UNIVERSITAS INDONESIA
Veritas, Probitas, Justitia
Est. 1849
Dr. Aries Subiantoro, ST. MSc.
State Space Modeling
A
state-space model represents a system by a series of
first-order differential state equations and algebraic
output equations
 State-space models are numerically efficient to solve,
can handle complex systems, allow for a more
geometric understanding of dynamic systems, and form
the basis for much of modern control theory
Figure 3.1
RL network
Control Systems Engineering, Fourth Edition by Norman S. Nise
Copyright © 2004 by John Wiley & Sons. All rights reserved.
State-Space Modeling

In general state-space models have the
following form (equations can be nonlinear and
time varying)
x1  f1 ( x1 , x2 , , xn , u1 , , um )
state
equations

xn  f n ( x1 , x2 , , xn , u1 , , um )
y1  h1 ( x1 , x2 , , xn , u1 , , um )
output
equations

y p  h p ( x1 , x2 , , xn , u1 , , um )
Coupled Tank System Model
dh1
A1
dt
dh
A2 2
dt
 Q1  a1sign (h1  h2 ) 2 g h1  h2
 a1sign (h1  h2 ) 2 g h1  h2  Qc
State-Space Modeling

For linear systems, can write as matrices

x

y

( nx 1)
( mx 1)
A x  ( nBxp ) u
( nxn ) ( nx 1)
( px 1)
state equations
( mxn ) ( nx 1)
( px 1)
output equation
C x  ( mDxp ) u
D
C
B
A
State-Space Modeling
There is a more intuitive way to find state-space
models
 Static system – current output depends only on
current input
 Dynamic system – current output depends on
current and past inputs (can be captured by initial
conditions)

State-Space Modeling
Question: What initial conditions do I need to
capture the system’s state?
 Definition: the state of a dynamic system is the
smallest set of variables (called state variables)
whose knowledge at t = t0 along with knowledge
of the inputs for t  t0 completely determines the
behavior of the system for t  t0.

Figure 3.5
Electrical network for
representation in state
space
Control Systems Engineering, Fourth Edition by Norman S. Nise
Copyright © 2004 by John Wiley & Sons. All rights reserved.
Figure 3.8
Electric circuit
for Skill-Assessment
Exercise 3.1
Control Systems Engineering, Fourth Edition by Norman S. Nise
Copyright © 2004 by John Wiley & Sons. All rights reserved.
Example-1
Consider the mechanical system shown in figure. We assume that the system is linear.
The external force u(t) is the input to the system, and the displacement y(t) of the mass
is the output. The displacement y(t) is measured from the equilibrium position in the
absence of the external force. This system is a single-input, single-output system.
From the diagram, the system equation is
Example-1
Then we obtain
Or
The output equation is
Example-1
• In a vector-matrix form,
 x1 (t )   0
 x (t )   k
 2   m
y ( t )  1
1   x (t )   0 
b   1    1 u (t )
   x2 (t )   
m
m
 x1 (t ) 
0 

 x 2 (t ) 
Example-1
The block diagram format:
Example-2
Obtain state equations of following mechanical translational system
and draw the state diagram. Where f(t) is input and x1 is output.
System equations are:
M1
d 2 x1
dt
2
dx1
D
 Kx1  Kx 2  0
dt
f (t )  M 2
d 2 x2
dt
2
 Kx2  Kx1
Example-2
dx1
 v1
dt
Now
d 2 x1
dt 2
dv1

dt
2
dx2
d
x2 dv2
 v2

2
dt
dt
dt
Choosing x1, v1, x2, v2 as state variables
dx1
 v1
dt
dx2
 v2
dt
dv1
M1
 Dv1  Kx1  Kx 2  0
dt
dv2
f (t )  M 2
 Kx 2  Kx1
dt
Example-2
In Standard form
dx1
 v1
dt
dv1
D
K
K

v1 
x1 
x2
dt
M1
M1
M1
dx2
 v2
dt
dv2
K
K
1

x2 
x1 
f (t )
dt
M2
M2
M2
Example-2
dx1
 v1
dt
dv1
D
K
K

v1 
x1 
x2
dt
M1
M1
M1
dx2
 v2
dt
dv2
K
K
1

x2 
x1 
f (t )
dt
M2
M2
M2
In Vector-Matrix form
 0
 x1   K
   
 v1    M 1
 x 2   0
   K
 v 2   M
2

1
D

M1
0
0
0
K
M1
0
K

M2
0
 0 
x


 1
0    0 
v1  



  0  f (t )
1  x2 
 1 



0  v2  

M2 
Figure 3.9
Translational
mechanical system
for Skill-Assessment
Exercise 3.2
Control Systems Engineering, Fourth Edition by Norman S. Nise
Copyright © 2004 by John Wiley & Sons. All rights reserved.
Figure 3-6 (p. 165)
Control Systems Engineering, Fourth Edition by Norman S. Nise
Copyright © 2004 by John Wiley & Sons. All rights reserved.
Example-3
State space representation of armature Controlled D.C Motor.
Ra
ea
La
ia
B 
eb
T
J

ea is armature voltage (i.e. input) and  is output.
di a
e a  R a i a  La
 eb
dt
T  J  B
Example-3
T  K t ia
J   B -K t i a  0
eb  K b
di a
La
 R a i a  K b  e a
dt
Figure P3.12
Motor and load
Control Systems Engineering, Fourth Edition by Norman S. Nise
Copyright © 2004 by John Wiley & Sons. All rights reserved.
Figure 3.11
Decomposing a
transfer function
Control Systems Engineering, Fourth Edition by Norman S. Nise
Copyright © 2004 by John Wiley & Sons. All rights reserved.
Figure 3.12
a. Transfer function;
b. decomposed
transfer function;
c. equivalent block diagram. Note:
y(t) = c(t)
Control Systems Engineering, Fourth Edition by Norman S. Nise
Copyright © 2004 by John Wiley & Sons. All rights reserved.
Homeworks
Submit Tugas: EMAS
Deadline: Senin, 19 September 2022 Pukul 07.00 WIB
Homeworks
Homeworks
Homeworks

Nise chapter 3:

Problems: 2, 3, 4, 19, 28
Next Lecture

Linear Sampled-Data Systems Phillips ch.
2-3