Transcript State Space Representation - Petra Christian University

State Space Representation
Hany Ferdinando
Dept. of Electrical Engineering
Petra Christian University
Overview
Introduction
State variable
Signal Flow Graph (revisited)
State Space (SS) matrix
Transfer function from SS
SS in Matlab
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Introduction
State Space (SS) represents a dynamic
system in matrices
It is MIMO (multi input multi output)
system (transfer function is SISO –
single input single output)
With SS, one can get future condition of
a dynamic system
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Examples
On-off switch:
There are only two positions
The state of the switch can assume one of
two possible states
if the present state is ‘on’, then one can
know the future state
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State Variable
Definition: is a set of variables such that
the knowledge of these variables and
the input functions will, with the
equations describing the dynamics,
provide the future state and output
of the system
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Model of A Dynamic System
Model of a dynamic system is in nthorder differential equation
The order depends on the number of
storage element in that system
SS will change one nth-order
differential equation into n first-order
differential equations
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Example
i(t)
dvC
iC  C
 i (t )  iL
dt
diL
uL  L
 vC  RiL
dt
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Example
dvC
1
1
 i (t )  iL
dt
C
C
1
1
vC  i (t )  iL
C
C
diL 1
R
 vC  iL
dt
L
L
iL  1 vC  R iL
L
L
If x1 = vC and x2 = iL then
1
1
x1  i (t )  x2
C
C
1
R
x2  x1  x2
L
L
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Example

0

x

 1
 x    1
 2 
L
1
  x   1 
C 1   i(t )
R   x2   C 
 
0
L
Matrix A
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Matrix B
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General Form of SS
x  Ax  Bu
y  C x  Du
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Signal Flow Graph (revisited)
Sum of the forward-path
factors
P

G ( s) 
1  L
k
k
N
q 1
q
Sum of the feedback loop
factors
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Signal Flow Graph State
From G(s) get the order of the equation
Make the denominator of G(s) in the
form of 1-(sum of the feedback loop) by
dividing G(s) with sn (n is the order of
G(s))
Draw the new SFG
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Example
b3 s  b2 s  b1 s  b0
G (s)  4
s  a3 s 3  a 2 s 2  a1 s  a 0
3
2
b3 s 1  b2 s  2  b1 s 3  b0 s  4
G (s) 
1
2
3
4
1  a3 s  a 2 s  a1 s  a 0 s
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Example
b3
1/s
U(s)
x4
-a3
1/s
-a2
b2
1/s
x3
b1
1/s
x2
x1
b0
Y(s)
-a1
-a0
Phase Variable Format
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Example
x2  x3
x1  x 2
x3  x4
x4  u  a3 x4  a2 x3  a1 x2  a0 x1
 x1   0
 x   0
 2  
 x 3   0
  
 x 4   a 0
1
0
0
1
0
0
 a1
 a2
0   x1  0
0   x 2  0
  u (t )
1   x 3  0 
   
 a3   x 4  1
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Example
b3
b2
b1
1/s
U(s)
x4
1/s
b0
1/s
x3
-a2
-a0
1/s
x2
-a3
x1
Y(s)
-a1
Input Feedforward Format
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Example
x1  a3 x1  x2  b3u
x 2  a2 x1  x3  b2 u
 x1    a3
 x   a
 2   2
 x 3    a1
  
 x 4   a 0
x 3  a1 x1  x4  b1u
x 4  a0 x1  b0 u
1 0 0  x1  b3 
0 1 0  x 2  b2 

u (t )
0 0 1  x3   b1 
   
0 0 0  x 4  b0 
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TF from SS
G( s)  C( s)B
1
(s) 
sI  A
CB
G (s) 
sI  A
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SS in Matlab (basic)
Use two functions:
tf2ss to convert transfer function model
into state space model
ss2tf to convert state space model into
transfer function model
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