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4 Finite-Frequency Robust Filtering for Multi-model Jumping System
4.1 Robust Finite Frequency Filter Design for Multi-model
Jumping System
4.1.1 System Description
For a given probability space ((, F, P), we consider the following continuous-time
multi-model jumping system:
⎧
⎪ ⎪ ⎨
⎪ ⎪ ⎩
˙
x(t) = A(r t )x(t) + B d (r t )ω(t),
y(t) = C 1 (r t )x(t) + D d (r t )ω(t),
z(t) = C 2 (r t )x(t),
x(t) = x 0 , r t = r 0 , t = 0.
(4.1)
where x(t) ∈ R
n is the state, y(t) ∈ R
m is the measured output, z(t) ∈ R
r is the
signal to be estimated, ω(t) ∈ L
m
2 [0, ∞] is unknown disturbance with known finitefrequency ranges. A(r t ), B d (r t ), C 1 (r t ), C 2 (r t ), D d (r t ) are known mode-dependent
matrices with appropriate dimensions. The random form process {r t , t ≥ 0} is a
continuous-time discrete-state Markov stochastic process taking values in a finite set
= {1, 2, . . . , N } with the following transition probability:
P i j = P {r t+t = j | r t = i} =
π i j t + o((t),
i = j
1 + π ii t + o((t), i = j
(4.2)
where t > 0 and lim t↓0 o((t)//t → 0. π i j ≥ 0 is the transition probability rates
from mode i at time t to mode j (i = j) at time t + t, and
N
j=1, j =i π i j = −π ii .
The transition rate matrix is denoted by =
π i j , i, j ∈
.
For presentation convenience, we denote A(r t ), B d (r t ), C 1 (r t ), C 2 (r t ), D d (r t ), as
A i , B di , C 1i , C 2i , D di , respectively.
The main objective of robust finite frequency filtering is to design a filter to
simultaneously satisfy the stochastic stability of multi-model jumping system (4.1)
and the finite frequency performance. In order to fully analyze the effects of transition
probabilities on the finite frequency specifications, a derandomisation technique is
introduced in the sequel. Firstly, the indicator function 1 A is defined as:
1 A (ω) =
1,
if ω ∈ A
0,
otherwise
and denote
q i (t) = E
x(t)1 {r (t)=i}
.
(4.3)
Combining Eqs. (4.1) and (4.4) results in the following expressions: