Candidate: Aidan O Dwyer, B.E., M.Eng., C. Eng. M.I.E.I. Ph. D. thesis. University Name: Dublin City University. Supervisor: Dr.

Transcription

1 itle: he estiation and coensation of rocesses with tie delays Candidate: Aidan O Dwyer, B.E., M.Eng., C. Eng. M.I.E.I. Ph. D. thesis University Nae: Dublin City University Suervisor: Dr. John Ringwood School: School of Electronic Engineering Month and Year of Subission: August 996 otal Nuber of Volues: i

2 I hereby certify that this aterial, which I now subit for assessent on the rogra of study leading to the award of Ph.D. is entirely y own work and has not been taken fro the work of others save and to the extent that such work has been cited and acknowledged within the text of y work. Signed: I.D. No.: Date: ii

3 Acknowledgeents I would like to exress y gratitude to y suervisor, Dr. John Ringwood, for his encourageent and advice over the ast four years. hanks, John. I would also like to thank y friends at the Dublin Institute of echnology, Kevin St., for their interest in y work; in addition, I would like to gratefully acknowledge the artial financial suort of the Euroean Social Fund raining for rainers rogra, adinistered through the D.I.. iii

4 Contents Abstract Chater : Introduction. Background to research work. hesis layout 4.3 hesis contributions 4 Chater : Aroaches to odel araeter and tie delay estiation 6. Introduction 6. ie doain ethods for araeter and tie delay estiation 8.. Off-line estiation ethods 8... Methods using known rocess araeters 8... Exeriental oen loo ethods Exeriental closed loo ethods Other ethods.. Methods based on ultile odel estiation..3 On-line estiation ethods..3. Methods that use rational aroxiations for the tie delay Overaraeterisation of the rocess in the discrete tie doain 4..4 Gradient ethods of araeter and tie delay estiation Introduction Gradient algoriths for estiation based on the Newton- Rahson, Gauss-Newton and steeest descent ethods..4.3 Other gradient algoriths for tie delay estiation 3..5 ie delay estiation in the absence of other rocess araeters 3.3 Frequency doain ethods for araeter and tie delay estiation 6.3. Frequency resonse estiation 6.3. Paraeter and tie delay estiation using higher order sectra 3 iv

5 .3.. Introduction Paraeter estiation techniques using higher order sectra Conclusions Model araeter estiation using frequency resonse data 35.4 Other ethods of rocess araeter and/or tie delay estiation 39.5 Conclusions 43 Chater 3: Oen loo tie doain gradient ethods of araeter and tie delay estiation Introduction Rational olynoial aroxiation of the tie delay variation Convergence of the non-delay odel araeters Convergence of the odel tie delay Convergence of the odel tie delay - Case Convergence of the odel tie delay - Case he tie delay as an integer ultile of the sale eriod he tie delay as a real ultile of the sale eriod Conclusions Convergence of the odel tie delay - Case Convergence of the full araeter set Convergence of the full araeter set - Case Convergence of the full araeter set - Case he tie delay as an integer ultile of the sale eriod - white noise inut he tie delay as a real ultile of the sale eriod - white noise inut Conclusions he tie delay as an integer ultile of the sale eriod - square wave inut he tie delay as a real ultile of the sale eriod - square wave inut Conclusions 03 v

6 3.6 Conclusions 04 Chater 4: Frequency doain ethods of odel araeter and tie delay estiation Introduction Process frequency resonse easureent Introduction Process frequency resonse identification in oen loo Frequency udating 4..4 Process frequency resonse identification in closed loo Use of ower sectral ethods for identifying the rocess frequency resonse Conclusions 4.3 Model araeter estiation using frequency resonse data Introduction he estiation of the araeters of an arbitrary order odel (with tie delay) Estiation using an analytical aroach Estiation using a gradient aroach Case studies FOLPD odel araeter estiation SOSPD odel araeter estiation (with no zero) Estiating the araeters of a third order odel (with tie delay) and no zeroes Estiating the araeters of a second order odel (with tie delay) and one zero Model structure selection Recursive estiation of the odel araeters Other issues he choice of the learning rate, µ Noralising used in the cost function Other ethods of calculating the initial odel araeter vi

7 values he choice of odel araeter estiation ethod Conclusions 74 Chater 5: he coensation of rocesses with tie delay Introduction Paraeter otiised controllers he design of PID araeter otiised controllers Introduction to the PID controller he secification of the controller araeters Iterative ethods uning rules he iniisation of a erforance criterion Direct synthesis Robust controllers he design of lead, lag or lead-lag araeter otiised controllers Conclusions Structurally otiised controllers he Sith redictor and its variations Introduction he design of the Sith redictor in continuous tie Sith redictor odifications in the continuous tie doain he control of unstable rocesses using tie delay coensators he ileentation of the Sith redictor in discrete tie he analytical redictor algorith he use of the Internal Model Control (IMC) strategy Generalised Sith redictors for MIMO rocess odels Direct synthesis ethods Introduction Continuous tie doain Discrete tie doain 06 vii

8 Direct synthesis controller design ethods for MIMO rocess odels Otial controller design ethods Introduction Inut-outut design aroach State-sace design aroach Other otiisation strategies for SISO rocess odels Predictive controllers Other coensation strategies for rocesses with tie delays Feedforward control Other strategies Conclusions 7 Chater 6: he coensation of rocesses with tie delays by using an aroriately odified Sith redictor 8 6. Introduction 8 6. he Sith redictor and its odifications Introduction Otiising the servo and regulator resonses 6..3 he design of a realistic odified Sith redictor he design of the tie advance aroxiation Siulation results Sensitivity analysis Conclusions 4 Chater 7: Closed loo tie doain gradient ethods for araeter and tie delay estiation Introduction Algoriths based on a Gauss-Newton gradient aroach heoretical develoent of the Gauss-Newton () algorith heoretical develoent of the Gauss-Newton () algorith 49 viii

9 7..3 heoretical develoent of the Gauss-Newton (3) algorith Algorith reresentations Algoriths based on a Newton-Rahson gradient aroach heoretical develoent of the Newton-Rahson () algorith heoretical develoent of the Newton-Rahson () algorith heoretical develoent of the Newton-Rahson (3) algorith Algorith reresentations Paraetric estiation - siulation results ie delay estiation Estiation of the non-delay araeters Paraeter estiation in the odified Sith redictor Introduction Develoent of the gradient algoriths Paraeter estiation - siulation results Analytical exloration of the algoriths used Non-delay odel araeter estiation Model tie delay index estiation - non-delay araeters known Model tie delay index estiation - non-delay araeters unknown Model tie delay index estiation for a general odel Process and odel in SOSPD for Process and odel of arbitrary order Conclusions 97 Chater 8: Conclusions Gradient algoriths for araeter and tie delay estiation he use of the Sith redictor structure for identification and control Future direction of the field Glossary of essential ters and sybols used References 33 ix

10 List of Figures Chater.: Correlation analysis (Unbehauen and Rao (987)) 7.: Oen loo ileentation 7.3: Closed loo ileentation 8 Chater 3 3.: MSE surface (Pade) 50 3.: MSE surface (Product) : MSE vs. Model gain : MSE vs. Model tie constant : Noralised MSE vs. tie delay index - white noise inut : Noralised MSE vs. tie delay index - square wave inut : Udating of the odel tie delay index - Case a: ie delay index estiate b: e ( n) corresonding to Figure 3.8a 6 3.8c: ie delay index estiate d: e ( n) corresonding to Figure 3.8c 6 3.9: Noralised MPE vs. tie delay index - white noise inut : Noralised MPE vs. tie delay index - square wave inut 65 3.: Udating of the odel tie delay index - Case 66 3.a: ie delay index estiate - white noise excitation 66 3.b: e 3 ( n) corresonding to Figure 3.a a: ie delay index estiate - square wave excitation b: e 3 ( n) corresonding to Figure 3.3a : Noralised MPE vs. tie delay index - white noise excitation - g b = : Noralised MPE vs. tie delay index - white noise excitation - g b = : Noralised MPE vs. tie delay index - square wave excitation - g b = : Noralised MPE vs. tie delay index - square wave excitation - g b = x

11 3.8: Noralised MPE vs. tie delay index - white noise excitation - s = 0. s : Noralised MPE vs. tie delay index - white noise excitation - s = 0. 0 s : Udating of the odel tie delay index - Case a: ie delay index estiate - white noise excitation 76 3.b: e 6 ( n) corresonding to Figure 3.a 76 3.: Noralised MPE vs. tie delay index (conditions in equation (3.60) et) 8 3.3: Noralised MPE vs. tie delay index (conditions in equation (3.60) violated) 8 3.4: Udating the full araeter set - Case a: Gain estiate b: ie constant estiate c: ie delay index estiate d: e n ( ) a: Gain estiate b: ie constant estiate c: ie delay index estiate d: e n ( ) a: ie delay index estiate b: ie delay index estiate : Noralised MPE vs. tie delay index - white noise excitation 9 3.9: Noralised MPE vs. tie delay index - white noise excitation : Udating the full araeter set - Case 9 3.3a: Gain estiate 9 3.3b: ie constant estiate 9 3.3c: ie delay index estiate d: e n ( ) : Noralised MPE vs. tie delay index - white noise excitation - g b = : Noralised MPE vs. tie delay index - white noise excitation - g b = : Noralised MPE vs. tie delay index - square wave excitation : Noralised MPE vs. tie delay index - square wave excitation a: Gain estiate b: ie constant estiate c: ie delay index estiate 0 xi

12 3.36d: e n ( ) : Noralised MPE vs. tie delay index - square wave excitation - g b = : Noralised MPE vs. tie delay index - square wave excitation - g b = Chater 4 4.: Oen loo ileentation 08 4.: Magnitude estiate - oen loo - forgetting factor = a: Phase estiate - oen loo - forgetting factor = b: Beat frequency (exanded) 3 4.4: Magnitude estiate - oen loo - forgetting factor = : Closed loo reresentation 4 4.6: Block diagra of the closed loo syste ileentation 8 4.7: Magnitude, hase and frequency convergence 9 4.8: Exaination of J < 0 - K = : K µ = : µ = : τ µ = : Unit ste resonse of the rocess and the FOLPD odel 4 4.3: Polar lot of the rocess and the FOLPD odel : J a < 0 for five airs of K and τ values : J a < 0 for five airs of K and τ values : K µ = : a µ = : a µ = : τ µ = : Unit ste resonse of the rocess and the SOSPD odel : Polar lot of the rocess and the SOSPD odel : Flowchart suarising the algorith for odel araeter estiation 6 4.3: K µ = : µ = xii

13 4.33: τ µ = : Unit ste resonse of the rocess and the FOLPD odel : Polar lot of the rocess and the FOLPD odel : K µ = : a µ = : a µ = : τ µ = : Unit ste resonse of the rocess and the SOSPD odel : Polar lot of the rocess and the SOSPD odel : K µ = : a µ = : a µ = : τ µ = : Unit ste resonse of the rocess and the SOSPD odel : Polar lot of the rocess and the SOSPD odel : K µ = : a µ = : a µ = : τ µ = : Unit ste resonse of the rocess and the SOSPD odel : Polar lot of the rocess and the SOSPD odel : K µ = : a µ = : a µ = : τ µ = : Unit ste resonse of the rocess and the SOSPD odel : Polar lot of the rocess and the SOSPD odel : Recursive estiation algorith : K µ = : µ = : τ µ = xiii

14 Chater 5 5.: Block diagra of a SISO rocess controlled by an ideal PID controller 77 5.: Sith redictor ileentation : Alternative Sith redictor ileentation : Block diagra for the IMC structure 03 Chater 6 6.: Block diagra of the Sith redictor structure 9 6.: Block diagra of a generalised Sith redictor structure 0 6.3: Modified Sith redictor design considered : Servo and regulator resonses - Siulation : Servo and regulator resonses - Siulation : Servo and regulator resonses - Siulation : Servo and regulator resonses - Siulation 4 33 MSP SP 6.8: Sα Sα when the araeters defined in equations (6.38) to (6.4) are used 38 MSP SP 6.9: Sα Sα when the araeters defined in equations (6.38) to (6.40) are used; the exact value of the tie advance is used 38 MSP SP 6.30: Sα Sα when the araeters defined in equations (6.38) to (6.4) are used, with the disturbance resent on the rocess outut 40 MSP SP 6.3: Sα Sα when the araeters defined in equations (6.38) to (6.40) are used, with the disturbance resent on the rocess outut; the exact value of the tie advance is used 40 Chater 7 7.: Block diagra of the Sith redictor 43 7.: Grahical interretation of the algorith : Reresentation of the Gauss-Newton algoriths for tie delay estiation : Reresentation of the Gauss-Newton algoriths for odel gain or xiv

15 odel tie constant estiation : Reresentation of the sensitivity functions for the Gauss-Newton algoriths : Reresentation of the Gauss-Newton algoriths for siultaneous odel araeter estiation : Reresentation of the Gauss-Newton algoriths for siultaneous odel araeter estiation (of a general order odel) : Reresentation of the sensitivity functions for the Gauss-Newton algoriths (general order odel) : Reresentation of the Newton-Rahson algoriths for tie delay estiation : Reresentation of the Newton-Rahson algoriths for odel gain or odel tie constant estiation 68 7.: Reresentation of the sensitivity functions for the Newton-Rahson () algorith : ie delay udating - Case : ie delay udating - Case : ie delay udating - Case : ie delay udating - Case : ie delay udating - Case : ie delay udating - Case : Phase lots of rocesses and their odels (Case 6) : ie delay udating - Case : Polar lots of rocesses and their odels (Case 7) : ie delay udating - Case : Phase lots of rocesses and their odels (Case 8) : Gain udating : ie constant udating : ie delay udating - odified Sith redictor : Gain udating - odified Sith redictor : ie constant udating - odified Sith redictor 86 xv

16 itle: he estiation and coensation of rocesses with tie delays Author: Aidan O Dwyer ABSRAC he estiation and coensation of rocesses with tie delays have been of interest to acadeics and ractitioners for several decades. A full review of the literature for both odel araeter and tie delay estiation is resented. Gradient ethods of araeter estiation, in oen loo, in the tie and frequency doains are subsequently considered in detail. Firstly, an algorith is develoed, using an aroriate gradient algorith, for the estiation of all the araeters of an aroriate rocess odel with tie delay, in oen loo, in the tie doain. he convergence of the odel araeters to the rocess araeters is considered analytically and in siulation. he estiation of the rocess araeters in the frequency doain is also addressed, with analytical rocedures being defined to rovide initial estiates of the odel araeters, and a gradient algorith being used to refine these estiates to attain the global iniu of the cost function that is otiised. he focus of the thesis is subsequently broadened with the consideration of coensation ethods for rocesses with tie delays. hese ethods are reviewed in a corehensive anner, and the design of a odified Sith redictor, which facilitates a better regulator resonse than does the Sith redictor, is considered in detail. Gradient algoriths are subsequently develoed for the estiation of rocess araeters (including tie delay) in closed loo, in the Sith redictor and odified Sith redictor structures, in the tie doain; the convergence of the odel araeters to the rocess araeters is considered analytically and in siulation. he thesis concludes with an overview of the ethods develoed, and rojections regarding future develoents in the toics under consideration.

17 CHAPER Introduction. Background to research work he ai of the research is to review identification and coensation ethods for rocesses with tie delays, and to develo aroriate identification and coensation strategies. A tie delay ay be defined as the tie interval between the start of an event at one oint in a syste and its resulting action at another oint in the syste. ie delays are also known as transort lags, dead ties or tie lags; they arise in hysical, cheical, biological and econoic systes, as well as in the rocess of easureent and coutation. Secific exales of systes with tie delay are defined by Latour et al. (967), Sandoz (987), Paageorgiou and Messner (989), Gendron et al. (993), Menhaj and Hagan (994) and Igarashi et al. (994), aongst others. he estiation of tie delays also arises in signal rocessing alications, where a tie delay is also known as a tie difference of arrival (DOA) between two signals; such a easureent arises in underwater tracking alications, bioedicine, geohysics, astronoy, acoustics, seisology and telecounications (Silva (987), Salt et al. (993), Shen et al. (993), Sheridan (993), Laguna et al. (994) and Webster (994)). Generally seaking, in the signal rocessing alications, it is the estiation of ure tie delay that is required, rather than the estiation of tie delays in the resence of other rocess araeters. Process araeter and tie delay estiation techniques ay be broadly divided into off-line techniques and on-line techniques, with on-line estiation requiring recursive estiation in a closed loo environent. On-line estiation is often called adative araeter estiation or sequential araeter estiation (Ljung (987)) or identification by eans of couters in on-line oeration with the rocess (Iserann (99), Iserann et al. (99)). he choice of identification ethod for araeter estiation deends on the urose of the identification, which deterines the odel needed and the accuracy required; a trade-off exists between required accuracy and coutational effort when choosing the identification ethod. he choice of

18 identification ethod also deends on whether the rocess ay be interruted fro its noral oeration, or if identification ust take lace during noral closed loo oeration. Aarent tie delays ay result when high order rocesses are aroxiated by eans of lower order transfer functions (Seborg et al. (989)). he tie delay estiated in these alications ay be a cobination of a ure tie delay and contributions due to high order dynaic ters in the rocess transfer function. Bentley (989) gives a neuatic transission line as an exale of such a syste and odels it as a first order lag lus tie delay (FOLPD) odel. As entioned, the urose of the identification deterines the odel required. Newell and Lee (989) state that there is uch debate (in rocess control circles) over how colex a odel ay reasonably be identified fro exeriental data; they suggest that this deends on the data quality available (i.e. if the data is corruted by noise) and the analysis technique used. he authors suggest that a cautious aroach is to identify a FOLPD odel fro the exeriental data and that an otiistic aroach is to identify a second order syste lus tie delay (SOSPD) odel fro the data. Other authors that consider aroriate odelling ethods for real rocesses include Latour et al. (967), Pollard (97), Edgar et al. (98), Sith and Corriio (985), Morari and Zafiriou (989), Paageorgiou and Messner (989), Seborg et al. (989), Hang and Chin (99), De Carvalho (993), Gendron et al. (993), Hang et al. (994b), Kotob et al. (994), Readle and Henry (994), Schei (994), Sirthwaite et al. (994) and Yang (994); these authors work is exained in ore detail by O Dwyer (996a). A conclusion fro this work is that even if the rocess has no hysical tie delay, it ay be ossible to odel such a (ossibly high order) rocess by a low order odel lus tie delay; it also aears reasonable that either a FOLPD odel (for overdaed rocesses) or a SOSPD odel (for overdaed or underdaed rocesses) should be estiated, as either of these aroxiate rocess odels aears to be sufficiently accurate for any alications. However, if a riori inforation on the rocess is available (such as the rocess order), the estiation of the full order odel lus tie delay ay be indicated; in any case, the work in the thesis will concentrate on the identification of rocesses that are adequately odelled by a linear odel with tie delay. 3

19 . hesis layout he thesis will deal with the identification of rocess odel araeters, and tie delay, in both oen loo and closed loo environents, together with the coensation of such systes. Chater classifies and outlines the aroaches to odel araeter and tie delay estiation that have aeared in the literature; both off-line and on-line estiation techniques are treated. Chater 3 develos one such estiation technique, involving the use of a gradient ethod, to estiate the araeters in oen loo, in the tie doain. Chater 4 discusses frequency doain ethods of araeter and tie delay estiation; one such ethod uses a gradient aroach to estiate the araeters. he focus is broadened in Chater 5, in which coensation ethods for rocesses with tie delays are discussed in detail; one such coensation ethod, that involves the design of a odified Sith redictor, is treated in Chater 6. ie doain ethods that facilitate the estiation of the araeters in closed loo (in a Sith redictor structure) using aroriate gradient techniques are described in Chater 7. In each of these chaters, conclusions as to the efficacy of the techniques discussed are reached, and further work is suggested. he conclusions of the work are outlined in Chater 8, followed by a glossary of essential ters used in the thesis and a list of references. hirteen technical reorts, written by the author, are referenced throughout the thesis; these reorts rovide suleentary details on the toics discussed..3 hesis contributions Original work contained in this thesis includes the following toics: () he corehensive review of ethods defined in the literature for odel araeter and tie delay estiation (Chater ). () he forulation of the cost function (which is a function of the error between the rocess and the odel in the tie doain) with resect to the tie delay variation between the rocess and the odel, when a variety of olynoials are used as aroxiations for the tie delay variation. his work is detailed in Chater 3. (3) he develoent of seven theores that are concerned with the iniisation of the cost function, when the araeters of a first order discrete stable rocess with tie 4

