Input PRBS design for identification of multivariable systems

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1 Iput PRBS desig for idetificatio of multivariable systems Wisto Garcia-Gabi *, Michael Ludh * Abstract This paper presets a systematic procedure for desigig the iput sigals to idetify multivariable processes. The procedure is based o time domai specificatios ad ca be applied to multivariable processes with m-outputs ad - iputs, which ca be operatig i closed-loop. The desig of the iput sigals, which are pseudo radom biary sequeces, are based o the old iformatio about the process model ad the cotroller, together with the measures of the iput ad output variaces of the process. The method proposes excitatio i the frequecy iterval where the model eeds to be accurate for robust feedback cotrol. The method is illustrated usig the Wood & Berry distillatio colum model, which is a -iputs-- outputs bechmark i process cotrol. I. INTRODUCTION Empirical idetificatio is a well-established methodology to obtai multivariable process models, ofte iteded for cotrol but also for other purposes. The choice of iput sigals for the idetificatio has a large impact o the quality of the model [1]. However, i most cases iput sigals caot be freely chose with respect to plat performace costraits. There exist a rather extesive literature o optimal experimet desig [-8]. The problem optimal experimet desig whose objective is to desig the least costly idetificatio experimet while guarateeig a sufficietly accurate model. The mai cotributio of the paper is to provide a methodology orieted to practitioers for facig the problem of iput sigal desig for idetificatio for cases whe the curret model i a MPC cotroller eeds to be updated. The problem of the experimet desig is approached with the idea of producig a procedure for desigig the sigals for idetificatio of multivariable processes, miimizig the disturbace i the process, but keepig i mid that the methods for cotroller maiteace tools must be developed for a easy implemetatio for plat operators. I the proposed methodology for iput sigal desig, first, the required icremets of the output variaces are defied. The, the iformatio relate to the process model ad the cotroller is used for calculatig the excitatio sigals. This paper is focused maily i Pseudo Radom Biary Sequece. However, the method provides the variace ad badwidth of the excitatio sigals, which ca be used for characterizig other sigals, e.g. multi-sie sigals. The paper is orgaized as follows. Itroductio shows the presetatio of the problem faced o the paper. Sectio II develops the procedure for iput PRBS desig for idetificatio. A case study usig Wood & Berry distillatio colum model is preseted i Sectio III. Fially the coclusios are summarized i Sectio IV. II. INPUT PRBS DESIGN FOR IDENTIFICATION Closed-loop idetificatio methods requires excitatio sigals to be applied i the process iputs such that they produce chages i the process outputs. The resultig additioal variaces i the outputs are subject to a trade-off. They must be big eough to produce persistet excitatio for idetificatio, but also to disturb the ormal operatio of the process as little as possible. I this procedure, the omial variaces of the process outputs are take as referece for defiig the set of icremet i the outputs due to the excitatio sigals. The excitatio sigals ca be applied to closed-loop systems as it is show i Fig. 1, where, r(t) is the set poit, u c (t) is the cotroller output, u ex (t) is the excitatio sigal, u(t) is the maipulated variable, v(t) is the measured oise, y(t) is the measured output. However, the procedure could also be also applied for desigig the experimet i ope-loop. Fig. 1 Closed-loop idetificatio diagram Pseudo Radom Biary Sequeces (PRBS) are ofte used as excitatio sigals for idetificatio propose, because it has a fiite legth that ca be sythesized repeatedly with simple geerators while presetig favorable spectra for idetificatio propose. The spectrum at low frequecies are flat ad costat, at high frequecies the spectra drop off, cosequetly the spectra have a specific badwidth, which ca be utilized for excitig the processes i the required frequecies. The aalytical expressio for the power spectrum of a PRBS sigal is give by s(ω) = A (N+1)T cl [ si (ωt cl/) N ωt cl / ] (1) where A is the sigal amplitude, ω is the frequecy, t cl is the clock period (i.e. the miimum time betwee chages i level *ABB AB, Corporate Research, SE7178, Västerås, Swede. {wisto.garcia-gabi, michael.ludh}@se.abb.com.

