State Space Trajectories Analysis of Control Systems in
State Space
imtiaz.hussain@faculty.muet.edu.pk
Introduction to State Space
• The state space is defined as the n-dimensional space in
which the components of the state vector represents its
coordinate axes.
• In case of 2nd order system state space is 2-dimensional space
with x1 and x2 as its coordinates (Fig-1).
 x1 
 a11 a12   x1 
 b1 
  u(t)
 x   a



 2
 21 a22   x2 
b2 
x2
x1
Fig-1: Two Dimensional State space
State Transition
• Any point P in state space represents the state of the system
at a specific time t.
x2
P(x1, x2)
x1
• State transitions provide complete picture of the system
x2
t0
t6
t1
t2
t3
t5
t4
x1
Forced and Unforced Response
• Forced Response, with u(t) as forcing function
 x1   a11 a12   x1   b1 
  u(t )
 x   a



 2   21 a22   x2  b2 
• Unforced Response (response due to initial conditions)
 x1   a11 a12   x1( 0)


 x   a

 2   21 a22   x2 ( 0)
Solution of State Equations &amp; State Transition Matrix
• Consider the state space model
x(t )  Ax(t )  Bu (t )
• Solution of this state equation is given as
t
x(t )   (t ) ( 0)    (t   )Bu ( )d
0
• Where (t ) is state transition matrix.
 (t )   [( SI  A) ]  e
1
1
At
Example-1
• Consider RLC Circuit
iL
Vc
+
+
Vo
-
dvc
C
 u ( t )  iL
dt
-
diL
L
  Ri L  vc
dt
Vo  Ri L
• Choosing vc and iL as state variables
dvc
1
1
  iL  u(t )
dt
C
C
diL 1
R
 vc  iL
dt
L
L
Example-1 (cont...)

vc   0
 i    1
 L 
L
1
  v   1 
c
C
  C u(t )



R  iL   
 
0
L
R  3, L  1 and C  0.5
vc  0
 i   1
 L 
 2 vc  2
  u(t )



 3 iL  0
Example-1 (cont...)
vc  0
 i   1
 L 
 2 vc  2
  u(t )



 3 iL  0
• State transition matrix can be obtained as
  S
 (t )   1[( SI  A)1 ]   1  
  0
0  0  2  
 



S  1  3  

• Which is further simplified as
2
 S  3




1 ( S  1)( S  2 )
(
S

1
)(
S

2
)
 (t )   

1
S


 ( S  1)( S  2) ( S  1)( S  2) 
1
Example-1 (cont...)
2

 S  3



1 ( S  1)( S  2 )
)
2

S
)(
1

S
(
 (t )   

S
1


 ( S  1)( S  2) ( S  1)( S  2) 
• Taking the inverse Laplace transform of each element
( 2e  e ) ( 2e  2e )
 ( t )   t  2 t
t
 2t 
 ( e  e ) ( e  2e ) 
t
2t
t
2t
State Space Trajectories
• The unforced response of a system released from any initial
point x(to) traces a curve or trajectory in state space, with
time t as an implicit function along the trajectory.
• Unforced system’s response depend upon initial conditions.
x(t )  Ax(t )
• Response due to initial conditions can be obtained as
x(t )  (t )x(0)
Example-2
• For the RLC circuit of example-1 draw the state space trajectory
with following initial conditions.
• Solution
vc (0) 1
  
 i L ( 0 )   2
x(t )  (t )x(0)
t
2t
t
2t
t
2t
v



1
 c  ( 2e  e ) ( 2e  2e )    3e  3e 
  t
 i    t  2 t
t
 2t   
 2t 
 L   (e  e ) ( e  2e )  2   e  3e 
Example-2 (cont...)
• Following trajectory is obtained
State Space Trajectory of RLC Circuit
2
1.5
t--------&gt;inf
iL
1
0.5
0
-0.5
-1
-1
-0.5
0
0.5
Vc
1
1.5
2
Example-2 (cont...)
State Space Trajectories of RLC Circuit
2
1.5
0 
1 
 
1
iL
0.5
1 
0 
 
 1
0
 
0
-0.5
0
1
 
-1
-1.5
-2
-2
-1.5
-1
-0.5
0
Vc
0.5
1
1.5
2
Equilibrium Point
• The equilibrium or stationary state of the
system is when
x(t )  0
State Space Trajectories of RLC Circuit
2
1.5
1
iL
0.5
0
-0.5
-1
-1.5
-2
-2
-1.5
-1
-0.5
0
Vc
0.5
1
1.5
2