multi-loop pi based control strategies on the activated ... fileapplication to the asp 5...

OutlineThe Activated Sludge Process

Multivariable PI control approachesDecentralized PI control approach

MIMO Decoupling control plus Decentralized PIDecentralised PI control with Feedforward decoupling action

Multi-loop PI based control strategies on the

Activated Sludge Process

Dr. Ramon Vilanova

Dept. Telecomunications and Systems EngineeringUniversitat Autonoma de Barcelona

[email protected]

(with Prof. Reza Katebi from ICC, University of Strathclyde, Glasgow, Soctland)

March 31, 2009

Dr. Ramon Vilanova Multi-loop PI based control strategies on the Activated Sludge Process




1 The Activated Sludge ProcessActivated Sludge ProcessNon-linear four state modelLinearized model

2 Multivariable PI control approachesMultivariable PI/PID methods

Application to the ASP

3 Decentralized PI control approachControl of TITO ProcessesApplication to the ASP

4 MIMO Decoupling control plus Decentralized PIThe control configurationApplication to the ASP

5 Decentralised PI control with Feedforward decoupling actionFeedforward controlFeedforward control with Decoupling purposesApplication to the ASP





Activated Sludge ProcessNon-linear four state modelLinearized model












Activated Sludge Process Non-linear four state model

dX

dt= µ(t)X − D(1 + r)X − rDXr

dS

dt= −

µ(t)

YX − D(1 + r)S + DSin

dDO

dt= −

Koµ(t)

YX − D(1 + r)DO + KLa(DOs − DO) + DOinDO

dXr

dt= D(1 + r)X − D(β + r)Xr

µ(t) = µmax

S

kS + S

DO

kDO + DO






Biomass X (0)=215 mg/lSubstrate S(0)=55 mg/lDissolved Oxygen DO(0)=6 mg/lRecycled Biomass Xr (0) = 400 mg/l

Table: Initial Contitions

β = 0.2 Kc=2 mg/lr = 0.6 Ks=100 mg/lα = 0.018 KDO=0.5Y = 0.65 DOs = 0.5 mg/lµmax = 0.15 h−1

Table: Kinetic parameters

G (s) =

(

G11(s) G12(s)G21(s) G22(s)

)

⇒

(

S(t)DO(t)

)

= G (s)

(

D(t)W (t)

)






The Gij(s) transfer function components are given as:

Linear Model Transfer functions

G11(s) =n11(s)

d(s)=

134.0243s3 + 295.3529s2 + 53.5176s + .5855

s4 + 2.4617s3 + 0.9859s2 + 0.1107s + 0.0008

G12 =n12(s)

d(s)=

−0.0312s2 − 0.0062s − 0.0001

s4 + 2.4617s3 + 0.9859s2 + 0.1107s + 0.0008

G21 =n21(s)

d(s)=

−9.2834s3 − 15.0312s2 − 2.6325s − 0.0123

s4 + 2.4617s3 + 0.9859s2 + 0.1107s + 0.0008

G22 =n22(s)

d(s)=

0.0699s3 + 0.0340s2 + 0.0042s + 2.9e−5

s4 + 2.4617s3 + 0.9859s2 + 0.1107s + 0.0008






Output responses for a 2.5% input change

0 500 1000 1500 200041

41.5

42

42.5

43 Linear System (dashed) Non−Linear (solid)

Time (h)

Sub

stra

te

0 500 1000 1500 20000.0825

0.083

0.0835

0.084

0.0845

0.085

0.0855 Linear System (dashed) Non−Linear (solid)

Time (h)

Dilu

tion

rate

0 500 1000 1500 20006.06

6.08

6.1

6.12

6.14

6.16

6.18


Time (h)

Dis

solv

ed O

xiyg

en

0 500 1000 1500 200090

90.5

91

91.5

92


Time (h)

Aer

atio

n ra

te





Multivariable PI/PID methodsApplication to the ASP












Multivariable Control of a TITO process

G11(s)r1 u1

y1

G22(s)u2 y2r2

G12(s)

G21(s)

K(s)






Four design methods for multivariable PI/PID tuning are compared

Davidson Method

Penttinen -Koivo Method

Maciejowski Method

Combined Method






Davidson Method

The objective of this method reduces to finding steady state gain matrixof the plant for a step input. The expression for this controller is:

u(s) =1

sKie(s) Ki = ǫG (0)−1

The multiplier ǫ can be tuned and adjusted simultaneously so the closedloop plant has a maximum speed of response for a step input.






