Dual structure of dynamic linear systems
One can easily prove that the formulas related to the observability and constructability pro- blems can be obtained by “duality” from the similar formulas provided for the reachability and controllability problems.
The system S and the corresponding dual system SD are related as follows:
S ↔ SD A ↔ ATD B ↔ CTD C ↔ BTD D ↔ DTD
T ↔ (P−1)T
S ↔ SD R+ ↔ (OD−)T O− ↔ (R+D)T X+ ↔ ED−
E− ↔ XD+
Let us now show the most common correspondences between systems S and SD. Reachabnility matrix:
R+ = h B AB . . . An−1B i
Observability matrix:
O− =
C CA...
CAn−1
Reachability standard form:
A =
A11 A12
O A22
B =
B1
0
C = h C1 C2 i
Observability standard form:
A =
A11 0 A12 A22
B =
B1
B2
C = h C1 0 i
Reachable subsystem:
SR = (A11, B1, C1)
Observable subsystem:
SO = (A11, B1, C1)
Not reachable subsystem: Unobservable subsystem:
Transformation matrix T that brings the system into the reachability standard form:
x= Tx, T = h T1 T2 i
where
ImT1 = ImR+ = X+
and T2 makes matrix T not singular.
Transformation matrix P−1 that brings the system into the observability stan- dard form:
x = Px, P−1 =
P1
P2
where
ImPT1 = Im(O−)T
and P2 makes matrix P−1 non singular.
Transfer matrix H(s):
H(s) = C1(sI − A11)−1B1
depends only on the reachable subsy- stem.
Transfer matrix H(s):
H(s) = C1(sI − A11)−1B1
depends only on the observable subsy- stem.
Controllability canonical form (for reachable systems with only one input):
Ac =
0 1 0 . . . 0 0 0 1 . . . 0
... ... ...
−α0 −α1 . . . −αn−1
bc =
0 0...
1
cc = h β0 β1 . . . βn−1 i
where parameters α0, . . . , αn−1, are the coefficients of the monic characteristic polynomial of matrix A: ∆A(λ) = λn+ λn−1αn−1 + . . . + α0.
Observability canonical form (for observable systems with only one output):
Ao =
0 0 . . . 0 −α0
1 0 . . . 0 −α1
... ... ... ...
0 0 . . . 1 −αn−1
bo =
β0
β1
. . . βn−1
Transfer function G(s) for system in canonical form:
G(s) = cc(sI−Ac)−1bc = co(sI−Ao)−1bo = βn−1sn−1+ . . . + β1s+ β0 sn+ αn−1sn−1+ . . . + α1s+ α0
Transformation matrix Tc which brings the system S in the controllability canonical form (x = Tcxc):
Tc = R+(R+c )−1 = h b Ab A2b . . . An−1b i
α1 α2 α3 . . . αn−1 1 α2 α3 . . . 1 0 α3 . . . 0 0 ... ... ... ... ... ...
αn−1 1 . . . 0 0 1 0 . . . 0 0
Transformation matrix P−1c which brings the system S in the observability canonical form (x = Pcxc):
P−1c = (Oc−)−1O− =
α1 α2 α3 . . . αn−1 1 α2 α3 . . . 1 0 α3 . . . 0 0 ... ... ... ... ... ...
αn−1 1 . . . 0 0 1 0 . . . 0 0
c cA cA2
. . . cAn−1
Let αi be the coefficients of the characteristic polynomial ∆A(λ) of matrix A:
∆A(λ) = λn + αn−1λn−1+ . . . + α1λ+ α0
and let di be the coefficients of monic polynomial p(λ) chosen freely:
p(λ) = λn+ dn−1λn−1 + . . . + d1λ+ d0
If the couple (A, b) is reachable, the gain vector k of the static state feedback u = k x which locates arbitrarily the eigenvalues of matrix A + bk is the following:
k= kcT−1c = kc
h b, Ab, . . . , An−1b i
α1 α2 . . . αn−1 1 α2 . . . 1 0 ... ... ... ... ...
αn−1 1 . . . 0 0 1 0 . . . 0 0
−1
where kc = h α0 − d0, α1 − d1, . . . , αn−1 − dn−1 i.
If the couple (A, c) is observable, the gain vector l of an asymptotic sta- te observer which arbitrarily located the eigenvalues of matrix A + lc is the following:
l = Pclc =
α1 α2 . . . αn−1 1 α2 . . . 1 0 ... ... ... ... ...
αn−1 1 . . . 0 0 1 0 . . . 0 0
c cA cA2
. . . cAn−1
−1
α0 − d0
α1 − d1
...
αn−1 − dn−1
| {z }
lc
If the couple (A, b) is reachable, the gain vector k of the static state feedback u = k x which locates arbitrarily the eigenvalues of matrix A + bk can be computed using the following Ackerman formula:
k = −h 0 . . . 0 1 i(R+)−1
| {z }
qT
p(A) = −qTp(A)
where qT is the last row of the inverse of the reachability matrix and where p(A) is the matrix obtained from the polynomial p(λ) when parameter λ is substituted by matrix A.
If the couple (A, c) is observable, the gain vector l of an asymptotic state obser- ver which arbitrarily located the eigenvalues of matrix A + lc can be computed using the following Ackerman formula:
l = −p(A) (O−)−1
0...
0 1
| {z }
q
= −p(A)q
where q is the last column of the inverse della matrice di osservabilit`a and where p(A) is the matrix obtained from the polynomial p(λ) when parameter λ is substituted by matrix A.
