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6.3: Describing Function

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    58455
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    Describing functions provide a method for the analysis of nonlinear sys­tems that is closely related to the linear-system techniques involving Bode or gain-phase plots. It is possible to use this type of analysis to determine if limit cycles (constant-amplitude periodic oscillations) are possible for a given system. It is also possible to use describing functions to predict the response of certain nonlinear systems to purely sinusoidal excitation, al­though this topic is not covered here (G. J. Thaler and M. P. Pastel, Analysis and Design of Nonlinear Feedback Control Systems, McGraw-Hill, New York, 1962). Unfortunately, since the frequency response and transient response of nonlinear systems are not directly re­lated, the determination of transient response is not possible via describing functions.

    Derivation of the Describing Function

    A describing function describes the behavior of a nonlinear element for purely sinusoidal excitation. Thus the input signal applied to the nonlinear element to determine its describing function is

    \[v_I = E \sin \omega t \nonumber \]

    If the nonlinearity does not rectify the input (produce a d-c output) and does not introduce subharmonics, the output of the nonlinear element can be expanded in a Fourier series of the form

    \[v_O = A_1 (E, \omega ) \cos \omega t + B_1 (E, \omega ) \sin \omega t + A_2 (E, \omega ) \cos 2 \omega t + B_2 (E, \omega ) \sin 2\omega t + \cdots + \nonumber \]

    The describing function for the nonlinear element is defined as

    \[G_D (E, \omega ) = \dfrac{\sqrt{A_1^2 (E, \omega ) + B_1^2 (E, \omega )}}{E} \measuredangle \tan^{-1} \dfrac{A_1 (E, \omega )}{B_1 (E, \omega )}\label{eq6.3.3} \]

    The describing-function characterization of a nonlinear element parallels the transfer-function characterization of a linear element. If the transfer function of a linear element is evaluated for \(s = j\omega \), the magnitude of re­sulting function of a complex variable is the ratio of the amplitudes of the output and input signals when the element is excited with a sinusoid at a frequency co. Similarly, the angle of the function is the phase angle between the output and input signals under sinusoidal steady-state conditions. For linear elements these quantities must be independent of the amplitude of excitation.

    The describing function indicates the relative amplitude and phase angle of the fundamental component of the output of a nonlinear element when the element is excited with a sinusoid. In contrast to the case with linear ele­ments, these quantities can be dependent on the amplitude as well as the frequency of the excitation.

    截屏2021-08-11 下午8.34.23.png
    Figure 6.6 Relationship for a saturating nonlinearity. (\(a\)) Transfer characteristics for saturating element. (\(b\)) Input and output waveforms for sinusoidal excitation.

    Two examples illustrate the derivation of the describing function for nonlinear elements. Figure 6.6 shows the transfer characteristics of a satu­rating nonlinearity together with input and output waveforms for sinusoidal excitation. Since the transfer characteristics for this element are not de­pendent on the dynamics of the input signal, it is clear that the describing function must be frequency independent.

    If the input amplitude \(E\) is less than \(E_M\),

    \[v_O = Kv_1 \nonumber \]

    In this case,

    \[G_D = K \measuredangle 0^{\circ} \ \ \ E < E_M \nonumber \]

    For \(E \ge E_M\), the output signal over the interval \(0 \le \omega t \le \pi\) is

    \[v_O = Kv_I \ \ \ \ 0 \le \omega t < \alpha \ \ \ \ \text{ or } \ \ \ \ \pi - \alpha < \omega t \le \pi \nonumber \]

    \[v_O = KE_M \ \ \ \ \alpha \le \omega t \le pi - \alpha \nonumber \]

    where

    \[\alpha = \sin^{-1} \dfrac{E_M}{E}\nonumber \]

    The coefficient \(A_1\) and \(B_1\) are in this case,

    \[\begin{array} {rcl} {A_1} & = & {\dfrac{2}{\pi} \int_{0}^{\alpha} KE \sin \omega t \cos \omega t \ d \omega t + \dfrac{2}{\pi} \int_{\alpha}^{\pi - \alpha} KE_M \cos \omega t \ d\omega t} \\ {} & \ & {+ \dfrac{2}{\pi} \int_{\pi - \alpha}^{\pi} KE \sin \omega t \cos \omega t \ d \omega t = 0} \end{array} \nonumber \]

    \[\begin{array} {rcl} {B_1} & = & {\dfrac{2}{\pi} \int_{0}^{\alpha} KE \sin^2 \omega t \ d \omega t + \dfrac{2}{\pi} \int_{\alpha}^{\pi - \alpha} KE_M \sin \omega t \ d\omega t} \\ {} & \ & {+ \dfrac{2}{\pi} \int_{\pi - \alpha}^{\pi} KE \sin^2 \omega t \ d \omega t = 0} \\ {} & = & {\dfrac{2KE}{\pi} \left [\sin^{-1} \dfrac{E_M}{E} + \dfrac{E_M}{E} \sqrt{1 - \left (\dfrac{E_M}{E} \right )^2} \right ]} \end{array} \nonumber \]

    Using Equation \(\ref{eq6.3.3}\), we obtain

    \[G_D (E) = K \measuredangle 0^{\circ} \ \ \ E \le E_M \nonumber \]

    \[G_D (E) = \dfrac{2K}{\pi} \left (\sin^{-1} R + R \sqrt{1 - R^2} \right ) \measuredangle 0^{\circ} \ \ \ E > E_M \nonumber \]

    where \(R = E_M/E\).

