Improving performance and stability of MRI methods in closed-loop

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1 Preprints of the 8th IFAC Symposium on Advanced Control of Chemical Processes The International Federation of Automatic Control Improving performance and stability of MRI methods in closed-loop Alain Segundo Potts Rodrigo Alvite Romano Claudio Garcia Universidade de São Paulo, São Paulo, SP, , Brazil ( alain Instituto Mauá de Tecnologia, São Caetano do Sul, SP, , Brazil ( Universidade de São Paulo, São Paulo, SP, , Brazil ( Abstract: Two representative approaches for MRI methods are reported in the literature The first one is based on the solution of an optimal problem, while the second is based on the prefiltering of the system input and output signals Each method has advantages and disadvantages according to the process to identify, the length of the prediction horizon or its mathematical implementation Herein a new MRI method is proposed (C-EMPEM), based on the advantages of both algorithms and on some improvements The new method was developed to identify either closed-loop or open-loop systems A comparison is performed among some MRI and PEM methods and the new one proposed, considering a closed-loop system The results indicate that in the studied case, the performance of the new method is better Keywords: Process identification; Optimal prediction; Prediction error method; Model predictive control INTRODUCTION Model predictive controllers (MPC) are widely used in the process industry because they can handle and optimize multivariable systems subject to input and output constraints MPC needs a model able to accurately predict the behavior of a dynamic system over a prediction horizon (Shook et al, 99, 99; Rossiter and Kouvaritakis, 00; Huang et al, 003; Gopaluni et al, 004; Laurí et al, 00) In the literature many techniques for the identification of dynamic models of processes are reported using prediction error methods (PEM), but they are based on one-step ahead predictors The so-called MPC relevant identification (MRI) methods have emerged from the need of models suitable to generate multistep ahead predictions and, consequently, more appropriate to predictive control applications It is important to emphasize that the advantages of the MRI methods arise only when a certain amount of bias is expected in the process modeling Hence, the MRI problem may be thought of as a way of distributing this bias in a frequency range that is less important for control purposes (Gopaluni et al, 00) In recent years, distinct MRI techniques were proposed based on different principles One of them, conceived by Rossiter and Kouvaritakis (00) use multiple models to generate the optimal j-steps ahead predictions This multimodel technique employs an optimal j-step ahead prediction model for each j {,,, P }, where P is the MPC prediction horizon Then all the P models are simultaneously used for predictions In spite of providing an optimal multistep ahead prediction, the number of parameters involved can be quite large, specially for multiinput and multi-output processes It is known that the variance of the parameter estimates is proportional to the ratio between the number of parameters and the dataset length The main drawback of the multimodel approach is the amount of data required to estimate a reasonable model set Such amount of data may be prohibitive in practical situations In another MRI approach, only one model is estimated by minimizing a multistep ahead prediction error cost function, which is highly nonlinear in the model parameters For instance, in (Gopaluni et al, 004) the authors proposed a Two-step algorithm to identify an optimal disturbance model Ĝ L (q), based on a high order FIR (Finite Impulse Response) model of the process Other authors deal with the parameter estimation problem directly, minimizing the multistep ahead prediction error cost function using nonlinear optimization techniques, eg, Laurí et al (00) It is also possible to transform a multistep ahead predictor into a one-step ahead by filtering the inputoutput data (Huang and Wang, 999) In this way, standard one-step PEM algorithms can be employed to solve the parameter estimation problem in a MRI context It is important to mention that, in this alternative, the filter should be estimated using the disturbance model, which is not known a priori In the context of the identification for control, the closedloop identification has advantages over the open-loop In (Huang and Wang, 999) the authors show that the best open-loop identified model is inferior to the closed-loop IFAC, 0 All rights reserved 408

