Chapter 5

# 5.1 Equivalent Unit Feedback Loop

Consider a typical single feedback loop system:

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):

[latex]e(t) = r(t) - y(t)[/latex] [latex]E(s) = R(s) - Y(s)[/latex] |
Equation 5‑1 |

These signals can then be introduced into the system block diagram:

An equivalent unit feedback loop system will be then:

The steady state error analysis can then be performed on the equivalent system, for the system error signal *e(t)* (or *E(s)* in Laplace domain), and the reference signal *r(t)* (or *R(s)* in Laplace domain). However, in the physical system, the input is *u(t)* (or *U(s)* 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)*, in its forward path:

[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] |
Equation 5‑2 |