mathematics//dynamical systems//equilibrium and stability//discrete-time stability
Discrete-time stability is the stability of a system that advances by steps, \(x_{k+1}=Ax_k\), and it holds when every eigenvalue of \(A\) has modulus less than one; it is the test that applies to everything that runs on a computer, from a digital controller and an observer to a time-series model and a training loop. Each step multiplies every mode by its eigenvalue \(\mu\), so after \(k\) steps the mode carries \(\mu^k\): it dies if \(|\mu|<1\) and grows if \(|\mu|>1\), whatever the sign or phase of \(\mu\).
Discrete-time stability is the stability of a system that advances by steps, xk+1=Axkx_{k+1}=Ax_kxk+1=Axk, and it holds when every eigenvalue of AAA has modulus less than one; it is the test that applies to everything that runs on a computer, from a digital controller and an observer to a time-series model and a training loop. Each step multiplies every mode by its eigenvalue μ\muμ, so after kkk steps the mode carries μk\mu^kμk: it dies if ∣μ∣<1|\mu|<1∣μ∣<1 and grows if ∣μ∣>1|\mu|>1∣μ∣>1, whatever the sign or phase of μ\muμ.
The continuous and the discrete tests are the same test seen through the sampling. A continuous mode eλte^{\lambda t}eλt sampled every Δt\Delta tΔt becomes μ=eλΔt\mu=e^{\lambda\Delta t}μ=eλΔt, so the stable left half-plane Reλ<0\operatorname{Re}\lambda<0Reλ<0 maps into the unit disc, the imaginary axis onto the unit circle, and a time constant τ\tauτ onto ∣μ∣=e−Δt/τ|\mu|=e^{-\Delta t/\tau}∣μ∣=e−Δt/τ. A mass on a spring with eigenvalues −0.2±1.99j-0.2\pm1.99j−0.2±1.99j, sampled at 100 Hz, gives ∣μ∣=e−0.002≈0.998|\mu|=e^{-0.002}\approx0.998∣μ∣=e−0.002≈0.998: stable, and close to the circle because nothing happens in 10 ms.
One inequality governs a whole family of algorithms.
An explicit Euler step is stable when ∣1+λΔt∣<1|1+\lambda\Delta t|<1∣1+λΔt∣<1, gradient descent when its learning rate stays under 2/λmax2/\lambda_{\max}2/λmax of the Hessian, discrete consensus when its gain stays under 2/λN2/\lambda_N2/λN of the Laplacian: each one is a matrix applied once per step, and each one diverges the moment an eigenvalue leaves the unit circle.
A complex pair μ=re±iθ\mu=re^{\pm i\theta}μ=re±iθ is an oscillation read per step: the modulus rrr is how much survives each step and the angle θ\thetaθ how far it turns, so its frequency is θ/Δt\theta/\Delta tθ/Δt. Two thermally coupled zones updated each minute with eigenvalues 0.98 and 0.92 have time constants of −1/ln0.98≈50-1/\ln0.98\approx50−1/ln0.98≈50 and −1/ln0.92≈12-1/\ln0.92\approx12−1/ln0.92≈12 minutes, the common mode slow and the differential mode fast.
Discrete systems can do what continuous ones cannot. A negative real μ\muμ flips the sign of its mode every step, the signature of a step size or gain set too high; μ=0\mu=0μ=0 kills a mode in a finite number of steps (deadbeat control and observers). Both have no counterpart in continuous time.
The stability of the algorithm and of the system are separate questions. Sampling a stable plant exactly keeps it stable (zero-order hold), while an explicit integrator with too large a step can explode on a decaying system (numerical stability). In an observer the error evolves under A−LCA-LCA−LC, so its eigenvalues must sit inside the circle, comfortably faster than the loop it feeds (Luenberger observer).
The largest modulus is the spectral radius, and ρ(A)<1\rho(A)<1ρ(A)<1 is the whole test; how big the transient gets before the decay wins is a different question, answered by the singular values of AkA^kAk (non-normal matrix).