Modeling of Physical Systems
BME331
University of Toronto
Endocrine
System
System
Modeling
Lab 1
Cardiovascular
System
System
Response
Lab 2
Neuromuscular
System
System Control
Lab 3
Learning Objectives
• Obtain mathematical models of physiological
systems.
• Describe the relationship between the unit
impulse and the impulse response of a
system.
• Define a linear time-invariant system.
Reality
Conceptual
Model
Mathematical
Model
Mechanical Systems
• Translation:
x1(t)
x(t)
x(t)
x2(t)
C
K
F(t)
𝐹𝐹 𝑡𝑡 = −𝐾𝐾𝐾𝐾(𝑡𝑡)
Linear Spring
F(t)
m
F(t)
𝐹𝐹 𝑡𝑡 = −𝐶𝐶(𝑥𝑥1̇ (𝑡𝑡)-𝑥𝑥2̇ (𝑡𝑡))
Damper
𝐹𝐹 𝑡𝑡 = 𝑚𝑚𝑥𝑥(𝑡𝑡)
̈
Inertia
Mechanical Systems
• Rotation:
𝜃𝜃(𝑡𝑡)
𝜏𝜏(𝑡𝑡)
𝜃𝜃2(𝑡𝑡)
𝜃𝜃1(𝑡𝑡)
K
𝜏𝜏 𝑡𝑡 = −𝐾𝐾𝜃𝜃(𝑡𝑡)
Linear Spring
𝜏𝜏(𝑡𝑡)
C
𝜏𝜏 𝑡𝑡 = −𝐶𝐶(𝜃𝜃1̇ (𝑡𝑡)-𝜃𝜃2̇ (𝑡𝑡))
Damper
𝜃𝜃(𝑡𝑡)
I
𝜏𝜏(𝑡𝑡)
𝜏𝜏 𝑡𝑡 = 𝐼𝐼 𝜃𝜃̈ (𝑡𝑡)
Inertia
Example
• Spring-Mass-Damper system
Electrical Systems
i(t)
R
u(t)
𝑈𝑈 𝑡𝑡 = 𝑅𝑅𝑅𝑅(𝑡𝑡)
Resistance
i(t)
C
u(t)
𝑖𝑖 𝑡𝑡 = 𝐶𝐶 𝑢𝑢̇ (𝑡𝑡)
Capacitance
i(t)
L
u(t)
.
𝑢𝑢 𝑡𝑡 = 𝐿𝐿𝚤𝚤̇ (𝑡𝑡)
Inductance
Example
• Phase-lag circuit
General System Properties
• Resistance
– Energy dissipation
– E.g.: Ohm’s Law, mechanical damping coefficients,
fluid resistance…
• Capacitance
– Storage of potential energy
– E.g: capacitor, spring compliance
• Inertance
– Storage of kinetic energy
– E.g.: Inductor, Newton’s second of motion, fluid
inertia
Example: Muscle model
Adapted from:
M.C.K. Khoo, Physiological Control Systems: Analysis, Simulation, and Estimation, Wiley & Sons
Note: in this diagram springs are described in terms of
their compliance (1/stiffness)
• F: actual force
produced by the
muscle on the bone
(taking into account
active and passive
components)
• Fo: Force developed
by the active
contractile element
of the muscle
• R: viscous damping
of the tissue
• Cp: elastic storage
properties of the
sarcolemma
• Cs: elastic storage
properties of the
muscle tendons
Example: Muscle model
•Consider how
the force F
changes if an
external force
stretches the
muscle by an
incremental
length x.
Adapted from:
M.C.K. Khoo, Physiological Control Systems: Analysis, Simulation, and Estimation, Wiley & Sons
Note: in this diagram springs are described in terms of
their compliance (1/stiffness)
• Relate F to F0 and x:
• This expression reflects how the actual force
produced by a muscle depends both on the
strength of the contraction and the stretch of the
muscle.
Let’s consider further what characterizes a
system…
The Unit Impulse
The unit impulse or δ-function has the properties:
•
•
∞
∫−∞ 𝛿𝛿
∞
∫−∞ 𝑓𝑓
𝑡𝑡 𝑑𝑑𝑑𝑑
0+
= ∫0− 𝛿𝛿
𝑡𝑡 𝑑𝑑𝑑𝑑 = 1
𝑡𝑡 𝛿𝛿 𝑡𝑡 𝑑𝑑𝑑𝑑 = 𝑓𝑓(0)
The Impulse Response
http://cnx.org/contents/0c47b9ac-f5b5-4761-8340-d9639168dcf1@2/Continuous-Time-Impulse-Respon
The Convolution Sum
Delayed and
shifted
versions of
the impulse
response
The Convolution Sum (Cnt’d)
In discrete time, the process in the previous
slide is called the convolution sum and can be
expressed as:
+∞
𝑦𝑦 𝑛𝑛 = � 𝑥𝑥 𝑘𝑘 ℎ[𝑛𝑛 − 𝑘𝑘]
𝑘𝑘=−∞
h[n]: Impulse response; x[n]: System input; y[n]: System output
The Convolution Integral
In continuous time, the corresponding process is
the convolution integral, expressed as:
+∞
𝑦𝑦 𝑡𝑡 = �
−∞
𝑥𝑥 𝜏𝜏 ℎ 𝑡𝑡 − 𝜏𝜏 𝑑𝑑𝑑𝑑
• Note: Although discrete time was used for ease of
illustration in the example, for the rest of this
course we will use only continuous time
formulations.
h(t): Impulse response; x(t): System input; y(t): System output
Convolution (Cnt’d)
• Thus, the time response y(t) to an input x(t) is
𝑦𝑦 𝑡𝑡 = �
+∞
−∞
𝑥𝑥 𝜏𝜏 ℎ 𝑡𝑡 − 𝜏𝜏 𝑑𝑑𝑑𝑑 = 𝑥𝑥(𝑡𝑡) ∗ ℎ(𝑡𝑡)
• The output is the input convolved with the
impulse response function.
Linear Time-Invariant (LTI) Systems
• A linear system is one that has the following properties
(where y1(t) is the response to x1(t) and y2(t) is the
response to x2(t)):
– The response to x1(t) + x2(t) is y1(t) + y2(t)
– The response to ax1(t) is ay1(t), where a is any complex
constant
• A time-invariant system is one in which the behaviour and
characteristics of the system are fixed over time.
Most of the analysis techniques that you will learn in this class
assume that the system is linear and time-invariant.
An LTI system is fully characterized by its impulse
response function.
Key Points
• We can mathematically model a physiological
system (you already know how! It’s no different
than modeling other physical systems).
• We can describe a system using its dynamic
equation.
• We can examine how it would respond to an
impulse.
• If it is LTI, the impulse response fully characterizes
the system.