Control Engineering

Why do control systems oscillate?
August 6, 2007

Three things are needed for a system to oscillate. Whenever they’re all present, oscillations will occur. When any one of them is missing, they won’t.

The three things are a resonance, gain, and positive feedback.

Resonance is the ability of something to store energy at a certain frequency. There are two equivalent ways to create resonance in a system. The first is to install a resonant component, such as a violin string or an RF tank circuit, which engineering teams hardly ever do unintentionally. The second is incorporating frequency-dependent phase shifts into the feedback path that sum to zero or 360º.

Gain feeds energy into the oscillation at the resonant frequency to make up for inevitable losses. Electronic oscillators have an electronic amplifier to feed electrical energy into the resonance. Mechanical clocks use an escapement to feed energy into the pendulum or balance wheel oscillations. Oscillation requires that the gain and loss elements sum to unity or more around the feedback loop.

Positive feedback keeps the energy properly timed to reinforce the oscillation. This is equivalent to saying that the phase shifts sum to zero or 360º.

The double torsion pendulum described in the July 23, 2007, entry (What’s so great about mechatronics?) was an example of a resonant element. Since the structure itself had a natural resonance, all that was required to provide positive feedback through a gain stage (amplifier). In that case, positive feedback was accidentally applied.

Oscillations appear apparently from nowhere due to interactions between non-resonant system components that have frequency-dependent responses. In other words: filters. Simple filters come in low-pass and high-pass varieties. Every such filter has a characteristic turn-over or cutoff frequency, which roughly divides between frequencies the filter passes and those it does not. The filter response characteristic (output signal strength divided by input signal strength) has a positive slope for high-pass filters and negative for low-pass.

What matters more for oscillations, however, is the fact that filters necessarily introduce frequency-dependent phase shifts as well. These shifts’ absolute values vary between 0º and 90º. They are positive for high-pass filters and negative for low-pass filters. High-pass filters shift high frequencies very little, while they shift low frequencies much more. Low-pass filters shift low frequencies very little and high frequencies a lot.

Suppose, for example, an engineering team is building a fly-by-wire system for an unpiloted aerial vehicle (UAV). The rudder control, for example, uses a dc electric motor driving a threaded shaft through a flexible coupling, with the shaft connected to the rudder assembly through a nut captured at the end of a bell crank. A potentiometer whose wiper turns with the rudder hinge provides feedback to the system.

To start with, the flexible coupling acts as a low-pass filter, passing the static load while isolating vibrations. Let’s say the mechanical engineer designs the coupling to have a cutoff frequency of 578 Hz.

In the traditional design approach, the mechanical engineer then hands the design off to an electronics engineer, who designs the amplifier to drive the control loop. To keep noise generated by turbulence around the UAV’s tail from affecting the system, she includes an electronic low-pass filter. Wouldn’t you know it? She picks a cutoff frequency of 578 Hz, too!

Next comes the controls engineer, who gets to pick the motor. Motors of course, respond better to signals that drive them at a steady speed than rapidly varying speeds. In other words, they can act as a low-pass filter, too. And, son of a gun, the controls engineer picks a motor that represents a filter with a cutoff frequency of 578 Hz.

Of course, the controls engineer also wires the whole thing up for negative feedback, which introduces a phase shift of 180º. He also sets the loop gain for a conservative 0 dB.

When he’s done, the project goes to the pattern shop for the first prototype. The electronics shop builds the electronics. All the components go to the test department, where they assemble and fly it. Thirty seconds into the first flight, the prototype starts making an ungodly whining noise. Two minutes into the flight, the tail falls off.

What happened was that each low-pass filter introduced a phase shift of 60º at 1,000 Hz. Three shifts of 60º add up to 180º. Add another 180º for the negative loop feedback, and you’ve got 360º to make a perfect 1 kHz oscillator. During takeoff and climb, aerodynamic forces damped the motion, so the actual loop gain stayed below unity. Later, things changed. The loop gain rose above unity, and the oscillation started. Metal fatigue took care of the rest.

The moral of this story is that, had the design team used concurrent/mechatronic methods, they might have noticed that the control-loop phase shifts were stacking up in a dangerous way, and saved the prototype.

Posted by on August 6, 2007 | Comments (1)