2 of the sigal), which must be multiple of the samplig time T s. The sequece repeats itself after T=N t cl uits of time, where N=-1 ad is the umber of shift registers used to geerate the sequece. For low frequecies, the power spectrum has the approximate value of A (N+1)t cl N At =.8/t cl the power spectrum is reduced by half. Therefore, the frequecy rage [ω low, ω high ] of a PRBS sigal cosidered to be useful for excitatio here is π T () ω.8 t cl [Rad/s] (3) Thus, for desigig a PRBS sigal is ecessary to determie the frequecy rage ad the amplitude A of the sigals. Some differet approaches for estimatio of the frequecy rage are discussed below. A. Estimate the rage usig time domai iformatio The frequecy specificatio of the PRBS is based o the ideas i [9] ad [10]. They ca also be used for multi-sie desig as ca be see i [11]. Here, we propose the followig procedure to estimate the lower frequecy of iterest: 1. Obtai crude estimates of the time costats ad the time delays of the ope loop process (τ ij ol, td ij ol ) for all m outputs for each iputs usig the process model that is used i the model-based cotroller.. Approximate the settlig times for all the iputoutput pairs: t ij ol = 4τ ij ol + td ij ol 3. Calculate the lower value of frequecy as follows ω low = 1 S f max(t ij ol ) [rad/s] (4) A safety factor S f > 1 is itroduced, with the mai propose to augmet the badwidth of the excitatio sigal. It is carried out reducig the lower value ad icreasig the upper value of the frequecy rage. Based o simulatio tests, S f = 1 is eough to maage chages i the dyamic of the models aroud 30 % with regard to the iitial model. If it is assumed that the actual model chaged much more respect to the iitial model, it is coveiet icrease the safety factor. Values i the rage 1 to 4 are recommeded. The upper frequecy for the rage ca be estimated either usig ope loop iformatio or usig closed loop iformatio. Whe usig ope loop iformatio, estimate for each iputoutput pair the highest frequecy cotet usig the time costat ad the time delay (τ ij ol, td ij ol ) as follows ω cl ij = α S f τ ij where α = max (( tdij ol ol + ), 0.5) (5) τ ij ol The variable factor α, is a measure of how much faster the iteded closed-loop speed of respose will be relative to ope-loop. The determie the upper value of frequecy for the PRBS as follows ω high = max(ω cl ij ) [rad/s] (6) Whe usig closed loop iformatio, first obtai estimates of the settlig time without takig ito accout the time delay for the closed-loop step respose for all the outputs: t i cl. The determie the upper frequecy for the rage usig ω high = 4 S f mi(t i cl ) [rad/s] (7) where S f is the earlier itroduced safety factor ad the 4 is chose to be sure to capture the iterestig frequecies i the trasiets. The upper value of frequecy must be lower tha the Nyquist frequecy thus, ω high ω N. B. Frequecy respose of the sigular values of the closed-loop system Aother method for obtaiig the badwidth of the excitatio sigal is usig the sigular values of the output sesitivity fuctio. This is however, oly possible whe a liear model ad cotroller is available. I this case, the iformatio regardig the cotroller model ad process model is used to calculate the output sesitivity fuctio. The, the frequecy respose of the sigular values of the output sesitivity fuctio is draw, as it is show i Fig.. Magitude (db) low Bode Diagram From: dy To: y Frequecy (rad/s) p high Fig. Frequecy respose of the sigular values N The idea here is to cosider a iterval aroud the peak, ω p i the sesitivity fuctio. The width is determied usig the factor β, which i the rage (4 to 10). It is selected accordig

