Page 784 - The Mechatronics Handbook

0066_Frame_C24 Page 32 Friday, January 18, 2002 5:36 PM
The controllable subspace of a state space model is composed of all states generated through every
. The stability of this subspace is determined by the location
possible linear combination of the states in x c
.
of the eigenvalues of A c
On the other hand, the uncontrollable subspace is composed of all states generated through every
. The stability of this subspace is determined by the location
possible linear combination of the states in x nc
.
of the eigenvalues of A nc
Hence, the input will have no effect over the uncontrollable subspace, so the best we can hope is that
this uncontrollable subspace is stable, since then the state in this subspace will go to the origin. In this
case the state space model is said to be stabilizable.
A key feature of the descriptions (24.187) and (24.188) arises from the fact that the transfer function
is given by
−1
(
–
H s() = C c sIA c) B c + D (24.189)
Equation (24.189) says that the eigenvalues of the uncontrollable subspace do not belong to the set of
poles of the system transfer function. This implies that there is a cancellation of all poles corresponding
–
to the roots of (sIA nc ).
Controllability Canonical Form
Lemma 24.2 Consider a completely reachable state space model for a SISO system. Then, there exists a
similarity transformation which converts the state space model into the following controllability canonical
form:

00 … 0 – α 0 1
10 … 0 – α 1 0
A′ = 01 … 0 – α 2 , B′ = 0 (24.190)
O
00 … 1 – α n−1 0

n
n−1
where λ + α n−1 λ + ⋅⋅⋅ + α 1 λ + α 0 = det (λ I − A) is the characteristic polynomial of A.
Lemma 24.3 Consider a completely controllable state space model for a SISO system. Then, there exists a
similarity transformation which converts the state space model into the following controller canonical form:

– α n−1 – α n−2 … – α 1 – α 0 1
1 0 … 0 0 0
A″ = 0 1 … 0 0 , B″ = 0 (24.191)
O
0 0 … 1 0 0

n
n−1
where λ + α n−1 λ + ⋅⋅⋅ + α 1 λ + α 0 = det (λ I − A) is the characteristic polynomial of A.
Observability, Reconstructibility, and Detectability
If we consider the state space model of a system, one might conjecture that if one observes the output
over some time interval then this might tell us some information about the state. The associated model
property is called observability (or reconstructibility).

©2002 CRC Press LLC