An Introduction to Control Systems - Proportional Control

Control Systems Intro - Proportional Control

ON-OFF control systems are used for simple control tasks. However, one step beyond ON-OFF control systems are proportional control systems. Let's think our way through this. As we think about proportional control, keep this block diagram in mind. In this block diagram the control effort, c(t), drives the system to produce the output, y(t). (In the steady state, the output is proportional to the control effort if the system is linear.) The output, y(t), is subtracted from the desired output - i.e. the input, u(t) - to produce the error, e(t). The error, e(t), is multiplied by a proportional gain constant, K_p, to produce the control effort, c(t). (In the simulator, the proportional gain is called K_p to distinguish it from the gain in the integral control mode, K_i.)

First, in many situations the output of a system is proportional to the control effort you apply to the system.

An example of a system like that might be a motor with a power amplifier (the actuator) driving the motor (the plant). If you are trying to control the speed of the motor, you might find that the steady state motor speed is proportional to a constant voltage applied to the power amplifier.
Mathematically, you would have:

Steady State Motor speed = Constant x Applied Constant Voltage
Speed = K x Vapp
K is a constant associated with the motor.

Now, something interesting happens if the applied voltage is proportional to the error (which is what makes this proportional control).

Vapp = Kp x Error
Kp is the proportional gain.

That's the standard symbol for a proportional gain. Do not confuse it with the constant, K, in the system itself.

Since the error is the difference between the desired response and the measured response, we have the following.

Speed = K x Vapp
Speed = K x Kp x Error
Speed = K x Kp x (Desired Speed - Speed)

Amazingly, this can be solved for the Speed variable.

Speed = Desired Speed x (K x Kp)/[1 + (K x Kp)]

Speed = K x Kp x (Desired Speed - Speed)

And, you can also get an expression for the Error.

Speed = Desired Speed x (K x Kp)/[1 + (K x Kp)] = K x Kp x Error
Error = Desired Speed/[1 + (K x Kp)]
PerCent Error (Error%) = 100/[1 + (K x Kp)]

There are some conclusions you can draw from this expression for the Speed.

If you want the speed to match the desired speed closely, you need to have the product of the motor constant and the proportional gain to be very large. (K x Kp should be large compared to 1.0)
Low gains will produce inaccurate systems.

There are some cautions. In particular, we have ignored the dynamics of the system. Of course, to understand that you need to know a little about system dynamics. That link will get you started, but be prepared. It's a long haul, about one course's worth of material.

The Effect of System Dynamics

At this stage you might not know much about system dynamics - but then again, you might. In either case, you can see some of the things that might happen.

Here is a link to a some material about controlling a first order system. There you can simulate controlling a first order system. and see the effects of "closing the loop".

One thing that you should observe is that closing the loop speeds up the response in a first order system. However, other untoward things can happen if the system dynamics are more complex.
You should also observe that increasing the proportional gain makes the system more accurate. In other words, in the steady state the system gets closer to the desired output - although it will never get there all the way.

Here is a link to a some material about controlling a second order system. There you can simulate controlling a second order system. and see the effects of "closing the loop".

In this system you should observe that closing the loop can lead to oscillations and that those oscillations become larger as you try to make the system more accurate.
As in the first order system, increasing the proportional gain makes the sytem more accurate. However, here that comes at the expense of larger oscillations.

In both of these simulations, you should note that the control effort - the signal applied to the system to move it closer to the desired output - is larger when the measured output is far from the desired output. As the output moves closer to the desired output, the control effort goes down.

The Effects of Sampling

When you use digital control, you sample the signal. If you implement a digital version of proportional control, some interesting things can happen. Here is a block diagram that represents a digital control system.

In proportional control the control effort is proportional to the error. In digital control you would compute the control effort, but then you would hold the control effort constant until the next time your control loop ran and sampled the output again. If you hold the control effort constant as the error gets smaller - during the time the loop is waiting for the next sampling time - you might be exerting more control effort than you need, and you can get the possibility of oscillations.

This link will take you to a simulator for a digital control system where you can examine those effects. That link will also show a prototype digital control system.