Signal Processing for Control - Signal Processing Systems - page 265

Fig. 5 The plant and
controller. Notice that the
exogenous input of Fig. 5 has
been split into two parts with
d c denoting the exogenous
input to the controller and d p
that to the plant

d p

z

d

c

Controller

Plant

y

u

k p

u

d c

Proportional
Gain

k d d/dt

y

Derivative
Gain

1
s

k i

Integral Gain

Fig. 6 A simple academic PID Controller

some form of inertia. Thus, control systems are only required to come close to
this objective. There are a number of precise definitions of the term “close.”
In the simplest cases, the requirements/specifications are that the closed-loop system
be asymptotically stable and that | y ( t ) − d c ( t ) |< ε ∀ t ≥ T ≥ 0, where T is

some specified “settling time” and is some allowable error. In more demanding

applications there will be specified limits on the duration and size of the transient
response [ 21 ] . An interesting additional specification is often applied when the plant
is approximately linear and time-invariant. In this case, an easy application of the
final value theorem for Laplace transforms proves that the error in the steady-state
response of the closed-loop system to a step input (i.e., a signal that is 0 up to t

ε

= 0

at which time it jumps to a constant value provided the

open-loop system (controller in series with the plant with no feedback) has a pole
at the origin. Note that this leads to the requirement/specification that the open-loop
system have a pole at the origin. This is achievable whereas, because of inaccuracies
in sensors and actuators, zero steady-state error is not.
In many cases the signal processing required for control is straightforward
and undemanding. For example, the most common controller in the world, used
in applications from the toilet tank to aircraft, is the Proportional + Integral +
Derivative (PID) Controller. A simple academic version of the PID Controller is
illustrated in Fig. 6 . Notice that this simple PID Controller is a signal processing
system with two inputs, the signal to be tracked d c and the actual measured
plant output y and one output, the control signal u . This PID controller has
three parameters that must be chosen so as to achieve satisfactory performance.

α ) goes to zero as t → ∞