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Frequency domain approach for designing sampling rates for system identification
Automatica, Vol. 11, pp. 189--191. Pergamon Press, 1975. Printed in Great Britain
Brief Paper Frequency Domain Approach for Designing Sampling Rates for System Identification* Abordage du Domaine de Fr6quence pour la Conception de Taux d'l~chantillonage pour Identification par Syst~me Frequenaverfahren zum Entwurf von Testverh~iltnissen fiir die Systemidentifikation I'lpHMeneHrIe ~IacroTrioro MeTo~a )UI~ pacqera cropoerH Bm6opra CHCTeM H~eHTHq~HKaI_(HH R O B E R T L. P A Y N E , t G R A H A M
C. G O O D W I N ~ a n d M A R T I N B. Z A R R O P §
Summary--This paper considers the problem of joint determination of input spectra and sampling rate for linear system identification. The concept of an 'input spectrum preserving sampler' is introduced and it is shown that, for this sampler, the usual anti-aliasing filter is optimal, and that joint optimal design of input spectrum and sampling rate may be readily performi~d in the frequency domain.
2. Model structure Consider a constant coefficient, linear, single inputmultiple output dynamic system with coloured measurement noise :~(t) = Ax(t) + Bu(t), (2.1) y(t) = Cx(t) + Du(t) + to(t),
(2.2)
where t e T = [0, t/I, x : T--> R", is the state vector, u : T - ~ R x is the input vector, y : T - ~ R "~ is the output vector and t o : T - + R '~ is the measurement noise vector, assumed to have a zero mean normal distribution and power density spectrum ate(f) with full rank for all f E ( - - ~ , o0).
1. Introduction The problem of optimal experiment design for the case of static regression models has been extensively studied in the statistical literature. Many of the key results are summarized in the recent book by Fedorov [3]. The extension of these results to dynamic models was first considered by Viort [4]. Further work on dynamic systems has been described by Mehra [5]. These works are, however, concerned only with the design of optimal input spectra and do not consider the related problem of optimal design of the sampling strategy. The problem of optimal sampling rate determination for linear system identification has been discussed by Astr6m [1 ] and Gustavsson [21. To gain insight into the problem these authors considered the case where the system input was prespecified. However, in general, to achieve maximal return from an experiment, a coupled design of the presampling filter, sampling rate and test signal should be carried out. The general design problem, with nonuniform sampling intervals and presampling filters, can be formulated. However, the general solution is complex and offers little insight. In the current paper the sampling intervals are assumed to be uniform and it is shown that, for this case, the standard anti-aliasing filter [2] is optimal in a well-defined sense, It is further shown that optimal coupled design of test signals and sampling rate can be readily performed in the frequency domain. The resultant design procedure is straightforward and leads to analytic solutions in simple cases.
3. Information matrix Fisher's information matrix [7] for the parameters, 0, in ,4, B, C and D is given by Me = tt z(,lo,
(3.1)
where tt is the total experiment time and ~ o is the average information matrix given by ~i o =
I2
QT(_j27rf) ~F-1(f) Q(j2~rf) dE(f),
(3.2)
where 6(f) is the input spectral distribution function [6], satisfying
f
® dE(f)= 1
(3.3)
and Q is a matrix with the lth column.given by Q~(s) = (~/~OD [ C ( s l - A) -1 B + D],
(3.4)
where 0j denotes the lth element of 0. The quantity 6(fs)-6(f0, A>A> o, can be interpreted physically as half the input power in the frequency range (ft, f,]. In equation 0.2) the total input power has been taken to be unity for convenience.
* Received 25 March 1974; revised 29 July 1974. The original version of this paper was not presented at any IFAC meeting. It was recommended for publication in revised form by associate editor K. J. Astr6m. 1"Department of Systems and Control, University of New South Wales, Australia. ~/Department of Electrical Engineering, University of Newcastle, New South Wales, Australia. g-Department of Computing and Control, Imperial College of Science and Technology, London SW7 2BT.
4. Effect o f filtering and sampling on information Letfh be a frequency above which the input spectrum is zero, i.e. the input energy lies wholly in the band [-f~, fh] then equation (3.2) for tl;~"0 may equivalently be written as
~o = f'_~QTt-i2,,f) ~'-'(r) ~j2~/) 06(..
