Page 790 - Mechanical Engineers' Handbook (Volume 2)

5 Design of Linear State Estimators 781

bance signals affecting the system and the measurement noise, the observer gain matrix
selection problem can be formulated as an optimal-estimation problem.

5.2 The Optimal Observer
Optimization of observer design has been primarily performed assuming stochastic models
for the disturbance inputs and measurement noise, though the effect of deterministic model
30
errors on observer design is also important. Consider the continuous-time system 3
˙ x(t) A(t)x(t) B(t)u(t) S(t)w (t) t t 0 (90)
1
y(t) C(t)x(t) w (t) t t 0 (91)
2
where w (t) is the random disturbance input, w (t) is the random measurement noise, and
2
1
T
their joint probabilities are assumed to be known. The column vector [w (t) w (t)] T is as-
T
2
1
sumed to be a white-noise process with intensity
V(t) V (t) V (t)
1
12
T
V (t) V (t) (92)
12
2
that is, the expected value
E [w (t ) w (t )]
w (t)
1
1
1
2
w (t ) T 1 2 T 2 2 V(t )
(t t ) (93)
21
where
(t t ) is the Dirac delta function.
2
1
The initial state x(t ) is assumed to be a random variable uncorrelated with w and w 2
1
0
and its probability given by
E[x(t )] x 0 and E{[x(t ) x ][x(t x ]} Q 0 (94)
T
0
0
0
0
0
The observer form is given by Eq. (74) and Fig. 10. The optimal-observer problem consists
of determining L( ), t t, and the initial condition on the observer ˆx(t ) so as to
0
0
minimize the expected value E{[x(t) ˆx(t)] T W(t)[x(t) ˆx(t)]}, where W(t) is a symmetric
positive-definite weighting matrix.
If the problem as stated is nonsingular
det[V (t)] 0 t t 0 (95)
2
and if the disturbance and measurement noise are uncorrelated
V (t) 0 (96)
12
the optimal-observer gain matrix is given by Kalman and Bucy 31 as
T
1
L(t) Q(t)C (t)V (t) t t 0 (97)
2
where Q is a solution of the matrix Riccati equation:
˙
T
1
T
T
Q(t) A(t)Q(t) Q(t)A (t) S(t)V (t)S (t) Q(t)C (t)V (t)C(t)Q(t) t t (98)
1 2 0
with the initial condition
Q(t ) Q 0 (99)
0
and the observer initial condition
ˆ x(t ) x 0 (100)
0