Also, for large-scale systems nonlinear MPC. The basic idea is to: the computation time may be a significant fraction of the sample interval. Aufderheide , bequette rpi. Aufderheide, B. Multiple linear model approaches tions. Simulation results are presented in Section 5, followed by a discussion and conclusions in Section 6. Much of the early work in multiple linear model- based approaches was inspired by challenges in the aerospace industry.
Although aircraft dynamic behavior is known to be nonlinear, specifications are normally 1. Fundamental model approaches developed based on linear models at a number of equilibrium flight conditions.
A multiple model adaptive Various early versions of nonlinear model predictive control MMAC approach was developed by Athans et control were focused on die development of efficient al. More recent efforts, similar to the technique using multiple PI controllers in several the work in linear model predictive control, have simulation examples. A nice tutorial overview of 1.
The continuous schedule QDMC. In this NL-QDMC approach, a nonlinear of the parameters based on the steam load is obtained by model was integrated once over the prediction horizon, interpolation of a fixed set of models. Azimzadeh, then a linear perturbation model was used for the Palizban and Romagnoli fit local model para- optimization iterations. This resulted in a convex meters based on the responses of a fundamental model optimization problem and enabled the use of proven of a batch fermentation process.
This NL-QDMC approach was that relies on the use of a bank of linear models to extended by Gattu and Zafiriou , who used a describe the dynamic behavior over a wide operating steady-state Kalman filter for state estimation; this range. A recursive Bayesian scheme assigns weights to allowed the application to open-loop unstable systems.
The combined-weighted model is then used A further improvement was the EKF-based nonlinear for the predictions in the optimal control move calcula- model predictive controller NL-MPC strategy of Lee tion.
Our initial research efforts in this area were applied and Ricker Doyle and Wisnewski applied a similar control DMC , to handle different operating regions strategy to a Continuous Kamyr Digester.
This is done by using a multiple model framework of step response models. Two differ- ent model banks will be tested: one uses actual step 1. The based strategies have been applied in industry, primarily minimal knowledge bank requires only the range of due to the effort and difficulty in developing a good gains, dominant time constants, and time delay approx- nonlinear model.
A fundamental model based on first imations for minimum phase behavior of the system. These two controllers are compared with the terms of basic fundamental knowledge about a system.
Van de Vusse reaction system moves to the left half plane and the gain now becomes negative. The gain at the optimum point is zero. Control The Van de Vusse reaction consists of two decom- is very challenging since the desired operating point has position reactions of A taking place in parallel. The zero gain and the dynamics on either side are very desired product is the concentration of B, Cb mol B different.
The manipulated input, u, is the dilution rate in 3. The rate constants, k1, k2, and k3 have two sets of kinetic parameters used in this simulation study: 1st Fig. The first set is multiple model adaptive estimator coupled with a single the nominal set used in all cases.
The second set is used model predictive controller. The multiple model struc- as a test of model uncertainty.
The control objective is to nonlinear model is not practical or possible. The operate as near as possible to the optimum point to advantage of MMPC is that a large number of models maximize the concentration of B. Operating points on can be used and constraints explicitly handled. The the left side of the optimum are non-minimum phase. As issues for MMPC are determining the number and type the dilution rate is increased the right half plane zero of models to encompass the plant behavior and the need Fig.
The multiple model adaptive estima- The prediction and corrected prediction models are: tors used in this paper are banks of step response models N and are essentially an extension of DMC to a multiple X model framework. N is the model horizon and typically is prevalent model used in predictive control starting from equivalent to the settling time.
Although limited to describing only one fixed linearization of an open-loop umin 5u l 5umax stable plant, it is intuitive, very simple to implement, and requires no fundamental modeling whatsoever. Single model DMC optimization errors, r l is the desired output trajectory, u l is the vector of manipulated variables, and Q and R are the To better understand the implementation of the output and input weighting matrices.
Absolute and multiple model banks we will briefly give an overview velocity constraints on the manipulated variable are of DMC optimization that will be referred to later to included. There is a prediction horizon of P steps with M show exactly how existing DMC programs can be control moves. At the next sampling time the weighted the minimal knowledge model bank discussed in the model bank is updated and the optimization is done next section is a possibility.
This is an issue 3. Actual step response models as a model bank in any multiple model control framework. One ap- proach is to have a supervisory level which determines The following common process control issues can be when each model is used, such as traditional gain dealt with by using a multiple model bank of step scheduling where the actual input or output determines response models: when a model is used.
Each step response model then is this model is equivalent to the actual plant? In essence, included in the model bank. The resulting weights for each model The number of step responses will be based on the range are bounded between zero and one and the sum of all of operability desired and the allowable maximum the weights equals one.
The weighting scheme will be distance between adjacent models. To handle parameter discussed in more detail in Section 3. The mathema- disturbances requires that the new set of parameter s be tical implementation provided next can also be used for relatively constant over the identification of the step a process with multiple inputs and outputs without any response models.
For example, problems with feed modification beyond the inputs and outputs becoming concentrations to a reactor due to poor control of vectors. This calculates the predicted other identification techniques may have been used to bias, ybias from a step response model that was taken at develop the nominal step response model for the process a different steady state uss , yss than the process is at being operated by DMC.