(4.1)
Suppose now that the output, Yk, is sampled at greater than the Nyquist rate for fs, that is the sampling frequency, 189
190
Brief paper
fs>2ft. This form of sampler does not distort that part of the output spectrum arising from the input and hence is sailed here an 'input spectrum preserving sampler'. Note that the part of the output spectrum due to the noise will be distorted due to aliasing. Equation (4.1) for 2fir0is then replaced by
l~lss
f/" QT(-j2~r13 tFs-~(.f) Q(j2~rf) d~(13, (4.2)
and
aFv(13 = F(j2rrf) ~F(f) Fr(--j2zrf). If F(s) satisfies (4.8) then (4.10) becomes
~sF(13 = ~F(f)
|=1
f E (--fs/2,f~/2).
(4.3)
Theorem 4.1 The matrix (J~r0- Ms s) satisfies the inequality
Mo-- Mos >1O,
(4.4)
that is, is non-negative definite (NND).
Proof Since T'(13 is positive definite (PD) for all f, it follows from (4.3) that T'8(f)-~F(13 is PD and hence that ~F-1(13-~I~'8-1(13is PD. Hence
~o_ ~os ____f'-~QT(--j2~'13 {W-1(13 --~',-1(13} Q(j2~r13 d~(f)
(4.5)
is NND. For the purposes of comparing different experiments it is necessary to have a measure of the 'goodness' of the experiment. A suitable measure is a scalar function ~ of the information matrix where ~ has the property that
~(M~)<~(M2)
if ( M x - M 2) is NND,
(4.6)
where M1 and M s are any two information matrices. Typical examples of such cost functions are the trace and determinant of the inverse information matrix. Note that these cost functions can be given physical interpretations, e.g. the determinant is related to the volume of the highest probability density region of the parameters. It now follows from (4.4) that ~(J~0) < ~bC~roS),
(4.7)
that is, the cost with sampling is greater than or equal to the cost with continuous observations. It will now be shown that the increase in cost resulting from sampling can be eliminated by the inclusion of a suitable presampling filter.
Theorem 4.2 For a cost function ~ satisfying (4.6), and for sampling rate fa> 2fj, any presampling filter with transfer function F(s) satisfying
I F(j2~r13 1 = 0 v f ¢ (-fs/2,fs/2)
(4.8)
and invertible for f e [-fA, f d gives the same cost using the samples as with continuous observations. Furthermore, any filter satisfying (4.8) is optimal with respect to 4.
Proof It is readily verified that the expression for the information matrix 1171o a given by (4.2) when a presampling filter, F(s), is present becomes
Zlos, ffi f ~,QT(-j2,r13
FT(-j2~'13 ~Fs,-1(13
x F(j2~rf) 0(j2~13 d~:(f), (4.9) where
~zrs~r) = "Fp(13+ ~ {Wp(ifs+13+ WF(lfs--f)} lffil
(4.10)
vfs [-A,A],
(4.12)
and hence equation (4.9) leads to the same result as if continuous observations were taken, that is
J~4osr = 37[0.
where ~Fs(13 is the distorted noise spectrum and is given by
Vs(13 = ~/(13 + ~ {~/(lf8 +t3 + V(tA-13};
(4.1 1)
(4.1 3)
Hence any filter having the property (4.8) gives the same cost with samples as with continuous observations. Furthermore, from equation (4.9) and equation (4.12) it can be seen that the form of the filter f o r f ~ (-fs/2, fs/2) does not affect the cost. Hence from equation (4.7) it follows that any filter satisfying (4.8) is optimal with respect to 4.
Remark 4.1 In the design of an experiment where the sampling rate is fs > 2fA and the cost function has property (4.6) then the standard anti-aliasing filter is optimal. The standard anti-aliasing filter, [2], has a transfer function F(s) with the property
I F(j21rf) I = 1 V f ~ (-fs/2,fs/2), ] I F(j2~r13 1 = 0 V f ¢ (-fs/2,A/2), j
(4.14)
and thus, from Theorem 4.2, is optimal with respect to 4In practice the ideal anti-aliasing filter is unrealizable but can be closely approximated by a sharp cut-off filter.
5. Joint optimal design of input and sampling rate In this section the results of the previous sections are used to develop a design procedure for joint optimal determination of the input, presampling filter and sampling rate for the case of constrained input power and fixed total number of samples. The restriction on the number of samples is frequently met in practice due to the cost of data acquisition or computer storage restrictions on data analysis. Recall that the total input power is constrained to be unity; see equations (3.2) and (3.3). If the total number of samples is constrained to be N, say, and the sampling rate is fs, then it follows that the total experiment time is given by
tl = N/fs.
