EEE8013

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EEE8013 Revision lecture 1 Ordinary Differential
Equations
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Modeling
Model Ordinary Differential Equations (ODE)
Dynamics Properties of the system, we have to
solve/study the ODE.
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First order systems Study approaches
First order ODEs
Analytic Explicit formula for x(t) (a solution
separate variables, integrating factor) which
satisfies
INFINITE curves (for all Initial Conditions
(ICs)).
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First order linear equations
First order linear equations - (linear in x and
x)
General form
Numerical Solution k5, u0.5
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Analytic solution Step input
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Response to a sinusoidal input
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Response to a sinusoidal input
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Second order ODEs
Second order ODEs
So I am expecting 2 arbitrary constants
u0 gt Homogeneous ODE
Lets try a
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Overdamped system
Roots are real and unequal
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Critically damped system
Roots are real and equal
A2, B1, x(0)1, x(0)0 gt c1c21
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Underdamped system
Roots are complex
Underdamped system
rabj
Theorem If x is a complex solution to a real ODE
then Re(x) and Im(x) are the real solutions of
the ODE
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Underdamped system, example
A1, B1, x(0)1, x(0)0 gt c11, c21/sqrt(3)
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Undamped
Undamped system
A0, B1, x(0)1, x(0)0 gtc11, c20
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Stability
In all previous cases if the real part is
positive then the solution will diverge to
infinity and the ODE (and hence the system) is
called unstable.
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Natural frequency, damping frequency, damping
factor
Case 1
gt Overdamped system implies that
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Natural frequency, damping frequency, damping
factor
Case 2
gt Critically damped system implies that
Case 3
gt Underdamped systems implies
called damped frequency or pseudo-frequency
Case 4
No damping the frequency of the oscillations
natural frequency
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Summary
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Stability revised
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NonHomogeneous (NH) differential equations

• u0 gt Homogeneous gt x1 x2.
• Assume a particular solution of the
  nonhomogeneous ODE xp

If u(t)Rcosnt gt
Then all the solutions of the NHODE are
So we have all the previous cases for
under/over/un/critically damped systems plus a
constant R/B.
If complementary solution is stable then the
particular solution is called steady state.
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Example
x(0)1, x(0)0 gt c1-1, c2-1/sqrt(3)
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State Space
Very difficult to be studied gt so we use
computers Computers are better with 1st order
ODE 1 nth gt n 1st Powerful tools from the
linear algebra
Use sensors Output x gt
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State Space
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Block Diagram

• X is an n x 1 state vector
• U is an q x 1 input vector
• Y is an p x 1 output vector
• A is an n x n state matrix
• B is an n x q input matrix
• C is an p x n output matrix
• D is an p x m feed forward matrix (usually zero)

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State space rules
The state vector describes the system gt Gives
its state gt The state of a system is a
complete summary of the system at a particular
point in time. If the current state of the
system and the future input signals are known
then it is possible to define the future states
and outputs of the system.

• The choice of the state space variables is free
  as long as some rules are followed
• They must be linearly independent.
• They must specify completely the dynamic
  behaviour of the system.
• Finally they must not be input of the system.

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State space
The systems states can be written in a vector
form as

• A standard orthogonal basis (since they are
  linear independent)
• for an n-dimensional vector space called state
  space.

Matlab definition
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Solution
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Solution II
How can we solve that ???
(This last equation is the characteristic
equation of the system, why???).
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Solution III
I assume that a21 so a12
I assume that a12 so a2-2
Matlab example
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General Solution
The roots of this equation are called eigenvalues
negative eigenvalues gt stable positive
eigenvalues gt unstable
repeated eigenvalues gt eigenvectors are not
linearly independent.
Complex eigenvalues gt conjugate and the
eigenvector will be complex gtsolution will
consists of sines, cosines and exponential terms
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Properties of general solution
If we start exactly on one eigenvector then the
solution will remain on that forever. Hence if I
have some stable and some unstable eigenvalues it
is still possible (in theory) for the solution to
converge to zero if we start exactly on a stable
eigenvector.
Determines the nature of the time response
(stable, fast..)
Determines the extend in which each state
contributes to
Determines the extend in which the IC excites the
To find the eigenvalues and eigenvectors use the
command eig()
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Example
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Example
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Example
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State Transition Matrix
Until now the use of vector ODEs was not very
helpful.
We still have special cases
gt No special cases are needed then
4 ways to calculate it!!!
Use the command expm (not exp)!
X(0)1 2
X(5) ?
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State Transition Matrix
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State Transition Matrix
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Solution
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SS gt TF???
TF
Response to ICs
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SS gt TF???
CE of the TF
The TF is a matrix
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SS gt TF???
Example Find the TF of
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Basic properties of state space
State space transformations
State space representations are not unique
Same input/output properties, gt same
eigenvalues
T is an invertible matrix
Z is the new state vector
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Basic properties of state space
Do these two systems have the TF?
Matlab example
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Observability - Controllability
Observability - Controllability
Notice the structure of A and C
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Observability - Controllability
There is a pole zero cancellation
The cancellation is due to C3 0.
pzmap(ss_model)
Matlab verification
We can influence x2 through U but we cannot
observe how it behaves and hence there is no way
to feedback that signal to a controller!!!
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Observability - Controllability
In this case we can see how both states behave
but we can not change U in any way so that we
can influence x2 due to the form of B.
Unobservable uncontrollable
Minimal realisation.
Difficult task if the system is nonlinear!!!!
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Observability - Controllability
Check the rank
gtgt rank(obsv(A,C)) gtgt rank(ctrb(A,B))