    截屏2021-08-11 下午8.48.20.png
    Figure 6.7 Relationships for an element with hysteresis. (\(a\)) Transfer characteristics. (\(b\)) Input and output waveforms for sinusoidal excitation.

    The transfer characteristics of an element with hysteresis, such as a Schmitt trigger or a relay, are shown in Figure 6.7\(a\). The memory associated with this type of element produces a phase shift between the fundamental component of the output and the input sinusoid applied to it as shown in Figure 6.7\(b\). It is necessary for the peak amplitude of the input signal to ex­ceed \(E_M\) in order to have the output signal other than a constant.

    Several features of the output signal permit writing the describing func­tion for this element. The relevant relationships include the following.

    1. While there is phase shift between the input signal and the funda­ mental component of the output, neither the amount of this phase shift nor the amplitude of the output signal are dependent on the excitation frequency.
    2. The amplitude of the fundamental component of a square wave with a peak amplitude \(E_N\) is \(4 E_N/\pi \).
    3. The relative phase shift between the input signal and the fundamental component of the output is \(\sin^{-1} (E_M/E)\), with the output lagging the input.
    Table 6.1 Describing Functions
    Nonlinearity
    Input = \(v_I = E \sin \omega t\)
    Describing Function
    (All are frequency independent.)
    截屏2021-08-11 下午8.53.16.png

    \(G_D (E) = K \measuredangle 0^{\circ} \ \ E \le E_M\)

    \(G_D (E) = \dfrac{2K}{\pi} \left (\sin^{-1} R + R \sqrt{1 - R^2} \right ) \measuredangle 0^{\circ},\)

    \(E > E_M\)

    where \(R = \dfrac{E_M}{E}\)

    截屏2021-08-11 下午8.56.52.png \(G_D (E) = \dfrac{4E_N}{\pi E} \measuredangle 0^{\circ}\)
    截屏2021-08-11 下午8.58.14.png

    \(G_D (E) = 0 \measuredangle 0^{\circ} \ \ E \le E_M\)

    \(G_D (E) = K \left [1 - \dfrac{2}{\pi} \left (\sin^{-1} R + R \sqrt{1 - R^2} \right ) \right ] \measuredangle 0^{\circ},\)

    \(E > E_M\)

    where \(R = \dfrac{E_M}{E}\)

    截屏2021-08-11 下午8.59.45.png

    \(G_D (E) = 0 \measuredangle 0^{\circ} \ \ E \le E_M\)

    \(G_D (E) = \dfrac{4E_N}{\pi E} \sqrt{1 - R^2} \measuredangle 0^{\circ}, \ \ E > E_M\)

    where \(R = \dfrac{E_M}{E}\)

    截屏2021-08-11 下午9.01.17.png

    \(E\) must exceed \(E_M\) or a d-c term results.

    \(G_D (E) = \dfrac{4E_N}{\pi E} \measuredangle -\sin^{-1} R\)

    where \(R = \dfrac{E_M}{E}\)

    Combining these relationships shows that

    \[\begin{array} {rcl} {G_D (E)} & = & {\dfrac{4E_N}{\pi E} \measuredangle -\sin^{-1} \dfrac{E_M}{E} \ \ E \ge E_M} \\ {G_D (E)} & \ & {\text{undefined otherwise}} \end{array} \nonumber \]

    Table 6.1 lists the describing functions for several common nonlineari­ties. Since the transfer characteristics shown are all independent of the frequency of the input signal, the corresponding describing functions are dependent only on input-signal amplitude. While this restriction is not necessary to use describing-function techniques, the complexity associated with describing-function analysis of systems that include frequency-de­pendent nonlinearities often limits its usefulness.

    截屏2021-08-11 下午9.05.45.png
    Figure 6.8 Soft saturation as a combination of two nonlinearities. (\(a\)) Transfer characteristics. (\(b\)) Decomposition into two nonlinearities.

    The linearity of the Fourier series can be exploited to determine the de­scribing function of certain nonlinearities from the known describing func­tions of other elements. Consider, for example, the soft-saturation charac­teristics shown in Figure 6.8\(a\). The input-output characteristics for this ele­ment can be duplicated by combining two tabulated elements as shown in Figure 6.8\(b\). Since the fundamental component of the output of the system of Figure 6.8\(b\) is the sum of the fundamental components from the two non­linearities

    \[G_D(E) = K_1 \measuredangle 0^{\circ} E \le E_M \nonumber \]

    \[\begin{array} {rcl} {G_D (E)} & = & {\left [ \dfrac{2K_1}{\pi} \left (\sin^{-1} R + R \sqrt{1 - R^2} \right ) + K_2 - \dfrac{2K_2}{\pi} \left ( \sin^{-1} R + R \sqrt{1 - R^2} \right ) \right ] \measuredangle 0^{\circ}} \\ {} & = & {\left [ K_2 + \dfrac{2(K_1 - K_2)}{\pi} \left ( \sin^{-1} R + R \sqrt{1 - R^2} \right ) \right ] \measuredangle 0^{\circ}} \end{array} \nonumber \]

    for \(E > E_M\), where \(R = \sin^{-1} (E_M/ E)\).

