1. Inverse Response Systems
rev. 5.5 of May 4, 2017 by Michele MICCIO
see also §12.3 and §19.3 in Stephanopoulos, “Chemical process control: an Introduction to theory and practice”, ISBN , Prentice Hall, 1983 1

2. 2nd order systems with numerator dynamics
Problem
For the case of a transfer function with:
a single zero
a positive-gain,
overdamped second-order
where τa is called lead time and in general may be >0 or <0
calculate and plot the response to the step input of magnitude M.
Solution
The response of this system to a step change in input of magnitude M is:
Chapter 6 of Seborg (2nd ed.) 2 2

3. 2nd order systems with numerator dynamics: step response
Solution
Parametrically in τa, the plot of the response is:
Hypothesis:
1 > 2 > 0
overshoot
The
1st order like
inverse response 3 3

4. 2nd order systems with numerator dynamics: step response
Note that as expected.
Hence, the effect of including the single zero does not change the final value nor does it change the number or location of the response modes. But the zero does affect how the response modes (exponential terms) are weighted in the solution, Eq
A certain amount of mathematical analysis (see Exercises 6.4, 6.5, and 6.6 of Seborg) will show that there are three types of responses when 1 > 2 > 0:
Case a: τa > τ1 overshoot
Case b: < τa ≤ τ1 “conventional” overdamped response
Case c: τa < 0 “inverse” response
Chapter 6 of Seborg (2nd ed.) 4 4

5. Inverse-response pattern
There is an initial inversion in the response: the process starts moving away from its ultimate value
The process output eventually heads in the direction of the final steady state
This behavior is called inverse response or non-minimum phase response. 5 5

6. Inverse-response pattern
Inverse response is the net result of two
i) opposing dynamic modes of
ii) different magnitudes, operating on
iii) different time scales
the faster mode has a small magnitude and is responsible for the initial, “wrong way” response
the slower mode has a larger magnitude and is responsible for the long-term, dominant response
They usually originate from a "parallel"
GP1
GP2 + -
m(s)
y(s) 6 6

7. Example process: drum boiler (corpo cilindrico di caldaia)
Disturbance :
cold feedwater flowrate (e.g., step increase)
Manipulated variable :
heating rate
Output :
level in the drum boiler
Cold feedwater
Steam
Heating power
In the long run, the level is expected to increase, because we have increased the feed material without changing the heat supply
But immediately after the cold feedwater flowrate has been increased, a drop in the drum liquid temperature is observed, which causes the bubbles to collapse and the observed level to reduce
See also:
Ex.12.3 pag Stephanopoulos, “Chemical process control: an Introduction to theory and practice”, Prentice Hall, 1984 7 7

8. Inverse-response systems
In summary:
“strange”, unexpected shape of the open-loop response, which is non monotone
potential of closed-loop instability due to the presence of a right half-plane (RHP) zero o x
z = −1/τa 8 8

9. Overdamped Process with Inverse Response+ -
m(s)
y(s)
Introduction to Process Control Romagnoli & Palazoglu 9 9

10. Overdamped Process with Inverse Response
 When the system exhibits inverse response, its transfer function has a positive zero + -
m(s)
y(s)
Introduction to Process Control Romagnoli & Palazoglu 10 10

11. Overdamped Process with Inverse Response: case 1 with positive gains
0 < k1 < k2  k1 - k2 < 0  k1/k2 < 1
Hyp.:
Under this condition, the overall transfer function has:
a positive zero given by
and a negative gain given by
(k1 - k2)
 This process exhibits inverse response
Introduction to Process Control Romagnoli & Palazoglu 11 11

12. Overdamped Process with Inverse Response: case 2 with positive gains
k1 > k2 > 0  k1 - k2 > 0  k1/k2 > 1
Hyp.:
Under this condition, the overall transfer function has:
a positive zero given by
and a positive gain given by
(k1 - k2)
 This process exhibits inverse response
Introduction to Process Control Romagnoli & Palazoglu 12 12

13. Overdamped Process with Inverse Response: case 3 with negative gains
k1 < k2 < 0  k1 – k2 < 0  k1/k2 – 1 > 0  k1/k2 > 1
Hyp.:
Under this condition, the overall transfer function has:
a positive zero given by
and a negative gain given by
(k1 - k2)
 This process exhibits inverse response
Introduction to Process Control Romagnoli & Palazoglu 13 13

