Niederlinski

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. 

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

Unfortunately much of Ais interaction analysis work has clouded the issue of how to design an effective control system for a multivariable process. In most process control applications the problem is not setpoint responses but load responses. We want a sy stem that holds the process at the desired values in the face of load disturbances. Interaction is therefore not necessarily bad, and in fact in some systems it helps in rejecting the effects of load disturbances. Niederlinski [AIChE J 1971, Vol 17, p. 1261) showed in an early paper that the use of decouplers made the load rejection worse. 

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. 

Eliminate unworkable variable pairings. Eliminate pairings with negative Niederlinski indices or that have obvious poor dynamic relationships. 

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. 

Luyben has discussed two approaches to decoupling, including one he terms ideal, but that offers limited benefits and is more complex to implement. Later Luyben and Vinante tested decoupling experimentally on a semiworks column. Since the column had little interaction, benefits of decoupling were minimal. Niederlinski and Waller have also studied decoupling. 

Niederlinski, A., A Heuristic Approach to the Design of Line2ir Multi-variable Interacting Control Systems, Automatica, 1,691 (1971). 

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