    Stability Analysis with the Aid of Describing Functions

    Describing functions are most frequently used to determine if limit cycles (stable-amplitude periodic oscillations) are possible for a given sys­tem, and to determine the amplitudes of various signals when these oscil­lations are present.

    截屏2021-08-11 下午9.16.32.png
    Figure 6.9 System arranged for describing-function analysis.

    Describing-function analysis is simplified if the system can be arranged in a form similar to that shown in Figure 6.9. The inverting block is included to represent the inversion conventionally indicated at the summing point in a negative-feedback system. Since the intent of the analysis is to examine the possibility of steady-state oscillations, system input and output points are irrevelant. The important feature of the topology shown in Figure 6.9 is that a single nonlinear element appears in a loop with a single linear ele­ment. The linear element shown can of course represent the reduction of a complex interconnection of linear elements in the original system to a single transfer function. The techniques described in Sections 2.4.2 and 2.4.3 are often useful for these reductions.

    截屏2021-08-11 下午9.18.38.png
    Figure 6.10 Nonlinear system. (\(a\)) Circuit. (\(b\)) Zener-limiter characteristics.

    The system shown in Figure 6.10 illustrates a type of manipulation that simplifies the use of describing functions in certain cases. A limiter con­sisting of back-to-back Zener diodes is included in a circuit that also con­tains an amplifier and a resistor-capacitor network. The Zener limiter is assumed to have the piecewise-linear characteristics shown in Figure 6.10\(b\).

    The describing function for the nonlinear network that includes \(R_1, R_2, C\), and the limiter could be calculated by assuming a sinusoidal signal for \(v_B\) and finding the amplitude and relative phase angle of the fundamental component of \(v_A\). The resulting describing function would be frequency dependent. A more satisfactory representation results if the value of the Zener current \(i_A\) is determined as a function of the voltage applied to the network.

    \[i_A = \dfrac{v_B}{R_1} - \dfrac{v_A}{R_1 ||R_2} - C \dfrac{dv_A}{dt}\label{eq6.3.15} \]

    The Zener limiter forces the additional constraints

    \[v_A = + V_Z \ \ \ i_A > 0\label{eq6.3.16} \]

    \[v_A = - V_Z \ \ \ i_A < 0\label{eq6.3.17} \]

    截屏2021-08-11 下午9.24.09.png
    Figure 6.11 Modeling system of Figure 6.10 as a single loop. (\(a\)) Block-diagram representation of nonlinear network. (\(b\)) Block diagram representation of complete system. (\(c\)) Reduced to form of Figure 6.9.

    Equations \(\ref{eq6.3.15}\), \(\ref{eq6.3.16}\) and \(\ref{eq6.3.17}\) imply that the block diagram shown in Figure 6.11\(a\) can be used to relate the variables in the nonlinear network. The pleasing feature of this representation is that the remaining nonlinearity can be characterized by a frequency-independent describing function. Figure 6.11\(b\) illustrates the block diagram that results when the network is combined with the amplifier. The two linear paths in this diagram are combined in Figure 6.1\(c\), which is the form suggested for analysis.

    Once a system has been reduced to the form shown in Figure 6.9, it can be analyzed by means of describing functions. The describing-function ap­proximation states that oscillations may be possible if particular values of \(E_1\) and \(\omega_1\) exist such that

    \[a(j\omega_1) G_D (E_1, \omega_1) = -1\label{eq6.3.18} \]

    or

    \[a(j\omega_1) = \dfrac{-1}{G_D(E_1, \omega_1)}\label{eq6.3.19} \]

    The satisfaction of Equation \(\ref{eq6.3.18}\) and \(\ref{eq6.3.19}\) does not guarantee that the system in question will oscillate. It is possible that a system satisfying Equation \(\ref{eq6.3.18}\) and \(\ref{eq6.3.19}\) will be stable for a range of signal levels and must be triggered into oscillation by, for example, exceeding a particular signal level at the input to the non­linear element. A second possibility is that the equality of Equation \(\ref{eq6.3.18}\) and \(\ref{eq6.3.19}\) does not describe a stable-amplitude oscillation. In this case, if it is assumed that the system is oscillating with parameter values given in Equation \(\ref{eq6.3.18}\) and \(\ref{eq6.3.19}\), a small amplitude perturbation is divergent and leads to either an increasing or a decreasing amplitude. As we shall see, the method can be used to resolve these questions. The describing-function analysis also predicts that if stable-amplitude oscillations exist, the frequency of the oscillations will be \(\omega_1\) and the amplitude of the fundamental component of the signal applied to the nonlinearity will be \(E_1\).