2 one, when the evaluation criterion used is the tracking error between the set point of the system and its output In this sense, some authors such as Huang and Shah (997) discussed the importance of the data prefiltering in the closed-loop identification This paper aims at proposing a new method that takes into account the main advantages of the next two MRI approaches The new method is composed of five steps, the first three consist of a variation of the Two-step algorithm proposed in (Gopaluni et al, 004) The proposed variation is to replace the high order FIR model by a high order ARX (Auto Regressive with exogenous inputs) structure There are three reasons to employ this structure First, based on the asymptotic theory (Ljung, 985), high order ARX models yield unbiased estimates even in closed-loop, whereas high order FIR structures just minimize bias in open-loop Second, it also provides a better initial approximation of the disturbance model, than using high order FIR structures, since in (Gopaluni et al, 004) ĜL = is used Third, the necessary amount of parameters in these models does not depend on the sampling period and, in general, it is lower than in FIR ones, reducing the model variance errors One drawback of using high order ARX structures is that it is necessary to know or to have a good estimate of the process delay The last two steps are a revised version of the Multistep Prediction Error Method (MPEM) proposed by Huang and Wang (999) Using the MPEM method, a re-identification is made to obtain a final reduced order model MODEL RELEVANT IDENTIFICATION Consider a MISO (multi-input single-output) black-box model: y(t) = G P (q)u(t) + G L (q)e(t), () where G P is a vector of n u discrete-time transfer functions which represent the process model, u(t) R n u corresponds to the measured process inputs, G L denotes the disturbance model and e(t) is a zero mean white noise signal with variance σe The j-step ahead predictor for model () can be formulated as (Ljung, 999): where ŷ(t + j t) = F j (q)ĝ L (q)ĝp (q)u(t + j) ( ) + F j (q)ĝ L (q) y(t + j), () j F j (q) = ĝ i q i, (3) i=0 being ĝ i the i th impulse response coefficient of ĜL(q) Then, according to (), the j-step ahead prediction error is given by: ε(t + j t) = y(t + j) ŷ(t + j t) ) = F j (q)ĝ L (y(t (q) + j) ĜP (q)u(t + j) (4) Let J P be an objective function which quantifies the multistep ahead prediction errors from a dataset of length N: J P = N P P (N P ) t=0 P ε (t + j t) (5) j= The model estimated by minimizing J P should provide the best multistep ahead predictions However, note that (5) is nonlinear in the model parameters Then, a suitable optimization algorithm should be employed, with the purpose of minimizing convergence problems In practical applications, any identified model has bias and variance errors associated with the identification algorithm If it is assumed that the true process consists of a deterministic part driven by the manipulated variables and a stochastic part driven by white noise, the quality of the predictor depends on the quality of the deterministic and the stochastic parts of the model Disturbance model estimation in a MRI context After finding the disturbance model ĜL ARX (q) based on a high order ARX model, it is reduced to a final dimension chosen by the user Then, according to the Two-step algorithm proposed in (Gopaluni et al, 004), Ĝ LARX (q) is optimized to Ĝ L (q) Such MRI algorithm is summarized as follows: i Identify a high order ARX model of the system: y(t) = ĜP ARX (q)u(t) + ĜL ARX (q)e(t) (6) ii Reduce the disturbance model estimated in the previous step, creating ĜL red (q) iii Using the process model estimated in step i and the reduced disturbance model estimated in step ii, identify a new optimal disturbance model Ĝ L (q), by minimizing the multistep objective function J P (5) If the optimal disturbance model structure is adopted as: Ĝ L(q) = C(q) D(q) = + c q + + c nc q nc + d q + + d nd q n, (7) d the j th impulse response coefficient of this model, given by: ĝ j = f j (c,, c nc, d,, d nd ), (8) is obtained by polynomial long division of C(q) by D(q) Without loss of generality, ĝ 0 in (3) can be set to Then, defining the auxiliary variables: [ φ j (t, θ j ) υ(t + ),, υ(t + j ), υ(t + j) +y(t + j), ε(t + j t),, ε(t + j n c t), ξ(t + j ),, ξ(t + j n d )] T (9) υ(t + j) y(t + j) G PARX (q)u(t + j) (0) ξ(t + j) F j (q)υ(t + j) () θ j [ĝ j,, ĝ,, c,, c nc, d,, d nd ] T, () using (7), it is possible to rewrite (4) as: ε(t + j t) = y(t + j) φ T j (t, θ j ) θ j (3) Finally, the identification problem can be posed as: 409

