3. CONTROLLABILITY:
Full-state feedback design commonly relies on pole-placement
techniques. It is important to note that a system must be completely
controllable and completely observable to allow the flexibility to place all
the closed-loop system poles arbitrarily. The concepts of controllability and
observability were introduced by Kalman in the 1960s.
A system is completely controllable if there exists an unconstrained
control u(t) that can transfer any initial state x(t0) to any other desired
location x(t) in a finite time, t0≤t≤T.
4. For the system
Bu
Ax
x
we can determine whether the system is controllable by examining the
algebraic condition
n
B
A
B
A
AB
B
rank 1
n
2
The matrix A is an nxn matrix an B is an nx1 matrix. For multi input systems,
B can be nxm, where m is the number of inputs.
For a single-input, single-output system, the controllability matrix Pc is
described in terms of A and B as
B
A
B
A
AB
B
P 1
n
2
c
which is nxn matrix. Therefore, if the determinant of Pc is nonzero,
the system is controllable.
5. Example: Consider the system
u
0
x
0
0
1
y
,
u
1
0
0
x
a
a
a
1
0
0
0
1
0
x
2
1
0
1
2
2
2
2
2
2
1
0 a
a
a
1
B
A
,
a
1
0
AB
,
1
0
0
B
,
a
a
a
1
0
0
0
1
0
A
1
2
2
2
2
2
c
a
a
a
1
a
1
0
1
0
0
B
A
AB
B
P
The determinant of Pc =1 and ≠0 , hence this system is controllable.
6. Example:Consider a system represented by the two state equations
1
2
2
1
1 x
d
x
3
x
,
u
x
2
x
The output of the system is y = x2. Determine the condition of controllability.
u
0
x
1
0
y
,
u
0
1
x
3
d
0
2
x
d
0
2
1
P
d
2
0
1
3
d
0
2
AB
and
0
1
B
c
The determinant of pc is equal to d, which is
nonzero only when d is nonzero.system is
controllable
B
A
B
A
AB
B
P 1
n
2
c
AB
B
Pc
7. OBSERVABILITY:
All the poles of the 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 ability to estimate a state variable.
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 history y(t) given the control u(t).
Cx
y
and
Bu
Ax
x
Consider the single-input, single-output system
where C is a 1xn row vector, and x is an nx1 column vector. This
system is completely observable when the determinant of the
observability matrix P0 is nonzero.
8. The observability matrix, which is an nxn matrix, is written as
1
n
O
A
C
A
C
C
P
Example:
Consider the previously given system
0
0
1
C
,
a
a
a
1
0
0
0
1
0
A
2
1
0
9.
1
0
0
CA
,
0
1
0
CA 2
Thus, we obtain
1
0
0
0
1
0
0
0
1
PO
The det P0=1, and the system is completely observable. Note that
determination of observability does not utility the B and C matrices.
10. We can check the system controllability and observability using the Pc and
P0 matrices.
From the system definition, we obtain
2
2
AB
and
1
1
B
det Pc=0 and rank(Pc)=1. Thus, the system is not controllable.
2
1
2
1
AB
B
Pc
Therefore, the controllability matrix is determined to be
Example: Consider the system given by:
x
1
1
y
and
u
1
1
x
1
1
0
2
x
11. From the system definition, we obtain
1
1
CA
and
1
1
C
1
1
1
1
CA
C
Po
Therefore, the observability matrix is determined to be
det PO=0 and rank(PO)=1. Thus, the system is not observable.
If we look again at the state model, we note that:
2
1 x
x
y
However,
2
1
1
2
1
2
1 x
x
u
u
x
x
x
2
x
x
Thus, the system state variables do not depend on u, and the system is
not controllable. Similarly, the output (x1+x2) depends on x1(0) plus x2(0)
and does not allow us to determine x1(0) and x2(0) independently.
Consequently, the system is not observable.
12. The observability matrix PO can be constructed in Matlab
by using obsv command.
From two-mass system,
Po =
1 1
1 1
rank_Po =
1
det_Po =
0
clc
clear
A=[2 0;-1 1];
C=[1 1];
Po=obsv(A,C)
rank_Po=rank(Po)
det_Po=det(Po)
The system is not observable.