    The above discussion shows how closely the describing-function stability analysis of nonlinear systems parallels the Nyquist or Bode-plot analysis of linear systems. In particular, oscillations are predicted for linear systems at frequencies where the loop transmission is +1, while describing-function analysis indicates possible oscillations for amplitude-frequency combina­tions that produce the nonlinear-system equivalent of unity loop trans­mission.

    The basic approximation of describing-function analysis is now evident. It is assumed that under conditions of steady-state oscillation, the input to the nonlinear element consists of a single-frequency sinusoid. While this assumption is certainly not exactly satisfied because the nonlinear element generates harmonics that propagate around the loop, it is often a useful approximation for two reasons. First, many nonlinearities generate har­monics with amplitudes that are small compared to the fundamental. Second, since many linear elements in feedback systems are low-pass in nature, the harmonics in the signal returned to the nonlinear element are often attenuated to a greater degree than the fundamental by the linear elements. The second reason indicates a better approximation for higher-order low-pass systems.

    The existence of the relationship indicated in Equation \(\ref{eq6.3.18}\) and \(\ref{eq6.3.19}\) is often deter­ mined graphically. The transfer function of the linear element is plotted in gain-phase form. The function \(- 1/G_D(E, \omega )\) is also plotted on the same graph. If \(G_D\) is frequency independent, \(- 1/G_D(E)\) is a single curve with \(E\) a parameter along the curve. The necessary condition for oscillation is satisfied if an intersection of the two curves exists. The frequency can be determined from the \(a(j\omega )\) curve, while amplitude of the fundamental com­ponent of the signal into the nonlinearity is determined from the \(- 1/G_D(E)\) curve. If the nonlinearity is frequency dependent, a family of curves \(- 1/G_D(E, \omega_1), -1/G_D(E, \omega_2)\), ... , is plotted. The oscillation condition is satisfied if the \(-1/G_D(E, \omega_i)\) curve intersects the \(a(j\omega )\) curve at the point \(a(j\omega_i)\).

    The satisfaction of Equation \(\ref{eq6.3.18}\) and \(\ref{eq6.3.19}\) is a necessary though not sufficient condi­tion for a limit cycle to exist. It is also necessary to insure that the oscilla­tion predicted by the intersection is stable in amplitude. In order to test for amplitude stability, it is assumed that the amplitude \(E\) increases slightly, and the point corresponding to the perturbed value of \(E\) is found on the \(- 1/G_D (E, \omega)\) curve. If this point lies to the left of the \(a(j\omega )\) curve, the geometry implies that the system poles(The concept of a pole is strictly valid only for a linear system. Once we apply the describing-function approximation (which is a particular kind of linearization about an operating point defined by a signal amplitude), we take the same liberty with the definition of a pole as we do with systems that have been linearized by other methods.) lie in the left-half plane for an increased value of \(E\), tending to restore the amplitude to its original value. Alterna­tively, if the perturbed point lies to the right of the \(a(j\omega )\) curve, a growing- amplitude oscillation results from the perturbation and a limit cycle with parameters predicted by the intersection is not possible. These relationships can be verified by applying the Nyquist stability test to the loop transmis­sion, which includes the linear transfer function and the describing function of interest.

    It should be noted that the stability of arbitrarily complex nonlinear sys­tems that combine a multiplicity of nonlinear elements in a loop with linear elements can, at least in theory, be determined using describing functions. For example, numerous Nyquist plots corresponding to the nonlinear loop transmissions for a variety of signal amplitudes might be constructed to determine if the possibility for instability exists. Unfortunately, the effort required to complete this type of analysis is generally prohibitive.

    Examples

    Since describing-function analysis predicts the existence of stable-ampli­tude limit cycles, it is particularly useful for the investigation of oscillators, and for this reason the two examples in this section involve oscillator cir­cuits.

    The discussion of Section 4.2.2 showed that it is possible to produce sinusoidal oscillations by applying negative feedback around a phase-shift network with three identically located real-axis poles. If the magnitude of the low-frequency loop transmission is exactly 8, the system closed-loop poles are on the imaginary axis and, thus, resultant oscillations are stable in amplitude. It is possible to control the magnitude of the loop transmission precisely by means of an auxiliary feedback loop that measures the ampli­tude of the oscillation and adjusts loop transmission to regulate this ampli­tude. This approach to amplitude control is discussed in Section 12.1.4.

    截屏2021-08-11 下午9.37.33.png
    Figure 6.12 Phase-shift oscillator with limiting.

    An alternative and simpler approach that is often used is illustrated in Figure 6.12. The loop transmission of the system for small signal levels is made large enough (in this case 10) to insure growing-amplitude oscillations if signal levels are such that the limiter remains linear. As the peak amplitude of the signal VA increases beyond one, the limiter reduces the magnitude of the loop transmission (in a describing-function sense) so as to stabilize the amplitude of the oscillations.

