BIOMATH
Outline
EESA
Department of Applied Mathematics,
UNIVERSITEIT
Biometrics and Process Control
GENT
Department of Electrical Energy,
Systems and Automation
• Introduction: the SHARON process
Existence, uniqueness and stability of the
equilibrium points of a
SHARON bioreactor model
Eveline I.P. Volcke, Mia Loccufier, Peter A. Vanrolleghem and
Erik J.L. Noldus
Benelux Meeting on Systems and Control
• Problem statement - Mathematical model
• Criterion for a unique equilibrium point
• Calculation of the unique equilibrium states
• Local asymptotic stability of the unique equilibrium
• Conclusions – Future work
Helvoirt, The Netherlands, 17 March 2004
UGent-Biomath, Coupure 653, 9000 Gent, Belgium (e-mail [email protected])
Outline
2 - Eveline Volcke – 23rd Benelux Meeting on Systems and Control - 17/03/2004
Introduction: a wastewater treatment plant
influent
• Introduction: the SHARON process
effluent
Settler
A-step
COD oxidation
Settler
B step
NH4+-oxidation
• Problem statement - Mathematical model
SHARON
Anammox
process
• Criterion for a unique equilibrium point
• Calculation of the unique equilibrium states
Dewatering
• Local asymptotic stability of the unique equilibrium
• Conclusions – Future work
Sludge
digestor
Sludge
3 - Eveline Volcke – 23rd Benelux Meeting on Systems and Control - 17/03/2004
4 - Eveline Volcke – 23rd Benelux Meeting on Systems and Control - 17/03/2004
Introduction: the SHARON process
Introduction: SHARON - Anammox
SRTmin versus temperature (pH=7):
NH4+
(NH4+→NO2-)
NO2-
100% NH4+
50% NH4+
N2
50% NO2-
NO3(NO2-→NO3-)
SHARON
SHARON
• CSTR (chemostat)
• no biomass retention:
SRT=HRT
• pH=7 ; T = 35°C
Anammox
Simplified stoichiometry:
at 35°C: ammonium oxidizers grow faster
than nitrite oxidizers
⇒ nitrite oxidizers are washed out
by keeping retention time low
⇒ partial nitrification to nitrite is achieved
5 - Eveline Volcke – 23rd Benelux Meeting on Systems and Control - 17/03/2004
+
−
NH4 + NO2 → N2 + 2H2O
⇒ Optimal NH4+/NO2- ≈ 1/1
6 - Eveline Volcke – 23rd Benelux Meeting on Systems and Control - 17/03/2004
1
Outline
Problem statement
Is NH4+/NO2- produced in SHARON reactor
• unique ?
• stable ?
for constant input variables :
• Introduction: the SHARON process
• Problem statement - Mathematical model
• Criterion for a unique equilibrium point
• Calculation of the unique equilibrium states
• Local asymptotic stability of the unique equilibrium
• Conclusions – Future work
7 - Eveline Volcke – 23rd Benelux Meeting on Systems and Control - 17/03/2004
•
•
•
•
•
dilution rate : Q/V = u0 =1/HRT
(total) influent ammonium concentration : TNHin=u1
(total) influent nitrite concentration : TNO2in=u2
(total) influent concentration of ammonium oxidizers: Xamm,in=u3
(total) influent concentration of nitrite oxidizers: Xnit,in=u4
OR
constant inputs
⇒ unique and stable equilibrium states?
