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From Data to Nonlinear Predictive Control. 2. Improving Regulatory Performance Using Identified Observers Meka Srinivasarao, Sachin C. Patwardhan,* and Ravindra D. Gudi Department of Chemical Engineering, Indian Institute of Technology, Bombay, Powai, Mumbai 400 076, India
The development of schemes for effectively examining unmeasured disturbances and plant-model mismatch in nonlinear model predictive control (NMPC) is an important area, which has attracted considerable attention over the past decade. In this work, we propose NMPC formulations for improving regulatory performance using nonlinear state space models identified from input-output data. The NARX-type MISO and MIMO nonlinear models identified in Srinivasarao et al.17 are used to develop the state estimators and the predictors in the proposed NMPC formulations. We show that these models provide information about the covariances of the fast changing disturbances that are necessary for developing an extended Kalman filter (EKF) used in an NMPC scheme (EKF-NMPC). Rather than formulating an EKF to reject these disturbances, which we show has certain drawbacks, we proceed to propose an alternate prediction scheme based on the direct use of the identified noise model for future trajectory predictions. We show that the resulting NMPC formulation (DC-NMPC) is relatively easy to tune to achieve disturbance rejection as well as offset removal. The efficacy of the proposed control schemes is demonstrated using simulation studies on two benchmark control problems: (a) regulatory control of a continuously stirred tank reactor (CSTR) exhibiting input multiplicity at a singular point, and (b) servo control of a CSTR exhibiting output multiplicities. The analysis of simulation results reveals that the proposed EKF-NMPC and DC-NMPC schemes consistently generate significantly better regulatory performances, when compared to the performance of a conventional NDMC-type formulation, even in the presence of significant plant-model mismatch. In particular, the DC-NMPC performs significantly better than the EKF-NMPC formulation and can be tuned to achieve excellent regulatory performance without causing excessive input variability using parameters of a single disturbance filter. The conclusions reached through simulation studies are validated by conducting servo and regulatory control experiments on a laboratoryscale heater-mixer setup. 1. Introduction Many key unit operations in chemical plants exhibit strongly nonlinear and complex dynamics. Development of nonlinear model predictive control (NMPC) schemes for achieving tight control of such systems has been an active area of research over the past two decades.1-5 In particular, the development of schemes for effectively addressing unmeasured disturbances and plant-model mismatch is an important area that has attracted considerable attention over the past decade. One of the prime concerns in model predictive control (MPC) as well as NMPC formulations has been the removal of steady-state offset in the presence of plant-model mismatch and unmeasured disturbances. In the early formulations of NMPC, the offset problem has been handled by designing an ad hoc disturbance estimator that is similar to those used in early linear DMC formulations, which gives the controller an implicit integral action. The simplest method to incorporate integral action in NMPC formulations is by generating the output targets by shifting the setpoints, using the disturbance estimates.1,6 The disturbance model in this approach assumes that the plant model mismatch is due to step disturbances in the output and the disturbance remains constant throughout the prediction horizon. Although these assumptions regarding the nature of the disturbances rarely hold true in practice, this method has been shown to eliminate steady-state offset for many cases.6 Meadows and Rawlings6 have also described an alternate approach that was based on steady-state target optimization for incorporating the integral action. * To whom correspondence should be addressed. Tel.: +91-2225767211. Fax: +91-22-25726895. E-mail:
[email protected].
Because steady-state offsets are instances of low-frequency disturbances, the schemes to remove the offset can address only very slowly varying unmeasured disturbances. Thus, prediction schemes formulated with the sole intention of offset removal cannot optimally reject unmeasured disturbances changing at a faster time scale. Fast-changing unmeasured disturbances can be handled by formulating state estimation and predictions based on extended Kalman filtering (EKF).7 These schemes are typically based on mechanistic models and assume that the functional relationships that relate the physical sources of unmeasured disturbances (such as feed concentration) with state and output dynamics are known a priori. The ability of the NMPC formulation to reject fast disturbances can be significantly improved if such a mechanistic model is available, together with the information on the disturbance characteristics (covariance matrices). However, because of plant-model mismatch, an NMPC scheme that uses an EKF based on a raw mechanistic model cannot alone solve the problem of the offset removal. To achieve offset removal, the mechanistic model is typically augmented with additional states (such as bias in inputs or outputs), which behave as integrated white noise processes. Alternatively, some of the model parameters or unmeasured disturbance variables are assumed to be changing and their behavior is modeled as a random walk.7 This augmented system can then be used to design an EKF and NMPC scheme, which can reject fast-changing unmeasured disturbances and guarantee offset removal. In the latter case, because the model parameters have a physical significance, the uncertainties in parameters can be addressed systematically by incorporating robustness at the controller design stage.8
10.1021/ie051285x CCC: $33.50 © 2006 American Chemical Society Published on Web 04/11/2006