20 delay are being estiated, in oen loo, in the tie doain. he theores involve the calculation of the cost function and, in ost cases, the roving that the cost function is uniodal with resect to the rocess tie delay (as the odel tie delay varies), under defined oerating conditions, using the rincile of induction. hese theores are roved in Chater 3. (4) he develoent of algoriths for rocess frequency resonse easureent, in both oen loo and closed loo, that use the ratio of the Fourier transfors of the outut and inut to the rocess and use a ower sectral density aroach. his work is detailed in Chater 4. (5) he develoent of analytical and iterative techniques to estiate the araeters of an arbitrary order odel lus tie delay fro the rocess frequency resonse (Chater 4). A odel order estiation technique is also develoed. (6) he full review of ethods defined in the literature for the coensation of rocesses with tie delays (Chater 5). (7) he develoent of a odified Sith redictor, that iroves the regulator resonse (when coared to the Sith redictor), while ensuring aroxiately the sae servo resonse. his work is reorted in Chater 6. (8) he develoent of five alternative gradient algoriths to the algorith defined by Marshall (979) and Bahill (983), for the udating of the odel araeters and tie delay, in closed loo, in the tie doain (Chater 7); two of the algoriths are based on a Gauss-Newton gradient ileentation and three of the algoriths are based on a Newton-Rahson gradient ileentation. hree theores are develoed that show that the cost function is uniodal with resect to the rocess tie delay as the odel tie delay varies, when the araeters of a nuber of rocess odels are to be estiated. 5

21 CHAPER Aroaches to odel araeter and tie delay estiation. Introduction A tie delay has been defined in Chater as the tie interval between the start of an event at one oint in a syste and its resulting action at another oint in the syste. his chater of the thesis will discuss tie delay estiation ethods (together with odel araeter estiation ethods, where aroriate) that have been roosed in the ublished literature; these ethods ay be broadly classified into tie doain and frequency doain techniques. ie doain estiation ethods will be treated first. A nuber of off-line estiation techniques are outlined, for single inut, single outut (SISO) and ultiinut, ulti-outut (MIMO) odel structures, in oen loo and in closed loo. A full discussion of ultile odel estiation techniques will then be carried out; these ethods tyically involve estiating a nuber of odels, each with a different value of the tie delay, and subsequently deterining the ost aroriate odel. However, these ethods tend to be coutationally intensive. A nuber of on-line estiation techniques will subsequently be treated, followed by a discussion of gradient ethods of araeter and tie delay estiation; the latter ethods ay be ileented in either oen loo or closed loo, and in either an off-line or on-line anner. he estiation of tie delays in the absence of other rocess araeters is also reviewed; such techniques are norally associated with signal rocessing alications. Frequency doain estiation techniques ay be classified in a siilar anner to tie doain estiation ethods. he use of the frequency doain, as a eans of estiating the araeters and tie delay of a rocess odel, has a certain intuitive aeal, since the tie delay contributes to the hase ter but not the gain ter of the frequency resonse. A colete discussion of frequency doain estiation ethods (which include higher order sectral algoriths) is rovided. 6

22 Other ossibilities for estiation are subsequently debated such as the use of neural networks, the use of rocess order identification ethods and the ileentation of the estiation strategies in the delta doain. In each of the sections of the chater, coarisons between the ethods reviewed and conclusions as to the alicability of various classes of ethods will be drawn, as aroriate. General conclusions fro the literature review will be drawn and aroaches to tie delay estiation where original work ay be usefully done will be outlined. he discussion in this chater is further detailed by O Dwyer (996a). 7

23 . ie doain ethods for araeter and tie delay estiation.. Off-line estiation ethods he off-line estiation techniques ay be broadly divided as follows: (a) Methods using known rocess araeters (b) Exeriental oen loo ethods (c) Exeriental closed loo ethods (d) Other ethods.... Methods using known rocess araeters Estiation ethods that use known rocess araeters are based on calculating an estiate of the araeters of a low order odel lus tie delay fro the known araeters of a high order rocess. he ethods are largely "rule of thub" based ethods that are unsuitable for the estiation of tie delays of unknown rocesses. he ethods, considered in detail by O Dwyer (99) reviously, will not be reconsidered in this chater.... Exeriental oen loo ethods hese ethods are based on estiating the araeters (including the tie delay) fro aroriate data gathered during tests while the rocess is in oen loo. yically, the inut to the rocess is in ste or ulse for. One of the first such ethods was described by Ziegler and Nichols (94), in which the tie constant and tie delay of a FOLPD rocess odel are obtained by constructing a tangent to the ste resonse at its oint of inflection. he intersection of the tangent with the tie axis at the ste origin rovides an estiate of the tie delay; the tie constant is estiated by calculating the intersection of the tangent with the value of the steady state outut divided by the odel gain. Other such tangent and oint ethods for estiating the araeters of a FOLPD odel are described by Cheng and Hung (985) and De Carvalho (993), aong others. he ethod ay also be used to deterine the 8

24 araeters of a SOSPD odel; Sith (957), Perlutter (965), Meyer et al. (967), Csaki and Kis (969), Sundaresan et al. (978) and Huang and Cleents (98) describe such aroaches. he ajor disadvantage of all these ethods is the difficulty of deterining the oint of inflection in ractice. Soe ethods that eliinate this disadvantage use two oints on the rocess ste resonse, to estiate the FOLPD odel araeters, such as those described by Sunderesan and Krishnasway (978) and Cheng and Hung (985), or use two, three or ore oints on the rocess ste resonse, to estiate the araeters of a SOSPD odel, such as those described by Huang and Cleents (98), Huang and Huang (993), Huang and Chou (994) and Rangaiah and Krishnasway (994a), (996). An alternative exeriental ethod involves calculating the araeters of an aroriate odel fro the area under the ste resonse outut curve (Nishikawa et al. (984), Arzen (987)). Exeriental oen loo tests are a straightforward ethod of calculating the odel araeters; however, the araeters of a FOLPD odel aroxiation, deterined by using actual ste resonse data, ay vary considerably deending on the oerating conditions of the rocess, the size of the inut ste change and the direction of the change, with these variations being usually attributed to rocess nonlinearities (Seborg et al. (989)). Harris and Mellicha (980) declare that a ajor drawback of an aroach that involves the introduction of a ste change is that the rocess ust be sufficiently disturbed by the change to obtain reasonably accurate dynaic inforation; such a disturbance ay well force the rocess outside the region of (aroxiately) linear behaviour. Arzen (987) oints out that ethods to deterine the dynaics of a rocess by exaining its resonse to a deterinistic signal such as a ste or ulse inut are conditioned on no drastic disturbances influencing the rocess. he tie scale of the rocess ust also be known in advance in order to deterine when the transient resonse has been coleted. Morari (988) akes the iortant oint that the ethod of judging odel quality by coaring the rocess ste resonse to the odel ste resonse is not necessarily the best eans of otiising the odel quality fro the oint of view of control syste design; the author shows that three rocesses that have ractically identical oen loo resonses ay behave very differently under feedback. 9

25 ...3 Exeriental closed loo ethods hese ethods are based on estiating the araeters (including the tie delay) of a odel fro aroriate data gathered during the closed loo oeration of the controlled rocess. he data is generally obtained fro the closed loo ste resonse; the ethods tyically involve the analytical calculation of the araeters of an aroriate rocess odel fro outut easureents (such as the steady state value of the resonse and the first and second eak resonse values), of a unity feedback closed loo syste under roortional control. yically, the tie delay is aroxiated in an aroriate anner (Yuwana and Seborg (98), Jutan and Rodriguez (984), Lee (989), Jutan (989), Bogere and Ozgen (989)), though this is not absolutely necessary (Sung et al. (994)). Chen (989) and Lee et al. (990) calculate the ultiate gain and frequency of a unity feedback closed loo syste under roortional control, fro the ste resonse, and use these easureents to calculate the araeters of an aroriate oen loo odel. Hwang (993), Hwang and seng (994) and Hwang and Shiu (994) use a cobination of the ethods based on ste resonse easureents, and easureents of the ultiate gain and frequency, to deterine the best rocess odel; the latter two aers also outline siilar identification strategies in closed loo when a PI or PID controller is used. Hwang (995) brings together this work by outlining ethods for the identification of a SOSPD rocess odel in closed loo, by using the P, PI or PID controllers, and alying either a ste, ulse or iulse test inut signal in setoint. In a ore recent alication, Kavdia and Chidabara (996) use the ethod of Yuwana and Seborg (98) to calculate the araeters of a FOLPD odel for an unstable rocess. Refineents to the ublished algoriths are ossible, as detailed by O Dwyer (996a); however, as entioned in this reort, the robustness of any of the estiation ethods to noise on the rocess resonse is questionable. One ethod for which this coent does not aly is the characteristic areas ethod of Nishikawa et al. (984), in which the area under the ste resonse outut curve is used to calculate the odel araeters. 0

26 ...4 Other ethods Other off-line estiation ethods do not naturally fall into any of the categories described earlier. Exales of ethods that ay be used to estiate the araeters and tie delay of a linear SISO odel include the following: (a) Aroxiating the tie delay by a Laguerre olynoial and using the standard (offline) least squares estiation ethod to identify the araeters of the resulting odel (De Souza et al. (987), (988), Salgado et al. (988)) and (b) Defining a state sace odel for the rocess and ileenting a axiu likelihood estiate for the rocess araeters and the tie delay based on this araeterisation (Nagy and Ljung (99)). Reresentative ethods that have been used to deterine the araeters and tie delay of a linear MIMO odel are (a) he extension of a ethod to estiate the odel order, defined for SISO systes, which is based on insecting the near singularity of the inforation atrix, to also estiate tie delays, if the ranges of the tie delays are known (Mancher and Hensel (985)) and (b) he resolving of outut signals of MIMO rocesses into a set of indeendent outut signals for SISO rocesses by using ersistently exciting Walsh function inut signals; the Walsh functions ay then be used to estiate the araeters and the tie delay of each of the SISO rocesses (Bohn (985))... Methods based on ultile odel estiation hese ethods are based on the estiation of a nuber of different rocess odels, for different values of the tie delay. he odel araeters chosen are those that iniise a cost function that deends on the difference between the rocess and the odel oututs. One of the best exales of the aroach is given by Baur and Iserann (978), who use recursive correlation analysis with least squares araeter estiation to detect ( d d ) searate odels, where ax = the axiu ax ax in odel order and the tie delay index (which is the integer value of the tie delay divided by the sale tie) lies between d in and d ax. A loss function V(,d), based on the residuals, is iniised as odel order is varied; the otiu estiates of odel

27 order,, and tie delay index, d, are deterined if V(+,d) and V(,d+) do not decrease significantly in relation to V(,d). Other authors that also estiate the odel order, araeters and tie delay index using a ultile odel ethod include Gabay and Merhav (976), Bokor and Keviczky (984), Peterka (989), Heerly (99), Musto and Lauderbaugh (99), Warwick and Kang (993) and uch et al. (994). Soe authors concentrate on estiating the tie delay and rocess araeters only; the tie delay is estiated by iniising the loss function as the tie delay index is varied, with the rocess araeters estiated using other ethods. Aong the authors that discuss such techniques are Hsia (969), Rao and Sivakuar (979), Rao and Palanisway (983), Hansen (983), Pearson and Wuu (984), Wuu and Pearson (984), Cheng and Hung (985), Abrishaakar and Bekey (985), (986), Batur (986), Agarwal and Canudas (987), Jiang (987), Juricic (987), Ki et al. (987), Unbehauen and Rao (987), Peter and Iserann (988), Casted (989), Co and Ydstie (990), Zheng and Feng (990), Ferretti et al. (99), Schei (99), Lublinsky and Fradkov (993), Chen and Loaro (993), Leva et al. (994), Readle and Henry (994), Ferretti et al. (995) and Wang and Cleents (995). he ultile odel estiation technique ay also be used to estiate the araeters of ultile-inut, single outut (MISO) or MIMO rocess odels with tie delays. Authors that estiate the odel order, araeters and tie delay indices using such ethods for these alications include Blessing et al. (978), Bokor and Keviczky (984), Mancher and Hensel (985), Xu (989) and Haest et al. (990). he attraction of ultile odel estiation ethods is that the grid searching used will facilitate the estiation of the araeters corresonding to a global iniu of a cost function, even in the resence of local inia, rovided enough odels are estiated. he ethod is relatively crude coared to the use of gradient search ethods (discussed in Section..4), and it is also ore coutationally intensive; however, the latter ethods do not guarantee the estiation of the araeters corresonding to the global iniu, in the resence of local inia...3 On-line estiation ethods On-line tie delay estiation requires recursive estiation of the tie delay in a closed loo environent. he techniques ay be classified as follows: (a) Methods that use rational aroxiations for the tie delay, followed by

28 (b) recursive identification of the odel araeters and Methods that involve overaraeterisation of the rocess in the discrete tie doain...3. Methods that use rational aroxiations for the tie delay he following rational aroxiations ay be used for the tie delay: (a) (b) (c) (d) (e) (f) (g) (h) he aylor s series exansion he Pade aroxiation he Laguerre aroxiation he Product aroxiation (or Paynter delay line) he direct frequency resonse aroxiation technique he Bessel aroxiation A transfer function aroxiation (fro Marshall (979)) and Nuerical otiisation (e.g. the equirile forula); this is defined by Piche (990). hese aroxiations have been detailed by O Dwyer (996a); for exale, the first order aylor s series aroxiation for the tie delay, e sτ, is sτ. Seborg et al. (989) declare that when the tie delay is less than one tenth of the tie constant (in a FOLPD rocess odel structure), then a first order Pade aroxiation for the tie delay is accurate to within engineering accuracy, considering that ost rocesses behave like low ass filters; corresondingly, the second order Pade aroxiation is accurate to within engineering accuracy when the tie delay is less than one fifth of the (reeated) tie constant of a ore general rocess odel structure. When the tie delay is aroxiated by a rational olynoial, the resulting odel araeters are norally estiated in a discrete tie environent using an algorith based, for instance, on recursive least squares (RLS); the tie delay ay then be deduced fro the odel araeters identified. Such an aroach is outlined by Roy et al. (990), (99a), (99b), (99c), (993a), (993b), Boje and Eitelberg (99), Bai and Chyung (993), Fernandes and Ferriers (994) and Yasterbov and Grzywaczewski (994). However, the ethod defined by Roy et al. (990), (99a), (99b), (99c), in which the tie delay is odelled by a zero in the continuous tie doain, with the araeters of the odel being identified using the RLS algorith in 3

29 the discrete tie doain, did not work for siulations taken by Kelly (99) or O'Dwyer (99) i.e. it was not ossible to estiate the tie delay fro the resulting rocess araeters identified...3. Overaraeterisation of the rocess in the discrete tie doain he ethod of overaraeterisation involves subsuing the tie delay ter into an extended (or overaraeterised) z doain nuerator olynoial. he corresonding araeters are estiated using a recursive estiation schee, and the tie delay is calculated based on the araeters identified; for a noise free syste, all nuerator araeters whose indices are saller than the tie delay index should be identified as zero. Only values of the tie delay that are integer ultiles of the sale eriod are directly estiated by the ethod. he art of the tie delay that is a fraction of the sale eriod ay be calculated fro the nuerator araeters identified, for rocesses that can be odelled by a FOLPD odel (O'Dwyer (99), (993)) and for rocesses that ay be odelled by a SOSPD odel (hoson et al. (989)); however, the robustness of these ethods of estiation in the resence of noise is questionable. Many overaraeterisation ethods have been defined to calculate the nuerator (and denoinator) araeters, and subsequently the tie delay, for rocesses that ay be odelled in SISO for or MIMO for. Kurz (979) and Kurz and Goedecke (98), for exale, define a robust ethod for estiating the SISO odel araeters that is equivalent to deterining the best atch between the iulse resonse of the overaraeterised odel and the iulse resonse of a nonoveraraeterised odel with a ure tie delay; the ethod suffers fro the disadvantage of having a heavy coutational load. Other ethods offer various tradeoffs between robustness and coutational load, such as those described by Biswas and Singh (978), Astro and Zhou (98), Friedlander (98), Wong and Bayoui (98), Haberayer and Keviczky (985), Haberayer (986), Batur (986), De Keyser (986), Koivo et al. (988), Hu et al. (988), Keviczky and Banyasz (988), Naji et al. (988), Xu (988), eng and Sirisena (988), Landau (990), eng (990), Guez and Pioviso (99), Lundh and Astro (994) and Readle and Henry (994). Other authors describe a recursive ethod to estiate the araeters, order and tie delay index for both a stochastic syste and a deterinistic syste, using an overaraeterised ethod to estiate the tie delay (Chen and Zhang (990) and 4

30 Zhang and Chen (990)). In an interesting aer, Keviczky and Banyasz (99) identify the tie delay index using overaraeterisation in the delta doain (see Section.4). Other authors identify MIMO rocess odels (with tie delays) using the ethod of overaraeterisation; Gurubasavaraj and Brogan (983), for instance, extend the ethod of Kurz and Goedecke (98) to estiate the tie delay for each inut-outut air of a MIMO rocess. Siulation results resented by the authors show that the tie delays ay be estiated in 0 sale eriods, for a x MIMO rocess with a axiu tie delay index of 4; the rocess order is however assued known a riori. Other authors that use overaraeterisation for this alication include Song and Xu (985) and Zhang and Chen (990). he attractiveness of the ethod of overaraeterisation as a eans of estiating odel araeters and tie delay is that it is a natural extension of ethods used in delay-free identification alications. However, the ethod has any disadvantages. () he coutational burden of the RLS algorith increases with the square of the nuber of estiated araeters (De Keyser (986), Glentis and Kaloutsidis (99), Ferretti et al. (995)). () he ersistent excitation condition (a condition for araeter convergence) is ore difficult to satisfy for overaraeterised odels (Ki et al. (987), Duont et al. (993)). (3) he adative caability of the corresonding controller is degraded, as it takes a long tie for the araeters to be retuned if a change in the rocess dynaics occurs (Ki et al. (987)). However, it is ossible by introducing a erturbation signal into the regressor vector, when the araeters of the odel with delay are being estiated, to achieve a siilar convergence rate for the araeters of an overaraeterised odel as for the araeters of a non-overaraeterised odel (Xia et al. (987), Xia and Moore (989)). (4) he resence of a high order nuerator olynoial increases the likelihood of coon factors in the nuerator and denoinator olynoials in the estiation odel, rendering identification ore difficult (Duont et al. (993)). (5) he overaraeterisation ethod is not robust if a load disturbance is resent, or if easureent noise is significant (Lee and Hang (985)). However, Xia et al. (987) and Xia and Moore (989) state that injecting an excitation signal into the regressor vector (for RLS estiation (987) or recursive extended least squares (RELS) 5

31 estiation (989)) allows the araeters of a odel of one order of overaraeterisation (987) or arbitrary degree of overaraeterisation (989) to have the sae guaranteed convergence as the araeters of a non-overaraeterised odel (i.e. ill-conditioning is avoided for the overaraeterised odel) for both odels with white noise excitation (987) and coloured noise excitation (989). Fro this discussion, the biggest disadvantage of the overaraeterisation ethod for the identification of a rocess with tie varying delay in closed loo, erturbed by a seudo-rando binary signal (PRBS), is the extra coutational burden associated with identifying a greater nuber of nuerator araeters. In an attet to reduce the coutational burden associated with the overaraeterisation ethod, the following ideas ay be worth considering: (a) If the tie constants of the rocess do not change significantly, then the denoinator araeters need not be estiated on-line; as well as reducing the coutational burden in the estiation stage, other advantages of this schee are that excessive fluctuation of the denoinator araeters is avoided and the denoinator araeter estiates cannot drift into or near an undesirable region. his suggestion was ade by Vogel and Edgar (98). A further suggestion ade by Seborg et al. (986) is that selective udating of certain odel araeters be eloyed when the nuber of araeters of the rocess to be estiated is large; such selective udating could be achieved by only udating those araeters that give a significant iroveent in the residual of the odel fit. (b) he saling interval could be adated to reduce the nuber of araeters to be estiated. o this end, Seborg et al. (986) suggest that the saling eriod be chosen so that the tie delay index has a value of two or three; such slow saling, the authors suggest, has the additional advantage of increasing the robustness of the corresonding adative controller. his advice ay be relevant only for sall values of the tie delay as otherwise it ay conflict with the ost often quoted rule of thub that the saling eriod should be between one fifth and one fifteenth of the 95% rise tie of the rocess ste resonse (Iserann (989)). he ethods of tie delay estiation using the overaraeterised odel that aear ost robust are those of Kurz (979) and eng and Sirisena (988). For a ractical alication, the ethod of eng and Sirisena (988) sees to be ost roising, because of its relative coutational silicity. he ethod of overaraeterisation as a eans of estiating tie delays ay be extended in 6