8 - Eveline Volcke – 23rd Benelux Meeting on Systems and Control - 17/03/2004
Mathematical model
Outline
Assume SHARON reactor controlled at constant pH
⇒ simplified reactor model:
• Introduction: the SHARON process
x& ( t ) = (u − x( t )) ⋅ u 0 + M ⋅ ρ(x( t ))
⎡ u1 ⎤ ⎡ TNH in ⎤
⎡ x1 ( t ) ⎤ ⎡ TNH ( t ) ⎤
⎡− a
⎢u ⎥ ⎢TNO 2 ⎥
⎢ x ( t ) ⎥ ⎢TNO 2( t ) ⎥
⎢
Q
in
2
2
⎥ M=⎢ c
⎥ u =
⎥=⎢
x( t ) = ⎢
u=⎢ ⎥=⎢
0
⎢ u 3 ⎥ ⎢ X amm,in ⎥
⎢ x 3 ( t ) ⎥ ⎢ X amm ( t ) ⎥
⎢1
V
⎥
⎢ ⎥ ⎢
⎥
⎥ ⎢
⎢
⎢
⎣0
⎣u 4 ⎦ ⎣ X nit ,in ⎦
⎣ x 4 ( t ) ⎦ ⎣ X nit ( t ) ⎦
x1
c1
⎤
⎡
a1 ⋅
⋅
⋅ x3
⎥
⎡ ρ ( x) ⎤ ⎢
b1 + x1 c1 + x 2
⎥
ρ=⎢ 1 ⎥=⎢
x
x
d
e2
2
1
2
⎣ρ 2 (x) ⎦ ⎢a 2 ⋅
⋅
⋅
⋅
⋅ x4 ⎥
b 2 + x 2 c 2 + x 1 d 2 + x 2 e 2 + x1
⎦⎥
⎣⎢
− b⎤
− d ⎥⎥
0⎥
⎥
1⎦
• Problem statement - Mathematical model
• Criterion for a unique equilibrium point
• Calculation of the unique equilibrium states
• Local asymptotic stability of the unique equilibrium
• Conclusions – Future work
u constant ⇒ xe unique and stable?
9 - Eveline Volcke – 23rd Benelux Meeting on Systems and Control - 17/03/2004
Calculation of equilibrium states
The equilibrium states are obtained from
0 = (u − x e ) ⋅ u 0 + M ⋅ ρ(x e )
Case u0 = 0 (i.e. no flow) :
• xe1=0 ; xe2, xe3, xe4 arbitrary
• xe2 = xe3 = 0 ; xe1, xe4 arbitrary
• xe3 = xe4 = 0 ; xe1, xe2 arbitrary
⇒ number of equilibrium solutions: ∞3 + 2 ∞2
Case u0 ≠ 0 :
1
x e = u + ⋅ M ⋅ ρ(x e )
u0
⇒ principle of contraction mappings
11 - Eveline Volcke – 23rd Benelux Meeting on Systems and Control - 17/03/2004
10 - Eveline Volcke – 23rd Benelux Meeting on Systems and Control - 17/03/2004
Criterion for a unique equilibrium
Contraction mapping theorem :
every contraction mapping ϕ , defined in a complete metric space X,
has one and only one fixed point in X, i.e. for which x = ϕ(x)
Case u0 ≠ 0:
xe = u +
1
⋅ M ⋅ ρ(x e ) ≡ ϕ(x e )
u0
Define mapping ϕ in the complete metric space R4x1
with the Euclidian norm as distance: l( x, w ) = x − w
IF ϕ is a contraction mapping
i.e. there exists a K < 1 such that
l(ϕ( x), ϕ(w )) ≤ K ⋅ l( x, w )
∀x, w in R 4×1
THEN ϕ possesses a unique fixed point xe = ϕ(xe)
that is the unique equilibrium point for the SHARON model
12 - Eveline Volcke – 23rd Benelux Meeting on Systems and Control - 17/03/2004
2
Criterion for a unique equilibrium
Is ϕ a contraction mapping?
Criterion for a unique equilibrium
Is ϕ a contraction mapping?
1
T
l 2 [ϕ(x), ϕ(w )] = 2 [ρ(x) − ρ( w )] M T M[ρ(x) − ρ(w )]
u0
s0
T
⋅ [ρ(x) − ρ(w )] [ρ(x) − ρ(w )]
2
u0
s
with s0 the largest eigenvalue
MT TM : T
≤ 02 ⋅ [x −of w
] J J[x − w ]
u0 2
(α + β) + (α − β) + 4 ⋅ γ 2
⎤
s 0 ⎡=∂ρ1
∂ρ1
∂ρ1
∂ρ1
⎥
⎢
∂x 2 x = x̂ 2 ∂x 3 x = x̂
∂x 4 x = x̂ ⎥ ⎡ δ1 δ 2 δ3 0 ⎤
⎢ ∂x1 x = x̂
J=⎢ 2
2
2
2
α ⎢=∂ρa2 + c ∂+ρ21 ; β =∂ρb2 + d ∂+ρ21 ; γ⎥⎥== ⎢⎣aδ⋅5 b δ−6 c ⋅0d δ8 ⎥⎦
l 2 [ϕ(x), ϕ(w )] ≤
≤
11
∂x
⎣⎢ 1
x = x̂ 21
12
∂x 2
x = x̂ 22
14
13
∂x 3
x = x̂ 23
∂x 4
x = x̂ 24
⎦⎥
13 - Eveline Volcke – 23rd Benelux Meeting on Systems and Control - 17/03/2004
s 0 ⋅ s1 2
⋅ l [x, w ]
2
u0
with s1IFthe largest eigenvalue of JT J :
YES,
s ⋅s
0
1
≤1
(µ + ν )u+ 2 (µ − ν) 2 + 4 ⋅ ξ 2
0
s1 =
2
u 20 ≥ s2 0 ⋅ s1 ≡ u204,dim
crit 2
2
2
µ = δ1 + δ 2 + δ3 ; ν = δ5 + δ 6 + δ8 ; ξ = δ1 ⋅ δ5 − δ 2 ⋅ δ 6
= criterion for a unique equilibrium (‘4dim’) !!!
