- Email: [email protected]
A nonlinear industrial model predictive controller using integrated PLS and neural net state-space model
Control Engineering Practice 9 (2001) 125}133
A nonlinear industrial model predictive controller using integrated PLS and neural net state-space model 夽 Hong Zhao *, John Guiver , Ramesh Neelakantan , Lorenz. T. Biegler Aspen Technology Inc., 1293 Eldridge Parkway, Houston, TX77077-1670, USA Department of Chemical Engineering, Carnegie Mellon University, Pittsburgh, PA 15232, USA Received 12 October 1999; accepted 9 June 2000
Abstract Model predictive control (MPC) technology has been well developed and successfully applied in the re"nery and petrochemical process industries over the last 20 years. Recent development has been focused on nonlinear MPC and robust MPC technologies because new challenges have been encountered in the polymer and chemical industries where many processes show strong nonlinearity and uncertainty. This paper presents a nonlinear industrial model predictive controller, recently developed by Aspen Technology Inc. This MPC controller uses a nonlinear, state-space, integrated partial least-squares (PLS) and neural net model (Zhao, Guiver and Sentoni, American control conference, Philadelphia, PA, USA, 1998), and a multi-step, constrained, Newton-type optimization algorithm (Oliveira and Biegler, Automatica, 31 (2) (1995) 281}286). It results in a robust and cost-e!ective industrial nonlinear MPC controller. A pH reactor example and a successful industrial application in NO emission control of a power plant are V presented to demonstrate the capability of this controller. 2001 Elsevier Science Ltd. All rights reserved. Keywords: Nonlinear control; Predictive control; State-space model; Neural nets; Industrial control
1. Introduction In the past two decades, model predictive control (MPC) technologies have been widely applied in the re"nery, petrochemical and other process industries. MPC has been recognized as an e$cient means to improve operating e$ciency and pro"tability (Yamamoto, & Hashimoto, 1991). However, industrial practice has encountered new challenges with MPC in many new areas such as the polymer industry and some chemical processes where the processes show strong nonlinear dynamic behavior. Recent research and development e!orts have been made towards addressing nonlinear process model-based MPC and robust MPC design. The next generation industrial MPC, as envisioned by several
夽
An early and shorter version of this paper by the authors, entitled &&Nonlinear industrial model predictive controller using integrated PLS and neural net state space model'' was presented at the 1999 IFAC World Congress in Beijing, P.R. China, July 1999. * Corresponding author. Tel.: #1-281-504-3269; fax: #1-281-5844329. E-mail address: [email protected] (H. Zhao).
authors (Qin & Badgwell, 1997; Froisy, 1994; Rawlings, Meadows & Muske, 1994; Morari & Lee, 1991, 1997) is likely to have the following features: nonlinear process models, state-space representation, multiple objectives, on-line adaptation and enhanced user interfaces. In the modeling and control of nonlinear processes, "rst-principle model-based MPC is a good approach and it is capable of working in a relatively broad operating range. However, developing "rst-principle models is expensive and they are not always available. Industrial practice has proven that the empirical models obtained through plant tests and identi"cation are more feasible and in most case are su$cient for MPC control. For nonlinear process modeling, neural nets have been introduced and investigated for process modeling by many researchers (Thibault & Grandjean, 1992). However, neural nets alone are not a panacea. Several concerns have been raised with regard to their application in the process industries. For example, they have been criticized for their weakness in extrapolation. To put neural nets into industrial MPC applications, some key issues such as model validity and extrapolation have to be addressed. This paper presents a new industrial nonlinear MPC
0967-0661/01/$ - see front matter 2001 Elsevier Science Ltd. All rights reserved. PII: S 0 9 6 7 - 0 6 6 1 ( 0 0 ) 0 0 1 0 1 - 5
126
H. Zhao et al. / Control Engineering Practice 9 (2001) 125}133
controller, Aspen Target2+, which uses an integrated PLS and neural net model to provide many advantages to the process industry.
2. Process model An MPC controller consists of two essential components: one is a process model which predicts the process future behavior, the other is an on-line optimizer which solves an open-loop optimal control problem at each control time interval and determines the control action for the next step. In industrial applications, usually an economic optimizer is also added to the top of the MPC controller to provide the optimal operating setpoints for controlled variables (CVs) as well as manipulated variables (MVs) to obtain the maximum pro"t without violating operating constraints. Obviously, the process model plays a very important role. The proposed state-space model structure consists of three parts as illustrated in Fig. 1(a). The "rst part of this model contains a number of internally grouped "lters that are represented by a set of linear state equations. The following two parts involve an explicit linear component and a nonlinear neural net. They are connected in parallel and are described by a linear output equation and
a nonlinear output error equation, respectively. The "nal model outputs are obtained by summing the outputs from the linear component with those from the nonlinear neural net component. The model can be written in the discrete-time state-space formulation as x(t#1)"Ax(t)#B u(t!q)#B v(t!1), S T y(t)"Cx(t)#f [x(t)], (1) ,, where y3RK is the vector of outputs, u3RP and v3RJ are, respectively, the manipulated input vector and disturbance input vector, q and 1 are dead-times of the manipulated input and disturbance input, x3RL is the vector of states, and f ( ) ) is a vector of nonlinear functions rep,, resenting the nonlinear neural net. Model (1) is an extended Wiener model, i.e. it consists of a linear dynamic element followed in series by a static nonlinear element. The only di!erence of model (1) from that of a classic Wiener model is that the linear state variables, which are usually greater than the output variables in number, are taken as inputs of the static nonlinear element. This structure o!ers more degree of freedoms to model nonlinearities in steady state as well as dynamics. Recently, Winner model has been investigated for modeling and MPC applications. Kapoor, McAvoy and Marlin (1986) demonstrated that a Winner model
Fig. 1. (a) State-space model using PLS and neural net. (b) Subsystem and corresponding state variables.
H. Zhao et al. / Control Engineering Practice 9 (2001) 125}133
can adequately represent many of the nonlinearities commonly encountered in processes such as distillation. Norquay, Palazoglu and Romagnoli (1998, 1999) have developed a MPC using Wiener models and applied it to an industrial C2-splitter. For the proposed model (1) structure, Sentoni, Biegler, Guiver and Zhao (1998) have recently proven that this speci"c model structure (decoupled A}B net), which is composed of a decoupled linear dynamic system followed by a neural net, is able to approximate every nonlinear, causal, discrete time invariant, multi-input single-output system with fading memory (Boyd & Chua, 1985). From an application point of view, model (1) structure provides several advantages over a traditional "nite impulse response (FIR) or nonlinear autoregressive with extra input (NARX) model. First, it is a parametric model with fewer parameters than a FIR or Volterra series model. Second, the nonlinear element can be implemented with feedforward neural nets (FNN) while being obtained through output-error identi"cation using an algorithm proposed by Zhao, Guiver and Sentoni (1998, 1999). Therefore, it is able to produce accurate long-term (multi-step) predictions for the MPC, which is di$cult for an NARX model using FNN to generate. Third, because the output equation is represented by an explicit combination of a linear map and a nonlinear map, the model identi"cation, then can be done in two steps: identifying a linear state-space model "rst and then a residual neural net model. In addition, with this structure, the model extrapolation capability can be enhanced by an autocombination of the two maps as well as a constrained on-line model adaptation.
