problem with the rank of controllability of a system

Question

Kamran 2022-3-25

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https://ww2.mathworks.cn/matlabcentral/answers/1680494-problem-with-the-rank-of-controllability-of-a-system

编辑： Sachin Lodhi 2023-12-19

I have this system given below

A=[-3.87695312500000e-06,-0.000377259521484375,-1.54541015625000e-06,-2.47435565751281e-05,0;
    -0.000383031005859375,-0.0438680297851563,-0.000171501464843750,-0.00287715774129662,0;
    -1.68273925781250e-06,-0.000184232788085938,-3.42102050781250e-06,-1.20840625134364e-05,0;
    -0.0288955468750000,-3.30957816406250,-0.0129358593750000,-0.452124041742296,0.00180860517844498;
    0,0,0,119.959269051020,-1.23669349537147]
B=[-0.000202364643587302;-0.0235307725101629;-9.88292445427022e-05;-1.42966622638009;0];
C=eye(5);
D=zeros(5,1);

the rank of the controlablity matrix gives me the following matrix

co=ctrb(A,B);

>> co=[-0.000202364643587302,4.42531724232830e-05,-1.98625050253011e-05,1.70702903536278e-05,-2.16416261767110e-05;-0.0235307725101629,0.00514571834133656,-0.00230959374267344,0.00198491754360988,-0.00251646821219010;-9.88292445427022e-05,2.16119945020915e-05,-9.70028879196148e-06,8.33665147162803e-06,-1.05691645905590e-05;-1.42966622638009,0.724270529374834,-0.654670725177827,0.844370048676126,-1.19908774023253;0,-171.501715503486,298.978019307210,-448.277993397705,655.672492400879]

Rank (co) returns 4 where as it seems it should be 5?

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Kamran 2022-3-25

For the given system, I can develop an lqr with following paratmers

R=diag([1e1,1e3,1e1,1e2,10]);

Q=1e2;

K=lqr(A,B,R,Q);

So I dont understand the rank of controlablity martix be 4 and not 5.

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Answer 1

Sachin Lodhi 2023-12-19

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此回答的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/1680494-problem-with-the-rank-of-controllability-of-a-system#answer_1373872

编辑：Sachin Lodhi 2023-12-19

Hi Kamran,

The controllability matrix 'co' has a rank of 4 instead of 5 because the first row is just the third row multiplied by 2. It means that these two rows are linearly dependent.

In terms of rank, which measures the dimension of the row or column space (the maximum number of linearly independent rows or columns), having two linearly dependent rows reduces the rank of the matrix by at least one.

So, in matrix ‘co’ of [5 x 5] dimension, if two rows are the same, the number of linearly independent rows would be '5-1 = 4', and thus the rank of the matrix would be 4.

I hope this helps.

Best Regards,

Sachin