32 alication fro SISO rocesses to MISO rocesses and MIMO rocesses. Surrisingly, the ethods that have been well docuented for the estiation of the rocess tie delay in the SISO environent have not been widely alied to the identification of tie delays in MIMO rocesses in the available literature (one of the few excetions is the ethod outlined by Gurubasavaraj and Brogan (983)). his toic is discussed ore fully by O Dwyer (996l)...4 Gradient ethods of araeter and tie delay estiation..4. Introduction Gradient ethods of araeter estiation are based on udating the araeter vector (which includes the tie delay) by a vector that deends on inforation about the cost function to be iniised. he gradient algoriths considered norally involve exanding the cost function as a second order aylor's exansion around the estiated araeter vector. he cost function is given by N J( n) = 05. e ( n j) j= 0 (.) with J(n) = cost function and e = error = rocess outut inus odel outut. A second order aylor s series exansion of the cost function ay be deterined fro equation (.) to be J( n + ) J( n + ) J( n + ) = J( n) + ( θ( n) θ ( n)) ( θ( n) θ ( n)) θ( n) θ ( n) ( θ( n) θ ( n) ) * * * (.) with θ( n) n R, θ( n ) = araeter vector and θ * ( n ) = otiu araeter vector. An estiate of the araeter vector is deterined by iniising J( n + ) with resect to the araeter vector. A silified udating strategy based on this iniisation is θ( n + ) = θ ( n) + µφ( n) (.3) 7

33 with J( n) J( n) φ( n) = θ ( n) θ( n) (.4) and with φ( n) n R and µ = learning rate; the default value of µ =.0. he artial derivative of the cost function with resect to the araeter vector ay be deterined recursively (fro equation (.)) to be J( n + ) θ( n) J( n) e( n + ) = + e( n + ) θ( n) θ( n) (.5) with the starting value of the artial derivative of the cost function with resect to the araeter vector assued zero. he calculation of the second artial derivative of the cost function with resect to the araeter vector deterines the nature of the otiisation algorith. Ljung (987) divides these otiisation algoriths into three classes: () he udating vector is a function of the cost function, the artial derivative of the cost function with resect to the araeter vector and the second artial derivative of the cost function with resect to the araeter vector. he Newton-Rahson algorith is an exale; under these circustances, the second artial derivative of the cost function with resect to the araeter vector (labelled the Hessian atrix), calculated using equation (.5), is given by J( n + ) θ ( n) J( n) e( n + ) e( n + ) e( n + ) = + e( n + ) + θ ( n) θ ( n) θ( n) θ( n) (.6) () he udating vector is a function of the cost function and the artial derivative of the cost function with resect to the araeter vector; in this case, an estiate of the second artial derivative of the cost function with resect to the araeter vector is used. he Gauss-Newton algorith (also called the ethod of scoring, the odified Newton-Rahson algorith or the quasilinearisation algorith), the Levenberg- Marquardt algorith and the steeest descent algorith are exales; the second artial derivative of the cost function with resect to the araeter vector for the 8

34 Levenberg-Marquardt algorith is J( n + ) J( n) e( n + ) e( n + ) = + δi θ ( n) θ ( n) θ( n) θ( n) + (.7) with δ being a ositive constant and the identity atrix, I R nxn. he udating vector, φ( n ), in this case is given by (fro equations (.4) and (.7)) and φ( n) = J( n) e( n ) e( n ) + + θ ( n) θ( n) e( n + ) J( n) e( n ) λ( n) + θ( n) + θ ( n) θ( n) (.8) µ φ J( n + ) J( n) ( n) e( n + ) J( n) θ ( n) = λ( n) θ ( n) + e( n + ) θ( n) θ ( n) δi + (.9) with λ( n ) = forgetting factor and θ( 0 ) = known starting values. he Gauss-Newton algorith oits the addition of the δi ter. hese two algoriths are secial cases of the Newton-Rahson algorith in which the following conditions are fulfilled: (a) the artial derivative of the cost function with resect to the araeter vector is assued to be zero at the current araeter vector (this is obviously an aroxiation, as the artial derivative will only be zero at the otiu araeter vector) and (b) the error ultilied by the second artial derivative of the error with resect to the araeter vector ay be neglected (Soderstro and Stoica (989) state that this is valid close to the otiu araeter vector). In all these cases, the starting value of the second artial derivative of the cost function with resect to the araeter vector is given as a ultile of the identity atrix. he Hessian atrix for the steeest descent algorith equals the identity atrix; the udating vector, φ( n ), is given by equation (.8), with the aroriate substitution. (3) he udating vector is a function of the cost function only; these algoriths either for gradient estiates by difference aroxiations and roceed as in (for exale) 9

35 the Gauss-Newton algorith, or have other secific search atterns. Other gradient algoriths would not naturally fall into these classes; one exale would be the least ean squares (LMS) algorith defined by Widrow and Stearns (985): θ( n + ) = θ ( n) µ ( e( n) θ( n)) (.0) he choice of the gradient algorith for a articular alication deends on the desired seed of tracking and the coutational resources available. Draer and Sith (98) declare that the Gauss-Newton algorith cobines the best features of the Newton-Rahson ethod and the steeest descent ethod, though the convergence of the algorith is slower than that of the Newton-Rahson algorith. he authors declare that the steeest descent ethod, though straightforward, often converges very slowly to the otiu araeter vector after raid initial rogress. Sith and Friedlander (985) agree, declaring that while the recursive Gauss-Newton algorith is quadratically convergent near a local iniu of the cost function, the steeest descent algorith is only linearly convergent in the sae situation. he Gauss-Newton algorith has the advantage over the Newton-Rahson algorith of being less coutationally intensive; Ljung (987) also states that the aroxiation used to deterine the Gauss-Newton algorith ensures that the Hessian atrix is ositive seidefinite, which eans that convergence is guaranteed to a stationary oint. On the other hand, Soderstro and Stoica (989) declare that the convergence of the Newton- Rahson algorith is quadratic, whereas in ractice the convergence of the Gauss- Newton algorith is linear but fast. In a ore general oint about the use of gradient ethods, it is iortant that the error surface in the direction of the tie delay (and indeed the other araeters) should be uniodal. he existence of a ultiodal error surface in the direction of the tie delay has serious consequences for the use of the gradient algorith; indeed, it aears that the task of deterining a global iniu in the resence of local inia is a very knotty roble. Vanderlatts (984) states that the estiation rocess ust be started fro various initial estiates to see if a consistent otiu ay be obtained under these conditions; reasonable assurance is then felt that this otiu oint corresonds to the true global iniu. Rekliatis et al. (983) state that the only ractical strategy for locating global inia in these situations is a ethod called 0

36 "ultistart with rando saling". his strategy involves ultile otiisation runs, each initiated at a different starting oint. he starting oints are selected by saling fro a unifor distribution. he global iniu is then the local iniu with the lowest cost function value aong all the local inia that ay be identified. Scales (985) suggests that in ractice, one usually has to assue that it is ossible to ake a guess at the osition of the global iniu that is sufficiently good so that no extraneous local inia interfere with the iniisation rocess. Ferretti et al. (996) declare that the use of a filter on the data increases the range of tie delay over which the cost function is uniodal; the bandwidth of the filter is related to an initial estiate of the tie delay uncertainty. However, the seed of convergence of any gradient algorith used is reduced by the inclusion of a filter in this anner. It ay be ossible to irove the chances that the global iniu of the error surface ay be deterined, even if the error surface is ultiodal, by adating techniques defined by Deuth and Beale (977), aongst others, that irove backroagation in neural networks. One technique defined by these authors is that of learning with oentu; the authors declare that oentu acts like a low ass filter on the error surface, allowing the ossibility of sliding through local inia. he idea is that a change in the araeter will be ut equal to the oentu constant (tyically 0.95) ties the revious change in the araeter lus 0.05 ties the resent change in the araeter. A further ossibility defined is to use an adative learning rate; the authors roose that the learning rate should be decreased by a factor of 0.7 if the new error exceeds the old error by a factor of.04 and increased by a factor of.05 if the new error is less than the old error. However, no theoretical basis is given for these two suggestions. A nuber of other authors have defined adative learning algoriths; aong the are Ho and Hsu (99), Kna and Wang (99) and Qiu et al. (99). hese algoriths have been develoed fro a trial and error aroach. Silva and Aleida (990) and Sato (99) also discuss the use of oentu and learning rate ters in the alication. On a ractical level, since all of the gradient ileentations ay identify araeters corresonding to a local iniu rather than a global iniu, it is iortant to coence iterations at good initial values of the araeters, which ay be obtained by hysical insight for a hysically araeterised odel structure. A further advantage in starting off at good initial values is that the nuber of iterations required for good identification is lower and the total couting tie required is less.

37 ..4. Gradient algoriths for estiation based on the Newton-Rahson, Gauss- Newton and steeest descent ethods A nuber of authors have defined gradient algoriths based on the Newton- Rahson ethod, for estiating rocess araeters; Liu (990), for exale, defines a araeter udating schee for an n th order rocess odel lus tie delay based on the algorith. Other algoriths for aroriate araeter udating based on the Newton- Rahson aroach include the ethod defined by Zhao and Sagara (990). he use of the Gauss-Newton algorith to estiate rocess araeters was erhas first roosed by Marshall (979), who uses such an algorith to identify the araeters of a FOLPD odel, in a Sith redictor structure. A nuber of assutions are ade in this analysis; Bahill (983) subsequently used these assutions to facilitate the develoent of an equation for the required change in the odel tie delay as a result of the change in the rocess tie delay. A nuber of odifications of the algoriths defined above have also been considered, including those ileented by Kaya and Scheib (984) (who ileent Marshall's (979) schee to udate the tie delay estiate, and estiate the araeters of a first order lag (FOL) odel of the non-tie delayed rocess using the RLS algorith), Roagnoli et al. (988) and O Connor (989). he Gauss-Newton algorith has also been used in oen loo to estiate rocess araeters; Durbin (984a), (984b), (985), for instance, uses the algorith to estiate the araeters of a FOLPD odel of the rocess. Siulation results quoted by Durbin (985) show that a change in tie delay index fro 0 to 4 is followed in about 5 sales, with a change in tie delay index fro 4 to 0 followed in about 40 sales. Other such gradient algoriths are defined by Wong (980), Brewer (988) and Banyasz and Keviczky (988), (994). Other authors, such as Sith and Friedlander (985) and Pak and Li (99), concentrate on estiating the tie delay only using the algorith. he straightforward nature of the steeest descent algorith has otivated a nuber of authors (such as Elnagger et al. (989), (990a), (990b), (99), (99), (993)) to aly it to the estiation of rocess araeters. hese authors (990a) estiate the non-delay araeters using the RLS algorith, and estiate the delay using the steeest descent algorith. Other authors, such as Robinson and Soudack (970), concentrate on estiating the tie delay only using the algorith.

38 ..4.3 Other gradient algoriths for tie delay estiation As entioned in Section..4., there are other gradient algoriths that ay be used for odel araeter and tie delay estiation; Gawthro and Nihtila (985), for instance, estiate a ure tie delay in a noise free environent by udating the tie delay based on the artial derivative of the error squared with resect to the tie delay. Gawthro et al. (989) use the sae technique to estiate the araeters of a continuous tie SISO rocess with tie delay. Other algoriths of the tye under discussion for estiating the odel araeters and tie delay are defined by Pueikis (985), Shah et al. (988) (who use the LMS algorith), Boudreau and Kabal (99), (993), Hwang and Chuang (994) and Li and Macleod (995). Algoriths of this tye that estiate the tie delay only are described by Chan et al. (980), (98), Etter and Stearns (98), Reed et al. (98), Feintuch et al. (98), Youn et al. (98), (983), David and Stearns (983), Duttweiler (983), Youn and Matthews (984), Messer and Bar-Ness (987), Ching and Chan (988), Vasilev and Aideirski (990), Ho et al. (990), (99), (993), Ching et al. (99), Dokic and Clarkson (99), Clarkson (993), So and Ching (993), So et al. (994) and Ching and So (994), aongst others...5 ie delay estiation in the absence of other rocess araeters In this section of the chater, the estiation of ure tie delays is considered. Such tie delays arise ainly in signal rocessing alications; in these alications, it is coon to use the ter tie difference of arrival (DOA) rather than tie delay. A nuber of classes of ethods for estiating this araeter ay be identified:. Methods for the estiation of a single tie delay based on the cross-correlation of two signals.. Methods for ultile tie delay estiation (in a MIMO environent) based on the cross-correlation of two signals. 3. Other tie delay estiation ethods. he estiation of ure tie delays using gradient ethods, considered in Section..4, are not reconsidered in this section. he cross-correlation of two signals ay be used to estiate the tie delay 3

39 between the two signals, as the tie at which the cross-correlation ter is axiised corresonds to the tie delay estiate. Most ethods of this tye are off-line in nature; aong the authors who discuss the cross-correlation ethod, and variations on the ethod, are Faure and Evans (969), Kna and Carter (976), Cabot (98), Carter (98), Scarborough et al. (98), Haas and Lindquist (98), Hassab and Boucher (98), Boucher and Hassab (98), Stein (98), Al-Hussaini and Kassa (984), Azaria and Hertz (984), Bradley and Kirlin (984), Schwartzenbach and Gill (984), Abatzoglou (986), Fertner and Sjolund (986), Hertz (986), Al-Hussaini and El- Gayaar (987), Gabr (987), Weiss and Stein (987), Krolik et al. (988), Zheng and Feng (988), Cusani (989), Avitzour (99), Kollar (99), Gardner and Chen (99a), (99b), Gardner and Sooner (99), Bar-Shalo et al. (993), Clarkson (993), Carter and Robinson (993), Jacovitti and Scarano (993), Kuar and Bar- Shalo (993), Shen et al. (993), Fong et al. (994) and Izzo et al. (994). Meyr and Sies (984) estiate the tie delay in closed loo using the ethod, and also define an algorith for the tracking of a randoly varying tie delay between two stochastic signals. During and Jansson (993) define a variation on the cross-correlation algorith that is suitable for an on-line ileentation; the authors state that the estiation tie of the algorith ileented on a exas Instruents MS300 signal rocessor is aroxiately s. Other authors use the technique to estiate tie delays in ulti-inut, ultioutut environents or between ultile sensors and ultile targets. hese algoriths are ostly off-line in nature, with exales of such algoriths described by Friedlander (980), Ng and Bar-Shalo (98), (986), reblay et al. (987) and Pallas and Jourdain (99). Segal et al. (99) and Antoniadis and Hero (994) develo on-line, iterative algoriths for solving the ultichannel tie delay estiation roble. Finally, other algoriths have been defined for the estiation of ure tie delays. One exale of such an off-line algorith is defined by Kenefic (98), in which the tie delay between two sensors ay be found by deterining the axiu of the robability density function (.d.f.) of the delay fro a given rior distribution. Nehorai and Morf (98), Hertz and Reiss (98), Azenkot and Gertner (985), Chiu (987), George and Goodan (988), Jesus and Rix (988), Moddeeijer (989), Chaagne et al. (99), Jane et al. (99), Yaada et al. (99), Lourtie and Moura (99), Boudreau and Kabal (99), El-Hawary and Mbaalu (993), Laguna et al. (994), Manickan et al. (994) and Koenig (995) define other such off-line tie delay 4

40 estiation algoriths. Less attention aears to have been aid to the on-line ileentation of non-cross correlation based algoriths, though one such algorith is defined by Bethel and Rahikka (987), who calculate recursively the.d.f. of the tie delay, fro which an otiu estiate of the tie delay ay be deterined. Algoriths based on the sae aroach are defined by Bethel and Rahikka (990) and Bethel et al. (995). Other on-line algoriths are defined by Naazi and Stuller (987), Feder and Weinstein (988), Naazi and Biswal (99) and Blackowiak and Rajan (995). he latter authors investigate the erforance of a siulated annealing algorith in the estiation of the alitude scaling factors and the tie delays of the searate arrivals in a signal coosed of closely saced arrivals with added noise. he ethod is articularly interesting as the cost function to be iniised has local inia that ake the alication of calculus based iniisation techniques (such as the Newton-Rahson gradient algorith) difficult; the authors declare that the siulated annealing algorith has the ability to slide through local inia. O Dwyer (996a) discusses the algoriths outlined above that aear to erit further investigation; overall, however, the algoriths that estiate ure tie delays only are less useful, at least in control alications, than algoriths that facilitate the estiation of both the tie delay and non-tie delay odel araeters. 5

41 .3 Frequency doain ethods for araeter and tie delay estiation In this section, both ethods of estiating the frequency resonse of a rocess and ethods for the subsequent estiating of the odel araeters (including tie delay) are considered, together with ethods for the direct estiation of the odel araeters based on the use of higher order sectra..3. Frequency resonse estiation he ethods that have been defined for rocess frequency resonse estiation, in both oen loo and closed loo environents, ay be classified as follows:. he resonse to a sine wave inut. he resonse to a ulse inut 3. Correlation analysis 4. Sectral analysis 5. Methods based on the ratio of Fourier transfors 6. Otiisation ethods 7. Cestral analysis 8. he use of a relay in series with the rocess in closed loo and 9. Other ethods. Of course, the frequency range over which the odel should be estiated needs to be defined. Generally, good frequency doain atching between the rocess and the odel over a wide range of frequencies about the frequency where the hase lag of the rocess equals 80 degrees is desirable, articularly for controller design (Harris and Mellicha (980), Edgar et al. (98), Wittenark and Astro (984), Lee et al. (990), Hang and Chin (99) and Eskinat et al. (993)). he frequency resonse of a rocess (in oen loo) at any frequency ay be deterined by calculating the agnitude and hase of the rocess fro its outut when an aroriate sine wave is inut to the rocess; however, the estiate obtained is sensitive to disturbances (Soderstro and Stoica (989), Larsen (994)). he frequency resonse of a rocess ay also be found by deterining the resonse of the rocess, in oen loo, to a ulse inut (Cleents and Schnelle (963)). 6

42 Good fitting of the agnitude resonse is found in exeriental work carried out by these authors; the goodness of fit, however, does aear to worsen at higher frequency values. Other ulse resonse techniques are defined by Rajakuar and Krishnasway (975), Harris and Mellicha (985), Seborg et al. (989) and Sirthwaite et al. (994). he frequency resonse ay be deterined directly by correlation (Rake (980), Unbehauen and Rao (987), Soderstro and Stoica (989), Larsen (994)). his aroach ay be reresented in block diagra for, as shown in Figure.. Figure.: Correlation analysis (Unbehauen and Rao (987)) R( ω) sin( ωt) Multilier Low-ass filter sine wave generator G ( jω) d( t) + + cos( ωt) Multilier Low-ass filter I( ω) he rocess frequency resonse at frequency ω, G ( jω ), equals R( ω) + ji( ω) (with d(t) being a disturbance). Larsen (994) declares that the ethod is insensitive to ste and white noise disturbances (due to the resence of the low-ass filters). However, long exerient ties are often required to deterine the rocess frequency resonse. Sectral analysis techniques ay also be used to calculate an estiate of the frequency resonse in both oen loo and closed loo environents. In oen loo, the rocess is reresented as shown in Figure.. Figure.: Oen loo ileentation n(t) G ( s) + + d(t) y(t) 7

43 In this case, n(t) and d(t) are uncorrelated. he technique involves deterining an estiate of the frequency resonse of the rocess, G ( jω ), as follows: G ( jω) S ( jω) S ( jω) (.) yn n with Syn( jω ) equal to the cross ower sectral density of y(t) with resect to n(t) and S ( n jω ) equal to the ower sectral density of n(t). he ower sectral densities ay be estiated using either the eriodogra (sale sectru) aroach, which involves estiating the ower sectral density in ters of the square of the corresonding discrete Fourier transfor (Unbehauen and Rao (987), Johannson (993)) or the correllelogra aroach, which involves estiating the relevant covariance functions, and calculating the estiates of the ower sectral densities fro the discrete Fourier transfors of these covariance functions (Unbehauen and Rao (987)). Alternative ethods defined by Schwartzenbach and Gill (984) and Unbehauen and Rao (987) ay be used to estiate the hase resonse of the rocess, which is iortant for tie delay estiation in articular. Chan et al. (978), Hannan and hoson (98), Friedlander and Porat (98), (984), Chan and Miskowicz (984) and achibana (984) also use ower sectral density techniques to calculate the odel araeters and/or the tie delay. he closed loo syste considered ay be reresented as shown in Figure.3. Figure.3: Closed loo ileentation r( t) + ( t) + G ( c s ) + G ( s) u( t) n( t) + + d( t) y( t) Wellstead (986) shows that, if r(t), (t) and d(t) are uncorrelated, then or 8 G ( jω) S ( jω) S ( jω) (.) y n

