强跟踪滤波器讲解
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http://www.tsinghua.edu.cn/publish/au/
1
Contents
1. Background 2. Strong tracking filters (STF) theory 3. Applications of STF 4. Conclusions
2
1. STF Theory
x(k 1) f (k, x(k), (t), u(k)) (k)v(k) y(k) h(k, (t), x(k)) e(k)
3
1.1 Main disadvantages of the EKF
1) Poor robustness against model/plant uncertainties;
2) Losing tracking ability to the sudden changes of the state when it has reached steady-state;
3) Can’t be used to estimate time-varying parameters.
(Ljung L., IEEE T-AC, 24(1): 36-50, 1979) 4
11
1.7 Published papers on STF
[1] D.H. Zhou, et al.. Extension of Friedland‘ separate-bias estimation to randomly time- varying bias for nonlinear systems. IEEE Trans. on Automatic Control, 1993, 38(8): 1270-1273.
12
[3] D. H. Zhou, Y. G. Xi and Z. J. Zhang, A suboptimal multiple fading extended Kalman filter. Chinese Journal
of Automation, 1991, 17(6):689-695.
[4] Y. Liang, D.X. An, D H Zhou, Estimation of time-varying time delay and parameters of a class of Jump Markov nonlinear stochastic systems. Computers &
1.2 Orthogonality Principle
E[x(k) xˆ(k | k)][x(k) xˆ(k | k)]T min
E[ (k) T (k j)] 0; j 1,2,...
xˆ(k 1| k 1) f (k, xˆ(k | k), u(k)) K(k 1) (k 1) (k 1) y(k 1) h(k 1, xˆ(k 1| k))
M(i 1) F(i)P(i | i)FT(i)HT(i 1)H(i 1) (M jl )
γ(1)γ T (1)
;i 0
V0
E[γ (i
1)γ T (i
பைடு நூலகம்
1)]
V0 (i)
γ(i 1)γ T
1
(i
1) ;
i
1
1.4 Some properties
EKF: open-loop filtering; K(k+1) independent of the predicted output residual (k) ;
K (k 1) g(xˆ(k | k))
STF: closed-loop filtering, being an adaptive filter; dependent on the predicted output residual.
K(k 1) g(xˆ(k | k), (k 1))
1.5 State and parameter estimation
x(k 1) f (k, x(k), (k), u(k)) (k)v(k) y(k 1) h(k 1, x(k 1), (k 1)) e(k 1)
xe (k ) [ xT (k ) T (k )]T
10
1.6 Separate-bias estimation
x(t) f (t, x(t), b(t), u(t)) (t)v(t) y(t) h(t, x(t), b(t)) e(t)
5
1.3 Fading factors
K(k 1) P(k 1| k)H T (k 1)(H (k 1) P(k 1| k)H T (k 1) R(k))1 P(k 1| k) (k 1)F(k)P(k | k)FT (k) T (k)Q(k)(k)
diag { 1 2 n }
(i 1) diag[1 2 n ]
[2] D. H. Zhou, P.M. Frank. Strong tracking filtering of nonlinear time-varying stochastic systems with colored noise: application to parameter estimation and empirical robustness analysis. Int. J. Control, 1996, 65(2):295-307.
Strong Tracking Filters (STF): Theory and Applications
Zhou Donghua (周东华), Professor
Dept. Automation, Tsinghua University email: zdh@mail.tsinghua.edu.cn
j
j d (i 1); j d (i 1) 1
1
; j d (i 1) 1
(5-23)
d (i 1) tr[N (i 1)]
n j 1
jM
jj
(i
1)
(5-24) (5-25) (5-26) (5-27) (5-28)
N(i 1) V0(i 1) R(i 1) H (i 1)Γ (i)Q(i)Γ T (i)H T (i 1)
1
Contents
1. Background 2. Strong tracking filters (STF) theory 3. Applications of STF 4. Conclusions
2
1. STF Theory
x(k 1) f (k, x(k), (t), u(k)) (k)v(k) y(k) h(k, (t), x(k)) e(k)
3
1.1 Main disadvantages of the EKF
1) Poor robustness against model/plant uncertainties;
2) Losing tracking ability to the sudden changes of the state when it has reached steady-state;
3) Can’t be used to estimate time-varying parameters.
(Ljung L., IEEE T-AC, 24(1): 36-50, 1979) 4
11
1.7 Published papers on STF
[1] D.H. Zhou, et al.. Extension of Friedland‘ separate-bias estimation to randomly time- varying bias for nonlinear systems. IEEE Trans. on Automatic Control, 1993, 38(8): 1270-1273.
12
[3] D. H. Zhou, Y. G. Xi and Z. J. Zhang, A suboptimal multiple fading extended Kalman filter. Chinese Journal
of Automation, 1991, 17(6):689-695.
[4] Y. Liang, D.X. An, D H Zhou, Estimation of time-varying time delay and parameters of a class of Jump Markov nonlinear stochastic systems. Computers &
1.2 Orthogonality Principle
E[x(k) xˆ(k | k)][x(k) xˆ(k | k)]T min
E[ (k) T (k j)] 0; j 1,2,...
xˆ(k 1| k 1) f (k, xˆ(k | k), u(k)) K(k 1) (k 1) (k 1) y(k 1) h(k 1, xˆ(k 1| k))
M(i 1) F(i)P(i | i)FT(i)HT(i 1)H(i 1) (M jl )
γ(1)γ T (1)
;i 0
V0
E[γ (i
1)γ T (i
பைடு நூலகம்
1)]
V0 (i)
γ(i 1)γ T
1
(i
1) ;
i
1
1.4 Some properties
EKF: open-loop filtering; K(k+1) independent of the predicted output residual (k) ;
K (k 1) g(xˆ(k | k))
STF: closed-loop filtering, being an adaptive filter; dependent on the predicted output residual.
K(k 1) g(xˆ(k | k), (k 1))
1.5 State and parameter estimation
x(k 1) f (k, x(k), (k), u(k)) (k)v(k) y(k 1) h(k 1, x(k 1), (k 1)) e(k 1)
xe (k ) [ xT (k ) T (k )]T
10
1.6 Separate-bias estimation
x(t) f (t, x(t), b(t), u(t)) (t)v(t) y(t) h(t, x(t), b(t)) e(t)
5
1.3 Fading factors
K(k 1) P(k 1| k)H T (k 1)(H (k 1) P(k 1| k)H T (k 1) R(k))1 P(k 1| k) (k 1)F(k)P(k | k)FT (k) T (k)Q(k)(k)
diag { 1 2 n }
(i 1) diag[1 2 n ]
[2] D. H. Zhou, P.M. Frank. Strong tracking filtering of nonlinear time-varying stochastic systems with colored noise: application to parameter estimation and empirical robustness analysis. Int. J. Control, 1996, 65(2):295-307.
Strong Tracking Filters (STF): Theory and Applications
Zhou Donghua (周东华), Professor
Dept. Automation, Tsinghua University email: zdh@mail.tsinghua.edu.cn
j
j d (i 1); j d (i 1) 1
1
; j d (i 1) 1
(5-23)
d (i 1) tr[N (i 1)]
n j 1
jM
jj
(i
1)
(5-24) (5-25) (5-26) (5-27) (5-28)
N(i 1) V0(i 1) R(i 1) H (i 1)Γ (i)Q(i)Γ T (i)H T (i 1)