From the left half-plane to the unit circle
Keep the same continuous mode and change the sampling period. Does a smaller discrete pole mean faster motion in seconds?
First, picture it
A continuous exponential mode is multiplied by exp(sT) every T seconds. Calling this multiplier z gives x[k+1]=zx[k] for the complex mode. The real trace here is exp(σt)cos(ωt); the two crosses are a conjugate pair. We plot its exact samples, not a numerical approximation or a redesigned controller. Start at σ=−0.5, ω=4 rad/s and T=0.2 s.
What the model says
Magnitude is exp(σT). Negative σ gives a factor below one, so repeated multiplication decays. Positive σ gives growth. At σ=0, the factor is exactly one: this isolated mode does not decay and is not asymptotically stable. For a full finite-dimensional LTI realization every eigenvalue must lie strictly inside the unit circle for asymptotic stability. A repeated defective boundary pole can grow even without a radius above one.
s = σ ± jω → z = exp(sT), |z| = exp(σT), x(t) = exp(σt) cos(ωt)Angles differing by 2π give the same z, so frequencies separated by 2π/T rad/s cannot be distinguished by these samples alone. Exact modal sampling preserves decay; implementing a feedback controller with a hold, computational delay or an approximate integrator is another problem. Do not infer that any slow sampled controller is safe from this picture. The bench has no actuator or feedback loop.
One mode, two clocks
A continuous exponential mode is multiplied by exp(sT) every T seconds. Calling this multiplier z gives x[k+1]=zx[k] for the complex mode. The real trace here is exp(σt)cos(ωt); the two crosses are a conjugate pair. We plot its exact samples, not a numerical approximation or a redesigned controller. Start at σ=−0.5, ω=4 rad/s and T=0.2 s.
Why the boundary is a circle
Magnitude is exp(σT). Negative σ gives a factor below one, so repeated multiplication decays. Positive σ gives growth. At σ=0, the factor is exactly one: this isolated mode does not decay and is not asymptotically stable. For a full finite-dimensional LTI realization every eigenvalue must lie strictly inside the unit circle for asymptotic stability. A repeated defective boundary pole can grow even without a radius above one.
Compare seconds, not only samples
Record a prediction that |z| decreases, then change T from 0.2 to 0.5 s. The radius changes from about 0.905 to 0.779. Yet the decay envelope after one second is exp(−0.5) in both recordings: the physical mode has not changed. Larger intervals contain more decay per step but fewer steps per second. Pin both traces and inspect the same physical time.
A map is not a digital controller design
Angles differing by 2π give the same z, so frequencies separated by 2π/T rad/s cannot be distinguished by these samples alone. Exact modal sampling preserves decay; implementing a feedback controller with a hold, computational delay or an approximate integrator is another problem. Do not infer that any slow sampled controller is safe from this picture. The bench has no actuator or feedback loop.
Make it concrete
Keep the same continuous mode and change the sampling period. Does a smaller discrete pole mean faster motion in seconds?
Open this experimentAngles differing by 2π give the same z, so frequencies separated by 2π/T rad/s cannot be distinguished by these samples alone. Exact modal sampling preserves decay; implementing a feedback controller with a hold, computational delay or an approximate integrator is another problem. Do not infer that any slow sampled controller is safe from this picture. The bench has no actuator or feedback loop.
Check your understanding+
Keep the same continuous mode and change the sampling period. Does a smaller discrete pole mean faster motion in seconds?
Record a prediction that |z| decreases, then change T from 0.2 to 0.5 s. The radius changes from about 0.905 to 0.779. Yet the decay envelope after one second is exp(−0.5) in both recordings: the physical mode has not changed. Larger intervals contain more decay per step but fewer steps per second. Pin both traces and inspect the same physical time.