EC-7.5 Poles, Zeros and Stability

What this is and why it exists

There is a diagram that summarises a system more compactly than anything else in engineering. It is a set of crosses and circles on a plane. From it you can read whether the system is stable, how fast it settles, whether it rings, and what its frequency response looks like.

That diagram is the subject of this topic, and three later subjects argue about it constantly. Control systems asks how to move the marks. Filter design asks where to place them. Amplifier work asks which one is limiting the bandwidth. All three assume a reader who can look at the picture and say what the system will do.

The Laplace transform was defined in the transforms module. Here it is used, and what it is used for is turning a system into that picture.

The vocabulary

• Transfer function — the Laplace transform of the impulse response, usually a ratio of two polynomials.
• Pole — a root of the denominator, where the transfer function grows without bound.
• Zero — a root of the numerator, where the transfer function vanishes.
• Complex plane — the plane on which poles and zeros are marked. Its horizontal axis is the real part, its vertical axis the imaginary part.
• Imaginary axis — the vertical axis, which separates stable pole positions from unstable ones.
• Left half-plane — the region to the left of the imaginary axis.
• Region of convergence — the set of values for which the transform integral has a finite answer.
• Transient — the part of the response that dies away.
• Steady state — the part that persists.
• Damping — how quickly an oscillation loses amplitude.

The mental model

For most systems of interest, the impulse response is built from exponentials and sinusoids, and transforming such a thing gives a ratio of two polynomials. That ratio is the transfer function, and it holds exactly the same information as the impulse response, in a form you can factor.

Factor it. The roots of the denominator are the poles and the roots of the numerator are the zeros. Mark the poles with crosses and the zeros with circles on the complex plane. That picture is the system. Everything below is read off it.

Take stability first, because it is the cleanest result in the module. A causal continuous-time system is stable when every pole lies strictly to the left of the imaginary axis. A pole on the axis gives a response that neither grows nor decays. A pole to the right of it gives one that grows without limit. Zeros do not affect stability at all.

Why the left half-plane? A pole at some position contributes an exponential to the impulse response, and the real part of the position is the exponent's rate. Negative real part, decaying exponential. Positive real part, growing exponential. Stability is that one sentence, plus the requirement that it hold for every pole, because one growing term is enough to ruin the total.

Now the shape of the response, which is read from the same marks. The horizontal distance of a pole from the imaginary axis sets how fast its contribution decays: further left, faster decay. Its vertical distance sets how fast that contribution oscillates: further from the horizontal axis, faster oscillation. Poles come in matched pairs above and below the horizontal axis whenever the system is real, and such a pair is a decaying oscillation.

This gives you a working intuition worth keeping. A pair of poles close to the imaginary axis is ringing — an oscillation that takes a long time to die away. That is exactly the overshoot you see on an oscilloscope when a circuit is tapped. Move the pair further left and the ringing damps out sooner. A single pole on the horizontal axis gives a plain decay with no oscillation at all.

Zeros do not decide stability but they shape the response strongly. A zero close to a pole partly cancels that pole's contribution, which is how a filter is made to reject one particular frequency. A zero in the right half-plane causes a response that initially moves the wrong way, which is a genuine nuisance in control design.

There is one subtlety that is usually glossed over and should not be. A transfer function written as a ratio of polynomials is ambiguous on its own. The same algebraic expression can belong to a causal unstable system or to a non-causal stable one. Which of the two it is depends on the region of convergence. The region is the set of values for which the defining integral converges. For a causal system it lies to the right of the rightmost pole. Stating the region is part of stating the answer, not a technicality, and it matters again in the last topic of this module.

Finally the split that the control work will formalise. The response of a system to an input separates into two parts. One is determined by the system's own poles, and dies away for a stable system. The other is determined by the input, and persists. The first is the transient and the second is the steady state.

For a stable system, waiting long enough leaves only the second part. The long-run behaviour then depends on the input and not on how the system started. That is the property that makes the steady-state response a meaningful thing to specify. For an unstable system there is no steady state at all, which is why stability has to be settled before any of it is discussed.

What you should now be able to explain or do

• Obtain a transfer function from an impulse response and factor it into poles and zeros.
• Mark poles and zeros on the complex plane and state the stability condition for a causal continuous-time system.
• Explain why the left half-plane is the stable region, in terms of the exponentials the poles contribute.
• Read decay rate and oscillation rate off a pole position, and recognise a pole pair close to the imaginary axis as ringing.
• Say what zeros do and do not affect.
• Explain why a transfer function needs its region of convergence stated to be unambiguous.
• Separate a response into transient and steady state and say why only a stable system has the second.

Check yourself