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There are two problems with the EKF-based formulation of NMPC previously discussed. To begin with, the functional relationships between all the sources of unmeasured disturbances and state dynamics may not be known exactly. Second, the parameters of the unmeasured disturbance model (i.e., covariance matrices of the unmeasured disturbances and the artificially introduced disturbance states) are typically unknown and have been treated as tuning parameters of the EKF. The performance of the state estimator, and, in turn, the NMPC formulation, are highly sensitive to the choice of these tuning parameters. Also, the number of additional states is restricted from observability considerations by the number of measured outputs. For the case when the number of unmeasured disturbances are larger than the number of measurements, the choice of the covariances of the additional states is not straightforward. Thus, when the system dimension is large, choosing an appropriate subset of parameters for state augmentation and tuning their covariances by trial and error is not a straightforward task.9 Recently, Valappil and Georgakis10 have proposed two methods for online estimation of a time-varying state noise covariance matrix based on (a) successive linearization and (b) Monte Carlo simulations. The latter approach is promising; however, it is computationally expensive and requires repeated use of a nonlinear ordinary differential equation (ODE) solver at each sampling instant to generate the required covariance information. The strategy that is based on successive linearization, with respect to the parameter vector, on the other hand, is computationally much less expensive. However, this approach is suitable when a good mechanistic model is available and the plant-model mismatch can be attributed to the variations of a small subset of parameters. In a situation where the exact characteristics of the unmeasured disturbances and their relationships with the state dynamics are unknown, a preferred option is to generate this information directly from input-output data. Many NMPC schemes proposed in the literature use nonlinear time-series models for prediction. From the viewpoint of the structures used for modeling unmeasured disturbance effects on process outputs, various black-box models proposed in the NMPC literature can be broadly classified into two classes: (a) nonlinear output error (NOE) or nonlinear moving average (NMAX) models11-13 and (b) nonlinear ARX (NARX) models.14,15 As indicated in the companion paper,17 these NMPC formulations have the following difficulties: (1) The NOE/NMAX model structure does not facilitate modeling of fast-changing unmeasured disturbances. In fact, the effect of all disturbances, which typically enter nonlinearly in the system dynamics, is lumped into a simple linear additive term in the outputs. Thus, disturbance modeling in the NMPC schemes based on these models solely focuses on offset removal rather than the rejection of fast-changing disturbances. (2) The NARX structure can explicitly capture the effect of unmeasured disturbances through nonlinear coupling between known inputs and unknown disturbances. However, in most cases, this model form has been used to capture the nonlinearity, with respect to manipulated (or known) inputs, and not much literature is available on how to translate its unmeasureddisturbance modeling capability into better regulatory performance of NMPC. Thus, to formulate an NMPC scheme that optimally rejects fast-changing unmeasured disturbances and as well as slowly developing plant-model mismatch, it is necessary to have (i) reasonably accurate model for unmeasured disturbance dynamics and (ii) a strategy to tune the characteristics of the additional
states introduced for offset removal. The NMPC schemes proposed in the present work are based on NARX-type MISO and MIMO nonlinear observers developed in the previous work,17 which explicitly models the effect of unmeasured disturbances. We have shown that these models provide information about the covariances of the fast-changing disturbances necessary for developing an EKF in the NMPC formulation. However, when the EKF is developed with a state space model augmented with artificially introduced states for offset removal, the covariance matrix of the additional states remains as a tuning parameter. As a consequence,the above-mentioned tuning limitations of EKF still exist and have a significant influence on the performance of NMPC. Therefore, taking motivation from two degrees of freedom structure in linear IMC formulation,16 we proceed to propose an alternate prediction scheme based on direct use of the identified noise model, which is relatively easy to tune and can be effectively used to achieve disturbance rejection as well as offset removal. The efficacy of the proposed control schemes is demonstrated using simulation studies on two benchmark CSTR control problems: (1) Control of the CSTR at a singular point.19,20 This system exhibits input multiplicity and change in the steady-state gain in the operating region. (2) Control of the CSTR at an unstable operating point.21 This reactor exhibits output multiplicities. The validity of the conclusions reached through the simulation studies is established by conducting servo and regulatory control experiments on a laboratory-scale heater-mixer setup. This paper is arranged in four sections. In the next section, we discuss two different ways of using the models identified in our previous work17 for future trajectory predictions in NMPC. In section 3, we present NMPC formulations based on these prediction schemes. The results of simulation studies are analyzed in section 4. Section 5 presents the results of experimental verification. The concluding remarks reached from simulation and experimental studies are presented in the last section. 2. Future Predictions Using NARX Type Observers Given a mechanistic model, formulation of an NMPC scheme based on EKF is well reported in the literature. However, when a black box mode is used in NMPC formulation, the state estimation and prediction schemes can differ considerably depending on the structure of the model. In this section, we show how the structure of the NARX type state space model developed in the Part I (Srinivasarao et al.17) can be used for formulating prediction schemes. The internal model used for predictions in our NMPC formulation is the MIMO nonlinear state space observer developed in Part I of this research.17
X(k + 1) ) F[X(k)] + Γu(k) + Ke(k)
(1)
y(k) ) Ω[X(k)] + e(k)
(2)
F[X(k)] ) [ΦX(k) + KΩ(X(k))] where X ∈ RN represents the state space vector, y ∈ Rr represents vector of measured outputs, u ∈ Rm represents vector of computed manipulated inputs and e(k) is a zero-mean white noise sequence with a covariance matrix of innovation sequence Ve. At each sampling instant, the internal model is used to predict future behavior of the plant over a finite future time horizon of length p (defined as the prediction horizon) starting from the current time instant k. Let us assume that, at any instant
