2.21: Feedback

Learning Objectives: Feedback

1. Understand how the closed-loop gain relates to the feedforward and feedback system functions
2. Implement stabilization using negative feedback
3. Design systems to compensate for nonideal amplifiers

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Feedback

Feedback Stabilization Example

Summary

Feedback → using the output of a system to control or modify your input

└ Adaptive Optics

Figure 2.21.1: Example of an open-loop system.

Figure 2.21.2: Example of a closed-loop system.

Whereas the open-loop system does not allow us to make corrections, a closed-loop system allows us to have

• Better control of system components
• Being able to detect any misalignment

Linear Feedback Systems

Assumption: \(H(s)\) is either unilateral or bilateral transform → \(H(s)\) causal and \(ROC_H\) to the right of the rightmost pole

Figure 2.21.3: Example of a linear feedback system design.

\(r(t)\): signal fed back

\(e(t)=x(t)-r(t)\): error between input signal and actual response

\(H(s)\): system function of the forward path
\(G(s)\): system function of the feedback path
\(Q(s)\): closed-loop system function

\[ Q(s)=\frac{Y(s)}{X(s)}=\frac{H(s)}{1+G(s)H(s)} \]

include both forward and feedback paths

Inverse System Design

Figure 2.21.3: Example of an inverse system design.

\(H(s)=k\)

\(P(s)\): system function of feedback path \(=G(s)\)

\[ Q(s)=\frac{H(s)}{1+H(s)P(s)}=\frac{k}{1+kP(s)}\approx \frac{k}{kP(s)}=\frac{1}{P(s)} \]

\(k\gg\) large and \(1 \ll kP(s)\)

\(Q(s)=k\) is the inverse of the feedback system

Compensation for NonIdeal Elements

Use of feedback systems to correct some non-ideal properties/measurements of the open-loop system

\(G(s)=K\) → \[ Q(s)=\frac{H(s)}{1+G(s)H(s)}=\frac{H(s)}{1+K\,H(s)}\approx \frac{H(s)}{K\,H(s)}=\frac{1}{K}=Q(s) \]

CONDITION: the closed loop gain \(\frac{1}{K}\) is substantially less than the open-loop gain \(|H(j\omega)|\)

KEY POINT: If \(K\ll 1\), \(|H(j\omega)|\) is an amplifier but without erratic gain

WATCH OUT: Find right \(K\) so it is small for high \(|Q|\) but big enough so \(|KH| \gg 1\).

Stabilization of Unstable Systems

Use of feedback systems to stabilize systems that are a-priori unstable

Example

(1) control trajectory of a rocket; (2) regularization of nuclear reactions in a nuclear power plant; (3) stabilization of an aircraft;

(4) natural and regulatory control of animal populations

\[ H(s)=\frac{b}{s-a} \] 1st-order system \(a>0\) → \(h(t)\) unstable

\[ G(s)=k \] constant gain

\begin{align*}
Q(s) &= \frac{H(s)}{1 + G(s)H(s)}
= \frac{\dfrac{b}{s-a}}{1 + k\,\dfrac{b}{s-a}}
= \frac{b}{s-a+kb}.
\end{align*}

pole @ \(s = a-kb\)

The close-loop system is stable if the pole \(s<0\), so \(kb>a\)

proportional feedback system

\[ H(s)=\frac{b}{s^2+a} \]

\(a>0\) → 2nd-order system → \(H(s)\) has imaginary poles and \(h(t)\) is sinusoidal

\(H(s)=\dfrac{b}{s^2+a}\), \(a<0\) → \(H(s)\) has one pole in the left-side plane and one in the right-half plane → \(H(s)\) is unstable.

\[ G(s)=k \]

\[ Q(s)=\frac{H(s)}{1+G(s)H(s)} =\frac{\dfrac{b}{s^2+a}}{1+k\,\dfrac{b}{s^2+a}} =\frac{b}{s^2+a+kb} \]

\[ \frac{b}{s^2+a+kb} =\frac{\omega_n^2}{s^2+2\zeta\omega_n s+\omega_n^2} \]

We cannot stabilize the system because we do not have any damping \(\zeta\).

\[ G(s)=k_1+k_2s \quad \text{(proportional-plus-derivative)} \]

\[ Q(s)=\frac{b}{s^2+bk_2s+(a+k_1b)} \]

where \(bk_2>0\) and \(a+k_1b>0\)

Sampled-data feedback systems

The output of a CT signal is sampled

Figure 2.21.4: Representation of the stabilization system for the given example.

\[ z(t)=d[n] \quad \text{for } nT\le t<(n+1)T \quad \text{only a period} \]

\[ d[n]=p[n]*g[n] \quad (\text{in TL}) \]

\[ p[n]=y(nT) \]

Tracking systems

Figure 2.21.5: Representation of a tracking system without (top) and with (bottom) accounting for disturbance \(d[n]\).