Control Performance Monitoring of State-Dependent Nonlinear Processes

Transcription

1 Control Performance Monitoring of State-Dependent Nonlinear Processes Lis F. Recalde*, Hong Ye Wind Energy and Control Centre, Department of Electronic and Electrical Engineering, University of Strathclyde, Glasgow, UK G11RD ( Abstract: This paper presents a novel approach to monitor control performance of nonlinear processes that can be described as state-dependent models (SDMs). A discrete Kalman filter (KF) is established to estimate the SDM parameters. A covariance control formlation is introdced to split the system closed-loop variance/covariance into two terms, one term to accont for the minimm expected qadratic loss bond (eqivalent to the minimm variance performance bond bt in state space formlation), and another to accont for performance deviations from the minimm variance bond. Simlation stdies have been condcted on several nonlinear process systems inclding a cold rolling mill model with roll eccentricity and a steel making system with real time oxyfel slab reheating frnace control data. The case stdy reslts demonstrate the comptational efficiency of the proposed strategy in real time monitoring and control of systems with fast, nonlinear and time-varying dynamics. Keywords: State-dependent model (SDM), parameter estimation, Kalman filter (KF), control performance monitoring, covariance control, steel indstry 1. INTRODUCTION Control performance assessment/monitoring (CPA/CPM) provides atomated monitoring, evalation and diagnosis of possible nder-performing control systems. The assessment is normally condcted on models fitted to closedloop operational data throgh the estimation of a minimm variance performance bond (MVPB) when Gassian stochastic distrbance signals are considered. A comprehensive review of CPA technologies and their impacts in indstry can be fond in (Jelali, 2006). Performance assessment of nonlinear systems remains an important and challenging topic which has not been flly investigated. One difficlty is that estimation of nonlinear model parameters from operating data is comptationally demanding. Frthermore, to inclde statistical properties, nonlinear processes are estimated as Volterra time series. The se of Volterra time series, albeit general, can also be intractable, limiting the assessment to only special classes of nonlinear models. In the work of (Harris and Y, 2007) and (Y et al., 2010), for example, the performance assessment is restricted to nonlinear models with additive linear or partially nonlinear distrbances and with explicit inpt/otpt representations. A more general strategy is reqired for performance monitoring of nonlinear systems. State-dependent models (SDM) are general nonlinear models with finite dimensionality, amenable to statistical analysis and relatively easy to be fitted to data with no prior assmptions abot the process nonlinearity (Priestly, 1988). This motivates or se of SDM to allow the closed- Special Thanks to The Eropean Commission, Research Fnd for Coal and Steel, Project acronym: Cognitive Control, Proposal No: RFS-PR loop performance information being inclded in the states covariance for general nonlinear systems. The extraction of MVPB from the reslting SDM model is therefore made by means of covariance control theory. The soltion to the covariance control problem provides a controller which assigns a given state covariance to the closed-loop system (Skelton et al., 1998). More specifically, the covariance control algorithm can provide the MVPB within the actal closed-loop performance. In this work, state-dependent modelling from operational data for CPM is carried ot throgh a discrete KF since the SDM can be formlated as trncated Volterra series with state-dependent coefficients. This SDM representation is locally linear. The main advantage of sing KF is that its innovations error covariance (the covariance of the difference between the available observations and their optimal estimates) contains the overall otpts variance. A performance index can therefore be formlated based on the difference between the otpts variance/covariance and the extracted MVPB. The developed index has two advantages against linear indexes, it can accommodate the time varying behavior encontered in process indstries, and can qantify the effects of nonlinearities on the systems performance. The proposed algorithm is examined with application stdies to steel processing lines and other nonlinear processes. Steel processing lines are highly nonlinear, fast speed and setpoint-dependent processes where the final prodct depends on cstomer-order specifications. The remaining paper is organised as follows. The SDM description and its parameter estimation throgh KF are presented in Section 2.1. Calclation of the MVPB sing covariance control and the derivation of the performance