3. Identi5cation of the state-space model 3.1. Identixcation of matrices (A, B, C) The identi"cation of linear part (A, B, C) uses several techniques including principal component analysis (PCA) and internal balancing-based model-order determination, engineering-oriented quasi-canonical form, and partial least-squares (PLS) parameter estimation. The PCA and internal balancing ensure the linear statespace model is a well-conditioned lower-order realization. The use of quasi-canonical form will result in a minimum of parameterization and also make state variables with engineering sense. The PLS algorithm prevents the singularity problem that may occur due to the co-linearity among multiple inputs. This guarantees robust parameter estimation. The identi"cation usually needs a few iterations depending on the nonlinearity of the process. The proposed approach assumes that the process can be represented by m MISO subsystems. For each MISO subsystem, the identi"cation process consists of the following major steps:
(i)
(ii)
127
Specify a dominant time constant value ¹ (not G critical) for each input}output pair of the process, and then construct either a group of "rst-order "lters (Zhao, Guiver & Klimasauskas, 1997) or a Laguerre system (Sentoni, Biegler, Guiver & Zhao, 1998) for each input. The initial model order is set su$ciently high to ensure that a broad range of dynamics are covered. Use the initial group of "lters or the Laguerre system to form matrices (A, B) so that the matrix A is block-diagonal and each (the ith) block corresponds to an (the ith) input, as shown below. Then +X(t) " t"1,2, K,, the state-variable vectors are generated by feeding the available input data into this initial model.
A"
A 0
0
$
0
0
2 0
A
2 0
0
$
\
0 A P
,
where (A , b , cT) corresponds the ith input and repG G G resent the ith subsystem in a quasi-canonical form as shown in Fig. 1(b). (iii) Take the state variables +X(t) " t"1,2, K, as input, and +y (t) " t"1,2, K, as target outputs to H estimate the output equation coe$cients ci using PLS algorithm. This results in a linear MISO state-space model for the jth output. (iv) For each input}output subsystem corresponding to each diagonal-block matrix A , perform a model G order reduction on +A , b , cT, using PCA and interG G G nal balancing algorithm. (v) Use the reduced subsystems to re-construct MISO system matrices (A, B), generate state-variable vectors +X(t) " t"1,2, K, by feeding the available input data into this model. (vi) Check model convergence by a given threshold. If the MISO model has converged, go to step (vii), otherwise go back to step (iii). (vii) Re-construct each diagonal subsystem into a quasi-canonical form shown in Fig. 1(b) and estimate the matrix C with the PLS algorithm, until all outputs ( j"1,2, m) are modeled. 3.2. Neural net training As shown in Fig. 1(a), the neural net is attached to the linear output model in parallel. This hybrid structure provides a #exibility to combine the linear model with a nonlinear model in an adjustable extent that enhances the capability of the whole model to extrapolate. It should be noticed that the f ( ) ) in model (1) is a static ,, nonlinear element, it is supposed to map the linear state variables to process output in partial contribution. In
128
H. Zhao et al. / Control Engineering Practice 9 (2001) 125}133
fact, the neural net is trained to capture nonlinearities re#ected from the linear model output errors, i.e., to learn how to bias the linear model. Therefore, a feedforward neural net is chosen and the time series of prediction error +e(t) " t"1,2, N, between the linear model and the process output data are used as output target to train the neural net. In the actual net training, the PLS algorithm is used again to ensure a robust identi"cation. It also minimizes the internal model's dimension, which enables more e$cient training than other conventional regression and neural net training algorithms for the given amount of data. 3.3. Model conxdence index (MCI) For nonlinear processes, most identi"ed process models have a local nature because the available data for model identi"cation may not re#ect all operating conditions. It is known that a neural net model can perform well only within the range covered by training data. When such an identi"ed neural net model is used for MPC on-line control, it is necessary to monitor its performance and validity. To address this issue, the MCI, a model validity statistic, is built up when a model is developed. The Q and R statistics of the PLS model for inputs X(t) and output Y(t) are both calculated with a 95% con"dence and recorded as the MCI. Later in the on-line mode, the MCI is computed for each set of new data to check the model validity and used to adjust the combination of the two outputs, i.e., that from the linear model and from the nonlinear model. This strategy works together with an adaptive algorithm, referred to as model prediction error tracking (described below) to enhance the model's validity and robustness. 3.4. Model prediction error tracking An extended Kalman "lter (EKF) is designed to track and compensate for plant-model mismatch errors adaptively. In the model combination part, an output predictor gain and a prediction error for each process output are "ltered by the EKF. The bene"t is to e!ectively prevent the MPC controller from responding to random disturbances. Instead of using a constant output error feedback for the MPC, the hybrid model prediction will be further "ltered by the adapted `dynamica o!set and an slowly adapted `dynamica model gain. These two additional model parameters as well as their corresponding EKF, as a part of the hybrid model, are incorporated into the whole process model. As described above, the identi"cation scheme has integrated many strategies to address the common concerns of using neural net models for industrial MPC application. This scheme ensures the feasibility and robustness of the nonlinear dynamic process identi"cation.
4. The nonlinear MPC controller 4.1. Control objective The nonlinear MPC controller is based on the minimization of the following open-loop quadratic objective performance index: min J(u, y)" . #y(t#i)!y (t#i)#QWG G 1. uR>H # +\#*u(t#j)#RSH # +\#u(t#j)!u (t#j)#QSH H H P (2a) s.t. x(t#1)"Ax(t)#B u(t!q)#B v(t!f), (2b) S T y(t)"Cx(t)#f [x(t)], (2c) ,, u )u(t#j))u , (2d) * 3 y )y(t#i))y . (2e) * 3 Here, u represents a reference trajectory for the inputs, P their values are usually determined by an economic optimizer in order to obtain maximum pro"t and minimum costs; y de"nes a reference trajectory for the outputs; 1. u and u are lower and upper operating limits for u and * 3 y and y are lower and upper operating limits for y. * 3 This general objective function consists of three penalty terms among which the "rst term represents the controlled process future outputs, de"ned by the deviations from a desired response vector over a prediction horizon of length P. The second term, a penalty on the rate of change of the inputs in which *u(t#j)"u(t#j) !u(t#j!1), is often referred to as `control move suppressiona to prevent aggressive control actions. The third term, i.e. the di!erence between the control input vector and its target value, is used to control future input behavior over a control horizon of length M. The relative importance of each of these three objective functions is controlled by setting the time-dependent weight matrices Q , R and Q , of which Q and Q are positive WG SH SH WG SH semi-de"nite and R must be positive de"nite. SH 4.2. Multi-step Newton-type algorithm The control problem (2) needs to be solved on-line within each control step interval. This requires a computationally cost-e!ective nonlinear programming solver. The multi-step Newton-type algorithm for nonlinear MPC proposed by Li and Biegler (1989) and Oliveira and Biegler (1995) is found to be an e$cient method and it has been used to calculate the control output. De"ning the following vectors:
*U" *u(t)2 *u(t#1)2 2 *u(t#M!1)2 U" u(t)2 u(t#1)2 2 u(t#M!1)2
2 ,
2 ,
H. Zhao et al. / Control Engineering Practice 9 (2001) 125}133
U " u (t)2 u (t#1)2 2 u (t#M!1)2 P P P P
2 ,
s.t. U )L*U)U , *" 3" Y )S L*U)Y *" K 3"
2 Y" y(t#1)2 y(t#2)2 2 y(t#P)2 ,
with
Y " y (t#1)2 y (t#2)2 2 y (t#P)2 1. 1. 1. 1.
