James B. Rawlings Research Group


Brian Odelson
B.S. Ch.E., Purdue University, 1998
Ph.D., University of Wisconsin-Madison, 2003
odelson[AT]bevo[DOT]che[DOT]wisc[DOT]edu

Current Work

Traditional MPC requires a number of parameters to be specified by the user. The performance of the controller will depend largely on these parameters which are often unknown, or change over time. We propose a new MPC monitoring layer to use normal closed-loop input-output data to compute these quantities or at least detect when they need to be re-identified.

Proposed MPC monitoring layer

Most of our research in this area has been focused on the estimator, specifically the determination of the noise characteristics. The adaptive filtering literature (Mehra 1972) generally uses open-loop data. We have extended this field to the full MPC problem including feedback control, target tracking, disturbance modeling, and constraints. This method is based on the Markov property of the driving noise sequences. For example, in the simple open-loop case:

\begin{eqnarray*}
y_{k} &=& Cx_{k} + v_{k} \\
y_{k+1} &=& CAx_{k} + Cw_{k} + v_...
...} &=& CAAAAx_{k}+CAAAw_{k}+CAAw_{k+1}+CAw_{k+2}+Cw_{k+3}+v_{k+4}
\end{eqnarray*}

It is easy to see that the expectation of the second moment of the outputs can be expressed in terms of the driving noise covariances.

\begin{equation*}
E\left[y_{j}y_{k}^{T}\right] = CA^{j}P_{k}(A^{T})^{k}C^{T} +
...
...\begin{array}{cc}
0 & j \neq k \\
R_{v} & j = k
\end{array}
\end{equation*}

This simple result can be modified and extended to include the closed-loop cases mentioned previously. For example, we simulate the estimation of a $2\times2$ covariance matrix for a closed-loop constrained MPC problem. The best representation of a 3-dimensional probability distribution is to look at a slice of constant probability, as shown below. This results in a 2-dimensional ellipse which illustrates the dynamic behavior.

Evaluating 3-D probability distributions

Here is a recent presentation given at Eastman Chemical in Kingsport, TN. The following image is linked to an MPEG file which gives an animated depiction of the process.

The plot is divided into three sections. The bottom third are the two inputs of the system, one of which is constrained until $k=1000$. The middle third are the two outputs of the system and their setpoints. Notice that the setpoints are unreachable while the input is saturated. Finally, the upper third are the noise covariances. The red ellipse is the actual noise covariance in the plant, and the blue ellipse is the estimated covariance.

Animated example

A copy of my preliminary report for my dissertation is also available.

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

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University of Wisconsin
Department of Chemical Engineering
Madison WI 53706