Introduction to Stability

1
Introduction to Stability

• The concept of stability
• Critical points
• Linear stability analysis
• Biochemical reactor model
• Stability analysis of the bioreactor model

2
The Concept of Stability

• Imprecise definition
• Consider a nonlinear system with the origin as a
  steady-state point
• Does the system return to the origin if perturbed
  away from the origin? If so, the system is
  stable. Otherwise, the system is unstable.

• Precise definition
• Stability produce a bound e on y(0) such that
  y(t) remains within a given bound d
• Asymptotic stability stable y(t) converges to
  the origin
• Commonly known as Lyapunov stability

3
Critical Points of a Linear System

• Two-dimensional system
• Divide equations
• 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

4
Types of Critical Points

• Center
• Two imaginary eigenvalues
• Spiral point
• Two complex eigenvalues
• Degenerate node
• No eigenvector basis exists (see text)

• Proper node
• Two identical real eigenvalues
• Improper node
• Two different real eigenvalues
• Saddle point
• Two real eigenvalues with different signs

5
Linear Stability Analysis

• General solution form
• Procedure
• Compute the eigenvalues of A
• The system is asymptotically stable if and only
  if Re(li) lt 0 for i 1, 2, , n
• The origin is unstable if Re(li) gt 0 for any i
• Stability allows zero eigenvalues

6
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
• Nonlinear model linearization about origin

7
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) lt 0 for i 1, 2, , n
• The steady state is unstable if Re(li) gt 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 periodic solutions can be
  stable
• Each stable state has a certain domain of
  attraction

8
Continuous Biochemical Reactor
9
Cell Growth Modeling

• Specific growth rate
• 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
• S concentration of rate limiting substrate
• Ks saturation constant
• mm maximum specific growth rate (achieved when
  S gtgt Ks)

10
Continuous Bioreactor Model

• Assumptions
• Sterile feed
• Constant volume
• Perfect mixing
• Constant temperature 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

11
Mass Balance Equations

• Cell mass
• VR reactor volume
• F volumetric flow rate
• D F/VR dilution rate
• Product
• Substrate
• S0 feed concentration of rate limiting
  substrate

12
Steady-State Solutions

• Simplified bioreactor model
• Steady-state equations
• Two steady-state points

13
Model Linearization

• Biomass concentration equation
• Substrate concentration equation
• Linear model structure

14
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
• Linear model coefficients (units h-1)

15
Stability Analysis

• Matrix representation
• Eigenvalues (units h-1)
• Conclusion
• Non-trivial steady state is asymptotically stable
• Result holds locally near the steady state

16
Washout Steady State

• Steady state
• Linear model coefficients (units h-1)
• Eigenvalues (units h)
• Conclusion
• Washout steady state is unstable
• Non-trivial steady state may be globally stable