2
U "U !U , U "U !U , *" * 3" 3 Y "Y !Y !S (U !U ), *" * K
the MPC control problem (2) can then be expressed as min J "(Y !Y)2Q1 (Y !Y)#*U2R*U QN QN U #(U!U )2Q2 (U!U ) (3a) P P s.t. Y"YH#S U, (3b) K U )U)U (3c) * 3 Y )Y)Y . (3d) * 3 Here YH corresponds to the system response for zero input; S is the system dynamic matrix, which represents K the sensitivity of the process to input changes around a nominal trajectory, i.e. S "*Y/*U"U U ; K Q "diag+Q ,, R"diag+R ,, and Q "diag+Q , WG SH SH are weighting matrices. The Newton-type algorithm assumes that the nonlinear optimization problem is iterated starting from a nominal input trajectory until convergence. The nominal input trajectory U is used to generate state and output trajectories X and Y , and also as a reference with respect to which future control moves are evaluated. At each iteration, the algorithm requires a linearized approximation of the system around the nominal trajectory, i.e. Y) "Y #S (U!U ). K
129
De"ning U " u(t!1)2 u(t!1)2 2 u(t!1)2 Then
(4)
2
.
U"U #L*U, Y) "Y #S (U !U )#S L*U, K K where L is a lower triangular matrix with elements of one on and below the diagonal. Further de"ne Ey "Y !Y !S (U !U ) 1. K and E "U !U . S P The MPC control will be achieved by solving the following quadratic program (QP) iteratively: min J "(E !S L*U)2Q1 (E !S L*U) W K W K U #*U2R*U#(L*U!E )2Q2 (L*U!E ) (5) S S
Y "Y !Y !S (U !U ). 3" 3 K The detailed control algorithm is described by Li and Biegler (1989) and Oliveira and Biegler (1995). One of the features of this control algorithm is that it uses analytical process model derivatives. This allows making full use of the sparse state-space model (1) to calculate the derivatives more e$ciently, which speeds the on-line computation signi"cantly. According to several simulated industrial MPC problems, it has been shown that the multi-step Newton-type algorithm is 50}100 times faster than a standard nonlinear programming solver such as feasible successive quadratic programming (FSQP). This result mostly attributes to the sparse, input-decoupled and block-diagonal state-space model structure. 4.3. Constraint handling As seen from Eq. (2d, 2e), both the MV constraints and CV constraints have been considered as position bounds and handled as `hard constraintsa in the above discussion. Oliveira and Biegler (1994) showed that active CV constraints can introduce extra feedback terms in the MPC controller. This can lead to instability of the constrained closed-loop system with certain active sets, independent of the choice of tuning parameters. To cope with these problems, the l * (exact) penalty treatment, which is a `soft constrainta handling with signi"cant features (Oliveira & Biegler, 1994) has been implemented. In many existing industrial MPC products, the CV constraints are commonly handled by adding a quadratic term, i.e. the l * penalty function, to the control objec tive (Qin & Badgwell, 1997, 1998). One of the properties of this type of treatment, however, is that a violation of the original constraints is unavoidable (Oliveira & Biegler, 1994). In contrast, the l * penalty treatment elimin ates the necessity of increasing the penalty weight to in"nity to recover the original constrained solution, and therefore allows better control of the errors resulting from constraint handling. In particular, the MPC control with l * penalty treatment has stability characteristics identical to the corresponding unconstrained case and therefore is well suited for industrial application. The modi"cation of CV constraints handing by using the
130
H. Zhao et al. / Control Engineering Practice 9 (2001) 125}133
l * penalty treatment is straightforward for the nonlin ear MPC described above. The control problem will be augmented as
s.t. U )L*U)U , *" 3" Y !e)S L*U)Y #e, *" K 3" e*0;
min J "J #o2e ? U C
where o"[o o 2 o]2 is a vector of penalty parameters.
(6)
Fig. 2. (a) Two-tank CSTR pH neutralization process. (b) Model predictions vs. measurements. (c) Nonlinear MPC control vs. linear MPC control.
H. Zhao et al. / Control Engineering Practice 9 (2001) 125}133
5. Examples In this section, two examples will be presented to illustrate the application and performance of the nonlinear MPC. The "rst one is a pH control problem with a two-tank CSTR system and the second is an application in a pulverized coal-"red boiler in a power plant. 5.1. pH control problem Fig. 2(a) shows a two-tank continuous stirring tank reactor (CSTR) neutralization system. This is a 6;4 system consisting of four MVs, two DVs and four CVs. The "rst-principle model of this process is described by Zhao et al. (1997). As is well known, the pH neutralization process is a highly nonlinear dynamic process that has been widely used as a benchmark problem for modeling and control. For this two-tank system, it is known that h (CV1) and h (CV3), the levels of tank T1 and tank T2, are slightly nonlinear, whereas the pH values, pH (CV2) and pH (CV4), are highly nonlinear functions of the inputs. A nonlinear state space is identi"ed from simulated process data by using the approach described by Zhao et al. (1998). Fig. 2(b) illustrates the model predictions of the four outputs from the identi"ed linear and hybrid nonlinear state-space models. As anticipated, the linear model portion gives poor predictions
131
on the pH and pH , while the hybrid nonlinear model produces accurate predictions on the pH and pH . In this case, the hybrid nonlinear model (dash line, marked as `combineda in Fig. 2(b)) produces accurate predictions for all four outputs. The MPC controller described in this paper is applied to the simulated process and the control results are shown in Fig. 2(c). Initially, the pH value of Tank 2 was held at 7.0, a step change of 1.5 was applied to the pH setpoint of CV at the 55th sampling step, and a step disturbance of 10 l/min HAC appeared at the 100th step. It is shown that the nonlinear MPC controller resulted in excellent control performance whereas the linear controller led to an unstable closed-loop system with oscillations. In this case, even if the linear MPC controller was conservatively tuned, the closed-loop oscillatory responses were unavoidable due to the strong nonlinearity of the process. 5.2. MPC application to a pulverized coal-xred boiler The nonlinear MPC has been successfully applied to a pulverized coal "red boiler with steam capacity of 650 tons/h in a 200 MW power plant (Arbas, Chomiak, Domanski, Swirski & Neelakantan, 1997). Fig. 3(a) shows the process where the control problem is formulated as a 27;6 system. The control objectives are to (i) improve boiler e$ciency, (ii) reduce NO emissions, V
Fig. 3. (a) Pulverized coal "red boiler of Ostroleka power plant. (b) NO control test in an advisory mode. (c) MPC closed-loop control test (10 h). V
132
H. Zhao et al. / Control Engineering Practice 9 (2001) 125}133
Table 1 Inputs and outputs of the pulverized coal "red boiler model MVs Total air #ow Total air dampers (x2) Secondary air dampers (x6) OFA damper (x2)
DVs 1 2 6 2
CVs
O concentration (x2) Combustion chamber temperature (x2) Net power generated Combustion chamber pressure Flue gas fan motor power (x2) Total air fan motor power (x2) Energy produced in steam (x2) Mills data (x4)