3 to sharpess of the peak. For A arrow peak a small value of β is adequate, o the other had, for smoother peak large value is coveiet. The upper value of the frequecy rage is the miimum value betwee β times the peak frequecy ad the geometric mea of peak frequecy ad Nyquist frequecy, as it is show below ω high = mi (βω p, ω p ω N ) [rad/s] The lower value of frequecy is as follows ω low = 1 β ω p [rad/s] of a sum of sie terms with differet frequecies. Thus, the variace of the multi-sie (9) is = ( A k k=1. (10) σ uex ) Cosiderig that the amplitude A k of all terms of the multisie have costat amplitude A. The variace of the multisie (10) is give by σ uex = A. (11). Cosider the closed loop system G from the excitatio sigals to the process outputs as show i Figure 3. The lower value of frequecy is of little iterest i case of PRBS. However, it is useful for desigig multi-sie excitatio sigals. C. PRBS clock period ad the umber of shift registers The clock period t cl ad the umber of shift register r of the PRBS ca ow be determied from t cl.8 ω high N = r 1 π t cl ω low (8) D. PRBS amplitudes Oce the frequecy rages of the excitatio sigals are defied, the amplitudes are yet to be defied. Although several authors defie methods to calculate the badwidth of the excitatio sigals [1,10]. A systematic procedure is ot provided for calculatig the amplitude of the excitatio sigals. Most of the time, literature merely idicates that the amplitudes of the PRBS sigals are chose such that they will geerate data with good eough sigal-to-oise ratio but will ot disturb the product quality. Normally, most of the iformatio eeded for determiig the amplitudes of PRBS sigals ca be obtaied by iterviewig experieced operators ad operatio egieers [1,9,1]. Below is a systematic way proposed, how to obtai the amplitudes of the excitatio sigals: A multi-sie is cosidered as base sigal for developig this procedure. A multi-sie ca be described as a weighted sum of siusoidal sigals u ex (t) = k=1 A k si(ω k t + φ u k ) (9) Where A k is the amplitude of the k-term of the multi-sie at the frequecy ω k ad φ k u is its phase. is the umber of terms of the multi-sie. The sigal variace of a multi-sie sigal is equal to the sigal power with mea removed, whe the multi-sie is composed The output sigal y(t)is Fig. 3 Liear model. y(t) = G k A k si(ω k t + φ y k=1 k ) (1) where G k is the gai of the system at the frequecy ω k ad φ k y is phase of the output sigal. The output variace ca be calculated as the power of the sigal as σ y = ( G ka k k=1, σ y = k=1 A for A k = A (13) ) G k Equatio (13) ca be rewritte for the multivariable case as [ σ yp G 11 ] = 1 k=1 G k k=1 1qk A 1 [ ] [ ] k=1 G p1k k=1 G pqk A q The equatio above ca be writte i compact form as λ 1 ψ 11 ψ 1q ζ 1 [ ] = 1 [ ] [ ] Λ = ΨΖ (14) λ p ψ p1 ψ pq ζ q where Λ is a vector cotaiig of the variaces of the p outputs of the system, Ψ is a matrix where each elemet has the sum of the square gais of the ij-output-iput pair for the sequece of frequecies ω k, ad Ζ is a vector with the square of the amplitude of multi-sie sequeces of the q iputs. Notice that i the multivariable case each multi-sie excitatio sigal must have a differet frequecy distributio sequeces [ω 1, ω ω k ]. This must be take ito accout if

