Control Systems Intro - ON/OFF Control Systems

In many applications, it is not enough to be able to measure a quantity.  Often you want to control a quantity.  Either you want the quantity (for example, temperature) to stay constant at some fixed value (like you expect when you set the wall thermostat) or you want to make the quantity vary in some predetermined fashion (for example, making an antenna point to a satellite as it crosses from horizon to horizon).  Consider what you need to do to control a quantity.  We'll use temperature as an example, but it could be any physical quantity.

• First, if you are trying to control a temperature, you need to know what the temperature is.
• Thus, the first thing you have to do to control a quantity (temperature) is to measure that quantity to see what value it has.
• Secondly, after you measure the temperature, you have to compare the measured temperature with the temperature you want to have - otherwise known as the desired temperature.  Is it too high?  Is it too low?  How much higher (lower) is it compared to what you want?
• Finally, after you have compared where you are at (i.e. measured the temperature) and compared it with what you want (i.e. the desired temperature) you should take control action.  There are numerous ways you can control a variable.
Some of the ways that you can control a variable include the following.
• You can turn a controller ON or OFF.  For example, the wall thermostat measures a room temperature and compares it to a set temperature (the desired temperature).  Then, if the temperature is too high, the thermostat turns the furnace/heater OFF.  If the temperature is too low, the thermostat turns the furnace/heater ON.
• You might make the control effort (the amount of fuel fed to the furnace, for example) proportional to the error (the difference between where you are - the measured temperature - and where you want to be - the desired temperature.
• And, there are other algorithms that use the error, or some function of the error (like the integral of the error, for example) to calculate the control effort.
There is one fly in the ointment.  In the schemes above, the control effort should really take into account the dynamics of the system you are trying to control - but that's a more advanced topic.

When you build a control system it is helpful to have a picture of how the signals are produced and affect things in your system.  Control system designers usually use a block diagram to show how signals flow in a control system.  Here is a block diagram of a typical control system.

Interpreting this block diagram, we have the following.

• The output of the system is y(t).
• The output is measured and fed back to be compared with the input, u(t).
• The result of the comparison is an error signal, e(t) = u(t) - y(t).
• The error signal is multiplied by a constant, K, to produce the control effort, c(t).
• K is referred to as the proportional gain.
• The control effort, c(t), is applied to the system.
• The system is represented here with a transfer function, G(s).
• Using a transfer function representation is done to remind us that the system will have dynamics, and that applying a control effort will not result in an immediate response in the output.  Rather, we have to wait for the system to respond to the input.

Simulating an ON-OFF System.

We have simulators available for ON-OFF control.  The first simulator controls a second order system.  You can use those simulators to gain some insight into the behavior of ON-OFF control systems.

Second Order System
• First, get the simulator for a second order system.  Click here to get the simulator.
• Run the simulator an observe what happens.  You should note the following.
• Initially, the simulator turns the control to ON, and the output of the system begins to increase.
• When the system output reaches the desired value the control turns OFF, and the system output begins to drop.
• As time goes on, the system oscillates around the desired output.

Third Order System
• First, get the simulator for a third order system.  Click here to get the simulator.
• Run the simulator an observe what happens.  You should note the following.
• Initially, the simulator turns the control to ON, and the output of the system begins to increase.
• When the system output reaches the desired value the control turns OFF, and the system output begins to drop.
• As time goes on, the system oscillates around the desired output.  However, because there are more time constants in this system, the system does not respond in the same way as the second order system.