BENG 221
Lecture 2
Tutorial
BENG 221
Mathematical Methods in Bioengineering
Overview
Example: Ring
Oscillator
Dynamics
Analytic ODE
Solution
Numerical
Verification
Numerical
Simulation
Lecture 2
Tutorial
Analytic and Numerical Methods in ODEs
Further Reading
Gert Cauwenberghs
Department of Bioengineering
UC San Diego
2.1
BENG 221
Lecture 2
Tutorial
Summary
Overview
Example: Ring
Oscillator
Dynamics
Analytic ODE
Solution
Numerical
Verification
Numerical
Simulation
Further Reading
We will review analytic and numerical techniques for solving
ODEs, using pencil and paper, and implemented in Matlab.
These simple techniques lay the foundations for solving more
complex systems of PDEs in the coming weeks.
By way of example we study the dynamics of a ring oscillator, a
circular chain of three inverters with identical capacitive loading.
2.2
BENG 221
Lecture 2
Tutorial
Today’s Coverage:
Overview
Example: Ring
Oscillator
Dynamics
Analytic ODE
Solution
Example: Ring Oscillator Dynamics
Numerical
Verification
Numerical
Simulation
Analytic ODE Solution
Further Reading
Numerical Verification
Numerical Simulation
2.3
BENG 221
Lecture 2
Tutorial
Some circuit elements
Overview
Example: Ring
Oscillator
Dynamics
Analytic ODE
Solution
Numerical
Verification
Numerical
Simulation
Further Reading
Figure : An inverting
transconductor (inverter)
converts and input voltage to an
output current, with gain −G.
iout = g(vin ) ≈ −Gvin
2.4
(1)
Figure : A capacitor converts
charge, or integrated current, to
voltage with gain 1/C.
Q
1
v=
=
C
C
Z
t
i dt
−∞
(2)
BENG 221
Lecture 2
Tutorial
Ring Oscillator
Overview
Example: Ring
Oscillator
Dynamics
Analytic ODE
Solution
Numerical
Verification
Numerical
Simulation
Figure : A 3-inverter ring oscillator with capacitive loading.
Further Reading
dv1
dt
dv2
C
dt
dv3
C
dt
C
2.5
= g(v3 ) ≈ −G v3
v1 (0)
= g(v1 ) ≈ −G v1 (3)
= g(v2 ) ≈ −G v2
=
v10
v2 (0) =
v20 (4)
v3 (0)
v30
=
BENG 221
Lecture 2
Tutorial
Eigenvalues
Ring oscillator ODE dynamics in matrix notation:
dv
= Av
dt
Overview
Example: Ring
Oscillator
Dynamics
Analytic ODE
Solution
Numerical
Verification
Numerical
Simulation
Further Reading
v(0) = v0
(5)
with


v1
v =  v2 
v3

0
A =  −1
0

0 −1
0
0 
−1
0


v10
v(0) =  v20  (6)
v30
where G/C ≡ 1 with no loss of generality.
Eigenvectors xi and corresponding eigenvalues λi of A satisfy
A xi = λi xi , or det(A − λi I) = 0, which reduces to λi 3 + 1 = 0,
with three solutions:
1
3
λi = (−1) =
2.6



+jπ/3
e

 e−jπ/3
−1
√
= 12 + j √23
= 12 − j 23
(7)
BENG 221
Lecture 2
Tutorial
Overview
Eigenvectors
The corresponding eigenvectors are:
Example: Ring
Oscillator
Dynamics
Analytic ODE
Solution

λ1
= −1 :
x1
Numerical
Verification
Numerical
Simulation
λ2
= e+jπ/3 =
1
2
=
√
+j
3
2
:
x2
=
Further Reading
λ3
= e−jπ/3 =
1
2
√
−j
3
2
:
x3
=

1
√1  1 
3
 1
1
√1  e +j2π/3
3
−j2π/3
 e
1
√1  e −j2π/3
3
e+j2π/3


(8)


The eigenvectors form a complex orthonormal basis:
xi ∗ xj = δij ,
i, j = 1, . . . 3
where xi ∗ is the complex conjugate transpose of xi .
2.7
(9)
BENG 221
Lecture 2
Tutorial
Eigenmodes
The general solution is the superposition of eigenmodes (see
Lecture 1):
Overview
Example: Ring
Oscillator
Dynamics
Analytic ODE
Solution
Numerical
Verification
Numerical
Simulation
Further Reading
2.8
v
=
3
X
ci xi eλi t
i=1