44 G ( jω) S ( jω) S ( jω) (.3) ry rn Aroxiations for the ower sectral densities ay be deterined by using the discrete Fourier transfor (DF), for instance. he frequency resonse of a rocess ay also be obtained by using ethods based on the ratio of Fourier transfors. In oen loo (Figure.), an estiate of G ( jω ) ay be exressed as G ( jω) F[ y( t)] F[ n( t)] (.4) with F[ ] being the Fourier transfor of the relevant signal. he Fourier transfor ters ay be aroxiated by using the DF (when the resulting aroxiation is called the eirical transfer function estiate (EFE)), by using the discrete tie Fourier transfor (DF) or by using a nuerical integration ethod, such as the Adas- Moulton ethod. he alicability of such aroxiations is discussed in detail by Wellstead (98), Ljung (987), Unbehauen and Rao (987), Johannson (993) and Guillaue et al. (996), aong others. Other ethods based on using Fourier transfors to estiate the tie delay and/or the odel araeters are defined by Hertz and Reiss (98), Azenkot and Gertner (985), Nagai (986), Chiu (987) and Boudreau and Kabal (99). In closed loo (Figure.3), and if r(t) and (t) are uncorrelated, with F[d(t)] = 0, it will be roved in Chater 4 that G ( jω) F[ y( t)] F[ n( t)] (.5) As before, the Fourier transfor ters ay be aroxiated by using the DF, the DF or an alternative nuerical integration of the Fourier transfor. he ethod for deterining the frequency resonse in equation (.5), when the Fourier transfor ters are aroxiated by using DF's, is also used by Laaire et al. (99) as a eans of deriving a robust estiator of the rocess frequency resonse. Band-ass filters could be ut on the inut and outut of the rocess so that F[d(t)] could be ore reasonably assued as zero, at one or ore frequency values (Hagglund and Astro (99), Ho et al. (994)). A related ossibility is to lace a nuber of band-ass filters 9

45 on the inut and outut of the rocess to deterine the frequency resonse at a nuber of frequencies corresonding to the centre frequencies of the band-ass filters (Goberdhansingh et al. (99)). Other authors that use Fourier transfors in closed loo as a eans of estiating the frequency resonse of the rocess include Harris and Mellicha (980), Krishnasway et al. (987), Koganezawa (99), Hang and Sin (99) and Hang et al. (994b). he frequency resonse of the rocess in oen loo ay be deterined fro the iniisation of a ossibly ultiodal cost function whose variables include either the DF of the inut and outut signals to the rocess (Marshand and Fu (985), Schoukens et al. (988), Pintelon and Schoukens (990), Pintelon and Van Biesen (990), Kollar (99)) or colex logarithic frequency resonse data (Banos and Goez (995), Guillaue et al. (995)). In closed loo, the axiu likelihood estiate of the rocess araeters ay siilarly be deterined by the iniisation of a ultiodal cost function whose variables include the DF of the inut and outut signals to the rocess (Pintelon et al. (99)). he inut and outut signals to the rocess are assued to be correlated through a rocess noise ter. Hassab and Boucher (976) estiate the tie delay of a delayed and attenuated relica of a signal by the use of the natural logarith of the agnitude squared of the outut signal (called the ower cestru of the signal). he authors state that when the technique is successful, the cestru yields a doinant eak away fro the origin corresonding to the desired tie delay. Barrett and Moir (986) use cestral ethods for restoring the unknown hase-frequency inforation fro the alitude-frequency inforation that ay be rovided by the ower sectral density techniques. he relay autotuning ethod, develoed first by Astro and Hagglund (984), ay be used to deterine one or ore oints on the frequency resonse of the rocess. he ethod involves the introduction of a relay eleent in arallel with the controller; the relay is switched in when rocess araeter estiation is required. he liit cycle rovoked at the rocess outut, as a result of the introduction of the relay eleent, ay be analysed to deterine aroxiations for the agnitude and frequency of the rocess at a rocess hase lag of 80 degrees. It is ossible, as the authors suggest, to deterine aroxiations for the agnitude and frequency of the rocess when the hase lag is 90 degrees, if an integrator is introduced in series with the relay. he authors also show that aroxiations for the agnitude and frequency ay be obtained when the hase lag is between 90 and 80 degrees, by the introduction of 30

46 aroriate hysteresis on the relay eleent. he ethod is develoed further by Arzen (987), Astro and Hagglund (988), Chiang and Yu (993), Frian and Waller (995) and Hwang (995). Other related aroaches using the relay autotuning ethod are roosed by Hagglund and Astro (989), (99), Schei (99), Astro et al. (993), Ho et al. (994), Lundh and Astro (994), Lee et al. (995b), Voda and Landau (995b) and Shen et al. (996a), (996b), (996c). In addition, the ethod ay be alied to the estiation of the araeters of MIMO rocess odels lus tie delays, as detailed by Loh et al. (993), Wu et al. (994) and Frian and Waller (994). Other ethods of estiating the frequency resonse of the rocess include estiating the agnitude and frequency of the rocess at a hase lag of 80 degrees (in closed loo), which is described by Balchen and Lie (987); in this ethod, the syste deviation signal is correlated with the excitation signal. In conclusion, techniques that directly estiate the frequency resonse both in oen loo and in closed loo have been well docuented in the literature. he robustness of any of the techniques when closed loo identification is required, with rocess noise added to both the inut and the outut, is questionable; soe authors address this roble by aroriate filtering of the rocess inut and outut signals rior to identification..3. Paraeter and tie delay estiation using higher order sectra.3.. Introduction Higher order sectra (or olysectra) are defined in ters of the higher order statistics (or cuulants) of a signal. he general otivations for the use of higher order sectral techniques are () to suress additive, ossibly coloured Gaussian noise that ay be resent on signals () to allow recovery of hase inforation fro signals and (3) to detect and quantify nonlinearities in tie series (Nikias and Petroulu (993)). he use of higher order sectra is exained with secial reference to the identification of the araeters of a SISO rocess odel with a tie delay, in both oen loo and closed loo environents. An iortant frequency doain aroach to the identification of such a rocess odel is to base the identification of the odel araeters on the agnitude and hase 3

47 resonse of the rocess. he use of second order statistics, which involve the calculation of the ower sectral densities of the inut and outut signals to the rocess, gives rise to identifiability robles when both the inut and outut records are containated by even white and utually uncorrelated noise sources (Delooulos and Giannakis (994)). However, because the higher order sectru of (coloured) Gaussian signals is identically zero, adding coloured Gaussian noise of unknown sectru to the rocess inut or outut does not affect the rocess frequency resonse estiation, if higher order sectral techniques are used. he ost coon higher order sectra of a signal that are calculated are the third order sectru (also called the bisectru) and the fourth order sectru (also called the trisectru), as defined by Nikias and Petroulu (993) and exlored in detail by O Dwyer (996a). Cross-cuulants and the cross-bisectru or cross-trisectru ay also be defined in a siilar anner, using relevant rocess inut and outut signals (O Dwyer (996a)). he bisectru and trisectru are secial cases of the n th order sectru of a signal. Generally seaking, for coutational reasons, the bisectru of a signal is the ost often calculated; the trisectru of the signal ay be calculated if the signal had zero (or very sall) third order cuulants and larger fourth order cuulants. A syetrically distributed rando variable has a third order cuulant equal to zero, for instance (Mendel (99)). he cestru of higher order sectra ay also be defined, as discussed by O Dwyer (996a)..3.. Paraeter estiation techniques using higher order sectra It has been entioned in Section.3.. that an iortant eans of deterining the araeters of the rocess odel in the frequency doain is to first deterine the agnitude and hase variation of the rocess odel with frequency. An interediate stage ay be to deterine the bisectru or trisectru agnitude and hase estiates of the rocess. Nikias and Petroulu (993) discuss a nuber of ethods that have been defined for deterining the bisectru (or trisectru) agnitude and hase estiates fro the inut and outut data of a rocess; all of the ethods involve the use of fast Fourier transfors. One of the difficulties about deterining higher order sectra in this way is that only a finite set of data is used. Nikias and Petroulu (993) and 3

48 Matsouka and Ulrych (984) also exlain a nuber of the ethods develoed (by Bartlet et al. (984) and Li and Ding (994), aongst others) for the estiation of the rocess agnitude and hase (subsequently referred to as the Fourier agnitude and Fourier hase) fro the bisectral agnitude and hase estiates of the rocess deterined. Other aers in which details of these algoriths are rovided include those by Haniff (99), Matson (99), Rangoussi and Giannakis (99), Cheng and Venetsanooulos (99) and Li and Ding (994). Pan and Nikias (987) discuss the reconstruction of the Fourier hase fro the corresonding trisectru. It is also ossible to deterine the bicestru and tricestru of the inut and outut data, as an interediate stage to deterining the Fourier gain and hase of the rocess. his is discussed by Alshebeili and Cetin (990), Alshebeili et al. (99) and Brooks and Nikias (993). he direct estiation of the rocess odel araeters and the tie delay using higher order sectral techniques (without first estiating the Fourier agnitude and Fourier hase of the rocess) does not aear to have been addressed in the literature. he estiation of the tie delay between two signals (i.e. the estiation of a tie delay ter only, with no other dynaics considered) has been exlored in detail by Nikias and Petroulu (993), aong others. he authors divide the ethods used into the following categories: (i) Conventional tie delay estiation techniques based on third order statistics that involve estiating the tie delay fro the bisectral and cross-bisectral hases of the inut and outut signals to the rocess; Hinich and Wilson (99), for exale, estiate the tie delay as the scaled difference between these hase estiates. Sato and Sasaki (977), Sasaki et al. (977), Nikias and Raghuveer (987), Nikias and Pan (988), Zhang and Raghuveer (99) and Nikias and Mendel (993) also outline ethods of this tye. (ii) Paraetric tie delay estiation techniques, which involve odelling the tie delay by a olynoial and estiating the olynoial coefficients; Nikias and Pan (988), ugnait (99) and Delooulos and Giannakis (994) also outline these ethods. In a ore recent aer, Delooulos and Giannakis (996) extend the ethod of Delooulos and Giannakis (994) to the estiation of a rocess odel (in rational olynoial for) in a closed loo environent, when both inut and outut data to the rocess is containated by additive noise having unknown cross-sectral characteristics. (iii) ie delay estiation techniques based on the cestru of higher order sectra; 33

49 Petroulu et al. (988) and Reddy and Rao (987) discuss such ethods in detail. (iv) Adative tie delay estiation based on the araetric odelling of higher order cross-cuulants, which use third order cuulants and a gradient-like algorith to estiate the tie delay, where the additive noises on the signals are of satially correlated Gaussian for with unknown correlation functions (Chiang and Nikias (990)) Conclusions he following conclusions about the use of higher order sectral techniques for rocess araeter estiation ay be drawn:. Conventional aroaches for rocess frequency resonse estiation (based on the ower sectru, for instance) have a lower coutational intensity and a requireent for a saller nuber of data oints than do the higher order sectral aroaches. However, the higher order sectral aroaches are robust to the resence of ossibly utually correlated, coloured Gaussian noise (or non-gaussian noise, with a syetric.d.f.) added to both the rocess inut and outut.. he roble of rocess identification in closed loo using higher order sectra has not been coletely resolved. he signals encountered in closed loo oeration do not fit the requireent for the signals secified for rocess identification in all details; nevertheless, identification of the rocess araeters ay be ossible in certain situations in a closed loo environent (e.g. if a PRBS driving signal is added to the inut of the rocess). Delooulos and Giannakis (996), in a recent aer, show that rocess identification in closed loo is ossible using the third order cuulants of the rocess inut and outut signals. It aears that the critical factor in the decision as to whether it is aroriate to use higher order sectra for rocess araeter estiation is the agnitude and nature of the additive noise resent on both the inut and outut signals to the rocess. Johnson (985) states that, realistically, noise ters either ay have a known or estiated ean, covariance and distribution, or ay have a constant bias coonent and a stochastic coonent having either a zero ean or being a filtered version of a white noise signal. For identification and control uroses in the self-tuning literature, the added noise is often considered to be odelled as the filtered version of a white noise signal. It is ossible to reduce the effect of noise ters by re-treatent of data 34

50 before identification (e.g. if the noise ter is drift on the inut or outut signals to the rocess, then the aroriate data could be filtered before identification); Ljung (987) discusses a nuber of aroaches in this area. In a closed loo rocess environent, there sees to be less justification for the use of higher order sectral techniques if a PRBS driving signal ust be added at the rocess inut, as such a signal will be uncorrelated to any noise signal and thus less coutationally intense ethods of rocess identification ay be aroriate. Overall, the use of higher order sectral techniques in syste identification sees suited to a restrictive range of robles, in which noise signals on the inut and outut to the rocess cannot be effectively dealt with by re-rocessing..3.3 Model araeter estiation using frequency resonse data he aroaches to estiate the araeters of an aroriately ordered rocess odel lus tie delay, ay be classified as follows:. Model araeter estiation using a grahical aroach. Model araeter estiation using an analytical aroach 3. Model araeter estiation using a least squares aroach; the estiation of the araeters of a high order odel lus tie delay and the araeters of a low order odel lus tie delay will be considered searately and 4. Model araeter estiation based on relay identification. he odel araeter and tie delay estiates ay be deterined grahically, fro the Bode lots of the rocess; Deshande and Ash (983) and Seborg et al. (989) aly the ethod to the estiation of the araeters of a FOLPD odel and a SOSPD odel, with Luyben (983) fitting higher order transfer functions with tie delay to the Bode lots. Seborg et al. (989) identify the disadvantages of the ethod as the tediousness of the rocedure, the introduction of errors in fitting araeters for second order odels (and, by extension, higher order odels) using a trial and error aroach and that the ethod does not easily facilitate the identification of ore general transfer function odels, such as those with nuerator dynaics. he araeters ay also be estiated analytically, fro the frequency resonse of the rocess. Iserann et al. (974), for instance, analytically deterine the tie constant and the tie delay of a ultile ole rocess odel. he odel transfer function is 35

51 G ( s) = sτ Ke ( + s ) n (.6) with K = odel gain, = odel tie constant and τ = odel tie delay. hen, the authors rovide an estiate for the odel tie constant and tie delay as follows: = ω K n G j ( ω) (.7) and τ [ j n ] = ω φ ( ω ) tan ( ω ) (.8) with G ( jω ) = agnitude of the rocess transfer function and φ ( jω) = hase of the rocess transfer function at a frequency ω. Secial cases of these ileentations are discussed by O'Dwyer (99), (993) and Hang et al. (993), (994b). Sundaresan and Krishnasway (978), Koganezawa (99) and O Dwyer (99) also consider other analytical ethods of calculating the araeters of FOLPD and SOSPD odels fro the rocess frequency resonse. Alternatively, the odel araeters and tie delay ay be estiated by iniising the squared error between the rocess and the odel in the frequency doain. For an arbitrary order odel lus tie delay, any of the techniques defined require the aroxiation of the tie delay by an aroriate rational olynoial; the tie delay as such is consequently not identified. Exales of such ethods are discussed by Levy (959), Whitfield (986), (987), Unbehauen and Rao (987) and Hakvoort and Van den Hof (994). Other authors, such as Dos Santos and De Carvalho (990) exlicitly estiate the araeters of an n th order odel lus tie delay. hese authors iteratively deterine the estiates of the odel order and the ole and zero values fro the estiate of the tie delay, with the estiate of the tie delay calculated based on a least squares rocedure using the hase lot. In a different ethod, Seborg et al. (989) suggest that the araeters and the tie delay of a rocess odel could be estiated by selecting the value of the tie delay iteratively and using the ethod of Levy (959), for exale, to deterine the reaining rocess 36

52 odel araeters. Siilar ethods based on this ultile odel estiation technique have been well exlored in the tie doain. However, the ultile odel estiation ethod is coutationally intensive. In a ore recent aer, Young et al. (995) estiate the odel araeters and tie delay of a linear rocess using a recursive nonlinear estiation technique in the frequency doain. he authors ention that it is ossible that the araeters (and delay) identified ay corresond to a local iniu of the cost function used, rather than a global iniu. It is also ossible to fit a low order odel lus delay to the rocess resonse, in a least squares sense. Lilja (988), for instance, calculates a FOLPD odel of a high order rocess fro four oints on the frequency resonse of the rocess. he non-tie delay araeters are deterined by iniising an aroriate quadratic cost function; the tie delay is deterined searately by iniising a ultiodal cost function using a odified Newton-Rahson algorith, though convergence of the tie delay estiate to the correct value of the tie delay is consequently not guaranteed. Nevertheless, the author gives advice on strategies to deterine the best estiate of the tie delay. A siulation result rovided by the author shows that a rocess of order 6 is well aroxiated by a FOLPD odel, in a frequency range corresonding to a hase lag range of 0 to 80 degrees. Other authors that describe algoriths of this tye include Seborg et al. (989) and Palor and Blau (994). Finally, the araeters of the rocess odel ay also be identified by analysing the rocess outut when a relay is switched into the closed loo coensated syste in lace of the controller. It is ossible to aroxiate the liit cycle outut as a sinusoid (this is the basis of the aroach of Astro and Hagglund (984) for controller tuning). However, it is also ossible to analyse the liit cycle outut without any such aroxiation being taken. Lee and Sung (993), for instance, calculate the tie constant and tie delay of a FOLPD odel using this aroach, as follows: = 05. t0 + ( a Kd) ln[ ( a K d) ] (.9) t 0 + e and τ = ln[ ] (.0) 37

53 with t 0 = eriod of oscillation of the liit cycle, a = alitude of the oscillation and d = relay alitude ( K would need to be known a riori). Arzen (987), Li et al. (99), Chang et al. (99), Leva (993), Benouarts and Atherton (994) and Egan (994) also describe algoriths of this tye. Indeed, it aears reasonable that further work in this area is ossible, as any authors use such relay coensator techniques for autotuning rather than for odel araeter estiation; to this end, a develoent of the ethod defined by Egan (994) is outlined briefly in Chater 8 and is discussed in detail by O Dwyer (996k). In conclusion, the aroaches for rocess odel araeter and tie delay estiation that have been roosed in the literature in the frequency doain have been briefly docuented above. he range of frequency values over which the rocess is estiated can be secified, deending on the alication. 38

54 .4 Other ethods of rocess araeter and/or tie delay estiation he revious sections of the chater have described in detail well-defined ethods of rocess araeter and tie delay estiation. Other ethods do not easily fall into the categories discussed earlier; in this section, araeter and/or tie delay estiation using neural networks, using rocess order identification ethods, using the delta oerator and using genetic algoriths are outlined. Neural networks ay be used for the identification and control of non-linear rocesses (Narendra and Parthasarathy (990)). he identification and control of tie delay rocesses using neural networks is a subject of recent research. Bhat and McAvoy (99), for instance, roose a detailed ethod to stri a back roagation neural network (BPN) to its essential weights and nodes to give it its silest ossible structure; the authors show that the striing algorith is caable of identifying the tie delay and order of a FOLPD rocess (in the discrete tie doain). Other authors that discuss the identification and control of rocesses using neural networks include Megan and Cooer (99), who resent a neural network aroach to adative control by analysing the relationshi between the error attern and the corresonding adjustent needed in the gain and tie constant of a first order lag (FOL) odel of a rocess, and Hinde and Cooer (994), (995) who exlore the use of a assive adative algorith which udates the rocess odel and the controller in closed loo by taking advantage of naturally occurring dynaic events, rather than injecting erturbations into the syste to create dynaic events. Cheng et al. (995) identify a non-linear dynaic rocess with unknown and ossibly variable tie delay using an internal recurrent neural network. However, it is true to say that the use of neural networks for the identification of rocesses with tie delays is in its infancy. Process order estiation strategies ay also be used to estiate the tie delay of a rocess (in the discrete tie doain), since the tie delay aears as an increase in the odel order of the nuerator transfer function. Process order identification strategies for SISO systes have been described by Unbehauen and Gohring (974), Van den Boo and Van den Enden (974), Wellstead (978), Stoica et al. (986), Unbehauen and Rao (987), Soderstro and Stoica (989), Niu et al. (990), O'Donnell (99) and Liang et al. (993). Process order identification strategies for 39