If the one time cost of testing initially u0. By keeping For a single step response model, the underlying the bias term, the steady state information is preserved assumption is that the process is always initiated at the and the models will have values that are correct for their exact same steady state conditions of the step response specific region of state space.
Likewise for a single model so no bias term is required. This term maintains the step models and individual weights wj: response model at steady state into the future when time Nm has exceeded the model horizon of the step response model. The total number of models is for nine inputs, disturbance term together to form a pseudo-additive three feed concentrations, and for two sets of kinetic disturbance term: parameters for a combination of 54 models. P where P is the prediction horizon.
Much of this work has disturbance term. Received : 24 January Accepted : 16 July Issue Date : July Anyone you share the following link with will be able to read this content:. Sorry, a shareable link is not currently available for this article. Provided by the Springer Nature SharedIt content-sharing initiative. Skip to main content. Search SpringerLink Search. References Cutler, C. Article Google Scholar Maiti, S.
However, simple Chemical Process Modeling and Control nonlinear output transformations suggested by the fundamental design Research Center and equations of the process give models that are quite insensitive to non- Department of Chemical Engineering Lehigh University linearities of the process. The complexity of DMC when it is applied on systems with very large time constants is also addressed.
Suggestions to overcome this problem are zyxwvutsrqp proposed. Introduction purities 0. They reported that it is very difficult to The strong nonlinearities of many chemical processes such as obtain a representative process model for moderate and high- high-purity distillation columns limit the application of model- purity columns.
To overcome the nonlinearity problem, they based linear multivariable controllers, such as DMC Cutler, suggested the use of on-line multivariable gain and time con- and IMC Garcia and Morari, , to relatively small stant scheduling techniques to update the process model.
This ranges of the operating conditions. Imprecision of the process results in nonlinear DMC that gives better performance than model makes it difficult to design model-based controllers even the standard DMC. However, they did not report any systematic for fixed operating conditions. Maurath et al. In this paper, multivariable predictive control, based on DMC principles, to we suggest a simpler approach to overcome the strong nonlinear- two-point composition control of a low-purity column and ity of the high-purity columns.
This paper focuses on the application of DMC columns to high-purity columns, the regulatory performance of to different column designs with product purities ranging from DMC becomes worse than the conventional diagonal propor- 10, ppm to 10 ppm. It compares the regulatory response tional-integral PI controllers. Then, nonlinear output transfor- feed composition disturbance of conventional diagonal control mations, often suggested by an analysis of the fundamental structures LV with that of multivariable DMC controller.
These transformations the linear model inaccurate. The the- oretical basis of these output nonlinear transformations is found in the work of Koung and Harris More elaborate nonlin- Present addrcss A. Gmrgiou is Chemical Engineering Dept. Princeton ear transformations for high-purity columns have also been sug- NJ Correspondencemnccrning this paper should be a d d r e s d to W. The objectives of this paper are: a to compare conventional diagonal control with the D M C design for moderate and high purity columns; and b to show that the performance of DMC can be significantly improved by the use of nonlinear transfor- mation of the composition measurements.
The tuning factor K affects directly the performance and robustness of the system. A theo- retical analysis of the effect of the design parameters on the robustness and performance of DMC is given by Ogunnaike b. The extension of DMC to multiloop systems is straightfor- ward by partitioning matrices. Based on a linear pro- and very-high-purity columns. Column A is a moderate-purity col- for a specified performance index.
For a detailed description, see 10,ppm impurity at the top and bottom, respectively. Col- Cutler The dynamic simulation was based on rigorous tray-to-tray calculations by making the assumption of negligible pressure drop in the column, but by taking into account the effects of nonequimolal overflow and nonideal vapor-liquid equilibrium. The composition analyzer dynamics are treated as process time delays equal to 6 minutes.
Column C is a very-high-purity ppm column studied by Fuentes and Luyben and Luyben Table lbgives The DMC algorithm minimizes the square of the deviation steady-state design parameters for the column. Assumptions between the predicted output trajectory and the setpoint values were constant relative volatility, theoretical trays, total condens- at R future sampling periods by solving the constrained least er, partial reboiler, equimolal overflow, saturated liquid feed squares minimization problem and reflux, and perfect level controllers constant levels in the.
K is the suppression factor or tun- ing parameter that penalizes the objective function for changes 3 Table la. Distillate composition mol frac. Column A In Bottoms composition mol frac. Reflux ratio 0. Steady-State Design for Column C High-purity columns For the high-purity columns, pulse testing results were diffi- Composition mol frac. Distillate, x, 0. Depending on the size and direction of the input Bottom, x, 0. Open-loop Distillate, D They reported that the process gains, time constants, Reflux, L These large dif- Trays from bottom Total, NT 40 ferences in the time constants, deadtimes, and gains illustrate Feed, NF 16 the difficulty of determining suitable models for columns B and Holdups kmol Base 5 min C.
This method was tested in several high-purity columns and it 0. The ATV method is 0. Both columns are modeled as second-order sys- Transmitter spans Compositions 50 ppm tems with deadtime. Select a Web Site. Choose a web site to get translated content where available and see local events and offers. Based on your location, we recommend that you select:. Select the China site in Chinese or English for best site performance. Other MathWorks country sites are not optimized for visits from your location.
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