(5.1)
It follows from equation (3.1) that Fisher's information matrix is given by Me = (N/fs) J~os (5.?~ where 2¢1os is the average information matrix corresponding to an input spectrum having highest frequency component fn and samples collected at frequency f8 > 2fn. From Remark 4.1 the usual anti-aliasingfilter is optimal and its inclusion prior to sampling ensures that subsequent sampling will not deteriorate the information matrix. Hence equation (5.2) can be written as
Me = (N/fs) 1~1o,
(5.3)
where /~0 is the information matrix corresponding to continuous observations. It should be noted that /~'0 in equation (5.3) can be calculated from equation (4.1) because the anti-aliasing filter has ensured that there is no information loss due to sampling. It follows from equations (3.2) and (3.3) that the set of all possible information matrices arising from powerconstrained inputs is the convex hull of the set of information matrices corresponding to single frequency power constrained inputs. This fact, together with a classical theorem of Carath6odory [3], leads to the conclusitm that any information matrix can be achieved from a design comprising not more than 1 = p(p+ 1)/2+ 1 lines in the
Brief paper input spectrum where p is the total number of parameters. Furthermore, optimal designs for cost functions satisfying (4.6) can be achieved with not more than p ( p + 1)/2 lines. The details of this development for static systems are contained in the book by Fedorov [3]. From equations (5.3) and (4.1) and using the fact that only I lines are required in the input spectrum, Me can be written as :~ ?t, Re [QT(_j2~f,) W-l(f~) Q(j2~f,)] (5.4)
Ms = ~
J~
iffil
with I
~At--- 1, ~>~0,
i = 1..... 1,
(5.5)
and 0~
(5.6)
where Re [.] denotes the real part of. The quantities Ax..... A~ have the physical interpretation of being the fraction of the total power at the corresponding frequencies A . . . . . A. The optimal coupled design of input and sampling rate can now be achieved by choosing (,~1..... Az), (fl ..... fz) and fs to minimize a scalar function ~ of Mo subject to equations (5.5) and (5.6).
191
References [1] K. J. ASTROM: On the choice of sampling rates in parameter identification of time series. Inform. Sei. 1, 273-278 (1969). [2] I. GUSTAVSSON:Survey of Applications of Identification in Chemical and Physical Process~. Paper 5.3, Preprints 3rd IFAC Symposium, The Hague/Delft, June (1973). Also in Automatica 11, January (1975). [3]V. V. F ~ o R o v : Theory of Optimal Experiments. Academic Press, New York (1971). [4] B. V[ORT: D-optimal Designs for Dynamic Models-Part I: Theory. Technical Report No. 314, Department of Statistics, University of Wisconsin, October (1972). [5] R. K. MEtmA: Frequency Domain Synthesis of Optimal Inputs for Linear System Parameter Estimation. Technical Report No. 645, Division of Engineering and Applied Physics, Harvard University, July (1973). [6] E. WONG: Stochastic Processes in Information and DynamiealSystems. McGraw-Hill, New York (1970). [7] S. D. SnowY: Statistical Inference. Penguin (1970).
Appendix ,4--,4 simple first-order example Model:
fs = 2(l+e)fh,
(6.1)
where e (> 0) is related to the cut-off characteristics of the presampling filter and should be small. Substitution of (6.1) into (5.3) gives N ~'fo, (l+e)A
Ms = ~
(6.2)
where 33"ois the'average information matrix (per unit time) with continuous observations. Again using property (4.6), the cost is apparently decreased when f~ is decreased. However, since fa is the highest frequency in the input, decreasing fh compresses the input spectrum towards the low frequency end. This compression has a tendency to increase the cost arising from the term ~r 0. Thus an optimal compromise will be achieved for some value of fh which will, in general, be less than the highest frequency in the optimal continuous observation design with constrained total experiment time rather than fixed number of samples. An example is given in Appendix A illustrating the above points.
7. Conclusions The concept of an 'input spectrum preserving sampler' has been introduced and it has been shown that for this sampler the optimal presampling filter for identification is the standard anti-aliasing filter. This form of sampling system has considerable practical appeal since it is readily implementable, and allows the filtered output signal to be reconstructed from the samples. Furthermore, it has been shown that with this form of sampling system the joint optimal design of input, presampling filter and sampling rate can be readily carried out in the frequency domain.
u(s)+e(s),
~(f)
=
1
aS(os+ 1,
¢o=2~rf.
reviewers for their helpful comments on an earlier version of this paper.