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k, we are free to choose only q future manipulated input moves. The degrees of freedom (q) for future trajectory manipulation are typically fewer than the prediction horizon and are spread across the horizon through input blocking as follows:
u(k + j|k) ) u(k|k)
(for j ) 1, ..., m1 - 1)
u(k + j|k) ) u(k + m1|k)
(3)
(for j ) m1 + 1, ..., m2 - 1) (4) l
u(k + j|k) ) u(k + mq-1|k)
(for j ) mq-1 + 1, ..., p - 1) (5)
where mj are selected such that
0 < m1 < m2 < .... < mq-1
pensation and NMPC formulation based on this scheme is called DC-NMPC. 2.2. Extended Kalman Filtering with State Augmentation. When a mechanistic model is used to develop an NMPC scheme, the integral action can be incorporated by estimating some of the important model parameters (which are assumed to behave similar to integrated white-noise processes) simultaneously with the states.7 When a black-box model is used, it is difficult to isolate a few parameters for simultaneous estimation with the states, because the parameters do not have any physical meaning. The other possibility is to augment the state space model (eqs 1-2) with artificial states similar to the linear MPC formulation18 and use the augmented model to formulate an EKF. Defining state noise as w(k) ) Ke(k), we can represent the model (eqs 1-2) as follows:
(6)
To use the model defined by eqs 1-2 to perform predictions in NMPC formulation, we consider following two strategies to address plant-model mismatch: (i) direct use of an identified noise model and (ii) extended Kalman filtering with state augmentation. 2.1. Direct Use of an Identified Unmeasured Disturbance Model. In the absence of plant-model mismatch, the model defined by eqs 1-2 directly yields an optimal state estimator of the form
X ˆ (k + 1|k) ) F[X ˆ (k|k - 1)] + Γu(k) + Ke(k)
(9)
X ˆ (k + j + 1|k) ) F[X ˆ (k + j|k)] + Γu(k + j|k) + K(k + j|k) (10) (k + j + 1|k) ) (k + j|k)
(11)
yˆ (k + j|k) ) Ω[X ˆ (k + j|k)] + e(k)
(12)
j ) 1, 2, ..., p - 1
(13)
where (k + j|k) represents an artificially introduced state vector for offset removal. The initial values for vector (k|k) can be estimated as follows:
e(k) ) y(k) - Ω[X ˆ (k|k - 1)]
(14)
ef(k) ) Φdef(k - 1) + [I - Φd]e(k)
(15)
(k|k) ) ef(k)
(16)
Here, the matrix Φd is used as a tuning parameter, which can be tuned to achieve the desired disturbance rejection and closedloop characteristics. The filter matrix Φd can be chosen to be a diagonal matrix of the form
Φd ) diag[γ1 γ2 ‚‚‚ γr ]
(17)
such that 0 e γi < 1. Note that this filter is similar to the disturbance filter used in the internal model control scheme with two degrees of freedom. In the remainder of the text, we refer to this bias estimation and prediction scheme as direct com-
R1 ) E[w(k)w(k)T] ) KVeKT
(20)
R12 ) E[w(k)e(k) ] ) KVe; R2 ) Ve
(21)
T
We can augment the aforementioned model as
X(k + 1) ) F[X(k)] + Γu(k) + Γββ(k) + w(k) (22)
(8)
In the presence of sustained plant-model mismatch, the expected value of innovations e(k) is not zero. Following Ricker,23 we propose a prediction scheme that uses an identified gain matrix K for future trajectory predictions as follows:
(18) (19)
where
X ˆ (k|k - 1) ) F[X ˆ (k - 1|k - 2)] + Γu(k - 1) + Ke(k - 1) (7) e(k - 1) ) y(k - 1) - Ω[X ˆ (k - 1|k - 2)]
X(k + 1) ) F[X(k)] + Γu(k) + w(k) y(k) ) Ω[X(k)] + e(k)
β(k + 1) ) β(k) + wβ(k)
(23)
η(k + 1) ) η(k) + wη(k)
(24)
y(k) ) Ω[X(k)] + Γηη(k) + e(k)
(25)
where β ∈ Rs and η ∈ Rt represent artificially introduced input and output disturbance vectors while the vectors wβ ∈ Rs and wη ∈ Rt are zero-mean white-noise sequences with covariances Qβ and Qη, respectively. Note that the total number of artificial states introduced cannot exceed the number of measurements. The model coefficient matrices (Γβ, Cη) and noise covariances matrices (Qβ, Qη) are treated as tuning parameters, which can be chosen to achieve the desired closed-loop disturbance rejection characteristics. The above set of equations can be combined into an augmented state space model of the form
where
Xa(k + 1) ) F a[Xa(k)] + Γuau(k) + wa(k)
(26)
y(k) ) Ωa[Xa(k)] + e(k)
(27)
[ ] [ ] [ ] [] [ ]
w(k) X(k) Xa(k) ) β(k) ; wa(k) ) wβ(k) wη(k) η(k)
(28)
F[X(k)] Γ ; Γua ) u Fa[Xa(k)] ) β(k) [0] η(k)
(29)
Ωa[Xa(k)] ) Ω[X(k)] + Γηη(k)
(30)
R1 [0] [0] R1a ) E[wa(k)wa(k) ] ) [0] Qβ [0] [0] [0] Qη
(31)
T
R12a ) E[wa(k)v(k)T] )
[ ] KVe [0]
(32)
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Note that the state and measurement noise are correlated in the above model. Thus, EKF is developed using the Kalman filter form given by Astrom and Wittenmark22 for the case where state and measurement noise are correlated. The resulting set of equations are as follows: State propagation:
X ˆ a(k|k - 1) ) Fa[X ˆ a(k - 1|k - 1)] + Γuau(k - 1)
(33)
the set-point vector. The filter matrix Φs can be chosen to be a diagonal matrix of the form
Φr ) diag[R1 R2 ‚‚‚ Rr ]
where 0 e Ri < 1,which can be selected to achieve the desired closed-loop tracking performance. Defining the future prediction error vector ef(k + i|k) as
ef(k + i|k) ) yr(k + i|k) - yˆ (k + i|k)
CoVariance propagation:
Pˆ (k|k - 1) ) Λ(k)Pˆ (k - 1|k - 1)Λ(k)T + R1a Ψ(k) )
(34)
[ ] [ ] ∂Ωa(X) ∂Xa
Λ(k) )
∂Fa ∂Xa
(35)
Xa)X ˆ a(k|k-1)
(46)
(47)
the MPC problem at the sampling instant k is defined as a constrained optimization problem whereby the future manipulated input moves u(k|k), u(k + m1|k), ..., u(k + mq-1|k) are determined by minimizing an objective function p
∑ef(k + i|k)TWEef(k + i|k) |k) i)1
min
(36)
Xa)X ˆ a(k|k-1)
u(k|k),...,u(k+mq-1 q
For Kalman gain estimation:
+
{∆u(k + mi|k)TWU∆u(k + mi|k)} ∑ i)1
+
{∆us(k + mi|k)TWUS∆us(k + mi|k)} ∑ i)1
L(k) ) (Pˆ (k|k - 1)Ψ (k) + R12a)[Ψ(k)Pˆ (k|k - 1)Ψ(k) + T
T
q
R2a]-1 (37) To determine the innoVation and state update:
ˆ a(k|k - 1)] ea(k) ) y(k) - Ωa[X
(38)
X ˆ a(k|k) ) X ˆ a(k|k - 1) + L(k)ea(k)(k)
(39)
yL e yˆ (k + i|k) e yH
(for i ) 1, 2, ..., p)
u e u(k + j|k) e u
To determine the coVariance update:
H
∆u e ∆u(k + j|k) e ∆u L
(40)
Under the assumption that the expected value of future innovations ea(k + i) is zero, the future predictions can be generated as follows:
ˆ a(k + j|k)] + Γuu(k + j|k) (41) X ˆ a(k + j + 1|k) ) F[X yˆ (k + j|k) ) Ωa[X ˆ a(k + j|k)]
(42)
(for j ) 1, ..., p - 1)
(43)
In the above formulation, (Qβ, Qη) are typically chosen as diagonal matrices and tuning of the EKF must be performed by trial and error. Note that the performance of the EKF and the NMPC scheme formulated using the EKF is sensitive to the choice of these matrices. In the remainder of the text, we refer to the NMPC formulation based on the augmented state space model and EKF as the EKF-NMPC scheme.
