CONTROL LAB
Experiment no.-3
Objective:
To study the three term PID controller and its effects on the feedback loop response.
Investigate the characteristics of Proportional (P), Integral (I) and Derivative (D) Controls.
Apparatus/ Tools:
MATLAB, SIMULINK.
Theory:
PD, PI and PID controllers are the basic form of controllers widely used in industrial systems, as
they are relatively simple and are often able to provide good performance.
C(t)
Disturbance Peak Decrement = A/B
Overshoot (A)
1.05 (B) ess
1.0
r (t) e (t) Controller u (t) y (t) 0.95
Plant 0.9
Reference (PID) Output 0.5
Rise time tr
0.1
Delay time t d t
Feedback (1) Peak time t p
Settling time ts
Fig. 1 Block Diagram Fig.2 Typical Step response of a Control System
The control system block diagram consists of Plant, Controller and a feedback path. The plant is
desired to be run at the reference set point (SP) irrespective of disturbances. Feedback is
measurements of process value (PV) obtained through sensors, continuously being fed back to the
controller. The PV tries to follow the SP reference while minimizing the error supplied for the
controlling action (i.e. steady state error ess).
y( s)
Steady state error ess lim e(t ) lim se(s) where e(s) r ( s ) y ( s ) 1 r (s)
t s 0
r (s)
y(s) G( s)
For a unit feedback system, T ( s) which includes controller and plant together.
r ( s) 1 G( s)
s r (s)
ess lim se(s) lim lim s r (s) 1 T (s)
s 0 1 G ( s )
s 0
s 0
When this error signal is sent to the controller, it computes the proportional, integral and derivative
t
error correction terms given by the controller output u (t ) kP e(t) ki e(t)dt kd d
dt e(t) .
0
ki kd s2 k p s ki
i.e. TFController = kP s kd s s
This signal u(t) will be sent to the plant and the new output y(t) will be obtained and the process will
be repeated characterizing reduced error ess under all probable operational dynamics.
Here, Proportional Controller gain kp reduces the rise time but never eliminates steady-state error;
Integral Controller gain ki eliminates steady-state error but worsens transient response; Derivative
Controller gain kd improves transient response, reduces overshoot and imparts system stability.
1
CONTROL LAB
Plant:
x
b (10Ns/m)
m F mx bx kx F
(1kg) TFPlant
X (s)
F (s)
ms2 1bsk
k (20N/m)
Fig.3 Mass Spring and Damper System
Adding controllers to this plant,
y(s) kp
with P Controller, T ( s) r (s)
ms2 b s ( k k p )
y(s) kd s k p
with PD Controller, T ( s) r (s)
ms2 (b kd ) s ( k k p )
y(s) k p s ki
with PI Controller, T ( s) r (s)
ms bs2 ( k k p ) s ki
3
y(s) kd s2 k p s ki
with PID Controller, T ( s)
r (s) ms3 (b kd ) s2 ( k k p ) s ki
Observation:
From the plant step response, find out the ess, tr and ts values. [0.95, 1.0 S, 1.5 S approx]
Use different controllers to reduce those parameters.
Further observe the output responses by varying Kp, Ki, Kd gains of PD, PI and PID Controllers.
Changing Gain Rise time Overshoot Settling time SteadyState Error
Increase Kp decrease increase small change decrease
Decrease Ki increase decrease decrease eliminate (by tuning)
Increase Kd small change decrease decrease small change
However, these correlations may not be exactly accurately verifiable, as Kp, Ki, and Kd are inter-
dependent and changing one of these variables can change the effect of the other two.
Manual Tuning:
The proportional term intends to compensate the steady state error. Higher Kp speeds up the
response to error generally but can cause instability.
The Derivative term looks at the rate of change of error (how quickly the error term is changing) as
a way of predicting the future and reacting more urgently to rapid changes in the system’s
equilibrium. However, the sensor measured signals are generally noisy, and when differentiated the
noise is amplified. Therefore though Kd imparts stability, too much Kd will also over-compensate,
and the system will be unstable.
2
CONTROL LAB
Integral looks at how long the system has been away from equilibrium (its ‘set point’). The longer
the system is away from set-point, the more it responds - it accumulates error over time. Integral is
what can let the system operate much closer to equilibrium set-point and stay there (i.e. eliminate
error). But it can cause big oscillations if the system responds slowly (usually with smaller
proportional gain) and error builds up. Unlike the Proportional or Differential control, Integral
control doesn’t need a constant error term to operate.
-with slow or week output response, provide higher controller stiffness by increasing Kp.
-with oscillations, reduce it by decreasing Kp and Ki.
-with overshoot, restrict it by reducing Ki.
-with process receiving an external disturbance, the maximum deviation of the response from the set
point is reduced by Kd. Therefore Kd imparts efficiency and stability.
Simulation:
Graph:
P PD PI PID
Step Resp: (a)Kp=300 (b) Kp=300, Kd=10 (c) Kp=30, Ki=70 (d)Kp=350,Ki=200,Kd=25
Fig. 5. Step Response with different Controllers
Results and Comments:
Use at least 12 different combination of Kp, Ki, Kd values to comment on their influence on
desired response.
