UCK303E Automatic Control
Lecture 3 and 4 : Time Domain
Analysis of Dynamic Systems
and Introduction to Laplace
Transformation
Previous Lecture
• Dynamic modeling of mechanical and electrical
(analog) devices
– based on basic physical principles (Newton’s law, Kirchhoff’s
current and voltage law…).
– using Mass/Inductance (Inertia), Spring/Capacitance
(Energy-storage), Damper/Resistance (Energy-dissipation)
elements.
– resulting in second order ODEs, higher order ODEs, system
of differential equations.
• Two interesting sub-topics:
– state-space representation of dynamic systems
– linearization of non-linear equations of motions
III. Suspension of a Car
Output of interest: Vertical position of chassis
Plant: Vehicle chassis (including tires), some actuators, and their dynamics
Inputs: Altitude of car chassis with bumps in the road
Suspension of a Car
Applying Newton’s Law to each mass
When we rearrange these formulas:
Free-body diagram for
suspension system 4
IV. Rotational Pendulum
Follow the same steps :
Small theta assumption :
: natural frequency…
5
In this lecture
• We explore the time-domain understanding of the
transformation by solving a second-order ODE
• Explicit time-domain solution by finding the homogenous and
forced response.
• Modes of the system and the homogenous and forced response
– Homogenous Solution => Basic Stability Understanding (complex plane
plots of the natural frequencies).
– Forced Solution => Frequency Response (gain and phase plots).
• We explore the both the stability notion and the forced
response characteristics via examples.
We will continue with…
• Laplace Transformation (Frequency Domain).
• Dirac Delta, Convolution
=> Inherit tie between time domain and frequency domain…
• Transfer Function representation of dynamic systems (will
continue in the next lecture)
• And we enter into automatic control systems via analyzing the
stability and the time-response of second order dynamic systems
(will continue in the next lecture)…
Lecture Outline
• Later we will look at linear time-invariant differential
equations.
– Higher-order ODEs and system of differential equations
• State-space representation : set of first order linear differential
equations
Textbook References and Reading Assignment
• In this lecture we will cover the topic mainly from Ogata and
FPE , and we will use some excerpts from Kuo, Nise, Goodwin,
Dorf, Oishi (UCB) – thank you!!!.
• Did you do your last week’s reading assignment?
– FPE : Chapter 2 or
– Kuo : Chapter 4– relevant portions
• Textbook follow-up (this week’s reading assignment) :
– Ogata : Chapter 2 main text + Example problem and solutions
– FPE : Chapter 2
– Kuo : Chapter 2
Solving an ODE
Trailer Example
The car comes to a stop : how is the behavior of the trailer?
Almost like a wall when it stops…
Let’s denote b= Bh+Bt and k=Kh and analyze the system…
Mass-spring-damper system
Step 1 : Mass-spring-damper (friction) system Step 2 : Free-body diagram
Step 3 : Equations of Motion Denote u(t):=f(t) as the
input to the system
Let’s look at the solution of this ODE
Complex Number
Real and Imaginary
Portion
Carry “Information”
real part Complex part
Recall Complex Numbers and Euler’s Theorem
Real and Imaginary Portion
Carry “Information”
Homogenous solution on the complex plane
Focusing on the complex root case
Homogenous solution on the complex plane
Initial State Response
displacement
Complex pair time
Real Pair
Response Types in Complex Plane
Stable and unstable regions in the complex plane
S domain
Forced Response
Uo=Fo (recall slide 12)
The solution
Generic Second Order System
Generic Input – Output Set
Assuming that the initial conditions are all zero
Roots of this are called “the poles”
of the system….
G(s) is called the “transfer function”
Generic Second Order System
Homogenous Response
Frequency Domain Time Domain
Ties to Future Lectures…
Forced Response in the frequency plane
Let’s just wait for another second on
What the wn and ksi means…
This is a very important characteristic
of linear systems.
Signals and Systems Point of View
Into the depths of Laplace
Transformation
Laplace Transformation Fundamentals
Laplace, himself…
Laplace Transformation
Laplace and Inverse Laplace Transformation
Inverse Laplace Transformation:
Note : Laplace Transformation Existence
Example I : Exponential
Example II : Step
Example III : Sinusoidal
Laplace Transform Properties
Laplace transform theorems
Important Laplace Transformation Pairs
Second Order Systems
Laplace transform table
From time domain
to frequency domain -
The convolution perspective...
State-Space Formulation
Two key properties of linear constant systems
• The response of a linear
constant (i. e. not time
varying system
parameters such as k, b,
m…) system
– Obeys the principle of
For the sake of this discussion
superposition Imagine this….
– Can be expressed as the
convolution of the input
with the unit impulse
response of the system
Superposition Principle
• General Signals –
decompose it into a sum
of elementary signals :
– Impulse, exponential, …
In Class Example :
Dirac Delta Function
Physical Understanding: Impulse
Norm bounded ~ “Finite Energy”
If a function f(x) is continuous at x :
Notice that the Dirac Delta is acting like a “sampling” operator at x=0 !
First step to convolution : Sampling
f(t)
• For an arbitrary T and
again assuming f(t) is
continuous at t=T :
Notice that the Dirac Delta is acting like a “time shift” sampling operator now!
How about making T not a constant, but the free time variable t?
Convolution
Representing f(t) as a sum of impulses :
Example : How about u(t)? (i.e. an input signal…)
H(t,tau) : Impulse Response of a Dynamic System
: identical
convolution
In Class Example : Use Dirac Delta
As u
Let’s find the impulse response…
Complex Numbers and Euler’s Theorem
Real and Imaginary Portion
Carry “Information”
Transfer Function
Notice how the analysis turned into a simple algebraic multiplication…
Time domain vs. Frequency domain
Laplace Transformation Fundamentals
Laplace Transformation
Forced Response Characteristics
Trailer Example
Almost like a wall when it stops…
Mass-spring-damper system
IF y2 –y1 = y
Mass-spring-damper (friction) system Free-body diagram
State diagram
Example to come…
Second-
order
systems’
step
responses
Second-
order
step
response
components
generated
by
complex
poles
Step
responses
for
second-
order
system
damping
cases
Second-
order
response
as a
function
of
damping
ratio
Second-order
underdamped
responses for
damping ratio
values
Second-order
underdamped
response
specifications
Percent
overshoot
vs.
damping
ratio
Normalized rise
time vs.
damping
ratio for a
second-order
underdamped
response
Into the depths of Laplace
Transformation
Example I : Mechanical Systems
Force-velocity, force-displacement, and impedance
translational relationships for springs, viscous dampers, and
mass
Example II : Mechanical Systems
Torque-angular velocity, torque-angular displacement,
and impedance rotational relationships for springs, viscous
dampers, and inertia
Example III : Electrical Systems
Voltage-current, voltage-charge, and impedance relationships
for capacitors, resistors, and inductors
V(s)/q(s) I(s)/V(s)
Basic time-domain test signals for control systems
Step function Ramp-function
Parabolic function
