from the state space matrices using Matlab functions. The rank of each matrix is then checked to determine if the system is controllable/observable or not.
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Week 16 controllability and observability june 1 final
- 1. Week 16 Controllability and Observability Prof CharltonS. Inao Defence University College of Engineering 16/3/2016
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- 8. State Controllability • ControllabilityMatrix 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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- 19. State Observability • ObservableMatrix (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 40C 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
- 23. Output Controllability • Outputcontrollability 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
- 24. 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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- 29. 296/3/2016 If determinant iszero , i.e singular… the system is non observable N=rank=3 System order=full rank, there fore it is observable
- 30. Finding the determinant 6/3/201630 Down (+) UP (-)
- 31. Unobservability via ObservabilityMtrix 316/3/2016 If determinant of the observability matrix is zero , the system is unobservable
- 32. Calculation of Determinant 6/3/201632 If determinant of the observability matrix is zero , the system is unobservable
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- 41. 416/3/2016 All zero column System order=2 Rank=1 Not equal , therfore UNOBSERVABLE
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- 44. Controllability and Observability UsingMatlab Prof Charlton S. Inao
- 49. • % StateSpace 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 ----------------