Lecture A3 : State space models for
continuous-time systems
Jan Swevers
July 2006
Aim of this lecture :
Teach you the procedure to derive a state space model starting from the system
equations:
• using the natural state variables approach
• using the direct simulation diagram
0-0
Lecture A3 : State space models for continuous-time systems
Contents of this lecture
• The state space model of a continuous-time system
• The state of a system
• How to derive a state space model :
– using the natural state variables
– using the direct simulation diagram
• Example
Control Theory [H04X3a]
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Lecture A3 : State space models for continuous-time systems
The state space model of a continuous-time
system
• It is the description of the system dynamic as a set of first order differential
equations.
• They are derived by introducing a set of state variables.
• The number of state variables is (in most cases) equal to the order of the
differential equation.
•
ẋ = Fx + Gu : state equation
y
= Hx + Ju : input/output − equation
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Lecture A3 : State space models for continuous-time systems
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Lecture A3 : State space models for continuous-time systems
The state of a system
• If we know the state of a system at a certain time instance and its future
input, we can calculate its future output unambiguously. The state of a
system is the minimal information that is required of its past
history/trajectory to predict its future. How the system arrived in this
state is not relevant.
• Examples :
– the voltage over a capacitor,
– the current through an inductance,
– the force through a spring,
– the velocity of a mass.
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Lecture A3 : State space models for continuous-time systems
• The choice of the state variables in not unique.
• We can choose the system variables that are responsible for the energy
storage in the system: the natural state variables.
• The number of state variables is equal to the number of independent energy
storing elements in the system. For certain systems, called degenerated
systems, this can be different from the number of energy storing elements
in the system. In such cases, an arbitrary specification of the values of the
variables that determine the energy storage in these elements in not
possible because of the geometry of the system (interconnection of certain
elements).
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Lecture A3 : State space models for continuous-time systems
How to determine the state space equations
using the natural state variables approach
Example :
An electro-mechanical system:
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Lecture A3 : State space models for continuous-time systems
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• Loop equation for the electrical part of the system:
di
u(t) = Ri + L + eb , with eb = Kω
dt
• Node equation for node 4 yields:
dω
I
+ Cω + T = 0, with T = −Ki
dt
• Natural state variables:
• This yields:




ẋ1
ẋ2


 = 
y
−R
L
+K
I


−K
L

x1
x2
=
i
ω


x1



+
− CI
x2


h
i x
1

 + 0 . u(t)
=
0 1
x2
Control Theory [H04X3a]
1/L
0

 u(t)
Lecture A3 : State space models for continuous-time systems
How to determine the state space equations from
the direct simulation diagram
Example
• Consider the following system:
ÿ(t) + 7ẏ(t) + 12y(t) = u̇(t) + 2u(t)
• Introduce the following operator s = d/dt:
(s2 + 7s + 12)y(t) = (s + 2)u(t)
of
B(s)
s+2
u(t) = 2
u(t)
y(t) = G(s)u(t) =
A(s)
s + 7s + 12
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Lecture A3 : State space models for continuous-time systems
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• Remark that:
d
1
s=
→
=
dt
s
Z
t
dτ
−∞
• Introducing an auxiliary variable y1 (t):
µ
¶
B(s)
1
y(t) =
u(t) = B(s)
u(t) = B(s)y1 (t)
A(s)
A(s)
and consequently :
A(s)y1 (t) = (s2 + 7s + 12)y1 (t) = ÿ1 (t) + 7ẏ1 (t) + 12y1 (t) = u(t)
..
.
y1(t)
u(t)
+
-
-
y1(t)
1/s
y1(t)
1/s
7
12
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Lecture A3 : State space models for continuous-time systems
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• This yields :
y(t) = B(s)y1 (t) = (s + 2)y1 (t) = ẏ1 (t) + 2y1 (t)
1
..
.
y1(t)
u(t)
+
-
-
y1(t)
1/s
+
y1(t)
1/s
2
7
12
Control Theory [H04X3a]
+
+
y(t)
Lecture A3 : State space models for continuous-time systems
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• Introducing the state variables x1 and x2 : the outputs of both integrator
elements:
1
u(t)
+
-
1/s
-
x1(t)
x2(t)
1/s
2
+
+
+
7
12
• This yields the following state space description:
ẋ1 (t) = −7x1 (t) − 12x2 (t) + u(t)
ẋ2 (t) = x1 (t)
y(t) = x1 (t) + 2x2 (t)
Control Theory [H04X3a]
y(t)
Lecture A3 : State space models for continuous-time systems
• This yields the control canonical form of the state space model.


 
−7 −12
1
, G =  
F = 
1
0
0
i
h
H =
1 2 , J=0
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Lecture A3 : State space models for continuous-time systems
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General form for the control canonical state space model :
B(s)
b1 sn−1 + b2 sn−2 + . . . + bn
u(t)
y(t) =
u(t) =
A(s)
sn + a1 sn−1 + . . . + an
yields :

F =










G =
−a1







−a2
1
..
.
. . . −an
0
...
0
0
1
0
0
1
0
..
.
0
0








...
Control Theory [H04X3a]










Lecture A3 : State space models for continuous-time systems
H =
J
=
h
0
b1
b2
. . . bn
14
i
In Matlab :
B = [0 1 2]; A = [1 7 12]; [F,G,H,J]=tf2ss(B,A);
Control Theory [H04X3a]
Lecture A3 : State space models for continuous-time systems
Example: quarter car
• Equations of motion
(1)
kw (r − x) − m1 ẍ − ks (x − y) − b(ẋ − ẏ) = 0
(2)
ks (x − y) + b(ẋ − ẏ) − m2 ÿ = 0
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Lecture A3 : State space models for continuous-time systems
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• Natural state variables
x1
= x
x2
= ẋ
x3
= x−y
x4
=
ẏ
• This yields the following equations:
ẋ1
ẋ2
ẋ3
ẋ4
= x2
b
ks
kw
kw
= ẍ = −
ẋ3 −
x3 −
x1 +
r
m1
m1
m1
m1
= x2 − x4
b
ks
= ÿ =
ẋ3 +
x3
m2
m2
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Lecture A3 : State space models for continuous-time systems
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• or



ẋ1
0



 ẋ 
 − kw
2




 =  m1
 ẋ3 
 0



0
ẋ4
1
0
− mb1
ks
−m
1
1
0
b
m2
ks
m2

y
=
h
1
0



x1
0


 
b
  x   kw 
  2   m1 
m1
r

+


 
−1 
  x3   0 
− mb2
0
x4

0
x1



i x 
 2 

−1 0 
 x3 


x4
Control Theory [H04X3a]
Lecture A3 : State space models for continuous-time systems
Aim of this lecture
• You must be able to derive a state space model starting from the system
equations:
– using the natural state variables approach
– using the direct simulation diagram
Control Theory [H04X3a]
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