An Integrated Frequent RTO and Adaptive Nonlinear MPC Scheme

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Process Systems Engineering

An Integrated Frequent RTO and Adaptive Nonlinear MPC Scheme based on Simultaneous Bayesian State and Parameter Estimation Jayaram Valluru, and Sachin C. Patwardhan Ind. Eng. Chem. Res., Just Accepted Manuscript • DOI: 10.1021/acs.iecr.8b05327 • Publication Date (Web): 29 Jan 2019 Downloaded from http://pubs.acs.org on February 4, 2019

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Industrial & Engineering Chemistry Research

An Integrated Frequent RTO and Adaptive Nonlinear MPC Scheme based on Simultaneous Bayesian State and Parameter Estimation Jayaram Valluru, Sachin C. Patwardhan* Department of Chemical Engineering, Indian Institutue of Technology Bombay, Mumbai, 400076, India. (*) Corrresponding author, Email: [email protected] Abstract Changes in process parameters and measured/unmeasured disturbances shift the optimal point at which the economic bene…ts are maximized. Combinations of real time optimization (RTO) techniques, that periodically determine economically optimum operating point using steady state optimization, and model predictive control (MPC) have been widely employed in the process industry for operating process plants optimally in the face of drifting disturbances and/or parameters. Due to long wait time between two successive RTO invocations and model inconsistency between the RTO and MPC layers, the conventional RTO schemes can end up operating the plant sub-optimally if the process parameters/unmeasured disturbances change signi…cantly during the wait time. Recently proposed frequent RTO approaches attempt to address this di¢culty by increasing the frequency of RTO invocation. On-line update of the steady state model employed by RTO layer using dynamic operating data is a major concern in implementation frequent RTO. Use of dynamic model based state and parameter estimation for maintenance of the model used in frequent RTO has not found much attention in the literature. In this work, it is proposed to develop a novel integrated frequent RTO and adaptive nonlinear MPC (NMPC) approach for operating a unit operation in a economically optimal manner. A dynamic mechanistic model based simultaneous state and parameter estimation scheme is used as a common link between the RTO and the NMPC components. Estimates of the drifting unmeasured disturbances / parameters generated by the state and parameter estimator are used to update the steady state model used for frequent RTO and the dynamic model used for predictions in NMPC. Use of a single model for carrying out RTO as well as control tasks eliminates di¢culties that arise due to mismatch between models used for RTO and control. E¢cacy of the proposed integrated optimizing control scheme is demonstrated by conducting simulation studies on benchmark systems from the literature. Analysis of simulation results reveals that the proposed integrated online optimizing control scheme maintains these systems at their respective economically optimal operating points in the presence of drifting disturbances/parameters. Keywords: Real-time Optimization, Adaptive NMPC, Bayesian State and Parameter Estimators.

1

Introduction

Due to increase in global competition and tightening of product quality requirements, it is necessary to operate a process plant close to economically optimal operating conditions while satisfying safety and product quality constraints.1 In general, changes in process parameters and measured/unmeasured disturbances, 1