1
1
= c1 e−t √  1  +
3
1


1
√
3
1
1
c2 e 2 t ej 2 t √  e+j2π/3  +
3
e−j2π/3


1
√
3
1
1
c3 e 2 t e−j 2 t √  e−j2π/3 
3
e+j2π/3
(10)
v(t) is real, and so c2 and c3 must be complex conjugate.
Therefore, the second and third eigenmodes are oscillatory with
an exponentially rising carrier. The first eigenmode is a decaying
exponential common-mode transient.
BENG 221
Lecture 2
Tutorial
First Eigenmode– Common-mode Decaying Exponential
IC: (1 1 1)
1.2
v1
v2
v3
Overview
Example: Ring
Oscillator
Dynamics
Analytic ODE
Solution
1
0.8
Numerical
Verification
Numerical
Simulation
0.6
Further Reading
0.4
0.2
0
0
1
2
3
4
5
t
6
7
8
9
10
Figure : Ring oscillator ODE solution for v(0) = (1, 1, 1)T .
2.9
BENG 221
Lecture 2
Tutorial
Overview
Example: Ring
Oscillator
Dynamics
Second/third Eigenmode– Exponentially Rising Three-phase
Oscillations
IC: (1
−0.5
−0.5)
4
5
t
6
150
v1
v2
v3
100
Analytic ODE
Solution
Numerical
Verification
50
Numerical
Simulation
Further Reading
0
−50
−100
−150
0
1
2
3
7
8
9
10
Figure : Ring oscillator ODE solution for v(0) = (1, − 21 , − 12 )T .
2.10
BENG 221
Lecture 2
Tutorial
Overview
Example: Ring
Oscillator
Dynamics
Initial Conditions
The IC constrained solution is obtained by virtue of the
orthonormality of the eigenvectors (see also Lecture 1):
Analytic ODE
Solution
v=
x∗i v(0) xi eλi t
(11)
i=1
Numerical
Verification
Numerical
Simulation
n
X
which, using the identity e+jα + e−jα = 2 cos(α), leads to:
Further Reading
v1
=
e−t
(v10 + v20 + v30 ) +
(12)
3
√
2 et/2
3
(v10 cos(
t) +
3
2
√
√
3
2π
3
2π
v20 cos(
t+
) + v30 cos(
t−
))
2
3
2
3
and identical expressions for v2 and v3 under ordered
permutation of the indices (consistent with the ring symmetry).
2.11
BENG 221
Lecture 2
Tutorial
Matlab Implementation
Overview
Example: Ring
Oscillator
Dynamics
Analytic ODE
Solution
Numerical
Verification
Numerical
Simulation
Further Reading
Using the eigenvector-eigenvalue decomposition of A in matrix
form:
AX=Xs
(13)
where X = (x1 , x2 , x3 ) and s = diag(λ1 , λ2 , λ3 ), the solution (11)
can be expressed in matrix form:
v = X diag(X∗ v(0))ediag(s)t
for efficient matlab implementation:
[X, s] = eig(A);
V = X * diag(X’ * V0) * exp(diag(s) * t);
2.12
(14)
BENG 221
Lecture 2
Tutorial
Initial Conditions
IC: (1 0 0)
100
v1
v2
v3
Overview
Example: Ring
Oscillator
Dynamics
Analytic ODE
Solution
80
60
40
Numerical
Verification
Numerical
Simulation
Further Reading
20
0
−20
−40
−60
−80
0
1
2
3
4
5
t
6
7
8
9
10
Figure : Ring oscillator ODE solution for v(0) = (1, 0, 0)T .
2.13
BENG 221
Lecture 2
Tutorial
Euler Integration
Euler numerical integration produces approximate solutions to:
Overview
dx
= f(x(t), t)
dt
Example: Ring
Oscillator
Dynamics
Analytic ODE
Solution
Numerical
Verification
Numerical
Simulation
Further Reading
(15)
at discrete time intervals t = n∆t, by finite difference
approximation of the derivative:
dx
1
(t) =
(x(t + ∆t) − x(t)) + O(∆t 2 )
dt
∆t
(16)
leading to the recursion:
x(t + ∆t) ≈ x(t) + ∆t f(x(t), t).
Matlab Euler example (ring oscillator):
Ve = V0; % Euler approximation, initialize to IC
Vs = V0; % Euler state variable, initialize
for te = tstep:tstep:trange
Vs = Vs + A * Vs * tstep;
Ve = [Ve, Vs];
end
2.14
(17)
BENG 221
Lecture 2
Tutorial
Euler Integration
IC: (1 0 0)
150
v1
v2
v3
Overview
Example: Ring
Oscillator
Dynamics
100
Analytic ODE
Solution
Numerical
Verification
50
Numerical
Simulation
Further Reading
0
−50
−100
0
1
2
3
4
5
t
6
7
