EE611
Deterministic Systems
Controllability and Observability of
Continuous-time Systems
Kevin D. Donohue
Electrical and Computer Engineering
University of Kentucky
Controllability Observability Example
For the circuit below with current source u as input,
voltage y as output, and state defined as the voltage on
the capacitors as indicated. Derive the state-space
model and discuss the ability to control (affect) the state
from the input and observe the state from the output.
u(t)
+ x1 -
+ x2 -
1Ω
1Ω
1F
[ ][
2Ω
1F
][ ] [ ]
+
y(t)
-
ẋ1
−1 0 x1
1
=
 ut 
0 −1 x 2
0
ẋ2
[]
x1
y t =[ 0 −1 ]
[ 2 ] ut 
x2
Controllability
The state equation:
ẋ=A xB u
is controllable if for any pair of states x(t1) and x(t0), ∃
an input u(t) that drives state x(t0) to x(t1) in a finite
time.
If the system is controllable, then an input to transfer
state x(t0) to x(t1) over the time interval [t0, t1] is given
by:
−1
ut =−B ' expA ' t 1 −t W c t 1, t 0  [ expAt 1−t 0 x t 0 −x t 1  ]
where
t
W c t , t 0 =∫ expA B B ' expA ' d 
t0
Conditions for Controllability
For an n state and p input system: ẋ=A xB u
This system is controllable for t > 0 if any one of the
equivalent conditions are met:
1. The n x n matrix Wc is nonsingular for all t > 0
t
t
W c t=∫ expA  B B ' expA '  d =∫ exp A t− B B' expA ' t−d 
0
0
2. The n x np controllability matrix C has full row rank
(n):
C=[ B A B A B A B ... A
2
3
n−1
B]
Conditions for Controllability
3. The n x (n+p) matrix [(A-λI) B] has full row rank
for every eigenvalue λ of A.
4. All eigenvalues of A have negative real parts, and
the unique solution Wc is positive definite.
A W c W c A '=−B B '
where Wc is the controllability Gramian:
∞
W c t =∫ expA B B ' expA ' d 
0
Controllability Examples
Given systems matrix A, determine which input vectors
result in controllability
[
−1 1
0
A= 0 −1 0
0
0 −2
] [] [] []
0
ba = 0
1
1
bb= 0
1
0
bc = 1
1
Given systems matrix A, determine which input
matrices result in controllability
[
−1 0
A=
0 −1
]
[]
ba =
0
1
[]
bb=
1
1
[ ]
bc = 1 0
0 1
Controllability Examples
Given system matrix A and input vector b determine an
input to bring initial state x(0) = 0 to x(4) = [-.2 .8 .1]'
[
−1 1
0
A= 0 −1 0
0
0 −2
]
[]
0
bc = 1
1
Hint: Do not do by hand. Plot input from 0 to 4 s when
finished. Also consider changing the state to the desired
value over a shorter time interval and describe how the
input changes. For example try
x(0) = 0 to x(2) = [-.2 .8 .1]'
Controllability Index
Consider an n state x p input system. The
controllability matrix is given by:
C=[ B A B A B ... A
2
m−1
B ... A
n−1
B]
where B can be expressed as a concatenation of column
vectors:
B=[ b1 b2 ... bm ... bp ]
Let µm be the number of linearly independent vectors in
C associated with with column m. Then C has rank n if
1 23... p =n
Then {1,  2, 3, ... , p} are called the controllability
indices and the maximum value of the set is call the
controllability index.
Controllability and Equivalence Transformations
The controllability property is invariant under under
any equivalence transformation and reordering of
columns of B.
This can be shown by substituting
similarity/equivalence transformation matrices P and
Q to establish the equality below:
P [ B A B A B ... A
2
n−1
B
 A
 A2 B
 ... An−1 B
]
B ]= [ B
And since P is nonsingular, its product with
controllability matrix has the same rank as the
controllability matrix.
Observability
ẋ=A xB u
y=C xD u
is observable if for any unknown initial state x(t0), there
exists a finite time t1 – t0 ∋ knowledge of input u(t) and
output input y(t) over [t0, t1] is all that is required to
uniquely determine x(t0).
If the system is observable, an estimator/observer to
compute state x(t0) is given by:
t
The state equation:
1
xt 0 =W −1
y dt
o t 1, t 0 ∫ expA ' t C' 
t0
t
W o t , t 0 =∫ expA ' C'C expA d 
t0
t
y
 t , t 0 =y t −C∫ expAt−B u  d −Dut 
t0
Conditions for Observability
For an n state and q output system: ẋ=A xB u
y=C xD u
1. This system is observable for t > 0 iff the nxn matrix
Wo is nonsingular fort all t > 0.
W o t =∫ expA ' C'C expA d 
0
2. The nq x n observability matrix O has full column
rank (n):
[ ]
C
CA
O= C A 2
⋮
C A n−1
Conditions for Observability
3. The (n+q) x n matrix [(A-λI)' C']' has full column
rank for every eigenvalue λ of A.
4. All eigenvalues of A have negative real parts, and
the unique solution Wo is positive definite
A Wo Wo A '=−C' C
where Wo is the observability Gramian:
∞
W o t =∫ expA ' C' C expA d 
0
Observability Example
Given system matrix A and input vector bo, output
vector co, with unit step input and x(0) = [-1, 5, 0]',
estimate this initial state from the input and observing
the output over interval t=[0 4] seconds.
[
−1 1
0
A= 0 −1 0
0
0 −2
]
[]
0
bo= 1
1
c o= [ 1 1 2 ]
Hint: Do not do by hand. Verify that observer did
indeed estimate the proper state value.
Observability Index
Consider an n state x q output system. The
observability matrix is given by:
O=[ C ' C' A ' C' A 2 ' ... C' A 
m
−1
' ... C' A n−1 ' ] '
where C can be expressed as a concatenation of row
vectors:
C=[ c1 ' c2 ' ... cm ' ... cq ' ] '
Let µm be the number of linearly independent vectors in
O associated with with row m. Then O has rank n if
1 23...q =n
Then {1, 2, 3, ... , q} are called the observability indices
and the maximum value of the set is call the
observability index.
Observability and Equivalence Transformations
The observability property is invariant under under any
equivalence transformation and any reordering of rows
of C.
Note: Rows k and m of an nxn matrix A can be
interchanged by pre-multiplying A by the identity
matrix with the kth and mth rows interchanged. For
example to interchange rows 2 and 4 of a 5x5 matrix,
multiply by:
1 0 0 0 0
[ ]
0
E 2,4 = 0
0
0
0
0
1
0
0
1
0
0
1
0
0
0
0
0
0
1
To interchange columns 2 and 4 post-multiply by the
same matrix.
Lecture Note Homework U8.1
Given system matrix A and input vector b plot an
input required to bring initial state x(0) = 0 to x(10)
= [-1, 1, -1]' and compute the energy of the input.
[
−1 0
0
A= 0 −2 0
0
0 −3
]
[]
1
b= 1
1
Repeat the previous problem for a final state of x(10)
= [-0.1, 0.1, -0.1]' Do you see a relationship between
the distance between the states and the required
energy?
Hand in a printout of the commented code used in
addition to the plots and results..
Lecture Note Homework U8.2
Given system matrix A, input vector b, output vector
c, input a unit step, and output
8
1
y t =2− exp−0.5 t− exp−t 
5
10
for t ≥ 0. Determine the original state at t=0.
[
A=
−1
0
1 −0.5
]
[]
b=
0
1
c=[ 1 1 ]
Hand in a printout of the commented code used in
addition to the result.