Nonlinear feedback control of unstable processes



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In the last twenty years substantial efforts have been devoted to studies of dynamic behavior and stability of chemical processes. Since several investigations have shown that process performance may be optimal at an unstable steady state, a control system to provide stable operation near that state is important. Examination of the conventional approach for control at an unstable steady state illustrates several possible difficulties in its implementation. With these motivations, a new control policy for process operation near an unstable steady state is proposed and developed in the present work. The new control strategy employs a nonlinear feedback controller with hysteresis which leads to a stable small amplitude oscillation in the neighborhood of the unstable steady state. This technique requires relatively small changes in the control parameter, thus avoiding the physical constraints sometimes faced with the conventional approach. A mathematical framework is developed to design the relay and to obtain quantitative estimates of the oscillation's characteristics and stability. The procedure first requires process linearization about the unstable steady state. Describing function (DF) theory or Tsypkin's method can then be employed to analyze the resulting nonlinear feedback control system. While DF theory provides an approximate analysis for general control nonlinearities, Tsypkin's technique is an exact method restricted to the class of relay nonlinearities. Application of Tsypkin's technique to systems with low order unstable linear parts such as those of concern in this work requires several extensions to the method. This new theory has general application in nonlinear system analysis beyond the specific problem emphasized here. The mathematical formalisms are successfully applied to several cases for both a one dimensional and a two dimensional chemical reactor model. In all examples considered, both routes of analysis, DF and Tsypkin, were used to obtain approximate solutions for the limit cycles arising from the prescribed control. Exact simulations were also carried out for all examples. Comparisons among the two approximate solutions and the exact simulation results reveal that system linearization causes small errors for these examples. Errors introduced by approximating the nonlinear control element in DF analysis, however, are significant. These applications demonstrate the potential utility of the new control philosophy and the associated design and analysis procedures.