2 index are presented in Section 3. In Section 4, simlations stdies are condcted on several indstrial examples. Conclsions and ftre work are discssed in Section DEVELOPMENT OF STATE-DEPENDENT MODELS 2.1 Model Description In a discrete time setting, fitting a time series into a nonlinear model in the form of SDM can be described by the relationship between the observation, y t, and the past history of the series as follows: y t = f (y t 1,, y t m, e t 1,, e t n ) + e t (1) The nonlinear fnction f describes the information of y t contained within its past history {y t i } m i=1 of order m, and the innovations process {e t j } n j=1 of order n. Any past observation, y t i, is, by itself a fnction of the innovation seqence {e t i j } n j=1, ths giving the notion of the state vector, i.e. at time t 1: x t 1 = [e t n,, e t 1, y t m,, y t 1 ] T (2) Here the state vector is not in the context of standard linear state space formlation, since it does not provide a minimal realisation, bt rather being sed as a set of qantities that contain all the necessary information abot the process (Priestly, 1988). If f is analytical, it can be expressed as a Volterra first-order expansion as follows: y t = m a i (x t 1 ) y t i + e t + i=1 +µ (x t 1 ) n g j (x t 1 ) e t j (3) j=1 where µ (x t 1 ) is the mean vale of the state vector x t 1. The coefficients a i and g j are first order derivatives of f (x t 1 ) with respect to x t 1. To estimate the SDM parameters throgh a seqential algorithm sch as KF, the otpt observation y t can be re-written as follows: y t = X t 1 Γ t + e t (4) where: X t 1 = [ 1, x T ] t 1 = [ Xt 1, 1 Xt 1, 2, Xt 1 m+n+1 ], Γ t = [µ, g n,, g 1, a m,, a 1 ] T For simplicity, the state vector notation x t 1 has been omitted in vector Γ t and the vector X t 1 is an extended state vector that incldes an element for the mean vale µ. The innovations term, e t, is expected to be zero mean white noise, with innovations variance Σ e, sch that the KF provides the optimal estimates. With this formlation, the entries of the past observations and the innovations appear as locally linear. In the presence of delays, the innovations shold be regarded as independent from the feedback effects. This is a necessary condition for the formlation of the feedback invariance for systems with delays (Harris and Y, 2007). To obtain the MVPB, a state space approach is sed. In state space form, the minimm expected qadratic loss is eqal to the minimm variance from its eqivalent polynomial form. Under the minimm expected qadratic loss, the expectation of otpt, E { } yt 2, is eqal to the expectation of the innovation term, E { } { e 2 t, or to the seqence, E, when a controllable inpt is d 1 } delayed k=1 e2 k with d time instances. From (4), we have: E { yt 2 } { = E Xt 1 Γ t Γ T t X T } { } t 1 + E e 2 t The presence of the expectation term in (5) from the past observation data sggests that a non-minimm expected qadratic loss feedback controller has been sed, which is actally more realistic from real application point of view. In practice, controllers achieving the minimm variance or the minimm qadratic loss may be difficlt to implement de to physical and energy efficiency constrains. Frthermore, a well-tned controller may be nder-performing in real processes, ths leading to non-minimm design performance. 