Signal that destabilises a positive feedback path with nonunity feedback

I am just starting to learn about feedback systems. There are a few questions that have me stumped, this being one of them. Could someone please help? Thanks!

Question: Find a signal that destabilizes a system with a delayed positive feedback path and nonunity feedback gain.

My understanding so far is that we are reintroducing the output as an input into the system with a positive coefficient. So if the input is positive, the signal with be amplified. Nonunity feedback gain means only a fraction of the output is being fed back into the system. Is that correct so far?

Now we want to find a signal that is destabilised by this feedback system. That means the original signal should be stable.

A sinusoidal wouldn't work, because the negative inputs cancel out the positives. The square of a sinusoidal is already unstable. How about the absolute value of a sinusoidal? Would that work?

• Just to know where to start from, have you covered Laplace transforms yet and does the concept of $H(s) = \frac{1}{s+1}$ mean anything? You don't yet mention phase and restrict yourself to positive and negative (0 and 180 degrees), so want to also change if the idea of "phase" for your system is possibly also a source of confusion? Mar 7, 2021 at 21:02
• I have covered Laplace transforms but not in much detail. I'll add that to my list to expand on. I haven't come across $H(s) = 1/{s+1}$ yet. I think what you're trying to say with your last question is to consider not just 0 and $\pi$ phase shifts, but others too? I don't think I really even considered that in the delay quantity. Please excuse by n00bness, I have taken in a lot in a short space of time. Mar 7, 2021 at 21:32
• I teach a course that is coming up that may interest you that would cover these missing details; depending on your starting background. You can check out the course outline here: ieeeboston.org/event/digital-signal-processing-webinar As long as you have a basic understanding of complex signals, the Fourier Transform and working with waveforms in both the time and frequency domain (and what the magnitude and phase would be in frequency etc) this would be helpful to you. If not then there would likely be some prerequisite study first. In the meantime I can give a quick answer below Mar 7, 2021 at 21:59
• Thanks. That's still a while out, so I think I'll continue trying my best and I'll see where I'm at when your course starts. When's the deadline for applying? I'm forming a decent grasping of the FT, eg I understand magnitude and phase. Some of it does go over my head at times, but I think that comes down to not having someone to ask when I'm stuck. Mar 7, 2021 at 22:20
• I just watched a video explaining the Laplace transform. Mar 7, 2021 at 22:23

1 Answer

I am going to provide some initial information as simply as I can that should be helpful in the OP's attempt to get a basic understanding of control systems. Further study in Laplace transforms (and z-transforms for discrete-time systems) will be helpful in these frequency domain modelling approaches to control systems.

First understand the concept of a transfer function as the frequency response for a basic two port (one input / one output) linear system; the transfer function defines the magnitude and phase versus frequency of the output compared to the input. This is often given in terms of the variable "s", but to avoid going down that path I am going to present it as a function of frequency only ($$H(f)$$). Below shows an example system with a plot of the magnitude and phase versus frequency as given by the $$H(f)$$ equation. This shows us for any particular frequency at the input, what the amplitude of it will be at the output and what the phase shift will be between the input and output: every frequency will be transformed by the system by changing its amplitude and phase shift only, as long as it is a linear system no other frequencies will be created so this is a nice compact description that we will ultimately use to determine stability in a control system.

Now consider the simplest case of a system with feedback as diagrammed below with input given as $$X$$ and output given as $$Y$$ with the output fed back and subtracted from the input:

The relationship between $$Y$$ and $$X$$ is easy to show with simple algebra, with our goal to describe a new transfer function of this new closed-loop system. The new transfer function would be given as $$Y/X$$ (output divided by input). I don't show it below to not clutter the simplicity of the equation but $$X$$ and $$Y$$ are also functions of frequency as $$X(f)$$ and $$Y(f)$$, so would be the Fourier Transforms of time domain signals:

$$(X-Y)H(f) = Y$$

$$\frac{Y}{X} = \frac{H(f)}{1+H(f)}$$

The above equation would adequately describe our new closed-loop system as long as it follow the Nyquist Stability Criterion. To really show this completely to cover all cases, the OP needs to first understand the Laplace Transform and the generalized transfer function as a function of s (as in $$H(s)$$), but here in this simple introduction we can reduce it to a condition that is covered in most cases: that is if the gain of the system given by $$|H(f)|$$ is less than one before the phase of the system given by $$\angle (H(f))$$ exceeds 180°, the system will be stable and will have its own closed loop transfer function given by the expression above. We can see this through an iteration of a simple case with H(f) = -.8 at a particular frequency (gain = 0.8 and phase = 180°).

Consider the feedback system as an iteration with the input starting at 1: the output will be -.8, which fed back to the input as a subtraction will add 0.8 to the input, resulting in 1.8 at the output. This is then multiplied by -.8 resulting in 1.44 being added to the input resulting in 2.44 at the output...

With closer inspection we see that this is a geometric series:

$$1 + \alpha + \alpha^2 + \ldots = \sum_{n=0}^\infty(\alpha^n) = \frac{1}{1-\alpha}$$

Which converges only if $$-1 < \alpha < 1$$.

Where here $$\alpha$$ was 0.8 (due to the negative feedback in the loop) which would converge to $1/(1-.8) = 5. If the gain of $$H(f)$$ exceeded one with the phase 180°, we would see how the same geometric series would not converge, as an example of an unstable loop. This gives a very basic simplified explanation to help aid in an intuitive understanding, but further proper study would be to understand the Laplace Transform and complex systems, and the meaning of poles and zeros in the s-plane, and how a stable system must have all of its poles in the left-half plane, and ultimately the full details of the Nyquist Stability Criterion. • Thanks! I understand that, except I'm confused when you mention "with the phase 180°". I really appreciate your help. Mar 7, 2021 at 23:21 • The phase shift is dependent on frequency (so there may be a frequency where the phase shift equals 180° which would change the sign of the signal:$\sin(\omega t + \pi) = -\sin(\omega t)$where$\pi\$ is 180° in radians. So what you do in a simplistic approach to check stability is observe the gain at the point where the phase crosses 180°, if the gain is less than 1, in most simpler cases the loop will be stable. There are cases where this doesn't hold (but for most simple loops it will) which is why ultimately you want to go by the Nyquist Stability Criteria. Mar 7, 2021 at 23:43