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such as ‡uctuations in feed ‡ow rates, variations in the feed composition, deactivation of catalysts over a period of time, change in overall heat transfer coe¢cients due to fouling in heat exchangers, and, changes in market conditions, shift the optimal point at which the economic bene…ts are maximized.2 Use of online optimization or Real time optimization (RTO) techniques, which create link between the advanced control layer and optimization of the economic performance of the process under consideration, have been advocated in the literature for operating a plant optimally in the face of drifting disturbances and/or parameters.1, 3, 4 Typically, the RTO is invoked once in few hours or even once in a day. Engell,2 Tatjewski5 and, more recently, Darby et al.4 have presented excellent reviews of theoretical developments in the area of RTO and their industrial implementations. They have pointed out following two major issues in implementation of the conventional RTO ² Long wait time: The drawback of the conventional RTO approach is that, it generates optimal performance only when model parameters and/or disturbances are changing abruptly but rarely. When parameters/disturbances in a plant change frequently and/or drift slowly, the plant rarely operates at a steady state, and therefore …nding the system steady state is a di¢cult task. Moreover, the setpoints speci…ed by infrequently invoked RTO no longer represent the optimum plant operation. The conventional RTO is not a good choice for optimally controlling such a system. Even when the changes in parameters/disturbances occur rarely, the RTO invocation has to be delayed till the plant settles to a new steady state. As a consequence, the plant operates suboptimally during the wait time. ² Model Consistency: RTO layer employs a mechanistic steady state model while MPC is typically implemented using a linear dynamic model, which is rarely updated. This structural mismatch between RTO and MPC layers may lead to unrealizable setpoint targets and/or operating constraints as their steady state gains may di¤er. The literature focussed on addressing these issues can be classi…ed into two broad categories. Merging of RTO and control layers into a single layer online optimizing control is one dominant direction of work.2, 6–10 The other broad direction is to make modi…cations while maintaining a clear separation of RTO and controller layers, i.e. two layer online optimizing control.11–13 An alternative to retaining the conventional two layer structure is merging of the economic and control performance objectives into a single optimizing control layer. Over the last decade, a signi…cant number of single layer dynamic real-time optimizing control (DRTO) schemes or economic linear/nonlinear model predictive control (EMPC/ENMPC) schemes have been proposed in the literature that merge economic and control objectives.6, 9, 10, 14, 15 EMPC achieves integration by including the economic cost criteria into the linear MPC objective function.7, 16–18 These approaches, however, do not eliminate the structural mismatch between the models used for optimization and control. ENMPC, on the other hand, achieves even closer integration by including the economic cost criteria into the NMPC objective function.6, 14, 15, 19 As a consequence, there is no model structure mismatch while meeting the economic and control objectives. The wait time issue is also addressed as ENMPC directly computes manipulated input pro…les that are optimal from the economic viewpoint. A signi…cant research e¤orts in the area of ENMPC appears to be directed towards establishing the nominal and robust stability conditions, which can be guaranteed only under some restrictive conditions.6 The resulting dynamic optimization problem is a complex constrained nonlinear programming problem that has to be solved within a sampling interval, which is a challenging task for a complex chemical process.11 Moreover, maintenance of this single layer approach can be a di¢cult exercise due to lack of transparency to the operators.4