    The describing function for the limiter in Figure 6.12 is (see Table 6.1)

    \[G_D (E) = 1 \measuredangle 0^{\circ} \ \ \ E \le 1\label{eq6.3.20} \]

    \[G_D (E) = \dfrac{2}{\pi} \left (\sin^{-1} \dfrac{1}{E} + \dfrac{1}{E} \sqrt{1 - \dfrac{1}{E^2}} \right ) \measuredangle 0^{\circ} \ \ \ E > 1\label{eq6.3.21} \]

    截屏2021-08-11 下午9.42.35.png
    Figure 6.13 Describing-function analysis of the phase-shift oscillator.

    This function decreases monotonically as \(E\) increases beyond one. Thus the quantity \(-1/G_D(E)\) increases monotonically for \(E\) greater than one and has an angle of \(-180^{\circ}\). The general behavior of \(-1/G_D(E)\) and the transfer function of the linear portion of the oscillator circuit are sketched on the gain-phase plane of Figure 6.13.

    The intersection shown is seen to represent a stable-amplitude oscilla­tion when the test proposed in the last section is used. An increase in \(E\)

    from the value at the intersection moves the \(-1/G_D(E)\) point to the left of the \(a(j\omega)\) curve. The physical significance of the rule is as follows. As­sume the system is oscillating with the value of \(E\) necessary to make \(G_D(E) a(j \sqrt{3}) = -1\). An incremental increase in the value of \(E\) decreases the magnitude of \(G_D(E)\) and thus decreases the loop transmission below the value necessary to maintain a constant-amplitude oscillation. The amplitude decreases until \(E\) is restored to its original value. Similarly, an incremental decrease in \(E\) leads to a growing-amplitude oscillation until \(E\) reaches its equilibrium value.

    The magnitude of \(E\) under steady-state conditions can be determined directly from Equation \(\ref{eq6.3.20}\) and \(\ref{eq6.3.21}\). The magnitude of \(a(j\omega)\) at the frequency where its phase shift if \(-180^{\circ}\), (\(\omega = \sqrt{3}\)), is 1.25. Thus oscillations occur with \(G_D(E) = 0.8\). Solving Equation \(\ref{eq6.3.20}\) and \(\ref{eq6.3.21}\) for the required value of \(E\) by trial and error results in \(E \simeq 1.45\), and this value corresponds to the amplitude of the fundamental component of \(v_A\).

    The validity of the describing-function assumption concerning the purity of the signal at the input of the nonlinear element is easily demonstrated for this example. If a sinusoid is applied to the limiter, only odd harmonics are present in its output signal, and the amplitudes of higher harmonics decrease monotonically. The usual Fourier-series calculations show that

    the ratio of the magnitude of the third harmonic to that of the fundamental at the output of the limiter is 0.14 for a 1.45-volt peak-amplitude sinusoid as the limiter input. The linear elements attenuate the third harmonic of a \(\sqrt{3}\) radian-per-second sinusoid by a factor of 18 greater than the funda­mental. Thus the ratio of third harmonic to fundamental is approximately 0.008 at the input to the nonlinear element. The amplitudes of higher harmonics are insignificant since their magnitudes at the limiter output are smaller and since they are attenuated to a greater extent by the linear ele­ment. As a matter of practical interest, the attenuation provided by the phase-shift network to harmonics is the reason that good design practice dictates the use of the signal out of the phase-shift network rather than that from the limiter as the oscillator output signal.

    截屏2021-08-11 下午9.48.50.png
    Figure 6.14 Function generator. (\(a\)) Configuration. (\(b\)) Waveforms.

    Figure 6.14\(a\) shows another oscillator configuration that is used as a second example of describing-function analysis. This circuit, which com­bines a Schmitt trigger and an integrator, is a simplified representation of that used in several commercially available function generators. It can be shown by direct evaluation that the signal at the input to the nonlinear element is a two-volt peak-to-peak triangle wave with a four-second period and that the signal at the output of the nonlinear element is a two-volt peak-to-peak square wave at the same frequency. Zero crossings of these two signals are displaced by one second as shown in Figure 6.14\(b\). The ratio of the third harmonic to the fundamental at the input to the nonlinear ele­ment is 1/9, a considerably higher value than in the previous example.

    Table 6.1 shows that the describing function for this nonlinearity is

    \[G_D (E) = \dfrac{4}{\pi E} \measuredangle -\sin^{-1} \dfrac{1}{E} \ \ E \ge 1 \nonumber \]

    截屏2021-08-11 下午9.52.32.png
    Figure 6.15 Describing-function analysis of the function generator.

    The quantity \(-1/G_D(E)\) and the transfer function for the linear element are plotted in gain-phase form in Figure 6.15. The intersection occurs for a value of \(E\) that results in the maximum phase lag of \(90^{\circ}\) from the nonlinear ele­ment. The parameters predicted for the stable-amplitude limit cycle im­plied by this intersection are a peak-to-peak amplitude for vA of two volts and a period of oscillation of approximately five seconds. The correspond­ence between these parameters and those of the exact solution is excellent considering the actual nature of the signals involved.

    Conditional Stability

    截屏2021-08-11 下午9.53.42.png
    Figure 6.16 Conditionally stable system.

