James B. Rawlings Research Group


Murali R. Rajamani
B. Chem. Engg., University of Mumbai-Institute of Chemical Technology, 2002
Ph.D., University of Wisconsin-Madison, 2007
rajamani[AT]bevo[DOT]che[DOT]wisc[DOT]edu

Current Work

Model Predictive Control (MPC) is one of the most widely used control techniques in the chemical industry. As its name suggests, MPC uses a model to predict the inputs that will give optimal control. MPC casts the control problem in the form of an optimization, which makes it convenient to handle constraints and nonlinear models explicitly. The three components of MPC are the regulator/controller, the estimator and the target calculator. A simple block representation of MPC is shown in Figure 1.

Figure 1. Model Predictive Control-block diagram

It is usually convenient to start with a discrete time state space model of the following form (may be a minimal realization of a transfer function):

$\displaystyle x_{k+1}$ $\displaystyle = Ax_k + Bu_k + w_k$    
$\displaystyle y_k$ $\displaystyle = Cx_k + Du_k + v_k$    

in which, $ A$, $ B$, $ C$ and $ D$ are known; $ x$, $ u$ and $ y$ respectively represent the states, inputs and outputs; and $ w_k$ and $ v_k$ are the state and output noises.

The estimator part of the MPC uses the past outputs and inputs to calculate the optimal estimate of the current state of the plant. The regulator on the other hand predicts the optimal inputs over a future horizon. The operator must choose the parameters for the regulator ($ Q$ and $ R$) and the estimator ($ Q_w$ and $ R_v$). The tuning parameters for the regulator are straightforward to choose. The parameters for the estimator on the other hand are not intuitive and require the knowledge of the disturbances entering the plant.

Usually only closed loop data is available from the plant, in which case an autocovariance least squares (ALS) technique can be used to find the optimal parameters for the estimator.

Example of industrial application

The technique was applied to data from a gas-phase reactor at Eastman chemical company to update the estimator. The process is represented in Figure 2.

Figure 2. Eastman gas phase reactor control

By calculating the new parameters for the estimator from data, the error in the predicition of the states is reduced, which can be seen in Figures 3 and 4.

Figure 3. Composition prediction error frequency-original and updated estimator.

Figure 4. Temperature prediction error frequency-original and updated estimator.

Example of laboratory scale CSTR

Figure 5. Laboratory setup to verify technique

Figure 5 shows a setup of a laboratory CSTR where acetic anhydride mixes with water to give acetic acid. This CSTR follows a non linear model due to the temperature effects. The above technique was also used to find the optimal parameters for the estimator when uncharacterised disturbances entered the plant. An example of the improved performance with the updated filter gain is shown in Figure 6 for the CSTR.

Figure 6. Output while rejecting a
deterministic output disturbance.

The goal of our research is to extend the technique to non-linear models so that parameters for the Extended Kalman Filter (EKF) and the Moving Horizon Estimator (MHE) can be estimated. Validation of the technique will be done using the laboratory CSTR.

Publications

1
Murali R. Rajamani.
Data-based Techniques to Improve State Estimation in Model Predictive Control.
PhD thesis, University of Wisconsin-Madison, October 2007.

2
Murali R. Rajamani and James B. Rawlings.
Estimation of the disturbance structure from data using semidefinite programming and optimal weighting.
Technical Report 2007-02, TWMCC, Department of Chemical and Biological Engineering, University of Wisconsin-Madison (Available at http://jbrwww.che.wisc.edu/tech-reports.html), June 2007.

3
James B. Rawlings and Murali R. Rajamani.
A hybrid approach for state estimation: Combining moving horizon estimation and particle filtering.
In Sandia CSRI Workshop, Large-Scale Inverse Problems and Quantification of Uncertainty, Santa Fe, New Mexico, September 2007.

4
Murali R. Rajamani, James B. Rawlings, and Tyler A. Soderstrom.
Application of a new data-based covariance estimation technique to a nonlinear industrial blending drum.
Submitted for publication in IEEE Transactions on Control Systems Technology, September, 2007.

5
Murali R. Rajamani, James B. Rawlings, and S. Joe Qin.
Achieving state estimation equivalence for misassigned disturbances in offset-free model predictive control.
Submitted for publication in AIChE Journal, August, 2007.

6
Murali R. Rajamani and James B. Rawlings.
Estimation of the disturbance structure from data using semidefinite programming and optimal weighting.
Accepted for publication in Automatica, January, 2008.

7
Wu Zhuang, Murali R. Rajamani, James B. Rawlings, and Jakob Stoustrup.
Application of autocovariance least-squares method for model predictive control of hybrid ventilation in livestock stable.
In Proceedings of the American Control Conference, New York, USA, July 11-13 2007.

8
Murali R. Rajamani and James B. Rawlings.
Improved state estimation using a combination of moving horizon estimator and particle filters.
In Proceedings of the American Control Conference, New York, USA, July 11-13 2007.

9
Wu Zhuang, Murali R. Rajamani, James B. Rawlings, and Jakob Stoustrup.
Model predictive control of thermal comfort and indoor air quality in livestock stable.
In Proceedings of the European Control Conference, Kos, Greece, 2-5 July 2007.

10
Murali R. Rajamani and James B. Rawlings.
Estimation of noise covariances and disturbance structure from data using least squares with optimal weighting.
In AIChE Annual Meeting, San Francisco, California, November 2006.

11
Murali R. Rajamani, James B. Rawlings, and S. Joe Qin.
Equivalence of MPC disturbance models identified from data.
In Proceedings of Chemical Process Control 7, Lake Louise, Alberta, Canada, January 2006.

12
Brian J. Odelson, Murali R. Rajamani, and James B. Rawlings.
A new autocovariance least-squares method for estimating noise covariances.
Automatica, 42(2):303-308, February 2006.

Personal Web Page: jbrwww.che.wisc.edu/~rajamani

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