(A.2)
For this example, a suitable cost function is I[M. Following the discussion of Section 5, an optimal input spectrum can be found comprising one line (p(p+l)/2 withp = 1). The information matrix with continuous observations and one-line input is given by (as cos+ 1) (oz M=
tI ( ~ o j S + l )
s .
(A.3)
Hence it follows that, for continuous observations and fixed experiment time, ts, the input which minimizes 1/M has frequency co* given by oJ* = ( ¢ 2 - 2 ~ ) - t
if2a2<~,
= oo if 2a*>~.
(A.4) (A.5)
If the noise is wide band (a ~ 0) the intuitively pleasing result ¢o* = 1#" (A.6) is obtained. Now if the output is filtered using an anti-aliasing filter and then sampled, the information matrix (A.3) for a fixed number of samples (N) becomes M = N(a2 ws+ 1) wI f s (~z ~ z + I p '
(A.7)
where fs is the sampling frequency. Now following the development of Sections 5 and 6 of the paper, the output is sampled at just above twice the input frequen~ (that is, f~--w#r). The information matrix becomes .. (a 2 oJS+l) ~o M = ~v~r 0. s ojs+ 1)2
(A.8)
and the frequency, ¢o*, minimizing I/M in the case of a fixed number of samples satisfies the equation - a 2 ~ oJ*4+3(aS-~-s) (o*S+l ----0,
(A.9)
which, for wide band (a -~ 0) noise, yields ca. ----1#'~/3.
Acknowledgements---The authors wish to thank the
(A.t)
where e(s) denotes coloured measurement noise with spectral density
6. Discussion of the designprocedure It can be seen from equation (5.4) that Mo is inversely proportional to fs. It follows from property (4.6) of the cost function that the cost is decreased when fs is decreased. The limit to how far fs can be decreased is dictated by the constraint fs > 2fh. Thus, in practice, the optimal choice offs is
1
y(s) = ~
(A.10)
This is a somewhat lower frequency than in the case with continuous observations. This result complies with the general observations made in Section 6 of the paper.
Brief Paper Frequency Domain Approach for Designing Sampling Rates for System Identification* Abordage du Domaine de Fr6quence pour la Conception de Taux d'l~chantillonage pour Identification par Syst~me Frequenaverfahren zum Entwurf von Testverh~iltnissen fiir die Systemidentifikation I'lpHMeneHrIe ~IacroTrioro MeTo~a )UI~ pacqera cropoerH Bm6opra CHCTeM H~eHTHq~HKaI_(HH R O B E R T L. P A Y N E , t G R A H A M
C. G O O D W I N ~ a n d M A R T I N B. Z A R R O P §
Summary--This paper considers the problem of joint determination of input spectra and sampling rate for linear system identification. The concept of an 'input spectrum preserving sampler' is introduced and it is shown that, for this sampler, the usual anti-aliasing filter is optimal, and that joint optimal design of input spectrum and sampling rate may be readily performi~d in the frequency domain.
2. Model structure Consider a constant coefficient, linear, single inputmultiple output dynamic system with coloured measurement noise :~(t) = Ax(t) + Bu(t), (2.1) y(t) = Cx(t) + Du(t) + to(t),
(2.2)
where t e T = [0, t/I, x : T--> R", is the state vector, u : T - ~ R x is the input vector, y : T - ~ R "~ is the output vector and t o : T - + R '~ is the measurement noise vector, assumed to have a zero mean normal distribution and power density spectrum ate(f) with full rank for all f E ( - - ~ , o0).
1. Introduction The problem of optimal experiment design for the case of static regression models has been extensively studied in the statistical literature. Many of the key results are summarized in the recent book by Fedorov [3]. The extension of these results to dynamic models was first considered by Viort [4]. Further work on dynamic systems has been described by Mehra [5]. These works are, however, concerned only with the design of optimal input spectra and do not consider the related problem of optimal design of the sampling strategy. The problem of optimal sampling rate determination for linear system identification has been discussed by Astr6m [1 ] and Gustavsson [21. To gain insight into the problem these authors considered the case where the system input was prespecified. However, in general, to achieve maximal return from an experiment, a coupled design of the presampling filter, sampling rate and test signal should be carried out. The general design problem, with nonuniform sampling intervals and presampling filters, can be formulated. However, the general solution is complex and offers little insight. In the current paper the sampling intervals are assumed to be uniform and it is shown that, for this case, the standard anti-aliasing filter [2] is optimal in a well-defined sense, It is further shown that optimal coupled design of test signals and sampling rate can be readily performed in the frequency domain. The resultant design procedure is straightforward and leads to analytic solutions in simple cases.
