Controllability and Observability
Basic information
 Controllability: Possibility of changing the states from the input.
 Observability: Possibility of estimating the states from the output.
 Linear dependence (text p. 45):
o Definition: A set of vectors {x1, x2, …, xm} in  n is linearly independent iff
1 x1   2 x 2  ...   m x m  0  1   2  ...   m  0
o Examples:
 x1 = [1 0]T and x2 = [1 2]T
1(1) + 2(1) = 0
1(0) + 2(2) = 0  1 = 2 = 0

 they are linearly independent
x1 = [1 1]T and x2 = [2 2]T
1(1) + 2(2) = 0
1(1) + 2(2) = 0  1 = -22

 they are linearly dependent
Rank and nullity of a matrix (text p. 48)
o Definition: The rank of a matrix A is the number of linearly independent
columns in A. The nullity of a matrix A is the difference between the total
number of columns and the number of linearly independent columns in the
matrix A. You are required to learn how to use Matlab to find the rank of a
matrix (p.49)
o Example:
1 2 3
rank 4 5 6
7 8 9

1 2 3
2
rank 4 5 5
3
7 8 9
Diagonal form and Jordan form (text p. 55): A square matrix A can be transformed
into a diagonal or block diagonal form using the following transformation (see p. 61
for Matlab usages – eig(a) and Jordan(a)):
1
Aˆ  Q A Q
where Q  [q1 q 2 ... q n ] and q1 q 2 ... q n are eigenvectors of A.
o Distinct real s: Diagonal matrix with all real elements, every element
corresponds to an eigenvalue.
o Distinct complex s: Diagonal matrix with all real or complex elements, every
element corresponds to an eigenvalue. Additional transform can remove the
imaginary parts but the transformed matrix becomes block diagonal matrix
(modal form).
o Repeated s (text p. 60): If the eigenvalues are not all distinct, the resulting
matrix may comprise blocks of upper triangular matrices along the diagonal
(Jordan form):
Continuous-time systems
 Controllability: Since the output does not affect the controllability, only the internal
state equation is considered here.
x’ = A x + B u
o Definition: A system is controllable iff there exists an input that can transfer
any initial state to any final state in a finite time.
o Requirements: A is n  n and B is n  p
 The n  n matrix
t
W c (t )   e
0

A
T
A(t   )
T
T A (t   )
B B e A  d   e
BB e
d
t
T
0
is nonsingular for any t > 0.
The n  np controllability matrix
C  [ A0 B A1 B A2 B ... An1 B]
has rank n (full row rank).
Examples:
 1 0 
1
1
 1
1  1
A
B     I B   , A B     C  


 0  1
0
0
0
0 0 
  (C )  1  Uncontroll able

The n  (n + p) matrix [A – I B] has full row rank at every  of A.

The unique solution of A Wc + WcAT = -B BT is positive definite if the
s of A all have negative real parts. The solution is called the
controllability Gramian and is given by:
Examples:

T
W c (t )   e A B BT e A  d
0
 See p. 61 for Matlab usages – ctrb and gram
o The controllability indices (1, 2, …, n) are the ranks of the sub-matrices of
the controllability matrix C . The controllability index  is the maximum of
(1, 2, …, n).
o A is n  n, B is n  p, and (B) = P
o The controllability matrix C can be reduced to a lesser degree (A is n  n, B is
n  p, and (B) = P): The system is controllable iff
0
1
2
n p
C n p1  [ A B A B A B ... A B]
has rank n or the n  n matrix C n p1 C n p1 is nonsingular.
T
Example 6.5 (text, p. 151)
o The controllability property is invariant under any equivalence transformation.
o The controllability indices are invariant under any equivalence transformation
and any reordering of the columns of B.

Observability:
o Observability is a property of the pair (A, C) and is independent of B and D.
o Definition: A system is observable iff any unknown initial states can be
determined uniquely from the knowledge of the input and the output over a
finite time interval.
o Duality: The pair (A, B) is controllable  The pair (AT, BT) is observable.
o Requirements: A is n  n and C is q  n
 The n  n matrix
A
T
W o (t )   e A  C C e d
t
T
0

is nonsingular for any t > 0.
The nq  n observability matrix O
C A 0 


C A1 


C A 2 


O  .

