Development of Second Order Pls Time Delay (SOPTD) Model from Orthonormal Basis Filter (OBF) Model Lemma D. Tfa*, M. Ramasamy*, Sachin C. Patwardhan **, M. Shhaimi* *Chemical Engineering Department, Universiti Teknologi PETRONAS, 375, Tronoh Perak, MALAYSIA (Tel: +65 368 7585; e-mail: marappagonder@petronas.com.my) **Department of Chemical Engineering, Indian Institte of Technology Bombay Powai, Mmbai 76 INDIA Abstract: A novel method to determine the parameters of a second order pls time delay (SOPTD) model from a step response is presented. The method is niqely effective in developing SOPTD models from Orthonormal Basis Filter (OBF) model. A noise free OBF model can be easily developed from a noisy response data and any type of inpt with a crde estimate of time constants and no-prior knowledge of time delay. The OBF model developed in this manner can captre the dynamics of a process with only a few nmbers of terms (parsimonios in parameters) and do not have the problem of inconsistency which is commonly encontered in ARX models. In addition, the OBF model gives the liberty to se any type of inpt seqence for identification so that we can design the best possible inpt seqence. However, the time delay in OBF models is estimated by a non-minimm phase zero and crrent methods of developing SOPTD model from a step response cannot be applied effectively. In this paper, an effective method to identify SOPTD systems or for approximating higher order systems by SOPTD model from OBF models is proposed. The efficacy of the proposed method is demonstrated throgh simlation stdies. Key words: - Orthonormal Basis Filter Model, System identification, Second Order Pls Time Delay model.. INTRODUCTION In developing linear dynamic pertrbation models for controller synthesis, Finite Implse Response (FIR) and Ato Regressive with Exogenos Inpt (ARX) model strctres are most commonly sed in process indstry. However, since these models are non-parsimonios in parameters, large data sets are reqired to minimize variance errors in model parameters. Orthonormal Basis Filter (OBF) models are parsimonios in parameters reqiring less nmber of parameters. These models can be considered as a generalization of FIR models in which the filters q -, q -, (delays) are replaced with more realistic, orthonormal basis filters, like Lagerre Filter, Katz Filters, Markov-OBF and general orthonormal basis filters (Nelles, ; Patwardhan, 5; Van Den Hof, 995). The parameters of OBF models can be determined sing least sqare method for both SISO and MIMO systems. Unlike ARX and ARMAX models, OBF models do not reqire a prior knowledge of time delay and they generally reslt in parameters which are consistent and nbiased. OBF models can generally be formlated either in otpt error or ARX strctre. System Identification sing OBF models can be copled with developing redced complexity transfer fnction models like first order pls time delay (FOPTD) and second order pls time delay (SOPTD) models for two reasons. Firstly, to se OBF models for identification effectively we need a good estimate of the dominant time constants or poles of the system (Patwardhan, 5). If the estimated pole is not good enogh, the reslting OBF model needs large nmb er of terms to captre the system dynamics (not parsimonios) and we loose one of the major advantages of OBF models. However, if identification of OBF models with development of FOPTD or SOPTD models is copled, an iterative techniqe can be sed with a crde initial estimate of time constant. In each iteration, a better estimate of the dominant time constant can be obtained sing the transfer fnction model which can be sed for developing more accrate and parsimonios OBF model. Secondly, there are many instances in control system design, sch as PID tning, in which it is satisfactory to obtain FOPTD or SOPTD models (Ljng, ). One instance is when it is intended to identify a system from closed-loop data sing the controller otpt and the plant otpt as the inpt and otpt seqences, respectively, of the identification problem.. Development of OBF model Two filters, L m and L n, are said to be orthonormal if they satisfy the property (Van Den Hof, 995) ( m n) L ( q), L ( q) () m n ( m n) where <,> represents inner prodct defined on the set of all stable transfer fnctions. Ths, a stable system, G(q), can be approximately represented by a finite length generalized Forier series expansion as:
G( q) c L ( q) () n k k k where q shift operator c the model parameters L k (q) the orthonormal basis filters for the system G(q) In time domain, the response, y(k), of an LTI system for an inpt, (k), can be described as y k ) c L ( q ) ( k ) + c L ( q ) ( k ) +... (3) + c L ( q ) ( k ) v ( k ) ( m m + where v(k) describes the white noise. The first step in the OBF model development is the selection of an appropriate type of orthonormal basis filter. The selection is based on a prior knowledge of the behavior of the process, i.e., whether the process is well damped or not. The varios types of Orthonormal Basis Filters are discssed below: Lagerre Filters The Lagerre filters are first-order lag filters with one real pole. They are, therefore, more appropriate for well damped processes (Nelles, ). L n ( pq) ( p ) n ( q p) n where p pole (estimated), p < () If an estimate of the time constant τ is available the corresponding pole can be calclated by: p exp( τ / T ) (5) where T s is the sampling interval. s If a good estimate of the time constant is not available, an iterative techniqe can be employed in which a crde estimate of a time constant is sed as a starting point and better estimates are obtained from the OBF model ths developed. A detailed discssion is available in Lemma et al., 7. Katz Filter Katz