Stability Loss Under Discretization

For

$$x[k+1]=A_dx[k],$$

asymptotic stability requires spectral radius

$$\rho(A_d)=\max_i|\lambda_i(A_d)|<1.$$

Poles on the unit circle are not asymptotically stable. They can be bounded only under additional semisimplicity and input conditions; repeated unit-circle poles create polynomial growth. BIBO stability of a minimal proper transfer function likewise requires all poles strictly inside the circle.

Exact sampling maps $s$ to $z=e^{sT_s}$ and wraps frequency. The bilinear map sends the left half-plane one-to-one into the disk but warps frequency.

Exact pole mapping #

For exact sampling of a continuous autonomous mode $e^{st}$,

$$z=e^{sT_s}.$$

If $s=\sigma+j\omega$, then

$$|z|=e^{\sigma T_s},\qquad \angle z=\omega T_s\pmod{2\pi}.$$

Thus:

• $\sigma<0$ maps inside the unit circle;
• $\sigma=0$ maps to the circle;
• $\sigma>0$ maps outside;
• continuous frequencies differing by $2\pi/T_s$ map to the same angle.

Exact ZOH discretization of a stable continuous plant preserves its pole stability. It does not guarantee that a discretized controller, added delay, sampling zero, or approximate simulation preserves closed-loop stability.

Approximation can change the stability region #

Forward Euler maps

$$z=1+sT_s.$$

A stable real pole $s=-a$ maps to $z=1-aT_s$ and is stable only when

$$|1-aT_s|<1\quad\Rightarrow\quad 0<aT_s<2.$$

So a numerically integrated stable mode becomes unstable when the step is too large. Backward Euler maps $z=1/(1-sT_s)$ and is A-stable for the open left half-plane, but introduces different numerical damping. Tustin maps

$$z=\frac{1+sT_s/2}{1-sT_s/2}$$

and preserves the half-plane/disk stability boundary while warping frequency.

Numerical stability is not accuracy. A method can keep poles inside the unit circle while badly distorting time constants, phase, and zeros.

Characteristic polynomial tests #

Given real polynomial

$$D(z)=z^n+a_1z^{n-1}+\cdots+a_n,$$

the Jury test determines whether all roots lie inside the unit circle using coefficient recursions analogous to a discrete Routh test. For a monic second-order polynomial

$$D(z)=z^2+a_1z+a_2,$$

strict Schur stability is equivalent to

$$|a_2|<1,$$

$$1+a_1+a_2>0,$$

$$1-a_1+a_2>0.$$

These conditions expose parameter regions without explicitly solving roots. Equality marks the boundary: roots at $z=1$, $z=-1$, or a conjugate pair on the circle depending on which condition is active.

For higher order, use a validated Jury/Schur algorithm rather than an improvised table, and cross-check roots with adequate numerical precision.

Bilinear reduction to a Hurwitz problem #

The map

$$w=\frac{z-1}{z+1}$$

sends the open unit disk to the open left half-plane; $z=-1$ maps to infinity. Substitute

$$z=\frac{1+w}{1-w}$$

into $D(z)$ and multiply by $(1-w)^n$ to obtain a polynomial in $w$. Continuous Routh–Hurwitz conditions on that polynomial become discrete stability conditions. Track exceptional roots at $z=-1$ separately and avoid coefficient blow-up through scaling.

Delay and sample period #

One computation interval adds $z^{-1}$ and raises the characteristic order. A controller designed assuming immediate $u[k]$ but implemented with $u[k+1]$ can lose its entire phase margin. Fractional delay varies continuously with frequency and should not be rounded without analysis.

Changing $T_s$ changes:

• mapped plant poles and zeros;
• number of delay samples for the same wall-clock latency;
• controller coefficients;
• Nyquist frequency and phase available at crossover;
• quantization increments per update;
• the physical meaning of every discrete pole.

Coefficients copied unchanged to a new sample rate generally implement a different controller.

Sampling zeros and nonminimum phase #

ZOH discretization can introduce zeros that have no direct continuous counterpart. As $T_s$ changes, they move; for plants with higher relative degree, some can approach or leave the unit circle. A stable continuous minimum-phase plant can therefore have a discrete model that cannot be stably inverted at the chosen rate.

Inspect zeros as well as poles before direct digital synthesis. Deadbeat and model-matching designs are especially sensitive because they are tempted to cancel them.

Finite precision moves poles #

Quantizing coefficients perturbs the characteristic polynomial. High-order direct-form realizations can be extremely sensitive: a small coefficient change may move a pole across the unit circle. Cascaded second-order sections, balanced state-space realizations, and explicit scaling reduce—not eliminate—this risk.

Test the quantized coefficients and actual arithmetic. Stability of double-precision design coefficients is not evidence for a fixed-point binary.

Robust discrete stability check #

1. Form the actual sampled plant with hold, delay, and sample convention.
2. Form the implemented controller realization, not only an ideal $D(z)$.
3. Compute internal closed-loop poles and inspect distance to the unit circle.
4. Apply Jury/Schur or a state-space Lyapunov test as an independent check.
5. Sweep $T_s$, delay, plant uncertainty, and quantized coefficients.
6. Simulate nonlinear constraints and intersample motion.
7. Measure target timing and repeat the pole/margin calculation with the observed worst case.

Discretization is a model transformation plus a runtime contract. Either part can move a stable paper design outside the unit circle.