Time Domain Analysis

1
Time Domain Analysis

• Linear differential equation systems
• Matlab example
• Linear stability analysis
• Nonlinear stability analysis
• Biochemical reactor example

2
Linear Differential Equations

• Coupled linear differential equations
• Matrix representation

3
Matrix Diagonalization

• Coordinate transformation
• Columns of M are the eigenvectors of A
• Transformed equations
• Solution of decoupled equations

4
Solution Properties

• Coordinate transformation
• Solution
• Solution properties determined by the eigenvalues
• Eigenvalues in time domain equivalent to poles in
  Laplace domain

5
Matlab Example

• Transfer function model
• Compute poles
• gtgt gtf(2 1,4 2 6 1)
• gtgt ppole(g)
• p
• -0.1634 1.1903i
• -0.1634 - 1.1903i
• -0.1732

• Convert to state-space model
• gtgt sysss(g)
• Compute eigenvalues
• gtgt lambdaeig(sys.a)
• lambda
• -0.1634 1.1903i
• -0.1634 - 1.1903i
• -0.1732
• gtgt ppole(sys)
• p
• -0.1634 1.1903i
• -0.1634 - 1.1903i
• -0.1732

6
Linear Stability Analysis

• Linear dynamic system
• Stability analysis
• The origin is the only steady-state point if A is
  full rank
• Determine stability of origin based on
  eigenvalues of A
• Solution form
• Eigenvalues can be real and/or complex numbers

7
Eigenvalue Analysis

• Real eigenvalues
• If li lt 0 then exp(lit) ? 0
• If li gt 0 then exp(lit) ? infinity
• If li 0 then exp(lit) ? 1
• Complex eigenvalues
• li aijbi
• exp(lit) exp(ait)sin(bit)
• If ai lt 0 then exp(lit) ? 0
• If ai gt 0 then exp(lit) ? infinity
• If ai 0 then exp(lit) ? sin(bit)
• Stability determined by the real parts of the
  eigenvalues Re(li)

8
Eigenvalues in the Complex Plane
9
Nonlinear Stability Concepts

• Nonlinear dynamic model
• Steady-state points
• Nonlinear models can have multiple steady states
• Stability must be determined for each steady
  state
• Nonlinear deviation model
• Can assume origin is the steady-state point

10
Types of Nonlinear Stability

• Stable
• Given bound on x(t)
• Produce bound on x(0)
• Origin is locally stable
• Otherwise origin is unstable

• Asymptotically stable
• Given bound on x(t) given x(t) ? 0
• Produce bound on x(0)
• Origin is locally asymptotically stable

11
Stability Theorems

• Linear system stability
• 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 slightly less restrictive
• Nonlinear system stability
• Linearize model about steady state to determine A
• Compute the eigenvalues of A
• The origin is locally asymptotically stable if
  Re(li) lt 0 for i 1, 2, , n
• The origin is unstable if Re(li) gt 0 for any i
• More advanced methods needed if Re(li) 0

12
Biochemical Reactor Example

• Continuous 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, mmax 0.48 h-1, YX/S 0.4 g/g
• D 0.15 h-1, Si 20 g/L
• Steady-state concentrations
• Linear model coefficients (units h-1)

15
Stability of the Non-Trivial Steady State

• Matrix representation
• Eigenvalue computation (units h)
• 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)
• Eigenvalue computation (units h)
• Conclusion
• Washout steady state is unstable
• Suggests that non-trivial steady state is
  globally stable