MATLAB can't compute a closed-loop final-value theorem on a controller.

Hi, I have an assignment where I need to design a state-feedback controller in the form: u(k) = K x[k]+ Sy¯[k], with a sampling period pi = 0.05, such that the (discrete-time) closed-loop poles are placed in 0:8 ± 0:1i and the static gain from y¯ to y is 1.

I need to plot a response of the open-loop system to a step of u(t), (which I did)

And 1 plot of the closed-loop response to a step of y¯(t), (doesn't accept S-value, calls it infinite or NaN)

The image below is the initial state-space model:

Here's my code so far:

Phi = [-0.12 0; 5 0]

Gamma = [2.25; 0]

C = [0 1]

D = 0

Pd = ss(Phi, Gamma ,C, D, 0.05)

step(Pd) %plotting the open-loop u(t)

K = [0.45333 -0.00071] %solved K by hand, also had to write down functions so paper is my go to.

PhiCL = Phi+Gamma*[k1 k2]

GammaCL = Gamma *[ s1 ];

syms s1 z;

%MATLAB doesn't compute Z-transform either, tried it by hand, the whole function yields 0

Zg = (C+D*K) * inv(z*eye (2) - PhiCL ) * GammaCL + D*[ s1 ]

Zlim = subs (Zg , z, 1)

s_val = solve (Zlim ==1 , s1) %can't solve fo s1, says: "Empty sym: 0-by-1"

y_bar = ss(PhiCL, GammaCL, (C+D*K), D*s1, 0.05)

%Error using ss (line 284)

%The "B" matrix must be a numeric array with no Inf's or NaN's.

I plotted the y_bar by removing the s1's from D*s1 and GammaCL, and I'm not sure if it converges to 0, I'll attach the plot below. I'm kind of cheating with this plot, and I have to get full points on this assignment or I can't enter the exam.

Thank you in advance!

[Marjo Toska]

My big K [k1 k2] was solved for poles [0.8 0.1] since I can't really compute +-0.1i by hand. The results are slightly off by yours I know, but that's not my concern at all. Because Z-transform is always 0 no matter if I use k2=0.1, k2=0.1i or k2=+-0.1i.

I'm concerned with plotting the closed-loop response, and I'm not sure how to do that, have a feeling that Z transform should be non-zero to do that, or solve K, setpoint S and observer L for that. shrug

[Paul]

Not sure what you meant by: "no matter if I use k2=0.1, k2=0.1i or k2=+-0.1i. " The feedback gain matrix has to be real. What's keeping you from solving for the K by hand? The problem would be to find K such that the characteristic equation of Phi + Gamma*K has roots at z = 0.8 +- 0.1*i.

Answer 1

[Paul]

Is that feedback gain matrix correct? Note that here I'm using K, but the code above used k1 and k2, which were not defined.

Phi = [-0.12 0; 5 0];
Gamma = [2.25; 0];
C = [0 1];
D = 0;
K = [0.45333 -0.00071];
eig(Phi + Gamma*K)  % doesn't match specifcation 0.8 +- 0.1i, using positive feedback consistent witht the question as asked

ans = 2×1

0.8910 0.0090