Asymptotically Optimum Recursive Prediction Error Methods in ...
Automat, JI
(2.9b)
~k + , = Fk?k + Gkuk + Kki$
(2.10a)
~k = Yk – H&
(2.10b)
(2.lb)
with n-vector state .x[, scalar input Ukand scalar noise Wkwhich is zero mean and white. More precisely for the theory of the next section we assume
+ yjRj-l@jti~
where 61 is introduced merely to ensure that R[ 261 for some 6>0 and all k, if R~ >61. The step size y; is chosen typically with y;>k -‘ initially and},~ + k-’ as k + W. The prediction error ti~ is given from
(2.la)
~(@)Wk
H(O’)G(O’) # O
W,k,
& = ~.l
where F(&) is abbreviated measure as is the matrix
as Fk, etc. The vector ~~ is a sensitivity Wk, as follows (2.lla)
E[W’, IF”’] =0,
E[w:l F-’]
E[w:ll--’] 1. The parameter and state estimates and predictions associated with the best scheme of ELS, HPE, RPE according to which has the minimum averaged prediction error squared is then selected for subsequent application, as for example in an adaptive controller. The
0, =8,.,
performance
measures are thus ~~ Ilti,ll’, ~ ~ Ilfi/1’ and ~ ~ [1~11’ 1 1 respectively for ELS, HPE and ~PE. An obvious refinement for the schemes is to exploit any a priori information on the range 0’ and to project HPE (and possibly RPE) estimates into this range should they stray out. Remark. The complexity of the TPE scheme is, in general, approximately the sum of the complexity of two ELS and one RPE scheme. For the case when the signal models are black box input–output models and HPE is equivalent to ELS, then the complexity is only approximately that of one ELS and one RPE scheme. Should one not wish to take advantage of the adaptive prewhitening and associated asymptotic efficiency potential of RPE, then switching between ELS and HPE is an acceptable double parameter estimation scheme (DPE), m in Fig. 2. (v) Adaptiue control. The parameter and state estimates can be used in adaptive mimmum variance controllers for the case of minimum phase plants, or in adaptive pole assignment, adaptive linear optimal quadratic gaussian (LQG) controllers or more robust schemes (details omitted). In a practical scheme, improved performance may be achieved by suitably weighting the controls from the ELS and RPE schemes, using insights from Friedlander ( 1982), rather than by switching from one to the other. Such modifications are beyond the scope of this paper.
“ik.
H, =
Brief Paper
‘--””-7 1
L–––––_–––––––_–
Adaptive Control
lc, r
L FIG. 2. A double estimator
3. Concwgtmw results Global stability and convergence results for the TPE algorithms are derived building on known theory in the technical report on which this paper is based, First, ELS results are refined, then extended to cope with switching to alternative estimators according to a switching rule which selects the estimator with the least averaged prediction error squared. The case when an alternative estimator is an HPE scheme is then studied, and finally convergence results for the full TPE scheme, which Includes an RPE estimator appropriately reinitialized from HPE, is studied. Detads are available from the authors. Remark 1. Based on the convergence theory omitted here, we claim that the complexlt y of TPE is rationalized as follows. If one accepts an RPE algorithm with an AR MAX model as a starting point, then adding ELS and reinitialization buys us the ability to project RPE into a closed loop stability domain, and is to our knowledge the first such result, Exploiting a priori information without nonlinear programming is bought at the cost of introducing HPE. To guarantee that HPE and thereby RPE are asymptotically optimal, the sufficient conditions on model parametrization are introduced (omitted here). These however are not necessary m order to achieve such asymptotic properties. Whether or not TPE is faster than ELS would in practice depend on initial conditions, noise sequences and specific parametrizdtions, Remark 2. It would be preferable, of course, if the power of a TPE algorithm could be gained by a simpler scheme no more complex than either ELS or RPE, but at present no such algorithm aPPears to blendall theprOperties established of this paper for the TPE scheme.
