Nonlinear control of a continuous solution polymerization

Almost all chemical process systems are nonlinear. Some are only slightly nonlinear, and so can be analyzed and controlled effectively with linear techniques. Others are so nonlinear that some form of nonlinear control may 

The subset of the state vector containing the first three state variables (monomer, temperature and initiator levels) of Subsystem I (x) will be controlled. This may be written as  

The vector x, consists of the desired values of monomer concentration, temperature and initiator concentration. A is chosen to be of the form k where I is the identity matrix, and k is some scalar, k 0.ln this approach, the value of k determines the speed of response of the system to a change in setpoint or load, and is specified by the designer. Defining the error as e = (x — Xr), and combining this definition with equations (5.24) and (5.25), the dynamics of the error are given by 

The important point here is that equation (5.28) can be solved exactly for the control u required to make the error system e asymptotically stable. By equating the term in the square brackets to zero, we find  

The design matrix, A controls the exponential decay of the error, and hence, the aggressiveness of the control. For a value of A which does not saturate the controls, the approach of the controlled variables to a new setpoint will be linear and first order despite the fact that the controlled system is nonlinear. 

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