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I going to show how to create a Minimum Variance Controller(MVC) and then ask what's the applications of MVC.

First! Let's say that we have a stochastic transfer function model, ARMAX in other words.

$$y = \frac{B}{A}u + \frac{C}{A}e$$

To find the control law

$$u = -\frac{G}{BF}y$$

We need to solve the diophantine equation

$$z^{d_0 -1}CB = ABF+BG$$

Where $$n = deg(A)$$ $$ m = deg(B)$$ $$d_0 = n - m$$

And

$$F = z^{d_0-1} + f_1 z^{d_0 -2} + ... + f_{d_0 - 1}$$ $$G = g_0z^{n-1} + g_1 z^{n -2} + ... + g_{n - 1}$$

Example:

If we got this second order difference equation:

$$az^2 + a_0 z + a_1 = bz + b_0 + cz^2 + c_0z + c_1$$

Then $$n = deg(A) = deg(C) = 2$$ $$m = deg(B) = 1$$ that will give $$d_0 = n - m = 1$$

The diophantine equation:

$$z^{d_0 -1}CB = ABF+BG$$

$$(1)(z^2 + c_0z + c_1)(bz + b_0) = (z^2 + a_0 z + a_1)(bz + b_0)(1) + (bz + b_0)(g_0z + g_1)$$

Because $z^0 = 1$

Simplify - remove the B, 1 and $z^2$

$$c_0z + c_1 = a_0 z + a_1 + g_0z + g_1$$

$$g_0 = c_0 - a_0$$ $$g_1 = c_1 - a_1$$

That will give us the controll law:

$$u = -\frac{g_0z + g_1}{bz + b_0}y$$

So if we say that our difference equation is:

$$z^2 - 1.9507754 z + 0.9704455 = 0.0049421z + 0.0048929 + z^2 + 0.1z + 0.3$$

With the sampling rate of $h=0.1$

And we simulate the system without feedback and with feedback. The random generator gives a white noise of $\mu = 0$ and $\sigma = 1$

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And the results are:

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Question:

What is the applications of MVC?

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