Input multiplicities in process control occur when more than one set of manipulated parameters m can produce the desired steady state outputs c. They are directly related to the stability and desirability of any intended loop pairing of m's and c's. A chemical reactor example illustrates the phenomena.
L. B. KOPPEL School of Chemical EngineeringPurdue University West Lafayette, IN 47907
SCOPEThis work is a study of the significance of the singularity condition of catastrophe theory in the design of nonlinear multivariable control systems. It explores the relationships among three concepts: the catastrophe condition; a widely used index of interaction developed by Bristol; and, the stability of a limiting version of typical industrial control systems.Recent studies of multiple steady-state phenomena have produced interesting examples of such behavior in chemical engineering systems. In the typical framework, the process is assumed to be described by an equation of the form f(x,a) = 0, where a is a set of parameters and x is a set of process state variables. For example, in a stirred-tank reactor a may consist of parameters such as residence time, coolant temperature, activation energy, reaction velocity constant, etc., while x consists of the resulting steady-state concentration and temperature. Catasrophe theory has been shown to be very useful in delineating values of the parameters set a which can produce multiple steady states; i.e., at which more than one value of x will satisfy f(x,a) = 0.In the process control framework, elements of the variables x are typically noted as output or controlled variables c, while those elements of the parameters a which are adjustable during operation are regarded as the manipulated variables m. In this nomenclature, f(c,m) = 0. In most applications, c and rn each contain the same number of variables. These are paired, according to some control strategy, into feedback loops in which one of the manipulated variables mj is manipulated to hold one of the controlled variables ci at a desired value. Reset (integral) action is usually included in each loop. Under these circumstances, it is operationally impossible for the process to reach more than one steady state value of c, despite the existence of such alternate steady states. Alternate steady-states are eliminated by the integral action in the feedback controllers, which can come to rest only at the desired value of c.In this work, we reverse the original question and ask whether more than one set of parameters a can produce the same values of the process state variables x. Mathematically, this is simply an interchange of nomenclature between a and x. However, in the process control framework, we are now asking whether more than one set of manipulated variables m can produce the same values of the controlled variables c. This is a conceptually different question. If such multiplicity can exist, the ramifications for process control are of great importance. The typical feedback control scheme described above can conceivably oper...