3. Information matrix Fisher's information matrix [7] for the parameters, 0, in ,4, B, C and D is given by Me = tt z(,lo,
(3.1)
where tt is the total experiment time and ~ o is the average information matrix given by ~i o =
I2
QT(_j27rf) ~F-1(f) Q(j2~rf) dE(f),
(3.2)
where 6(f) is the input spectral distribution function [6], satisfying
f
® dE(f)= 1
(3.3)
and Q is a matrix with the lth column.given by Q~(s) = (~/~OD [ C ( s l - A) -1 B + D],
(3.4)
where 0j denotes the lth element of 0. The quantity 6(fs)-6(f0, A>A> o, can be interpreted physically as half the input power in the frequency range (ft, f,]. In equation 0.2) the total input power has been taken to be unity for convenience.
* Received 25 March 1974; revised 29 July 1974. The original version of this paper was not presented at any IFAC meeting. It was recommended for publication in revised form by associate editor K. J. Astr6m. 1"Department of Systems and Control, University of New South Wales, Australia. ~/Department of Electrical Engineering, University of Newcastle, New South Wales, Australia. g-Department of Computing and Control, Imperial College of Science and Technology, London SW7 2BT.
4. Effect o f filtering and sampling on information Letfh be a frequency above which the input spectrum is zero, i.e. the input energy lies wholly in the band [-f~, fh] then equation (3.2) for tl;~"0 may equivalently be written as
~o = f'_~QTt-i2,,f) ~'-'(r) ~j2~/) 06(..
(4.1)
Suppose now that the output, Yk, is sampled at greater than the Nyquist rate for fs, that is the sampling frequency, 189
190
Brief paper
fs>2ft. This form of sampler does not distort that part of the output spectrum arising from the input and hence is sailed here an 'input spectrum preserving sampler'. Note that the part of the output spectrum due to the noise will be distorted due to aliasing. Equation (4.1) for 2fir0is then replaced by
l~lss
f/" QT(-j2~r13 tFs-~(.f) Q(j2~rf) d~(13, (4.2)
and
aFv(13 = F(j2rrf) ~F(f) Fr(--j2zrf). If F(s) satisfies (4.8) then (4.10) becomes
~sF(13 = ~F(f)
|=1
f E (--fs/2,f~/2).
(4.3)
Theorem 4.1 The matrix (J~r0- Ms s) satisfies the inequality
Mo-- Mos >1O,
(4.4)
that is, is non-negative definite (NND).
Proof Since T'(13 is positive definite (PD) for all f, it follows from (4.3) that T'8(f)-~F(13 is PD and hence that ~F-1(13-~I~'8-1(13is PD. Hence
~o_ ~os ____f'-~QT(--j2~'13 {W-1(13 --~',-1(13} Q(j2~r13 d~(f)
(4.5)
is NND. For the purposes of comparing different experiments it is necessary to have a measure of the 'goodness' of the experiment. A suitable measure is a scalar function ~ of the information matrix where ~ has the property that
~(M~)<~(M2)
if ( M x - M 2) is NND,
(4.6)
where M1 and M s are any two information matrices. Typical examples of such cost functions are the trace and determinant of the inverse information matrix. Note that these cost functions can be given physical interpretations, e.g. the determinant is related to the volume of the highest probability density region of the parameters. It now follows from (4.4) that ~(J~0) < ~bC~roS),
(4.7)
that is, the cost with sampling is greater than or equal to the cost with continuous observations. It will now be shown that the increase in cost resulting from sampling can be eliminated by the inclusion of a suitable presampling filter.
Theorem 4.2 For a cost function ~ satisfying (4.6), and for sampling rate fa> 2fj, any presampling filter with transfer function F(s) satisfying
I F(j2~r13 1 = 0 v f ¢ (-fs/2,fs/2)
(4.8)
and invertible for f e [-fA, f d gives the same cost using the samples as with continuous observations. Furthermore, any filter satisfying (4.8) is optimal with respect to 4.
