Understanding the Describing Function Method in Control Systems

$$A_1 = \frac{2}{\pi} \int_{0}^{\pi} y \sin(\omega t) \, d(\omega t)$$ $$B_1 = \frac{2}{\pi} \int_{0}^{\pi} y \cos(\omega t) \, d(\omega t)$$

Where the amplitude is $\gamma_1 = \sqrt{A_1^2 + B_1^2}$ and the phase is $\phi_1 = \tan^{-1}(\frac{B_1}{A_1})$.

Exam Tip: When performing these derivations, always check if the nonlinearity is symmetric. If the nonlinearity is an odd function (like a relay or deadzone), the output will have no even harmonics and no DC component ($A_0 = 0$), and the phase shift $\phi_1$ will typically be zero.

1. Relay (On-Off) Nonlinearity

A relay nonlinearity is a switching element that jumps between two fixed values, $+m$ and $-m$. This is commonly seen in thermostat controls or simple bang-bang controllers.

When a sinusoidal input $x(t) = X \sin(\omega t)$ is applied, the output $y(t)$ switches between $m$ and $-m$ at the points where the input crosses zero. To find the describing function, we calculate the fundamental Fourier coefficient $A_1$ using the integral:

$$A_1 = \frac{2}{\pi} \int_{0}^{\pi} y \sin(\omega t) \, d(\omega t)$$

For a relay switching between $\pm m$, the integration yields a fundamental amplitude of $Y_1 = \frac{4m}{\pi}$. Therefore, the describing function for a relay is:

$$N(X, \omega) = \frac{4m}{\pi X}$$

Note that the gain of a relay decreases as the input amplitude $X$ increases, because the "switching" happens less frequently relative to the period of the sine wave.

2. Deadzone Nonlinearity

A deadzone nonlinearity is characterized by a region where the input produces zero output. This occurs when the input magnitude is less than a threshold, often denoted as $D/2$. Once the input exceeds this threshold, the output follows a linear relationship with slope $k$.

To derive the describing function, we must integrate the output only over the intervals where the input is outside the deadzone. Let $\alpha$ be the angle where the input reaches the deadzone threshold, such that $\sin(\alpha) = \frac{D}{2X}$.

After solving the definite integrals for the linear segments, the describing function is found to be:

$$N(X, \omega) = k \left[ 1 - \frac{2}{\pi} \left( \sin^{-1}\left(\frac{D}{2X}\right) + \frac{D}{2X} \sqrt{1 - \left(\frac{D}{2X}\right)^2} \right) \right]$$

This formula shows that as $X$ becomes very large compared to $D$, the term in the brackets approaches 1, and the nonlinearity behaves like a pure linear gain $k$. However, as $X$ approaches $D/2$, the effective gain drops to zero.

3. Saturation Nonlinearity

Saturation occurs when a system's output increases linearly with the input up to a certain limit $S$, but then remains constant (saturates) regardless of further increases in the input.

In this case, the output $y(t)$ is $kx(t)$ when $|x(t)| < S$, and $y(t) = \pm kS$ when $|x(t)| \geq S$. To find the describing function, we define $\beta$ as the angle where the input reaches the saturation limit, such that $\sin(\beta) = \frac{S}{X}$.

The fundamental amplitude $Y_1$ is calculated by integrating the linear part and the saturated part separately:

$$A_1 = \frac{4}{\pi} \left[ \int_{0}^{\beta} k X \sin(\omega t) \sin(\omega t) \, d(\omega t) + \int_{\beta}^{\pi/2} k S \sin(\omega t) \sin(\omega t) \, d(\omega t) \right]$$

Solving these integrals leads to the describing function:

$$N(X, \omega) = \frac{2k}{\pi} \left[ \sin^{-1}\left(\frac{S}{X}\right) + \frac{S}{X} \sqrt{1 - \left(\frac{S}{X}\right)^2} \right]$$

As the input amplitude $X$ increases, the ratio $S/X$ decreases, causing the effective gain of the describing function to decrease. This reflects the fact that a larger portion of the input signal is spent in the "saturated" region where the gain is effectively zero.

Summary Table of Describing Functions

< D/2$|$k [ 1 - \frac{2}{\pi} (\sin^{-1}(\frac{D}{2X}) + \frac{D}{2X}\sqrt{1-(\frac{D}{2X})^2}) ]$| | Saturation | Constant output for$|x| > S$|$\frac{2k}{\pi} [ \sin^{-1}(\frac{S}{X}) + \frac{S}{X}\sqrt{1-(\frac{S}{X})^2} ]$ |

By replacing the complex nonlinear block with its corresponding describing function $N(X, \omega)$, engineers can use frequency-domain techniques like Nyquist or Bode plots to analyze the stability and limit cycle behavior of nonlinear control systems.