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The need for real time optimization arises mainly due to drifting disturbances and parameters. The disturbances typically have fast and slow transient modes, and, improving/maintaining the economic performance requires responding to the slow modes11 and/or slowly drifts in model parameters. The high frequency ‡uctuations in the disturbances do not in‡uence the economic performance signi…cantly and can be dealt separately. This time scale separation inherent in most process plants has prompted many researchers to propose changes in the conventional two layer structure. To deal with slow as well as frequent parameter changes/external disturbances, several variants of the two layer RTO structure have been proposed in the literature.2, 5, 10–13, 17, 20 Frequent RTO approach proposed by Sequiera et al.12 resorts to frequent implementation of RTO (at a sampling rate smaller than the process settling time). This can lead to implementation of the setpoint changes as ramps rather than step changes.6 However, they propose to use the steady state data for updating the model used for optimization, which requires waiting for the steady state to be reached each time a change is introduced. Use of two layer MPC structure has been proposed where an additional linear steady-state model based frequent economic optimization step is introduced (which is referred to as LP-MPC/QP-MPC) to reduce the gap between RTO and control layers.13, 21, 22 These approaches e¤ectively address the wait time issue in the conventional formulation while dealing with the modeling uncertainties. While these approaches attempt to reduce steady state gain mismatches between the models used for RTO and for control, inconsistencies arising from the structural mismatch remain as linear model based MPC is used in the control layer. To reduce this gap between RTO and control layers, two layer DRTO scheme has been proposed.10, 11, 23, 24 Unlike the conventional approach that employs a steady state model, DRTO periodically optimizes the economic performance over a shifting time window using a nonlinear mechanistic dynamic model as constraints in the optimization formulation. The computed optimal setpoint pro…les are then implemented either using an auxiliary controller11 or an NMPC that deals with the fast disturbances using a quadratic cost function.10, 11 In particular, Lyapunov based ENMPC formulations developed by Elis and Christo…des10 at both the layers guarantees closed loop stability along with improvements in the economic performance. DRTO approach e¤ectively addresses the wait time issue and inconsistencies in model used in optimization layer and MPC layer. One di¢culty with this approach is solving a dynamic optimization problem at frequent intervals is computationally expensive especially for large dimensional systems. Further, it is assumed that the variations of disturbances/parameters are measured and the initial state of the system is known. It may be argued that, ability to handle uncertainties in the model parameters, is the most crucial aspect of the conventional two layer formulation. However, as pointed out by Ellis et al.,6 majority of the available ENMPC or single layer DRTO formulations do not explicitly handle model parameter uncertainties. Typically a dynamic model with nominal set of parameters is used for ENMPC formulation.6 The disturbance variations are often assumed to be measured or known.11, 14, 15, 25, 26 Moreover, majority of the available algorithms even assume the states to be perfectly measured at each sampling instant. Few notable exceptions to this are adaptive optimizing control formulation developed by Jang et al.20 and Hashemi et al.,27 adaptive ENMPC formulation developed by Guay and coworkers,8, 28 and adaptive DRTO by Dunnebier et al.24 These formulations explicitly employs an observer for simultaneous state and parameter estimation and use the estimated parameters for real time optimization and control. Thus, for implementing the online optimizing approaches in an e¤ective manner, the system states and the drifting process parameters need to be known with a reasonable accuracy. Engell2 points out that, a state estimation scheme is a key ingredient of a directly optimizing control scheme as not all states are measured in practice. While discussing open issues in the area of RTO, Engell2 states that "an accurate state estimation is at least as critical for the performance of the closed loop system as exact tuning of the opti-

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mizer, more attention should be paid to the investigation of the performance of the state estimation schemes in realistic situations with non-negligible model plant mismatch". The accuracy of the optimum operating policy computed by any online optimizing scheme depends on the accuracy of the estimated parameters. The error in the parameter estimates may shift the optimum operating point to a lesser pro…table operating region.29 General industrial practice is to estimate the process parameters of the steady state model oine4 or using steady state operating data collected on-line12, 30 . The use of mechanistic dynamic models online for monitoring and control purpose has received a signi…cant attention in the last two decades31–33 . Soroush,34 in his review paper, has suggested that dynamic model based simultaneous state and parameter estimators can be used as a link between the real time optimization (RTO) and control layer that are used together for achieving adaptive and economically optimal operation in the presence of drifting disturbances/parameters. These schemes, however, have found very little application in the conventional two layer RTO or frequent RTO schemes barring a few recent exceptions. Valluru et al.35 proposed an adaptive optimizing control scheme that combined RTO layer and NMPC layer using simultaneous state and parameter estimation of a dynamic mechanistic model. The model parameters were updated using extended Kalman …lter (simultaneous EKF) and used for development of frequent RTO and adaptive NMPC schemes. Use of simultaneous EKF for developing frequent RTO has also been proposed later by Krishnamurthy et al.36 and Matias & Roux.37 They, however, restrict the use of updated model parameters for developing frequent RTO schemes and do not make the control layer adaptive. A distinct advantage of frequent RTO approach over ENMPC/DRTO is the ease of maintenance due to clear separation between optimization of economic and control objectives. However, on-line update of the steady state model employed by RTO layer using dynamic operating data is a major concern in implementation of frequent RTO. In this work, it is proposed to develop a novel integrated frequent RTO and adaptive NMPC approach for operating a unit operation in a economically optimal manner.38 A dynamic mechanistic model based simultaneous state and parameter estimation scheme is used as a common link between the RTO and the NMPC components. Use of dynamic model for parameter/unmeasured disturbance estimation facilitates frequent invocation of RTO and development of an adaptive controller that optimally tracks the setpoint trajectories generated by the RTO. Moreover, use of a single model for carrying out RTO as well as control tasks eliminates di¢culties that arise due to mismatch between models used for steady state optimization and control. The preliminary version of adaptive optimizing control scheme proposed by Valluru et al.35 has been enhanced as follows: ² in addition to the simultaneous extended Kalman …lter (EKF), recently developed shifting window state and parameter estimator39 has been used for achieving integration between frequent RTO and NMPC ² e¢cacy of the proposed integrated optimizing control scheme has been demonstrated by conducting simulation studies on the benchmark Williams-Otto reactor system40 and a CSTR system41 with irregularly sampled multi-rate measurements and uncertain measurement delays, in addition to ideal reactive distillation (RD) column42 example. This paper is organized in four sections. Details of the proposed integrated on-line optimizing control scheme are presented in the Section 2, Section 3 presents the simulation case studies, and the conclusions reached through analysis of the simulation results are summarized in Section 4.