Proof It is readily verified that the expression for the information matrix 1171o a given by (4.2) when a presampling filter, F(s), is present becomes
Zlos, ffi f ~,QT(-j2,r13
FT(-j2~'13 ~Fs,-1(13
x F(j2~rf) 0(j2~13 d~:(f), (4.9) where
~zrs~r) = "Fp(13+ ~ {Wp(ifs+13+ WF(lfs--f)} lffil
(4.10)
vfs [-A,A],
(4.12)
and hence equation (4.9) leads to the same result as if continuous observations were taken, that is
J~4osr = 37[0.
where ~Fs(13 is the distorted noise spectrum and is given by
Vs(13 = ~/(13 + ~ {~/(lf8 +t3 + V(tA-13};
(4.1 1)
(4.1 3)
Hence any filter having the property (4.8) gives the same cost with samples as with continuous observations. Furthermore, from equation (4.9) and equation (4.12) it can be seen that the form of the filter f o r f ~ (-fs/2, fs/2) does not affect the cost. Hence from equation (4.7) it follows that any filter satisfying (4.8) is optimal with respect to 4.
Remark 4.1 In the design of an experiment where the sampling rate is fs > 2fA and the cost function has property (4.6) then the standard anti-aliasing filter is optimal. The standard anti-aliasing filter, [2], has a transfer function F(s) with the property
I F(j21rf) I = 1 V f ~ (-fs/2,fs/2), ] I F(j2~r13 1 = 0 V f ¢ (-fs/2,A/2), j
(4.14)
and thus, from Theorem 4.2, is optimal with respect to 4In practice the ideal anti-aliasing filter is unrealizable but can be closely approximated by a sharp cut-off filter.
5. Joint optimal design of input and sampling rate In this section the results of the previous sections are used to develop a design procedure for joint optimal determination of the input, presampling filter and sampling rate for the case of constrained input power and fixed total number of samples. The restriction on the number of samples is frequently met in practice due to the cost of data acquisition or computer storage restrictions on data analysis. Recall that the total input power is constrained to be unity; see equations (3.2) and (3.3). If the total number of samples is constrained to be N, say, and the sampling rate is fs, then it follows that the total experiment time is given by
tl = N/fs.
(5.1)
It follows from equation (3.1) that Fisher's information matrix is given by Me = (N/fs) J~os (5.?~ where 2¢1os is the average information matrix corresponding to an input spectrum having highest frequency component fn and samples collected at frequency f8 > 2fn. From Remark 4.1 the usual anti-aliasingfilter is optimal and its inclusion prior to sampling ensures that subsequent sampling will not deteriorate the information matrix. Hence equation (5.2) can be written as
Me = (N/fs) 1~1o,
(5.3)
where /~0 is the information matrix corresponding to continuous observations. It should be noted that /~'0 in equation (5.3) can be calculated from equation (4.1) because the anti-aliasing filter has ensured that there is no information loss due to sampling. It follows from equations (3.2) and (3.3) that the set of all possible information matrices arising from powerconstrained inputs is the convex hull of the set of information matrices corresponding to single frequency power constrained inputs. This fact, together with a classical theorem of Carath6odory [3], leads to the conclusitm that any information matrix can be achieved from a design comprising not more than 1 = p(p+ 1)/2+ 1 lines in the
Brief paper input spectrum where p is the total number of parameters. Furthermore, optimal designs for cost functions satisfying (4.6) can be achieved with not more than p ( p + 1)/2 lines. The details of this development for static systems are contained in the book by Fedorov [3]. From equations (5.3) and (4.1) and using the fact that only I lines are required in the input spectrum, Me can be written as :~ ?t, Re [QT(_j2~f,) W-l(f~) Q(j2~f,)] (5.4)
Ms = ~
J~
iffil
with I
~At--- 1, ~>~0,
i = 1..... 1,
(5.5)
and 0~
(5.6)
where Re [.] denotes the real part of. The quantities Ax..... A~ have the physical interpretation of being the fraction of the total power at the corresponding frequencies A . . . . . A. The optimal coupled design of input and sampling rate can now be achieved by choosing (,~1..... Az), (fl ..... fz) and fs to minimize a scalar function ~ of Mo subject to equations (5.5) and (5.6).