Penttinen -Koivo Method

In this method, the plant is diagonalised at very high frequencies. Theexpression for the controller is slightly altered from previous method toform:

u(s) = (Kp +1

sKi )e(s)

where

Kp = (CB)−1ρ Ki = ǫG (0)−1

Matrices C and B are taken from the corresponding state-spacerealisation of G (s):

G (s) :

{

x = Ax + Bu

y = Cx + Du






Maciejowski method

The technique proposed is to diagonalise the system near bandwith, ωb.In keeping with the earlier proportional - integral controller design, thiscontributes three terms that of proportional, integral and derivative:

K (s) = Kp + Ki

1

s+ Kd s

where

Kp = ρG (jωb)−1 Ki = ǫG (jωb)

−1 Kd = δG (jωb)−1






Combined method

This method has been proposed in [Wahab & Katebi UKCC, 2006] andproposes the extension of the three studied controller methods tocombine all together to form:

Kp = ρG (jωb)−1 Ki = ǫG (0)−1 Kd = δ(CB)−1






Comparison of Multivariable PID approaches applied to the ASP

0 20 40 60 80 10040

45

50

55 Substrate

Time (h)

0 20 40 60 80 1000.08

0.09

0.1

0.11

0.12

0.13

0.14

0.15

0.16 Dilution rate

Time (h)

0 20 40 60 80 1003.5

4

4.5

5

5.5

6

6.5 DO

Time (h)

0 20 40 60 80 10040

50

60

70

80

90

100

110 Aeration rate

Combined

Davidson

Maciejowski

Penttinen−Koivo





Control of TITO ProcessesApplication to the ASP












Decentraliced control block diagram for a TITO System

G11(s)K1(s)r1

u1y1

G22(s)K2(s)u2 y2r2

G12(s)

G21(s)






The transfer functions seen by each local controller:

G1(s) = G11(s)

(

1 −G12(s)G21(s)

G11(s)G22(s)T2(s)

)

= G11(s) (1 − I (s)T2(s))

G2(s) = G22(s)

(

1 −G12(s)G21(s)

G11(s)G22(s)T1(s)

)

= G22(s) (1 − I (s)T1(s))

where

I (s) =G12(s)G21(s)

G11(s)G22(s)






T1(s) =K1(s)G11(s)

1 + K1(s)G11(s)≈ T d

1 (s)

T2(s) =K2(s)G22(s)

1 + K2(s)G22(s)≈ T d

2 (s)






The desired input-output transfers will obey to (Tc1 = 2, Tc2 = 0.5):

T d1 (s) =

1

Tc1s + 1T d

2 (s) =1

Tc2s + 1

FO approximation of the effective transfer functions

0 5 10 15 20 25 300

100

200

300

400

500

600

Step response of G11

(s)

Time (h)0 1 2 3 4 5

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

Step response of G22

(s)

Time (h)

0 5 10 15 20 25 300

100

200

300

400

500

600

Step resp. of G1(s) (dash−dot) FO (solid) G

11(s) (dasehd)

Time (h)0 1 2 3 4 5

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

Step resp. of G2(s) (dsah−dot) FO (solid) G

22(s) (dasehd)

Time (h)






The desired input-output transfers will obey to:

T d1 (s) =

1

Tc1s + 1T d

2 (s) =1

Tc2s + 1

with Tc1 = 2 and Tc2 = 0.5. The resulting G1(s) and G2(s) first orderapproximations:

Gapp1 (s) =

500

(4s + 1)G

app2 (s) =

0.0355

(0.5s + 1)

TITO Decentraliced design

Local controllers will be designed on the basis of Gapp1 (s) and G

app2 (s)






u(s) = Cr (s)r(s) − Cy (s)y(s)

where

Cr (s) = Kc

[

β +1

Ti s

]

Cy (s) = Kc

[

1 +1

Ti s+ Td s

]

2-DOF PI Control System






Applied 2-DoF PI Tuning

κc = KcKp =2 − τc

τc

(1)

τi =Ti

T= τc(2 − τc) (2)

β =1

2 − τc

0 < τc ≤ 1






Time responses for step changes on substrate and DO for two tunings ofthe Substrate controller: τc1=0.9 & τc1=0.5, whereas the DO controlleris set to τc2=1.8

0 50 100 15040

42

44

46

48

50

52

54 Substrate:

Time (h)

0 50 100 1500.08

0.09

0.1

0.11

0.12

0.13 Dilution rate

Time (h)

0 50 100 1503.5

4

4.5

5

5.5

6

6.5

Time (h)

DO

0 50 100 15040

50

60

70

80

90

100

110 Aeration rate

Time (h)

τc1=0.9 & τ

c2=1.8

τc1=0.5 & τ

c2=1.8






Regulation performance for a periodical discharge on Sin of about 10%during one hour every 24h. Tunings are τc1=0.9 & τc1=0.5, whereas theDO controller is set to τc2=1.8

