Control Systems and Adaptive
Process. Design, and control
methods and strategies
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Controllability and observability
Controllability
• Consider the system of n states and p inputs

x  Ax  Bu
with constant matrices A  nxn and B   . The states
equation, or the pair (A, B), is said to be controllable if for any
n
initial state x(0)  x0  n and any final state x1   , there is
an input that transfers the state x from x0 to x1 in finite time.
Otherwise, the equation (1.1), or the pair (A, B), is said noncontrollable.
nxp
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Controllability and observability
Controllability
• You can determine if the system is controllable by examining
the algebraic condition:


rank B AB A2 B  An1B  n
• Matrix A has dimension n x n and B n x 1. For systems with
multiple input matrix B is n × m, where m is the number of
inputs.
• For a system of single-input single-output, controllability
matrix Pc is described in terms of A and B as:


Pc  B AB A2 B  An 1B
which is an n x n matrix, therefore, if the determinant of Pc is
not zero, the system is controllable.
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Controllability and observability
Controllability
• Example: consider the system
 0
x (t )   0

  a 0
1

 0
A 0

  a 0
0
 a1
1
0
 a1
0 
 0
1  x   0 u

 
 a 2 
1
0 
1 

 a 2 
y  1 0 0x  0u
 0
B   0
 
1
 0 
AB   1 



a
 2 
from which we have that

Pc  B
AB
0
0
A 2 B  0
1

1  a 2

1


A2 B    a 2 


2
(
a

a
)
 2
1 


 a2 

2
( a 2  a1 )
1
The determinant of Pc = 1 ≠ 0, so that the system is
controllable.
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Controllability and observability
Controllability test
• The following statements are equivalent:
1. The pair ( A, B), A  nn , B  n p , is controllable
2. The controllability matrix

C  B AB A2 B  An 1B

C  nnp
is of rank n (full row rank).
3. Matrix n x n
t
Wc (t )   e BB e
A
T
AT 
0
t
d   e A( t  ) BBT e A
T
( t  )
d
0
is nonsingular for all t > 0.
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Controllability and observability
Minimum energy control
• Control spending minimum energy to bring the system
from state x0 to state x1 at time t1, in the sense that, for
other control ũ(t) to make the same transfer, is always
true that:
t1

0
t1
 t1 A( t  ) T AT ( t  )  1
2
2
T AT ( t1 )
T
1
~
u ( ) d   u( ) d  ( x0 e
 x1 )Wc (t1 )  e 1 BB e 1 d Wc (t1 )(e A( t1 ) x0  x1 ) 


0
o

 ( x0T e A
T
( t1 )

1
2
2
 x1T )Wc1 (t1 )(e A( t1 ) x0  x1 )  Wc (t1 )(e A( t1 ) x0  x1 )
It is observed that the minimum control power is greater
when the distance between x0 and x1 is greater, and the
transfer time t1 is lower x0 y x1.
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Controllability and observability
Controllability PBH tests
•
The Popov-Belevitch Hautaus (PBH) tests have interesting
geometric interpretations used to analyze the controllability
in the form of Jordan. There are two types of test, of
eigenvectors and of rank.
1. Eigenvectors test: The pair (A,B) is not controllable if and
1n
v


only if there is a left eigenvector
of A such that
vB  0
2. Rank PBH test: Pair (A,B) is controllable if and only if
ranksI
 A B  n
for every s
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Controllability and observability
Controllability PBH tests
• Controllability and similarity transformation: invariance
theorem regarding controllability coordinate changes..
Controllability is an invariant property with respect to
equivalence transformations (coordinate changes).
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Controllability and observability
Observability
• All poles of a closed-loop system can be placed
arbitrarily in the complex plane if and only if the
system is observable and controllable. Observability
refers to the possibility of estimating a state variable.
• According to R. Dorf, a system is completely
observable if and only if there exists a finite time T
such that the initial state x(0) can be determined
from the observation of history y(t) given the control
u(t).
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Controllability and observability
Observability
• Considering the system of one input and one output

x  Ax  Bu
and
y  Cx
where C is a row vector 1 x n and x is a column vector n x 1.
This system is fully observable when the determinant of the
observability matrix Po is nonzero, where
 C 
 CA 

Po  
  
 n 1 
CA 
which is a matrix of n x n.
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Controllability and observability
Observability
• Example: Consider the system
 0
A 0

  a 0
1
0
 a1
0 
1 

 a 2 
Therefore
CA  0
Is thus obtained
1 0
C  1 0 0
and
CA2  0 0 1
1 0 0 
Po  0 1 0


0 0 1
the determinant of Po = 1 and the system is fully
observable. Note that the determination of the
observability matrix does not use matrices B and D.
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Controllability and observability
Observability
• The concept of observability is dual to controllability. Tries to
find out the possibility of estimating the system state from the
knowledge of the output. Consider the steady linear system

x  Ax  Bu
A  nn ;
y  Cx  Du
B  n p ;
C  qn ;
D  q p
This state equation (1.2) is observable if for any unknown
initial state x(0), there is a finite time t1 such that the
knowledge of the input u and the output y on the interval
[0,t1] is sufficient to determine uniquely the initial state x(0).
Otherwise the system is not observable.
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Controllability and observability
State variables
• For a given system, they exist plenty of possible sets
of state variables. However, all possible sets must
consist of the same number of state variables and
the defined variables must be fully independent.
Understanding as independent variable that whose
value cannot be expressed in terms of the other
variables; which implies that the initial values of each
of the chosen state variables may be assigned freely.
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Controllability and observability
State variables
•
Example, in a system such as shown in figure 3.1 may be
taken as state variables the speed ẏ(t) of the mass M and the
force ky(t) in the spring; the strength in the spring and the
displacement y(t) of the mass may not be taken, since the
former is equal to the second multiplied by the constant K.
Another valid alternative would be to take as state variables
of the system the speed ẏ(t) and the displacement y(t) of the
mass.
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Controllability and observability
State variables
• General methods for the selection of the state variables of a
system:
-Method of physical variables: the selection of the state
variables is performed based on the energy storage
elements existing in the system.
-Method of phase variables.
-Jordan canonical form.
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Controllability and observability
State variables
• Linear systems with variable parameters: in a system whose
dynamic behavior is characterized by
This equation can be represented by the following state and output
equations
Calculating coefficients Bi(t) by means of
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Controllability and observability
State variables
• Obtaining the transfer function from the state equations:
– The transfer matrix or function of a linear time-invariant
system can be obtained from the state equations of the
system by applying the Laplace transform.
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Bibliography
• R. Dorf, R. Bishop: Modern control systems.
• Class notes ETSII. UNED
Interesting links
• http://iaci.unq.edu.ar/materias/control2/web/clases/Cap6.pdf
• http://www.slideshare.net/IsRrItA/variables-de-estado
• http://www.virtual.unal.edu.co/cursos/ingenieria/2001619/lecciones/esta
do/node4.html#SECTION00631000000000000000
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