Cost function to minimize variance of error

I need to design a cost function to minimize the variance of error input to my controller in order to find out optimum values of my controller parameters. I have the transfer function of my plant and my reference signal is a constant 0. My controller is a simple lead lag filter with a gain. How do I design a cost function for this task?

Answer 1

[Sam Chak]

@Rishabh Pachchhao,

Since you didn't provide info about your system. here is an example that you can study. However, I don't think that miniming the error variance works.

In the example, if you let , then the variance .

k0 = 1;
k  = fminunc(@costfun, k0)

Local minimum found.

Optimization completed because the size of the gradient is less than
the value of the optimality tolerance.

k = 532.3080

s   = tf('s');
Gp  = 1/(s*(s + 5)*(s + 10));
Gc  = k*((s + 4)*(s + 0.1))/((s + 7.3)*(s + 0.01));
Gcl = feedback(Gc*Gp, 1);
t   = linspace(0, 10, 1001);
step(Gcl, t)

function J = costfun(k)
    s   = tf('s');
    Gp  = 1/(s*(s + 5)*(s + 10));
    Gc  = k*((s + 4)*(s + 0.1))/((s + 7.3)*(s + 0.01));
    Gcl = feedback(Gc*Gp, 1);
    t   = linspace(0, 10, 1001);
    y   = step(Gcl, t);
    err = 1 - y;
    % J   = var(err);
    J   = trapz(t, err.^2);
end