SC · Laplace

The s-Plane, Poles and Zeros

A single pole position on the s-plane fixes almost all of a system’s behavior. The pole’s real part says how fast the response decays, its imaginary part how fast it oscillates. Move the pole and watch the response change.

Move the pole, change the response

On the left is the s-plane, with × marking a pole pair s=σ±jω. The green region is the stable left half; the central vertical line is the imaginary axis. Below is that pole’s natural response e^(σt)cos(ωt). Use the two sliders to move the pole into the left half, where the response decays to 0.

s-plane (× = pole)

natural response e^(σt)cos(ωt)

real part σ0.40

imaginary part ω2.00

Pole position and response

s = 0.40 + j2.00

The pole is right of the imaginary axis, its real part σ positive. The natural response e^(σt)cos(ωt) grows without bound. A pole in the right half — an unstable system.

Unstable (right half)

Poles are infinity, zeros are zero

A transfer function is written as a ratio H(s) = N(s)/D(s). At the s values that make the denominator D(s) zero, H shoots to infinity; these are the poles, marked × on the plane. At the s values that make the numerator N(s) zero, H becomes 0; these are the zeros, marked ○. The shape of a system’s natural response is set almost entirely by the poles, while the zeros add weighting to that shape.

Observepole p = σ + jω

A pole makes the denominator zero.

Chooseresponse e^σt, σ < 0 → ?

The real part σ is the decay rate.

Fill inoscillation cos(?t)

The imaginary part is the oscillation frequency.

On your ownstable ⟺ poles in ?

Stable means all poles in the left half.

Real part is decay, imaginary is oscillation

The natural response a pole s = σ + jω creates has the form e^(σt)cos(ωt). The real part σ is the decay rate of the exponential e^(σt): if σ is negative the response decays, and the more negative, the faster. The imaginary part ω is the angular frequency of the cosine: the farther from the real axis, the faster it oscillates. So a pole’s horizontal position on the plane shows how fast it dies away, its vertical position how fast it shakes. A real pole (ω=0) is simple decay with no oscillation.

Left half-plane means stable

If every pole of a system lies in the left half-plane — real part negative — all natural response modes decay and the system is BIBO stable. If even one pole is in the right half, that mode diverges and the system is unstable; on the imaginary axis it oscillates without dying, marginally stable. This generalizes the a < 0 stability condition of B4 to a single position on the s-plane, complex poles included. Stability is decided by the poles; the zeros cannot change it.

Back to the first screen

Dragging the pole left of the imaginary axis made the response decay to 0 (stable); pushing it right made the amplitude grow and diverge (unstable); placing it on the axis made it oscillate forever at the same size (marginal). The farther up or down from the real axis, the faster it oscillated. The pole’s horizontal position (real part) was the decay rate, its vertical position (imaginary part) the oscillation frequency. A system’s whole behavior was written in the location of one point on the s-plane.

A transfer function H(s) = N(s)/D(s) is characterized by its poles (D(s)=0, H→∞, marked ×) and zeros (N(s)=0, H=0, marked ○). The natural response a pole s = σ ± jω makes is e^σtcos(ωt): the real part σ is the decay rate (σ<0 decays), the imaginary part ω is the oscillation frequency. All poles in the left half (Re<0) means stable; right half unstable; imaginary axis marginal (generalizing B4’s a<0 to complex poles). Zeros add weighting to the response shape but cannot change stability.

On to the next unit

If poles and zeros draw the system, then the function H(s) that binds them is the system’s complete identity card. The next unit, the transfer function, Laplace-transforms a system’s differential equation to build H(s) = output/input, and multiplies that H(s) by the transform of the input to get the output. That convolution in time becomes multiplication in the s-domain is exactly what we saw in C4. Poles and zeros are simply the factored roots of the denominator and numerator of that H(s).