11
and (iii) reduce lose of ignition. Listed in Table 1 are the model inputs and outputs. It is known from the industrial practice that improving boiler e$ciency and reducing NO emissions can be V di$cult in power plants "red with pulverized coal, especially those equipped with low-NO combustion sysV tems. During coal combustion, moisture and oxygen are believed to dominate the formation of NO and the V relations between them are nonlinear. To develop a process model for the nonlinear MPC, plant tests and identi"cation were performed in the plant and an open-loop model validation test indicated that the identi"ed nonlinear model is surprisingly accurate within the operating limits for predicting NO emission. To ensure a safe and V successful application, the nonlinear MPC controller was "rst run in an advisory mode on top of a WDPF distributed computer system so that the operators can compare the MPC control output with local PID loops' setpoints and make corresponding manual manipulation on the local MV setpoints. The advisory mode tests with the operators transferring the MPCs MV setpoints to regulatory controllers was conducted for many days. All mill con"gurations were tested (with 4 or all 3 combinations running). Under a normal operation with various disturbances, the MPC control running in advisory mode in Ostroleka showed very promising results: the model was su$ciently accurate to predict NO and other outputs, V the controller's behavior was as good as expected (advisory set points never occurred unreliable). To illustrate the result, a test record for a duration of 10 h, on a test day is presented in Fig. 3(b). As can be seen, the NO emission V was below the limit (460 mg/Nm) when the advisory mode was on, even reaching the value of 300 mg/Nm. An increase in NO of almost 30% was observed after V the advisory was turned o!. During this testing, no increase in the loss of ignition (LOI) was observed. The net power varied between 160 and 180 MW. A closed-loop MPC control test result was even more encouraging, which resulted in signi"cant improvement on the boiler operation. Fig. 3(c) shows a 10 h testing record and it can be seen that whenever the nonlinear
2 2 1 1 2 2 2 4 16
NO emission level (x2) CO emission level (x2) Outlet #ue gases Temperature (x2)
2 2 2
6
MPC was turn on, NO reduced signi"cantly and the V boiler e$ciency increased. The statistics from this test indicated that the NO emission was reduced 15}25%, V the boiler average e$ciency increased 0.1}0.3%, the loss of ignition was decreased by 2%. In addition, the MPC control system shown to be robust under mill changes and rapid load changes within the operating limits of 135}200 MW.
6. Conclusions A nonlinear industrial MPC controller that was recently developed by Aspen Technology Inc. is presented. It uses a state-space model with integrated PLS and neural net and a multi-step Newton-type algorithm (Oliveira & Biegler, 1994, 1995). Many features are introduced to meet the needs of industrial application and to address some important issues regarding the use of neural net model application. Key features include the use of a nonlinear state-space process model, cost-e!ective Newton-type optimization algorithm, l * (exact) out put constraints handling and many other aspects described in this paper as well. Two examples demonstrate the control performance with strong nonlinear pH process and the real application in a power plant with a signi"cant operational and economic results.
References Arbas, J., Chomiak, T., Domanski, P. D., Swirski, K., & Neelakantan, R. (1997). Pulverized coal "red boiler optimization and emissions control using neural nets and fuzzy logic. AspenWorld 97, Boston, MA, USA. Boyd, S., & Chua, L. (1985). Fading memory and problem of approximating nonlinear operators with Volterra Series. IEEE Transactions on Circuits and Systems, 32(11), 1150}1161. Froisy, J. B. (1994). Model predictive control: Past, present and future. ISA Transactions, 33(2), 235}243. Kapoor, N., McAvoy, T. J., & Marlin, T. E. (1986). E!ect of recycle structure on distillation tower time constants. A.I.Ch.E. Journal, 39(3), 446}455.
H. Zhao et al. / Control Engineering Practice 9 (2001) 125}133 Li, W. C., & Biegler, L. T. (1989). Multistep, Newton-type control strategies for constrained nonlinear processes. Chemical Engineering Research Design, 67, 562}577. Morari, M., & Lee, J. H. (1991). Model predictive control: The good, the bad, and the ugly. In Y. Arkun, & W. H. Ray (Eds.), Chemical process control-CPC IV, the fourth international conference on chemical process control (pp. 419}444). Amsterdam: Elsevier. Morari, M., & Lee, J. H. (1997). Model predictive control: Past, present and future. ADCHEM+97, Ban!, Canada. Norquay, S. J., Palazoglu, A., & Romagnoli, J. A. (1998). Model predictive control using Wiener models. Chemical Engineering Science, 53(1), 75}84. Norquay, S. J., Palazoglu, A., & Romagnoli, J. A. (1999). Application of Wiener model predictive control (WMPC) to an industrial C2splitter. Journal of Process Control, 9(6), 461}473. Oliveira, N. M. C., & Biegler, L. T. (1994). Constraint handling and stability properties of model-predictive control. A.I.Ch.E. Journal, 40(7), 1138}1155. Oliveira, N. M. C., & Biegler, L. T. (1995). An extension of Newton-type algorithms for nonlinear process control. Automatica, 31(2), 281}286. Qin, S. J., & Badgwell, T. A. (1997). An overview of industrial model predictive control technology. In: J. C. Kantor, C. E. GarcmH a, & B. Carnahan (Eds.) Fifth international conference on chemical process control CACHE, AIChE (pp. 232}256). Qin, S. J., & Badgwell, T. J. (1998). An overview of nonlinear model predictive control applications. In: F. AllgoK wer, & A. Zheng (Eds.)
133
Nonlinear model predictive control, Progress in system and control theory, Vol. 26 (pp. 369}392). Rawlings, J. B., Meadows, E. S., & Muske, K. R. (1994). Nonlinear model predictive control: A tutorial and survey, ADCHEM+94 Proceedings, Kyoto, Japan (pp. 203}214). Sentoni, G., Biegler, L., Guiver, J., & Zhao, H. (1998). State-space nonlinear process modeling: Identi"cation and universality. A.I.Ch.E. Journal, 44(10), 2229}2239. Thibault, J., & Grandjean, B. P. A. (1992). Neural networks in process control: A survey. In: K. Najim, & E. Dufour (Eds.) Advanced Control of Chemical Processes. IFAC symposium series, 8 (pp. 251}260). Yamamoto, S., & Hashimoto, I. (1991). Present status and future needs: the review from Japanese industry. In: Y. Arkun, & W. H. Ray (Eds.), Chemical Process Control-CPC IV, the fourth International conference on chemical process control, Amsterdam, The Netherlands (pp. 1}28). Zhao, H. Guiver, J., & Klimasauskas, C. (1997). NeuCOP II: A nonlinear multivariable process modeling, control and optimization package. The A.I.Ch.E. 1997 Spring National Meeting, Houston, TX, USA. Zhao, H. Guiver, J. P., & Sentoni, G. B. (1998). An identi"cation approach to nonlinear state-space model for industrial multivariable model predictive control. American control conference, Philadephia, PA, USA. Zhao, H. Guiver, J. P., & Sentoni, G. B. (1999). Identi"cation approaches for industrial nonlinear model predictive control. 14th IFAC World Congress, Beijing, China.