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2

Model for Estimation, Predictive Control and RTO

The proposed integrated approach assumes that a reliable mechanistic or grey-box model is available for the system under consideration. Integration of estimation, control and steady state optimization is achieved by using this model at di¤erent time scales. The model and associated modeling assumptions are brie‡y described in this section. Consider a process represented by a set of index one semi-explicit di¤erential algebraic equations (DAEs) x 

= f (x() z() m() µ)

(1)

0£1

= G[x() z ()]

(2)

y

= h(x() z())

(3)

where x 2 R represents di¤erential state variables, z 2 R represents algebraic state variables, m 2 R represents manipulated inputs, µ 2 R represents slowly varying model parameters / unmeasured disturbances and y 2 R represents true measured variables. Here, f (), G() and h() are known non-linear function vectors of dimension ( £ 1)  ( £ 1) and ( £ 1)  respectively. 0£1 represents a  £ 1 vector of zeros. Let  represents the sampling interval. The manipulated inputs (m) are held piecewise constants over a sampling interval ( ). Similar to the modelling assumptions considered by Valluru et al.,39 the true values of the manipulated inputs (m) are related to the known/computed manipulated inputs (u) as m = u +w  where, w 2 R , is a zero mean Gaussian process i.e., w »N (0£1  Q ) Under the assumptions considered by Valluru et al.,39 the true system dynamics represented by equations (1-2) can be represented in discrete form as follows : x+1 0£1

e (x  z  u  wu  µ) + wx = F = G[x+1  z+1 ]

(4) (5)

where wx 2 R is a zero-mean Gaussian white noise with covariance Qx . To simplify the notation of measurement and state noise e¤ecting the system dynamics, a combined state noise vector is de…ned as follows h i   w = wu (6) wx h i  [w ] = 0(+)£1 and Q =  [w ] =   Qu Qx and the discrete model equations (4)-(5) are represented in standard discrete form as follows: x+1

= F[x  z  u  µ w ]

(7)

0£1

= G[x+1  z+1 ]

(8)

Measurements (y) are available at a regular sampling interval ( ) i.e., y = h (x  z ) + v

(9)

where, v 2 R  represents the measurement noise, which is modeled as zero mean Gaussian white noise process with covariance matrix R i.e., v »N (0 R ). In the scenario when measurements are sampled at multiple rates, the dimensions of the measurement vector,   and dimensions of the covariance matrix R are time dependent

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Page 6 of 35

The control variables of interest, denoted by y 2 R (which, in general, need not correspond to the measured outputs) are assumed to be related to the states as follows y = H (x  z )

(10)

To carry out steady state real time optimization, steady state model of the form f (x  us  z  µ) = 0£1

(11)

G(x  z ) = 0£1

(12)

derived from the dynamic model (equations(1-2)) is employed. The subscript  here denotes the steady state quantities.

