Describing Function: Analysis of Nonlinear Systems

The describing function is an approximate frequency-domain method for a restricted class of nonlinear feedback problems in control engineering. A linear system satisfies the principle of superposition: the response to a weighted sum of inputs equals the same weighted sum of the individual responses. A nonlinear system does not satisfy that property over the operating range being analysed.

Nonlinear systems can show amplitude-dependent frequency response, multiple equilibria, limit cycles, jumps and other behaviour that a single linear model cannot represent. The standard Nyquist stability criterion and pole-zero tests apply directly to linear models. They may still support a local linearisation or a describing-function approximation, but they do not by themselves prove global nonlinear stability. Nonlinear elements may provide useful design effects:

1. Saturation can represent or enforce a physical actuator limit.
2. Relay or switching control can implement a simple control law with discrete outputs.
3. Dead bands and hysteresis can reduce unwanted switching in the presence of noise.

Physical systems become nonlinear outside a sufficiently small operating region. A designer may also add saturation, switching, hysteresis or a dead band deliberately. Whether this improves performance, safety or cost depends on the plant, controller and operating requirements.

A thermostat is a familiar example. It turns heating on below one threshold and off above another, with hysteresis between the thresholds to avoid rapid switching. Two introductory analysis methods are listed below. Neither replaces a nonlinear model and simulation when the approximation assumptions are weak.

1. Describing function method in control system
2. Phase plane method in control system

Common Nonlinearities

Most types of control systems include nonlinear effects. A memoryless static nonlinearity maps the current input to the current output without an internal dynamic state. A dynamic nonlinearity has memory, so its output also depends on past input or internal state. Hysteresis and backlash are history-dependent even when shown as input-output plots.
The following sections introduce common non-linearities in a control system:

1. Saturation nonlinearity
2. Friction nonlinearity
3. Dead zone nonlinearity
4. Relay nonlinearity (ON OFF controller)
5. Backlash nonlinearity

Saturation Nonlinearity

Saturation limits output magnitude once the input leaves a central operating range. Magnetic saturation can appear in the field system of a DC motor. The following curve shows a piecewise approximation:

Within the central region, output is approximately proportional to input. Beyond the breakpoints, the incremental gain decreases and output approaches its limit.
An amplifier or actuator can show the same saturation non linearity when a commanded output exceeds the available voltage, current, force or travel.

Friction Nonlinearity

Friction opposes relative motion and creates a nonlinear relation between velocity and force or torque. An electric motor drive can include bearing friction, seal friction, brush contact and load friction.

Common modelling terms include:

1. Static Friction: The force that must be overcome to start relative motion.
2. Dynamic Friction: Friction during relative motion, often modelled with Coulomb and velocity-dependent components.
3. Limiting Friction: The maximum static friction immediately before sliding starts.
   Sliding and rolling contact produce different loss mechanisms. Sliding friction acts between surfaces that move across each other, while rolling resistance acts during rolling contact.
   Control models commonly separate static friction, Coulomb friction and viscous friction.

Dead Zone Nonlinearity

A dead zone can occur in motors, DC servo motors, valves and actuators. With dead zone non linearities, the output remains zero or unchanged while input stays within a band around the origin. Output responds only after the input magnitude exceeds the relevant boundary.

Relays Nonlinearity (ON/OFF Controller)

Relay control maps a continuous input to two or more discrete output states. A two-state version is an on/off controller.

The plots show (a) an ideal on/off characteristic, (b) on/off control with hysteresis and (c) on/off control with a dead zone. Hysteresis means the switching threshold depends on the previous output state. A dead zone means no output transition occurs while input remains inside a defined band. Real electromechanical relays also have finite operating and release times, contact dynamics and coil thresholds.

Backlash Nonlinearity

Mechanical transmissions such as gear trains and linkages can have clearance that produces backlash nonlinearities. When input direction reverses, the driving member moves through the clearance before it contacts the opposite face and transmits motion. The resulting input-output path depends on motion history.

In the diagram, tooth A starts between teeth B₁ and B₂. Clockwise input first takes up half the clearance before A contacts B₁. Output then follows the input along the engaged segment under the stated unity-ratio assumption. When input reverses, contact with B₁ is lost and the output remains fixed while the clearance is crossed, assuming negligible inertia and a friction-held load.
After the full clearance has been traversed, A contacts B₂ and drives the output in the opposite direction. Another reversal repeats the lost-motion interval. This closed input-output path is a form of mechanical hysteresis, but it should not be confused with magnetic hysteresis.

