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Week 16 Controllability

and ObservabilityProf Charlton S. InaoDefence University

College of Engineering

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Instructional Objectives

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CONTROLLABILITY

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State Controllability• Controllability Matrix CM

• System is said to be state controllable if

BABAABBCM n 12

)( nCMrank

State Controllability (Example)• Consider the system given below

• State diagram of the system is

xy

uxx

2101

3001

1

1

)(sU

)(sY1

-1s

3-1s

2

1x

2x

State Controllability (Example)•

ABBCM

0011

CM

01

B

01

A B

System order(state variable) is 2 but rank is

1, therefore not controllable

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Workout Exercise

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OBSERVABILITY

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State Observability• Observable Matrix (OM)

• The system is said to be completely state observable if

1

2MMatrix ity Observabil

nCA

CACAC

O

nOMrank )(

n= system order ,based on the number of state variable

State Observability (Example)• Consider the system given below

• OM is obtained as

• Where

xy

uxx

4010

2010

CAC

OM

40C

12020

1040

CA

State Observability (Example)•

12040

MO

1)(sU -1s -1s 1x2x

2

4

)(sY

Rank =1n=system order =2

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 12O

Work Out Exercise

• Check the state controllability, state observability and output controllability of the following system

10,10

,0110

CBA

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Reference/Basis

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Reference/Basis

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If determinant is zero , i.e singular… the system is non observable

N=rank=3

System order=full rank, there fore it is observable

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Finding the determinant

Down (+)

UP (-)

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Unobservability via Observability Mtrix

If determinant of the observability matrix is zero , the system is unobservable

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Calculation of Determinant

If determinant of the observability matrix is zero , the system is unobservable

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All zero column

System order =2Rank=1Not equal , therfore UNOBSERVABLE

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Identical but negated(opposite sign)

Identical but negated(opposite sign)

Controllability and Observability Using Matlab

Prof Charlton S. Inao

• % 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 ----------------