3

Online Optimizing Control Scheme

3.1

Integrated RTO and Adaptive MPC Scheme

A schematic representation of the proposed integrated RTO and Adaptive Predictive Control scheme is shown in Figure 1. At the core of the proposed integrated scheme is a state and parameter estimator, which is used to estimate drifting unmeasured disturbances / model parameters along with the state variables. The estimated model parameters are used to update the dynamic prediction model employed by NMPC scheme and the steady state model used by the RTO. The RTO is executed either when changes in estimated unmeasured disturbances/parameters cross a prede…ned threshold or at a user de…ned frequency. The optimum setpoints computed by RTO are sent to the NMPC scheme, which moves the plant along the desired optimum trajectory. Distinguishing features of the proposed integrated scheme are as follows: ² A single model is used to perform RTO and multivariable control tasks ² Online update of parameters/unmeasured disturbances in the dynamic model used for prediction, which amounts to employing an adaptive NMPC formulation ² Economic optimum tracking is carried out without waiting for the system to attain a steady state thereby eliminating the delays involved in tracking the optimum ² Multirate measurements which are available with constant/variable delays are handled and are used for state and parameter estimation The proposed integrated scheme has three components: (a) Nonlinear state and parameter estimator (b) Real Time Optimizer and (c) Adaptive NMPC scheme, which are brie‡y described in the sub-sections that follow.

3.2

On-line Parameter Estimation

The proposed integrated approach is developed using simultaneous state and parameter estimation schemes available in the literature. The available approaches can be classi…ed into two categories: (a) conventional approaches based on the random walk model and (b) recently developed moving window based approaches39 . Two representative schemes of each category are brie‡y discussed in this subsection.

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Unmeasured Disturbances/ Time Varying Parameters Input Noise Adaptive NMPC (Dynamic Prediction Model) Setpoints r ( k RTO ), u s ( k RTO )

RTO (Steady State Model)

m(k ) 



Measurement Noise

u(k ) Plant

m(k )





y (k )

θ(k ) Online Parameter Estimator

θ(k RTO )

Figure 1: Schematic Representation of the proposed integrated RTO and Adaptive NMPC Formulation

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3.2.1

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Conventional Approach to Parameter Estimation

By this approach, variations of the unmeasured disturbances/parameters are modelled as a random walk process. Assumption 1: It is assumed that µ +1 = µ  + w

(13)

where w is assumed to be zero mean Gaussian white noise processes with covariance, Q . The state dynamics is augmented with the arti…cially introduced random walk model. Thus, de…ning h h i i   , the an augmented state vector, X = x µ  and an augmented noise vector, W = wx w augmented DAE model can be represented as follows

where

X+1

= F[X  z  u + w ] + W

(14)

0£1

= G[X+1  z+1 ]

(15)

y

= h(X  z ) + v

(16)

# " F[x  z  u + wu  µ ] F[] = µ

The augmented model is then used to develop a recursive Bayesian estimator, such as DAE-EKF42 or DAEUKF,43 that simultaneously estimates the states and parameters/disturbances. Alternatively, a moving horizon estimation (MHE) based simultaneous state and parameter estimator can be developed using the random walk model.44 Typically, the matrix Q is assumed to be diagonal and the individual variances are treated as the tuning parameters. Alternatively, Q can be estimated from operating data.45 3.2.2