A nonlinear industrial model predictive controller using integrated PLS and neural net state-space model 夽 Hong Zhao *, John Guiver , Ramesh Neelakantan , Lorenz. T. Biegler Aspen Technology Inc., 1293 Eldridge Parkway, Houston, TX77077-1670, USA Department of Chemical Engineering, Carnegie Mellon University, Pittsburgh, PA 15232, USA Received 12 October 1999; accepted 9 June 2000
Abstract Model predictive control (MPC) technology has been well developed and successfully applied in the re"nery and petrochemical process industries over the last 20 years. Recent development has been focused on nonlinear MPC and robust MPC technologies because new challenges have been encountered in the polymer and chemical industries where many processes show strong nonlinearity and uncertainty. This paper presents a nonlinear industrial model predictive controller, recently developed by Aspen Technology Inc. This MPC controller uses a nonlinear, state-space, integrated partial least-squares (PLS) and neural net model (Zhao, Guiver and Sentoni, American control conference, Philadelphia, PA, USA, 1998), and a multi-step, constrained, Newton-type optimization algorithm (Oliveira and Biegler, Automatica, 31 (2) (1995) 281}286). It results in a robust and cost-e!ective industrial nonlinear MPC controller. A pH reactor example and a successful industrial application in NO emission control of a power plant are V presented to demonstrate the capability of this controller. 2001 Elsevier Science Ltd. All rights reserved. Keywords: Nonlinear control; Predictive control; State-space model; Neural nets; Industrial control
1. Introduction In the past two decades, model predictive control (MPC) technologies have been widely applied in the re"nery, petrochemical and other process industries. MPC has been recognized as an e$cient means to improve operating e$ciency and pro"tability (Yamamoto, & Hashimoto, 1991). However, industrial practice has encountered new challenges with MPC in many new areas such as the polymer industry and some chemical processes where the processes show strong nonlinear dynamic behavior. Recent research and development e!orts have been made towards addressing nonlinear process model-based MPC and robust MPC design. The next generation industrial MPC, as envisioned by several
夽
An early and shorter version of this paper by the authors, entitled &&Nonlinear industrial model predictive controller using integrated PLS and neural net state space model'' was presented at the 1999 IFAC World Congress in Beijing, P.R. China, July 1999. * Corresponding author. Tel.: #1-281-504-3269; fax: #1-281-5844329. E-mail address: [email protected] (H. Zhao).
authors (Qin & Badgwell, 1997; Froisy, 1994; Rawlings, Meadows & Muske, 1994; Morari & Lee, 1991, 1997) is likely to have the following features: nonlinear process models, state-space representation, multiple objectives, on-line adaptation and enhanced user interfaces. In the modeling and control of nonlinear processes, "rst-principle model-based MPC is a good approach and it is capable of working in a relatively broad operating range. However, developing "rst-principle models is expensive and they are not always available. Industrial practice has proven that the empirical models obtained through plant tests and identi"cation are more feasible and in most case are su$cient for MPC control. For nonlinear process modeling, neural nets have been introduced and investigated for process modeling by many researchers (Thibault & Grandjean, 1992). However, neural nets alone are not a panacea. Several concerns have been raised with regard to their application in the process industries. For example, they have been criticized for their weakness in extrapolation. To put neural nets into industrial MPC applications, some key issues such as model validity and extrapolation have to be addressed. This paper presents a new industrial nonlinear MPC
0967-0661/01/$ - see front matter 2001 Elsevier Science Ltd. All rights reserved. PII: S 0 9 6 7 - 0 6 6 1 ( 0 0 ) 0 0 1 0 1 - 5
126
H. Zhao et al. / Control Engineering Practice 9 (2001) 125}133
controller, Aspen Target2+, which uses an integrated PLS and neural net model to provide many advantages to the process industry.
2. Process model An MPC controller consists of two essential components: one is a process model which predicts the process future behavior, the other is an on-line optimizer which solves an open-loop optimal control problem at each control time interval and determines the control action for the next step. In industrial applications, usually an economic optimizer is also added to the top of the MPC controller to provide the optimal operating setpoints for controlled variables (CVs) as well as manipulated variables (MVs) to obtain the maximum pro"t without violating operating constraints. Obviously, the process model plays a very important role. The proposed state-space model structure consists of three parts as illustrated in Fig. 1(a). The "rst part of this model contains a number of internally grouped "lters that are represented by a set of linear state equations. The following two parts involve an explicit linear component and a nonlinear neural net. They are connected in parallel and are described by a linear output equation and
a nonlinear output error equation, respectively. The "nal model outputs are obtained by summing the outputs from the linear component with those from the nonlinear neural net component. The model can be written in the discrete-time state-space formulation as x(t#1)"Ax(t)#B u(t!q)#B v(t!1), S T y(t)"Cx(t)#f [x(t)], (1) ,, where y3RK is the vector of outputs, u3RP and v3RJ are, respectively, the manipulated input vector and disturbance input vector, q and 1 are dead-times of the manipulated input and disturbance input, x3RL is the vector of states, and f ( ) ) is a vector of nonlinear functions rep,, resenting the nonlinear neural net. Model (1) is an extended Wiener model, i.e. it consists of a linear dynamic element followed in series by a static nonlinear element. The only di!erence of model (1) from that of a classic Wiener model is that the linear state variables, which are usually greater than the output variables in number, are taken as inputs of the static nonlinear element. This structure o!ers more degree of freedoms to model nonlinearities in steady state as well as dynamics. Recently, Winner model has been investigated for modeling and MPC applications. Kapoor, McAvoy and Marlin (1986) demonstrated that a Winner model
Fig. 1. (a) State-space model using PLS and neural net. (b) Subsystem and corresponding state variables.
H. Zhao et al. / Control Engineering Practice 9 (2001) 125}133
can adequately represent many of the nonlinearities commonly encountered in processes such as distillation. Norquay, Palazoglu and Romagnoli (1998, 1999) have developed a MPC using Wiener models and applied it to an industrial C2-splitter. For the proposed model (1) structure, Sentoni, Biegler, Guiver and Zhao (1998) have recently proven that this speci"c model structure (decoupled A}B net), which is composed of a decoupled linear dynamic system followed by a neural net, is able to approximate every nonlinear, causal, discrete time invariant, multi-input single-output system with fading memory (Boyd & Chua, 1985). From an application point of view, model (1) structure provides several advantages over a traditional "nite impulse response (FIR) or nonlinear autoregressive with extra input (NARX) model. First, it is a parametric model with fewer parameters than a FIR or Volterra series model. Second, the nonlinear element can be implemented with feedforward neural nets (FNN) while being obtained through output-error identi"cation using an algorithm proposed by Zhao, Guiver and Sentoni (1998, 1999). Therefore, it is able to produce accurate long-term (multi-step) predictions for the MPC, which is di$cult for an NARX model using FNN to generate. Third, because the output equation is represented by an explicit combination of a linear map and a nonlinear map, the model identi"cation, then can be done in two steps: identifying a linear state-space model "rst and then a residual neural net model. In addition, with this structure, the model extrapolation capability can be enhanced by an autocombination of the two maps as well as a constrained on-line model adaptation.
