EE 529 Circuit and Systems Analysis Lecture 9

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Lecture 9
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State vector
a listing of state variables in vector form
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State equations
System dynamics
Input vector
State vector
Measurement Read-out map
Output vector
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xn-vector (state vector)
up-vector (input vector)
ym-vector (output vector)
n
Anxn
System matrix
n
p
Bnxp
Input (distribution) matrix
n
n
Cmxn
Output matrix
m
p
Dmxp
Direct-transmission matrix
m
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Solution of state eqns
Consists of
Free response
(Homogenous soln)
(particular soln)
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Homogenous solution
Homogenous equation
has the solution
State transition matrix
X(0)
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State transition matrix
An nxn matrix ?(t), satisfying
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Determination of ?(t) transform method
Laplace transform of the differential equation
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Determination of ?(t) transform method
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Determination of ?(t) time-domain solution
Scalar case
?
where
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Determination of ?(t) time-domain solution
For vector case, by analogy
?
where
Can be verified by substitution.
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Properties of TM
?(0)I
F(t)
F(-t)
?-1(t) ?(-t)
F(t2-t0)
F(t1-t0)
F(t2-t1)
?(t2-t1)F(t1-t0) F(t2-t0)
F(t)
F(t)
F(t)
F(t)
F(t)
F(t)
F(kt)
F(t)k F(kt)
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General solution
Scalar case
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General solution
Vector case
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General solution transform method
L
?
?
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Inverse Laplace transform yields
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For initial time at tt0
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The output
y(t)Cx(t)Du(t)
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Example

• Obtain the state transition matrix ?(t) of the
  following system. Obtain also the inverse of the
  state transition matrix ?-1(t) .

For this system
the state transition matrix ?(t) is given by
since
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Example
The inverse (sI-A) is given by
Hence
Noting that ?-1(t) ?(-t), we obtain the inverse
of transition matrix as
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Exercise 1

• Find x1(t) , x2(t)
• The initial condition

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Exercise 1 (Solution)
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Example 2
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Exercise 2

• Find x1(t) , x2(t)
• The initial condition
• Input is Unit Step

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Exercise 2 (Solution)
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Matrix Exponential eAt
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Matrix Exponential eAt
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The transformation where
?1,?2,,?n are distinct eigenvalues of A. This
transformation will transform P-1AP into the
diagonal matrix
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Example 3
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• Method 2

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Matrix Exponential eAt
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Matrix Exponential eAt
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Example 4
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Laplace Transform
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