Moving Window Parameter Estimator39

This parameter estimation scheme is based on a signi…cantly di¤erent modeling assumption regarding variations of unmeasured parameters/disturbances. Assumption 2: It is assumed that the variation of parameters/unmeasured disturbances (µ) occur at a signi…cantly slower rate or at a lower frequency than the rates at which states change over a time window, T ´ [ ¡  , ]in the recent past. Thus, it is assumed that the model parameters remain constant over the time window, [ ¡ , ]. Consider output measurement data of the recent past i.e. say  + 1 data points over the time window [ ¡  ] Y = fy¡ y g At the  instant, the parameter/unmeasured disturbance vector µ is estimated by minimizing the negative of the log-likelihood function i.e., 2 3  X b = arg min 4 5 µ log(det(§ )) + " §¡1 (17)  " 

=¡

8


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Subject to : Prediction step : bj¡1 x 0£1

= F[b x¡1j¡1  b z¡1j¡1  u¡1  µ 0] = G[b xj¡1  b zj¡1 ]

(18) (19)

Pj¡1 = ©¡1 P¡1j¡1 ©¡1 + ¡w¡1 Q¡w¡1 Correction step : L =

"

Lx Lz

#

= Pj¡1 C [§ ]¡1

(20)

§ = C Pj¡1 C + R

(21)

xj¡1  b zj¡1 ) " = y ¡ h (b

(22)

bj x

0£1

bj¡1 + Lx " = x = G[b xj  b zj ]

Pj = (I ¡ L C )Pj¡1

(23) (24) (25)

µ 2   =  ¡   ¡  + 1 Details of maximum a-priori (MAP) version of the moving window parameter estimator can be found in Valluru et al.39 Also, alternate simultaneous state and parameter estimation schemes based on Assumption 2 are available in the literature.46, 47 At each instant , after invoking the chosen simultaneous state and parameter estimation scheme, we set ^j and the updated parameter vector µ is used in NMPC and RTO formulations discussed in the µ = µ subsequent sub-sections. Remark 1 When a conventional approach for simultaneous state and parameter estimation is used, the maximum number of parameters and unmeasured disturbances that can be estimated cannot exceed the dimension of the measurement vector ().48 In many applications, however, the dimension  (i.e. ) can exceed the dimension of the measurement vector. To alleviate di¢culties arising from such situations, under the assumption that maximum  elements of  can change simultaneously, Deshpande et al.48 have developed an intelligent state estimation approach that uses generalized likelihood ratio based diagnosis tests to …nd the subset of actively changing parameters and only updates the parameters from the subset. Thus, in a situation where    it is possible to employ an intelligent state estimation scheme that can intelligently switch between subsets of  and overcome limitations arising from observability of the augmented state vector. A similar strategy can also be adopted for the moving window based state and parameter estimation.

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3.3

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Steady State Economic Optimization

Since the model parameters/unmeasured disturbances change at a slow rate, online optimization can be carried out at a low frequency. Thus, it is proposed to invoke the optimization step in any of the following ways : (a) Fixed time interval: By this approach, RTO step is invoked after every   samples, i.e. at   =    2    (b) Parameter change crosses a threshold: Let  =  represent the last sampling instant when the steady state optimization was invoked. The steady state optimization is invoked at a subsequent instant  =   where any one of the following criteria is satis…ed jµ  ¡µ  j ¸ ¢

for

 = 1 2 

(26)

Here, ¢ :  = 1 2  represent pre-speci…ed thresholds. Given updated estimates of the parameter vector, µ at  =   , this component of the proposed integrated scheme solves a steady state economic optimization problem, where a suitable economic cost function (y  us ) is optimized, subject to the steady state nonlinear plant model (11)-(12) and bounds on the decision variables, states and outputs. The steady state optimization problem at instant  is formulated as follows: max (y us ) (27) us = us subject to f (x  us  z  µ  ) = 0£1

(28)

G(x  z ) = 0£1

(29)

y = H (x z ) us min · us · us max

(30)

ys min · ys · ys max

(31)

xs min · xs · xs max

(32)

The optimum solution to this problem yields the optimum setpoint, r = y . After completion of the optimization step, we set  =    and for  ¸   r = r and us = us

(33)