.



.


n 1 
C
A




has rank n (full column rank).


A  I 
The (n + q)  n matrix 
 has full column rank at every  of A.
 C 
The unique solution of AT Wo + WoA = -CT C is positive definite if the
s of A all have negative real parts. The solution is called the
observability Gramian and is given by:

A
T
W o (t )   e A  C C e d
T
0
o (A is n  n, C is q  n, and (C) = q) The observability indices (1, 2, …, n)
are the ranks of the sub-matrices of the observability matrix O . The
observability index  is the maximum of (1, 2, …, n).
o The observability matrix O can be reduced to a lesser degree (A is n  n, C is
q  n, and (C) = q).The system is observable iff
O n  q 1
C A 0

1
C A

2
C A
 .

.
.

C A n  q













has rank n or the n  n matrix O nq1 O nq1 is nonsingular.
o The observability property is invariant under any equivalence transformation.
o The observability indices are invariant under any equivalence transformation
and any reordering of the row of C.
T

Jordan-Form: If a state equation is transformed into Jordan form, its controllability
and observability can sometimes be checked by inspection (Ex. 6.11, P. 169).
o A single-input system is controllable iff in the Jordan form matrix there is one
Jordan block associated with each distinct eigenvalue and every entry of B
corresponding to the last row of each Jordan block is nonzero.
o A single-output system is observable iff in the Jordan form matrix there is one
Jordan block associated with each distinct eigenvalue and every entry of C
corresponding to the first column of each Jordan block is nonzero.

Kalman decomposition theorem: A state-space equation can be transformed into the
form shown in text p. 167.
o This formulation has 4 different parts.
o Although normally we only consider the input-output relationship but the
internal states are important in terms of stability.
Controllable
Not Observable
Controllable
Observable
u(t)
y(t)
Not Controllable
Observable
Not Controllable
Not Observable
 x co '  A co

 
x c o '   A 21
 x co '  0

 
x c o ' 0

0
A co
0
0

A13
A 23
A co
A 43
0   x co  B co 
 
 

A 24  x c o  B c o 


u
0   x co  0 
 
 

A c o  x c o  0 
y  Cco Cco 0 0  x  D  u
o If we pick out the co part, we have a controllable/observable system:
x co '  A co x co  B cou
y  C co x co  Du
ˆ ( s )  C co ( sI  A co ) 1 B co  D
G
o It has the same transfer matrix as the original system.
o MATLAB usage: minreal can reduce any state equation to a controllable and
observable state equation (6.46, P. 163/4).
o Example:
0
0  x1  0
 x1 '   1 1 0
 x '  0  1 1
0
0  x2  1
 2 
 x3 '   0 0  1
0
0   x3   0  u
  
    
1  x4  0
 x4 '  0 0 0  2
 x5 '  0 0 0
0  2  x5  1
 x1 
x 
 2
 0 1 1 0 1   x3 
y
 
 x4 
 x5 
The above Jordan form is uncontrollable (why???). Rearrange the order we
have:
0
0 0  x1  0
 x1 '   1 1
 x '  0  1
0
0
1  x2  1
 2 
 x4 '   0 0  2
1 0    x 4   0   u
  
    
0  2 0  x5  1
 x5 '  0 0
 x3 '  0 0
0
0  1  x3  0
y

0
1
1
0
Discarding x3 we have:
0
0
 x1 '   1 1
 x '  0  1
0
0
 2 
 x4 '   0 0  2
1
  
0 2
 x5 '  0 0
  
 x1 
x 
 2
1   x 4 
 
 x5 
 x3 
  x1  0
  x  1
  2  
   x 4   0   u
    
  x5  1
    
 x1 
x 
 2
 0 1 0 1   x 4 
y
 
 x5 
 
The new system is now controllable but not observable (why???). Rearrange
the order we have:
0 0
0   x2  1
 x2 '  1
 x '  0  2 0
0   x5  1
 5 
 x1 '    1
0 1
0    x1   0  u
  
    
1 0  2   x 4  0 
 x4 '  0
  
    
 x2 
x 
 5
 1 1 0 0   x1 
y
 
 x4 
 
Again, discarding everything except the co part we have:
 x2 '  1 0   x2  1
 x '   0  2   x   1  u
  5  
 5 
x 
y  1 1  2 
 x5 
This system is both controllable and observable and it has the same transfer
function as the original system.

