Chapter 6 Selecting State-Specific Key Variables
6.2.3 Methodology for finding each type of key variable
Since it is not easy to demarcate two adjacent states clearly, we need to use some simple operations for separation. State-indication variables are used for this purpose.
Most of the time, the state-indication variable can be identified from the standard operation procedure (SOP) which specify the operations to be performed in different states. Examples include status of valve, equipment status, etc. Special valve or equipment status could also be use for this class of key variables.
State-differentiation Variable
The state-differentiation variable is determined based on signals that have different values in each state. In this chapter, we use a method motivated by the Analytical Hierarchy Process (AHP) popular in decision-making, to find the best subset of variables for separating the different states of a process transition.
First the differentiability of a single variable is measured. Consider a variable x whose signalT ={ , ,..., ,... }x x1 2 xi xm corresponds to all states of the process’ operation.
Let T' { ,= x xi i+1,..., }xj be the signal from a specific state MS for which the suitability of variable x to serve as a key variable is being considered. The state-differentiability
ϑ of signal T during MS is defined based on the average overlap of the max and min values of every non-extreme three-sample window in 'T with T:
1
1 1 1 1
1 1 1 1 1
Total number{min( , , ) ' max( , , )}
1
1 Total number{min( , , ) max( , , )}
j
n n n n n n
n i n n n n n n
x x x T x x x
j i x x x T x x x
ϑ − − + − +
= + − + − +
≤ ≤
= − − ∑ ≤ ≤ (6-1)
1 1
2 j i
m− − < ≤ϑ
−
Here the operation Total number( )calculates the number of times the condition in the operand is satisfied by the given signal. A small ϑ indicates that variable x cannot be used for positively identifying state MS, while a value of ϑ=1indicates that the variable can conclusively locate when the process is in MS. While ϑ can indicate the state-differentiability of a single variable, it is necessary to identify the state- differentiability of a set of variables. The extension to the multivariate case where the differentiability of a set of variables is considered next.
Let the process be operated in s states. A subset q key variables from the total of Q measurements has to be selected so as to provide best overall differentiability of every state.
The Q x s dimensioned variable differentiability matrix A is first calculated by calculating the state-differentiability of each variable for each state by following the above procedure. Let the relative importance of differentiating any state be κ.
0≤ ≤κ 1 The relative importance matrix for all the states is constructed asB=[ ... ]κ κ1 2 κs . One way to estimate the relative importance of a state is based on its extent relative to other states. The correlation among the Q variables is measured using their covariance C. The following decision procedure is then used to select the q key variables using A, B, and C:
1. Calculate the overall differentiability matrix W =A B* and select the variable whose coefficient in W is the largest. This yth variable is deemed to be a State- differentiation key variable.
2. Each element of the variable differentiability matrix A is then updated using the covariance matrix C as follows: Am n, = Am n, *(1−C y m( , )) . Step 1 is then repeated to find the overall differentiability matrix and select the next best State- differentiation key variable. This is repeated until max (W) < a user specified threshold or q variables have been selected.
The above procedure provides a systematic way to discount variables that are correlated and is necessary since many variables offer only similar information.
State-progression Variable
These key variables are identified based on the variable’s ability in indicating the progress of a state. These can be identified from the nature of the state. If a variable shows different values during the progression of a state, it is suitable for use as a state- progression variable. Consider a signalT ={ , ,..., ,... }x x1 2 xi xn . The variable’s ability to indicate the progress of a state MS iscalculated based on its uniqueness index ςwithin that state:
1
2 1 1 1 1
1 1
2 {min( , , ) max( , , )}
n
i i i i i i i
n Total number x x x T x x x
ς −
= − + − +
= − ∑ ≤ ≤ (6-14)
As in the case of state-differentiation variable, state-progression is calculated based on how many other points are located in the [min(xi−1, ,x xi i+1),max(xi−1, ,x xi i+1)] range during the given state. 1/n< ≤ς 1. A small value of ς reveals that the variable has a poor capability to serve as a state-progression key variable while ς =1indicates that every value of the variable is unique through the state; thus the variable can
exactly reveal the progression of the state. The method described above for identifying sets of variables that together have high differentiation ability can also be used here to find a set of state-progression variables.
External-affect Variable
The external-affect variable is identified based on the definition of the plant sections. It should be noted that a variable that is the input to a given section would be an output from another section. Identification of these kinds of variables has to be based on the process flow sheet and the division of the process into sections.
Active Variable
In this work, signals that are active in a state are identified from the number of singular points. A signal that has more singular points within a given period (state) is more active than another variable with fewer singular points. The algorithm for identifying singular points has been reported earlier. Other analysis techniques such as PCA and ICA may also be used for evaluating signal activity.
Important-balance Variable
Identification of such variables is based on the heat and mass balance of the process. All transient process states – normal transitions and all abnormal operations – occur when a steady-state balance is broken. Process knowledge is important to find the group of variables that affect a specific balance.
In the next Section, we describe a systematic methodology for selecting key variables in real-time.