Nonlinear Control Elements in VS .NET

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Nonlinear Control Elements
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FIG 5.3. A limit cycle develops, whose amplitude is that at which loop gain is 1.0.
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low controller gain. Where the process line crosses the set point is the load. Initiated by a load change, the error is converted to a change in output, which in turn is reflected by the process as a new error, which is negative. The disturbance proceeds around the loop, each cycle being attenuated because the gain product of process and controller is less than 1.0. Because the gain product of the two linear clcments in Fig. 5.2 is 0.5, the cycle is damped uniformly to >i-amplitude. Observe the graph of a pH process and a linear controller with an arbitrary gain, as displayed in Fig. 5.3. Because a pH curve exhibits high gain to small signals and low gain to large signals, loop gain may cross 1.0. If it does, a small error will be amplified, as shown in E ig. 5.3, until the amplitude is reached where the loop gain is 1.0; the loop will then oscillate uniformly at this point. Notice also that a large error will be attenuated by the pH curve above the point of cycling, so that the limit cycle will be approached from without as well as from within. The limit cycle is, then, the normal condition for this control loop. Its amplitude can be changed by adjusting the gain of the controller. A limit cycle can also be developed by the combination of a linear process and a nonlinear controller. When the proportional band of a linear controller is set too low, causing loop gain to exceed 1.0 in the linear region, the loop will eventually cycle at the limits of the controller output. A limit cycle always indicates the presence of a nonlinear element. Whether a process can tolerate a limit cycle is up to the judgment of the engineer. In order to render a decision, two factors must be determined: 1. The period of a limit cycle is found in the same way as the natural period of a linear loop. It is the period at which the phase lags of all the elements in the loop total 180 .
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FIG 5.4. A plot showing square-loop hysteresis in a valve.
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2. The amplitude of a limit cycle depends principally upon the gain of the process at its natural period. Having found this gain, an inputoutput graph or an amplit,ude-gain graph may be constructed, from which the error amplitude of the limit cycle may be measured. NONLINEAR DYNAMIC ELEMENTS
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Principal among nonlinear dynamic elements is the hysteresis 100~. In process control, the most serious form of hysteresis is encountered in control valves bothered with friction, and in on-off operators. The stem position of a control valve whose motion is opposed by friction is related to controller output in the manner described by Fig. 5.4. The part icular characteristic shown is that of square-loop hysteresis, the most severe form. Less severe Ioops wiI1 be somewhat rounded, but the worst case deserves prime consideration. The amount of hysteresis H encountered in a valve is the change in controller output required to reverse the direction of stem travel. When driven by a sine wave, a valve with hysteresis produces both phase shift and distortion. The former characteristic classifies it as a dynamic element,, whiIe the Iatter distinguishes it as being nonIinear. Controller output and stem position are plotted vs. time for a sinusoidal forcing function in Fig. 5.5. If the controller output is oscillating with a peak-to-peak ampIitude A, its unsteady-state component is 0.5A sin 4. The controller output leads valve position in amplitude by 0.5H. The phase angle of stem position
FIG 5.5. Hysteresis causes both phase lag and distortion.
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Nonlinear Control Elements
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FIG 5.6. Phase and gain both vary with the ratio of amplitude to hysteresis.
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is zero in Fig. 5.5 when the output amplitude has reached 0.5H. phase lag is then zero minus the phase of the output at that point:
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