4 the multivariable system will be excited at the same time i all the iputs with multi-sie sigals. The amplitude of the iputs ca be obtaied as a solutio of the eq. (14). This is fidig a miimum of a costraied multivariable fuctio as follows J = arg mi ζ (Ζ(ζ)) T Ζ(ζ) (15) of the PRBS. This is equivalet to apply a delayed sigle PRBS sigal i each excitatio iput [1,13]. III. CASE STUDY The Wood & Berry distillatio colum was cosidered as - iputs--outputs process model for testig the procedure. The distillatio colum model i [14] is give by Subject to: ΨΖ Λ (16) G(s) = [ s+1 e s s+1 e 3s s+1 e 7s 19.4 (1) e 3s] 14.4s+1 Equatio (15) attempts to fid a costraied miimum of a scalar fuctio give by the sum of the output powers. The solutio is subject to a set of iequalities with the geeral form (16). This meas that the variaces of the p-outputs must be higher tha a threshold Λ = [λ 1 λ p ] T defied a priori, as high eough to achieve a persistece of excitatio. The process was cotrolled with a discrete time MPC (Matlab toolbox) with samplig iterval T s = 1 ad a white oise with a variace of 0.1 was added at each output. Figure 4 shows the frequecy respose of the process i ope loop (blue) ad the output sesitivity fuctio (red). Iequality sets with the form I Ψ [ 0 0 ] Ζ Λ (17) 0 0 I q ca be added to guaratee that each iput produces eough persistece of excitatio i all the outputs. For each iput i where this is required there will oe costrait of the type i (17) with I i = 1 ad I j = 0 j i. Oce Ζ = [ζ 1 ζ q ] T is obtaied as solutio of (15). The amplitude A i of the multi-sie iputs ca be calculated as follows A i = ζ i. (18) Fially, the variace of the excitatio sigals are σ uex = ζ i = A i i (19) Thus, the iput desig is characterized by the variace of the excitatio sigals (19) ad their frequecy rages. These iput sigals ca be applied to the process as multi-sie sigals or other excitatio sigals, for example a PRBS, which must have the variace obtaied i (19). I the last case, the PRBS amplitude must switch betwee two levels ±A PRBS, this value is defied as the square root of the variace, as is show below A PRBSi = σ uex (0) i I a case whe excitatio sigals are applied at the same time i a multivariable process, it is importat to have a low crosscorrelatio betwee the excitatio sigals. This ca be accomplished by differet iitializatios of the shift register Figure 4. Frequecy respose of process (blue), output sesitivity fuctio (red). Table 1 shows the variace of the outputs i closed loop before applyig the idetificatio sigals. Table 1 Variaces i the outputs ad iputs i closed loop CV1 CV MV1 MV σ Based o these output variaces i closed-loop were defied as σ y = 0., the icremet i the output variaces required for the idetificatio proposes. A. Frequecy specificatios The procedures i sectio II-A, were used to determie the frequecy rage. The time costats ad the time delays for the ope loop process were obtaied directly from (1) here. Estimates of the closed loop rise time was obtaied from simulatios. The results are summarized i Table.

5 Table Frequecy rage obtaied ω low [rad/s] ω high [rad/s] Ope- loop Closed-loop The upper frequecy that will be used below is obtaied usig the closed-loop iformatio (0.18 rad/s). However, the approximated upper frequecy calculated usig the ope loop iformatio is reasoable good compared with the closedloop value. B. Amplitude specificatios The procedure i sectio II-D, was used to determie the excitatio sigal amplitude. Here we used a multi-sie sigal with 10 frequecies logarithmically spread from ω low = 0.01 to ω high = The excitatio was required to geerate a additioal output variace of σ y1 = 0. ad σ y = 0.. The optimizatio problem i (15) - (17) the gave the variaces for the excitatio sigals σ uex1 = ad σ uex = C. Simulatio Results Table 3 shows the icremets i the output variaces whe the excitatio sigals are applied o the closed loop simulated process. The first row shows the expected theoretical value. The secod row shows the icremetal variace i the outputs obtaied i the simulatio usig a multi-sie as excitatio sigals. Fially, the third row shows the icremetal variace whe the PRBS sigal is applied. Table 3. Obtaied