8
9
10
Figure : Ring oscillator ODE simulation using Euler integration.
2.15
BENG 221
Lecture 2
Tutorial
Overview
Example: Ring
Oscillator
Dynamics
Analytic ODE
Solution
Numerical
Verification
Numerical
Simulation
Crank-Nicolson Integration
Better numerical ODE methods exist that use higher-order finite
difference approximations of the derivative. Crank-Nicolson is a
second order method that approximates (15) more accurately
using a centered version of the finite difference approximation of
the derivative:
∆t
1
dx
(t +
)=
(x(t + ∆t) − x(t)) + O(∆t 3 )
(18)
dt
2
∆t
leading to a recursion:
Further Reading
x(t + ∆t) ≈ x(t) + ∆t f(x(t +
≈ x(t) +
∆t
∆t
), t +
)
2
2
∆t
(f(x(t), t) + f(x(t + ∆t), t + ∆t)).(19)
2
Matlab CN example (ring oscillator):
2.16
G = (eye(3) - A * tstep / 2) \ (eye(3) + A * tstep / 2);
Vc = V0; % CN approximation, initialize to IC
Vs = V0; % CN state variable, initialize
for te = tstep:tstep:trange
Vs = G * Vs;
Vc = [Vc, Vs];
end
BENG 221
Lecture 2
Tutorial
Crank-Nicolson Integration
IC: (1 0 0)
100
v1
v2
v3
Overview
Example: Ring
Oscillator
Dynamics
Analytic ODE
Solution
80
60
40
Numerical
Verification
Numerical
Simulation
Further Reading
20
0
−20
−40
−60
−80
0
1
2
3
4
5
t
6
7
8
9
Figure : Ring oscillator ODE simulation using Crank-Nicolson
integration.
2.17
10
BENG 221
Lecture 2
Tutorial
Higher-Order Integration
Overview
Example: Ring
Oscillator
Dynamics
Analytic ODE
Solution
Numerical
Verification
Numerical
Simulation
Further Reading
Crank-Nicolson (19) is implicit in that the solution at time step
t + ∆t is recursive in the state variable x(t + ∆t), rather than
forward e.g., given in previous values of the state variable x(t).
More advanced methods, such as Runge-Kutta and several of
Matlab’s built-in ODE solvers, are explicit and/or higher order,
allowing faster integration although possibly at the expense of
numerical stability.
Matlab ode23 example (ring oscillator):
OdeOptions = odeset(’RelTol’, 1e-9); % accuracy, please
dVdt = @(t,V) A * V;
[tm, Vm] = ode23(dVdt, [0 trange], V0);
2.18
BENG 221
Lecture 2
Tutorial
Explicit Forward Second-Order Integration
IC: (1 0 0)
100
v1
v2
v3
Overview
Example: Ring
Oscillator
Dynamics
Analytic ODE
Solution
80
60
40
Numerical
Verification
Numerical
Simulation
Further Reading
20
0
−20
−40
−60
−80
2.19
0
1
2
3
4
5
t
6
7
8
9
10
Figure : Ring oscillator ODE simulation using an explicit version of
Crank-Nicolson integration using a forward series expansion up to
second order.
BENG 221
Lecture 2
Tutorial
Matlab Built-in ode23 Solver
IC: (1 0 0)
100
v1
v2
v3
Overview
Example: Ring
Oscillator
Dynamics
Analytic ODE
Solution
80
60
40
Numerical
Verification
Numerical
Simulation
Further Reading
20
0
−20
−40
−60
−80
0
1
2
3
4
5
t
6
7
8
9
10
Figure : Ring oscillator ODE simulation using the Matlab built-in ode23
numeric ODE solver.
2.20
BENG 221
Lecture 2
Tutorial
Bibliography
Overview
Example: Ring
Oscillator
Dynamics
Analytic ODE
Solution
Numerical
Verification
Numerical
Simulation
Further Reading
R. Haberman, Applied Partial Differential Equations with
Fourier Series and Boundary Value Problems, 4th Ed., 2004,
Ch. 6.
Wikipedia, Conjugate Transpose, http:
//en.wikipedia.org/wiki/Conjugate_transpose.
Wikipedia, Numerical Ordinary Differential Equations,
http://en.wikipedia.org/wiki/Numerical_
ordinary_differential_equations.
Wikipedia, Runge Kutta Methods, http:
//en.wikipedia.org/wiki/Runge-Kutta_methods.
2.21