Question

consider the system

X(t) = ax(1) + bu(t) with a = 0.001,b= 1,x(0) = 5. (a) Simulate this system using the Matlab command initial (b) Now use u(t) = -kx(t) where k is found as the optimal gain by minimizing the performance index J= ax (1) + ru (1) dt Use q=1, r=1 to simulate this system.

Answer 1

(a) Since a > 0, initial condition response will blow up as clear from the following expression for x(t).

                                                    

(7) 0)= (7)

So plot has been shown with value of x(t) on dB scale.

MATLAB code is:

a = 0.001;
b = 1;
sys = ss(a,b,[],[]);
x0 = 5;
t = 0:1:1e5;
[y,t,x] = initial(sys,x0,t);
plot(t,20*log10(x));
xlabel('time(s)');
ylabel('x(t)(dB)');
grid on;

Plot of x(t) is shown below;

700.... 600... α(τ)(dB) ..... . 10 time(s)

(b) This part is little confusing. I have taken two scenarios and written code for both.

      (1) We already have x(t) from part (a). Now carry out the integral to obtain J(k) for different values of k, with x(t) already obtained. Then find the value of "k" which gives minimum J.

MATLAB code for this case is shown:

a = 0.001;
b = 1;
q = 1;
r = 1;
x0 = 5;
t = 0:0.01:10;
k = (-1:0.1:10);
sys = ss(a,b,[],[]);
    [y,t,x] = initial(sys,x0,t);

for i = 1:length(k)

        u(i,:) = -k(i)*xf;
        J(i) = sum(q*xf(i,:).*xf(i,:)+r*u(i,:).*u(i,:))*(t(2)-t(1));
end

Plot of J(k) vs k, is shown below:

J(k) in dB : 3:0 .. V: 48.05 ... .. 2

Clearly minimum of J(k) occurs at k = 0.

(2) Change the system state space equation to

                                                     

i(t) = (a - kb)x(t)

Solve this system with given initial condition and get x(t) for each k. Then do the integral for each k to find J(k).

MATLAB code is shown:

a = 0.001;
b = 1;
q = 1;
r = 1;
x0 = 5;
t = 0:0.01:10;
k = (-1:0.1:10);
a1 = a - k;
for i = 1:length(k)
    sys = ss(a1(i),b,[],[]);
    [y,t,x] = initial(sys,x0,t);
    xf(i,:) =x;
    u(i,:) = -k(i)*xf(i,:);
    J(i) = sum(q*xf(i,:).*xf(i,:)+r*u(i,:).*u(i,:))*(t(2)-t(1));
end
index = find(abs(J)==min(abs(J)));
kmin = k(index)
Jmin = J(index)

Plot of J(k) is shown:

J(k) in dB .......... 60........ V: 28.05 o 2