    The system shown in block-diagram form in Figure 6.16 combines a satu­rating nonlinearity with linear elements. The negative of the loop trans­mission for this system, assuming that the amplitude of the signal at \(v_A\) is less than \(10^{-5}\) volts so that the nonlinearity provides a gain of \(10^5\), is determined by breaking the loop at the inverting block, yielding

    \[-L(s) = 10^5 a(s) = \dfrac{5 \times 10^5 (0.02s + 1)^2}{(s + 1)^3 (10^{-3} s + 1)^2}\label{eq6.3.23} \]

    A Nyquist diagram for this function is shown in Figure 6.17. The plot re­veals a phase margin of \(40^{\circ}\) combined with a gain margin of 10, implying moderately well-damped performance. The plot also shows that if the mag­nitude of the low-frequency loop transmission is lowered by a factor of between 8 and \(6 \times 10^4\), the system becomes unstable. Systems having the property that a decrease in the magnitude of the low-frequency loop trans­mission from its design-center value converts them from stable to unstable performance are called conditionally stable systems.

    The nonlinearity can produce the decrease in gain that results in insta­bility. The system shown in Figure 6.16 is stable for sufficiently small values of the signal \(v_A\). If the amplitude of \(v_A\) becomes large enough, possibly be­cause of an externally applied input (not shown in the diagram) or because of the transient that may accompany the turn-on, the system may start to oscillate because the describing-function gain decreases.

    The common characteristic of conditionally stable systems is a phase curve that drops below \(-180^{\circ}\) over some range of frequencies and then recovers so that positive phase margin exists at crossover. These phase characteristics can result when the amplitude falls off more rapidly than \(1/\omega^2\) over a range of frequencies below crossover. The high-order rolloff is used in some systems since it combines large loop transmissions at moderate frequencies with a limited crossover frequency. For example, the transfer function

    \[-L'(s) = \dfrac{5 \times 10^5}{(2.5 \times 10^3 s + 1)(10^{-3} s + 1)^2}\label{eq6.3.24} \]

    has the same low-frequency gain and unity-gain frequency as does Equation \(\ref{eq6.3.23}\). However, the desensitivity associated with Equation \(\ref{eq6.3.23}\) exceeds that of \(\ref{eq6.3.24}\) at frequencies between \(4 \times 10^4\) radians per second and 50 radians per second because of the high-order rolloff associated with Equation \(\ref{eq6.3.23}\). The gain advantage reaches a maximum of approximately 10 at one radian per second. This higher gain results in significantly greater desensitivity for the loop transmission of Equation \(\ref{eq6.3.23}\) over a wide range of frequencies.

    Quantitative information about the performance of the system shown in Figure 6.16 can be obtained using describing-function analysis. The describing-function for the nonlinearity for \(E > 10^{-5}\) is

    \[G_D (E) = \dfrac{2 \times 10^5}{\pi} \left (\sin^{-1} \dfrac{10^{-5}}{E} + \dfrac{10^{-5}}{E} \sqrt{1 - \dfrac{10^{-10}}{E^2}} \right ) \measuredangle 0^{\circ} \nonumber \]

    截屏2021-08-11 下午10.09.10.png
    Figure 6.17 Nyquist diagram of conditionally stable system.
    截屏2021-08-11 下午10.10.19.png
    Figure 6.18 Describing function analysis of conditionally stable system.

    where \(E\) is the amplitude of the (assumed sinusoidal) signal \(v_A\). The quan­tities \(-1/G_D(E)\) and \(a(j\omega)\) are plotted in gain-phase form in Figure 6.18, and two intersections are evident. The intersection at \(\omega \simeq 50\) radians per sec­ond, \(E \simeq 10^{-4}\) volt does not represent a stable limit cycle. If the system is assumed to be oscillating with these parameters, an incremental decrease in the amplitude of the signal \(v_A\) leads to a further decrease in amplitude and the system returns to stable operation. This result follows from the rule mentioned in Section 6.3.2. In this case, a decrease in \(E\) causes the \(-1/G_D(E)\) curve to lie to the left of the \(a(j\omega)\) curve, and thus the system poles move from the imaginary axis to the left-half plane as a consequence of the perturbation. The same conclusion is reached if we consider the Nyquist plot for the system when the amplitude of \(v_A\) is \(10^{-4}\) volt. The gain attenuation of the limiter then shifts the curve of Figure 6.17 downward so that the point corresponding to \(\omega = 50\) radians per second intersects the - 1 point. An incremental decrease in \(E\) moves the curve upward slightly, and the resulting Nyquist diagram is that of a stable system.

    Similar reasoning shows that a small increase in amplitude at the lower intersection leads to further increases in amplitude. Following this type of perturbation, the system eventually achieves the stable-amplitude limit cycle implied by the upper intersection with \(\omega \simeq 1.8\) radians per second and \(E \simeq 0.73\) volt. (The reader should convince himself that the upper inter­section satisfies the conditions for a stable-amplitude limit cycle.)

    It should be noted that the concept of conditionally stable behavior aids in understanding the large-signal performance of systems for which the phase shift approaches but does not exceed ­\(-180^{\circ}\) well below crossover, and then recovers to a more reasonable value at crossover. While these systems can exhibit excellent performance for signal levels that constrain operation to the linear region, performance generally deteriorates dra­matically when some element in the loop saturates. For example, the recovery of this type of system following a large-amplitude step may include a number of large-signal overshoots, even if the small-signal step response of the system is approximately first order.