3. Identi5cation of the state-space model 3.1. Identixcation of matrices (A, B, C) The identi"cation of linear part (A, B, C) uses several techniques including principal component analysis (PCA) and internal balancing-based model-order determination, engineering-oriented quasi-canonical form, and partial least-squares (PLS) parameter estimation. The PCA and internal balancing ensure the linear statespace model is a well-conditioned lower-order realization. The use of quasi-canonical form will result in a minimum of parameterization and also make state variables with engineering sense. The PLS algorithm prevents the singularity problem that may occur due to the co-linearity among multiple inputs. This guarantees robust parameter estimation. The identi"cation usually needs a few iterations depending on the nonlinearity of the process. The proposed approach assumes that the process can be represented by m MISO subsystems. For each MISO subsystem, the identi"cation process consists of the following major steps:
(i)
(ii)
127
Specify a dominant time constant value ¹ (not G critical) for each input}output pair of the process, and then construct either a group of "rst-order "lters (Zhao, Guiver & Klimasauskas, 1997) or a Laguerre system (Sentoni, Biegler, Guiver & Zhao, 1998) for each input. The initial model order is set su$ciently high to ensure that a broad range of dynamics are covered. Use the initial group of "lters or the Laguerre system to form matrices (A, B) so that the matrix A is block-diagonal and each (the ith) block corresponds to an (the ith) input, as shown below. Then +X(t) " t"1,2, K,, the state-variable vectors are generated by feeding the available input data into this initial model.
A"
A 0
0
$
0
0
2 0
A
2 0
0
$
\
0 A P
,
where (A , b , cT) corresponds the ith input and repG G G resent the ith subsystem in a quasi-canonical form as shown in Fig. 1(b). (iii) Take the state variables +X(t) " t"1,2, K, as input, and +y (t) " t"1,2, K, as target outputs to H estimate the output equation coe$cients ci using PLS algorithm. This results in a linear MISO state-space model for the jth output. (iv) For each input}output subsystem corresponding to each diagonal-block matrix A , perform a model G order reduction on +A , b , cT, using PCA and interG G G nal balancing algorithm. (v) Use the reduced subsystems to re-construct MISO system matrices (A, B), generate state-variable vectors +X(t) " t"1,2, K, by feeding the available input data into this model. (vi) Check model convergence by a given threshold. If the MISO model has converged, go to step (vii), otherwise go back to step (iii). (vii) Re-construct each diagonal subsystem into a quasi-canonical form shown in Fig. 1(b) and estimate the matrix C with the PLS algorithm, until all outputs ( j"1,2, m) are modeled. 3.2. Neural net training As shown in Fig. 1(a), the neural net is attached to the linear output model in parallel. This hybrid structure provides a #exibility to combine the linear model with a nonlinear model in an adjustable extent that enhances the capability of the whole model to extrapolate. It should be noticed that the f ( ) ) in model (1) is a static ,, nonlinear element, it is supposed to map the linear state variables to process output in partial contribution. In
128
H. Zhao et al. / Control Engineering Practice 9 (2001) 125}133
fact, the neural net is trained to capture nonlinearities re#ected from the linear model output errors, i.e., to learn how to bias the linear model. Therefore, a feedforward neural net is chosen and the time series of prediction error +e(t) " t"1,2, N, between the linear model and the process output data are used as output target to train the neural net. In the actual net training, the PLS algorithm is used again to ensure a robust identi"cation. It also minimizes the internal model's dimension, which enables more e$cient training than other conventional regression and neural net training algorithms for the given amount of data. 3.3. Model conxdence index (MCI) For nonlinear processes, most identi"ed process models have a local nature because the available data for model identi"cation may not re#ect all operating conditions. It is known that a neural net model can perform well only within the range covered by training data. When such an identi"ed neural net model is used for MPC on-line control, it is necessary to monitor its performance and validity. To address this issue, the MCI, a model validity statistic, is built up when a model is developed. The Q and R statistics of the PLS model for inputs X(t) and output Y(t) are both calculated with a 95% con"dence and recorded as the MCI. Later in the on-line mode, the MCI is computed for each set of new data to check the model validity and used to adjust the combination of the two outputs, i.e., that from the linear model and from the nonlinear model. This strategy works together with an adaptive algorithm, referred to as model prediction error tracking (described below) to enhance the model's validity and robustness. 3.4. Model prediction error tracking An extended Kalman "lter (EKF) is designed to track and compensate for plant-model mismatch errors adaptively. In the model combination part, an output predictor gain and a prediction error for each process output are "ltered by the EKF. The bene"t is to e!ectively prevent the MPC controller from responding to random disturbances. Instead of using a constant output error feedback for the MPC, the hybrid model prediction will be further "ltered by the adapted `dynamica o!set and an slowly adapted `dynamica model gain. These two additional model parameters as well as their corresponding EKF, as a part of the hybrid model, are incorporated into the whole process model. As described above, the identi"cation scheme has integrated many strategies to address the common concerns of using neural net models for industrial MPC application. This scheme ensures the feasibility and robustness of the nonlinear dynamic process identi"cation.
4. The nonlinear MPC controller 4.1. Control objective The nonlinear MPC controller is based on the minimization of the following open-loop quadratic objective performance index: min J(u, y)" . #y(t#i)!y (t#i)#QWG G 1. uR>H # +\#*u(t#j)#RSH # +\#u(t#j)!u (t#j)#QSH H H P (2a) s.t. x(t#1)"Ax(t)#B u(t!q)#B v(t!f), (2b) S T y(t)"Cx(t)#f [x(t)], (2c) ,, u )u(t#j))u , (2d) * 3 y )y(t#i))y . (2e) * 3 Here, u represents a reference trajectory for the inputs, P their values are usually determined by an economic optimizer in order to obtain maximum pro"t and minimum costs; y de"nes a reference trajectory for the outputs; 1. u and u are lower and upper operating limits for u and * 3 y and y are lower and upper operating limits for y. * 3 This general objective function consists of three penalty terms among which the "rst term represents the controlled process future outputs, de"ned by the deviations from a desired response vector over a prediction horizon of length P. The second term, a penalty on the rate of change of the inputs in which *u(t#j)"u(t#j) !u(t#j!1), is often referred to as `control move suppressiona to prevent aggressive control actions. The third term, i.e. the di!erence between the control input vector and its target value, is used to control future input behavior over a control horizon of length M. The relative importance of each of these three objective functions is controlled by setting the time-dependent weight matrices Q , R and Q , of which Q and Q are positive WG SH SH WG SH semi-de"nite and R must be positive de"nite. SH 4.2. Multi-step Newton-type algorithm The control problem (2) needs to be solved on-line within each control step interval. This requires a computationally cost-e!ective nonlinear programming solver. The multi-step Newton-type algorithm for nonlinear MPC proposed by Li and Biegler (1989) and Oliveira and Biegler (1995) is found to be an e$cient method and it has been used to calculate the control output. De"ning the following vectors:
*U" *u(t)2 *u(t#1)2 2 *u(t#M!1)2 U" u(t)2 u(t#1)2 2 u(t#M!1)2
2 ,
2 ,
H. Zhao et al. / Control Engineering Practice 9 (2001) 125}133
U " u (t)2 u (t#1)2 2 u (t#M!1)2 P P P P
2 ,
s.t. U )L*U)U , *" 3" Y )S L*U)Y *" K 3"
2 Y" y(t#1)2 y(t#2)2 2 y(t#P)2 ,
with
Y " y (t#1)2 y (t#2)2 2 y (t#P)2 1. 1. 1. 1.
