Colorado School of Mines CHEN403
Direct Synthesis Controller Tuning
First Order Process............................................................................................................................................... 2
Second Order Process.......................................................................................................................................... 3
Third Order & Higher Processes ..................................................................................................................... 3
Processes with Dead Time................................................................................................................................. 4
FOPDT Example ..................................................................................................................................................... 7
Direct synthesis methods are based upon prescribing a desired form for the system’s response and then finding a controller strategy & parameters to give that response.
L
′ G
L
Y sp
′
+
-
ε ′
G c
P
′
G v
M
′
G p
+
+
Y
′
Y
′ m
G m
For the feedback control loop above the overall transfer functions between the output Y
′ and the set point & disturbance are:
Y
′
Y sp
′
=
G G G c v p
1
+
G G G G c v p m and:
Y
L
′
′
=
1
+
G
L
G G G G c v p m
.
Note that for the set point transfer function, we can manipulate it to give:
G c
=
G G v p

1
(
Y Y sp
−
(
)
G m
′ ′ sp
)
.
One implication is that if we pick a desired form for the response to a set point change,
′ ′ sp
, then we have set out the desired form for the controller. For example, we might think that it would be great to have the output to immediately track the set point change,
Direct Synthesis Controller Tuning - 1 - December 21, 2008

Colorado School of Mines CHEN403 i.e., Y
′ =
Y
′ sp
. However, a more practical response would be a first order decay into the final value, or:
Y
Y sp
′
′
= c s
1
τ +
1
.
We can get this type of response with a controller strategy of the form:
G c
=
G G v p



1


 c s
1
τ +
−
G m


1



1
τ + c s 1





=
G G v p
[ c s
1
G m
]
.
The direct synthesis equations are shown for
′ ′ sp
but they really make more sense for looking at the dynamic response of the measured variable, Y m
′
Y sp
′
. Then the transfer function with respect to the set point and the implied controller strategy is:
Y m
′
Y sp
′
=
G G G G m c v p
1
+
G G G G c v p m
⇒
G c
=
G G G v p m
(
Y Y m
′ ′ sp

1
−
(
)
Y Y m
′ ′ sp
) (
1
G G G v p m
) ( )
.
Note that the product of the three transfer functions is simply the open-loop transfer function, represented usually as G
OL
or some approximation of the actual transfer function, ɶ :
Y m
′
Y sp
′
=
1
+ c c
⇒
G c
=
1
(
Y Y m
′ ′ sp
−
(
)
Y Y m
′ ′ sp
)
1 ɶ
1
G
τ c s
.
Notice that this shows there is an integral action to the Direct Synthesis controller strategy.
First Order Process
Let’s look at the Direct Synthesis controller strategy for a first order process:
=
K p
τ + p s 1
.
Then the controller strategy is:
Direct Synthesis Controller Tuning - 2 - December 21, 2008

Colorado School of Mines CHEN403
G c
=
1
K p s 1
⋅
τ
1 c
=
τ + p s 1
= s K s K
τ p

 1
+
1
τ p s

 .
Notice that this is simply PI control with the settings:
K c
=
τ p
K
τ p c
and
τ = τ
I p
.
Second Order Process
Let’s create a 2 nd order process by assuming the process & actuator (valve) are both first order. Then:
= p
K s p
1
⋅ v
K s v
τ + τ +
1
G c
= s
K p
1
⋅ s
K v
1
⋅
τ
1
= c s
τ τ p v s
2
( p v
) s
+
1
K K
τ p v c s
=
τ + τ v
K K
τ p v c

 1
+
1
⋅ +
τ + τ p v s
τ + τ p v
⋅ s

 .
Notice that this is PID control with the settings:
K c
=
τ + τ p v
K K
τ p v c
,
τ = τ + τ
I p v
, and
τ =
D
τ τ p v
τ + τ p v
.
Further note that integral & derivatives times are only dictated by the process & not the desired controller settling time.
Third Order & Higher Processes
Processes that are 3 rd order and higher will lead to controller strategies that are different from PID controllers. However we can get PID controller settings by eliminating terms that correspond to high order derivatives. For example:
=
K a s n + a s n n
− 1 n
−
1 +
⋯
+ a
0
Direct Synthesis Controller Tuning - 3 - December 21, 2008