in the full version
4. Conclusions The paper has proposed a novel TPE controller scheme for plants with prior information. The algorithms involve parallel processing of three estimation schemes so as to extract as much to the potential advantages of each, and compensate for the potential dangers of applying either separately. The parallel processing is orchestrated so as to yield appropriate information flow between estimators and to select the best estimate at any given time in minimum prediction error sense. With the advent of parallel processing microprocessor technology, the trade-off between improved performmrce and computational cost could be
y+,
Dt,sirccl
Outp\)L
Trajectory
scheme
attractive, For the cue where theonginalm odelsareb lack-box input–output models, then thescheme simplifies eliminating the need for oneofthe components, the HP E estimator, Otherwise the scheme can only be simplified with the risk of loss of performance. The TPE scheme, tncluding ELS, RPE and HPE algorithms, is suitable in its unmodified form for globally convergent adaptive control of plants with fixed but unknown parameters. Such algorithms are the first which exploit u priori plant information while still guaranteeing global convergence, They are the first which are motivated and justified by the appropriate blending together of the asymptotic ODE analysis techniques and global convergence theories based on convergence of positive supermartingales.
References Friedlander, B. ( 1982). A modified prefilter for some recursive parameter estimation algorithms. IEEE Trans. Auf. Control, AC-27, 232-235, Kumar, R. and J, B. Moore ( 1982). Convergence of adaptive minimum variance algorithms via weighting coefficient selection. IEEE Trans. Auf. Confro/, AC-27, 146– 153. Ljung, L. and T. Soderstrom ( 1983). Theory and Pracrice oj Recursive Idenf ificu[ion, MIT Press, Cambridge, Massachusetts, Luenberger, D. G. ( 1973). Introduction to A4athemarica/ Nonlinear Programming, p. 161. Addison-Wesley, London. Moore, J. B. ( 19S3a), Persistency of excitation in extended least squares. IEEE Trans. Auf. Confro/, AC-28, 60-68. See also Proc. 20th Con/. Decision and Conrrol, pp. 539-540. Moore, J. B. ( 19S31>). Sidestepping t he positive real restriction for stochastic adapt!ve schemes. Ricerche di ,4uromatica, 13, 56-84. Moore, J, B, and H, Weiss ( 1980a). Recursive prediction error methods for adaptive estimation. IEEE Trans. .Syst., Man & Cyhernet., SMC-9, 197-205, Moore, J. B, and H, Weiss ( 1980 b). Recursive prediction errors without a stability test. Automatic, 16, 683–688, Sin, K. S. and G. C. Goodwin ( 1982). Stochastic adaptive control using a modified least squares algorithm. Auromatica, 18, 315-322.
(2.9b)
~k + , = Fk?k + Gkuk + Kki$
(2.10a)
~k = Yk – H&
(2.10b)
(2.lb)
with n-vector state .x[, scalar input Ukand scalar noise Wkwhich is zero mean and white. More precisely for the theory of the next section we assume
+ yjRj-l@jti~
where 61 is introduced merely to ensure that R[ 261 for some 6>0 and all k, if R~ >61. The step size y; is chosen typically with y;>k -‘ initially and},~ + k-’ as k + W. The prediction error ti~ is given from
(2.la)
~(@)Wk
H(O’)G(O’) # O
W,k,
& = ~.l
where F(&) is abbreviated measure as is the matrix
as Fk, etc. The vector ~~ is a sensitivity Wk, as follows (2.lla)
E[W’, IF”’] =0,
E[w:l F-’]
E[w:ll--’] 1. The parameter and state estimates and predictions associated with the best scheme of ELS, HPE, RPE according to which has the minimum averaged prediction error squared is then selected for subsequent application, as for example in an adaptive controller. The
0, =8,.,
performance
measures are thus ~~ Ilti,ll’, ~ ~ Ilfi/1’ and ~ ~ [1~11’ 1 1 respectively for ELS, HPE and ~PE. An obvious refinement for the schemes is to exploit any a priori information on the range 0’ and to project HPE (and possibly RPE) estimates into this range should they stray out. Remark. The complexity of the TPE scheme is, in general, approximately the sum of the complexity of two ELS and one RPE scheme. For the case when the signal models are black box input–output models and HPE is equivalent to ELS, then the complexity is only approximately that of one ELS and one RPE scheme. Should one not wish to take advantage of the adaptive prewhitening and associated asymptotic efficiency potential of RPE, then switching between ELS and HPE is an acceptable double parameter estimation scheme (DPE), m in Fig. 2. (v) Adaptiue control. The parameter and state estimates can be used in adaptive mimmum variance controllers for the case of minimum phase plants, or in adaptive pole assignment, adaptive linear optimal quadratic gaussian (LQG) controllers or more robust schemes (details omitted). In a practical scheme, improved performance may be achieved by suitably weighting the controls from the ELS and RPE schemes, using insights from Friedlander ( 1982), rather than by switching from one to the other. Such modifications are beyond the scope of this paper.
“ik.
H, =
Brief Paper
‘--””-7 1
L–––––_–––––––_–
Adaptive Control
lc, r
L FIG. 2. A double estimator
3. Concwgtmw results Global stability and convergence results for the TPE algorithms are derived building on known theory in the technical report on which this paper is based, First, ELS results are refined, then extended to cope with switching to alternative estimators according to a switching rule which selects the estimator with the least averaged prediction error squared. The case when an alternative estimator is an HPE scheme is then studied, and finally convergence results for the full TPE scheme, which Includes an RPE estimator appropriately reinitialized from HPE, is studied. Detads are available from the authors. Remark 1. Based on the convergence theory omitted here, we claim that the complexlt y of TPE is rationalized as follows. If one accepts an RPE algorithm with an AR MAX model as a starting point, then adding ELS and reinitialization buys us the ability to project RPE into a closed loop stability domain, and is to our knowledge the first such result, Exploiting a priori information without nonlinear programming is bought at the cost of introducing HPE. To guarantee that HPE and thereby RPE are asymptotically optimal, the sufficient conditions on model parametrization are introduced (omitted here). These however are not necessary m order to achieve such asymptotic properties. Whether or not TPE is faster than ELS would in practice depend on initial conditions, noise sequences and specific parametrizdtions, Remark 2. It would be preferable, of course, if the power of a TPE algorithm could be gained by a simpler scheme no more complex than either ELS or RPE, but at present no such algorithm aPPears to blendall theprOperties established of this paper for the TPE scheme.
in the full version
4. Conclusions The paper has proposed a novel TPE controller scheme for plants with prior information. The algorithms involve parallel processing of three estimation schemes so as to extract as much to the potential advantages of each, and compensate for the potential dangers of applying either separately. The parallel processing is orchestrated so as to yield appropriate information flow between estimators and to select the best estimate at any given time in minimum prediction error sense. With the advent of parallel processing microprocessor technology, the trade-off between improved performmrce and computational cost could be
y+,
Dt,sirccl
Outp\)L
Trajectory
scheme
attractive, For the cue where theonginalm odelsareb lack-box input–output models, then thescheme simplifies eliminating the need for oneofthe components, the HP E estimator, Otherwise the scheme can only be simplified with the risk of loss of performance. The TPE scheme, tncluding ELS, RPE and HPE algorithms, is suitable in its unmodified form for globally convergent adaptive control of plants with fixed but unknown parameters. Such algorithms are the first which exploit u priori plant information while still guaranteeing global convergence, They are the first which are motivated and justified by the appropriate blending together of the asymptotic ODE analysis techniques and global convergence theories based on convergence of positive supermartingales.
References Friedlander, B. ( 1982). A modified prefilter for some recursive parameter estimation algorithms. IEEE Trans. Auf. Control, AC-27, 232-235, Kumar, R. and J, B. Moore ( 1982). Convergence of adaptive minimum variance algorithms via weighting coefficient selection. IEEE Trans. Auf. Confro/, AC-27, 146– 153. Ljung, L. and T. Soderstrom ( 1983). Theory and Pracrice oj Recursive Idenf ificu[ion, MIT Press, Cambridge, Massachusetts, Luenberger, D. G. ( 1973). Introduction to A4athemarica/ Nonlinear Programming, p. 161. Addison-Wesley, London. Moore, J. B. ( 19S3a), Persistency of excitation in extended least squares. IEEE Trans. Auf. Confro/, AC-28, 60-68. See also Proc. 20th Con/. Decision and Conrrol, pp. 539-540. Moore, J. B. ( 19S31>). Sidestepping t he positive real restriction for stochastic adapt!ve schemes. Ricerche di ,4uromatica, 13, 56-84. Moore, J, B, and H, Weiss ( 1980a). Recursive prediction error methods for adaptive estimation. IEEE Trans. .Syst., Man & Cyhernet., SMC-9, 197-205, Moore, J. B, and H, Weiss ( 1980 b). Recursive prediction errors without a stability test. Automatic, 16, 683–688, Sin, K. S. and G. C. Goodwin ( 1982). Stochastic adaptive control using a modified least squares algorithm. Auromatica, 18, 315-322.