55 MIMO systes have been defined by Guidorzi (975), (98), se and Weinert (975), Lin and Wu (98), Van Overbeek and Ljung (98), Stoica (983), Zhang et al. (985), Li (985), Chen and Guo (987), Guo et al. (989), Chen and Zhang (990), Zhang and Chen (990), Guillaue et al. (99), Gu and Misra (99), Glentis and Kaloutsidis (99) and Niu and Fisher (994). he estiation of the tie delay using these strategies would, however, be conditioned on the order of the non-delay art of the rocess being known a riori. It is also ossible to estiate the rocess araeters using the delta oerator rather than the z (or shift) oerator. he delta oerator (also known as the Euler oerator) is defined as follows: δ = z s (.) where s equals the sale tie. Wellstead and Zarro (99) show that the region of stability for z doain oles (i.e. the unit circle) translates into a circle of centre ( s, 0) and radius s in the delta doain. hus, as the saling rate is increased, the stability region defined in the delta doain aroaches that of the continuous tie doain. he following advantages are claied for the delta oerator reresentation over the shift oerator reresentation: (a) he reresentation of discrete systes in the delta doain avoids the robles of coefficient sensitivity in recursive digital filters at high sale frequencies, seen in the z doain (Goodall (990)). (b) A related advantage is that the delta oerator allows suerior finite word length coefficient reresentation (Middleton and Goodwin (986), (990)) under the assution that the sale tie is chosen according to the norally quoted rules of thub. (c) A further advantage of the delta oerator is that it "alost always" has less roundoff noise associated with it than does the corresonding z oerator (Middleton and Goodwin (986), (990)). (d) Middleton and Goodwin (986), (990) and Goodwin et al. (988), (99) declare that for araeter estiation, the least squares solving of a set of linear equations is better conditioned for odels reresented by the delta transfor. 40

56 errett and Downing (993), (994) use the delta oerator for syste identification on a fixed oint DSP using the RLS algorith. he authors show that the delta oerator is nuerically ore robust than the shift oerator, at the cost of an additional coutational requireent (this oint is also ade by Goodwin et al. (99)). Other authors that discuss the use of the delta oerator for syste identification uroses include Goodwin et al. (988), Middleton and Goodwin (990), Wilkinson et al. (99) and Jabbari (99). he use of the delta doain for the estiation of the tie delay as well as the odel araeters has not been considered in the literature, with the excetion of the ethod defined by Keviczky and Banyasz (99), who identify the tie delay index as the su of the roduct of each nuerator araeter identified and its index, divided by the su of the identified nuerator araeters, where both sus are taken u to a defined axiu tie delay index. he rocess is odelled as a SOSPD odel, with identification of the tie delay taking lace in the delta doain. here does see to be scoe to estiate the tie delay (and the other odel araeters) in the delta doain, using techniques siilar to those used in the z doain (e.g. the overaraeterisation of the rocess odel); the ethod of Keviczky and Banyasz (99), for instance, is an analogue of a ethod defined by these authors (Keviczky and Banyasz (988)) in the z doain. he use of the delta doain for syste identification is also exlored by O Dwyer (996a), (996l). Finally, the use of genetic algoriths for rocess identification (including tie delay identification) is beginning to attract interest. Genetic algoriths search fro a oulation of oints, use inforation about the cost function (rather than its derivative or other auxiliary knowledge used by gradient algoriths) and have a rando coonent (quantified as a utation rate) that hels drive the odel araeters towards values corresonding to the global iniu of the aroriate cost function. hese algoriths tend to be very coutationally intensive; Kristinsson and Duont (99) declare, for instance, that a genetic algorith is erhas fifty ties ore coutationally intensive than is a recursive instruental variable (RIV) syste identification algorith. Genetic algoriths are considered to be one extree solution to the exloitation-exloration trade-off, as described by Renders and Flasse (996); genetic algoriths trade-off large coutation tie, and oor accuracy of the global iniu, with reliability in calculating the global iniu. he authors consider the use of gradient algoriths to be another solution to the exloitation-exloration tradeoff; other solutions, such as the use of ultile odel estiation ethods, are of course 4

57 also ossible. In an interesting recent alication, Yang et al. (996) use a genetic algorith to estiate the denoinator araeters and tie delay of a reduced order rocess odel, while using the less coutationally intensive least squares algorith to subsequently deterine the nuerator araeters (which is a linear roble). 4

58 .5 Conclusions his chater has considered a wide variety of ethods for tie delay and odel araeter estiation, in both the continuous tie and discrete tie doains. he ethods have been discussed in detail by O Dwyer (996a), in which coarisons have been drawn where aroriate between ethods. he wide sectru of ethods covered eans that an overall conclusion as to the best ethod to use is not aroriate. However, it is ossible to indicate the areas and ethods in which original work ay be done.. A ajor section of the chater has been devoted to the use of gradient ethods for odel araeter and tie delay estiation. It has been decided to investigate fully the ethods defined by Durbin (984a), (984b), (985)), which facilitate identification of the odel araeters and the tie delay in oen loo, because of the otential of the ethods to estiate the araeters quickly, even in the resence of bias and noise ters. Alternative olynoial aroxiations to the tie delay than those taken by the author will also be considered. his work is carried out in Chater 3. In addition, it has been decided to investigate closed loo ethods for estiating the odel araeters and tie delay, based on the work done by Marshall (979), (980) and Bahill (983). Alternative udating algoriths for the araeters will be defined, considering fewer assutions on syste behaviour than are considered by the above authors; in addition, the araeter udating strategies will be extended to the estiation of the araeters of higher order odels than the strategies considered reviously. his work is reorted in Chater 7.. he frequency doain is a very natural doain for tie delay estiation, as was entioned in Section.. It has been decided to estiate the rocess frequency resonse, both in oen loo and in closed loo, using aroaches based on the ratio of Fourier transfors, and the ratio of aroriate ower sectral densities. It has also been decided to investigate a ethod that cobines an analytical aroach that gives initial estiates of the araeter and tie delay values, with a least squares aroach using a gradient algorith that udates these estiates to ore accurate odel araeter estiates. Both of these toics are discussed in Chater Other areas in which original work ay be done are (a) he estiation of the rocess araeters (including tie delay) based on an 43

59 analytical descrition of the actual rocess outut, rather than on araeters calculated assuing a sinusoidal rocess outut, when a relay is introduced in series with the rocess in closed loo. his work is detailed by O Dwyer (996k). (b) he use of relevant discrete tie algoriths for the estiation of the araeters lus tie delay of MIMO rocess odels; in articular, the alication of overaraeterisation ethods, such as the algorith defined by Wong and Bayoui (98). Soe reliinary work in this area is detailed by O Dwyer (996l). 44

60 CHAPER 3 Oen loo tie doain gradient ethods of araeter and tie delay estiation 3. Introduction Gradient ethods of araeter estiation are based on udating the araeter vector using a vector that deends on inforation about the cost function to be iniised (which is equal to the su of the squared error between the rocess and odel oututs). he use of gradient algoriths for odel araeter and tie delay estiation is discussed in detail in Chater. he review and analysis of the available literature has revealed the close relationshi between any of the ethods used for tie delay estiation using gradient ethods. For control alications, the estiation of non-tie delay araeters as well as the tie delay is frequently required (e.g. for coensator design). herefore, it has been decided to concentrate on ethods that intrinsically estiate both odel araeters and tie delay. he best odel to use for identification uroses is a vexed question as it deends, aongst other factors, on the data quality available (see Chater ); a cautious aroach, which has been ileented in this chater, is to identify the araeters of a FOLPD odel (Newell and Lee (989)). It was decided to investigate fully the ethod defined by Durbin (984a), (984b), (985) because of its otential to estiate the araeters quickly, even in the resence of bias inuts and noise ters. In this ethod, the rocess is assued to be odelled by a FOLPD odel. he rocedures develoed ay be alied to the estiation of the araeters of higher order odels of a general order rocess, though the colexity of the develoent is greatly increased; this alication will be discussed further in the conclusions of the chater. he gradient algoriths are ileented by deterining the artial derivatives of the error between the rocess outut and the discretized odel outut, with resect to the gain, tie constant and tie delay. hese artial derivatives are subsequently used to udate the odel araeters. However, rior to calculating the artial

61 derivative of the error with resect to the tie delay, the tie delay variation (which equals the rocess tie delay inus the odel tie delay) is aroxiated by a rational olynoial. Such an aroxiation will be valid for sall values of the tie delay variation; the aroach is aroriate, as the use of the Gauss-Newton gradient algorith, for exale, deends on the difference between the estiated araeters and the actual araeters being sall (as the Gauss-Newton algorith is derived fro a second order aylor s series exansion of the cost function about the otiu araeter vector). he ost aroriate rational olynoial to use ay be deterined by finding the relationshi between the ean squared error (MSE) function between the rocess and odel oututs and the tie delay; this relationshi, which ay be deterined both analytically and in siulation, ust be uniodal about the rocess tie delay for successful alication of the gradient descent algoriths, as ust the corresonding relationshi of the MSE function to the rocess gain and tie constant values. he chater considers, both analytically and in siulation, the convergence of the araeters of a FOLPD odel to corresonding rocess araeters. he convergence of the non-tie delay odel araeters is considered first, when the tie delay is assued known. Subsequently, the convergence of the odel tie delay is discussed, when the non-delay odel and rocess araeters are identical. A nuber of theores are develoed; one theore considers convergence of the odel tie delay index in the idealised case, when it is assued that the rocess tie delay is an integer ultile of the sale eriod, and is known a riori. Subsequent work assues, ore realistically, that the rocess tie delay is unknown a riori; convergence is considered when the rocess tie delay is an integer ultile of the sale eriod, and a real ultile of the sale eriod, as well as when the revious odel outut is used to calculate the new odel outut. Finally, the convergence of all of the odel araeters is considered, when all of the rocess araeters are unknown, in the idealised and realistic cases entioned above. his structure allows a corehensive exloration of the issues. he author has considered the use of four gradient algoriths: the Levenberg- Marquardt algorith, the Gauss-Newton algorith, the steeest descent algorith (Ljung (987)) and the least ean square (LMS) algorith (Widrow and Stearns (985)). he ileentation of these algoriths is described in Chater. O Dwyer and Ringwood (994a), (994b) show the estiation of the araeters of a FOLPD

62 odel using these algoriths, with each algorith facilitating the udating of the araeters in a broadly siilar anner (at least for the siulations taken). he siulation results quoted in this chater will use the Levenberg-Marquardt gradient algorith for udating the araeters, with no loss of generality, as the rocedures develoed to facilitate convergence of the odel araeters to the rocess araeters are aroriate for the use of any of the gradient algoriths entioned.

63 3. Rational olynoial aroxiation of the tie delay variation It has been stated in Section 3. that rior to calculating the artial derivative of the error with resect to the tie delay, the tie delay variation, r, is aroxiated by a rational olynoial. he two first order aroxiations to the tie delay variation considered are as follows: sr aylor: e sr (3.) Pade: e sr sr sr (3.) he MSE function between the rocess and odel oututs was calculated analytically, when the tie delay variation was reresented by each of these aroxiations in turn; it is assued that the rocess tie delay is an integer ultile of the sale eriod. It was deterined that the MSE erforance surface was uniodal with resect to the odel tie delay index when the first order aylor s series aroxiation was used. hese calculations are done in the discrete tie doain, as integer values of the rocess tie delay aear as aroriate ower ters on the nuerator transfer function of the rocess, in this doain; in addition, a standard rocedure has been defined to calculate the MSE surface, by Widrow and Stearns (985), in the doain. hese calculations are erfored in subsequent sections of this chater. he use of the first order Pade aroxiation roduced a non-quadratic MSE erforance surface, which is non-uniodal in the odel tie delay index. his develoent is given by O Dwyer (996e). he relationshi between the MSE function and the odel gain and tie constant ters searately is uniodal; no aroxiation is used for the tie delay variation when these araeters are being udated. he use of higher order aroxiations for the tie delay variation is ossible; soe of the second order aroxiations that ay be used in these circustances are as follows: sr aylor: e sr s r (3.3) Pade: e sr sr s r sr s r (3.4)

64 Marshall (979) : e sr s r s r (3.5) Product (Piche (990)): e sr sr s r sr s r (3.6) Laguerre (Piche (990)): e sr sr s r sr s r (3.7) Paynter (Robinson and Soudack (970)): e sr + sr s r (3.8) Product (Gradshteyn and Ryzhik (980)): e sr sr s r sr s r (3.9) Direct Frequency Resonse (Stahl and Hie (987)): e sr sr s r sr s r (3.0) here are an infinite nuber of higher order aroxiations that ay be used for the tie delay variation; just one of these aroxiations is the third order aroxiation defined by Marshall (979): e sr sr ( 067. ) + ( 067. sr) 3 3 (3.) he use of the second order aylor s series aroxiation deends on the use of a higher order odel for the rocess than a FOLPD odel; the other aroxiations ay be used with a FOLPD rocess odel. It is shown by O Dwyer (996e) that when the MSE surface was calculated versus odel tie delay index, over a large nuber of sales, for the aroxiations in equations (3.3) to (3.), uniodality was achieved only when a second order aylor s series aroxiation was used for the tie delay wo exales of a non-uniodal MSE surface are rovided below, when a second order Pade aroxiation and a second order Product aroxiation (as defined by

65 Piche (990)) is used for the tie delay variation (Figures 3. and 3., resectively); the oint x arks where r = 0 (or g = g ) in each case. Figure 3.: MSE surface (Pade) Figure 3.: MSE surface (Product) he condition that only the use of either a first order aylor s series aroxiation or a second order aylor s series aroxiation for the tie delay variation will guarantee uniodality of the resulting MSE function versus odel tie delay is related to the z doain odels calculated (using the zero order hold equivalence aroach) when various aroxiations are used (O Dwyer (996e)). he oles of the z doain odel are always within the unit circle when either a first order aylor s series aroxiation or a second order aylor s series aroxiation for the tie delay variation is used, but one or ore oles are either on or outside the unit circle when any other aroxiation is used, for at least soe values of the odel tie delay index. It is erhas not surrising that the resultant generation of an unstable discrete doain odel does not facilitate convergence of the odel tie delay index to the rocess tie delay index. Even in cases where uniodality of the MSE surface is achieved over a large range of odel tie delay index values, an infinite sike always exists on the MSE surface when r = 0 for all aroxiations excet the aylor s series aroxiations taken (e.g. Figures 3., 3.). hus, an exact estiate of the rocess tie delay, using the gradient ethod, will not be ossible in these circustances. Unfortunately, heore 3. (Section 3.3) roves that satisfactory values of the nontie delay odel araeters will not be estiated unless an exact estiate of the rocess tie delay is deterined.

66 3.3 Convergence of the non-delay odel araeters his section deals with the convergence of the non-delay odel araeter estiates to the non-delay rocess araeter estiates using gradient ethods; it is desired to rove that when the odel tie delay index equals the rocess tie delay index, then the gradient algoriths ay rovide successful convergence of the odel gain and tie constant values to the rocess gain and tie constant values, resectively. heore 3.: For a first order discrete stable syste, the MSE erforance surface is iniised when the odel gain equals the rocess gain and the odel tie constant equals the rocess tie constant, under the following conditions: (a) he odel tie delay index equals the rocess tie delay index (b) Measureent noise is assued uncorrelated with the rocess inut and outut and (c) he inut to the rocess and the odel is assued to be a white noise inut. Proof: he rocess difference equation is s s y (n) = e y (n ) + K ( e )u(n g ) + w( n) (3.) with = rocess tie constant, K = rocess gain and τ = g s, s = sale eriod, g = rocess tie delay index; w(n) = easureent noise. he odel difference equation is (assuing the revious rocess outut is used in its calculation) s s y ( n) = e y( n ) + K( e ) u( n g ) (3.3) with K = odel gain, = odel tie constant and g = odel tie delay index. he difference between the rocess and odel outut, e ( n), is (fro equations (3.) and (3.3)) s s e ( n) = y ( n) y ( n) = ( e e ) y ( n ) s s + K ( e ) u( n g ) K ( e ) u( n g ) + w( n) (3.4) he rocedure defined by Widrow and Stearns (985) ay be used to calculate the

67 MSE erforance surface as follows: E e n r r G z z G z z G z dz [ ( )] = y y ( 0) + ww( 0) + [ ( ) uu( ) ( ) y u( ) ( )] j Φ Φ π z (3.5) with Φ uu n n ( z) = ruu( n) z, Φy u( z) = ry u( n) z, r n r n y y ( ), uu( ) and r ( ww n ) n= n= being the autocorrelation functions of y ( n), u( n) and w ( n ) resectively; r n y u( ) is the cross-correlation of y ( n) and u(n). he odel G ( z) corresonds to the outut y ( n ). Using the residue theore to calculate the closed curve integral, the MSE function is calculated (fro equation (3.5)) to be (O Dwyer (996)): s K ( e ) E[ e ( n)] = s ( e ) s K ( e ) + s ( e ) s s KK( e )( e ) e s s ( e e ) s ( g g ) +r ww ( 0 ) (3.6) he MSE function is iniised when E[ e ( n)] K and E[ e ( n)] ( / ) equal zero siultaneously. he required calculations, deterined by artially differentiating equation (3.6) (O Dwyer (996)), show that (assuing g = g ) s s E[ e ( n)] K( e )( e ) = 0 K = s s s K ( e )( e e ) (3.7) and s s E[ e ( n)] K( e ) ( e ) = 0 K = s s s ( / ) ( e ) ( e e ) (3.8) If both E[ e ( n)] K and E[ e ( n)] ( / ) equal zero siultaneously, then it ay be deduced fro equations (3.7) and (3.8) that = and K = K.

68 A corollary to this theore is that if g iniised when K g, then the MSE function is not = K and =. his eans that the odel tie delay index ust converge to the rocess tie delay index before convergence of the odel gain and tie constant values to the rocess gain and tie constant values, resectively, is ossible. A further corollary to this theore is that the MSE function is not iniised when K = K unless g = g and =, and the MSE function is not iniised when = unless g = g and K = K. hese conclusions have been constant features of the siulation results taken for odel araeter and tie delay udating using gradient ethods. he uniodality of the MSE function with resect to the araeters individually was deonstrated, in tyical siulation results, by lotting the MSE function versus the corresonding araeter, as shown in Figures 3.3 and 3.4. For Figure 3.3, g = and = g with K = 00 and for Figure 3.4, K = K and g = g with = 00; for both of these lots, the MSE is calculated based on equation (3.6), with easureent noise assued absent. Figure 3.3: MSE vs. Model gain Figure 3.4: MSE vs. Model tie constant Model gain Model tie constant hese lots confir that the MSE function is quadratic with resect to the odel gain, though it is not quadratic with resect to the odel tie constant (equation (3.6)).

69 3.4 Convergence of the odel tie delay his section of the chater will consider the use of the gradient algorith for udating the tie delay only, with the non-tie delay rocess and odel araeters ut equal. he gradient algorith used, deending as it does on the artial derivative of the cost function with resect to the araeter value (equation (.4)), will be a function of the error between the rocess and the outut, and the artial derivative of the error between the rocess and the outut with resect to the araeter value. he cases outlined below are considered; these cases are chosen to corehensively cover the ileentations ossible.. he error and the artial derivative of the error with resect to the tie delay variation are calculated by using a first order aylor s series aroxiation for the tie delay variation. his will be referred to as Case in subsequent work in this section of the chater. his is an idealised ileentation, as the rocess tie delay is assued known a riori (and is assued to be an integer ultile of the sale eriod).. he artial derivative of the error with resect to the tie delay variation is calculated by using a first order aylor s series aroxiation for the tie delay variation and the error is calculated based on using a FOLPD rocess odel. In this case, udating of the odel tie delay when it is both an integer ultile of the sale eriod, and a real ultile of the sale eriod, is considered. his will be referred to as Case in subsequent work in this section of the chater. his case rovides a ore realistic ileentation than Case above, as the rocess tie delay is not assued known a riori. 3. he artial derivative of the error with resect to the tie delay variation is calculated by using a first order aylor s series aroxiation for the tie delay variation and the error is calculated based on using a FOLPD rocess odel. he revious odel outut is used to calculate the new odel outut. his will be referred to as Case 3 in subsequent work in this section of the chater; as in Case, the tie delay is not assued known a riori, though it is assued that the rocess tie delay is an integer ultile of the sale eriod.