Colorado School of Mines CHEN403
G c
= a s n n + a n − 1 s n − 1
K
+
⋯
+ a
0 ⋅
τ c
1 s
=
= a s n n + a n − 1 s n − 1
K
τ c s
+
⋯
+ a
0
K a
1

 1
+ a
0
1 a
2 s a
3 s ⋯
+ a n ⋅ a s a
1 1 a
1 a
1 s n − 1



So the appropriate PID control settings are:
K c
= a
1
K
τ c
,
τ = a
1 a
0
, and
τ = a
2 a
1
.
Again note that integral & derivatives times are only dictated by process parameters & not the desired controller settling time.
Processes with Dead Time
Let’s consider a 1 st order system with dead time (FOPDT):
=
K e p
−θ p s
τ + p s 1
.
If we keep the same desired form for the response to a set point change, then the controller strategy is:
G c
=
τ + p
K e p s
−θ p
1 s
⋅
τ
1 c s
=
K
τ p

 1
+
1
τ p s

 e
θ p s
.
This does not correspond to a standard controller strategy (because of the exponential term). In fact, this exponential term shows a requirement for prediction of what is going to happen, so this cannot be physically realized in a controller strategy.
If we really wanted to do this though, we could approximate the exponential term with a truncated Taylor series expansion: e
θ p s p s then:
Direct Synthesis Controller Tuning - 4 - December 21, 2008

Colorado School of Mines CHEN403
G c
≈
K
τ p

 1
+
1
τ p s


(
1
+ θ p s
)
=
K
1 
 p p s p p s


=
τ + θ p p
K

 1
+
1
τ + θ p
⋅ + s
τ θ p p
τ + θ p
⋅ s


.
Now we have PID control with the settings:
K c
=
K
τ p c p
,
τ = τ + θ
I p p
, and
τ =
D
τ θ p
τ + θ p p
.
However, we could change our desired form for the response to a set point change to something involving the same process dead time: m
′
Y sp s
−θ p s
Y
′
= e
λ +
1 then:
G c
=

1
(
Y Y m
′ ′ sp
−
(
)
Y Y m
′ ′ sp
)
G s c and for this FOPDT process:
G c
= s
K e p
1
−θ p s
⋅ s e
−θ p s e
−θ p s e
−θ p s e
−θ p s
= s
K p
(
τ + − c s
1 e
−θ p s
) .
Now we can make a truncated Taylor series approximation: e
−θ p s ≈ − θ p s to get:
G c
≈
K p
( c s
τ + p s 1 p
τ + p s 1 s
)
=
K p
(
τ + θ c p
) s
=
K p
(
τ c p p
)


1
+
1
τ p s


Direct Synthesis Controller Tuning - 5 - December 21, 2008

Colorado School of Mines CHEN403 and we’re back to having PI control with the settings:
K c
=
K p
τ p
( p
) and
τ = τ p
.
We used a simple truncated Taylor series expansion for the dead time term here. What happens if we use a Padé approximation instead? Using a 1 st order Pade approximation the controller strategy for a FOPDT is:
G c
G c
G c
G c
=
=
=
=
K
K
K p p


( s
)
K
( p p


τ + θ
2 p
) s 1
2


(
(
τ + θ
τ + θ
2 p
2
) s p
)
1 s
2
1
2 c p s
2 p p
− s
2
1
− θ
2 p p
(

 c s
τ + p s 1
− θ
2 p
(
τ + p s
)
1
1
(
+ θ
2
( p s s


+ θ
2
) ( p s
)
+ θ
2 p s
−
1
− θ
2 p s
) p p


(
τ + θ p
) s
+ τ θ
2 c p s
2 s
2


) ( ) s


If we assume that the dead time is much less than the controller settling time (
θ << τ c then we can neglect the quadratic term in the denominator.
)
G c
=
(
τ + θ p 2
K p
) p s 1
(
τ + θ c p
)
2 s p p s
2
G c
=
K p
τ + θ p 2
(
τ + θ c p p
) 