3 N P min Θ t= P ε (t + j t, θ j ) (4) j= subject to: ĝ j (θ j ) f j (θ j ) = 0, j =,,, P, where θ j = I j Θ and I j = diag ( ) 0,, 0,,, }{{}}{{} P j n c +n d +j It should be stressed that the formulation described in (9) to (4) is slightly different from the one presented in (Gopaluni et al, 004), in order to deal with generic disturbance model structures 3 STABILITY OF THE DISTURBANCE MODEL From the estimation point of view, the main objective of the MRI methods is to generate accurate multistep ahead predictions Equation () can be considered a one-step ahead prediction when the input and the output signals are filtered by F j (q) Moreover, such filter coefficients as well as the model predictions depend on the estimate disturbance model Ĝ L (q) The prefiltering is an important artifice in the estimation phase It affects the bias distribution of the resulting model and may also remove disturbances of high or low frequencies, that one does not want to include in the modeling (Ljung, 999) As the inverse of (7) acts as a filter in (4), Ĝ L must be stable In this work, it is proposed that this requirement is imposed as an additional constraint to the parameter estimation problem (4) A straightforward manner to include the stability requirement is through the Jury criterion (Ogata, 995) The Jury criterion is the discrete-time analogous of the Routh-Hurwitz one It allows determining whether all roots of a polynomial are inside the unit circle, without the need to explicitly calculate its roots Considering C(q) in (7), the Jury table is constructed as (note that c 0 ): where: c 0 c c m c m c m c 0 0 m m c () 0 c () 0 = c 0 c m c 0 c m = c c m c 0 c m i = c i c m c 0 c m i c (k) i = c (k ) i c(k ) m k+ c (k ) 0 c (k ) m i k+ (5) for i = 0,, m k and k = 0,, m In accordance with the Jury table, if c 0 > 0, then all the roots will be inside the unit circle if and only if all terms of the first column of the odd lines are positive If any term in the first column of the odd lines is null, the number of roots outside the unit circle is equal to the number of negative elements Then, it is possible to formulate the Jury criterion as a set of constraints to the optimization problem: N P P min ε (t + j t, θ j ) (6) Θ t= j= subject to: ĝ j (θ j ) f j (θ j ) = 0, s k (θ j ) > 0, where s k = c (k) 0 j =,,, P k = 0,, m Although the additional constraints in (6) are nonlinear, if one restricts the search space to disturbance models which yield stable predictors, it is supposed to attenuate convergence problems to local minima The constrained nonlinear least-squares problem (6) can be treated as a Sequential Quadratic Program (SQP) The idea of SQP is to model this problem at the current point Θ k by a quadratic subproblem and to use the solution of this subproblem to find the new point Θ k+ SQP corresponds to the application of the Newton method to the Karush Kuhn Tucker (KKT) optimality conditions (Nocedal and Wright, 999) 3 Example One of the most common disturbance models used to identify industrial processes is the integrator Suppose that one wants to identify a system using the disturbance model with the form: Ĝ L (q) = + c q q (7) If the proposed algorithm is applied to (7), the impulse response coefficients of ĜL(q) have the structure: ĝ j = + c j =,, P (8) and according to the Jury criterion: (c ) > 0 (9) Then, the optimization problem (6) can be solved subject to constraints (8) and (9) and the parameter vector Θ is given by: Θ = [ ĝ P,, ĝ, ĝ,, c, ] 4 REDUCED ORDER MODEL ESTIMATION Although the high order ARX models proposed in this paper provide (practically) unbiased estimates in closedloop scenarios, the high number of parameters leads to an increase in their variance Moreover, the majority of commercial MPC uses reduced order models, essentially 40

4 due to the fact that high order models have many states and large state matrices may introduce numerical errors in the calculation of the predictions A multistep prediction error method (MPEM) based on reduced order models was developed by Huang and Wang (999) Instead of directly minimizing the objective function J P, in this algorithm the input-output dataset is filtered by: P L(e jω ) = F j (e jω ), (0) j= then the model parameters are estimated using standard one-step prediction error methods Such an approach extends to a multistep context the fact that the difference between one and j-step ahead predictions is the filter F j (q), defined in (3) Spectral factorization algorithms (eg Ježek and Kučera (985)) are frequently used, in order to obtain the prefilter L(q) from (0) Nevertheless, F j (q) depends on G L (q), which is unknown In (Shook et al, 99), G L (q) and L(q) are estimated using an iterative procedure Huang and Wang (999) use a one-step PEM to get an estimate of G L (q) In this work, the authors propose that the disturbance model estimated by solving the optimization problem (6) is used to create the prefilter This strategy aims at deriving the prefilter L(q) through an estimate of the disturbance model obtained from a