PROPORTIONAL
INTEGRAL
AND
DERIVATIVE
CONTROLLER
Unit step Definitions of Transient-Response Specifications
The time-domain specifications just given are quite important,
since most control systems are time-domain systems; that is,
they must exhibit acceptable time responses.
(This means that, the control system must be modified until the
transient response is satisfactory.)
R e u Y
+ controller Plant
-
220 volt
Proportional, derivative and integral controller
This introduction will show you the characteristics of the each
of proportional (P), the integral (I), and the derivative (D) controls, and how to use them
to obtain a desired response.
we will consider the following unity feedback system:
R e u Y
+ controller Plant
-
Plant: A system to be controlled
Controller: Provides the excitation for the plant;
Designed to control the overall system behavior
Proportional, derivative and integral controller
Proportional controller:
A simple form of control where the controller response is proportional to control
error ‘e’. It is the linear relationship between controller out put and error ‘e’.
OR:
The change in controller output from the set point (desired) value is proportional
to error, given as:
u e(t ) u = K p e(t ) where Kp is proportional constat
R e u Y
controller Plant
+
-
Proportional, Derivative and integral controller
Derivative controller
The change of controller output from the set value is proportional
to the rate of change of error with respect to time.
de de
u u = Kd where K d is derivative constat
dt dt
R e u Y
controller Plant
+
-
Proportional, derivative and Integral controller
Integral controller
In a controller with integral action, the value of controller output is change at the
rate of
proportional to the actuating error signal
du
e u e.dt u = K i e.dt where Ki is inegral constat
dt
R e u Y
controller Plant
+
-
Proportional, derivative and integral controller
Proportional controller:
A simple form of control where the controller response is proportional to control error ‘e’. It is the linear
relationship between controller out put and error ‘e’.
OR: The change in controller output from the set point (desired) value is proportional to error, given as:
u e(t ) u = K p e(t ) where Kp is proportional constat
Derivative controller
The change of controller output from the set value is proportional to the rate of change
of error with respect to time.
de de
u u = Kd where Kd is derivative constat
dt dt
Integral controller R Y
In a controller with integral action, the value e u
controller Plant
of controller output is change at the rate of +
proportional to the actuating error signal -
du
e u e.dt u = K i e.dt where Ki is inegral constat
dt
Proportional plus derivative plus integral controller
The combination of proportional action control, integral action control and
derivative action control is termed as:
Proportional plus Derivative plus Integral controller PID controller
or
three term controller.
The combined action has the advantage of these individual control actions
de
u = K pe + K i e.dt + K d
dt
The transfer function of the PID controller looks like the following:
Ki K d s 2 + K p s + Ki
u (s) = K p + + Kd s =
s s
Kp = Proportional gain
Ki = Integral gain
Kd = Derivative gain
The three-term controller
First, let's take a look at how the PID controller works in a closed-loop
system using the schematic shown above.
The variable (e) represents the tracking error, the difference between
the desired input value (R) and the actual output (Y).
This error signal (e) will be sent to the PID controller, and the
controller computes both the derivative and the integral of this
error signal.
R e u Y
+ PID controller Plant
-
The three-term controller
The signal (u) just past the controller is now equal to the:
proportional gain (Kp) times the magnitude of the error
plus the integral gain (Ki) times the integral of the error
plus the derivative gain (Kd) times the derivative of the error.
de
u = K p e + K i e.dt + K d
dt
This signal (u) will be sent to the plant, and the new output (Y) will be obtained.
This new output (Y) will be sent back to the sensor again to find the new error signal (e).
The controller takes this new error signal and computes its derivative and its
integral again.
This process goes on and on.
The characteristics of P, I, and D controllers
A proportional controller (Kp) will have the effect of:
reducing the rise time ,
but
never eliminate, the steady-state error.
The characteristics of P, I, and D controllers
An integral control (Ki) will have the effect of:
eliminating the steady-state error,
but it may
make the transient response worse.
The characteristics of P, I, and D controllers
A derivative control (Kd) will have the effect of:
increasing the stability of the system,
reducing the overshoot, and improving the transient
response.
Effects of each controllers Kp, Ki and Kd on closed loop system
CL RESPONSE RISE RIME OVER SHOOT SETTLING TME S-S ERROE
Kp Decrease Increase Small change Decrease
KI Decrease Increase Increase Eliminate
Kd Small change Decrease Decrease Small change
Note that these correlations may not be exactly accurate, because Kp, Ki,
and Kd are dependent of each other.
In fact, changing one of these variables can change the effect of the other
two.
For this reason, the table should only be used as a reference when you are
determining the values for Kp, Ki, and Kd.
Effects of each controllers Kp, Ki and Kd on closed loop system
CL RESPONSE RISE RIME OVER SHOOT SETTLING TME S-S ERROE
Kp Decrease Increase Small change Decrease
KI Decrease Increase Increase Eliminate
Kd Small change Decrease Decrease Small change
