RLC Circuit of February 10, 2011

The problem was set up in class starting on February 10, 2011. The Octave script below simulates the control system using a full order observer for the states, and assuming that the output voltage is all that is available for measurement.

% This script simulates the RLC circuit from the notes of February 10,
% 2011.  It uses a full order observer.  It also shows how to reorder
% the states.  It includes a simulation of the system which is very
% interesting because it shows how there is a big error in the observer
% at first.

c=2 %Capacitance   
R=3 %Resistance
L=4 %Inductance
A=[[0 -1/L];[1/c -1/(R*c)]]
B=[1/L; 0]
C=[0 1]
D=0
sys1=ss(A,B,C,D);
T=fliplr(eye(2)); %Transformation Matrix for swapping states
Ar=T*A*T^(-1); %Transformed system
Br=T*B;
Cr=C*T^(-1);
sysr=ss(Ar,Br,Cr,0);
Rr=obsv(sysr); %Observability Matrix for swapped states
R=obsv(sys1); %Not swapped Observability Matrix
rank_r=rank(R); %Is the system observable
rank_rr=rank(Rr)
g=place(sys1,[-1.5 -2.5]); %Place the poles for the controller
printf("Check to see if the place did the right thing.")
printf("The poles after feedback are at: ")
controller_poles = eig(A-B*g)
K=place(A',C',[-33,-30.5])' %Place poles for the observer
% Note: you can see the observer problem with poles [-100,-105]
% and initial conditions [1 2 3 4]
Ahat=A-K*C;
printf("Check to see if the place did the right thing.")
printf("The poles of the observer are: ")
observer_poles = eig(Ahat)
Bhat=B;
%Make a concatinated system of the observer and system
Atot=[[A, -B*g];[K*C, Ahat-Bhat*g]]; 
Btot=[B;Bhat];
Ctot=eye(size(Atot));
systot=ss(Atot,Btot,Ctot);
polestot=pole(systot)
t=0:.0001:.5;
[X,Y]=lsim(systot, zeros(size(t))', t, [5 5 1 1]');
plot(t,X)
legend('x_1','x_2','xhat_1','xhat_2')
title('System and Observer Simulation')
xlabel('Time (seconds)')
% Check the transformed to error terms total system.
T=[[1 0 0 0];[0 1 0 0];[1 0 -1 0];[0 1 0 -1]];
TAtotTI=T*Atot*T^(-1);
polestot_transformed=eig(TAtotTI);