2.2 Parameter Estimation of SDM The model parameters can be identified by fitting the closed-loop operating data to the SDM. In (Priestly, 1988), the evoltion of the SDM parameters is approximated to linear fnctions of the difference between the elements of the extended state vector X t and the elements of its predictor ˆX t : a i,t = a i,t 1 + g j,t = g j,t 1 + µ t = µ t 1 + X 1 t λ t X i+n+1 t ɛ i,t (5) X j+1 t ξ j,t (6) with X t = X t ˆX t and X 1 t = 1 so that the mean vale can also be pdated. ɛ, ξ and λ can be random locally to preserve nonlinearity and add non-stationarity to the model, that is: Γ t = Γ t 1 + X T t β t (7) β t = β t 1 + V t where β t = [λ t, ɛ 1,t,, ɛ m,t, ξ 1,t,, ξ n,t ] and V t is a seqence of compatible dimensions that contains independent random-valed elements with mltivariate normal distribtion. For simplicity, it is assmed that V t N (0, Σ V ) with noise covariance Σ V. Combining both eqations in (7), the estimation of the SDM parameters can be developed based on the following model: where: Φ t = F t Φ t 1 + W t (8) y t = H t Φ t + e t

3 X t 1 F t 1 = I (m+n+1) 2 diag({ X t} n+1 2 ) 0 (m+n+1) 2 I (m+n+1) 2 W t = [ [ ] ] T (m+n+1), V t, ΣW = 0 Σ V Φ t = [Γ t, β t ] T, H t = [ ] X t, 0 (m+n+1) and I (m+n+1) 2, 0 (m+n+1) 2 diag({ X t} m+n+1 n+2 ) are identity and zero matrices. Given the observations y t,, y t m and the innovations e t 1,, e t n, the best estimate of the SDM parameters vector Φ t can be generated by the KF algorithm as follows: ˆΦ t t 1 = F t ˆΦ t 1 t 1 P t t 1 = F t P t 1 t 1 F T t + Σ W ˆΦ t t = ˆΦ t t 1 + K t e t (9) K t = P t 1 t 1 H T [ t Ht P t 1 t 1 H T ] t + Σ e P t t = [I K t H t ] P t t 1 [I K t H t ] T + K t Σ e K T t where K t is the KF gain, Σ e is the innovations noise variance, Σ W is the extended noise covariance matrix for V t and P t t 1 is the parameters error covariance. It can be arged that the extended Kalman filter (EKF) is a rather simpler formlation, in the sense that both approaches, EKF and SDM estimation with KF, are eqally locally linear. Nonetheless, only in the above formlation, the model parameters are fnctions of the state vector and conseqently nonlinear dring the estimation. The accracy of the SDM parameters to fit the model otpt to the nonlinearity can be determined by the relative magnitde of Σ V. For instance, to estimate y t based on a short past history (m is a small nmber), a small vale of Σ V will make X t locally linear. On the contrary, for estimations based on long past history (m is a large nmber) sch vales of Σ V wold be ndesirable, ths leading to the selection of a higher vale of Σ V. 3. MINIMUM PERFORMANCE BOUND Consider that the otpt observation y t from (4) has a non-minimm open-loop state space realisation given by: with: X t+1 = A t X t + B t t + e t+1 (10) y t = CX t A t = µ g op n 1 gop 0 a op m 1 aop 0 B t = [ b m+n b m+n 1 b 0 ] T C = [ ] e t+1 = [ 0 e t+1 ] T t is the controllable inpt, A t and B t are matrix containing coefficients of X t and t, respectively. The sperscript op stands for open loop. The minimm expected qadratic loss is identical to the minimm variance only if an optimal estimator is applied over a state space model with noise-free otpt measrement signal as in (10) (Warwick, 1990). This assmption is valid for controller design, however, the problem nder stdy is model parameters identification. The lack of an optimal state estimator is circmvented by the fact that the optimality of the KF is added to the model state parameters and all the state vector elements are observable. Under these conditions, the closed-loop system shold be able to achieve the minimm variance throgh the following controller: t = J MV t X t (11) where J MV t is the static state minimm variance feedback gain (the sperscript MV stands for minimm variance): J MV t = (ˆB T t C T CˆB t ) 1 ˆB T t C T C t  t and ˆB t are optimal parameter estimates calclated by the KF. The controller in (11) also compensates the past innovations since X t is sed for feedback. It can now be stated that poor control performance from a state space approach occrs mainly when the system is not flly reachable in a finite period of time. It cold be arged that the system is not flly controllable bt it cold also be the case that the system was originally controllable and became nder-performing over time, ths an nreachable system is a more general assmption for CPA. 3.1 Covariance Control Formlation For an nder-performing system, the control objective now becomes finding a controller based on the actal closedloop variance that contains a minimm variance bond given by the controller in (11). This control objective is feasible by means of covariance control that ses