Introduction to Stability
1. The concept of stability
2. Critical points
3. Linear stability analysis
4. Biochemical reactor model
5. Stability analysis of the bioreactor model
6. Matlab tools for linear stability analysis
The Concept of Stability
Imprecise definition
» Consider a nonlinear system with the origin as a steady-state point:
dy
f (y ) f ( y ) 0
dt
» Does the system return to the origin if perturbed away from the origin?
If so, the system is stable. Otherwise, the system is unstable.
dy
f (y ) y (0) ε lim y (t ) 0
dt t
y2
Precise definition
» Stability: produce a bound e on y(0) d
such that y(t) remains within a given
e
bound d y1
y(0)
» Asymptotic stability: stable and y(t) y(t) 0
converges to the origin
» Commonly known as Lyapunov
stability
Critical Points of a Linear System
Two-dimensional system
dy1
dy a11 y1 a12 y2
Ax dt
dt dy2
a21 y1 a22 y2
dt
Divide equations
dy2 dy2 dt a21 y1 a22 y2
dy1 dy1 dt a11 y1 a12 y2
Critical point
» Point where dy2/dy1 becomes undetermined
» Only the origin for a homogeneous linear system
» Five types of critical points depending on the geometric shape of
trajectories near the origin and eigenvalues of A matrix
Types of Critical Points
Proper node Center
» Two identical real » Two imaginary
eigenvalues eigenvalues
Improper node Spiral point
» Two different real » Two complex
eigenvalues eigenvalues
Saddle point Degenerate node
» Two real eigenvalues » No eigenvector
with different signs basis exists (see
text)
Linear Stability Analysis
General solution form
for distinct eigenvalues
y (t ) c1x (1) e l1t c2 x ( 2) e l2t cn x ( n ) e lnt
Imaginary
Procedure Left-Half Plane Right-Half Plane
» Compute the eigenvalues
of A
» The system is
asymptotically stable if Stable Unstable
Real
and only if Re(li) 0 for any i
» Stability allows zero
eigenvalues
Nonlinear Systems
Steady-state points
» Nonlinear models can have multiple steady states
» Stability must be determined for each steady state
Consider origin as a generic steady-state point
dy
f (y ) f ( y ) 0
dt
dy dy
y' y y f (y y ) g (y)
dt dt
dy
g(y ) g (0) 0
dt
Nonlinear model linearization about origin
dy dy
f (y ) Ay
dt dt
Linearized Stability Analysis
Local analysis
» For linear systems stability analysis is global
» For nonlinear systems stability analysis is local
Procedure
» Linearize model about steady state to determine A
» Compute the eigenvalues of A
» The steady state is locally asymptotically stable if
Re(li) 0 for any i
» More advanced methods needed if Re(li) = 0
Comments
» Nonlinear systems may have more than one stable state
» Both steady states and periodic solutions can be stable
» Each stable state has a certain domain of attraction
Continuous Biochemical Reactor
Exit Gas Flow
Fresh Media Feed
(substrates)
Agitator
Exit Liquid Flow
(cells & products)
Cell Growth Modeling
Specific growth rate
1 dX
m X biomass concentrat (g/L)
ion
X dt
Yield coefficients
» Biomass/substrate: YX/S = -DX/DS
» Product/substrate: YP/S = -DP/DS
» Product/biomass: YP/X = DP/DX
» Assumed to be constant
Substrate limited growth
mm S
m (S )
KS S
» S = concentration of rate limiting substrate
» Ks = saturation constant
» mm = maximum specific growth rate (achieved when S >> Ks)
Continuous Bioreactor Model
Assumptions
Sterile feed
Constant volume
Perfect mixing
Constant temperature and pH
Single rate limiting nutrient
Constant yields
Negligible cell death
Product formation rates
» Empirically related to specific growth rate
» Growth associated products: q = YP/Xm
» Nongrowth associated products: q = b
» Mixed growth associated products: q = YP/Xmb
Mass Balance Equations
Cell mass
dX dX
VR FX VR mX DX mX
dt dt
» VR = reactor volume
» F = volumetric flow rate
» D = F/VR = dilution rate
Product
dP dP
VR FP VR qX DP qX
dt dt
Substrate
dS 1 dS 1
VR FS 0 FS VR mX D( S 0 S ) mX
dt YX / S dt YX / S
» S0 = feed concentration of rate limiting substrate
Steady-State Solutions
Simplified model equations
dX mm S
DX m ( S ) X f1 ( X , S ) m (S )
dt KS S
dS 1
D( S0 S ) m (S ) X f 2 ( X , S )
dt YX / S
Steady-state equations
mm S
DX m ( S ) X 0 m (S )
KS S
1
D( S0 S ) m (S ) X 0
YX / S
Two steady-state points
KS D
Non - Trivial : m ( S ) D S X YX / S ( S 0 S )
mm D
Washout : S S0 X 0
Model Linearization
Biomass concentration equation
dX f1
f1 ( X , S )
X X , S
X X f1 S S
S
dt X ,S
zero
mm X m m XS
m S D X 2
S
K S S K S S
Substrate concentration equation
dS f 2 f 2
f 2 ( X , S ) X X S S
dt
X X , S S X , S
zero
1 mm S 1 m X m m XS
X m
D S
YX / S K S S K S S
2
YX / S K S S
Linear model structure: dX
a11 X a12 S
dt
dS
a21 X a22 S
dt
Non-Trivial Steady State
Parameter values
» KS = 1.2 g/L, mm = 0.48 h-1, YX/S = 0.4 g/g
» D = 0.15 h-1, S0 = 20 g/L
Steady-state concentrations
KS D
S 0.545 g/L X YX / S ( S 0 S ) 7.78 g/L
mm D
Linear model coefficients (units h-1)
mm X m m XS
a11 0 a12 1.472
KS S K S S
2
1 mm S 1 mm X
m m XS D 3.529
a21 0.375 a22
YX / S KS S YX / S K S S K S S
2
Stability Analysis
Matrix representation
X dx 0 1.472
x x Ax
S dt 0.375 3.529
Eigenvalues (units h-1)
l 1.472
A lI l1 0.164 l1 3.365
0.375 3.529 l
Conclusion
» Non-trivial steady state is asymptotically stable
» Result holds locally near the steady state
Washout Steady State
Steady state: S Si 20 g/L X 0 g/L
Linear model coefficients (units h-1)
m maxS
a11 D 0.303 a12 0
KS S
1 m maxS 1 m max X
m max XS D 0.15
a21 1.132 a22
YX / S KS S YX / S K S S K S S
2
Eigenvalues (units h)
0.303 l 0
A lI l1 0.303 l1 0.15
1.132 0.15 l
Conclusion
» Washout steady state is unstable
» Non-trivial steady state may be globally stable
Matlab Tools for Stability Analysis
Matlab provides several functions for linear
stability analysis of nonlinear systems
» fsolve – finds steady-state point for nonlinear ODE
system
» linmod – linearizes nonlinear ODE system about
given steady state to generate linear ODE system
» Eigenvalue – computes eigenvalues of linear ODE
system
All the manual calculations needed for linear
stability analysis can be performed
numerically within Matlab
The function linmod requires Matlab add-ons
(Simulink and Control Sytems toolbox) to be
covered in ChE 446 this fall