Stability of Discrete-Time Systems

In modern engineering, from robotics to digital communication, control algorithms are implemented on microprocessors. These discrete-time systems operate on sampled data points rather than continuous signals, making their stability analysis fundamentally different from their continuous-time counterparts. Ensuring a system's stability—meaning its output remains bounded for a bounded input—is the first and most critical step in any control design.

The Fundamental Criterion: Poles Inside the Unit Circle

The stability of a linear, time-invariant discrete-time system is determined entirely by the location of its closed-loop poles. A pole is a root of the system's characteristic equation in the z-domain. For a system to be asymptotically stable (the output decays to zero after a disturbance), every pole must have a magnitude less than one. Geometrically, this means all poles must lie strictly inside the unit circle in the complex z-plane.

This is the discrete-time parallel to the continuous-time rule, where poles must lie in the left-half of the s-plane. The reason stems from the z-transform relationship $z=e^{sT}$, where $T$ is the sampling period. A pole in the s-plane at $s=\sigma +j\omega$ maps to a z-plane pole with magnitude $|z|=e^{\sigma T}$. For stability in continuous-time, we require $\sigma <0$, which directly translates to $|z|=e^{\sigma T}<1$. If any pole lies on the unit circle ($|z|=1$), the system is marginally stable, exhibiting sustained oscillation. Any pole outside the unit circle leads to an output that grows without bound, indicating instability.

Example: Consider a system with a characteristic equation $z^2-1.2z+0.35=0$. The roots are $z=0.6\pm j0.1$. The magnitude is $|z|=\sqrt{(0.6{)}^2+(0.1{)}^2}=\sqrt{0.37}\approx 0.608$. Since $0.608<1$, both poles are inside the unit circle, and the system is stable.

The Jury Stability Test

For high-order systems, finding poles analytically can be cumbersome. The Jury stability test provides a tabular method, analogous to the Routh-Hurwitz criterion for continuous systems, to determine the number of unstable poles without explicit root calculation. You apply it directly to the polynomial coefficients of the characteristic equation.

Given an n-th order characteristic polynomial: $$P(z)=a_n{z}^n+a_{n-1}{z}^{n-1}+...+a_{1}z+a_0,a_n>0$$

You construct the Jury table, which has $(2n-3)$ rows. The first two rows are the coefficients in forward and reverse order. Subsequent rows are calculated using determinants from the preceding two rows. The necessary and sufficient conditions for stability are:

1. $P(1)>0$
2. $(-1{)}^{n}P(-1)>0$
3. $|a_0|<a_n$
4. All "first column" elements in the odd-numbered rows of the Jury table are positive.

Worked Example: Test stability for $P(z)=z^3-1.1z^2-0.1z+0.2$.

• Condition 1: $P(1)=1-1.1-0.1+0.2=0$. This fails ($0>0$ is false). Therefore, the system is not strictly stable. A root is at $z=1$, indicating marginal stability.

This test is powerful because it handles the discrete domain directly, but its algebraic conditions must be checked meticulously.

Stability Analysis via Bilinear Transformation

Sometimes, it's advantageous to leverage the mature tools of continuous-time stability analysis. The bilinear transformation (also known as the Tustin transform or w'-transform) allows us to do this by mapping the interior of the unit circle in the z-plane to the left-half of a new, complex plane.

The most common transformation is $z=\frac{1+w}{1-w}$, which maps the unit circle ($|z|=1$) to the imaginary axis in the w-plane. You can solve for $w$ to get the transformative substitution: $w=\frac{z-1}{z+1}$.

The procedure is:

1. Take the discrete characteristic equation in $z$: $P(z)=0$.
2. Substitute $z=\frac{1+w}{1-w}$ into $P(z)$.
3. Simplify to obtain a new polynomial $Q(w)=0$ in the w-domain.
4. Apply the Routh-Hurwitz criterion to $Q(w)$. Stability in the w-plane (all roots in left-half plane) guarantees stability in the z-plane (all roots inside unit circle).

This method is particularly useful when you need to perform gain sweeps or assess relative stability metrics like phase and gain margin using Bode plots, which are naturally set in a continuous-like frequency domain.

Application Scenario: A control engineer has designed a digital compensator and derived the closed-loop characteristic equation. To quickly assess how changes in a gain parameter $K$ affect stability, they use the bilinear transformation. They substitute $z=(1+w)/(1-w)$ and apply the Routh-Hurwitz criterion to the resulting $Q(w)$ polynomial. This generates stability inequalities in terms of $K$, clearly showing the range of $K$ for which all w-plane roots are in the LHP, and thus all z-plane poles are inside the unit circle.

Common Pitfalls

1. Applying Continuous-Time Criteria Directly to the z-Plane: A common conceptual error is to look for poles in the "left-half" of the z-plane. The correct boundary for stability is the unit circle, not the imaginary axis. A pole at $z=-0.9$ is stable (inside the unit circle), while a pole at $z=+0.5+j0.9$ with magnitude $\sqrt{0.25+0.81}\approx 1.03$ is unstable (just outside the circle).

1. Misinterpreting Marginal Stability: A system with poles on the unit circle and no repeated poles there is marginally stable. In practice, this is often unacceptable, as parameter variations or nonlinearities can push these poles outside the circle, causing instability. Treat marginal stability as a fragile, design-limiting condition.

1. Algebraic Errors in the Jury Table: The Jury test is algebraically intensive. A single sign error in constructing the rows will invalidate the test. Always verify the initial conditions ($P(1)>0$, etc.) first, as they can provide a quick instability check and validate your subsequent table calculations.

1. Ignoring the Sampling Period in Transformation Methods: While the bilinear transformation maps stability regions correctly, the relationship between frequency in the w-plane and the actual digital frequency is warped. For precise frequency-domain analysis (e.g., bandwidth), you must account for this frequency warping, especially as the frequency of interest approaches the Nyquist frequency.

Summary

• The primary criterion for discrete-time stability is that all closed-loop poles (roots of the characteristic equation in z) must lie strictly inside the unit circle in the complex z-plane.
• The Jury stability test is a direct, tabular method for determining stability from polynomial coefficients, serving as the discrete-time equivalent of the Routh-Hurwitz criterion.
• The bilinear transformation $z=(1+w)/(1-w)$ maps the unit circle to the imaginary axis, allowing engineers to convert a discrete-time stability problem into a continuous-time one where tools like Routh-Hurwitz can be applied.
• Stability analysis is a prerequisite for any further digital control design objectives, such as improving transient response or steady-state error, and must be performed with careful attention to algebraic detail.