### PID Controllers - University of Toledo

```Professor Walter W. Olson
Department of Mechanical, Industrial and Manufacturing Engineering
University of Toledo
PID Control
Outline of Today’s Lecture
 Review
 Margins from Nyquist Plots
 Margins from Bode Plot
 Non Minimum Phase Systems
 Ideal PID Controller
 Proportional Control
 Proportional-Integral Control
 Proportional-Integral Derivative Control
 Ziegler Nichols Tuning
Margins
 Margins are the range from the current system design to the edge
of instability. We will determine
 Gain Margin
 How much can gain be increased?
 Formally: the smallest multiple amount the gain can be increased before the
closed loop response is unstable.
If the gain margin is expressed in dB, then the multiple gain is G  100.05gm
 Phase Margin
 How much further can the phase be shifted?
 Formally: the smallest amount the phase can be increased before the closed
loop response is unstable.
 Stability Margin
 How far is the the system from the critical point?
Gain and Phase Margin Definition
Nyquist Plot
1

gm
-1
m
Magnitude, dB
Gain and Phase Margin Definition
Bode Plots
0
Positive Gain Margin
Phase, deg
w
-180
Phase Margin
w
Phase Crossover Frequency
Stability Margin
 It is possible for a system to have relatively large gain and
phase margins, yet be relatively unstable.
Stability
margin, sm
Non-Minimum Phase Systems
 Non minimum phase systems are those systems which have poles
on the right hand side of the plane: they have positive real parts.
 This terminology comes from a phase shift with sinusoidal inputs
s 1
s 1
G
(
s
)

and
G

2
 Consider the transfer functions 1
s  s  2
s( s  2)
 The magnitude plots of a Bode diagram are exactly the same but the phase
has a major difference:
Another Non Minimum Phase System
A Delay
 Delays are modeled by the function which multiplies the T.F.
y (t )  x (t  T )
G( s)  eTs
Proportional-Integral-Derivative
Controller
Based on a survey of over eleven thousand controllers in the refining,
chemicals and pulp and paper industries, 97% of regulatory controllers
utilize PID feedback.
L. Desborough and R. Miller, 2002 [DM02].
 PID Control, originally developed in 1890’s in the form of motor
 The first theory of PID Control was published by a Russian
(Minorsky) who was working for the US Navy in 1922
 The first papers regarding tuning appeared in the early 1940’s
 Today. there are several hundred different rules for tuning PID
controllers (See Dwyer, 2003, who has cataloged the major methods)
 While most of the discussion is about the “ideal” PID controller,
there are many forms of the PID controller
PID Control
 Process independent
 The best controller where the specifics of the process can not be modeled
 Leads to a “reasonable” solution when tuned for most situations
 Inexpensive: Most of the modern controllers are PID
 Can be tuned without a great amount of experience required





Not optimal for the problems
Can be unstable unless tuned properly
Not dependent on the process
Hunting (oscillation about an operating point)
Derivative noise amplification
The Ideal PID Controller
 The input/output realtionship for the PID Controller is the Integral-Differential
Equation

dy
1
u(t )  k p y (t )  kt  y ( )d  kd
 k p  y (t ) 
0
dt
Ti

 The ideal PID controller has the transfer function
t

0
t
y ( )d  Td


ki
1
CPID ( s)  k p   kd s  k p 1 
 Td s 
s
 Ti s

 Structurally it would look like
CPID ( s)
++
kp
ki
s
++
+
kd s
-1
P( s)
dy 

dt 
The Ideal PID Controller
 The system transfer function is
G( s) 
CPID ( s)
R(s)
++
Y ( s)
C ( s ) P( s )

R( s ) 1  C ( s ) P( s )
kp
ki
s
++
+
kd s
-1
Y(s)
P( s)
Proportional Control
Y ( s)
k P( s )
 d
R( s ) 1  k d P ( s )
CPID ( s)
kp
Y(s)
R(s)
++
ki
s
++
+
kd s
-1
P( s)
Proportional Control
Monahemi, 1992, reports that the angel of attack,  , to the elevator angle,  , for an F-16 is
0.0072  s  23  s  0.05s  0.04 
 ( s)
P( s ) 

 ( s)  s  0.7  s  1.7   s 2  0.08s  0.04 
2
This system is unstable.
Can it be stabilized with a proportional controller?