additioal variaces u ex1 0, u ex = 0 u ex1 = 0, u ex 0 Theoretical = 0.0 = 0.0 σ y = 0.06 σ y = variace Multisie = 0.06 = 0.04 σ y = 0.0 σ y = σ MV1 = σ MV1 = σ MV = σ MV = PRBS = 0.66 = 0.39 σ y = 0.41 σ y = σ MV1 = σ MV1 = = 0.00 = σ MV σ MV u ex1 0, u ex 0 = 0.40 = σ y = = = = 0.08 = = 1.15 = = 0.07 σ y σ MV1 σ MV σ y σ MV1 σ MV It ca be observed i Table 3, that each output variace pass the threshold defied iitially σ y = 0.. This is because that was defied as costrait i the optimizatio process (15)- (17). It ca also be oticed that simultaeous excitatio i both iputs icreases the output variace more tha eeded. The geerated sigals have bee tested for excitatio i closed loop. The process iputs ad outputs were used for idetificatio usig Matlab s System Idetificatio toolbox which provided, as expected, a accurate model. IV. CONCLUSION This paper proposed a simple approach to determie how a multivariable model ca be excited durig a experimet to update the cotroller model. The method relies o that there exists some, o loger ideal, model for cotrol of the process. The method is orieted for practitioers ad operators usig basic cocepts. Also, it ca be easily coded i the cotrol system. The method proposes a frequecy rage ad variaces of the excitatio sigals. This iformatio was used i the example for desigig PRBS sigals, however, other sigals like multi-sie fuctios ca also be calculated. Further, this method ca be straightforward applied to o-square multivariable systems. The mathematical support of the method is ot limited for square systems. REFERENCES [1] Rivera, D. E., & Ju, K. S. (000). A itegrated idetificatio ad cotrol desig methodology for multivariable process system applicatios. Cotrol Systems, IEEE, 0(3), [] F. Pukelsheim, Optimal desig of experimets. Joh Wiley, [3] A. Atkiso ad A. Doer, Optimum experimet desig. Oxford: Claredo Press, 199. [4] G. Goodwi ad R. Paye, Dyamic System Idetificatio: Experimet Experimet Desig ad Data Aalysis. New York: Academic Press, [5] T. S. Ng, G. C. Goodwi, ad T. Söderström, Optimal experimet desig for liear systems with iput-output costraits, Automatica, vol. 13, pp , [6] R. K. Mehra, Optimal iput sigals for parameter estimatio i dyamic systems survey ad ew results, IEEE Trasactios o Automatic Cotrol, vol. AC-19, pp , [7] H. Hjalmarsso, From experimet desig to closed loop cotrol, Automatica, vol. 41, o. 3, pp , March 005. [8] Bombois, X., & Scorletti, G. (01). Desig of least costly idetificatio experimets: The mai philosophy accompaied by illustrative examples. Joural Europée des Systèmes Automatisés, 46(6-7), [9] Gaikwad, S. V., & Rivera, D. E. (1996). Cotrol-relevat iput sigal desig for multivariable system idetificatio: Applicatio to highpurity distillatio. IProc. IFAC World Cogress, Sa Fracisco (Vol. 1000, pp ). [10] Lee, H. (006). A plat-friedly multivariable system idetificatio framework based o idetificatio test moitorig (Doctoral dissertatio, Arizoa State Uiversity). [11] Lee, H., Rivera, D. E., & Mittelma, H. D. (003). Costraied Miimum Crest Factor Multisie Sigals for" Plat-Friedly" Idetificatio of Highly Iteractive Systems. I I: 13th IFAC Symp. o System Idetificatio. Rotterdam. [1] Zhu, Y. (1998). Multivariable process idetificatio for MPC: the asymptotic method ad its applicatios. Joural of Process Cotrol, 8(), [13] Yao, L., Zhao, J., & Qia, J. (006). A improved pseudo-radom biary sequece desig for multivariable system idetificatio (A16-395). I Itelliget Cotrol ad Automatio, 006. WCICA 006. The Sixth World Cogress o (Vol. 1, pp ). IEEE. [14] Luybe, W. L. (199). Practical distillatio cotrol (pp. 7-84). New York: Va Nostrad Reihold.

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