feedback control to achieve the desired system performance (Skelton et al., 1998). Assigning an exact performance is theoretically possible bt rather nrealistic for CPA, alternatively the control problem is oriented towards assigning a desired performance bond. Sppose that the state covariance that characterises the actal closed-loop performance is specified in terms of a performance bond, that is: lim E { y 2 } t < CΠX C T (12) t where Π X is a given positive-definite matrix that stabilises the closed-loop system nder controllability assmptions. The closed-loop system covariance is bonded by: ) T Π X > ( ˆBJ Π X ( ˆBJ) + Σe (13) and J is the state feedback gain that achieves the actal covariance. From (13), J can be developed to be: J = (ˆB T QˆB) 1 ˆB T Q + (ˆB T QˆB ) 1 2 LS 1 2 (14) where L is an arbitrary matrix sch that L < 1 and Q = (Π X Σ e ) 1 S = Π 1 X ÂT Q +  T QˆB (ˆB T QˆB) 1 ˆB T QÂ

4 Q and S are positive-definite matrices, J is a general soltion to (12) and can generate reqired covariance bond (Skelton et al., 1998). When lim t E { y 2 t } = CΣe C T, the second term in (14) becomes zero and J = J MV. Both terms from the state feedback gain can ths be re-written as: J = J MV + J (15) For CPM, J represents the change in the system s closed-loop variance/covariance over time. Using the state feedback controller from (15) into (10), the following closed-loop state space realisation is obtained: X t+1 = B J X t + e t (16) y t = CX t since A t = BJ MV t. Writing α t = B J, the model in (16) is the non-minimm closed-loop state space realisation of (4) and can be estimated from closed-loop operating data. A by-prodct of sing covariance control for CPM is that for strctred controllers the derivation of the feedback gain can provide a control re-tning methodology. That is, the closed-loop state covariance can be partitioned as follows: [ ] Πp Π pc Π X = Π T pc Π c where Π p, Π c and Π pc are the plant state covariance, dynamic controller state covariance, and the covariance between plant and controller states, respectively. This approach reqires measrements of the system controllable inpts and otpts. The details of this methodology and the reslts on PI/PID control re-tning will be presented in a ftre paper. 3.2 Performance Index A straightforward se of the Harris index can be considered for the closed-loop system performance as follows: = ΠMV Π X (17) where Π MV is the determinant of the system minimm covariance matrix Π MV. The index lies within [0, 1]. In practice, the control performance can be classified as < 0.5(poor)/ [0.5, 0.8] (good)/ > 0.8(optimal), and will depend on the applications at hand (Jelali, 2007). For CPM, the variance of the innovations may deviate from the white noise variance. It is therefore practical to take the determinant of the innovations as the metric and also to make the change in variance/covariance over time independent of the innovations dring the estimation of the SDM. That is: X 1 t X n+2 y t = α 1,t. + α 2,t ( J ) t. (18) Xt n+1 Xt m+n+1 where α 1,t = [µ, g n,, g 1 ], α 2,t ( J ) = [ a m,, a 1 ] and only α 2,t ( J ) is dependent on ( J ). This approach is cmbersome for feedback/feedforward controllers since some terms of the innovations can inclde the change in variance from the feedforward action, bt in no sense restrictive for feedback controllers since the innovation terms only inclde the minimm achievable variance. When time delays are present, the feedback invariance can be extracted by setting n > d. The selection of m and n vales is related to the smoothness of the system nonlinearity as mentioned in Section 2.2. In practice, these vales can be set to be m = n for feedback control, or m < n for feedback/feedforward control. 4. SIMULATION STUDIES The developed CPM algorithm is applied to several process scenarios inclding a simple model with nonlinear actator, a cold rolling mill model with roll eccentricity, and a steel making system with real time oxyfel frnace control data. The method described in (Harris and Y, 2007) is sed for performance comparison. 