multistep predictor As claimed in (Gopaluni et al, 004), when the model structure mismatches that of the actual system, the objective function (5) yields better models for MPC applications than other MRI approaches 4 The Multistep Prediction Error Method Consider a general parametric model structure: n u B i (q) Ĝ P (q) = A i (q) u i(t) () i= where u i (t) denotes the i th input and B i (q) = b i q + + b nb iq n b () A i (q) = + a i q + + a naiq na (3) Once L(q) is calculated from (0), (5) can be rewritten as (Huang and Wang, 999): J P = L(q)Ĝ L (y(t) (q) (q)u(t)) ĜP (4) t Then, defining the variables: y f (t) L(q)Ĝ L (q)y(t) (5) u f (t) L(q)Ĝ L (q)u(t) (6) Finally, the parameters of the process model () are obtained from: Ĝ P = min ( y f (t) ĜP (q)u f (t)) (7) t Note that (4) may be considered an output-error criterion The proposed model relevant identification algorithm is summarized in the flowchart of Fig Fig Flowchart of the MRI algorithm C-EMPEM It is worth mentioning that to reduce the order of the model, it is necessary to know what the suitable orders n a and n b are A straight manner to obtain this information is through the singular Hankel values, which provide a measure of the energy for each state in a system and they are the basis for many algorithms of model reduction, in which high energy states are retained while low ones are discarded However in practice, the user chooses the desired order for the system, which normally does not exceed the third order According to Fig, the authors suggest the name C- EMPEM (Closed-loop Enhanced Multi-step Prediction Error Method) for the new method proposed 5 AN APPLICATION EXAMPLE To test the proposed method, consider the following thirdorder process proposed by Clarke et al (987) and used in (Huang and Wang, 999): G P (q ) = q + 003q q 3 903q + 54q 058q 3 (8) with a random-walk disturbance transfer function: G L (q ) = q (9) The simulation is carried out in closed-loop and the controller is a QDMC (Quadratic Dynamic Matrix Control), with prediction horizon P = 50 and control horizon m c = 5 The sampling time is unitary It was collected 00 points The system is excited by a Pseudo Random Binary Sequence (PRBS) shown in Fig The signal has two different frequencies in order to excite the process in slow and faster modes The disturbance is a white noise signal and the signal-tonoise ratio in variance is SNR = 3 Note that in order to provide more realism to the disturbance, the SNR is very low, thus representing a process difficult to identify The structure of the proposed process model for this plant is (Huang and Wang, 999): Ĝ P (q) = b q 3 + b q 4 + a q (30) The closed-loop identification was performed using the direct approach method This method has the following 4

5 3 5 Huang and Wang (999) BJ-PEM OE-PEM Shook et al (999) Gopaluni et al (004) C-EMPEM JP P Fig Samples PRBS signal used to identify the system advantages, considering the two frequencies of the input signal and that the signal-to-noise ratio is low (Forssell and Ljung, 998): The bias of Ĝ P is small The estimate of the disturbance model is good 5 Results In the first step of the method, the system was identified by a high order ARX Next, an order reduction of the Ĝ LARX model is made to obtain a model with the required dimension Ĝ L The fact of using a good preliminary model of G L provides an adequate starting condition for the optimization problem (6) Two disturbance models with different dimensions were estimated for a prediction horizon P = 50 A final process model (Ĝ P i ) was identified in the MPEM stage (see Fig ), based on the structure OE (Output Error) The obtained models are presented in Table Table Results of the identification i Ĝ P i Ĝ L i Fig 3 Comparison of the methods using the objective function defined in (5) for P =,,, 50 JP 75 x Huang and Wang (999) BJ PEM C EMPEM P Fig 4 Comparison of the best methods using the objective function defined in (5) for P =,,, Set-point Huang and Wang (999) Gopaluni et al (004) C-EMPEM Shook et al (999) BJ-PEM OE-PEM 0037q q 4 006q 0860q q 0047q q q q 0968q 00976q The complete model was validated using a different dataset The summation of the j-step prediction errors, j =,, P for model i = is presented in Fig 3 The results achieved by the proposed algorithm are compared to the prediction errors provided by two PEM methods (Box-Jenkins and OE) and three MRI algorithms (Shook et al, 99; Huang and Wang, 999; Gopaluni et al, 004) Despite not being designed for long range predictions, the PEM structure is largely employed in commercial MPCs The iterative algorithm developed by Shook et al (99) presented the worst results for higher prediction horizons The method proposed by Gopaluni et al (004) as well as the model OE-PEM resulted in bad predictions for small P Fig 5 Step response of the estimated models but improved their performance for higher prediction horizons The model BJ-PEM had a good performance for all prediction horizons, but was inferior to the new algorithm (C-EMPEM) and to the one proposed by Huang and Wang (999), that showed similar results, with a slight advantage of the C-EMPEM, as observed in Fig 4 Furthermore, Fig 3 portrays the improvement of the prediction errors obtained by identifying a low order process model, through 4