filters allow the incorporation of a pair of conjgate complex poles; they are therefore effective for modelling weakly damped processes (Nelles, ). The Katz filters are defined by ( a )( b ) f ( a, b, q) q + a( b ) q b L n n,, (6a) ( b )( q a) f ( a, b, ) n,, (6b) q + a( b ) q b L n q where n bq + a( b ) q + f ( a, b, q) ( ) q + a b b (7) Re a + Re + Im (8) b ( Im + Re ) (9) a <, b < Re real part of the pole Im imaginary part of the pole Markov-OBF When a system involves time delay and an estimate of the time delay is available, Markov-OBF can be sed. The time delay in Markov-OBF is inclded by placing some of the poles at the origin. Van Den Hof, et al., (995) dis cssed Generalized Orthonormal Basis Filters and its relation with the other orthonormal basis fnctions in detail. To estimate the model parameters, the regression matrix, X, is formlated first. It is formed by filtering the inpts with the appropriate orthonormal basis filters (Nelles, ). L( m) L ( m + ). X.. L ( N ) L L L ( m ) ( m).. ( N )...... Lm () Lm ().. Lm ( N m) () where Li Li( q, p) ( k) are the inpts filtered with the appropriate orthonormal basis filters. The model parameters are then calclated by the linear least sqare formla: c k T T ( X X ) X y (). SOPTD model Once an OBF model is developed, a step change in its inpt is introdced and a noise free response is obtained. The OBF model developed in this way approximates the time delay by a non-minimm phase zero and this approximation appears as an inverse response in the step response of the OBF model. Figre depicts a typical step response of an OBF model. Becase of the approximate natre of the time delay, crrent methods of developing a SOPTD model from step response of OBF models are not effective. The two commonly sed methods are discssed below.
.6. -. Fig.. A typical step response of an OBF model The Smith method is a graphical method of determining the SOPTD parameters from the step response of a system (Smith, 97). The time at which the normalized step response reaches % (t ) and 6% (t 6 ), with apparent time delay removed, are first determined from the step response data. The vale of the damping coefficient, ζ, and τ /t 6 are then determined from a graph sing the ratio t /t 6. From τ /t 6 and t 6, the natral period, τ, is calclated. The method can be sed to identify both nderdamped and overdamped systems. Rangaiah and Krishnaswamy (99, 996) proposed varios methods for determining the parameters SOPTD models. For nderdamped systems they presented two different methods. The methods are based on finding three points that minimize the integral absolte error (IAE) between the actal response and the step response of a SOPTD model by which the process is to be approximated. Seborg, et al, () indicates that the methods works qite well for the range.77 ζ 3.. The Smith techniqe has major difficlties in its application. Removing the apparent time delay in finding t 6 and t is not a simple task and it is even more difficlt for OBF step responses. Graphical method for estimating the apparent time delay is sally inaccrate and the parameter estimation is seriosly affected. The Rangaiah and Krishnaswamy method gives good reslts only in a limited range and, in addition, it doesn t treat both the nderdamped and overdamped cases together.. PRESENT WORK In this work, a novel method for determining the parameters of the SOPTD model is presented. The method is niqely effective in developing SOPTD models from Orthonormal Basis Filter (OBF) models. It can be sed to identify both nderdamped and overdamped second order systems with or withot apparent time delay or to approximate a higher order system with a SOPTD model. It eliminates the need of estimating the apparent time delay separately and enables to determine all the parameters inclding the apparent time delay with high accracy. The transfer fnction of a second order system with time delay is given by (): Y ( s) G s) U( s) τ ( y(t) 6 t (sec) t Ke d ζτs s + s + () where K steady state gain t d time delay ζ damping coefficient τ natral period of oscillation To estimate the parameters, a step inpt is introdced into the OBF model and the step response is determined. The steady state gain can then be estimated by finding the ltimate response and dividing it by the step size, A. y K (3) A Once the steady state gain is estimated the normalized response is obtained by dividing the original response by the steady state gain. The inflection point of the normalized response crve is the point at which the tangent to the crve attains the maximm slope. It can be easily fond by determining the instant of maximm slope, i.e., by filtering the normalized response y(k) with the filter given by () and determining the vale of k that maximizes y. ( q ) y( k) y () It is fond, analytically, that the damping coefficient depends only on the vale of the normalized response at the inflection point. The dependence is shown in Fig.. Therefore, by determining the normalized response at the inflection point the damping coefficient can be obtained from the figre. Damping coefficent 8 6...6.8 Normalized response at inflection point Fig.. Damping coefficient as a fnction of the normalized response at the inflection point The most commonly sed method for determining the time delay is the maximm slope method. In this method, a tangent is drawn at the inflection point and the intersection point to the time axis is taken as the approximate vale of the time delay. However, this approximation for second order and higher order systems is not accrate enogh. In the present work, the time delay by the maximm slope method is divided into two parts: the apparent time delay and the contribted time delay (the tail of the sigmoid crve).