14 - Eveline Volcke – 23rd Benelux Meeting on Systems and Control - 17/03/2004
Criterion for unique equilibrium:alternative
The SHARON reactor model in equilibrium can be
rewritten in 2 state variables (instead of 4)
The equilibrium states are now obtained from
ye = w +
≤
s0
T
⋅ [x − w ] J T J[x − w ]
2
u0
Criterion for a unique equilibrium
The SHARON reactor possesses a unique equilibrium
for sufficiently high dilution rates:
u 0 ≥ s 0 ⋅ s1
1 ˆ
⋅ M ⋅ ρˆ (y e ) ≡ ϕˆ ( y e )
u0
≡ u 04,dim
crit
⎡x ⎤
⎡ u ⎤ ˆ ⎡− a − b⎤
y = ⎢ 1⎥ w = ⎢ 1⎥ M
=⎢
⎥
⎣ c − d⎦
⎣x 2 ⎦
⎣u 2 ⎦
= 120 day −1
or
The reactor has a unique equilibrium y e = ϕˆ (y e )
if ϕˆ is a contraction mapping
u 0 ≥ ŝ 0 ⋅ ŝ1
⇔ u 0 ≥ ŝ 0 ⋅ ŝ1 ≡ u 02,dim
crit
best criterion!
= ‘2 dim’ criterion for a unique equilibrium !!!
15 - Eveline Volcke –
23rd
Benelux Meeting on Systems and Control - 17/03/2004
16 - Eveline Volcke –
Criterion for a unique equilibrium
u0,crit in terms of TNHin (Xamm,in= Xnit,in 0.01
23rd
≡ u 02,dim
crit
= 10.63 day −1
Benelux Meeting on Systems and Control - 17/03/2004
Outline
mole/m3):
u1 = TNHin ↑
⇒ u0,crit ↑
2-dim criterion
gives best results
In practice:
u0 = 0.5 – 1 day-1
<< u0,crit
17 - Eveline Volcke – 23rd Benelux Meeting on Systems and Control - 17/03/2004
• Introduction: the SHARON process
• Problem statement - Mathematical model
• Criterion for a unique equilibrium point
• Calculation of the unique equilibrium states
• Local asymptotic stability of the unique equilibrium
• Conclusions – Future work
18 - Eveline Volcke – 23rd Benelux Meeting on Systems and Control - 17/03/2004
3
Calculation of unique equilibrium states
The unique fixed point xe = ϕ(xe) of a
contraction mapping ϕ is obtained by
the method of successive approximations:
xn+1= ϕ(xn)
Calculation of the equilibrium states
In terms of TNHin (Xamm,in= Xnit,in 0.01 mole/m3):
almost no conversion (TNHe ≈ TNHin)
⇒ equilibrium state : wash-out of the biomass
reactor doesn’t work!
n = 0,1,2,…
for an arbitrary starting value x0
e.g. for TNHin =
70 mole/m3:
Application to the SHARON process model:
• calculate xe for different values of u1 = TNHin
• choose x0=u (e.g.)
