4.3 DISTURBANCE RESISTANT

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In real systems we expect that certain events will occur that are not part of our system model.

In this case we assume that the system control is happening as expected, and we add in a new disturbance input.

The block diagram below shows one of these systems, with a disturbance injected between the controller and the process.

Notice that in the above form we are reducing the problem by finding differences (basically a partial differential solution) which will be a good approximation when the disturbance is not too fast or large.

We can develop an equation for the controller, based on the desired system response to the disturbance,

The closed form expression can be calculated by replacing the desired transfer function,

These systems are often called regulators, and can be used when a system is subject to unexpected noise. Examples of possible applications would include plumbing systems, electrical power supplies, etc.

4.3.1 Disturbance Minimization

We can use an approach similar to the deadbeat controller, but we still need to know the type of disturbance expected to develop a controller.

Consider a case where the disturbance is a step function

We can examine the previous controller for stability as well,

If we have a first order system we only need to have a sample time that is shorter than the system time constant.