Open-loop eigenvalues

The sensitivity matrix also involves the relative changes for the dynamic characteristics of the process as these are represented by the eigenvalues of the system. The design parameters can be then used to adjust the open-loop eigenvalues. This can be considered as a pole placement procedure where instead of the controller, the design itself provides the mechanism to alter the system dynamics. 

The roots of equation (8.95) are the open-loop poles or eigenvalues. For the closed-loop system described by equation (8.94), the characteristic equation is... 

Certain quantitative measures from linear control theory may help at various steps to assess relationships between the controlled and manipulated variables. These include steady-state process gains, open-loop time constants, singular value decomposition, condition numbers, eigenvalue analysis for stability, etc. These techniques are described in... 

We show the reactor and jacket temperatures (T and T ) along with CA, the concentration of component A in the reactor. Initially, when the simulation started, the heat transfer area was sufficiently large to maintain open-loop static and dynamic stability. However, a few minutes into the simulation, we reduced UA by 20 percent. This creates dynamic instability with complex eigenvalues as in Eq. (4.25). The reactor temperature and composition start oscillating with a growing amplitude. However, the amplitude growth stops roughly 5 hours after the onset of instability and the reaction enters into a limit cycle of constant period and amplitude. 

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