Describing Function Analysis of Nonlinear Systems

The describing function method in control system replaces a selected nonlinear element with an amplitude-dependent approximation based on the fundamental response to a sinusoidal input.

The describing function method can predict candidate self-sustained oscillations in a feedback loop that contains a suitable nonlinearity and linear dynamics. It extends frequency-response reasoning by using a gain and phase that may depend on input amplitude and, for dynamic nonlinearities, frequency. The result is approximate and does not establish general nonlinear stability.

The describing function method defines a complex ratio between the fundamental harmonic components of the nonlinear output and the sinusoidal input. The ratio gives amplitude-dependent gain and phase. Mathematically,

where:
N = describing function,
X = amplitude of the input sinusoid,
Y = amplitude of the fundamental output component,
φ₁ = phase shift of the fundamental output component.
The following block diagram separates the nonlinear element from the linear dynamics.
Here, G₁(s) and G₂(s) represent linear elements and N represents the nonlinear element.

Assume that input x to the nonlinear element is sinusoidal:

The nonlinear output y is generally a non-sinusoidal periodic function. Express it as a Fourier series:

For an odd-symmetric nonlinearity with a symmetric sinusoidal input, the mean value Y₀ is zero, giving:

The approximation retains only the fundamental component. This is most credible when G₁(s) and G₂(s) strongly attenuate higher harmonics, so those harmonics contribute little to the signal returning to N. If the loop does not have enough low-pass character, the approximation may be poor and nonlinear simulation is needed.
Under that assumption, consider y₁:

Write y₁(t) in the following form:

The corresponding phasor form is:

Coefficients A₁ and B₁ of the Fourier series are:

The describing function then follows from the fundamental output divided by the input:

The next sections apply this definition to several nonlinearities.

Describing Function for Saturation Non Linearity

We have the characteristic curve for saturation as shown in the given figure.

Let us take input function as

Now from the curve we can define the output as:

Let us first calculate Fourier series constant A₁.

On substituting the value of the output in the above equation and integrating the function from 0 to 2π we have the value of the constant A1 as zero.
Similarly we can calculate the value of Fourier constant B₁ for the given output and the value of B₁ can be calculated as,


The phase angle for the describing function can be calculated as

Thus the describing function for saturation is

Describing Function for Ideal Relay

We have the characteristic curve for ideal relay as shown in the given figure.

Let us take input function as

Now from the curve we can define the output as

The output periodic function has odd symmetry :

Let us first calculate Fourier series constant A₁.

On substituting the value of the output in the above equation and integrating the function from 0 to 2π we have the value of the constant A₁ as zero.
Similarly we can calculate the value of Fourier constant B₁ for the given output and the value of B₁ can be calculated as

On substituting the value of the output in the above equation y(t) = Y we have the value of the constant B₁

And the phase angle for the describing function can be calculated as

Thus the describing function for an ideal relay is

Describing Function for Real Relay (Relay with Dead Zone)

We have the characteristic curve for real realy as shown in the given figure. If X is less than dead zone Δ, then the relay produces no output; the first harmonic component of Fourier series is of course zero and describing function is also zero. If X > Δ the relay produces the output.

Let us take input function as

Now from the curve we can define the output as


The output periodic function has odd symmetry :

Let us first calculate Fourier series constant A₁.

On substituting the value of the output in the above equation and integrating the function from 0 to 2π we have the value of the constant A₁ as zero.
Similarly we can calculate the value of Fourier constant B for the given output and the value of B can be calculated as

Due to the symmetry of y, the coefficient B₁ can be calculated as follows,

Therefore, the describing function is

Describing Function for Backlash Non Linearity

We have the characteristic curve for backlash as shown in the given figure. Let us take input function as


Now from the curve we can define the output as

Let us first calculate Fourier series constant A₁.

On substituting the value of the output in the above equation and integrating the function from zero to 2π we have the value of the constant A₁ as

Similarly we can calculate the value of Fourier constant B for the given output and the value of B₁ can be calculated as

On substituting the value of the output in the above equation and integrating the function from zero to pi we have the value of the constant B₁ as

We can easily calculate the describing function of backlash from below equation