Файл:PID Compensation Animated.gif

Описание

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The PID controller has the standard form K_p+K_i/s+K_ds. Therefore, the closed loop transfer function of the whole system becomes

[math]\displaystyle{ \lt semantics\gt \lt mrow class="MJX-TeXAtom-ORD"\gt \lt mstyle displaystyle="true" scriptlevel="0"\gt \lt mi\gt G\lt /mi\gt \lt mo\gt =\lt /mo\gt \lt mrow class="MJX-TeXAtom-ORD"\gt \lt mfrac\gt \lt mrow\gt \lt msub\gt \lt mi\gt G\lt /mi\gt \lt mrow class="MJX-TeXAtom-ORD"\gt \lt mi\gt c\lt /mi\gt \lt /mrow\gt \lt /msub\gt \lt msub\gt \lt mi\gt G\lt /mi\gt \lt mrow class="MJX-TeXAtom-ORD"\gt \lt mi\gt p\lt /mi\gt \lt /mrow\gt \lt /msub\gt \lt /mrow\gt \lt mrow\gt \lt mn\gt 1\lt /mn\gt \lt mo\gt +\lt /mo\gt \lt msub\gt \lt mi\gt G\lt /mi\gt \lt mrow class="MJX-TeXAtom-ORD"\gt \lt mi\gt c\lt /mi\gt \lt /mrow\gt \lt /msub\gt \lt msub\gt \lt mi\gt G\lt /mi\gt \lt mrow class="MJX-TeXAtom-ORD"\gt \lt mi\gt p\lt /mi\gt \lt /mrow\gt \lt /msub\gt \lt /mrow\gt \lt /mfrac\gt \lt /mrow\gt \lt mo\gt =\lt /mo\gt \lt mrow class="MJX-TeXAtom-ORD"\gt \lt mfrac\gt \lt mrow\gt \lt msub\gt \lt mi\gt K\lt /mi\gt \lt mrow class="MJX-TeXAtom-ORD"\gt \lt mi\gt d\lt /mi\gt \lt /mrow\gt \lt /msub\gt \lt msup\gt \lt mi\gt s\lt /mi\gt \lt mrow class="MJX-TeXAtom-ORD"\gt \lt mn\gt 2\lt /mn\gt \lt /mrow\gt \lt /msup\gt \lt mo\gt +\lt /mo\gt \lt msub\gt \lt mi\gt K\lt /mi\gt \lt mrow class="MJX-TeXAtom-ORD"\gt \lt mi\gt p\lt /mi\gt \lt /mrow\gt \lt /msub\gt \lt mi\gt s\lt /mi\gt \lt mo\gt +\lt /mo\gt \lt msub\gt \lt mi\gt K\lt /mi\gt \lt mrow class="MJX-TeXAtom-ORD"\gt \lt mi\gt i\lt /mi\gt \lt /mrow\gt \lt /msub\gt \lt /mrow\gt \lt mrow\gt \lt msup\gt \lt mi\gt s\lt /mi\gt \lt mrow class="MJX-TeXAtom-ORD"\gt \lt mn\gt 3\lt /mn\gt \lt /mrow\gt \lt /msup\gt \lt mo\gt +\lt /mo\gt \lt mo stretchy="false"\gt (\lt /mo\gt \lt mn\gt 1.33\lt /mn\gt \lt mo\gt +\lt /mo\gt \lt msub\gt \lt mi\gt K\lt /mi\gt \lt mrow class="MJX-TeXAtom-ORD"\gt \lt mi\gt d\lt /mi\gt \lt /mrow\gt \lt /msub\gt \lt mo stretchy="false"\gt )\lt /mo\gt \lt msup\gt \lt mi\gt s\lt /mi\gt \lt mrow class="MJX-TeXAtom-ORD"\gt \lt mn\gt 2\lt /mn\gt \lt /mrow\gt \lt /msup\gt \lt mo\gt +\lt /mo\gt \lt mo stretchy="false"\gt (\lt /mo\gt \lt mn\gt 1\lt /mn\gt \lt mo\gt +\lt /mo\gt \lt msub\gt \lt mi\gt K\lt /mi\gt \lt mrow class="MJX-TeXAtom-ORD"\gt \lt mi\gt p\lt /mi\gt \lt /mrow\gt \lt /msub\gt \lt mo stretchy="false"\gt )\lt /mo\gt \lt mi\gt s\lt /mi\gt \lt mo\gt +\lt /mo\gt \lt msub\gt \lt mi\gt K\lt /mi\gt \lt mrow class="MJX-TeXAtom-ORD"\gt \lt mi\gt i\lt /mi\gt \lt /mrow\gt \lt /msub\gt \lt /mrow\gt \lt /mfrac\gt \lt /mrow\gt \lt /mstyle\gt \lt /mrow\gt \lt annotation encoding="application/x-tex"\gt {\displaystyle G={\frac {G_{c}G_{p}}{1+G_{c}G_{p}}}={\frac {K_{d}s^{2}+K_{p}s+K_{i}}{s^{3}+(1.33+K_{d})s^{2}+(1+K_{p})s+K_{i}}}}\lt /annotation\gt \lt /semantics\gt }[/math]<img alt="{\displaystyle G={\frac {G_{c}G_{p}}{1+G_{c}G_{p}}}={\frac {K_{d}s^{2}+K_{p}s+K_{i}}{s^{3}+(1.33+K_{d})s^{2}+(1+K_{p})s+K_{i}}}}" aria-hidden="true" class="mwe-math-fallback-image-inline mw-invert" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/672f02e08b2d8f513be9db18606700ead10e82ea" style="vertical-align: -2.671ex; width:56.723ex; height:6.676ex;"/>

As the proportional, integral and derivative terms of the PID compensator are increased, the evolution of the step response, from the initial uncompensated response (corresponding to PID coefficents Kp=1, Ki=0, Kd=0) to the final desired response is depicted. The following effects of PID compensation can be readily observed:

• The proportional term increases the speed of the system. It also decreases the residual steady state error of the step response, but can not eliminate it completely.

• The integral term eliminates the residual steady state error of the step response, but adds undesired "oscillations" to the transient response (overshoot).

• The derivative term "damps out" the undesired oscillations in the transient response.

| источник = Own work | время создания = 2015-05-28 | автор = Physicsch — Лицензия: CC0 (Creative Commons Zero, Public Domain Dedication) http://creativecommons.org/publicdomain/zero/1.0/deed.en }} Источник файла — сайт Wikimedia Commons, куда он был загружен под одной из свободных лицензий ( https://commons.wikimedia.org/wiki/File:PID_Compensation_Animated.gif ). Авторов, работавших над этим файлом см. в истории файла: https://commons.wikimedia.org/w/index.php?title=File:PID_Compensation_Animated.gif&action=history

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