17.3: The Practical Effects of an Open-Loop Transfer-Function Pole at s = 0 + j0

The development above demonstrates an important characteristic of an open-loop system with a transfer-function pole at the \(s\)-plane origin: frequency response of such a system increases in magnitude progressively as excitation frequency decreases, becoming, in theory, infinitely large for zero frequency. What is the significance of this type of response relative to experimental testing? The question is appropriate since, as asserted in the introduction to this chapter, frequency-response of the open-loop portion of a practical system (actual hardware) can indicate the stability or instability of the corresponding closed-loop system. Figure 17.2.2 and Equation 17.2.3 show clearly that the frequency-response magnitude for very low frequencies of excitation could indeed be impractically large. This might lead us to assume that, in order to avoid excessive output magnitudes, we would need only to excite the open-loop system at frequencies above some practical lower limit. The discussion in this section will show that this would be a potentially harmful assumption.

What is the nature of response associated with a system’s transfer-function pole at the origin of the \(s\)-plane? We know that a pole on the \(\operatorname{Re}(s)\) axis in the right-half \(s\)-plane means the system’s time response (to an impulse, for example) is essentially monotonically growing exponential, an instability. (See the discussion of Section 16.4.) Conversely, a pole on the \(\operatorname{Re}(s)\) axis in the left-half \(s\)-plane means the response is essentially monotonically decaying exponential, a stable response. Can we infer, then, that a pole on the \(\operatorname{Re}(s)\) axis at the origin, between the right- and left-half planes, means time response is neither unstable nor stable, but somehow neutral?

Let us try to answer that question, at least for the open-loop system of current interest that motivated this discussion, by deriving the unit-impulse response, \(h(t)\). From Equation 8.9.5, a system’s transfer function is also the Laplace transform of its unit-impulse response; accordingly, Equation 17.1.2 gives \(L[h(t)]=\Lambda \omega_{b} /\left[s\left(s+c_{\theta} / J\right)\left(s+\omega_{b}\right)\right]\). We could find the complete equation for \(h(t)\), but it is easier and equally relevant for this discussion to apply the final-value theorem [Equation 15.3.1] to find the steady-state response [which we can show follows the decay to zero of stable terms whose time functions are \(e^{-\left(c_{\theta} / J\right) t}\) and \(e^{-\omega_{b} t}\)]: \(\lim _{t \rightarrow \infty} h(t)=\lim _{s \rightarrow 0} s \times L[h(t)]=\Lambda J / c_{\theta}\). This constant-in-time response to an impulse excitation is unusual, since impulse response of a stable system normally decays to zero.

Let us probe further and find the response to a step input, which is a better gauge of stability than impulse response. Equation 8.9.10 shows that the unit-impulse response of any LTI-SISO system is the time derivative of the unit-step response, \(h(t)=d x_{H} / d t\). Therefore, the unit-step response as \(t \rightarrow \infty\) of the open-loop system of current interest must be \(\lim _{t \rightarrow \infty} x_{H}(t)=C+\left(\Lambda J / c_{\theta}\right) t\), \(C\) being a constant. This linear-function-of-time response to a constant input can increase monotonically, without bound. This is not an exponential monotonic instability or exponential oscillatory instability of the type we have observed previously; nevertheless, for all practical purposes, it is a genuine form of instability.

We can be even more specific relative to frequency response. With reference to the functional diagram that depicts testing of the open-loop system, Figure 17.1.2, suppose that the input signal, in addition to the sinusoidal excitation, has a slight constant voltage offset, \(E_{o f f}\). (Even with high-pass filtering, it is often nearly impossible to eliminate all such offsets from real circuitry used in experimental testing.) Therefore, with \(H(t)\) denoting the Heaviside unit-step function, the complete input signal is:

\[e_{i n}(t)=E_{i n} \sin \omega t+E_{o f f} H(t)\label{eqn:17.14} \]