The optimum setpoint, r  and optimal steady state manipulated inputs, us  are communicated to the NMPC for tracking till the RTO is invoked again. When RTO is invoked at a …xed interval, the choice of   depends on the rate at which the parameters/ unmeasured disturbances change. To tune    the …rst step is to …nd a subset of vector  that changes at a relatively fast rate with reference to the remaining parameters. The rate at which the parameters in this subset change can be used as a basis for selecting   . Since the optimal setpoints computed by the frequent steady state RTO is implemented using state and parameter estimator and NMPC, it is advantageous to select   comparable or higher than the closed loop settling time of NMPC and settling time of the state and parameter estimator. The second option, i.e. invoking RTO when parameter change crosses a threshold, makes   adaptive and can reduce computational load when elements of  are not changing signi…cantly. 10


Page 11 of 35 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


3.4

Adaptive Nonlinear Model Predictive Control

Tracking of optimal updated set-points, r  is achieved by employing observer error feedback based adaptive NMPC scheme. In this sub-section, modi…cations to observer error feedback based NMPC scheme originally developed by Purohit et al.42 for DAE systems are discussed brie‡y. Given updated parameter vector µ  and measurement, y  at sampling instant , the current state is estimated as follows bj¡1 x

= F[b x¡1j¡1  b z¡1j¡1  u¡1  µ  ¹ 0]

(34)

Pj¡1

=

(36)

0£1

K e

= G[b xj¡1  b zj¡1 ] =

©¡1 P¡1j¡1 ©¡1 "

K K

#

+ ¡w¡1 Qu ¡w¡1

= Pj¡1 C [C Pj¡1 C + R ]¡1

= y ¡ h (b xj¡1b zj¡1 ) bj x

(35)

(37) (38)

bj¡1 + K e = x

(40)

Pj = (I ¡ K C )Pj¡1

(41)

0£1

= G[b xj  b zj ]

(39)

The updated states (b xj  b zj ) are then used to initialize model predictions in MPC formulation. The innovation signal, e  contains information on model plant mismatch and high frequency noise. As suggested by Purohit et al.,42 a …ltered innovation signal is computed as follows e = © e¡1 + (I ¡ © )e

(42)

and further is used in carrying out model based predictions. In addition, when the controlled variables are subset of measured variables, the following additional …ltered error signal is used in the model predictions42 " ´

=

y ¡ H (e xj  e zj )

= © ´ ¡1 + (I ¡ © )"

(43)

Here, © matrix is parameterized as © = [1 2  ] where 0 ·  · 1 can be chosen to shape the regulatory response of NMPC in the presence of measured and unmeasured disturbances. At the  sampling instant, given updated parameter vector µ   and a set of future manipulated input moves U = fuj  u+1j  u+ ¡1j g model predictions are carried out as follows e++1j x

0£1

= F[e x+j  e z+j  u+j  µ  ¹ 0]+ Kx e = G[e x++1j  e z++1j ]

(44) (45)

 = 0 1 2 ¡ 1

ej = x x ^j and e zj = ^ zj 11


(46)


Page 12 of 35

where,  represents the prediction horizon. If the controlled variables are subset of measurement variables, then the controlled outputs are predicted as follows e++1j y



= H (e x++1j  e z++1j ) + ´ 

(47)

= 0 1 2 ¡ 1

where, ´ , is de…ned by equation (43). If the controlled outputs, y  are di¤erent from the measured outputs, y then the controlled outputs, y ~  are predicted as follows e++1j y



= H (e x++1j  e z++1j )

(48)

= 0 1 2 ¡ 1

The use of updated µ in the predictions makes the proposed NMPC formulation adaptive. Given the model predictions and setpoints computed from RTO, modi…ed NMPC objective function is formulated over the horizon [  +  ] as follows 8 9  ¡1 n

An Integrated Frequent RTO and Adaptive Nonlinear MPC Scheme

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