70 3.4. Convergence of the odel tie delay - Case heore 3.: For a first order discrete stable syste of known gain and tie constant, the MSE erforance surface versus odel tie delay index is uniodal with a iniu value of the MSE occurring when the odel tie delay index equals the rocess tie delay index, under the following conditions: (a) he tie delay variation is aroxiated by a first order aylor s series aroxiation (b) he easureent noise is uncorrelated with the rocess inut (c) he resolution on the rocess tie delay is assued to be equal to one sale eriod and (d) he error and the artial derivative of the error with resect to the tie delay variation are calculated based on using the first order aylor s series aroxiation for the tie delay variation. Proof: he rocess difference equation, y ( n ), is s s y (n) = e y (n ) + K( e )u(n g ) + w( n) (3.9) with = =, K = K = K. he corresonding odel difference equation, calculated by substituting a first order aylor s series aroxiation for the tie delay variation, is (assuing the revious rocess outut is used in its calculation) K( g g g g ) s ( ) s s s y ( n) = e y( n ) u( n g) K( e ) u( n g ) (3.0) herefore, fro equations (3.9) and (3.0), e ( n) = y ( n) y ( n) is given by ( g g ) s s e( n) = K[( e ) u( n g ) + u( n g ) ( g g ) s + ( e ) u( n g )] + w( n) s (3.) he MSE erforance surface, E[ e ( n)], ay then be calculated as

71 K ( g g ) s s E[ e ( n)] = E[ K ( e ) u ( n g ) + + E K ( [ g g ) s u ( n g )] s ( e ) u( n g ) u( n g )] ( g g ) s s + E[ K ( e ) u ( n g )] ( g g ) s s + E[ K( e ) u( n g ) w( n) + w ( n)] ( g g ) s s s + E[ K ( e )( e ) u( n g ) u( n g )] + E K ( [ g g ) ( g g ) ) u( n g ) u( n g )] s s s ( e K( g g ) s s + E[ K( e ) u( n g ) w( n) + u( n g ) w( n)] (3.) herefore, it ay be shown that (O Dwyer (996)) K ( g g ) s s E[ e ( n)] = K ( e ) [ ruu( 0) ruu ( g g )] + [ r ( 0) r ( )] s K ( e ) s + ( g g)[ ruu ( 0) ruu ( ) + ruu ( g g + ) ruu ( g g)] + rww ( 0) (3.3) uu uu herefore, E[ e ( n)] = r ww ( 0) for g = g. Now ruu ( 0) ruu ( n) n and for g < g, it ay be shown by coaring the sizes of the individual ters in equation (3.3) that E[ e ( n)] > r ww ( 0) for all values of g and g (O Dwyer (996)). For g > g, it ay also be shown by coaring the sizes of the individual ters that E[ e ( n)] > r ww ( 0) for all values of g and g (O Dwyer (996)). hus, the iniu value of the MSE function occurs at g = g and the easureent noise has no effect on the estiated rocess delay index. he only situation that arises for which E[ e ( n)] = r ww ( 0) for g g is when the inut has a flat autocorrelation function, which corresonds to a constant level inut. hus, any inut change is sufficient for correct rocess delay index estiation, if the rocess delay index is estiated by

72 deterining the iniu of the MSE erforance surface. However, if a gradient ethod is used to estiate g, then an additional restriction that the MSE function ust be uniodal about a iniu value when g = g, is iosed. he uniodality of the MSE function in equation (3.3) ay be roved by induction; an outline of the inductive roof (rovided in full by O Dwyer (996)) is as follows: (a) For g > g, it is required to rove that the MSE function at g = g is greater than the MSE function at g (3.3), rovided = g. It ay be roved that this is true, using equation s s s [( e ) ( s ) s ( e ) ][ ruu ( ) ruu( )] ( s )( e )[ ruu( ) ruu ( )] > 0 (3.4) Sile analysis shows that this exression is always true. (b) For g > g, it is required to rove that the MSE function at g = g n is greater than the MSE function at g = g n. Alying equation (3.3), it ay be roved that this is true, rovided ( ) s s s s e [ r ( n) r ( n + )] + [ + ( e ) ][ ( 0) ( )] r r uu uu uu uu s s + ( e ) [ ( ) ( + ) ( + ) + ( + ) ( + )] > 0 nr n n r n n r n uu uu uu (3.5) he condition in equation (3.5) is fulfilled by any excitation signals; one exale is a white noise signal. (c) For g < g, it is required to rove that the MSE function at g = g + is always greater than the MSE function at g this is true, rovided = g. Using equation (3.3), it ay be roved that s K ( e ) [ r ( 0) r ( )] > 0 s uu uu (3.6),

73 which is always true. (d) For g < g, it is required to rove that the MSE function at g = g + n + is greater than the MSE function at g = g + n. Alying equation (3.3), it ay be roved that this is always true, rovided ( ) s s s s e [ r ( n) r ( n + )] + [ ( e ) ][ ( 0) ( )] r r uu uu uu uu s s ( e ) [ ( ) + ( + ) ( ) ( + ) ( + )] > 0 nr n n r n n r n uu uu uu (3.7) As with equation (3.5), the condition in equation (3.7) is fulfilled by any excitation signals; one exale is a white noise signal. his theore is suerficially siilar to one develoed by Elnagger et al. (990a), for alication to the estiation of the tie delay of a FOLPD rocess odel, in which the tie delay is not aroxiated; these authors do not rove, however, that the corresonding MSE surface is uniodal. he uniodality of the MSE function (given by equation (3.3)) versus odel tie delay index is confired by reresentative siulation results given in Figures 3.5 and 3.6. For these siulations, K = K =. 0, = = 0. 7 seconds with g = 30. he noralised MSE (equal to the MSE divided by r uu ( 0 ) ) is lotted versus odel tie delay index in both cases, with r ww ( 0 ) ut to zero. he excitation signal used to roduce Figure 3.5 is white noise, with the excitation signal used to roduce Figure 3.6 being a square wave of eriod equal to 00 sales. Considering equations (3.9), (3.0) and (3.), a block diagra reresentation of the gradient ethod to udate the odel tie delay index is shown in Figure 3.7.

74 Figure 3.5: Noralised MSE vs. tie Figure 3.6: Noralised MSE vs. tie delay index - white noise inut delay index - square wave inut Model tie delay index Model tie delay index Model tie delay index Model tie delay index Figure 3.7: Udating of the odel tie delay index - Case. u(n)= white noise g s K( e ) z z e PROCESS s z y ( n ) K(e ( g g ) ( g g ) + z) z s s g s MODEL UPDAE ALGORIHM (g ) e s + + y ( n ) e ( n ) + g One reresentative siulation result corresonding to heore 3. is given in Figures 3.8a-3.8d. he tie delay indices and the rocess inus odel outut are lotted against sale nuber. At the beginning, the starting values of the rocess and

75 odel tie delay index were both equalised; a ste change was then ade to the rocess tie delay index value. In the siulation, the udate for the odel tie delay is a fractional ultile of the sale eriod; when the addition of these udates exceeds the value of the sale eriod (in either the ositive or negative direction), then an aroriate adjustent is ade in the odel tie delay index, with the udate for the odel tie delay reset to zero. he rocess and odel gain and tie constant araeters were ut equal to.0 and 0.7 seconds, resectively (i.e. the siulation conditions corresond to the conditions taken to calculate the MSE curves in Figures 3.5 and 3.6). he Levenberg-Marquardt gradient algorith (Ljung (987)) was used to udate the odel tie delay index; the sale tie was defined equal to 0. seconds. In the ileentation of this algorith, the starting value of the inverse Hessian atrix was defined equal to 5I, with δ = and λ( n ) = 0.95 (these values were deterined fro siulation results to be aroriate to the alication). Coloured easureent noise, generated by low-ass filtering a white noise signal, was added. he odel tie delay index was liited in variation to one sale eriod er iteration (which is a for of filtering on the tie delay index value); such filtering was found to be desirable in siulation. Fast convergence to the rocess tie delay index is seen, even in the resence of very substantial coloured easureent noise; this is true if the starting value of the odel tie delay index is either greater than or less than the rocess tie delay index, as exected fro heore 3.. he error, e ( n), in Figures 3.8b and 3.8d is non-zero due to the resence of the coloured easureent noise. Figure 3.8a: ie delay index estiate Figure 3.8c: ie delay index estiate = g -- = g 5 = g = g Sale nuber Sale nuber

76 Figure 3.8b: e ( n) corresonding to Figure 3.8d: e ( n) corresonding to Figure 3.8a Figure 3.8c he rocedures outlined deend on a riori knowledge of the rocess tie delay index, g (as ay be seen clearly in the odel in Figure 3.7, for instance). herefore, the ileentation ust be regarded as an idealised case. Section 3.4. rovides the develoent of a ore realistic ileentation Convergence of the odel tie delay - Case he tie delay as an integer ultile of the sale eriod It is necessary to odify the rocedure outlined in Section 3.4. if a riori knowledge of the rocess tie delay index, g, is not available (as will norally be the case). One ossibility is to calculate the error, e 3 ( n), based on using a FOLPD rocess odel. he odel difference equation in this case is (assuing that the revious rocess outut is used in its calculation, and that = =, K = K = K ) s s y 3( n) = e y( n ) + K( e ) u( n g ) (3.8) with y ( n) given by equation (3.9). herefore, fro equations (3.9) and (3.8), e ( n) = y ( n) y ( n) is given by 3 3

77 s e3( n) = K( e )[ u( n g ) u( n g )] + w( n) (3.9) he artial derivative of the error with resect to the tie delay variation ay then be calculated by using a first order aylor s series aroxiation for the tie delay variation. his error, e derivative entioned above is ( n), is given by equation (3.); the corresonding artial e ( n) ( g g ) Ks = [ ] u ( n g u n g ) ( ) (3.30) he udate vector (for udating the odel tie delay - equation (.4)), which deends on the roduct of the error ( e3 ( n)) ultilied by the artial derivative of the error with resect to the tie delay variation ( e( n) ( g g)) is then indeendent of g. he cost function that aroxiately corresonds to this udate vector will be referred to as the ean of the roduct of the errors (MPE) function; this function is defined as E[ e( n) e3 ( n)] in this case. he udate vector that exactly corresonds to this cost function deends on e ( n)[ e ( n) ( g g )] and e ( n)[ e ( n) ( g g )]. It is 3 3 assued that e ( n)[ e ( n) ( g g )] e ( n)[ e ( n) ( g g )]. his is a 3 3 reasonable assution, bearing in ind that the tie delay variation, which is aroxiated by a first order aylor s series aroxiation, is assued to be sall. he MPE cost function will equal the MSE cost function at g equation (3.0) reduces to equation (3.8)). = g (when If any other aroxiation to the tie delay variation is used rather than a first order aylor s series aroxiation, then it ay be shown (O Dwyer (996e)) that the artial derivative of the error with resect to the tie delay variation is a function of g. hus, if g is unknown a riori, then a first order aylor s series aroxiation for the tie delay variation is the only aroxiation of interest. It is desired to rove convergence of the odel tie delay index to the rocess tie delay index, with the rocess tie delay index being unknown, but with the other odel araeters being known a riori.

78 heore 3.3: For a first order discrete stable syste of known gain and tie constant, then the MPE erforance surface versus odel tie delay index is uniodal, with a iniu value of the MPE occurring when the odel tie delay index equals the rocess tie delay index, under the following conditions: (a) he tie delay variation is aroxiated by a first order aylor s series aroxiation (b) he easureent noise is uncorrelated with the rocess inut (c) he resolution on the rocess tie delay is assued to be equal to one sale eriod (d) he error is calculated based on using a FOLPD rocess odel; the artial derivative of the error with resect to the tie delay variation is calculated based on using the first order aylor s series aroxiation for the tie delay variation and (e) he rocess tie delay index is greater than the odel tie delay index, as the odel tie delay index converges. Proof: he rocess difference equation, y difference equation, y 3 ( n), is given by equation (3.9). he odel ( n), is given by equation (3.8). he odel difference equation for calculating the artial derivative of the error with resect to the tie delay variation, y ( n), is given by equation (3.0). he exressions e ( n) = y ( n) y ( n) and e ( n) = y ( n) y ( n) are given by equations (3.) and (3.9), resectively. he 3 3 MPE erforance surface, E[ e( n) e3 ( n)], ay then be calculated, using a rocedure siilar to that used in equations (3.) and (3.3), to be equal to (O Dwyer (996)) g g K s s ( e ) ( ) [ ruu( 0) ruu( ) + ruu( g g + ) ruu( g g)] + rww( 0) s + K ( e ) [ r ( 0) r ( g g )] (3.3) uu uu herefore, E[ e ( n) e3 ( n)] = r ww ( 0 ) for g = g. It ay be shown by coaring the sizes of the individual ters in equation (3.3) that E[ e ( n) e3 ( n)] > r ww ( 0 ) for g > g only (O Dwyer (996)). hus, the iniu value of E[ e( n) e3 ( n)] occurs at g = g (when g is restricted to be less than or equal to g ) and the easureent noise has no effect on the estiated rocess delay value. If g > g, then, fro equation (3.3), the only

79 situation that arises for which E[ e ( n) e3 ( n)] = r ww ( 0 ) for g g is when the inut has a flat autocorrelation function, which corresonds to a constant level inut. hus, any inut change is sufficient for correct rocess delay index estiation, rovided that the required condition on g is fulfilled, if the rocess delay index is estiated by deterining the iniu of the MPE erforance surface. However, if a gradient ethod is used to estiate g, then an additional restriction that the MPE function ust be uniodal for g value occurring at g g > g, with a iniu MPE = g, is iosed. he uniodality of the MPE function for > g ay be roved by induction; an outline of the inductive roof (rovided in full by O Dwyer (996)) is as follows: It ay be roved that the MPE function at g = g is greater than the MPE function at g = g (using equation (3.3)), rovided that s s ( e )[ r ( 0) r ( )] + [ ( 0) ( ) + ( )] > 0 r r r uu uu uu uu uu (3.3) Alying equation (3.3), it ay be roved that the MPE function at g = g n is greater than the MPE function at g = g n, rovided that ( s s e )[ r ( n) r ( n + )] + [ ( 0) ( )] r r uu uu uu uu s + [ > nr n n r n n r n uu ( ) ( ) uu ( ) ( ) uu ( )] 0 (3.33) Both of the conditions in equations (3.3) and (3.33) are fulfilled by any excitation signals; one exale is a white noise signal. he behaviour of the MPE function (given by equation (3.3)) versus odel tie delay index is confired by Figures 3.9 and 3.0, in reresentative siulation results. For these siulations, K = K =. 0, = = 0. 7 seconds and g = 30; these conditions are identical to those used to calculate the noralised MSE curves in Figures 3.5 and 3.6. he noralised MPE (equal to the MPE divided by r uu ( 0 ) ) is lotted versus odel tie delay index in both cases, with r ww ( 0 ) ut to zero. he

80 excitation signal used to roduce Figure 3.9 is white noise, with the excitation signal used to roduce Figure 3.0 being a square wave of eriod equal to 00 sales. hese lots show that the MPE erforance surface is greater than r ww ( 0 ) for g > g only, and that when the conditions in equations (3.3) and (3.33) are fulfilled, the MPE function is uniodal for g > g, with a iniu MPE value occurring at g = g. Figure 3.9: Noralised MPE vs. tie Figure 3.0: Noralised MPE vs. tie delay index - white noise inut delay index - square wave inut Model tie delay index Model tie delay index Model tie delay index Model tie delay index Considering equations (3.9), (3.0), (3.), (3.8) and (3.9), a block diagra reresentation of the gradient ethod to udate the odel tie delay index is shown in Figure 3.. Reresentative siulation results corresonding to heore 3.3 are given in Figures 3. and 3.3, with the tie delay indices and the rocess inus odel outut lotted against sale nuber. he siulation conditions are identical to those used in Section 3.4. (and thus corresond to the conditions taken to calculate the MPE curves in Figures 3.9 and 3.0), with the addition that in the siulation in which a square wave is the excitation signal, the learning rate, µ, for the tie delay is ut to 0 and filtering on the tie delay udate is eloyed; these conditions were deterined to be aroriate for the alication.

81 Figure 3.: Udating of the odel tie delay index - Case u(n)=white noise PROCESS g s K( e ) z s z e y ( n ) z MODEL ( g g ) ( g g ) s K( e + z) z s s g e s + + y ( n ) - e ( n ) + g UPDAE ALGORIHM (g ) MODEL e s z K e s ( ) z g + + y ( 3 n ) - e ( 3 n ) + Figure 3.a: ie delay index estiate- Figure 3.3a: ie delay index estiatewhite noise excitation square wave excitation = g -- = g = g -- = g Sale nuber Sale nuber

82 Figure 3.b: e ( n) corresonding to Figure 3.3b: e ( n) corresonding to 3 Figure 3.a Figure 3.3a 3 Good convergence to the rocess tie delay index is seen for g > g. Other suleentary siulation results show no convergence to the rocess tie delay index when g < g. his verifies heore 3.3. However, the nature of the MPE functions (Figures 3.9 and 3.0) ean that convergence of the odel tie delay index could not be guaranteed, as the MPE goes negative when g > g. Convergence could only be guaranteed if g is always less than or equal to g. he convergence of the odel tie delay index in the siulations taken is due to the anner in which the araeter is being udated in the siulation, which tends to revent g going greater than g ; as Figures 3.b and 3.3b show, estiation is ossible in the resence of coloured easureent noise only when such noise is at a low alitude he tie delay as a real ultile of the sale eriod heore 3.3 in the revious section dealt with the estiation of tie delays that are integer ultiles of the sale eriod. For the estiation of tie delays that are real ultiles of the sale eriod (and assuing = =, K = K = K ), the FOLPD rocess difference equation is given as (O Dwyer (996e)): s g b s g b s s y3( n) = e y3( n ) + K( e ) u( n g) + K( e e ) u( n g ) + w( n) (3.34)

83 with g b = rocess tie delay inus the rocess tie delay index. he corresonding odel difference equation (assuing the revious rocess outut is used in its calculation) is s g a s ga s s y4( n) = e y3( n ) + K( e ) u( n g) + K( e e ) u( n g ) (3.35) with g a = odel tie delay inus odel tie delay index. he odel difference equation for calculating the artial derivative of the error with resect to the tie delay variation (and assuing that the revious rocess outut is used in its calculation) is K( g g g g + b a) s s y5( n) = e y3( n ) u( n g) ( g g + g g b a) s s K[ e ] u( n g ) (3.36) his equation ay be deduced fro equation (3.0). herefore, fro equations (3.34) and (3.35), g b s g b s s e4( n) = y3( n) y4( n) = K( e ) u( n g) + K( e e ) u( n g ) gas gas -s -K( - e ) u( n - g ) - K( e - e ) u( n - g - ) + w( n) (3.37) and, fro equations (3.34) and (3.36), g b s g b s s e5( n) = y3( n) y5( n) = K( e ) u( n g) + K( e e ) u( n g ) Ks ( g g + gb ga ) g g + g g s( b a ) s + u( n g) + K[ e ] u ( n g ) + w ( n ) (3.38) he MPE erforance surface, E[ e4( n) e5 ( n)], ay then be calculated, using a rocedure siilar to that used in equations (3.) and (3.3), to be equal to (O Dwyer (996))

84 g bs g bs s s g a s s K [( e ) + ( e e ) + ( e )( e e )] r ( 0) K s g a s s ( + e e ) ( + g g g g r b a) uu( 0) + K g g g [ ( e b s )( e b s e s ) + ( e s )( e a s ) g s ( e e ) ( + )] ( ) g g g g r b a uu a s s + K g g g g b a s e g b s e s e g b s e g a s [ ( + ) ( + ) ( )( ) - ( e g b s e s )( e s e g a s + )] r ( g g ) uu g g g g - K g b a s b s s g as ( e )[ ( + ) + ( e e ruu g g + )] ( ) + g g g g g b a s b s s g as K ( e e )[ ( + ) ( e )] ruu( g g + ) + rww( 0) (3.39) uu Now, using equation (3.39), it ay be shown that E[ e4( n) e5 ( n)] = r ww ( 0 ) if g = g and g b = g. he behaviour of the MPE function versus odel tie delay is given by a Figures 3.4, 3.5, 3.6 and 3.7, in reresentative siulation results. For these siulations, K = K =. 0, = = 0. 7 seconds and g = 5, with the tie delay taken in intervals of 0.0 ties the sale eriod. he noralised MPE (equal to the MPE divided by r uu ( 0 ) ) is lotted versus odel tie delay index for g b = 0. 0 and g b = 05. in Figures 3.4 and 3.5, when the excitation signal to the rocess is white noise. he noralised MPE is lotted versus odel tie delay index for g b = 0. 0 and g b = 05. in Figures 3.6 and 3.7, when the excitation signal to this rocess is a square wave of eriod equal to 00 sales. In Figures 3.4 and 3.6, the noralised MPE calculated fro equation (3.3) is sueriosed on the lots for coarison uroses. For all siulations, r ww ( 0 ) is ut to zero. Figures show the true ultiodal nature of the MPE function versus odel tie delay when the tie delay is a real ultile of the sale eriod. he estiation of the real value of the rocess tie delay is iossible using gradient ethods.

85 Figure 3.4: Noralised MPE vs. tie Figure 3.5: Noralised MPE vs. tie delay index - white noise excitation delay index - white noise excitation - g b = g b = 05. Figure 3.6: Noralised MPE vs. tie Figure 3.7: Noralised MPE vs. tie delay index - square wave excitation delay index - square wave excitation - g b = g b = 05.