Proportional Controller
0.0072  s  23  s  0.05s  0.04 
 ( s)
P( s ) 

 ( s )  s  0.7  s  1.7   s 2  0.08s  0.04 
2
0.0072  s  23  s 2  0.05s  0.04  k p
k p P( s )
Y ( s)
G( s) 


R( s ) 1  k p P( s )  s  0.7  s  1.7   s 2  0.08s  0.04   0.0072  s  23  s 2  0.05s  0.04  k p
kp=7.2
Proportional Controller
 With kp=7.2,
}
Error
Also: the response time is poor
 We could reduce the error with a prefilter:
CPID ( s)
R(s)
0.139
++
kp
ki
s
++
+
kd s
-1
Y(s)=(s)
P( s)
Proportional – Integral Controller
 Most controllers using this technology are of this form:
ki 

k

 p
 P( s )
k p s  ki P ( s )
Y ( s)
s
 

k 
R( s )

1  k p  s  ki P ( s )
1   k p  i  P( s ) 
s

CPID ( s)
kp
Y(s)
R(s)
++
ki
s
++
+
P( s)
kd s
-1
 This reacts to the system error and reduces it
Proportion-Integral
Control
 Applying PI control to the F-16 Elevator,
 0.0072  s  23  s 2  0.05s  0.04  
k 

k p s  ki 

2
k p  i  P( s)



s

0.7
s

1.7
s

0.08
s

0.04





Y ( s)
s





2
k 
R( s)

 0.0072  s  23  s  0.05s  0.04  
1   k p  i  P( s)
1  k p  s  ki   s  0.7  s  1.7  s 2  0.08s  0.04 
s






k p s  s  0.7  s  1.7   s 2  0.08s  0.04   ki 0.0072  s  23  s 2  0.05s  0.04 
Y ( s)

R ( s ) 1  k p  s  s  0.7  s  1.7   s 2  0.08s  0.04   ki 0.0072  s  23  s 2  0.05s  0.04 
with C ( s )  100 
k p  100
ki  50
50
s


Response time
improved with
no error
Proportional- Integral-Derivative
Control
 The derivative component is rarely used.
 Reduces overshoot
 May slows the response time depending on the system
 Sensitive to noise
 For the F-16 Elevator
with C ( s )  250 
650
 25s
s
PID Tuning
 Tuning is the choosing of the parameters kd, ki, and kp, for a PID




Controller
The oldest and most used method of tuning are the ZieglerNichols (ZN) methods developed in the 1940’s.
The first method is based on the assumption that the process
without its feedback loop performs with a 1st order transfer
function, perhaps with a transport delay
The second method assumes that a higher order system has
dominant poles which can be excited by gain to the point of steady
oscillation
In order to establish the constants for computing the parameters
simple tests are performed of the process
Ziegler-Nichols PID Tuning
Method 1 for First Order Systems
 A system with a transfer function of the form P( s ) 
has the time response to a unit step input:
K  st0
e
sa
 This response might also be generated from a higher order
system that is has high damping.
Ziegler-Nichols PID Tuning
Method 1 for First Order Systems
 The advice given is to draw a line tangent to the response curve through the
inflection point of the curve.
 However, a mathematical first order response doesn’t have a point of inflection as it is of
the form y(t )  e at(at no place does the 2nd derivative change sign.) My advice:
place the line tangent to the initial curve slope
 You also have to adjust for the gain K of the system by multiplying compensator by 1/K
Rise
Time
T
Lag L
Type
P
PI
PID
kp
T
L
0.9T
L
1.2T
L
Ti
Td
3.33L
2L
0.5L
Ziegler-Nichols PID Tuning
Method 1 for First Order Systems
 For this example,
Rise
Time
T
Lag L
L
0.1
T
0.2

1 
CPI ( s )  k p  1 

 Ti s 
0.9  2  
1 
CPI ( s ) 
1 

0.1  3.33Ls 
1
CPI ( s )  1.8(1 
)
0.333s
Type
P
PI
PID
kp
Ti
0.9T
L
1.2T
L
3.33L
T
L
2L
Td
0.5L
Ziegler-Nichols PID Tuning
Method 2 for Unknown Oscillatory System
 The form of the transfer function unknown but the system can be put
in steady oscillation by increasing the gain:
 Increase gain,K, on closed loop system until the gain at steady
oscillation, Kcr, is found
 Then measure the critical period, Pcr
 Apply table for controller constants and multiply by system gain 1/K
Type
kp
Ti
Pcr
P
0.5Kcr
1
Cycle
PI
0.45Kcr
Pcr
1.2
PID
0.6Kcr
0.5Pcr
Td
0.125Pcr
Ziegler-Nichols PID Tuning
Method 2 Example
k=1000
k=1500
k=2000
17.8  1.0
Pcr 
 2.8 sec
6
Type
1
Cycle
Kcr=1875
1.0
17.8
kp
Ti
Td
0.5Kcr
P
Pcr
k=1750
PI
0.45Kcr
Pcr
1.2
PID
0.6Kcr
0.5Pcr
0.125Pcr
1.2 

CPI ( s)  0.45*1875  1 

 2.8s 
 0.4286 
CPI ( s)  843.75  1 

s 

Ziegler-Nichols PID Tuning
Method 2 Example
1.2 

CPI ( s)  0.45*1875  1 

 2.8s 
 0.4286 
CPI ( s)  843.75  1 

s 

Pcr
1
Cycle
Kcr=1875
1.0
17.8
Summary
 Ideal PID Controller
 Proportional Control
 Proportional-Integral Control
 Proportional-Integral Derivative Control
 Ziegler-Nichols Tuning
Next Class: PID Controls Continued
```