(49)
subject to the following constraints:
L
Pˆ (k|k) ) (I - L(k)Ψ(k))Pˆ (k|k - 1)
(48)
(50) (51)
H
(52)
for j ) 0, m1, ..., mq-1
(53)
∆u(k + mj|k) ) u(k + mj|k) - u(k + mj-1|k)
(54)
∆us(k + mj|k) ) u(k + mj|k) - us(k)
(55)
where
Here, us(k) represents the solution of the following target optimization problem:
min [r(k) - ys(k)]T[r(k) - ys(k)]
(56)
uL e us(k) e uH
(57)
us(k)
subject to
where ys(k) is computed as 3. NMPC Formulation In this section, we briefly describe the nonlinear predictive control formulation, which uses either of the previously described prediction schemes. In addition to predicting the future output trajectory, at each instant, a filtered future set-point trajectory is generated using a reference system of the form28
Xr(k + j + 1|k) ) ΦrXr(k + j|k) + [I - Φr][r(k) - y(k)] (44) yr(k + j + 1|k) ) y(k) + Xr(k + j + 1|k) for j ) 0, 1, ..., p - 1
(45)
with the initial condition Xs(k|k) ) 0. Here, r(k) ∈ Rr represents
X ˆ s(k) ) F[X ˆ s(k)] + Γus(k) + Kef(k)
(58)
ˆ s(k)] + e(k) ys(k) ) Ω[X
(59)
for prediction with direct compensation and as
ˆ s(k)] + Γus(k) + Γββ(k) X ˆ s(k) ) F[X
(60)
ys(k) ) Ω[X ˆ s(k)] + Γηη(k)
(61)
when EKF with state augmentation is used. In the aforementioned formulation, WE represents the (positive definite) error weighting matrix, WU represents the (positive semidefinite) input move weighting matrix, and WUS represents the (positive
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semidefinite) terminal input weighting matrix. The desired closed-loop performance can be achieved by judiciously selecting the prediction horizon p, control horizon q, and the weighting matrixes WE,WU, and WUS. The resulting constrained nonlinear optimization problem can solved using standard nonlinear programming techniques, such as sequential quadratic programming (SQP). The controller is implemented in a moving horizon framework. Thus, after solving the optimization problem, only the first move uopt(k|k) is implemented on the plant, i.e.,
u(k) ) uopt(k|k)
(62)
and the optimization problem is reformulated at the next sampling instant, based on the updated information from the plant.
In the case of NDMC, we consider an NOE-type deterministic state space model of the form
Xd(k + 1) ) ΦdXd(k) + Γuu(k)
(65)
y(k) ) Ω[Xd(k)] + v(k)
(66)
where the matrices (Φd, Γu) are parametrized using generalized orthonormal basis filters (GOBFs).13,17 Future predictions, in this case, are generated by simple output correction as follows:
ˆ d(k + j|k) + Γuu(k + j|k) (67) X ˆ d(k + j + 1|k) ) ΦdX ˆ d(k + j|k)] + d(k) yˆ (k + j|k) ) Ωd[X
(68)
d(k) ) y(k) - Ω[X ˆ d(k|k - 1)]
(69)
In the case of DC-NMPC, we choose
Φd ) γI
4. Simulation Study In this section, we compare the performances of the proposed NMPC schemes with that of conventional NMPC formulation using simulation studies on two CSTR process models. These NMPC schemes are developed using NARX type L-MISO and L-MIMO models developed in our previous work.17 Note that the inputs (u) and outputs (y) used in the models and controller formulations are scaled variables, as defined in Srinivasarao et al.17 4.1. CSTR with Input Multiplicity. The CSTR system under consideration consists of a reversible exothermic reaction A h B. The dynamic model for simulating the CSTR system are given in our previous work,17 whereas the nominal parameters and the operating steady state used in the simulation studies can be found in Economou26 and Li and Biegler.27 This system exhibits input multiplicity and a change in the sign of steadystate gain in the operating region. The maximum conversion point of the system is selected as the operating point of the CSTR. The difficulties associated with controlling such systems at the optimum operating point have been discussed in detail by Patwardhan and Madhavan.20 In the present work, the output concentration (y1 ) Cb) and reactor level (y2 ) h) in the CSTR are the two measured outputs of the system. The inlet flow rate (u1 ) Fi) and inlet feed temperature (u2 ) Ti) are used as manipulated variables. The inlet concentration (Cai) is treated as an unmeasured disturbance and it is further assumed that its dynamics is governed by the following stochastic process:
1 w(k) δCai(k) ) 1 - 0.95q-1
(63)
h ai + δCai(k) Cai(k) ) C
(64)
where {w(k)} is a white noise sequence with a standard deviation of 0.02. The sampling interval (T) is chosen to be 0.1 min. In addition, the measurements of Cb and h are assumed to be corrupted with Gaussian white noise with standard deviations of 0.007 and 0.0025, respectively. The objective of this simulation case study is demonstration of the advantages of the proposed NMPC strategy in rejecting the effect of unmeasured disturbances on the measured outputs. We compare three methods of studying unmeasured disturbances: nonlinear dynamic matrix control (NDMC), NMPC with direct compensation (DC-NMPC), and NMPC with state augmentation and EKF (EKF-NMPC).
