Page 791 - Mechanical Engineers' Handbook (Volume 2)

782 Control System Design Using State-Space Methods

The resulting state estimator is called the Kalman–Bucy filter. The similarity of Eqs. (97)
and (98) to the corresponding Eqs. (25) and (26), respectively, for the LQR problem is a
result of the duality of the state estimation and state feedback control problems noted earlier.
One difference, however, is that the Riccati equation for the optimal observer can be imple-
mented in real time since Eq. (99) is an initial condition for Eq. (98). In contrast, for the
finite-time LQR problem, Eq. (27) gives the terminal condition for the Riccati equation (26).
The steady-state properties of the optimal observer for linear time-varying and time-
invariant systems parallel those of the optimal controller for the LQR problem and are
3
described by Kwakernaak and Sivan. If the time-varying system
˙ x(t) A(t)x(t) S(t)w (t) (101)
1
y(t) C(t)x(t) w (t) (102)
2
is uniformly completely controllable by w (t) and uniformly completely reconstructible, the
1
solution Q(t) of Eq. (98) converges to a steady-state solution Q (t)as t → for any
s
0
positive-semidefinite Q . The corresponding steady-state optimal observer
0
˙ ˆ x(t) A(t)ˆx(t) L (t)[y(t) C(t)ˆx(t)] (103)
s
where
1
T
L (t) Q (t)C (t)V (t) (104)
s
2
s
is exponentially stable. Also, if the system and the noise statistics are invariant, the matrix
Riccati differential equation (98) becomes an algebraic equation as t → :
0
T
T
T
1
AQ QA SV S QC V CQ 0 (105)
s
s
1
2
s
s
If the corresponding time-invariant system is completely controllable by the input w (t) and
1
completely observable, Eq. (105) has a unique positive-definite solution Q and the corre-
s
sponding steady-state optimal observer of Eqs. (103) and (104) is asymptotically stable. Note
that the measurable input u(t) has been omitted from Eqs. (101) and (102) for simplicity but
does not change the substance of the results.
3
The discrete-time version of the optimal linear observer follows. Consider the discrete-
time system
x(k 1) F(k)x(k) G(k)u(k) S(k)w (k) (106)
1
y(k) C(k)x(k) D(k)u(k) w (k) (107)
2
where w (k), w (k) are zero-mean, uncorrelated vector random variables representing distur-
1
2
bance and measurement noise, respectively. Their joint probabilities are assumed to be
known. The column vector [w (k) w (k)] T has the variance matrix
T
T
1
E [w (k) w (k)] V (k) V (k)
2
w (k)
12
1
1
T
T
T
w (k) 1 2 V (k) V (k) (108)
12
2
2
The initial state x(k ) is considered to be a random variable, uncorrelated with w and w 2
0
1
with
T
E[x(k )] x 0 and E{(x(k ) x )(x(k ) x )} Q 0 (109)
0
0
0
0
0
The observer form is given by Eq. (78) and Fig. 11. The optimal-observer problem consists
of determining L(k ), k k k, and initial condition on the observer ˆx(k ) so as to
0

0

T
minimize the expected value E{[x(k) ˆx(k)] W(k)[x(k) ˆx(k) ]}, where W(k) is a symmetric
positive-definite weighting matrix.