1
+ (
1
τ + θ
2 p
) s
+
τ θ p
2
τ + θ p p
This is in the form of a PID controller where: s




K c
=
K p
τ + θ p 2 p
(
τ + θ c p
) ,
τ = τ + θ
I p 2 p
, and
τ =
D
τ θ p
2
τ + θ p p
.
Direct Synthesis Controller Tuning - 6 - December 21, 2008

Colorado School of Mines CHEN403
To get the form reported by Smith & Corropio we have to make the further assumption that the dead time is also much less than the process time constant (
θ << τ p
):
G c
=
K p
(
τ c p p
)



1
+
τ
1
+ p s
θ
2 p s

 and this is still in the form of a PID controller but now the settings are:
K c
=
K p
τ p
( ) p
,
τ = τ
I p
, and
τ = θ
D 2 p
.
FOPDT Example
Let’s look at the response to a unit step change in the set point for a FOPDT process where
K p
=
0.25
, K
L
=
0.75
,
τ = τ = p L
10 , and
θ = θ = p L
1 .
Though it’s not needed for the Direct Synthesis method, let’s calculate the stability limit for
P-control. Under P-control the characteristic equation is:
1
+
G G G G c v p m
=
0
⇒
1
+
K c
K e p
−θ p s s 1
=
0 p s 1 K K e c p
−θ p s =
0 .
The stability limit can be determined by the direct substitution method: p u j K K e cu p
−θ ω u j =
0 p u j 1 K K cu p cos
( ) j sin
(
θ ω u
)

=
0 which leads to the two simultaneous equations:
1
+
K K cu p cos
(
θ ω = p u
)
0
τ ω − p u sin
(
θ ω p u
)
K K cu p
0

⇒ 



 K cu
= −
K p cos
1
(
θ ω p u
) tan
( )
0
The 2 nd equation must be solved numerically. For this particular problem the equation & solution are:
10
ω + u tan
( )
0
⇒ ω = u
1.632
⇒
P u
=
2
π
ω u
=
3.85
and K cu
=
65.40
.
Direct Synthesis Controller Tuning - 7 - December 21, 2008

Colorado School of Mines CHEN403
The following table gives controller settings from various methods. The response time is considered that time at which the final response stays within ±5% of the final value (0.95 to
1.05). Three of the responses are depicted in the figure following this table.
K c
τ
I
τ
D
Slight overshoot – max value
Direct Synthesis -
τ =
1
Direct Synthesis -
τ =
0.5
Direct Synthesis -
τ =
2 c
13.3
ZN Some Overshoot
20
26.7
Original Zeigler-Nichols 29.7
Original Zeigler-Nichols 38.5
21.6
10
10
10
3.2
1.9
1.9
—
—
—
—
0.5
1.3
1.04. Response time is actually before the first peak, about 3.4 min.
Even larger overshoot – max value 1.2. Response time is after the first peak, about 5 min.
No apparent overshoot –
Response time about 6.4 min.
Large overshoot – max value about 1.6. Response time after
1 st trough, about 8 min.
Multiple harmonics at early times. Again large overshoot – max value about 1.5. 1 st trough just outside range – response time about 8 min.
Multiple harmonics over an extended period of time.
Response time about 15 min.
ZN No Overshoot
Rule of Thumb
14.4
32.7
1.9
3.9
1.3
Long, broad cycling. Response time about 19 min.
τ =
1.32
Iu
. Large overshoot – max value 1.6. Response time 9 min.
Direct Synthesis Controller Tuning - 8 - December 21, 2008

Colorado School of Mines CHEN403
1.0
0.8
0.6
0.4
1.6
1.4
1.2
0.2
0.0
-0.2
0 5
Original ZN PI
"Rule of Thum b" PI
Direct Synthesis, λ =1
10
Tim e (t)
15 20 25
Direct Synthesis Controller Tuning - 9 - December 21, 2008