(70)
where 0 e γ < 1 is a single tuning parameter. The nominal value of γ is chosen as γ ) 0. In the case of EKF-NMPC, we choose to model extra states as bias in inputs, i.e., we set
Γβ ) Γ; Γη ) [0]
(71)
in eqs 22 and 25 while performing the state augmentation. This choice has been shown to be superior to other possibilities in the case of linear MPC.18,24 Thus, depending on the type of model used, we compare four NMPC formulations, which are referenced as (i) DC-NMPC (MISO), (ii) DC-NMPC (MIMO), (iii) EKF-NMPC (MISO), and (iv) EKF-NMPC (MIMO) in the remainder of the text. The NMPC control schemes based on EKF with state augmentation requires tuning of the covariances. It was observed that the closed-loop regulatory performance in EKF-NMPC formulations is quite sensitive to the choice of these parameters. The diagonal elements of the covariance matrix were tuned by trial and error using a single set of unmeasured disturbances and measurement noise realizations (referenced as nominal realizations in the remaining text) and the following matrices were selected:
EKF-NMPC (MISO): EKF-NMPC (MIMO):
[ [
-8 0 Qβ ) 1 × 10 0 0.6
]
-8 0 Qβ ) 1 × 10 0 0.12
]
We compare the regulatory performances of the above NMPC schemes using the following indices: (a) the sum of the square of errors (ISE), (b) the maximum absolute error (MAE), and (c) the performance index for manipulated input variations (PIMV), which is defined as Ns
PIMV )
∆u(i)TWU∆u(i) ∑ i)1
(72)
where WU is the input move weighting matrix used in the MPC objective function and Ns represents number of samples in the simulation run. To make a fair comparison, identical controller parameters are used for all the predictive control schemes under consideration. A prediction horizon of 40 and a control horizon of 3 has been selected for the simulations. The control horizon has
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Figure 1. CSTR with input multiplicity: comparison of regulatory performance of NDMC and DC-NMPC (MIMO) and unmeasured disturbance in inlet concentration (Cai). Table 1. CSTR with Input Multiplicity: Comparison of Regulatory Performances for Nominal Disturbance Set NMPC-DC NMPC-DC EKF-NMPC EKF-NMPC (MIMO) (MISO) (MISO) (MIMO)
NDMC ISE (Cb) 29.1007 MAE (Cb) 0.0539 17.980 PIMV
5.9733 0.026 90.148
8.6641 0.0232 118.48
16.5365 0.0380 7.539
15.0590 0.03489 36.692
been implemented by blocking the inputs with m1 ) 5 and m2 ) 15. The weighting matrices in the objective function are selected as follows:
WE )
[
]
1 0 ; WU ) I; WUS ) [0] 0 100
(73)
The manipulated input are estimated, subject to the following constraints:
0.4224 e Fi e 1.5775 300 e Ti e 510
(74)
The regulatory control problem is defined as the rejection of stationary as well as nonstationary disturbances in the unmeasured inlet concentration (Cai). The closed-loop simulations are performed for 250 sampling instants and a step jump with a magnitude of 0.15 is introduced at the 100th sampling instant (see Figure 1). The regulatory performances of different NMPC schemes when the nominal set of noise realizations case are used have been reported in Table 1. Note that we make this comparison based on the ISE and the maximum deviation of reactor concentration Cb, because the reactor level is not significantly affected by changes in inlet concentration. Figure 1 presents variations of controlled reactor concentrations for NDMC and DC-NMPC (MIMO) (with γ ) 0) for the case of nominal noise realizations. The corresponding manipulated plant inputs are presented in Figure 2. From these figures and Table 1, the following observations can be made by comparing different schemes on the basis of the maximum absolute deviation and the ISE values for Cb: (1) NMPC formulations that are based on direct compensation, as well as EKF, outperform the NDMC formulation. Thus, the inclusion of the unmeasured disturbance model (identified
Figure 2. CSTR with input multiplicity: comparison of regulatory performance-manipulated input. Table 2. CSTR with Input Multiplicity: Comparison of Regulatory Performance for Stochastic Simulations Cb formulation DC-NMPC (MIMO) DC-NMPC (MISO) NDMC EKF-NMPC (MIMO) EKF-NMPC (MISO)
statistics
ISE
MAE (× 102)
PIMV
mean (std) mean (std) mean (std) meana (std)a meanb (std)b
6.110 (0.90) 10.531 (8.37) 17.458 (6.145) 23.503 (10.594) 25.091 (12.332)
3.21 (0.87) 4.00 (1.82) 4.2 (1.31) 3.783 (0.796) 4.31 (1.14)
81.887 (16.396) 108.10 (25.67) 13.307 (1.584) 11.019 (5.939) 12.502 (3.869)
a Statistics are evaluated for 18 realizations. b Statistics are evaluated for 28 realizations.
only from input-output data) for future trajectory predictions significantly contributes toward improving the regulatory performance in the presence of unmeasured disturbances. (2) Both formulations that are based on direct compensation without disturbance filtering (γ ) 0) perform significantly better than the EKF-based NMPC formulation. (3) As can be expected from the model validation results from our previous work,17 the L-MIMO model-based NMPC schemes perform better than the L-MISO model-based NMPC schemes. To make our conclusions independent of a particular noise realization, we have further performed stochastic simulation studies and reported an average performance of each controller. The regulatory control problem is simulated using 50 realizations of inlet concentration disturbance and measurement noises. To make a fair comparison, we have used a fixed set of 50 noise realizations for testing the regulatory performance of each control scheme. The average values of the performance indices obtained through the stochastic simulations are presented in Table 2. Note that the values reported in this table for the EKFNMPC (MISO) formulation have been computed using only 18 noise realizations out of 50, and, for EKF-NMPC (MIMO), the computation uses only 28 noise realizations out of 50. For the remaining 32 noise realizations for EKF-NMPC (MISO) and 22 noise realizations for EKF-NMPC (MIMO), it was observed that the EKF-based formulation resulted in unacceptably large ISE values and failed to work. The cutoff for selecting
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these noise realizations is based on the worst-case performance of the DC-NMPC scheme (ISE cutoff ) 70). This indicates that the EKF-based formulations, even when tuned well for the nominal set of disturbance and noise realizations, may not produce consistent regulatory performance. It was found that, even when an accurate unmeasured disturbance model has been included, the tuning of co-variance matrix Qβ of artificially introduced states for offset removal has a crucial role in the performance of EKF-based formulation. Thus, the choice of Qβ, which yields good regulatory performance for a subset of noise realizations for EKF-NMPC (MIMO), may prove to be a wrong choice for the remainder. On the other hand, note that the direct-compensation-based NMPC schemes, which do not require any additional tuning (because γ is chosen to be zero in this case), performed consistently well for all of the realizations considered. Moreover, the results of the stochastic simulations are qualitatively similar to the results obtained using nominal realizations. Thus, the NMPC formulation based on the direct compensation scheme seems to be a better choice for rejecting stationary as well as nonstationary unmeasured disturbances. In a real plant, the characteristics of the unmeasured disturbances often change with time. Thus, it is important to investigate whether the advantages of including a noise model in NMPC formulations are retained when the noise characteristics change. To get some insight on this aspect, we further performed stochastic simulation studies by introducing plantmodel mismatch in the unmeasured disturbance model for the following two cases: changes in the disturbance variance and changes in the disturbance dynamics. For changes in the disturbance Variance, in process simulations, the variance of w(k) in eq 63 changes to 1.5σ{w(k)} or to 0.5σ{w(k)}:
δCai(k) )
1 w(k) 1 - 0.95q-1
(75)
In regard to changes in the disturbance dynamics, it is assumed that the dynamics of Cai(k) in a plant changes as
δCai(k) )
1 w(k) 1 - Rq-1
(76)
where R changes to either 0.99 (case A) or 0.85 (case B), while σ{w(k)} ) 0.02. Note that this change amounts to simultaneous changes in pole location and effective noise variance. The results of stochastic simulations for the aforementioned plant-model mismatch in the disturbance model are reported in Table 3 for the case of changes in the disturbance variance and in Table 4 for the case when the disturbance dynamics changes. These tables clearly show that the direct-compensationbased NMPC schemes outperform the NDMC, even when there is significant mismatch between the identified unmeasured disturbance model and the true characteristics of the unmeasured disturbances in the plant. In particular, the NMPC scheme performs better when the disturbance dynamics in the plant becomes faster than the nominal case. The comparison of performances of NMPC schemes based on controlled output variability alone can present a biased view about the efficacy of proposed schemes. Table 1 clearly shows that NDMC performs much better than the proposed NMPC schemes when compared on the basis of PIMV. This is not surprising because the disturbance rejection requires significantly higher variations in manipulated inputs. As a consequence, the values of PIMV are high for all NMPC formulations except
Table 3. CSTR with Input Multiplicity: Stochastic Simulation Results for Change in Disturbance Variance Cb MAE (× 102)
PIMV
For the 1.5σ Case mean 26.380 (σ) (9.631) mean 14.262 (σ) (6.011) mean 14.03 (σ) (8.85)
5.47 (2.03) 5.23 (2.5) 2.65 (1.02)
13.830 (1.893) 109.27 (24.26) 80.30 (14.37)
For the 0.5σ Case mean 12.863 (σ) (3.433) mean 8.547 (σ) (7.063) mean 6.221 (σ) (0.873)
3.45 (0.66) 2.96 (1.14) 1.982 (3.19)
12.787 (1.038) 101.191 (18.88) 78.54 (12.29)
ISE NDMC DC-NMPC (MISO) DC-NMPC (MIMO)
NDMC DC-NMPC (MISO) DC-NMPC (MIMO)
Table 4. CSTR with Input Multiplicity: Stochastic Simulation Results for Change in Disturbance Dynamics Cb ISE NDMC DC-NMPC (MISO) DC-NMPC (MIMO)
NDMC DC-NMPC (MISO) DC-NMPC (MIMO)
For Case A mean 37.07 (σ) (39.35) mean 32.26 (σ) (47.49) mean 15.043 (σ) (14.477) mean (σ) mean (σ) mean (σ)
For Case B 14.87 (3.22) 8.24 (5.45) 7.86 (4.38)
MAE (× 102)
PIMV
5.8 (3.2) 5.9 (3.7) 2.247 (0.502)
13.09 (1.034) 103.29 (19.47) 81.75 (14.38)
3.6 (0.6) 2.86 (0.99) 2.3 (0.83)
12.669 (2.078) 94.2657 (33.878) 70.01 (20.38)
EKF-NMPC (MISO). Also, PIMV values for DC-NMPC without disturbance filtering (i.e., γ ) 0) are significantly higher than EKF-NMPC formulations and this may be unacceptable from the viewpoint of an operator. However, it should be noted that the performance of EKF-NMPC has been tuned using trial and error. To investigate whether we can achieve a reduction in the input variability in DC-NMPC, we have performed simulations for different values of the disturbance filter γ using nominal disturbance and measurement noise realizations, and the results are presented in Figure 3. From this figure, it may be observed that the input variability decreases considerably when γ is changed from 0 to 0.9 with a marginal increase in the ISE values for Cb. Note that the resulting ISE values are still significantly better than those of the EKF-NMPC and NDMC formulations. However, for values of γ > 0.9, there is a sharp increase in the ISE values. Thus, the disturbance filter parameter γ can be effectively used to achieve good disturbance rejection using DC-NMPC without requiring large input variability. Moreover, when compared to the tuning diagonal elements of Qβ in the EKF-NMPC formulation, the disturbance filter parameter γ has a direct influence on the closed-loop performance and can be tuned relatively easily. Finally, the comparison of servo performance of the NDMC and DC-NMPC are presented in Figure 4. These results demonstrate that the proposed NMPC formulation generates satisfactory servo performance and it is marginally better than the servo performance of NDMC. 4.2. CSTR with Output Multiplicity. The second simulation case study considered is servo control of CSTR15,21 exhibiting output multiplicities. The dynamic model for the system and nominal model parameters is given in our previous work.17 It
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Figure 3. CSTR with input multiplicity: effect of the disturbance parameter (γ) on the regulatory performance of DC-NMPC.
Figure 4. CSTR with input multiplicity: comparison of servo performances of NDMC and DC-NMPC (MIMO).
is desired to control the (dimensionless) reactant concentration by manipulating the (dimensionless) cooling jacket temperature. The system under consideration exhibits two stable steady states and one unstable steady state. The main purpose of this section is to demonstrate the ability of the proposed NMPC scheme to control the process at an unstable operating point. The sampling time is chosen as 0.5 dimensionless time units. Measurements of the concentration are assumed to be corrupted with a zero mean Gaussian white noise signal with standard deviation of 0.02. In the predictive control scheme, the prediction and control horizons are taken as 5 and 1, respectively. The model developed in our previous work17 has been used to formulate the DC-NMPC scheme. The weighting matrices in the objective function are chosen as WE ) 1, WU ) 1, γ ) 0, and WUS ) 0. Figures 5 and 6 show the servo response of the system using the proposed NMPC scheme for a sequence of set-point changes from one operating point to another without and with measurement noise, respectively. It can be observed that the NMPC controller manages a quick transition to any of the setpoints, including the middle unstable operating point. However, with the current choice of the NMPC tuning parameters, the input variations required to keep the process at the unstable operating condition are significantly large when measurement noise is
Figure 5. CSTR with output multiplicity: servo performance in the absence of noise.
Figure 6. CSTR with output multiplicity: servo performance in the presence of noise.
present. These fluctuations can be reduced while maintaining the closed-loop stability by either choosing a nonzero value for the filter parameter (γ) or by increasing the input weighting W U. 5. Experimental Studies In this section, we present validation results on a experimental case study involving a heater-mixer setup developed at the Department of Chemical Engineering at the Indian Institute of Technology in Bombay. The heater-mixer setup consists of two stirred tanks in series, as shown in Figure 7. Details of the experimental setup can be found in our previous work.17 The temperatures in the first tank (T1) and the second tank (T2), and the liquid level in the second tank (h2), are measured variables, whereas the heat inputs to the first and second tank (m1 and m2) and the cold water flow to the second tank (m3) are treated as manipulated inputs. The cold water flow to the first tank (d) and the cold water temperature (Tc) are treated as unmeasured disturbances. We have developed a gray box model for this process as a benchmark for validating the identified models. Details of this model are given in the companion paper.17 Because the process is only mildly nonlinear, we have deliberately introduced nonlinear maps between manipulated inputs used for modeling
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Figure 7. Schematic diagram of the heater-mixer experimental setup.
