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 0131286293, Prentice Hall, 1983 1

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

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

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. 6-15. 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: 0 < τa ≤ τ1 “conventional” overdamped response Case c: τa < 0 “inverse” response Chapter 6 of Seborg (2nd ed.) 4 4

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

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

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. 216 - Stephanopoulos, “Chemical process control: an Introduction to theory and practice”, Prentice Hall, 1984 7 7

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

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

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

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

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

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

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

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

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 < 1 --> steady-state gain < 0') disp (' case 2: tau1/tau2 > K1/K2 > 1 --> steady-state gain > 0') disp ('For simplicity, the 2nd transfer function is assigned K2 = tau2 = 1') 16

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

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

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

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

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

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

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

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

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 syli@sjtu.edu.cn) 25 Introduction to Process Control Romagnoli & Palazoglu 25