Nonlinear Gain Compensation( 对象非线性增益补偿 )

Lei XieInstitute of Industrial Control,

Zhejiang University, Hangzhou, P. R. China

Problem Discussion for Nonlinearity Gain Compensation

(1) which kind of nonlinearity are there in the control loop for a heat exchanger? Give your methods to compensate there nonlinearity.

Steam

Condensate

ProcessStream

T1

T2

RV

RF

T2sp

TC

u (2) analyze why the PH reactor is one of the most difficulty controlled process

& provide improved schemes.

Heat Exchanger

Process Steady-state Model：

VVFp RHTTRc )( 12

FFp

V

Vp RRc

H

R

TK

12

Plant Gain:

Discussion: How to reduce the nonlinearity of process gain or make the compensated augmented plant become linear.

Steam

Condensate

ProcessStream

T1

T2

RV

RF

T2sp

TC

u

Compensation Method: Nonlinear Valves

VFVp RRR

TK

112

VV V

fK f

u

TC GV (s) GP (s)

GM (s)

u

+_

++ T2

T2sp

RV

Linear valve (线性阀 )

Equal-percentage valve( 等百分比阀或对数阀 )

V VR f

Vf cu

VV

ff

u

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

u

F/F

max

Linear valve

Equal-percentage valve

Characteristics for FC Valves

Linear valve

Vf cu

1

[1 ( 1) ]Vf R uR

Suppose 0 1

1, 1; 30V Vu u

f f RR

Equal-percentage valve

1uVf R 2

VV

fc f

u

How about FO valves ?

Compensation Method: Cascade Control

TCFC

Condensate

Steam

Process Stream

T2sp

T1

RV

T2

P1

RVsp

RF

FC

FFp

V

Vspp RRc

H

R

TK

12

Plant Gain:

Compensation Method: Ratio Control

TC

FC×

u

T2T1

Condensate

Process Stream

Steam

T2sp

RVsp

RF

For controller TC, the augmented controlled plant is nearly linear (why?)

Basic pH ControlAC

F2

F1

pHm

pHsp

Base

Acid

Neutralization Tank

Acid flow, % of Base

for a strong base-strong acid system

Existing problem ?

Base

Acid

Closed-loop Response for Basic pH Control

Experimental process

(1) pHsp: 6.5↑7.0 at 60 min

(2) F1: 30↓15 L/min at 110 min

(3) pH1: 5↓4.5 at 160 min

(4) pH2: 11↓10.5 at 210 min

pH Ratio Cascade Control

AC

F2

F1

pHm

pHsp

Base

Acid

u2

FC ×

NeutralizationTube

Advantages & Disadvantages ?

Base

Acid

Closed-loop Response for pH Ratio Cascade Control

Experimental process

(1) pHsp: 6.5↑7.0 at 60 min

(2) F1: 30↓15 L/min at 110 min

(3) pH1: 5↓4.5 at 160 min

(4) pH2: 11↓10.5 at 210 min

pH Control with Nonlinear Gain

F2

F1

pHm

pHsp

Base

Acid

F2sp

FC

NeutralizationTube

pHC

PID GFC (s) GP (s)

GM (s)pHm

+_

++pHsp pHF2

spF2

Nonlinear pH Controller

Problem: if the setpoint is not at pH=7.0, which will happen ?

Closed-loop Response for Nonlinear pH Control

Experimental process

(1) pHsp: 6.5↑7.0 at 60 min

(2) F1: 30↓15 L/min at 110 min

(3) pH1: 5↓4.5 at 160 min

(4) pH2: 11↓10.5 at 210 min

pH Control with Nonlinear Transformation

D

xsp

xm

PID GFC (s)

GM (s)pHm

+_

pHsppHF2

sp F2 pHNeutralization

Tubef(z)

f(z)

777 1010)(10 pHpHpHfHOHx

Note: x is nearly proportional to acid flow, F2, which results in a linear augmented plant. Besides, x is corresponding to pH. If pH=7, then x = 0; if pH>7, x > 0; and if pH<7, x < 0.

Closed-loop Response for pH Control

with Nonlinear Transformation

Experimental process

(1) pHsp: 6.5↑7.0 at 60 min

(2) F1: 30↓15 L/min at 110 min

(3) pH1: 5↓4.5 at 160 min

(4) pH2: 11↓10.5 at 210 min

Methods for Compensating Process Nonlinearity

Nonlinear Valves Cascade Control Variable Ratio Control Nonlinear Gain Compensation Nonlinear Transformation Identification + Adaptive

Control