14. Overdamped Process with Inverse Response: case 4 with negative gains
0 > k1 > k2  k1 – k2 > 0  k1/k2 – 1 < 0  k1/k2 < 1
Hyp.:
Under this condition, the overall transfer function has:
a positive zero given by
and a positive gain given by
(k1 - k2)
 This process exhibits inverse response
Introduction to Process Control Romagnoli & Palazoglu 14 14

15. Overdamped Process with Inverse Response:
Summary
To summarize,
just two conditions involving the parameters, if fulfilled, determine the occurrence of a positive zero in the transfer function:
and, in parallel, an inverse open loop response
Introduction to Process Control Romagnoli & Palazoglu 15 15

16. The Overdamped Process with Inverse Response in MatLab®
script file Inverse_Response.m
% Inverse Response caused by two 1st order systems in parallel
% rev. 2.6 by M.Miccio on April 21, 2016
. . .
disp ('Inverse Response caused by two 1st order systems in parallel.')
disp ('The two 1st order systems have positive gains. The positive ZERO exists if')
disp (' case 1: tau1/tau2 < K1/K2 < > steady-state gain < 0')
disp (' case 2: tau1/tau2 > K1/K2 > > steady-state gain > 0')
disp ('For simplicity, the 2nd transfer function is assigned K2 = tau2 = 1') 16

17. Dynamic linear systems exhibiting Inverse Response
 The previous case
is NOT discussed on Stephanopoulos
k2τ1 < k1 v.
§ 12.3
Stephanopoulos 17 17

18. Feedback Control of Inverse Response Systems
PID controller with suitable tuning
open loop tuning
2nd Ziegler-Nichols method tuning
Inverse Response Compensator 18

19. 1. PID controller with suitable tuning
GPID(s)
m(s)
ySP(s)
(s) +
y(s) − 19
Introduction to Process Control Romagnoli & Palazoglu 19

20. PID controller with adapted open loop tuning
Process reaction curve method
A FOPDT model is adopted, by fitting the step response after “ignoring” the inverse piece of the response curve and considering it like a dead time
 Ch. 6
D. Cooper, "Practical Process Control using Loop-Pro Software", PDF textbook 20 20

21. PID controller with closed loop tuning
Example 1
co= 0.55 radians/time
MR = AR/KKc = 0.5
ultimate gain: Ku =1/MR = 2
natural period:
Controller Kc i D P
0.5Ku PI
0.4Ku
0.8Pu
PID
0.6Ku
0.5Pu
0.125Pu
2nd Ziegler-Nichols method: Kc=1.2; I =5.7 and D=1.4 21 21

22. 2. Compensation
 §19.3
Stephanopoulos, “Chemical process control: an Introduction to theory and practice”, Prentice Hall, 1984 G c
(s)
ySP(s)
y(s) + −
(s)
Controller Mechanism
m(s)
y'(s)
Li Shaoyuan
 Minor loop provides a corrective signal to eliminate inverse response from the feedback loop.
GCOMP(s) always exhibits a zero in the origin!
Inverse response compensator
GCOMP(s)
Inverse response process 22
Introduction to Process Control Romagnoli & Palazoglu 22

23. Compensation 23 M(s)=y(s)/Ginv(s)
Introduction to Process Control Romagnoli & Palazoglu 23

24. Inverse response process
Compensation
 §19.3
Stephanopoulos, “Chemical process control: an Introduction to theory and practice”, Prentice Hall, 1984
Gcontr-mech(s)
m(s)
ySP(s)
(s) +
y(s) −
Controller Mechanism
Gcontr-mech(s)
Inverse response process
After compensation it is …
GOL(s) = Gcontr-mech(s)•GINV_RESP(s) 24
Introduction to Process Control Romagnoli & Palazoglu 24

25. Compensation The compensated open-loop transfer function GOL(s) is
In the case: K1 > K2 > 0
the zero of the resulting open-loop transfer function is non-positive for:
M(s)=y(s)/Ginv(s)
Rule of thumb:
(from Li Shaoyuan 25
Introduction to Process Control Romagnoli & Palazoglu 25