    Although a detailed analysis of such behavior is beyond the scope of this book, examples of the large-signal performance of systems that approach conditional stability are included in Chapter 13.

    Nonlinear Compensation

    As we might suspect, the techniques for compensating nonlinear systems using either linear or nonlinear compensating networks are not particu­larly well understood. The method of choice is frequently critically depend­ent on exact details of the linear and nonlinear elements included in the loop. In some cases, describing-function analysis is useful for indicating compensation approaches, since systems with greater separation between the \(a(j\omega)\) and \(- 1/G_D(E)\) curves are generally relatively more stable. This section outlines one specific method for the compensation of nonlinear systems.

    As mentioned earlier, fast-rolloff loop transmissions are used because of the large magnitudes they can yield at intermediate frequencies. Unfor­tunately, if the phase shift of this type of loop transmission falls below \(-180^{\circ}\) at a frequency where its magnitude exceeds one, conditional sta­bility can result. Nonlinear compensation can be used to eliminate the pos­sibility of oscillations in certain systems with this type of loop transmission.

    As one example, consider a system with a linear-region loop transmission

    \[-L(s) = \dfrac{200}{(s + 1)(10^{-3} s + 1)^2}\label{eq6.3.26} \]

    This loop transmission has a monotonically decreasing phase shift as a function of increasing frequency, and exhibits a phase margin of approxi­mately \(65^{\circ}\). Consequently, unconditional stability is assured even when some element in the loop saturates.

    In an attempt to improve the desensitivity of the system, series compen­sation consisting of gain and two lag transfer functions might be added to the loop transmission of Equation \(\ref{eq6.3.26}\), leading to the modified loop trans­mission

    \[-L'(s) = \left [\dfrac{200}{(s + 1)(10^{-3} s + 1)^2} \right ] \left [\dfrac{2.5 \times 10^3 (0.02s + 1)^2}{(s + 1)^2} \right ] \label{eq6.3.27} \]

    This loop transmission is of course the one used to illustrate the possibility of conditional stability (Equation \(\ref{eq6.3.23}\)).

    截屏2021-08-11 下午10.25.33.png
    Figure 6.19 Nonlinear compensating network.

    Consider the effect of implementing one or both of the lag transfer func­tions with a network of the type shown in Figure 6.19. If the magnitude of voltage \(v_c\) is less than \(V_B\), the diodes do not conduct and the transfer function of the network is

    \[\dfrac{V_o (s)}{V_i (s)} = \dfrac{R_2Cs + 1}{(R_1 + R_2)Cs + 1} \nonumber \]

    Element values can be selected to yield the lag parameters included in Equation \(\ref{eq6.3.27}\).

    The bias voltage \(V_B\) is chosen so that when the signal applied to the network is that which exists when the loop oscillates, the diodes clip the capacitor voltage during most of the cycle. Under these conditions, the gain of the nonlinear network (in a describing-function sense) is

    \[\dfrac{v_O}{v_I} \simeq \dfrac{R_2}{R_1 + R_2} \nonumber \]

    Note that if both lag transfer functions are realized this way, the loop transmission can be made to automatically convert from that given by Equation \(\ref{eq6.3.27}\) to that of Equation \(\ref{eq6.3.26}\) under conditions of impending instability. This type of compensation can eliminate the possibility of conditionally stable performance in certain systems. The signal levels that cause satura­tion also remove the lag functions, and thus the possibility of instability can be eliminated.

    PROBLEMS

    Exercise \(\PageIndex{1}\)

    截屏2021-08-11 下午10.34.43.png
    Figure 6.20 Nonlinear elements. (\(a\)) Limiter. (\(b\)) Deadzone.

    One of the difficulties involved in analyzing nonlinear systems is that the order of nonlinear elements in a block diagram is important. Demon­strate this relationship by comparing the transfer characteristics that result when the two nonlinear elements shown in Figure 6.20 are used in the order \(ab\) with the transfer characteristics that result when the order is changed to \(ba\).

    Exercise \(\PageIndex{2}\)

    截屏2021-08-11 下午10.36.55.png
    Figure 6.21 Positional servomechanism.

    Resolvers are essentially variable transformers that can be used as mechanical-angle transducers. When two of these devices are used in a servomechanism, the voltage obtained from the pair is a sinusoidal function of the difference between the input and output angles of the system. A model for a servomechanism using resolvers is shown in Figure 6.21.

    (a) The voltage applied to the amplifier-motor combination is zero for \(\theta_O - \theta_I = n \pi\), where \(n\) is any integer. Use linearized analysis to deter­mine which of these equilibrium points are stable.

    (b) The system is driven at a constant input velocity of 7 radians per sec­ond. What is the steady-state error between the output and input for this excitation?

    (c) The input rate is charged from 7 to 7.1 radians per second in zero time. Find the corresponding output-angle transient.

