Closed System Control: From Defense to Pharma

A differential equation representing some aspect of the reaction can then be considered a hyperplane within the process space. By “successively plotting” each equation’s hyperplane, an N dimensional object is created by the intersection of the hyperplanes.

 
 
This N dimensional polygon created by the hyperplane intersections is the reaction domain, RDn. The reaction domain is the geometric set of all possible states for the reaction. The reaction domain is a subspace of the process space, which in turn is a subspace of a set of real numbers. Due to conservation of mass and energy, negative quantities are not permitted, and because we are not working at the quantum level, imaginary numbers can be ignored. 

The reaction domain is a closed set, because all possible state vectors are contained within the domain and the result of any operation performed on any process variable is also contained within the domain. Because the reaction domain contains all possible states, some states or geographic regions are undesirable and need to be excluded from consideration. We accomplish this by constraining the reaction domain with additional explicit constraint equations or by applying the Lagrangian method to relevant equations.

The result is an N dimensional closed polygon of acceptable state vectors for the reaction.

The acceptable domain, “ ” provides the adaptability and flexibility that is desired for PAT. The variability of the process is captured by the breadth of the domain.

From a different perspective, we can consider the acceptable domain as representing the known reaction space, while state vectors outside the acceptable domain represent the unknown-unknowns.

 

This state vector space is closed under all operations as defined by the equations that describe the constrained N dimensional polygon. The role of a control system is to navigate the acceptable domain, given the stochastic nature of the process, and direct the reaction to the most desired state.

In performing this action, the control system may not permit the reaction to escape the acceptable domain, even during state transitions. As mentioned previously, the acceptable domain is closed for all state operations. However, a control system can cause a momentary violation if the trajectory of the state transition takes it outside the acceptable domain.



A control system move that is transitioning from two acceptable states creates a transient unacceptable state, or unknown-unknown. This situation violates our goal of creating an a priori PAT-compliant control system with no unknown-unknowns.

Currently the three traditional methods to address this problem are:
1.    Closest end-point, with single valid trajectory.
2.    Boundary skimming.
3.    Shortest in-boundary multistep route.

Each of the above will result in reaching either the terminal state or a point in close proximity without boundary violation. Unfortunately, these methods are computationally complex, time-consuming and must be determined in real time due to the stochastic nature of the variables.

This conflicts with our goal of a prior definition, design and automated creation of a reusable control system strategy.

Potential implementations
We propose an alternative method by which we transform the acceptable domain    into a convex hull:

 

In fact, according to Caratheodory’s theorem, if ADN is a subset of N dimensional vector space, or in this context, a subset of the process and reaction space, then combinations of most N+1 points are sufficient in the definition above.

By restricting the operable vector space to a convex hull, we know that any state transition will remain within the acceptable domain at all times, inclusive of the path. This is the definition of a convex hull. Any spatial coordinate within the convex hull can be reached from any other coordinate by a straight line. This also allows the closure of the reaction domain, as all states and transitions are both contained and uniquely defined.

Clearly, creating a convex hull comes at the cost of reducing the size of the acceptable domain and could pose performance penalties as N increases or the order of governing equations become more nonlinear. Luckily, this typically is not an issue in chemical and biological applications.

Because time is a nonreversible domain dimension, we can view the reaction space as evolving in time. Therefore, rather than creating a single convex hull subscribing the entire acceptable domain, we can transform the acceptable domain into a series of sequential convex hulls over time.  For example, consider the nonconvex vector space below: