4.1 POSITIONING CONTROLLERS

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In these systems we will focus on how the output relates to the input.

These techniques work best with simple systems that require a jump from one setpoint to another.

The general procedure to design a controller is,

1. Model the process (i.e., find Gp)

2. Determine the control objective (i.e. select the desired system response Gdr)

3. Substitute these into the equation below and solve for the controller transfer function,

4. Check the equation for practicality and stability

5. Develop the control equation for the computer

6. Write the computer program

Try to derive the expression given above,

4.1.1 Dead Beat Control

It is possible (for a first order process) to match the output function to the input function in one step.

The desired function is,

Given the process model below, design a deadbeat controller,

Trying to eliminate the error in one step can require an extremely high (power) gain. When this gain is excessive we may use a less powerful controller.

4.1.2 Programming Examples

The following programs are examples of methods for programming using a few common languages.

These examples are for the Computer Boards DAS08-AOM board (The boards we used in the lab with Labview).

Please note that these programs have not been debugged.

4.1.2.1 - BASIC

A simple example of the deadbeat controller is given below. This can be converted to many of the `modern' version of basic that have appeared. Comments have been added to help clarify the operation

4.1.2.2 - C

The example program below should implement the deadbeat controller example given in this section,

4.1.2.3 - Pascal

The example program below should implement the deadbeat controller example given in this section,

4.1.2.4 - 6811 Assembler

The example program below should implement the deadbeat controller example given in this section for a 6811 single chip microcontroller.

As an exercise, implement the controller on a Basic Stamp chip.

4.1.3 First Order Response

The deadbeat controller can call for extremely high gains, but this requires a high level of power not commonly found in engineered systems.

When we want to slow down the system response so that it occurs over a number of controller steps, we can replace the deadbeat (one time step) function with a gradual first order reduction (exponential decay) over a number of time steps.

This gradual response will use a time constant to produce the rate of response (recall that the difference between input and output is reduced 63% for each duration of the time constant).

The general form of this relationship is seen below,

Consider the example for the deadbeat controller,

Assume the response time constant should be 1.0 seconds, and the sampling time is 0.2 seconds, develop the final form of the controller.