8 Normalized response.5 y ip Inflection point Coefficients m and m 6 m m t da tdc t ip Time Fig. 3. The apparent and contribted time delay. The total time delay (t d ) determined by the maximm-slope method is the sm of the apparent time delay (t da ) and the contribted time delay t dc. Hence, the apparent time delay can be calclated by sbtracting the contribted time delay from the time delay determined by the maximm slope method. The contribted time delay, t dc, does not depend on the pre time delay therefore it can be calclated from the parameters of the second order transfer fnction withot the apparent time delay. t da t t (5) d dc The total time delay is estimated by (6) if the response time t ip, slope s ip and normalized response y ip all at the inflection point are determined as noted before. t y ip d tip (6) sip It was fond that the contribted time delay and the natral period τ are directly proportional to any difference of response time, t n - t m, and the coefficients of proportionality depend only on the damping coefficient and the specific vales of t m and t n chosen. In this paper, t m and t n are chosen as the time to reach 3% and % of the maximm response, respectively. Since we se time differences and not absolte time, all the calclations do not depend on the apparent time delay, and hence, nlike the Smith method it is not reqired to estimate the apparent time delay separately. The apparent time delay itself is calclated withot any difficlty sing (5), (6) and (7). t dc m (7) ( t t3) τ m ( t ) (8) 3 t The dependence of m and m on the damping coefficient is as presented in Fig.. 6 8 Damping coefficent Fig.. m and m as fnctions of the damping coefficient To generate the crves in Fig., introdce a step inpt into any known second order model with varios vales of the damping coefficient, ζ, determine the corresponding t 3 and t vales for each step response and calclate the corresponding m and m vales sing (7) and (8). 3. SIMULATION STUDY In this stdy, the proposed method is sed for both identifying a second order pls time delay system and approximating a higher order system by a SOPTD model from a noisy response data. In the first case, data from a closed loop system and in the second case data from open loop system are sed. Both data sets are generated sing SIMULINK. 3. Identification of a SOPTD system The data for identification is generated sing the closed loop system shown in Fig. 5. The transfer fnction of the process is given by (9). s e G ( s) (9) 6s + 6s + Repeating Seqence Stair3 PID PID Controller 6s+ Transfer Fcn s+ Transfer Fcn3 s+ Transfer Fcn Band-Limited White Noise Transport Delay Scope Fig. 5. The closed-loop block diagram for generating the identification data sing SIMULINK. An additive white noise with a signal to noise ratio (SNR) of 5.83 is added. SNR is defined as the ratio of the variance of the inpt signal to the variance of the noise signal. Proportional-only controller with proportional controller gain, K c, is sed. Set point changes as shown in Fig. 6 are introdced to the system.
