Nonlinear Control Elements in .NET

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lements with nonlinear properties appear both in processes and in their cont,rol systems. Up to this point an effort has been made to compensat e for severe nonlinear elements naturally occurring in the process, so as to maintain a const~ant loop gain. But in this chapt er the effects of variable loop gain will be thoroughly explored in a search t,o improve performance and economy. It was pointed out that even linear controllers have nonlinear regions, i.e., beyond the proportional band. These areas are ordinarily of no consequence. But in situations where they are, methods must be available to deal with them and with other nonlinearities, similarly incidental to the prime cont rol function. Sometimes nonlinear devices are used to keep costs down. For example, a const>ant-speed motor is a less expensive final operator than a variable-speed motor. But it has only three out put conditions: plus full speed, minus full speed, and stop; heme it is a nonlinear element. By the same token, a thermostat is a simpler device than a three-mode con124
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Nonlinear Control Elements
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troller. It is worthwhile to examine the applications where t hese nonlinear dcviccs may be satisfactorily employed to take advant age of their economy. Perhaps the most interesting aspect of nonlinear control devices is their intentional int roduction intd an otherwise linear loop in order to improve performance. Enough has been presented about linear controllers to promote an apprcciation for the limitat ions to which they are subject in the regulation of difficult processes. Although nonlinear control devices are not new, only recently have methods been developed which allow their performance to be evaluated in the closed loop. Consequently their employment on difficult applicat,ions can be thought of as a new technique which has not yet been widely exploited in the process industries, principally because it is not well understood. NONLINEAR ELEMENTS IN THE CLOSED LOOP Three basic forms of nonlinear clemer~ts are commonly encountered. First, there is the continuous nonlinear function, such as a pH curve or the characteristic of a control-valve plug. Second is the discontinuous fun&ion, typical of saturating types of control elements. Third is the dynamic nonlinearity, whose phase shift and gain vary wit h signal amplitude, as cont,rast,ed to linear dynamic elements, whose phase and gain vary with period. Devices exhibiting hysteresis are members of this category.
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Variable Loop Gain
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A linear cont,rol loop is identified by its constant dynamic gain, which applies the same damping to dist,urbanres of all magnitudes. This statement, holds true whether the loop consists entirely of linear clcments or includes a nonlinear function intentionally introduced to compensate another function naturally occurring in the process. In a nonlincnr cont,rol loop, gain varies with the amplitude of the oscillation. Whether loop gain varies directly or inversely with amplitude is a determining factor. Where gain increases with amplitude, small disturbances will be more heavily damped t,han large ones. Stability of the loop is then conditioned on the product of cont roller gain and amplitude of t,hc disturbxncc. A disturbance sufficiently large to cause loop gain to exceed 1.0 will trigger regenerative oscillation. Quit c another characteristic appears where loop gain varies inversely with amplitude. Small disturbances will be amplified and large ones attenuated such that the loop c~onvcrgcs to uniform oscillation where its gain is 1.0. Bccausc loop gain is 1.0 at a specific amplitude, the loop will always oscillate with that, amplitudePit is called a limit cycle.
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FIG 5.1. Whether loop gain increases or decreases with amplitude is a determining factor.
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The amplitude of the limit cycle can be changed by adjusting the gain of the controller, but it cannot he damped. Figure 5.1 illustrates both variations of nonlinear loops.
The Input-Output Graph
h tool which is of great help in envisioning what happens in a loop cont~aining a nonlinear element is the input-out put graph. It is simply a plot of the dynamic gains of both process and controller at the period of oscillation of the loop. A signal path is then formed as the wave reflected between these two halves of the control loop. Figure 5.2 is a plot of a linear process and a linear controller adjusted for s/4-amplitude damping. The coordinates are the manipulated (11~) and controlled (c) variables, the respective input and output of the process. The slope of the line representing the process is its gain at the period of oscillation. The slope of the controller line is its inverse gain at that period, opposite in effect to the process because the process output is the controller input. The lines of Fig. 5.2 indicate a high process gain and a comparatively
FIG 5.2. The cycloidal path in the input-output graph is actually a projection of the sinusoidal signal variation.
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