4.1 Actator Nonlinearity A typical sorce of nonlinear oscillations in process control is valve stiction, which is the friction a valve needs to overcome before the actator changes its position. A simple feedback/feedforward control system with He s model (Jelali and Hang, 2010) of valve stiction is simlated in Matlab/Simlink. The control system is given by (Desborogh and Harris, 1993): y t = t ( ) 0.2q 1 1 q 1 ω 0,t + q q 1 1 q 1 ω 1,t where y t is the process otpt, ω 0,t and ω 1,t are the nknown and known zero mean, Gassian distrbance terms, respectively. q 1 is the backward shift operator. Time delays of 3s and 5s were sed for the controllable inpt and the known distrbance, respectively. The feedback/feedforward controllers are given as: K ff t = 1 q 1 ω 1,t + K fb 1 q 1 (y sp y t ) with feedback gain K fb = and feedforward gain K ff = y sp is the setpoint signal for the otpt. The flow chart for valve stiction in He s model is given in Fig. 1. Vales of 0.35 and 0.25 were chosen for f d and f s, respectively, to generate a strong stiction. The controllers are properly tned for setpoint tracking bt not for distrbance rejection, casing the system otpt with and withot valve stiction to be very noisy, see Fig 2. The performance of the system withot valve stiction is very close to the lower bond for good performance, see Fig. 3. Valve stiction makes the index oscillatory at time intervals when the He s model is active. Sch oscillations wold have not been picked p by a linear, noniterative method even thogh they make the index vale go below 0.5. The degradation in performance index is not significant bt infers an oscillatory distrbance. The mean vales of the calclated performance indexes are comparable to the vales obtained of, and , sing the comparison method for sitations withot and with valve stiction, respectively. The vale of m was set to be 8 whereas the vale of n was set to be 10 (m < n) to make the KF more sensitive to stiction oscillations.

5 Controller otpt 10-3 t () 5.5 yes [ () t - ( -1) ] cm = + t r cm > f s no strip thickness [m] () t =t ()- sign( cm - fs) r =sign( cm - f s )f d v f d () t = ( t-1 ) v r v =cm h sp 1 h sp 2 h sp 3 h1 h2 h3 h est 1 h est 2 h est Fig. 4. Estimated otpt strip thickness vs measred strip thickness for stands 1-3 Fig. 1. He s valve stiction model flow chart h 1 h 2 h 3 y 4 original process process with valve stiction performance index Fig. 2. Feedback/feedforward control system with and withot valve stiction performance index no stiction stiction Fig. 3. CPM for feedback/feedforward control system with and withot valve stiction 4.2 Cold Rolling Mill with Roll Eccentricity A cold rolling mill system is simlated in Matlab sing the model from (Kodati, 2010). The model consists of 3 stand rolling mills. A mlti-loop architectre is implemented to control the process sing a combination of PID and high order controllers. Strip thickness, mass flow, looper angle (inter-stand) and speed control are implemented in every stage. Strip thickness at the exit of every stand is assessed. The time delay to the thickness sensor is varying bt measrable. An initial calibration of setpoints is sed to represent a new slab entering the rolling mill. Controllers Fig. 5. CPM of otpt strip thickness for stands 1-3 are properly tned bt not set to compensate roll eccentricity which is added to each stand with freqencies of 4.679Hz and 4.997Hz for lower and pper rolls, respectively. Roll eccentricity refers to any conditions cased by axial deviations between the roll barrel and the roll necks that reslts in irreglarities in the mill rolls. The estimated strip thickness (h est ), measred strip thickness (h) and thickness setpoints (h sp ) of the 3 stands are compared (Fig. 4). The estimation process accrately accommodates roll eccentricity variations in all the stands with vales of m and n both set to be 7 since the delay is varying bt always smaller than 3. The performance index from each stand deteriorates when the strip moves forward along the processing line de to the accmlated roll eccentricity from the previos stand (Fig. 5). As expected, changes in setpoints are picked as peaks in the corresponding stand (de to controller transient), and as drops in other stands since the accmlated eccentricity is passed. The comparison method calclates performance vales of to be , and , respectively, for the 3 stands. The process is performing reasonably well since all performance vales are within [0.5, 0.8]. 4.3 CPM wiht Real Oxyfel Frnace Data Real oxyfel frnace data provided by Swerea MEFOS (One of the partners of the Cognitive Control Project completed in or previos work (Recalde, Katebi, and Taro, 2013; Recalde, Katebi, and Ye, 2014) was employed in this CPM stdy. The frnace is sed to reheat the slabs coming from the rolling mill and consists of 2 parts being