6 Ĝ L (q), when compared to the results produced by the prediction based on a high order FIR model as proposed by Gopaluni et al (004) In Fig 5 the step response of the different models is shown As can be observed, the results are similar to the ones obtained in Fig 3, in which the best methods were those proposed in (Huang and Wang, 999) and the C-EMPEM To validate the results obtained in Fig 3, Fig 4 and Fig 5, a Monte Carlo test was implemented, changing the variance of the white noise without varying the signal-tonoise ratio of the test (SNR = 3) In Fig 6 and Fig 7 the results of 00 simulations, using the six aforementioned algorithms, are shown The graphs display the number of times that each presented model has the lowest summation of prediction horizons The proposed C-EMPEM method was the best in the majority of the achievements for both models in Table Note that the BJ-PEM model, despite being a structure based on a one-step prediction method, had a good performance for long range predictions C-EMPEM Huang and Wang (999) BJ-PEM Gopaluni et al (004) 5% Fig 6 Monte Carlo simulations for model in Table 5% 67% 0% 84% C-EMPEM Huang and Wang (999) BJ-PEM Gopaluni et al (004) Fig 7 Monte Carlo simulations for model in Table 3% % 6 CONCLUSIONS A new MRI method was proposed in this paper It consists of five steps and it is formed by the enhancement and fusion of two known MRI methods, based on (Gopaluni et al, 004) and (Huang and Wang, 999) To ensure the stability of the disturbance model and to reduce the bias of the preliminary process model in closedloop, the algorithm of Gopaluni et al (004) was revised, including new restrictions to the original formulation and substituting the original high order FIR by an ARX structure Then, with the optimal MPEM disturbance 5% model found, the system input and output are prefiltered by the filter L(q)Ĝ L (q) Thus the MPEM proposed in (Huang and Wang, 999) was enhanced for the proposed method The employed multistep validation criterion showed that the j-step ahead predictions generated by the C-EMPEM method were better than the other MRI methods cited in the paper, as observed in Fig 6 and Fig 7 ACKNOWLEDGEMENTS The authors are grateful to PETROBRAS for its support to this research REFERENCES Clarke, D, Mohtadi, C, and Tuffs, P (987) Generalized predictive control - parts and Automatica, 3(), Forssell, U and Ljung, L (998) Closed-loop identification revisited - update version Technical report, Department of Electrical Engineering Linkping University Gopaluni, R, Patwardhan, R, and Shah, S (00) Bias distribution in MPC relevant identification In 5th Triennial World Congress IFAC Gopaluni, R, Patwardhan, R, and Shah, S (004) MPC relevant identification-tuning the noise model Journal of Process Control, 4, Huang, B, Malhotra, A, and Tamayo, E (003) Model predictive control relevant identification and validation Chemical Engineering Science, 58, Huang, B and Shah, S (997) Closed-loop identification: a two step approach Journal of Process Control, 7(6), Huang, B and Wang, Z (999) The role of data prefiltering for integrated identification and model predictive control In HF Chen, DZ Cheng, and JF Zhang (eds), Proceedings of the 4th World Congress of IFAC Ježek, J and Kučera (985) Efficient algorithm for matrix spectral factorization Automatica, (6), Laurí, D, Martínez, M, Salcedo, J, and Sanchis, J (00) PLS-based model predictive control relevant identification: PLS-PH algorithm Chemometrics and Intelligent Laboratory Systems, 00, 8 6 Ljung, L (985) Asymptotic variance expressions for identified black-box transfer function models IEEE Transactions on Automatic Control, 30(9), Ljung, L (999) System Identification: Theory for the user Prentice Hall Nocedal, JN and Wright, SJ (999) Numerical Optimization Springer Ogata, K (995) Discrete-Time Control Systems Prentice Hall, nd edition Rossiter, J and Kouvaritakis, B (00) Modelling and implicit modelling for predictive control International Journal of Control, 74(), Shook, DS, Mohtadi, C, and Shah, SL (99) Identification for long-range predictive control IEE Proceedings-D, 38(), Shook, D, Mohtadi, C, and Shah, S (99) A controlrelevant identification strategy for GPC IEEE Transactions on Automatic Control, 37(7),