Page 75 - Understanding Automotive Electronics

2735 | CH 2 Page 62 Tuesday, March 10, 1998 10:55 AM

2 THE SYSTEMS APPROACH TO CONTROL AND INSTRUMENTATION

dynamic response is also affected by G. Raising the gain too far will cause the
system response to become oscillatory. Often if the gain is too large, the system
becomes unstable. This property of having a steady error in response to a
steady disturbance is fundamental to all control systems incorporating a
proportional controller, but can be eliminated by use of a proportional-integral
controller.
Concept of Integration
Before beginning the discussion of proportional integral (PI) control, it is
worthwhile to discuss the concept of integration briefly. Of course those readers
having a background in and familiarity with integral calculus can skip this
discussion. The concept of an integrator can perhaps best be understood with
reference to the block diagram of Figure 2.25a. In mathematical terms this
system is denoted:

d
y = ∫ x t

and the output y is said to be the integral of the input x with respect to time. In
more practical terms the integrator can be thought of as a device that
continuously “adds” or accumulates the input such that the input is the rate of
change of the output.
A good practical example of an integrator is depicted in Figure 2.25b. In
this figure the integrator is a storage tank into which fluid is flowing. The
output for this example is the total volume of fluid that has accumulated at any
time (until the tank is full). The input to this integrator is the volume flow rate
x of fluid flowing into the tank. For example, if the volume flow rate into a tank
were 10 gal/min then every minute the volume of fluid in the tank (y) would
increase by 10 gallons, that is to say, the volume of fluid in the tank would be
10 multiplied by the time (from empty) that the fluid flows into the tank. This
is a rather straightforward concept when the input is constant. However, if the
input volume flow rate changes continuously, then the volume of fluid is given
by the integral (with respect to time) of the input flow rate.
Another device that acts as an integrator is a capacitor into which a
current is flowing, as depicted in Figure 2.25c. The charge stored in the
capacitor Q is the integral with respect to time of the current:

Q = ∫ i t
d
This property of a capacitor is used to implement the integral part of an
analog proportional integral control system. (See the discussion in Chapter 3 of
operational amplifiers.)

62 UNDERSTANDING AUTOMOTIVE ELECTRONICS