2
U "U !U , U "U !U , *" * 3" 3 Y "Y !Y !S (U !U ), *" * K
the MPC control problem (2) can then be expressed as min J "(Y !Y)2Q1 (Y !Y)#*U2R*U QN QN U #(U!U )2Q2 (U!U ) (3a) P P s.t. Y"YH#S U, (3b) K U )U)U (3c) * 3 Y )Y)Y . (3d) * 3 Here YH corresponds to the system response for zero input; S is the system dynamic matrix, which represents K the sensitivity of the process to input changes around a nominal trajectory, i.e. S "*Y/*U"U U ; K Q "diag+Q ,, R"diag+R ,, and Q "diag+Q , WG SH SH are weighting matrices. The Newton-type algorithm assumes that the nonlinear optimization problem is iterated starting from a nominal input trajectory until convergence. The nominal input trajectory U is used to generate state and output trajectories X and Y , and also as a reference with respect to which future control moves are evaluated. At each iteration, the algorithm requires a linearized approximation of the system around the nominal trajectory, i.e. Y) "Y #S (U!U ). K
129
De"ning U " u(t!1)2 u(t!1)2 2 u(t!1)2 Then
(4)
2
.
U"U #L*U, Y) "Y #S (U !U )#S L*U, K K where L is a lower triangular matrix with elements of one on and below the diagonal. Further de"ne Ey "Y !Y !S (U !U ) 1. K and E "U !U . S P The MPC control will be achieved by solving the following quadratic program (QP) iteratively: min J "(E !S L*U)2Q1 (E !S L*U) W K W K U #*U2R*U#(L*U!E )2Q2 (L*U!E ) (5) S S
Y "Y !Y !S (U !U ). 3" 3 K The detailed control algorithm is described by Li and Biegler (1989) and Oliveira and Biegler (1995). One of the features of this control algorithm is that it uses analytical process model derivatives. This allows making full use of the sparse state-space model (1) to calculate the derivatives more e$ciently, which speeds the on-line computation signi"cantly. According to several simulated industrial MPC problems, it has been shown that the multi-step Newton-type algorithm is 50}100 times faster than a standard nonlinear programming solver such as feasible successive quadratic programming (FSQP). This result mostly attributes to the sparse, input-decoupled and block-diagonal state-space model structure. 4.3. Constraint handling As seen from Eq. (2d, 2e), both the MV constraints and CV constraints have been considered as position bounds and handled as `hard constraintsa in the above discussion. Oliveira and Biegler (1994) showed that active CV constraints can introduce extra feedback terms in the MPC controller. This can lead to instability of the constrained closed-loop system with certain active sets, independent of the choice of tuning parameters. To cope with these problems, the l * (exact) penalty treatment, which is a `soft constrainta handling with signi"cant features (Oliveira & Biegler, 1994) has been implemented. In many existing industrial MPC products, the CV constraints are commonly handled by adding a quadratic term, i.e. the l * penalty function, to the control objec tive (Qin & Badgwell, 1997, 1998). One of the properties of this type of treatment, however, is that a violation of the original constraints is unavoidable (Oliveira & Biegler, 1994). In contrast, the l * penalty treatment elimin ates the necessity of increasing the penalty weight to in"nity to recover the original constrained solution, and therefore allows better control of the errors resulting from constraint handling. In particular, the MPC control with l * penalty treatment has stability characteristics identical to the corresponding unconstrained case and therefore is well suited for industrial application. The modi"cation of CV constraints handing by using the
130
H. Zhao et al. / Control Engineering Practice 9 (2001) 125}133
l * penalty treatment is straightforward for the nonlin ear MPC described above. The control problem will be augmented as
s.t. U )L*U)U , *" 3" Y !e)S L*U)Y #e, *" K 3" e*0;
min J "J #o2e ? U C
where o"[o o 2 o]2 is a vector of penalty parameters.
(6)
Fig. 2. (a) Two-tank CSTR pH neutralization process. (b) Model predictions vs. measurements. (c) Nonlinear MPC control vs. linear MPC control.
H. Zhao et al. / Control Engineering Practice 9 (2001) 125}133
5. Examples In this section, two examples will be presented to illustrate the application and performance of the nonlinear MPC. The "rst one is a pH control problem with a two-tank CSTR system and the second is an application in a pulverized coal-"red boiler in a power plant. 5.1. pH control problem Fig. 2(a) shows a two-tank continuous stirring tank reactor (CSTR) neutralization system. This is a 6;4 system consisting of four MVs, two DVs and four CVs. The "rst-principle model of this process is described by Zhao et al. (1997). As is well known, the pH neutralization process is a highly nonlinear dynamic process that has been widely used as a benchmark problem for modeling and control. For this two-tank system, it is known that h (CV1) and h (CV3), the levels of tank T1 and tank T2, are slightly nonlinear, whereas the pH values, pH (CV2) and pH (CV4), are highly nonlinear functions of the inputs. A nonlinear state space is identi"ed from simulated process data by using the approach described by Zhao et al. (1998). Fig. 2(b) illustrates the model predictions of the four outputs from the identi"ed linear and hybrid nonlinear state-space models. As anticipated, the linear model portion gives poor predictions
131
on the pH and pH , while the hybrid nonlinear model produces accurate predictions on the pH and pH . In this case, the hybrid nonlinear model (dash line, marked as `combineda in Fig. 2(b)) produces accurate predictions for all four outputs. The MPC controller described in this paper is applied to the simulated process and the control results are shown in Fig. 2(c). Initially, the pH value of Tank 2 was held at 7.0, a step change of 1.5 was applied to the pH setpoint of CV at the 55th sampling step, and a step disturbance of 10 l/min HAC appeared at the 100th step. It is shown that the nonlinear MPC controller resulted in excellent control performance whereas the linear controller led to an unstable closed-loop system with oscillations. In this case, even if the linear MPC controller was conservatively tuned, the closed-loop oscillatory responses were unavoidable due to the strong nonlinearity of the process. 5.2. MPC application to a pulverized coal-xred boiler The nonlinear MPC has been successfully applied to a pulverized coal "red boiler with steam capacity of 650 tons/h in a 200 MW power plant (Arbas, Chomiak, Domanski, Swirski & Neelakantan, 1997). Fig. 3(a) shows the process where the control problem is formulated as a 27;6 system. The control objectives are to (i) improve boiler e$ciency, (ii) reduce NO emissions, V
Fig. 3. (a) Pulverized coal "red boiler of Ostroleka power plant. (b) NO control test in an advisory mode. (c) MPC closed-loop control test (10 h). V
132
H. Zhao et al. / Control Engineering Practice 9 (2001) 125}133
Table 1 Inputs and outputs of the pulverized coal "red boiler model MVs Total air #ow Total air dampers (x2) Secondary air dampers (x6) OFA damper (x2)
DVs 1 2 6 2
CVs
O concentration (x2) Combustion chamber temperature (x2) Net power generated Combustion chamber pressure Flue gas fan motor power (x2) Total air fan motor power (x2) Energy produced in steam (x2) Mills data (x4)