86 Conclusions In suary, the gradient ethod will allow the estiation of rocess tie delays that are integer ultile of the sale eriod, in the case where the rocess tie delay is the only unknown araeter, rovided the rocess tie delay is always greater than the odel tie delay, as the odel tie delay converges to the rocess tie delay. An alternative non-gradient ethod that involves estiating the rocess tie delay index by deterining the iniu ositive value of the MPE erforance surface would allow the estiation of the rocess tie delay index under the sae conditions as the gradient ethod. Unfortunately, it is not ossible to estiate rocess tie delays that are not integer ultiles of the sale eriod using the gradient ethod, though it aears fro Figures that it ay be ossible to do so using a non-gradient ethod based on deterining the iniu ositive value of the MPE erforance surface (at least for g > g ) Convergence of the odel tie delay - Case 3 In heore 3.3, the revious rocess outut is used to calculate the new odel outut. It was decided to investigate the convergence attern of the odel tie delay to the rocess tie delay if the revious odel outut is used to calculate the new odel outut. he tie delay is not assued known a riori. heore 3.4: For a first order discrete stable syste of known gain and tie constant, the MPE erforance surface versus odel tie delay index is uniodal, with a iniu value of the MPE occurring when the odel tie delay index equals the rocess tie delay index, under the following conditions: (a) he tie delay variation is aroxiated by a first order aylor s series aroxiation (b) he easureent noise is uncorrelated with the rocess inut and outut (c) he resolution on the rocess tie delay is assued to be equal to one sale eriod

87 (d) he error is calculated based on using a FOLPD rocess odel; the artial derivative of the error with resect to the tie delay variation is calculated based on using the first order aylor s series aroxiation for the tie delay variation (e) he conditions rovided in the theore are observed on the odel araeters and (f) he revious odel outut is used to calculate the new odel outut. Proof: he rocess difference equation, y ( n), is given by equation (3.9). he corresonding FOLPD odel difference equation is (assuing the revious odel outut is used in its calculation) s s y 6( n) = e y 6( n ) + K( e ) u( n g ) (3.40) he odel difference equation for calculating the artial derivative of the error with resect to the tie delay variation (and assuing that the revious odel outut is used in its calculation) is K( g g ) s ( g g s ) s s y 7( n) = e y 7( n ) u( n g) K( e ) u( n g ) (3.4) herefore, fro equations (3.9) and (3.40), s e ( n) = y ( n) y ( n) = e [ y ( n ) y ( n )] s + K( e )[ u( n g ) u( n g )] + w( n) (3.4) and, fro equations (3.9) and (3.4), s s e ( n) = y ( n) y ( n) = e [ y ( n ) y ( n )] + K( e ) u( n g ) K g g s g g s s [ ( - ) ( - ) - u( n - g ) + ( e -- ) u( n - g - )] + w( n) (3.43) he MPE erforance surface, E[ e6 ( n) e7 ( n)], (which will aroxiately corresond to the udate vector fored fro the roduct of e ( 6 n ) and e 7( n ) ( g g ), and will thus be indeendent of g ) ay then be calculated, using a rocedure siilar to that used in equations (3.) and (3.3), to be equal to (O Dwyer (996) 7

88 g g K s s ( e ) ( ) [ ruu( ruu ruu g g ruu g g rww 0 ) ( ) + ( + ) ( )] + ( 0 ) s s + K ( e ) [ r ( 0) r ( g g )] + e [ r ( 0) + r ( 0) r ( 0) r ( 0)] uu uu y y y 6y 7 yy 6 yy 7 s Ke ( s s s e ) r ( g ) + Ke ( e )[ r ( g ) r ( g )] uy 6 uy 7 uy 7 K e g g r g Ke e ( g g s ) s s s s ( ) uy ( ) [ ] ruy ( g ) 6 6 s s s s + e K( e )[ r g r g + e K g g r g r g uy ( ) uy ( )] ( ) [ uy ( ) uy ( )] (3.44) with ruy x ( n) being the cross-correlation function between u(n) and y ( x n ), r n y y ( ) being the autocorrelation function of y ( n ) and r n y y ( ) being the cross-correlation function between y ( x n ) and y (O Dwyer (996)): x x x ( n ). hese ters ay be calculated as follows r y y ( 0) = s K ( e ) r uu( 0) (3.45) s e r yy 6 ( 0) = s s s Ke ( e ) ruy ( g) + K ( e ) ruu( g g 6 ) (3.46) s ( e ) r yy 7 ( 0) = ( g g ) s s s s s Ke ( e ) ruy ( g) K [ e ]( e ) ruu( g g 7 ) s ( e ) (3.47) r y 6y 7 ( g g ) s s s s s Ke ( e )[ ruy ( g ) + ruy ( g)] K [ e ]( e ) r 7 6 uu( 0) ( 0) = s ( e ) K( g g ) + s e s s [ ruy ( g) ruy ( g )] / ( e ) (3.48) 6 6 s n s and ruy ( n) = ruy ( n ) + K( e ) ( e ) r 6 6 uu( 0) (3.49) 73

89 ( g g ) s s s ruy ( n) = e ruy ( n ) K[ e ] r 7 7 uu( 0) (3.50) ( n ) s s r ( n) = e K ( e ) r ( 0), n > g and r ( n) = otherwise. (3.5) uy uu uy 0 If a gradient ethod is used to deterine g, when g > g, then the MPE function (equation (3.44)) ust be uniodal for g at g > g, with a iniu MPE value occurring = g. he uniodality of the MPE for g > g ay be roved by induction; an outline of the inductive roof (rovided in full by O Dwyer (996)) is as follows: It ay be roved, using equation (3.44), that the MPE function at g = g is always greater than the MPE function at g = g. Siilarly, it ay be roved that the MPE function at g = g n is greater than the MPE function at g = g n, rovided that ( g n ) s s s s e > e [ ( g ) e ( e )] (3.5) his is a sufficient condition. he behaviour of the MPE function, given by equation (3.44), versus tie delay index is confired by Figures 3.8 and 3.9, in reresentative siulation results. For these siulations, K = K =. 0, = = 0. 7 seconds and g = 30. he noralised MPE (equal to the MPE divided by r uu ( 0 ) ) is lotted versus odel tie delay index in both cases, with r ww ( 0 ) ut to zero. he excitation signal in both cases is white noise, with the sale eriod taken to equal 0. seconds for Figure 3.8 and 0.0 seconds for Figure 3.9 (this eans that the MPE function in Figure 3.8 ay be directly coared with that in Figure 3.9). Non-uniodal behaviour is seen in the latter case (for g at values of n when the conditions for convergence are violated. > g ) It is obvious fro Figures 3.8 and 3.9 (without the necessity of a roof by induction) that convergence of the odel tie delay index to the rocess tie delay index is not ossible when g < g. Figure 3.8: Noralised MPE vs. tie Figure 3.9: Noralised MPE vs. tie 74

90 delay index - white noise excitation - delay index - white noise excitation - s = 0. seconds s = 0. 0 seconds Considering equations (3.9), (3.40), (3.4), (3.4) and (3.43), the block diagra reresentation of the gradient ethod to udate the odel tie delay index is as shown in Figure 3.0. A reresentative siulation result corresonding to heore 3.4 is shown in Figures 3.a and 3.b, with the tie delay indices and the rocess inus the odel outut lotted against sale nuber. he siulation conditions are identical to those used in Section 3.4., and thus corresond to the conditions taken to calculate the MPE curve in Figure 3.8; the results ay be directly coared to those shown in Figures 3.a and 3.b. he results in Figures 3.a and 3.b show that convergence of the odel tie delay index to the rocess tie delay index is ossible if the conditions for convergence are fulfilled. However, the nature of the MPE function (Figure 3.8) eans that such convergence could not be guaranteed, and the convergence of the odel tie delay index in this case is due to the anner in which the araeter is being udated in the siulation (which revents g going greater than g ). Figure 3.b shows the low level of coloured easureent noise used in the siulation. 75

91 Figure 3.0: Udating of the odel tie delay index - Case 3. u(n) = white noise g s K( e ) z y ( n ) s z e PROCESS MODEL ( g g ) ( g g ) s K( e + z) z s ( z e ) s s g y ( 7 n ) - + e K z e MODEL s s z g UPDAE ALGORIHM (g ) y ( 6 n ) - e ( 6 n ) g + Figure 3.a: ie Delay Index estiate Figure 3.b: e6 ( n) corresonding to - white noise excitation Figure 3.a = g -- = g Sale nuber Sale nuber Overall, the use of the revious odel outut to calculate the new odel outut, rather than the use of the revious rocess outut to calculate the new odel outut, does not aear to be beneficial, because of the narrower conditions for convergence of the latter ileentation, coared to the forer ileentation. 76

92 3.5 Convergence of the full araeter set his section of the chater will consider the use of the gradient algorith for udating all of the odel araeter values. he gradient algorith used is a function of the error between the rocess and the outut, and the artial derivative of the error between the rocess and the outut. he following cases are considered, to corehensively cover the ileentations ossible:. he error and the artial derivative of the error with resect to the tie delay variation are calculated by using a first order aylor s series aroxiation for the tie delay variation. his will be referred to as Case in subsequent work in this section of the chater, and corresonds to Case in Section 3.4, where only the odel tie delay index is udated. As in Section 3.4, this is an idealised ileentation, as the rocess tie delay is assued known a riori (and is assued to be an integer ultile of the sale eriod).. he artial derivative of the error with resect to the tie delay variation is calculated by using a first order aylor s series aroxiation for the tie delay variation, and the error is calculated based on using a FOLPD rocess odel. his will be referred to as Case in subsequent work in this section of the chater (and it corresonds to Case in Section 3.4, so that it rovides a ore realistic ileentation than Case above, as the rocess tie delay is not assued known a riori). In this case, the following conditions are considered: (a) he rocess tie delay is an integer ultile of the sale eriod - white noise inut. (b) he rocess tie delay is a real ultile of the sale eriod - white noise inut. (c) he rocess tie delay is an integer ultile of the sale eriod - square wave inut. (d) he rocess tie delay is a real ultile of the sale eriod - square wave inut. In all cases, the odel gain and tie constant are udated assuing a FOLPD rocess odel. he case where the revious odel outut is used to calculate the new odel outut is not considered in detail, as the results in Section revealed that there was no benefit, when the odel tie delay index was being udated, in ileenting such a 77

93 rocedure coared to the rocedures in Case, Section 3.4. (when the revious rocess outut is used to calculate the new odel outut) Convergence of the full araeter set - Case heore 3.5: For a first order discrete stable syste of unknown araeters, the MSE erforance surface versus odel tie delay index is uniodal, with a iniu value of the MSE occurring when the odel tie delay index equals the rocess tie delay index, under the following conditions: (a) he tie delay variation is aroxiated by a first order aylor s series aroxiation (b) he easureent noise is uncorrelated with the rocess inut and outut (c) he resolution on the rocess tie delay is assued to be equal to one sale eriod (d) he error and the artial derivative of the error with resect to the tie delay variation are calculated based on using the first order aylor s series aroxiation for the tie delay variation (e) he excitation signal to the rocess and odel is assued to be white noise (f) he odel gain and tie constant are udated based on using a FOLPD rocess odel and (g) he conditions rovided in the theore are observed on the odel araeters. Proof: he rocess difference equation, y ( n), is given by equation (3.). he corresonding odel difference equation, y ( n), is given by equation (3.3) (assuing the revious rocess outut has been used in its calculation). he odel difference equation, calculated by substituting a first order aylor s series aroxiation for the tie delay variation, is (assuing the revious rocess outut is used in its calculation) K g g g g ( ) s ( ) s s s y 8 ( n) = e y( n ) u( n g ) K ( e ) u( n g ) (3.53) hen, fro equations (3.) and (3.53), e ( n) = y ( n) y ( n) equals

94 K( g g) ( g g ) s s u( n g) + K( e ) u( n g ) + w( n) + K ( e s ) u( n g ) + ( e s e s ) y( n ) (3.54) he MSE erforance surface, E[ e ( n)], ay then be calculated (using a rocedure 8 siilar to that used in equations (3.) and (3.3)) to be equal to (O Dwyer (996)) K ( g g ) ( g g ) s s s s s ( e e ) ry y ( 0) + ( e ) ruu ( ) K ( g g) s + [ ( g g ) s s s + K ( e ) + K ( e ) ] ruu( 0) K( g g ) s ( g g ) s s s K ( e )[ ruu ( g g + ) K ( e ) ruu( g g)] K ( g g ) ( g g ) s s s s s ( e e )[ ruy ( g ) K ( e ) r ( g )] uy s s s + ( e e ) K ( e ) r ( g ) + r ( 0) uy ww (3.55) For white noise excitation: r ( k) = r ( 0 ), k = 0 and r ( k) = 0 otherwise. (3.56) uu uu uu Also, it ay be shown that, for white noise excitation, s s r ( g + n) = ( e ) K ( e ) r ( 0), n (3.57) uy n uu and r ( uy g + n ) = 0 otherwise. (3.58) Reason: r ( uy g ) = E y n u n g [ ( ) ( )] s s = E[{ e y( n ) + K ( e ) u( n g )} u( n g )] s s = e r ( g ) + K ( e ) r ( ) uy s = e r ( g ) uy uu Reeated alication of this rocedure gives equation (3.58). 79

95 r r uy uy s s ( g + ) = e ruy ( g) + K ( e ) r uu ( 0) s = K ( e ) r ( 0) s s ( g + ) = e ruy ( g + ) + K( e ) r uu ( ) uu s s = K ( e ) e r ( 0) uu Reeated alication of this rocedure gives equation (3.57). For g = g, the value of the MSE (equation (3.55)) equals s s s s MSE ot = ( e e ) ry y ( 0) + [ K( e ) K( e )] ruu( 0) + rww ( 0) (3.59) By coaring the alitudes of the individual ters in equations (3.55) and (3.59), it ay be shown that E[ e ( n)] 8 > MSE ot for (a) g (b) g > g (for all values of the other rocess and odel araeters) and < g, rovided K K / and (O Dwyer (996)). he conditions in (b) above are sufficient, rather than necessary, conditions. However, if a gradient ethod is used to deterine g, then an additional restriction that the MSE function ust be uniodal about a iniu value when g = g, is iosed. he uniodality of the MSE function in equation (3.55) ay be roved by induction; an outline of the inductive roof (rovided in full by O Dwyer (996)) is as follows: (a) For g > g, it ay be roved, using equations (3.55) to (3.59), that the MSE function at g = g is always greater than the MSE function at g = g. Siilarly, using equations (3.55) to (3.58), it ay be roved that the MSE function at g = g n is always greater than the MSE function at g = g n. (b) For g < g, it ay be roved, using equations (3.55) to (3.59), that the MSE function at g = g + is always greater than the MSE function at g = g, rovided 80

96 and K / K / (3.60) hese are sufficient, rather than necessary conditions. Siilarly, using equations (3.55) to (3.58), it ay be roved that the MSE function at g = g + n + is greater than the MSE function at g = g + n, rovided K s s s [( n + ) ( e )] s ( n ) s s s s s s s + K ( e ) e ( e e ){( e ) e [( n + ) e n)]} > 0 (3.6) his is a necessary condition. his theore indicates that if K and are unknown, then convergence of the odel tie delay index to the rocess tie delay index ay only be coletely guaranteed if the value of the odel tie delay index is always less than or equal to the rocess tie delay index. he nature of tyical MSE functions versus odel tie delay index is shown by Figures 3. and 3.3. In Figure 3., K =. 0, K = 0., =. 0 seconds and = 0. seconds, so that the conditions in equations (3.60) and (3.6) are fulfilled. In Figure 3.3, K =. 0, K = 0., = 0. 7 seconds and = 0. seconds, so that the conditions in equation (3.60) are violated. In both siulations, g = 30. he noralised MSE (equal to the MSE divided by r uu ( 0 ) ) versus odel tie delay index is lotted in both cases, with r ww ( 0 ) ut to zero. As exected, the noralised MSE function is uniodal with resect to the odel tie delay index, with a iniu value at g = g when the conditions in equations (3.60) and (3.6) are fulfilled; when the conditions given by equation (3.60) are violated, the MSE curve is still uniodal, but it has a iniu value when g g. 8

97 Figure 3.: Noralised MSE vs. tie delay index (conditions in equation (3.60) et) Figure 3.3: Noralised MSE vs. tie delay index (conditions in equation (3.60) violated) Considering equations (3.), (3.3), (3.4), (3.53) and (3.54), a block diagra reresentation of the gradient ethod to udate both the odel araeters and the 8

98 odel tie delay index is as shown in Figure 3.4. he non-delay odel araeters are udated based on a FOLPD rocess odel. Figure 3.4: Udating the full araeter set - Case u(n) = white noise K ( e ) z PROCESS z e s g s y n ( ) z K ( e ( g g ) ( g g) s + z) z s s g K e s z g MODEL MODEL UPDAE ALGORIHM (g ) + + e s + e s + y ( n ) y ( 8 n ) + e ( 8 n ) z ( / ) - - g + K UPDAE ALGORIHM ( K, ) e ( n ) One set of reresentative siulation results corresonding to heore 3.5 are given in Figures 3.5a-3.5d and 3.6a-3.6d, with the araeters and the rocess inus odel outut lotted against sale nuber. At the beginning, the starting values of the gain, tie constant and tie delay index for both the rocess and odel are equal; a ste change was then ade to the rocess araeter values. he araeter values are taken as K =. 0, K = 0., =. 0 seconds and = 0. seconds, so that the conditions for the uniodality of the MSE function (given by equations (3.60) and (3.6)) are fulfilled; Figure 3. shows the corresonding MSE function. 83

99 Figure 3.5a: Gain estiate Figure 3.5b: ie constant estiate = K -- = K = -- = Sale nuber Sale nuber Figure 3.5c: ie delay index estiate Figure 3.5d: e n ( ) = g -- = g Sale nuber Figure 3.6a: Gain Estiate Sale nuber Figure 3.6b: ie constant Estiate = K -- = K = -- = Sale nuber Sale nuber Figure 3.6c: ie delay index Estiate Figure 3.6d: e n ( ) = g -- = g Sale nuber Sale nuber 84

100 he siulation conditions for udating the odel tie delay index are identical to those in Section 3.4. (with the excitation signal assued to be white noise). he odel gain and tie constant were udated in a siilar anner, with these araeter estiates filtered by a low ass filter. A lower liit was also ut on the odel tie constant of 0. ties the starting value of the odel tie constant (it was deterined in siulation that this lower liit is necessary, as it is the recirocal of the tie constant that is being udated). A lower liit of zero is laced on the odel gain and tie delay index value. Fast convergence to the rocess araeter values is seen for a relatively low level of coloured easureent noise, for both cases of interest (i.e. when the starting value of the odel tie delay index is in turn, greater than and less than the rocess tie delay index). his is as exected fro heore 3.5. If the level of coloured easureent noise is greater than is taken in the siulations corresonding to Figures 3.5a-3.5d and Figures 3.6a-3.6d, reasonable (if noisy) convergence to the correct value of the rocess tie delay index is observed (see Figures 3.7a and 3.7b). Figure 3.7a: ie delay index estiate Figure 3.7b: ie delay index estiate = g -- = g = g -- = g Sale nuber Sale nuber he siulation results discussed are interesting because they show that the odel gain and tie constant values converge to the rocess gain and tie constant values, in the resence of coloured easureent noise (which is redicted fro heore 3.). Generally seaking, the odel tie delay index ust converge to the rocess tie delay index before convergence of the other araeters is observed. his 85

101 was also exlored theoretically in heore 3.. Siulation work has revealed that there does see to be a (relatively low) level of coloured easureent noise above which the odel gain and tie constant estiates do not converge to the rocess gain and tie constant values, because of the noisy convergence of the odel tie delay index estiate; a greater level of filtering on this araeter could be helful. Of course, as in Section 3.4., the rocedures outlined deend on a riori knowledge of the rocess tie delay index, g. In heore 3.5, the revious rocess outut is used to calculate the new odel outut. he revious odel outut could be used to calculate the new odel outut; the odel difference equation under these circustances is (when a first order aylor s series is used to aroxiate the tie delay variation) K g g g g ( ) s ( ) s s s y 9 ( n) = e y 9 ( n ) u( n g ) K( e ) u( n g ) (3.6) Suleentary siulation results confir that convergence of the odel araeters to the rocess araeters was achieved, when a odel corresonding to equation (3.6) was used to deterine the gradient algorith, under the circustances discussed in this section of the chater. he uniodality of the MSE function versus odel tie delay index for this case ay also be roved by calculating the MSE function using the ethod of Widrow and Stearns (985). If the residue theore is used to calculate the required closed curve integral, it ay be deterined that the MSE erforance surface is quadratic in r when the MSE is calculated based on the odel in equation (3.6), and is thus uniodal in this variable i.e. if, fro equations (3.) and (3.6), e ( n) = y ( n) y ( n), then, 9 9 following the rocedure given in equations (3.5) and (3.6), it ay be roven that (O Dwyer (996)) E[ e n A r B r 9 ( )] = C + + (3.63) with s K ( e ) A = s e (3.64) 86