With use of \(OLTF(s)\) of Equation 17.1.2 and inverse Laplace transformation, we can find an algebraic equation for the response in time to input Equation \(\ref{eqn:17.14}\), assuming zero initial conditions. The full equation is quite lengthy. It includes transient-response terms whose time functions are \(e^{-\left(c_{\theta} / J\right) t}\) and \(e^{-\omega_{b} t}\), both of which decay quickly to insignificance. If we omit those transient-response terms, then we can express the “steady-state” response in the form (homework Problem 17.?):

\[e_{\text {out}}(t)=E_{\text {in}}\left(M R(\omega) \times \sin (\omega t+\phi(\omega))+\frac{\Lambda}{\omega\left(c_{\theta} / J\right)}\right)+E_{o f f} \frac{\Lambda J}{c_{\theta}}\left[t-\frac{\left(c_{\theta} / J+\omega_{b}\right)}{\left(c_{\theta} / J\right) \omega_{b}}\right]\label{eqn:17.15} \]

Open-loop frequency-response magnitude ratio \(\operatorname{MR}(\omega)\) and phase angle \(\phi(\omega)\) are the quantities defined generally by Equations 17.1.3 and 17.1.4, respectively, and represented for this specific system, from Equation 17.1.7, by the equations:

\[M R(\omega)=\left|\Lambda \frac{\omega_{b}}{j \omega\left(j \omega+c_{\theta} / J\right)\left(j \omega+\omega_{b}\right)}\right| \quad \text { and } \quad \phi(\omega)=\angle\left(\frac{\omega_{b}}{j \omega\left(j \omega+c_{\theta} / J\right)\left(j \omega+\omega_{b}\right)}\right) \nonumber \]

Ideally in experimental sinusoidal testing, there would be only steady-state sinusoidal response. Much less ideally, however, there are in Equation \(\ref{eqn:17.15}\) three types of time response: the sinusoidal term, two constant-in-time terms, and a term that increases linearly with time. The constant-in-time terms would be a nuisance, but they could be tolerable, provided they were not so large as to overcome mechanical or electrical limits. However, the unstable linear “drift”, \(E_{o f f}\left(\Lambda J / c_{\theta}\right) t\), would present a serious problem in the testing process. Consider, for example, the numerical solution of Equation \(\ref{eqn:17.15}\) for the open-loop system with the ideal frequency response of Figures 17.1.3 and 17.2.2, and for excitation at the neutral-stability frequency, \(\omega=\omega_{n s}\). For this case, Figures 17.1.3 and 17.2.2 show that \(M R\left(\omega_{n s}\right)=1\) and \(\phi\left(\omega_{n s}\right)=-180^{\circ}\). Completing the calculations in Equation \(\ref{eqn:17.15}\) with the remaining numerical values gives:

\[e_{o u t}(t)=E_{i n}\left[\left(-\sin \omega_{n s} t+2.309\right)+\frac{E_{o f f}}{E_{i n}}(400 t-51 / 3)\right]\label{eqn:17.16} \]

Equation \(\ref{eqn:17.16}\) shows, for example, that if the offset voltage is even a mere 1% of the input sinusoidal magnitude, \(E_{o f f} / E_{i n}=0.01\), then, after only one second of system response, the linear drift increases to four times the sinusoidal-response magnitude; and the linear drift just continues rising inexorably thereafter. Practical sinusoidal testing of this system would be possible only if the offset voltage were almost completely eliminated, in order to prevent, or at least minimize, this unstable drift.

The type of response represented in Equation \(\ref{eqn:17.15}\) and illustrated numerically in Equation \(\ref{eqn:17.16}\) would pose a formidable challenge for any type of standard experimental testing on any system whose mathematical model has a pole at the origin of the \(s\)-plane. This mathematical model represents a general physical deficiency of such a system: the absence of a passive mechanism that tends to restore the system toward a static equilibrium state after the system has been perturbed dynamically. For example, the damped-rotor plant of our system of current interest, Figures 16.3.2 and 17.1.2, is the prime contributor to the unstable drift; this plant lacks a restoring rotational spring, and so its transfer function, Equation 16.3.7, \(P T F(s)=1 /\left[s\left(J_{S}+c_{\theta}\right]\right.\), has a pole at the origin.