Quantitative Measures for Controllability and Resiliency

The quantitative assessment of the controllability and resiliency of chemical processes has generated considerable interest. The term resiliency was introduced by Morari (1983), who also pioneered qualitative measures for its assessment. Furthermore, Perkins (1989) presented an approach for the simultaneous design of processes and their control systems that addresses plantwide controllability directly. 

All of the linear C R measures use the approximations, F s and which describe the effects of the control variables and disturbances, respectively, on the process outputs. A commonly used controllability measure is the relative-gain array (RGA Bristol, 1966), which relies only on F s. The disturbance condition number (DCN Skogestad and Morari, 1987) and the disturbance cost (DC Lewin, 1996) are resiliency measures that require a disturbance model, in addition to P s. These C R measures are especially useful in 

Stages 2 and 3 of the design process (see Table 20.1) because they do not assume a controller structure or a specific controller design and tuning. 

It is assumed that each input variable is nominally at the midpoint of its range and is expressed in perturbation variable form, and scaled by dividing by its nominal value. For example, if Fj is an inlet flow rate, nominally at 500 Ibmol/hr, its operating range is 0 Fj 1,000 Ibmol/hr, in perturbation variable form, —500 F, 500, and in scaled form, -1 F, 1. Thus, F ir and F 5) are scaled by multiplying the gains in each column by the nominal value of the appropriate input variable. As a result, all of the scaled inputs vary over the same range [—1, 1]. Note, however, that the RGA is scale independent, whereas the DC is input scale dependent. 

When the controller Cj is put into manual operation, that is, when it is turned off, the process gain as seen by controller C2 is 

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This framework formed the basis for quantitative measures of dynamic resilience proposed in [3-5], A key feature of these methods is that the dynamic resilience metrics derived are plant-inherent and independent of specific controller type and tuning (within the class of linear, constant parameter controllers). A limitation is that the performance-limiting factors are considered individually, making it difficult to rank plants that exhibit combinations of these characteristics to varying degrees. 

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