Process Operability Algorithms#

There are two main operations that may be needed to perform an operability analysis:

  1. Operability sets quantification: Obtaining and quantifying the AIS, AOS, DOS, DIS, EDS, and so on. This yields insights into the achievability of a given process objectives. If the analysis is in low dimensions (\(\leq3\)), it can be performed visually. In higher dimensions, the OI still serves as a valuable metric to assess the operability of a process. Irrespective of dimensionality, computational geometry algorithms and polytopic calculations are necessary.

  2. Inverse mapping: From a DOS, evaluate the corresponding DIS. For a general nonlinear process model, frequently represented in mathematical terms as a nonlinear system of equations as vector-valued functions, this may be a non-trivial task.

Algorithms have been developed in the literature to address these challenges. For quantification of the operability sets and thus, the OI itself, the multimodel approach has been developed. For inverse mapping, the nonlinear programming-based (NLP-based) and the implicit mapping approaches have been successfully employed to evaluate the inverse map of a given process model. This section will go briefly over these methods but the reader is encouraged to go over the Bibliography for a more thorough explanation of them.

Multimodel Approach#

The multimodel approach [6, 7, 8] employs the paradigm of a series of polytopes being able to represent any nonlinear space. This approach simplifies the calculation of operability sets since polytopes are by definition convex and can be described by their half-space representation (\(\mathcal{H}-rep\)) or vertex representation (\(\mathcal{V}-rep\)), as a system of linear inequality constraints.

Illustrative Example

Let’s consider the schematic below, in which each point corresponds to a coordinate in the input space, and their respective coordinate in the output space is obtained through the process model \(M\). Paired polytopes (color coded) can be “drawn” to represent the AOS accordingly:

_images/multimodel_01.gif

Fig. 12 AIS-AOS representation using paired polytopes.#

In the animation above, \(P_1^u\) is paired with \(P_1^y\) and so on, in the general form \(P_k = \{P_k^u,P_k^y\}\)

Another example in which the non-convex AOS is approximated as a series of paired polytopes can be depicted in the animation below, in which one can see that the polytopes approximate the overall non-convex AOS region with relative accuracy, as well as the DOS and DIS:

_images/multimodel_02.gif

Fig. 13 AIS-AOS polytopic approximation#

Due to its intrinsic roots in computational geometry and linear programming, the multimodel approach is a suitable process operability algorithm for:

  1. Replacing non-convex regions with paired polytopes, allowing efficient OI computation and representation of the operability sets.

  2. OI evaluation, allowing to rank competing design and control structures.

Nonlinear Programming-Based (NLP-based) Approach#

The NLP-based approach [4, 5] converts the inverse mapping task into a nonlinear programming formulation. The premise of this algorithm is that the DOS can be discretized into a series of coordinate points in the output space (AOS/DOS) and that an objective function of error minimization nature (e.g., Euclidean distance) is posed between the feasible operation and desired operation (DOS). The solution to the nonlinear programming problem at each discretized point is the DIS that attains the operation of the DOS. Mathematically, this can be posed as the following NLP optimization problem:

\[\begin{split}\begin{gathered} \emptyset_{\mathrm{k}}=\min _{\mathrm{u}_k^*} \sum_{j=1}^n\left(\left(\mathrm{y}_{j, k}-\mathrm{y}_{j, k}^*\right) / \mathrm{y}_{j, k}\right)^2 \\ \text { s.t: process model } (M) \\ \mathrm{u}_k^{\min } \leq \mathrm{u}_k^* \leq \mathrm{u}_k^{\max } \\ \mathbf{c}_1\left(\mathrm{u}_k^*\right) \leq 0 \end{gathered}\end{split}\]

for \(j=1:n\) output variables in the AOS/DOS, and \(k\) discretized points. Lastly, nonlinear constraints \(c\) might be imposed to the inverse map problem if needed; these can be a result of product specifications, equipment limitations and so on.

Illustrative Example

In the animation below, the optimization problem is posed for each discretized DOS point. Then, for each point, a corresponding feasible DIS solution is obtained. Due to the nature of the objective function (error/distance minimization), the DOS points will be shifted as close as possible to enable feasible operation represented by DOS* and DIS*:

_images/nlp_01.gif

Fig. 14 Inverse mapping using NLP-based approach#

The main features of the NLP-based approach are

  1. Obtaining feasible operability sets, since the output points are shifted to be as close as possible to enable feasible operation.

  2. Searching for new AIS unexplored regions, giving insights about process feasibility. This is particularly useful in finding new designs and/or material properties based on operability analysis.

  3. Imposing constraints to the inverse mapping operation, allowing for searching for regions that might be limited to constraints related to market demands, product specifications and material limitations.

Since a successful solution of an NLP is always feasible, the DOS and DIS that achieve the error minimization between feasible and desired operation are named slightly differently as the Feasible Desired Output Set (DOS*) and Feasible Desired Input Set (DIS*).

Lastly, the NLP-based approach can be extended to encompass the search for intensified and/or modular designs, as proposed in the literature [4].