# Control Systems/Noise Driven Systems

## Noise-Driven Systems

Systems frequently have to deal with not only the control input *u*, but
also a random noise input *v*. In some disciplines, such as in a study
of electrical communication systems, the noise and the data signal can
be added together into a composite input *r = u + v*. However, in
studying control systems, we cannot combine these inputs together, for a
variety of different reasons:

1.  The control input works to stabilize the system, and the noise input
    works to destabilize the system.
2.  The two inputs are independent random variables.
3.  The two inputs may act on the system in completely different ways.

As we will show in the next example, it is frequently a good idea to
consider the noise and the control inputs separately:

## Probability Refresher

We are going to have a brief refesher here for calculus-based
probability, specifically focusing on the topics that we will use in the
rest of this chapter.

### Expectation

The expectation operatior, **E**, is used to find the *expected, or mean
value* of a given random variable. The expectation operator is defined
as:

$$E[x] = \int_{-\infty}^\infty x f_x(x)dx$$

If we have two variables that are independent of one another, the
expectation of their product is zero.

### Covariance

The **covariance** matrix, *Q*, is the expectation of a random vector
times it\'s transpose:

$$E[x(t)x'(t)] = Q(t)$$

If we take the value of the *x* transpose at a different point in time,
we can calculate out the covariance as:

$$E[x(t)x'(s)] = Q(t)\delta(t-s)$$

Where δ is the impulse function.

## Noise-Driven System Description

We can define the state equation to a system incorporating a noise
vector *v*:

$$x'(t) = A(t)x(t) + H(t)u(t) +B(t)v(t)$$

For generality, we will discuss the case of a time-variant system.
Time-invariant system results will then be a simplification of the
time-variant case. Also, we will assume that *v* is a **gaussian random
variable**. We do this because physical systems frequently approximate
gaussian processes, and because there is a large body of mathematical
tools that we can use to work with these processes. We will assume our
gaussian process has zero-mean.

## Mean System Response

We would like to find out how our system will respond to the new noisy
input. Every system iteration will have a different response that varies
with the noise input, but the average of all these iterations should
converge to a single value.

For the system with zero control input, we have:

$$x'(t) = A(t)x(t) + B(t)v(t)$$

For which we know our general solution is given as:

$$x(t) = \phi(t, t_0)x_0 + \int_{t_0}^t \phi(t, \tau)B(\tau)v(\tau)d\tau$$

If we take the **expected value** of this function, it should give us
the expected value of the output of the system. In other words, we would
like to determine what the expected output of our system is going to be
by adding a new, noise input.

$$E[x(t)] = E[\phi(t, t_0)x_0] + E[\int_{t_0}^t \phi(t, \tau)B(\tau)v(\tau)d\tau]$$

In the second term of this equation, neither φ nor B are random
variables, and therefore they can come outside of the expectaion
operation. Since *v* is zero-mean, the expectation of it is zero.
Therefore, the second term is zero. In the first equation, φ is not a
random variable, but x~0~ does create a dependancy on the output of
*x*(t), and we need to take the expectation of it. This means that:

$$E[x(t)] = \phi(t, t_0)E[x_0]$$

In other words, the expected output of the system is, on average, the
value that the output would be if there were no noise. Notice that if
our noise vector *v* was not zero-mean, and if it was not gaussian, this
result would not hold.

## System Covariance

We are now going to analyze the covariance of the system with a noisy
input. We multiply our system solution by its transpose, and take the
expectation: `<small>`{=html}(this equation is long and might break onto
multiple lines)`</small>`{=html}

$$E[x(t)x'(t)] = E[\phi(t, t_0)x_0 + \int_{t_0}^t\phi(\tau, t_0)B(\tau)v(\tau)d\tau]$$$E[(\phi(t, t_0)x_0 + \int_{t_0}^t\phi(\tau, t_0)B(\tau)v(\tau)d\tau)']$

If we multiply this out term by term, and cancel out the expectations
that have a zero-value, we get the following result:

$$E[x(t)x'(t)] = \phi(t, t_0)E[x_0x_0']\phi'(t, t_0) = P$$

We call this result *P*, and we can find the first derivative of P by
using the chain-rule:

$$P'(t) = A(t)\phi(t, t_0)P_0\phi(t, t_0) + \phi(t, t_0)P_0\phi'(t, t_0)A'(t)$$

Where

$$P_0 = E[x_0x_0']$$

We can reduce this to:

$$P'(t) = A(t)P(t) + P(t)A'(t) + B(t)Q(t)B'(t)$$

In other words, we can analyze the system *without needing to calculate
the state-transition matrix*. This is a good thing, because it can often
be very difficult to calculate the state-transition matrix.

## Alternate Analysis

Let us look again at our general solution:

$$x(t) = \phi(t, t_0)x(t_0) + \int_{t_0}^t \phi(t, \tau)B(\tau)v(\tau)d\tau$$

We can run into a problem because in a gaussian distribution, especially
systems with high variance (especially systems with infinite variance),
the value of *v* can momentarily become undefined (approach infinity),
which will cause the value of *x* to likewise become undefined at
certain points. This is unacceptable, and makes further analysis of this
problem difficult. Let us look again at our original equation, with zero
control input:

$$x'(t) = A(t)x(t)+B(t)v(t)$$

We can multiply both sides by *dt*, and get the following result:

$$dx = A(t)x(t)dt + B(t)v(t)dt$$

We can define a new differential, *dw(t)*, which is an infinitesimal
function of time as:

$$dw(t) = v(t)dt$$

Now, we can integrate both sides of this equation:

$$x(t) = x(t_0) + \int_{t_0}^t A(\tau)x(\tau)d\tau + \int_{t_0}^tB(\tau)dw(\tau)$$
 However, this leads us to an
unusual place, and one for which we are (probably) not prepared to
continue further: in the third term on the left-hand side, we are
attempting to integrate with respect to a *function*, not a *variable*.
In this instance, the standard Riemann integrals that we are all
familiar with cannot solve this equation. There are advanced techniques
known as **Ito Calculus** however that can solve this equation, but
these methods are currently outside the scope of this book.

