{"id":699,"date":"2019-07-26T02:13:24","date_gmt":"2019-07-26T02:13:24","guid":{"rendered":"https:\/\/pressbooks.library.ryerson.ca\/controlsystems\/?post_type=chapter&p=699"},"modified":"2021-01-12T19:25:36","modified_gmt":"2021-01-12T19:25:36","slug":"5-1-equivalent-unit-feedback-loop","status":"publish","type":"chapter","link":"https:\/\/pressbooks.library.torontomu.ca\/controlsystems\/chapter\/5-1-equivalent-unit-feedback-loop\/","title":{"raw":"5.1 Equivalent Unit Feedback Loop","rendered":"5.1 Equivalent Unit Feedback Loop"},"content":{"raw":"

Consider a typical single feedback loop system:<\/p>\r\n\r\n\r\n[caption id=\"attachment_700\" align=\"aligncenter\" width=\"498\"]\"Figure Figure 5-1: Typical Feedback Loop[\/caption]\r\n

In most cases, the system will be non-unit feedback. For example, y(t) may be a temperature signal, and b(t) will be a voltage signal out of a thermocouple (sensor). The input signal u(t) will also be a voltage signal. It is pointless to make comparisons between u(t) and y(t). Let us introduce the reference signal, r(t) (a desired level of output, and not a physical quantity), and the so-called system error, e(t):<\/p>\r\n\r\n\r\n\r\n\r\n
\r\n

[latex]e(t) = r(t) - y(t)[\/latex]<\/p>\r\n

[latex]E(s) = R(s) - Y(s)[\/latex]<\/p>\r\n<\/td>\r\n

\r\n

Equation 5\u20111<\/p>\r\n<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n

These signals can then be introduced into the system block diagram:<\/p>\r\n\r\n\r\n[caption id=\"attachment_701\" align=\"aligncenter\" width=\"500\"]\"Figure Figure 5-2: Modified Block Diagram with Reference Signal[\/caption]\r\n

An equivalent unit feedback loop system will be then:<\/p>\r\n\r\n\r\n[caption id=\"attachment_702\" align=\"aligncenter\" width=\"500\"]\"Figure Figure 5-3: Equivalent Unit Feedback Loop[\/caption]\r\n

The steady state error analysis can then be performed on the equivalent system, for the system error signal e(t)<\/em> (or E(s)<\/em> in Laplace domain), and the reference signal r(t)<\/em> (or R(s)<\/em> in Laplace domain). However, in the physical system, the input is u(t)<\/em> (or U(s)<\/em> in Laplace domain), and the controller input is the actuating error [latex]e_{a}(t)[\/latex] (or [latex]E_{a}(s)[\/latex] in Laplace domain). Note that the equivalent unity feedback loop has the \"open loop transfer function\", G(s)H(s)<\/em>, in its forward path:<\/p>\r\n\r\n\r\n\r\n\r\n
[latex]\\frac{Y(s)}{R(s)} = H(s) \\cdot \\frac{Y(s)}{U(s)} = H(s) \\cdot \\frac{G(s)}{1+ G(s)H(s)} = \\frac{G(s)H(s)}{1+G(s)H(s)} = \\frac{G_{open}(s)}{1+G_{open}(s)}[\/latex]<\/td>\r\n\r\n

Equation 5\u20112<\/p>\r\n<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n

<\/p>","rendered":"

Consider a typical single feedback loop system:<\/p>\n

\"Figure
Figure 5-1: Typical Feedback Loop<\/figcaption><\/figure>\n

In most cases, the system will be non-unit feedback. For example, y(t) may be a temperature signal, and b(t) will be a voltage signal out of a thermocouple (sensor). The input signal u(t) will also be a voltage signal. It is pointless to make comparisons between u(t) and y(t). Let us introduce the reference signal, r(t) (a desired level of output, and not a physical quantity), and the so-called system error, e(t):<\/p>\n\n\n\n
\n

[latex]e(t) = r(t) - y(t)[\/latex]<\/p>\n

[latex]E(s) = R(s) - Y(s)[\/latex]<\/p>\n<\/td>\n

\n

Equation 5\u20111<\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

These signals can then be introduced into the system block diagram:<\/p>\n

\"Figure
Figure 5-2: Modified Block Diagram with Reference Signal<\/figcaption><\/figure>\n

An equivalent unit feedback loop system will be then:<\/p>\n

\"Figure
Figure 5-3: Equivalent Unit Feedback Loop<\/figcaption><\/figure>\n

The steady state error analysis can then be performed on the equivalent system, for the system error signal e(t)<\/em> (or E(s)<\/em> in Laplace domain), and the reference signal r(t)<\/em> (or R(s)<\/em> in Laplace domain). However, in the physical system, the input is u(t)<\/em> (or U(s)<\/em> in Laplace domain), and the controller input is the actuating error [latex]e_{a}(t)[\/latex] (or [latex]E_{a}(s)[\/latex] in Laplace domain). Note that the equivalent unity feedback loop has the “open loop transfer function”, G(s)H(s)<\/em>, in its forward path:<\/p>\n\n\n\n
[latex]\\frac{Y(s)}{R(s)} = H(s) \\cdot \\frac{Y(s)}{U(s)} = H(s) \\cdot \\frac{G(s)}{1+ G(s)H(s)} = \\frac{G(s)H(s)}{1+G(s)H(s)} = \\frac{G_{open}(s)}{1+G_{open}(s)}[\/latex]<\/td>\n\n

Equation 5\u20112<\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

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