102 and s s s K e K K e e ( ) ( )( ) B = s s s e e e C K e s s s s K K e e K e ( ) ( )( ) ( ) = s + s s s e e e e (3.65) (3.66) If, in addition, K = K = K and = =, then the MSE function equals E[ e ( n)] (with e 7 of equations (3.63) to (3.66), to be 7 ( n) given by equation (3.43)), which ay be calculated, by the use 7 E[ e ( n)] = s K ( e ) r s (3.67) e his function is uniodal, and is iniised when r = 0 (i.e. when g = g ) Convergence of the full araeter set - Case In this case, the rocedures defined do not deend on a riori knowledge of the tie delay index, g (as in the rocedures defined in Section 3.4.) he tie delay as an integer ultile of the sale eriod - white noise inut heore 3.6: For a first order discrete stable syste of unknown araeters, the MPE erforance surface versus odel tie delay index is iniised when the odel tie delay index equals the rocess tie delay index, under the following conditions: (a) he tie delay variation is aroxiated by a first order aylor s series aroxiation (b) he easureent noise is uncorrelated with the rocess inut and outut (c) he resolution on the rocess tie delay is assued to be equal to one sale eriod 87

103 (d) he error is calculated based on using a FOLPD rocess odel; the artial derivative of the error with resect to the tie delay variation is calculated based on the first order aylor s series aroxiation for the tie delay variation (e) he conditions rovided in the theore are observed on the odel araeters and (f) he inut to the odel and the rocess is assued to be a white noise signal. Proof: he rocess difference equation, y ( n), is given by equation (3.). he corresonding odel difference equation, y ( n), is given by equation (3.3) (assuing the revious rocess outut is used in its calculation). he odel difference equation for calculating the artial derivative of the error with resect to the tie delay variation, y 8 ( n), is given by equation (3.53) (assuing that the revious rocess outut is used in its calculation). he error, e ( n) = y ( n) y ( n), is given by equation (3.4) and the error, e ( n) = y ( n) y ( n), is given by equation (3.54). he 8 8 MPE erforance surface, E[ e( n) e8 ( n)], ay then be calculated, using a rocedure siilar to that used in equations (3.) and (3.3), to be equal to (O Dwyer (996)) s s ( e e ) r ( 0) yy ( g g ) s s s s + [ K ( e ) + K ( e )( e + )] ruu( 0) g g g g s s s s + KK( e ){ ( ) ( ) ruu ( g g + ) [ ( e ) + ] ruu( g g)} g g s s s s + K( e e ){ ( ) [ ruy ( g ) ruy ( g )] ( e ) ruy ( g )} + rww ( 0) g g K s s ( e ) ( ) ruu( ) (3.68) with r ( uu n ) and r n uy ( ) rovided in equations (3.56), (3.57) and (3.58) resectively. For white noise excitation, at g (3.56) to (3.58) and equation (3.68)) = g, the value of the MPE equals (using equations yy s s s s MPE ot = ( e e ) r ( 0) + [ K ( e ) K ( e )] r ( 0) + r ( 0) uu ww (3.69) 88

104 By coaring the alitudes of the individual ters in equations (3.68) and (3.69), it ay be shown that E[ e( n) e8 ( n)] > MPE ot for (a) g and (b) g > g (for all values of rocess and odel araeters) < g, rovided K K ( g g ) and (O Dwyer (996)). he conditions in (b) are sufficient, rather than necessary conditions. However, if a gradient ethod is used to deterine g, then an additional restriction that the MPE function ust be uniodal with a iniu MPE value occurring at g = g, is iosed. he conditions for uniodality ay be roved by induction; an outline of the inductive roof (rovided in full by O Dwyer (996)) is as follows: (a) g > g : It ay be roved, using equations (3.56) to (3.58) and equations (3.68) and (3.69), that the MPE function at g = g is greater than the MPE function at g = g. Siilarly, using equations (3.56) to (3.58) and equation (3.68), it ay be roved that the MPE function at g = g n is always greater than the MPE function at g = g n. (b) g < g : It ay be roved, using equations (3.56) to (3.58) and equations (3.68) and (3.69), that the MPE function at g = g + is greater than the MPE function at g = g, rovided that the following sufficient conditions are obeyed: - - s s ( - ) K ( - e ) > K ( - e ) and > (3.70) s he nature of the MPE function eans that for a full inductive roof, it is necessary to rove that the MPE function at g = g + is greater than the MPE function at g = g + (this is because the MPE function in equation (3.68) deends on r ( g g + ) ). A necessary condition for this to be true, using equations (3.56) to uu (3.58) and equation (3.68), is if s K e -s K e -s [ ( - ) - ( - )] > 89

105 - - s s s - s - - s s s ( e - e ) K ( - e )[ ( - e - )( - e ) - ] (3.7) Siilarly, it ay be roved, using equations (3.56) to (3.58) and equation (3.68), that the MPE function at g = g + n + is greater than the MPE function at g = g + n, rovided that K -s s ( - e ) - -( n-) - < - s s s s K ( - e ) e ( e - e ) s s s s s s [( n + ) e - ( n + ) e + n] + ( - e )( - e ) e (3.7) his is a necessary condition. he theore indicates that if K and are unknown, then convergence of the odel tie delay index to the rocess tie delay index ay only be coletely guaranteed if the value of the odel tie delay index is always less than or equal to the rocess tie delay index. he behaviour of the MPE function (given by equation (3.68)) versus tie delay index is confired, in reresentative siulation results, by Figures 3.8 and 3.9. In Figure 3.8, K =. 0, K = 0., = 0. 7 seconds and = 0. seconds so that the conditions given in equations (3.70) and (3.7) (but not (3.7)) are fulfilled; in Figure 3.9, K =. 0, K = 3. 0, = 0. 7 seconds and = 05. seconds, so that none of the conditions in equations (3.70), (3.7) or (3.7) are fulfilled (the forer conditions are identical to those used to calculate the noralised MSE curve in Figure 3.3). he noralised MPE (equal to the MPE divided by r uu ( 0 ) ) is lotted versus tie delay index in both cases, with r ww ( 0 ) ut to zero and g = 30. he excitation signal in both cases is a white noise signal. he results are as exected fro the theore. 90

106 Figure 3.8: Noralised MPE vs. tie delay index - white noise excitation Figure 3.9: Noralised MPE vs. tie delay index - white noise excitation Considering equations (3.), (3.3), (3.4), (3.53) and (3.54), a block diagra reresentation of the gradient ethod to udate both the odel araeters and the odel tie delay index ay be reresented as shown in Figure 3.30; as in Section 3.5., the non-delay odel araeters are udated based on a FOLPD rocess odel. A reresentative siulation result corresonding to heore 3.6 is given in Figures 3.3a-3.3d, with the araeters and the rocess inus odel outut lotted against sale nuber. he siulation conditions for udating the tie delay are identical to those in Section 3.5., though the rocess araeter variations considered are different. It was also found necessary to liit the variation of the non-tie delay odel araeters; for the siulations taken, 0. 5 < K < 3. 0 and 0.5 seconds < < 30. seconds were the liits. he noralised MPE curve corresonding to these siulation results is given by Figure

107 Figure 3.30: Udating of the full araeter set - Case u(n) = white noise PROCESS K ( e ) z z s g e s y n ( ) z ( g g g g ) s ( ) K ( e + z) z s s g MODEL e s + + y ( 8 n ) - e ( 8 n ) + g MODEL UPDAE ALGORIHM (g ) e s z K e z e s s z g + + y ( n ) - + UPDAE ALGORIHM ( K, ) ( / ) e ( n ) K Figure 3.3a: Gain estiate Figure 3.3b: ie constant estiate = -- = = K -- = K Sale nuber Sale nuber 9

108 Figure 3.3c: ie delay index estiate Figure 3.3d: e n ( ) = g -- = g Sale nuber Sale nuber hese results confor with heore 3.6. In this theore, the revious rocess outut is used to calculate the new odel outut. he revious odel outut could also be used to calculate the new odel outut, with the odel difference equation being given by equation (3.6). Siulation results confir that convergence of the odel araeters to the rocess araeters also results in this case, under the circustances discussed in this section of the chater he tie delay as a real ultile of the sale eriod - white noise inut heore 3.6 has dealt with the estiation of rocess tie delays that are integer ultiles of the sale eriod. For the estiation of rocess tie delays that are real ultiles of the sale eriod, then the difference equation of a FOLPD rocess is (O Dwyer (996e)) s b s y 4( n) = e y 4( n ) + K ( e ) u( n g ) + g g bs s K ( e e ) u( n g ) + w( n) (3.73) he corresonding odel difference equation (assuing the revious rocess outut is used in its calculation) is s gas gas s y 0 ( n) = e y 4 ( n ) + K ( e ) u( n g ) + K ( e e ) u( n g ) (3.74) 93

109 he odel difference equation for calculating the artial derivative of the error with resect to the tie delay variation (and assuing that the revious rocess outut is used in its calculation) is K g g g g ( + b a ) s s y ( n) = e y4 ( n ) u( n g) ( g g + g g b a) s s K[ e ] u( n g ) (3.75) his equation ay be deduced fro equation (3.53). herefore, fro equations (3.73) and (3.74), s s b s e0( n) = y4( n) y 0( n) = ( e e ) y4( n ) + K( e ) u( n g) g ( ) b s s + K e e u( n g ) g as g a s s K ( e ) u( n g ) K ( e e ) u( n g ) + w( n) (3.76) g and, fro equations (3.73) and (3.75), s s b s e( n) = y4( n) y ( n) = ( e e ) y 4( n ) + K( e ) u( n g) g b s s + K ( e e ) u( n g ) Ks ( g g + gb ga) + u( n g) g g + g g s( b a) s + K[ e ] u ( n g ) + w ( n ) (3.77) g he MPE erforance surface, E[ e0 ( n) e ( n)], ay then be calculated, using a rocedure siilar to that used in equations (3.) and (3.3), to be equal to (O Dwyer (996)) g y 4y 4 uy 4 s s s s b s ( e e ) r ( 0) + ( e e ) K ( e ) r ( g ) g uy 4 s s b s s + ( e e ) K ( e e ) r ( g ) 94

110 K ( g g + g g ) s s b a s g a s + ( e e )[ K( e )] ruy ( g ) 4 ( g g + g g ) s s s s b a g as s + ( e e )[ K{ e K( e e )] ruy ( g) + 4 g bs g bs s ga s s [ K ( e ) K ( e e ) K ( e ) ( )] ( ) g g g g r + + b a uu 0 g a s s s s K e e e + g g g g r ( )[ ( b a)] uu( 0) + g K( g g + gb g ) b s a s g as [ K( e )[ K( e )] ruu( g g) + g g g g g bs ( + b a) s s s g as s K( e e )[ K{ e } K( e e )] ruu( g g ) g ( g g + g g b a ) b s s s s g as + K( e )[ K{ e } K( e + e )] ruu( g g ) g K( g g + gb ga) b s s s g a s + K( e e )[ K( e )] ruu( g g + ) + rww( 0) + [ g K bs g bs s g as s K ( e )( e e ) ( e e ) s ( g g + g b ga )] ruu ( ) + K s g a s ( e )( e ) r ( ) (3.78) uu with ry 4 y 4 ( n) being the autocorrelation function of y 4 ( n). he behaviour of the MPE function versus odel tie delay index is given, in reresentative siulation results, by Figures 3.3 and For these siulations, K =. 0, K = 0., = 0. 7 seconds, = 0. seconds and g = 5, with the tie delay taken in intervals of 0.0 ties the sale eriod. he noralised MPE (equal to the MPE divided by r uu ( 0 ) ) is lotted versus odel tie delay index for g b = 0. 0 and g b = 05.. In Figure 3.3 the noralised MPE calculated fro equation (3.68) is sueriosed on the lots for coarison uroses. For all siulations, r ww ( 0 ) is ut to zero. Clearly, the MPE function is ultiodal with resect to tie delay, when the tie delay is a real ultile of the sale eriod (as in Section 3.4..). he estiation of the real value of the tie delay is therefore iossible using gradient ethods. 95

111 Figure 3.3: Noralised MPE vs. Figure 3.33: Noralised MPE vs. tie delay index - white noise tie delay index - white noise excitation - g b = 0. 0 excitation - g b = Conclusions Overall, the gradient ethod will allow the estiation of rocess araeters, for white noise excitation, rovided that the rocess tie delay is an integer ultile of the sale eriod, and rovided that the rocess tie delay is always greater than the odel tie delay, as the odel tie delay converges to the rocess tie delay (or, if the rocess tie delay is less than the odel tie delay, the conditions in equations (3.70), (3.7) and (3.7) ust be fulfilled). his conclusion is broadly analogous to the conclusion in Section he use of a non-gradient ethod of estiating the rocess tie delay index that involves deterining the iniu ositive MPE value is also viable, at least when the rocess is excited by white noise. Unfortunately, it is not ossible to estiate the rocess araeters if the rocess tie delay is not an integer ultile of the sale eriod, using the gradient ethod. On the other hand, the indications fro Figures 3.3 and 3.33 are that a non-gradient ethod of estiating the rocess tie delay, based on calculating the iniu ositive value of the MPE surface, ay allow the estiation of the correct value of rocess tie delay, when the 96

112 rocess tie delay is not an integer ultile of the sale eriod (at least for g > g ). It is interesting that this latter ethod could erhas be used to estiate the rocess tie delay when g < g, if g is close to g (at least in the siulations taken) he tie delay as an integer ultile of the sale eriod - square wave inut heore 3.7: For a first order discrete stable syste of unknown araeters, the MPE erforance surface versus odel tie delay index is uniodal, with a iniu value of the MPE occurring when the odel tie delay index equals the rocess tie delay index, under the following conditions: (a) he tie delay variation is aroxiated by a first order aylor s series aroxiation (b) he easureent noise is uncorrelated with the rocess inut and outut (c) he resolution on the rocess tie delay is assued to be equal to one sale eriod (d) he error is calculated based on using a FOLPD rocess odel; the artial derivative of the error, with resect to the tie delay variation, is calculated based on the first order aylor s series aroxiation for the tie delay variation (e) he conditions rovided in the theore are observed on the odel araeters and (f) he excitation signal inut is a square wave with a half eriod greater than the axiu ossible rocess tie delay. Proof: he rocess difference equation, y ( n), is given by equation (3.). he corresonding odel difference equation, y ( n), is given by equation (3.3) (assuing the revious rocess outut is used in its calculation). he odel difference equation for calculating the artial derivative of the error with resect to the tie delay variation, y 8 ( n), is given by equation (3.53) (assuing that the revious rocess outut is used in its calculation). he error, e ( n) = y ( n) y ( n), is given by equation (3.4) and the error, e ( n) = y ( n) y ( n) is given by equation (3.54). he 8 8 MPE erforance surface, E[ e( n) e8 ( n)], is given by equation (3.68). For a square wave excitation signal of alitude ± : r ( k ) uu 97

113 It ay be shown that, for square wave excitation, s r ( g + ) = K ( e ) uy s g s g uu [( e ) r ( 0 g ) + ( e ) r ( g ) r ( g g )] (3.79) uu uu uy uy s s Reason: r ( ) = e r ( 0) + K ( e ) r ( 0 g ) s = K ( e ) r ( 0 g ) uy uy uu s s r ( ) = e r ( ) + K ( e ) r ( g ) s s = K ( e )[ e r ( 0 g ) + r ( g )] uy uy uu uu uu uu s s r ( 3) = e r ( ) + K ( e ) r ( g ) uu s s s = K ( e )[( e ) r ( 0 g ) + e r ( g ) + r ( g )] uu uu uu Reeated alication of this rocedure gives equation (3.79). For g = g, the value of E[ e( n) e8 ( n)] equals MPE ot, which is given by (using equations (3.68) and (3.79)) s s s s s [ K ( e ) K ( e )] r ( 0) + ( e e ) K ( e ) r ( g ) uu s s s s s g ( e e ) K K ( e )( e )[( e ) r ( 0 g ) r ( g g )] s s yy uy uu uu + ( e e ) r ( 0) + r ( 0 ) (3.80) ww By coaring the alitudes of the individual ters in equations (3.68) and (3.80), it ay be shown that E[ e( n) e8 ( n)] > MPE ot for (a) g and (b) g > g (for all values of other rocess and odel araeters) < g, rovided K K and (O Dwyer (996)). he conditions in (b) are sufficient, rather than necessary conditions. However, if a gradient ethod is used to deterine g, then an additional restriction that the MPE function ust be uniodal with a iniu MPE value occurring at g = g, is iosed. he conditions for uniodality ay be deterined by induction. It is assued that the excitation signal is a square wave signal of alitude 98

114 ± and of eriod equal to 00 sales (i.e. that g < 50 ). An outline of the inductive roof (rovided in full by O Dwyer (996)) is as follows: (a) g > g : It ay be roved, using equations (3.68), (3.79) and (3.80), that the MPE function at g = g is greater than the MPE function at g = g for all araeter values. Siilarly, using equations (3.68) and (3.79), it ay be roved that the MPE function at g = g n is always greater than the MPE function at g = g n, rovided s s s s s 0. 04{ K( e ) + K( e )[ ( e ) ]} s s s g s K ( e )( e e )( e ) ( g ) ( g ) [ e e ruu n g 0 04 e ] s s s s s s K( e e ) ( ) ( + ). ( ) >0 (3.8) his is a necessary condition. (b) g < g : It ay be roved, using equations (3.68), (3.79) and (3.80), that the MPE function at g = g + is greater than the MPE function at g = g, rovided that the following sufficient conditions are obeyed: K K and (3.8) Siilarly, it ay be roved, using equations (3.68) and (3.79), that the MPE function at g = g + n + is greater than the MPE function at g = g + n, rovided that s. { ( ) - ( - - s s 04 K e K e )[ ( - e - s s ) - ]} n s s ( + ) 0. 04K ( e e ){ [ ( ) ] e n e g n s s s 99

115 ns n ( g n) s ( ) ( + ) + [ ( e ) ][ e ( e ) ] [ ( ) ]} e n s s s s e g n s s s + K e e > e ( g e s ) s s r g n ( ) [ ( ) uu( )] 0 (3.83) his is a necessary condition. he behaviour of the MPE function, given by equation (3.68), versus odel tie delay index is shown for reresentative siulations in Figures 3.34 and In Figure 3.34, K =. 0, K = 0., = 0. 7 seconds and = 0. seconds, so that the conditions in equations (3.8), (3.8) and (3.83) are fulfilled; in Figure 3.35, K =. 0, K = 3. 0, = 0. 7 seconds and = 05. seconds, so that the condition in equation (3.8) is fulfilled, but the conditions in equations (3.8) and (3.83) are violated. he noralised MPE (equal to the MPE divided by r uu ( 0 ) ) is lotted versus odel tie delay index in both cases, with r ww ( 0 ) ut to zero and g = 30; the conditions taken are identical to those used to calculate the noralised MPE curves in Figures 3.8 and 3.9. he excitation signal used in the deterination of Figures 3.34 and 3.35 is a square wave signal of alitude ± and of eriod equal to 00 sales. he results are as exected fro the theore. Figure 3.34: Noralised MPE vs. tie Figure 3.35: Noralised MPE vs. tie delay index - square wave excitation delay index - square wave excitation 00

116 Considering equations (3.), (3.3), (3.4), (3.53) and (3.54), a block diagra reresentation of the schee to udate both the odel araeters and the odel tie delay index ay be drawn; this block diagra is the sae as Figure A reresentative siulation result corresonding to heore 3.7 is given in Figures 3.36a-3.36d, with the araeters and the rocess inus odel outut lotted against sale nuber. he siulation conditions for udating the tie delay are identical to those in Section (and thus, these siulation results ay be coared with those in Figures 3.3a to 3.3d), excet that the excitation signal is a square wave inut of eriod equal to 00 sales and alitude of ±. he noralised MPE corresonding to these conditions is given by Figure In addition, the learning rate for the odel tie delay is ut to 0 and filtering on the tie delay udate is eloyed. Figure 3.36a: Gain estiate Figure 3.36b: ie constant estiate = -- = = K -- = K Sale nuber Figure 3.36c: ie delay index estiate Sale nuber Figure 3.36d: e n ( ) = g -- = g Sale nuber Sale nuber 0