Figure 8. Heater-mixer setup: Typical variation of stationary and nonstationary disturbances.
and control (u) and the current inputs (m) to thyristor power controllers. Thus, the inputs u1-u3 used for modeling and control are related to the percentage of current inputs m1-m3 as follows:
m1 ) f1(u1) ) 12.5 +
{
75 1 + exp[-0.6(u1 + 3)]
[ ( )]
m2 ) f2(u2) ) 112.5 exp -
u2 6
2
- 25
m3 ) u3
}
(77) (78) (79)
The effective relation between manipulated inputs u1 and u2 and the heat inputs to the first and second tanks introduces the soft saturation and Gaussian-type nonlinearity. Note that the Gaussian-type nonlinear function relating u2 to m2 introduces input multiplicity behavior between u2 and T2. The nominal operating point is given in the previous paper.17 The model used in the controller formulation consists of (a) a SISO nonlinear observer relating T1 with u1, (b) an SISO nonlinear observer relating h2 with u3, and (c) an MISO nonlinear observer relating T2 with u. The choice of the MISO model for T2 is made based on the model validation results presented in our previous work.17 We have used this model to develop a DC-NMPC scheme and evaluate its regulatory and servo performances. The prediction horizon and the control horizon in DC-NMPC formulation were chosen as 40 and 1, respectively. The weighting matrices in the objective function were chosen as follows:
WE ) diag[10 10 50 ]
(80)
WU ) diag[10 15 15 ]; WUS ) [0]
(81)
To reduce excessive variability in manipulated inputs while rejecting unmeasured disturbances, the disturbance filter was selected as γ ) 0.9 based on the simulation results for CSTR with input multiplicity. The manipulated input moves are computed, subject to the following constraints:
-7 e u1 e 4.763 -8 e u2 e 8 9 e u3 e 15
(82)
Figure 9. Heater-mixer setup: Typical open-loop output variation in the presence of stationary and nonstationary disturbances.
The operating point is chosen as the peak steady state point, with respect to u2 (i.e., u1 ) u2 ) 0), which is a singular point where steady-state gain with respect to u2 is reduced to zero and changes its sign in the neighborhood. As discussed in in previous paper,17 the stationary component of the unmeasured disturbance d is generated by passing a random sequence through the low-pass filter, as given in eq 84:
δd(k) )
0.045 w(k) 1 - 0.99z-1
d(k) ) ds + δd(k)
(83) (84)
where w(k) represents the zero-mean Gaussian white noise with a standard deviation of 1. The nonstationary change in d(k) is introduced by abruptly changing ds. To provide a qualitative idea of possible deviations from the setpoint when these stationary and nonstationary disturbances are introduced under open-loop conditions, we have presented the results of variations of process outputs in Figures 8 and 9 when the manipulated inputs are held constant at their respective nominal steady-state values. To evaluate the regulatory and servo performances of the proposed DC-NMPC formulation, we have introduced a series of step changes in d and setpoints. An abrupt step change of magnitude 0.1 L/min at sampling instant 51 and of magnitude -0.2 L/min at sampling instant 250 is introduced in the
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corresponding to δT2 ) 0 and variations in T2 are restricted to (1 °C in the face of step jumps in d and within (0.75 °C when only the stationary component of unmeasured disturbance is acting. The transitions in level setpoint are also achieved smoothly, and it is controlled within ( 0.5 cm once the setpoint is reached. Note that disturbances of similar magnitude under open-loop conditions cause an ∼3 °C change in T2 and an ∼5-6 cm change in the level. Also, the set-point transitions and disturbance rejection are achieved with acceptable variations of the manipulated inputs (see Figure 12). Thus, as expected from the simulation studies, the proposed DC-NMPC exhibits excellent servo and regulatory performances on the experimental setup. 6. Conclusions Figure 10. Heater-mixer setup: Servo and regulatory performances unmeasured disturbance variation.
Figure 11. Heater-mixer setup: Servo and regulatory performancesvariation of controlled outputs.
Figure 12. Heater-mixer setup: Servo and regulatory performancesvariation of manipulated inputs.
unmeasured disturbance (d), as shown in Figure 10. The unmeasured disturbance in the cold water flow rate is also shown in Figure 10. The set-point changes were introduced simultaneously, and the resulting variations of controlled output are presented in Figure 11. The corresponding input moves are presented in Figure 12. From Figure 11, it may be observed that the variations of T1 are maintained within (1 °C, with a maximum perturbation of ∼2 °C when a large step jump is introduced in d. This is a significant reduction when compared to an open-loop variation of 6 °C. The DC-NMPC achieves smooth changes to and from the singular peak operating point
In this work, we have proposed nonlinear model predictive control (NMPC) formulations for improving regulatory performance using nonlinear state space models identified from inputoutput data. The nonlinear ARX (NARX)-type MISO and MIMO nonlinear models identified in our previous work17 have been used to develop the state estimators and the predictors in the proposed NMPC formulations. We show that these models provide information about the covariances of the fast-changing disturbances that are necessary for developing an extended Kalman filter (EKF) used in the NMPC scheme (EKF-NMPC). However, when the EKF is developed with a state space model augmented with artificially introduced states for offset removal, the covariance matrix of the additional states remains as a tuning parameter. Moreover, the choice of this matrix has a significant influence on the regulatory performance of NMPC. Therefore, we proceed to propose an alternate NMPC scheme based on direct use of the identified noise model for future trajectory predictions (DC-NMPC), which is relatively easy to tune to achieve disturbance rejection, as well as offset removal. The efficacy of the