    Exercise \(\PageIndex{3}\)

    截屏2021-08-11 下午10.40.10.png
    Figure 6.22 Square-rooting circuit.

    An analog divider was described in Section 6.2.2. Assume that the trans­fer function of the operational amplifier shown in Figure 6.2 is

    \[a(s) = \dfrac{3 \times 10^5}{(s + 1)(10^{-5} s + 1)^2 \nonumber \]

    Is the divider stable over the range of inputs \(-10 < v_A < + 10\), \(0 < v_B < +10\)?

    A square-rooting circuit using a technique similar to that of the divider is shown in Figure 6.22. What is the ideal input-output relationship for this circuit? Determine the range of input voltages for which the square-rooter is stable, assuming \(a(s)\) is as given above.

    Exercise \(\PageIndex{4}\)

    截屏2021-08-11 下午10.42.56.png
    Figure 6.23 Inverted pendulum

    Figure 6.23 defines variables that can be used to describe the motion of an inverted pendulum. Determine a transfer function that relates the angle 0 to the position \(x_B\), which is valid for small values of \(\theta\). Hint. You may find that a relatively easy way to obtain the required transfer function is to use the two simultaneous equations (or the corresponding block diagram) which relate \(x_T\) to \(\theta\) and \(\theta\) to \(x_B\) and \(x_T\).

    Assume that you are able to drive \(x_T\) as a function of \(\theta\). Find a transfer function, \(X_t(s)/\theta (s)\), such that the inverted pendulum is stabilized.

    Exercise \(\PageIndex{5}\)

    截屏2021-08-11 下午10.46.04.png
    Figure 6.24 Diode-capacitor network.

    A diode-capacitor network is shown in Figure 6.24. Plot the output voltage that results for a sinewave input signal with a peak value of \(E\). You may assume that the diodes have an ideal threshold of 0.5 volt (i.e., no conduction until a forward-bias voltage of 0.5 volt is reached, any forward cur­rent possible without increasing the diode voltage above 0.5 volt). Evalu­ate the magnitude and angle of \(G_D(1)\) for this network. (You may, of course, work out \(G_D (E)\) in general if you wish, but it is a relatively involved expression.)

    Exercise \(\PageIndex{6}\)

    截屏2021-08-11 下午10.50.08.png
    Figure 6.25 Nonlinear transfer relationship.

    Determine the describing function for an element with the transfer char­ acteristics shown in Figure 6.25.

    Exercise \(\PageIndex{7}\)

    截屏2021-08-11 下午10.51.34.png
    Figure 6.26 Nonlinear oscillator.

    Analyze the loop shown in Figure 6.26. In particular, find the frequency of oscillation and estimate the levels of the signals \(v_A\) and \(v_B\). Also calculate the ratio of third harmonic to first harmonic at the input to the nonlinear

    Exercise \(\PageIndex{8}\)

    截屏2021-08-11 下午10.52.46.png
    Figure 6.27 Nonlinear system.

    Can the system shown in Figure 6.27 produce a stable amplitude limit cycle? Explain.

    Exercise \(\PageIndex{9}\)

    Find a transfer function that, when combined with a limiter, can pro­duce stable-amplitude limit cycles at two different frequencies. Design an operational-amplifier network that realizes your transfer function.

    Exercise \(\PageIndex{10}\)

    截屏2021-08-11 下午10.54.36.png
    Figure 6.28 Controller transfer characteristics

    The transfer characteristics for a three-state, relay-type controller are illustrated in Figure 6.28.

    (a) Show that the describing function for this element is

    \[G_D (E) = \dfrac{2}{\pi E} \sqrt{2 + 2\sqrt{1 - \dfrac{1}{E^2}}} \measuredangle -\tan^{-1} \dfrac{1}{E \left (1 + \sqrt{1 - \dfrac{1}{E^2}} \right )}\nonumber \]

    (b) The controller is combined in a negative-feedback loop with linear elements with a transfer function

    \[a(s) = \dfrac{a_0}{(s + 1)(0.1s + 1)}\nonumber \]

    What is the range of values of \(a_0\) for stable operation?

    (c) For \(a_0\) that is twice the critical value, find the amplitude of the funda­mental component of the signal applied to the controller.

    Exercise \(\PageIndex{11}\)

    截屏2021-08-11 下午11.01.23.png
    Figure 6.29 \(R-L-C\) oscillator.

    One possible configuration for a sinusoidal oscillator combines a Schmitt trigger with an \(R-L-C\) circuit as shown in Figure 6.29. Find the relationship between \(E_M\), \(E_N\), and the damping ratio of the network that insures that oscillations can be maintained. (You may assume negligible loading at the input and output of the Schmitt trigger.)

    Exercise \(\PageIndex{12}\)

    Three loop-transmission values, given by Equations \(\ref{eq6.3.23}\), \(\ref{eq6.3.24}\), and \(\ref{eq6.3.26}\) were considered as part of the discussion of conditionally stable systems. Assume that three negative-feedback systems are constructed with \(f(s) = 1\) and loop transmissions given by the expressions referred to above. Com­pare performance by calculating the first three error coefficients for each of the three systems.


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