Find non singular transformation matrix

I am working on state space system in matlab script. here C =[0 0 0 0 0 0 1 0 1 0] but I want to observe 1st state i.e. required Cr = [1 0 0 0 0 0 0 0 0 0]. So to get this C I am using transformation matrix T = Cr/C.....but here i am getting singular T matrix

I want nonsingular T matrix

[Paul]

To make sure we're clear on the question:

You have a model

xdot = A*x + B*u

y = C*x + D*u

where C = [0 0 0 0 0 0 1 0 1 0]

The goal is find the similarity tranformation matrix T that defines a new state vector, z, so that

zdot = Anew*z + Bnew*u

y = Cnew*z + Dnew*u

where Cnew = [1 0 0 0 0 0 0 0 0 0]

Is my understanding correct?

[RJS]

yes

Answer 1

[Paul]

Assuming the goal is as described in this comment, one solution would be:

T1 = [0 0 0 0 0 0 1 0 1 0];
T = [T1;null(T1).']; 

See this Answer thread for much more discussion and other approaches.

[RJS]

Thank you @Paul ,after this transformation also i am getting the same result But I want to observe !st state or each state of the system ,

[Paul]

Are there now additional obejctives for the T matrix? If so, I don't understand what they are. Can you clarify this statement and explain how it relates to T: "I want to observe 1st state or each state of the system." ,

[RJS]

If the required C vector is [0 0 0 0 1 0 0 0 0 0], then how to find this transformation

[Paul]

Follow the same approach, but put the C vector from plant in the fifth row of T instead of the first. Here's an example:

sys = rss(10); % random system for example
sys.c

ans = 1×10

-0.2270 -1.7085 0 -1.4590 0.2800 0.3869 1.1516 -0.3147 0 -0.6773

N = null(sys.c).';
T = [N(1:4,:); sys.c; N(5:end,:)]; % sys.c in the fifth row, distribute N across the others
sysnew = ss2ss(sys,T);
sysnew.c  % verify desired result

ans = 1×10

0 0 0 0 1 0 0 0 0 0