Set point change - - 5 5 Time(sec) Fig. 6. Inpt changes in set point. The controller and the plant otpts for the set point changes are shown in Fig.7. 5 Plant otpt -5 5 5 Time (sec) - Controller otpt - 5 5 Time (sec) Fig. 7. Controller and plant otpt data from the closed loop system. An OBF model with eight Lagerre filter terms and a single pole is developed iteratively starting with an initial time constant of 3 s (p.967, and T s s) sing the controller otpt and the plant otpt as inpt-otpt data of the system identification problem. The vale of the pole, p, in the Lagerre filter converges to.886 in for iterations with the corresponding model parameters, c k, eqal to [.57,.97,.566, -.33,.36,.,.7]. The SOPTD model developed from the OBF model is given by (). -.9.9e G m ( s) () 6.5s + 6.57s + Fig. 8 shows the step response of the actal SOPTD system and the approximate SOPTD model identified from the noisy plant otpt and controller otpt data. OBF models with otpt error strctre developed from closed-loop data theoretically leads to biased parameter estimates. Amplitde.5.5 3 5 6 7 Time (sec) Fig. 8. Comparison of step responses of the original (ooo) and the identified ( ) systems. Phase (deg) Magnitde (abs) - - -7 - -6-3 - - Freqency (rad/sec) Fig. 9. Bode plot of the original ( ooo ) and the identified SOPTD ( )models. However, for the simlation example, it is observed that the bias in the estimates does not affect the reslt significantly. Even thogh there is some difference between the actal and the estimated parameters, the step responses and the Bode plots (Fig. 9) show insignificant differences especially for control system design. 3. Approximation of a forth order system by SOPTD model In this example, a forth-order system with a time delay given by () is considered. The open-loop response of the system with additive white noise, SNR5.8, for a Psedo Random Binary Seqence (PRBS) inpt is shown in Fig.. G s) 8.8s -6s 6.5e 3 + 68.s + 5.5s ( + 3.3s + () An inpt-otpt data is generated for the system sing SIMULINK. An OBF model with eight Lagerre filters is developed with an initial time constant of s (p.9753, and T s s).
Response 5-5 5 5 Time(sec) Fig.. The response of the forth order model with additive white noise (SNR5.8) for PRBS inpt In for iterations the pole, p, in the Lagerre filters converges to.86 with the model parameter estimate, c k, eqal to [.37,.5,.6787, -.3, -.7,.3,.587,.985]. A SOPTD model is obtained from the step response of the OBF model and the reslting transfer fnction is given by (). The step responses of the original forth order system and the SOPTD approximation are presented in Fig.. It is observed that the step response of the SOPTD model and that of the original forth order system are very close. G 6.5e s) 35.93s +.63s + m( Amplitde 7 6 5 3 7.97s 6 Time(sec) Fig.. The step responses of the original ( ooo ) and the approximate SOPTD models ( ) Magnitde (abs) Phase (deg) - -8-36 -5-3 - - () Freqency (rad/sec) Fig.. Bode diagram of the original forth order system (ooo) and the approximate SOPTD model ( ). The bode diagram of the original forth order and it s approximation by SOPTD model is presented in Fig.. It can be noted from the bode plot that the approximate SOPTD model gives good reslt in the freqency range sefl for control system design.. CONCLUDING REMARKS A novel method for estimating the parameters of an overdamped SOPTD model from OBF model is presented. Models for nderdamped systems can be estimated sing either generalised OBFs or Gatz filters. It is shown that the parameters of a SOPTD model can be effectively estimated withot the need of estimating the apparent time delay separately. The approximation of higher order systems by SOPTD model is shown to provide accrate reslts both in time and freqency domains. ACKNOWLEDGMENT The athors grateflly acknowledge the spport from the Universiti Teknologi PETRONAS for carrying ot this research work. REFERENCES Lemma, D. T., Ramasamy, M., Shhaimi, M., Patwardhan, S.C, (7) Control Relevant System Identification Using Orthonormal Basis Filters, Proceedings of IEEE 7 International Conference on Intelligent and Advanced Systems, KL convention centre, Kala Lmpr, Malaysia. Ljng, L. (). Identification for control: simple process models. Proceeding of the st IEEE conference on Decision and Control, 65-657, Las Vegas, Nevada USA. Nelles, O. (). Nonlinear System Identification. Springer-Verlag, Berlin Heidel Berg. Patwardhan, S.C., Shah, S.L. (5). From data to diagnosis and control sing generalized orthonormal basis filters, Part I: Development of state observers. Jornal of Process Control (5), 89-835. Rangaiah, G. P., Krishnaswamy, P.R. (99). Estimating Second-Order Pls Dead Time Model Parameters, Ind. Eng. Chem. Res., 33, 867. Rangaiah, G. P., Krishnaswamy, P.R. (996). Estimating Second-Order Dead Time Parameters from Underdamped Process Transients. Che. Eng. Sci., Vol. 5, No. 7, 9-55, Pergamon. Seborg, D.E., Edgar, T.F., Mellichamp, D.A. (), Process Dynamics and Control, nd ed. John Wiely & Sons. Smith, C.L., Digital Compter Process Control, Intext, Scranton, PA, 97. Van den Hof, P.M.J., P.S.C. Heberger and J. Bokor. (995). System Identification with Generalized Orthonormal Basis Fnctions. Atomatica, Vol 3, No., 8-83.