19 - Eveline Volcke – 23rd Benelux Meeting on Systems and Control - 17/03/2004
Outline
TNHe = 69.96
TNO2e = 0.035
Xamm,e = 0.122
Xnit,e = 0.010
[mole/m3]
20 - Eveline Volcke – 23rd Benelux Meeting on Systems and Control - 17/03/2004
LAS of the unique equilibrium
Linearization principle:
• Introduction: the SHARON process
• Problem statement - Mathematical model
The SHARON reactor model
x& ( t ) = (u − x( t )) ⋅ u 0 + M ⋅ ρ( x( t ))
= f (x)
is locally asymptotically stable if the eigenvalues of the Jacobian matrix
• Criterion for a unique equilibrium point
• Calculation of the unique equilibrium states
• Local asymptotic stability of the unique equilibrium
• Conclusions – Future work
∂f (x)
∂x x =x e
⎡ ∂f1
⎢
⎢ ∂x1 x = x e
=⎢ M
⎢
⎢ ∂f 4
⎢ ∂x1 x = x
e
⎣
⎤
∂f1
⎥
∂x 4 x =x ⎥
e
M ⎥
⎥
∂f 4
⎥
L
∂x 4 x =x ⎥
e ⎦
L
all have a strictly negative real part (are in the open left phase plane)
21 - Eveline Volcke –
23rd
Benelux Meeting on Systems and Control - 17/03/2004
LAS of the unique equilibrium
22 - Eveline Volcke – 23rd Benelux Meeting on Systems and Control - 17/03/2004
Outline
Eigenvalues of the Jacobian matrix:
s = −u0
(2×)
s + (2⋅ u0 − α − δ) ⋅ s + (u0 − α)(u0 − δ) − β⋅ γ = −u0
• Introduction: the SHARON process
2
all have a negative
part (are
∂ρ1 left phase plane) if
∂ρ1 in the open
∂ρreal
1
+
+ c⋅
α = −a ⋅
x1δx)= x> 0 ∂x 2
- sum = (2⋅ u0 − α∂−
e
x =x e
∂x 3
x =x e
product = (u0 − α
u − δ) −∂βρ⋅1γ > 0
∂ρ)(
1 0
β = −b ⋅
∂x1
−d⋅
x = xe
∂x 2
∂ρ 2
∂x1
−d⋅
x = xe
∂ρ 2
∂x 2
• Local asymptotic stability of the unique equilibrium
+
x = xe
• Criterion for a unique equilibrium point
• Calculation of the unique equilibrium states
x =x e
fulfilled for all values of TNHin
∂ρ
∂ρ
= −a ⋅ 2 equilibrium
+ c ⋅ 2 is
⇒ uniqueγ(wash-out)
∂x1 x = x
∂x 2
locally asymptoticallye stable ! x = xe
α = −b ⋅
• Problem statement - Mathematical model
∂ρ 2
∂x 4
• Conclusions – Future work
x = xe
23 - Eveline Volcke – 23rd Benelux Meeting on Systems and Control - 17/03/2004
24 - Eveline Volcke – 23rd Benelux Meeting on Systems and Control - 17/03/2004
4
Conclusions
Future work
• A simplified mathematical model for a SHARON reactor
with constant pH was constructed
For realistic (low) values of the dilution rate u0 :
• A criterion for a unique equilibrium point was deducted and
the equilibrium states were calculated
using a contraction mapping theorem
• calculate all equilibrium points
• A unique equilibrium point is only obtained
for high values of the dilution rate
corresponding with wash-out of the biomass
and almost no conversion
• define attraction regions for each equilibrium point
• show that the system converges to one of the equilibrium
points, regardless the initial condition
using Liapunov’s theory
• The unique equilibrium point is locally asymptotically stable
25 - Eveline Volcke – 23rd Benelux Meeting on Systems and Control - 17/03/2004
26 - Eveline Volcke – 23rd Benelux Meeting on Systems and Control - 17/03/2004
Verification of the calculated equilibrium
Thank you for your attention!
Simulation of the SHARON reactor in Simulink
Questions?
TNHin = 70 mole/m3; Xamm,in= Xnit,in 0.01 mole/m3;
27 - Eveline Volcke – 23rd Benelux Meeting on Systems and Control - 17/03/2004
28 - Eveline Volcke – 23rd Benelux Meeting on Systems and Control - 17/03/2004
Verification of the calculated equilibrium
System dynamics
The characteristic equation
s2 + (2⋅ u0 − α − δ) ⋅ s + (u0 − α)(u0 − δ) − β⋅ γ = −u0
TNHe = 69.96
TNO2e = 0.035
Xamm,e = 0.122
Xnit,e = 0.0100
can also be written as
s2 + 2 ⋅ ς ⋅ ωn ⋅ s + ωn = 0
2
ζ and ωn
determine the
system dynamics
• non-oscillating
transient behaviour
The same equilibrium values are obtained!
29 - Eveline Volcke – 23rd Benelux Meeting on Systems and Control - 17/03/2004
30 - Eveline Volcke – 23rd Benelux Meeting on Systems and Control - 17/03/2004
5