4.1: THE STABILITY PROBLEM

The discussion of feedback systems presented up to this point has tacitly assumed that the systems under study were stable. A stable system is defined in general as one which produces a bounded output in response to any bounded input. Thus stability implies that

\[\int_{-\infty}^{\infty} |v_O (t)| dt \le M < \infty \label{eq4.1.1} \]

for any input such that

\[\int_{-\infty}^{\infty} |v_I (t)| dt \le N < \infty \nonumber \]

If we limit our consideration to linear systems, stability is independent of the input signal, and the sufficient and necessary condition for stability is that all poles of system transfer function lie in the left half of the s plane. This condition follows directly from Equation \(\ref{eq4.1.1}\), since any right-half-plane poles contribute terms to the output that grow exponentially with time and thus are unbounded. Note that this definition implies that a system with poles on the imaginary axis is unstable, since its output is not bounded unless its input is rather carefully chosen.

The origin of the stability problem can be described in intuitively appeal­ing through nonrigorous terms as follows. If a feedback system detects an error between the actual and desired outputs, it attempts to reduce this error to zero. However, changes in the error signal that result from correc­tive action do not occur instantaneously because of time delays around the loop. In a high-gain system, these delays can cause a tendency to over­ correct. If the magnitude of the overcorrection exceeds the magnitude of the initial error, instability results. Signal amplitudes grow exponentially until some nonlinearity limits further growth, at which time the system either saturates or oscillates in a constant-amplitude fashion called a limit cycle.(The effect of nonlinearities on the steady-state amplitude reached by an unstable system is investigated in Chapter 6.) The feedback system designer must always temper his desire to provide a large magnitude and a high unity-gain frequency for the loop transmission with the certain knowledge that sufficiently high values for these quantities invariably lead to instability.

Figure 4.1 Block diagram of single-loop amplifier.

As a specific example of a system with potentially unstable behavior, con­sider a simple single-loop system of the type shown in Figure 4.1, with

\[a(s) = \dfrac{a_0}{(s + 1)^3} \nonumber \]

and

\[f(s) = 1 \nonumber \]

The loop transmission for this system is

\[-a(s) f(s) = \dfrac{-a_0}{(s + 1)^3} \nonumber \]

or for sinusoidal excitation,

\[-a(j\omega) f(j\omega) = \dfrac{-a_0}{(j \omega + 1)^3} = \dfrac{-a_0}{-j \omega^3 - 3 \omega^2 + 3j \omega + 1} \label{eq4.1.6} \]

If we evaluate Equation \(\ref{eq4.1.6}\) at \(\omega = \sqrt{3}\), we find that

\[-a (j \sqrt{3}) f(j \sqrt{3}) = \dfrac{a_0}{8} \nonumber \]

If the quantity \(a_0\) is chosen equal to 8, the system has a real, positive loop transmission with a magnitude of one for sinusoidal excitation at three radians per second.

We might suspect that a system with a loop transmission of +1 is capable of oscillation, and this suspician can be confirmed by examining the closed-loop transfer function of the system with \(a_0 = 8\). In this case,

\[A(s) = \dfrac{a(s)}{1 + a(s) f(s)} = \dfrac{8}{s^3 + 3s^2 + 3s + 9} = \dfrac{8}{(s + 3) (s + j \sqrt{3}) (s - j\sqrt{3})} \nonumber \]

This transfer function has a negative, real-axis pole and a pair of poles located on the imaginary axis at \(s = \pm j \sqrt{3}\). An argument based on the properties of partial-fraction expansions (see Section 3.2.2) shows that the response of this system to many common (bounded) transient signals includes a constant-amplitude sinusoidal component.

Further increases in low-frequency loop-transmission magnitude move the pole pair into the right-half plane. For example, if we combine the forward-path transfer function

\[a(s) = \dfrac{64}{(s + 1)^3} \nonumber \]

with unity feedback, the resultant closed-loop transfer function is

\[A(s) = \dfrac{64}{s^3 + 3s^2 + 3s + 65} = \dfrac{64}{(s + 5) (s - 1 + j 2\sqrt{3})(s - 1 - j2\sqrt{3})} \nonumber \]

With this value for \(a_0\), the system transient response will include a sinusoidal component with an exponentially growing envelope.

If the dynamics associated with the loop transmission remain fixed, the system will be stable only for values of \(a_0\) less than 8. This stability is achieved at the expense of desensitivity. If a value of \(a_0 = 1\) is used so that

\[a(s) f(s) = \dfrac{1}{(s + 1)^3} \nonumber \]

we find all closed-loop poles are in the left-half plane, since

\[A(s) = \dfrac{1}{s^3 + 3s^2 + 3s + 2} = \dfrac{1}{(s + 2) (s + 0.5 + j \sqrt{3}/2) (s + 0.5 - j \sqrt{3}/2)} \nonumber \]

in this case.

In certain limited cases, a binary answer to the stability question is sufficient. Normally, however, we shall be interested in more quantitative information concerning the "degree" of stability of a feedback system. Frequently used measures of relative stability include the peak magnitude of the frequency response, the fractional overshoot in response to a step input, the damping ratio associated with the dominant pole pair, or the variation of a certain parameter that can be tolerated without causing absolute instability. Any of the measures of relative stability mentioned above can be found by direct calculations involving the system transfer function. While such determinations are practical with the aid of machine computation, insight into system operation is frequently obscured if this process is used. The techniques described in this chapter are intended not only to provide answers to questions concerning stability, but also (and more important) to indicate how to improve the performance of unsatis­ factory systems.