6 temperatre [ C] T sp z3 T sp z4 T z3 T z4 T est z3 T est z samples Fig. 6. Termocoples temperatre vales for zone 3 and 4. Real frnace data provided by Swerea MEFOS z3 z samples Fig. 7. CPM of otpt temperatres for zone 3 and 4. Real frnace data provided by Swerea MEFOS divided into 6 zones. The steel strip travels across the zones via a spporting roll. A total of 35 brners, 25 thermocoples and 5 pyrometers are sed to control the frnace and steel strip temperatre. Model mismatch is low between the estimated temperatres (T est ) and the measred temperatres (T ) for zones 3 and 4, see Fig. 6. The vales of m and n were set to be 5. The temperatre setpoints (T sp ) for these zones are also added for illstration and comparison. There are sdden drops in temperatre in both zones when the slabs leave the zones and do not affect performance index. The performance vales are very close to the poor performance bond, see Fig. 7. The index in zone 3 is worse than the one in zone 4 bt does not present any oscillatory behavior that may come from a nonlinearity. Withot more information abot the process, poorly performing controller/s can be the sorce of the degraded performance. Frther assessment carried ot by MEFOS revealed that the controller in zone 3 was indeed badly tned and its effects were passed on to zone 4. The comparison method calclates the performance vales of to be and for zones 3 and 4, respectively. parameters can be carried ot sing KF or any other seqential identification algorithms. The KF is sed in this work to assre the optimal parameter estimation in the SDM. The flexibility of sing covariance control to achieve the desired performance specifications is the basis of the proposed CPM algorithm. A by-prodct of this covariance control application is the possibility to develop a control re-tning mechanism for strctred controllers sch as PI/PID control, widely sed in indstries. The sccessfl implementation of this strategy spports the proposed CPM approach to be applied in assessment and diagnosis for general nonlinear dynamic systems. REFERENCES Desborogh, L. and Harris, T. (1993). Performance assessment measres for nivariate feedforward/feedback control. Canadian Jornal of Chemical Engineering, 71, Harris, T. and Y, W. (2007). Controller assessment for a class of non-linear systems. Jornal of Process Contol, 17, Jelali, M. (2006). An overview of control performance assessment technology and indstrial applications. Control Engineering Practice, 14, Jelali, M. (2007). Performance assessment of control systems in rolling mills application to strip thickness and flatness control. Jornal Process of Control, 14, Jelali, M. and Hang, B. (2010). Detection and Diagnosis of Stiction in Control loops. Springer, London. Kodati, P. (2010). Modelling and control of mlti stage rolling mill process. MATLAB Central File Exchange. Priestly, M. (1988). Non-linear and Non-stationary Time Series Analysis. Academic Press INC., San Diego, CA. Recalde, L., Katebi, R., and Taro, H. (2013). PID based control performance assessment for rolling mills: A mltiscale PCA approach. IEEE International Conference of Control Applications. Recalde, L., Katebi, R., and Ye, H. (2014). Seqential control performance diagnosis of steel processes. IFAC World Congress Skelton, R., Iwasaki, T., and Grigoriadis, K. (1998). A Unified Algebraic Aprroach to Linear Control Design. Taylor & Francis, Great Britain. Warwick, K. (1990). Relationship between Åström control and the Kalman linear reglator - Caines revisited. Optimal Control Applications and Methods, 11, Y, W., Wilson, D., and Yong, B. (2010). Control performance assessment for nonlinear systems. Jornal of Process Control, CONCLUSION The CPM algorithm proposed in this work is capable of assessing nonlinear systems that can be fitted to a SDM and accommodating process variations in the state vector. It can track control performance changes over time and is comptationally efficient. The identification of the SDM