11
and (iii) reduce lose of ignition. Listed in Table 1 are the model inputs and outputs. It is known from the industrial practice that improving boiler e$ciency and reducing NO emissions can be V di$cult in power plants "red with pulverized coal, especially those equipped with low-NO combustion sysV tems. During coal combustion, moisture and oxygen are believed to dominate the formation of NO and the V relations between them are nonlinear. To develop a process model for the nonlinear MPC, plant tests and identi"cation were performed in the plant and an open-loop model validation test indicated that the identi"ed nonlinear model is surprisingly accurate within the operating limits for predicting NO emission. To ensure a safe and V successful application, the nonlinear MPC controller was "rst run in an advisory mode on top of a WDPF distributed computer system so that the operators can compare the MPC control output with local PID loops' setpoints and make corresponding manual manipulation on the local MV setpoints. The advisory mode tests with the operators transferring the MPCs MV setpoints to regulatory controllers was conducted for many days. All mill con"gurations were tested (with 4 or all 3 combinations running). Under a normal operation with various disturbances, the MPC control running in advisory mode in Ostroleka showed very promising results: the model was su$ciently accurate to predict NO and other outputs, V the controller's behavior was as good as expected (advisory set points never occurred unreliable). To illustrate the result, a test record for a duration of 10 h, on a test day is presented in Fig. 3(b). As can be seen, the NO emission V was below the limit (460 mg/Nm) when the advisory mode was on, even reaching the value of 300 mg/Nm. An increase in NO of almost 30% was observed after V the advisory was turned o!. During this testing, no increase in the loss of ignition (LOI) was observed. The net power varied between 160 and 180 MW. A closed-loop MPC control test result was even more encouraging, which resulted in signi"cant improvement on the boiler operation. Fig. 3(c) shows a 10 h testing record and it can be seen that whenever the nonlinear
2 2 1 1 2 2 2 4 16
NO emission level (x2) CO emission level (x2) Outlet #ue gases Temperature (x2)
2 2 2
6
MPC was turn on, NO reduced signi"cantly and the V boiler e$ciency increased. The statistics from this test indicated that the NO emission was reduced 15}25%, V the boiler average e$ciency increased 0.1}0.3%, the loss of ignition was decreased by 2%. In addition, the MPC control system shown to be robust under mill changes and rapid load changes within the operating limits of 135}200 MW.
6. Conclusions A nonlinear industrial MPC controller that was recently developed by Aspen Technology Inc. is presented. It uses a state-space model with integrated PLS and neural net and a multi-step Newton-type algorithm (Oliveira & Biegler, 1994, 1995). Many features are introduced to meet the needs of industrial application and to address some important issues regarding the use of neural net model application. Key features include the use of a nonlinear state-space process model, cost-e!ective Newton-type optimization algorithm, l * (exact) out put constraints handling and many other aspects described in this paper as well. Two examples demonstrate the control performance with strong nonlinear pH process and the real application in a power plant with a signi"cant operational and economic results.
References Arbas, J., Chomiak, T., Domanski, P. D., Swirski, K., & Neelakantan, R. (1997). Pulverized coal "red boiler optimization and emissions control using neural nets and fuzzy logic. AspenWorld 97, Boston, MA, USA. Boyd, S., & Chua, L. (1985). Fading memory and problem of approximating nonlinear operators with Volterra Series. IEEE Transactions on Circuits and Systems, 32(11), 1150}1161. Froisy, J. B. (1994). Model predictive control: Past, present and future. ISA Transactions, 33(2), 235}243. Kapoor, N., McAvoy, T. J., & Marlin, T. E. (1986). E!ect of recycle structure on distillation tower time constants. A.I.Ch.E. Journal, 39(3), 446}455.
H. Zhao et al. / Control Engineering Practice 9 (2001) 125}133 Li, W. C., & Biegler, L. T. (1989). Multistep, Newton-type control strategies for constrained nonlinear processes. Chemical Engineering Research Design, 67, 562}577. Morari, M., & Lee, J. H. (1991). Model predictive control: The good, the bad, and the ugly. In Y. Arkun, & W. H. Ray (Eds.), Chemical process control-CPC IV, the fourth international conference on chemical process control (pp. 419}444). Amsterdam: Elsevier. Morari, M., & Lee, J. H. (1997). Model predictive control: Past, present and future. ADCHEM+97, Ban!, Canada. Norquay, S. J., Palazoglu, A., & Romagnoli, J. A. (1998). Model predictive control using Wiener models. Chemical Engineering Science, 53(1), 75}84. Norquay, S. J., Palazoglu, A., & Romagnoli, J. A. (1999). Application of Wiener model predictive control (WMPC) to an industrial C2splitter. Journal of Process Control, 9(6), 461}473. Oliveira, N. M. C., & Biegler, L. T. (1994). Constraint handling and stability properties of model-predictive control. A.I.Ch.E. Journal, 40(7), 1138}1155. Oliveira, N. M. C., & Biegler, L. T. (1995). An extension of Newton-type algorithms for nonlinear process control. Automatica, 31(2), 281}286. Qin, S. J., & Badgwell, T. A. (1997). An overview of industrial model predictive control technology. In: J. C. Kantor, C. E. GarcmH a, & B. Carnahan (Eds.) Fifth international conference on chemical process control CACHE, AIChE (pp. 232}256). Qin, S. J., & Badgwell, T. J. (1998). An overview of nonlinear model predictive control applications. In: F. AllgoK wer, & A. Zheng (Eds.)
133
Nonlinear model predictive control, Progress in system and control theory, Vol. 26 (pp. 369}392). Rawlings, J. B., Meadows, E. S., & Muske, K. R. (1994). Nonlinear model predictive control: A tutorial and survey, ADCHEM+94 Proceedings, Kyoto, Japan (pp. 203}214). Sentoni, G., Biegler, L., Guiver, J., & Zhao, H. (1998). State-space nonlinear process modeling: Identi"cation and universality. A.I.Ch.E. Journal, 44(10), 2229}2239. Thibault, J., & Grandjean, B. P. A. (1992). Neural networks in process control: A survey. In: K. Najim, & E. Dufour (Eds.) Advanced Control of Chemical Processes. IFAC symposium series, 8 (pp. 251}260). Yamamoto, S., & Hashimoto, I. (1991). Present status and future needs: the review from Japanese industry. In: Y. Arkun, & W. H. Ray (Eds.), Chemical Process Control-CPC IV, the fourth International conference on chemical process control, Amsterdam, The Netherlands (pp. 1}28). Zhao, H. Guiver, J., & Klimasauskas, C. (1997). NeuCOP II: A nonlinear multivariable process modeling, control and optimization package. The A.I.Ch.E. 1997 Spring National Meeting, Houston, TX, USA. Zhao, H. Guiver, J. P., & Sentoni, G. B. (1998). An identi"cation approach to nonlinear state-space model for industrial multivariable model predictive control. American control conference, Philadephia, PA, USA. Zhao, H. Guiver, J. P., & Sentoni, G. B. (1999). Identi"cation approaches for industrial nonlinear model predictive control. 14th IFAC World Congress, Beijing, China.