Systematic approach:
o An uncontrollable system can be transformed into a controllable one that has
the same transfer matrix (zero-state equivalent) (P.160, Ex. 6.8).
1) Form the controllability matrix and test the controllability of the system.
2) Determine the matrix Q for the transformation:
a. If rank(A) is n1, the first n1 columns of Q are any n1 linearly
independent columns from the controllability matrix.
b. The remaining columns can be chosen arbitrarily as long as Q is
nonsingular.
3) Let P = Q-1 and perform the following similarity transformation:
o
1)
2)
3)

A  P A P 1 B  P B C  C P 1 D  D
An unobservable system can be transformed into an observable one that has
the same transfer matrix (zero-state equivalent).
Form the observability matrix and test the observability of the system.
Determine the matrix P for the transformation:
a. If rank(A) is n2, the first n2 rows of P are any n2 linearly independent
rows from the observability matrix.
b. The remaining rows can be chosen arbitrarily as long as P is
nonsingular.
Perform the following similarity transformation:
A  P A P 1 B  P B C  C P 1
Matlab usages (p. 161) – ctrbf and obsvf
DD
Discrete-time systems
 Controllability: Since the output does not affect the controllability, only the internal
state equation ((A, B) pair) is considered here.
o Definition: A system is controllable iff there exists an input that can transfer
any initial state to any final state in a finite time.
o Requirements: A is n  n and B is n  p
 The n  n matrix
n 1
 
W dc (n  1)    A B B A
m 0

m
T
T m
is nonsingular for any t > 0.
The n  np controllability matrix
0
1
2
n1
C d  [ A B A B A B ... A B]
has rank n (full row rank).


The n  (n + p) matrix [A – I B] has full row rank at every  of A.
The unique solution of Wdc – A WdcAT = B BT is positive definite if
the |s| of A all less than 1. The solution is called the discrete
controllability Gramian and is given by:

 
W dc    A B B A
m
T
m 0

T m
Observability:
o Observability is a property of the pair (A, C) and is independent of B and D.
o Definition: A system is observable iff any unknown initial states can be
determined uniquely from the knowledge of the input and the output over a
finite time interval.
o Duality: The pair (A, B) is controllable  The pair (AT, BT) is observable.
o Requirements: A is n  n and C is q  n
 The n  n matrix
n 1
 
W do (n  1)   A
T m
m 0



C C  A
T
m
is nonsingular.
The nq  n observability matrix
C A 0 
 1 
C A



2
C A 

O d  .


.

.



C A n 1 


has rank n (full column rank).
A  I 
The (n  q)  n matrix 
 has full row rank at every  of A.
 C 
The unique solution of Wdo – AT WdoA = CT C is positive definite if
the |s| of A all less than 1. The solution is called the observability
Gramian and is given by:

 
W do   A
m 0
T m
C C  A
T
m
o (A is n  n, C is q  n, and (C) = q) The observability indices (1, 2, …, n)
are the ranks of the sub-matrices of the observability matrix O . The
observability index  is the maximum of (1, 2, …, n).
o The observability matrix O can be reduced to a lesser degree (A is n  n, C is
q  n, and (C) = q): The system is observable iff
O n  q 1
C A 0

1
C A

2
C A
 .

.
.

C A n  q













has rank n or the n  n matrix O nq1 O nq1 is nonsingular.
o The observability property is invariant under any equivalence transformation.
o The observability indices are invariant under any equivalence transformation
and any reordering of the columns of B.
Controllability index is the shortest input sequence that can transfer any state to any
other state.
Observability index is the shortest input and output sequence needed to determine the
initial state uniquely.
Controllability after sampling: The input of a continuous-time system is piecewise
constant with a sampling period of T.
o If the continuous-time system is uncontrollable, then its discretized state
equation is also uncontrollable, regardless of the sampling period.
o If the continuous-time system is controllable, then its discretized state
equation is also controllable if |Im[i – j]|  2 m / T for m = 1, 2,…,
whenever Re[i – j] = 0.
o Example: 6.12, P.174
T


