Parameter Estimation for Time-Variant Processes
For many real processes, the parameters of the governing linear difference equations are not constant. They rather vary over time due to internal or external influences. Also, quite often non-linear processes can only be linearized in a small interval aro
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For many real processes, the parameters of the governing linear difference equations are not constant. They rather vary over time due to internal or external influences. Also, quite often non-linear processes can only be linearized in a small interval around the current operating point. If the operating point changes, also the linearized dynamics will change in this case. For slow changes of the operating point, one can obtain good results with linear difference equations that contain time-varying parameters. The method of recursive least squares (see Chap. 9) can also be used to identify time-varying parameters. Different methods are introduced in the following that allow to track the changes of time varying parameters with the method of least squares.
12.1 Exponential Forgetting with Constant Forgetting Factor In connection with the method of weighted least squares, a technique was suggested in Sect. 9.6 which allowed the identification of slowly time-varying processes by choosing the weights w.k/ as w.k/ D N
0 k
:
(12.1.1)
This particular way of choosing w.k/ to rate the error is termed exponential forgetting. The recursive estimation equations (9.6.11), (9.6.12), and (9.6.13) for the method of weighted least squares with exponential forgetting had been given as O O C 1/ D .k/ O (12.1.2) .k C .k/ y.k C 1/ T .k C 1/.k/ 1 P.k/ .k C 1/ (12.1.3) .k/ D T .k C 1/P.k/ .k C 1/ C 1 P.k C 1/ D I .k/ T .k C 1/ P.k/ : (12.1.4)
R. Isermann, M. Münchhof, Identification of Dynamic Systems, DOI 10.1007/978-3-540-78879-9_12, © Springer-Verlag Berlin Heidelberg 2011
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12 Parameter Estimation for Time-Variant Processes
The influence of the forgetting factor can be recognized directly from the inverse of the covariance matrix (9.6.6) P 1 .k C 1/ D P 1 .k/ C
.k C 1/
T
.k C 1/ :
(12.1.5)
P 1 is proportional to the information matrix J (11.2.60) given by J D
1 ˚ 1 ˚ T E D 2 E P 1 ; 2 e e
(12.1.6)
see (Eykhoff, 1974; Isermann, 1992). By taking < 1, the information of the last step is reduced or the covariances are increased respectively. This means a worse quality of the estimates is pretended, such that the new measurements get more weight. For D 1, one obtains ˚ lim E P.k/ D 0 (12.1.7) k!1 ˚ ˚ (12.1.8) lim E .k/ D lim E P.k C 1/ .k C 1/ D 0 : k!1
k!1
O C 1/. Then For large times k, the measurements have practically no influence on .k the elements of P 1 .k C 1/ tend to infinity, (12.1.5). If, however, one uses a forgetting factor < 1, then, from (12.1.5), follows P
1
k
.k/ D P
1
.0/ C
k X
ki .i /
T
.i / :
(12.1.9)
iD0
For large values ˛ of the initial matrix P.0/ D ˛I, the first term in (12.1.9) vanishes. As for < 1 k k1 X X ki D lim i < 1 (12.1.10) lim k!1
iD1
k!1
iD0
(convergent series with positive elements) and hence P 1 .k/ converges to fixed values ˚ (12.1.11) lim E P 1 .k/ D P 1 .1/ k!1
and does not approach infinity. Hence, ˚ lim E P.k/ D P.1/ k!1
as well as
˚ lim E .k/ D .1/
k!1
(12.1.12)
(12.1.13)
are
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