proposed control schemes is demonstrated using simulation studies on two benchmark control problems: (i) regulatory control of a continuously stirred tank reactor (CSTR) exhibiting input multiplicity at a singular operating point and (ii) servo control of a CSTR exhibiting output multiplicities. The analysis of simulation results reveals that the proposed EKF-NMPC and DC-NMPC schemes consistently generate significantly better regulatory performances, when compared to the performance of conventional nonlinear DMC-type formulations, even in the presence of significant plant-model mismatch. In particular, the DC-NMPC performs significantly better than the EKF-NMPC formulation and can be tuned relatively easily to achieve excellent regulatory performance without causing excessive input variability using parameters of a single disturbance filter. The conclusions reached through simulation studies are validated by conducting servo and regulatory control experiments on a laboratory-scale heatermixer setup. Literature Cited (1) Henson, M. A. Nonlinear Model Predictive Control: Current Status and Future Directions. Comput. Chem. Eng. 1998, 23, 187-202. (2) Morari, M.; Lee, J. H. Model Predictive Control: Past, Present and Future. Comput. Chem. Eng. 1999, 23, 667-682. (3) Qin, S. J.; Badgewell, T. A. An overview of nonlinear model predictive control applications. In Nonlinear Model PredictiVe Control; Allgo¨wer, F., Zheng, A., Eds.; Birkhauser: Basel, Switzerland, 2000. (4) Bartusiak, R. D. NLMPC: A platform for optimal control of feedor product-flexible manufacturing. In Pre-prints of International Workshop on Assessment and Future Directions of Nonlinear Model PredictiVe Control; Fruedenstadt-Lauterbad, Germany, August 26-30, 2005; Institute
Ind. Eng. Chem. Res., Vol. 45, No. 10, 2006 3603 for Systems Theory in Engineering, University of Stuttgart: Stuttgart, Germany, pp 13-14. (5) Camacho, E. F.; Bordons, C. Nonlinear Model Predictive Control: An Introductory Survey. In Pre-prints of International Workshop on Assessment and Future Directions of Nonlinear Model PredictiVe Control, Fruedenstadt-Lauterbad, Germany, August 26-30, 2005; Institute for Systems Theory in Engineering, University of Stuttgart: Stuttgart, Germany, pp 15-30. (6) Meadows, E. S.; Rawlings, J. B. Model Predictive Control. In Nonlinear Process Control; Henson, M. A., Seborg, D. E., Eds.; Prentice Hall: Upper Saddle River, NJ, 1997; Chapter 5. (7) Lee, J. H.; Ricker, N. L. Extended Kalman Filter Based Nonlinear Model Predictive Control. Ind. Eng. Chem. Res. 1994, 33, 1530. (8) Nagy, Z. K.; Braatz, R. D. Robust nonlinear model predictive control of batch processes. AIChE J. 2003, 49 (7), 1777-1786. (9) Lee, J. H.; Ricker, N. L. Nonlinear Modeling and State Estimation for Tennessee Eastman Challenge Process. Comput. Chem. Eng. 1995, 19 (9), 983-1005. (10) Valappil, J.; Georgakis, C. Systematic estimation of state noise statistics for extended Kalman filters. AIChE J. 2000, 46 (2), 292-308. (11) Doyle, F.; Pearson, R. K.; Ogunnaike, B. A. Identification and Control Using Volterra Models; Springer-Verlag: London, 2002. (12) Sentoni, G. B.; Guiver, J. P.; Zhao, H.; Biegler, L. T. A state space approach to nonlinear process modeling: Identification and universality. AIChE J. 1998, 44 (10), 2229. (13) Saha, P.; Krishnan, S. H.; Rao, V. S. R.; Patwardhan, S. C. Modeling and predictive control of MIMO nonlinear systems using WienerLaguerre models. Chem. Eng. Commun. 2004, 191 (8), 1083-1119. (14) Su, H. T.; McAvoy, T. J. Artificial neural networks for nonlinear process identification and control. In Nonlinear Process Control; Henson, M. A., Seborg, D. E., Eds.; Prentice Hall: Upper Saddle River, NJ, 1997; Chapter 7. (15) Hernandez, E.; Arkun, Y. Control of nonlinear systems using polynomial ARMA models. AIChE J. 1993, 39 (3), 446-460. (16) Morari, M.; Zafiriou, E. Robust Process Control; Prentice Hall: Upper Saddle River, NJ, 1989. (17) Srinivasarao, M.; Patwardhan, S. C.; Gudi, R. D. From Data to Nonlinear Predictive Control. Part I. Identification of Nonlinear State Observers. Ind. Eng. Chem. Res. 2006, 45 (6), 1989-2001.
(18) Muske, K.R.; Badgwell, T. A. Disturbance modeling for offsetfree linear model predictive control. J. Process Control 2002, 12, 617632. (19) Biegler, L. T.; Rawlings, J. B. Optimization approaches to nonlinear model predictive control. In Chemical Process ControlsCPC IV: Proceedings of the Fourth International Conference on Chemical Process Control, Padre Island, TX, February 17-22, 1991; Arkun, Y., Ray, W. H., Eds.; CACHE and AIChE: New York, 1991; pp 543-571. (20) Patwardhan, S. C.; Madhavan, K. P. Nonlinear Predictive Control Using Approximate Second-Order Perturbation Models. Ind. Eng. Chem. Res. 1993, 32, 334-344. (21) Uppal, A.; Ray, W. H.; Poore, O. B. On the dynamic behavior of continuous stirred tank reactors. Chem. Eng. Sci. 1974, 29, 967-985. (22) Astrom, K. J.; Wittenmark, B. Computer Controlled Systems; Prentice-Hall: Upper Saddle River, NJ, 1997. (23) Ricker, N. L. Model Predictive Control with State Estimation. Ind. Eng. Chem. Res. 1990, 29, 374-382. (24) Patwardhan, S. C.; Manuja, S.; Narasimhan, S.; Shah, S. L. From data to diagnosis and control using generalized orthonormal basis filters. Part II: Model predictive and fault tolerant control. J. Process Control 2006, 16 (2), 157-175. (25) Muske, K. R.; Edgar, T. F. Nonlinear State Estimation. In Nonlinear Process Control; Henson, M. A., Seborg, D. E., Eds.; Prentice Hall: Upper Saddle River, NJ, 1997; Chapter 6. (26) Economu, C. An Operator Theory Approach to Nonlinear Controller Design, Ph.D. Dissertation, California Institute of Technology, Pasadena, CA, 1985. (27) Li, W. C.; Biegler, L. T. Process Control Strategies for Constrained Nonlinear Systems. Ind. Eng. Chem. Res. 1988, 27, 142. (28) Richalet, J.; Rault, A.; Testud, J. L.; Papon, J. Model predictive heuristic controlsapplication to industrial processes. Automatica 1978, 14, 413-428.
ReceiVed for reView November 19, 2005 ReVised manuscript receiVed February 27, 2006 Accepted March 2, 2006 IE051285X
