Niederlinski index

A fairly useful stability analysis method is the Niederlinski index. It can be used to eliminate unworkable pairings of variables at an early stage in the design. The settings of the controllers do not have to be known, but it applies only when integral action is used in all loops. It uses only the steadystate gains of the process transfer function matrix. 

The method is a necessary but not sufficient condition for stability of a closedloop system with integral action If the index is negative, the system will be unstable for any controller settings (this is called integral instability ). If the 

Example 16.3. Calculate the Niederlinski index for the Wood and Berry column. 

Since the NI is positive, the closedloop system with the specified pairing may be stable. 

Notice that the pairing assumes distillate composition jtp is controlled by reflux R and bottoms composition Xg is controlled by vapor boilup V. 

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The RGA is useful for avoiding poor pairings. If the diagonal element in the RGA is negative, the system may show integral instability the same situation that we discussed in the use of the Niederlinski index. Very large values of the RGA indicate that the system can be quite sensitive to changes in the parameter values. 

For a 2 X 2 system, a negative Niederlinski index is equivalent to pairing on a negative RGA element. For example, in Example 16.7 the RGA element is positive (2.01). This says that the Xj, to i pairing is okay. However, the 12 element is negative (— 1.01), telling us that the Xp to V pairing is not okay. 

Suitable pairings should give an RGA number close to 0 at crossover frequency. Niederlinski index... 

A simple method for avoiding poor pairing is based on the computation of the Niederlinski index. Note that integral action must be used in all controllers. It needs only the knowledge of the steady-state gains K,. The formula is ... 

The necessary but not sufficient condition for stability is that NI for the selected pairing be positive. For a 2 x 2 system, the RGA and Niederlinski index are equivalent. This is not true, however, for larger systems. 

Eliminate unfeasible pairing that gives a negative Niederlinski index. 

Calculate the RGA, Niederlinski index, minimum singular value, and condition number of the following matrix of steady-state gains. 

After defining the sets of controlled and manipulated variables, steady state controllability indices like Niederlinski index, relative gain array, Morari resiliency index, condition number are determined (e.g. Luyben, 1990). The evaluation of the... 

Table 1 presents diagonal elements of the RGA matrix for the most promising pairings. All diagonal control structures are feasible at steady state, but with different pattern of interactions. Niederlinski index NI, used to assess the stability of pairings, is positive in all the control structures. 

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