191
References [1] K. J. ASTROM: On the choice of sampling rates in parameter identification of time series. Inform. Sei. 1, 273-278 (1969). [2] I. GUSTAVSSON:Survey of Applications of Identification in Chemical and Physical Process~. Paper 5.3, Preprints 3rd IFAC Symposium, The Hague/Delft, June (1973). Also in Automatica 11, January (1975). [3]V. V. F ~ o R o v : Theory of Optimal Experiments. Academic Press, New York (1971). [4] B. V[ORT: D-optimal Designs for Dynamic Models-Part I: Theory. Technical Report No. 314, Department of Statistics, University of Wisconsin, October (1972). [5] R. K. MEtmA: Frequency Domain Synthesis of Optimal Inputs for Linear System Parameter Estimation. Technical Report No. 645, Division of Engineering and Applied Physics, Harvard University, July (1973). [6] E. WONG: Stochastic Processes in Information and DynamiealSystems. McGraw-Hill, New York (1970). [7] S. D. SnowY: Statistical Inference. Penguin (1970).
Appendix ,4--,4 simple first-order example Model:
fs = 2(l+e)fh,
(6.1)
where e (> 0) is related to the cut-off characteristics of the presampling filter and should be small. Substitution of (6.1) into (5.3) gives N ~'fo, (l+e)A
Ms = ~
(6.2)
where 33"ois the'average information matrix (per unit time) with continuous observations. Again using property (4.6), the cost is apparently decreased when f~ is decreased. However, since fa is the highest frequency in the input, decreasing fh compresses the input spectrum towards the low frequency end. This compression has a tendency to increase the cost arising from the term ~r 0. Thus an optimal compromise will be achieved for some value of fh which will, in general, be less than the highest frequency in the optimal continuous observation design with constrained total experiment time rather than fixed number of samples. An example is given in Appendix A illustrating the above points.
7. Conclusions The concept of an 'input spectrum preserving sampler' has been introduced and it has been shown that for this sampler the optimal presampling filter for identification is the standard anti-aliasing filter. This form of sampling system has considerable practical appeal since it is readily implementable, and allows the filtered output signal to be reconstructed from the samples. Furthermore, it has been shown that with this form of sampling system the joint optimal design of input, presampling filter and sampling rate can be readily carried out in the frequency domain.
u(s)+e(s),
~(f)
=
1
aS(os+ 1,
¢o=2~rf.
reviewers for their helpful comments on an earlier version of this paper.
(A.2)
For this example, a suitable cost function is I[M. Following the discussion of Section 5, an optimal input spectrum can be found comprising one line (p(p+l)/2 withp = 1). The information matrix with continuous observations and one-line input is given by (as cos+ 1) (oz M=
tI ( ~ o j S + l )
s .
(A.3)
Hence it follows that, for continuous observations and fixed experiment time, ts, the input which minimizes 1/M has frequency co* given by oJ* = ( ¢ 2 - 2 ~ ) - t
if2a2<~,
= oo if 2a*>~.
(A.4) (A.5)
If the noise is wide band (a ~ 0) the intuitively pleasing result ¢o* = 1#" (A.6) is obtained. Now if the output is filtered using an anti-aliasing filter and then sampled, the information matrix (A.3) for a fixed number of samples (N) becomes M = N(a2 ws+ 1) wI f s (~z ~ z + I p '
(A.7)
where fs is the sampling frequency. Now following the development of Sections 5 and 6 of the paper, the output is sampled at just above twice the input frequen~ (that is, f~--w#r). The information matrix becomes .. (a 2 oJS+l) ~o M = ~v~r 0. s ojs+ 1)2
(A.8)
and the frequency, ¢o*, minimizing I/M in the case of a fixed number of samples satisfies the equation - a 2 ~ oJ*4+3(aS-~-s) (o*S+l ----0,
(A.9)
which, for wide band (a -~ 0) noise, yields ca. ----1#'~/3.
Acknowledgements---The authors wish to thank the
(A.t)
where e(s) denotes coloured measurement noise with spectral density
6. Discussion of the designprocedure It can be seen from equation (5.4) that Mo is inversely proportional to fs. It follows from property (4.6) of the cost function that the cost is decreased when fs is decreased. The limit to how far fs can be decreased is dictated by the constraint fs > 2fh. Thus, in practice, the optimal choice offs is
1
y(s) = ~
(A.10)
This is a somewhat lower frequency than in the case with continuous observations. This result complies with the general observations made in Section 6 of the paper.