0 50 100 15040

40.5

41

41.5

42

42.5

43

43.5

44 Substrate:

Time (h)

0 50 100 150

0.075

0.08

0.085

0.09 Dilution rate

Time (h)

0 50 100 1506

6.05

6.1

6.15

6.2

Time (h)

DO

0 50 100 15089

89.5

90

90.5

91

91.5

92

92.5

93 Aeration rate

Time (h)

τc1=0.9 & τ

c2=1.8

τc1=0.5 & τ

c2=1.8





The control configurationApplication to the ASP












Combination of a MIMO PI controller plus single-loop controllers

G11(s)r1 u1

y1

G22(s)u2 y2

r2

G12(s)

G21(s)

K(s)

K1(s)

K2(s)






Process outputs for the MIMO PID controlled process as well as theirapproximations as First-Order Systems (10% step ref. change

0 5 10 15 20 25 30 35 40 45 500

1

2

3

4

5 Step response on S: FO (dashed) Process output (solid)

Time (h)

0 5 10 15 20 25 30 35 40 45 500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8 Step response on DO: FO (dashed) Process output (solid)

Time (h)

First-Order Systems approximation for outer PI design

G1(s) =1

(4s + 1)G2(s) =

1

(4s + 1)






Process outputs for the MIMO PID plus outer PI controllers

0 50 100 15040

42

44

46

48

50

52

54 Substrate

0 50 100 1500.08

0.085

0.09

0.095

0.1

0.105

0.11

0.115

0.12 Dilution rate

0 50 100 1504

4.5

5

5.5

6

6.5 DO

0 50 100 15020

40

60

80

100

120 Aeration rate

MIMO+Decentraliced PI

MIMO






Process outputs for the MIMO+PID controlled process for a variable Sin

0 50 100 15040

40.5

41

41.5

42

42.5

43

43.5 Substrate

0 50 100 1500.07

0.072

0.074

0.076

0.078

0.08

0.082

0.084

Dilution rate

0 50 100 1506

6.05

6.1

6.15

6.2

6.25 DO

0 50 100 15087

88

89

90

91

92

93 Aeration rate

MIMO+Decentraliced PI

MIMO





Feedforward controlFeedforward control with Decoupling purposesApplication to the ASP












Classical Feedback + Feedforwardcontrol

P (s)K(s)r(s) u(s) y(s)

d(s)

Pd(s)−Kff (s)

uff (s)

Internal Model Control Feedback +Feedforward control


d(s)

Pd(s)−Kff (s)

P (s)

Pd(s)

uff (s)






Proposed Feedforward Control Architecture


d(s)

Pd(s)−Kff (s)

N(s)

Pd(s)

D(s)

uff (s)






Incorporation of Feedforward corrective actions on a decentralized TITOcontrol scheme.

G11(s)K1(s)

u1 y1

G22(s)K2(s)u2 y2

r2

G12(s)

G21(s)

Qff21

(s)Tff21

(s)






Reference signal based feedforward corrective actions on a decentralizedTITO control scheme.

G11(s)K1(s)

u1 y1

G22(s)K2(s)u2 y2

r2

G12(s)

G21(s)Q

ff21

(s)Tff21

(s)

r1

Q1(s)






Where does the Q1(s) block comes from?

IMC - Feeback controller interpretation

Ki (s) =Qi (s)

1 − Gi (s)Qi (s)

Qi (s) =Ki (s)

1 + Gi (s)Ki (s)






Expression for the FF controllers

Qff21 =

n22(s)

n21(s)

1

(λff21s + 1)

λff21 has to be in accordance with the expected control signal

bandwidth.

λff21 is chosen five times smaller than the corresponding fastest time

constant of Q1(s) (The IMC parameter for K1(s)).

Expression for the FF controllers

λff21 = 0.4 λff

12 = 0.2






Comparison of the reference-driven and control signal-driven feedforwardactions.

60 65 70 75 8051.2

51.4

51.6

51.8

52

52.2

52.4

Substrate:

Time (h)

60 65 70 75 800.08

0.085

0.09

0.095

0.1

Dilution rate

Time (h)

0 5 10 15 20 25 305.8

5.85

5.9

5.95

6

6.05

6.1

6.15

DO

Time (h)

0 5 10 15 20 25 3085

90

95

100

105 Aeration rate

Time (h)

Decentraliced PI

Decentraliced PI + Feedforward






Multi-loop PI based control strategies on the


Dr. Ramon Vilanova

Dept. Telecomunications and Systems EngineeringUniversitat Autonoma de Barcelona

[email protected]

(with Prof. Reza Katebi from ICC, University of Strathclyde, Glasgow, Soctland)

March 31, 2009


multi-loop pi based control strategies on the activated ... fileapplication to the asp 5...

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