Stable and Unstable Systems

Introduction

Stability determines whether a system produces reasonable, finite outputs for reasonable, finite inputs. An unstable system can produce outputs that grow without bound — potentially destroying hardware, causing oscillation, or making the system completely useless for its intended purpose. Every practical system must be designed to be stable under its expected operating conditions.

The most common stability definition in signal processing is BIBO (Bounded-Input Bounded-Output) stability: if every bounded input produces a bounded output, the system is stable. A bounded signal is one whose magnitude never exceeds some finite value. If there exists any bounded input that causes the output to grow unboundedly, the system is unstable.

Formal Definition (BIBO Stability)

A system is BIBO stable if and only if every bounded input produces a bounded output:

$$|x(t)| \leq M_x < \infty \quad \text{for all } t \implies |y(t)| \leq M_y < \infty \quad \text{for all } t$$

There must exist a finite bound $M_y$ (which may depend on $M_x$ and the system) such that the output never exceeds it.

Stability Condition for LTI Systems

For LTI systems, BIBO stability has an elegant necessary and sufficient condition:

A continuous-time LTI system is BIBO stable if and only if its impulse response is absolutely integrable:

$$\int_{-\infty}^{\infty} |h(t)| dt < \infty$$

A discrete-time LTI system is BIBO stable if and only if its impulse response is absolutely summable:

$$\sum_{n=-\infty}^{\infty} |h[n]| < \infty$$

Proof (Discrete-Time)

For bounded input $|x[n]| \leq M_x$: $$|y[n]| = |\sum_k h[k]x[n-k]| \leq \sum_k |h[k]||x[n-k]| \leq M_x \sum_k |h[k]|$$

If $\sum|h[k]| = S < \infty$, then $|y[n]| \leq M_x \cdot S = M_y < \infty$. The output is bounded.

Conversely, if $\sum|h[k]| = \infty$, one can construct a bounded input that produces unbounded output (choose $x[n-k] = \text{sgn}(h[k])$).

Transfer Function Stability Criteria

Continuous-Time (s-domain)

A causal LTI system with transfer function $H(s)$ is stable if and only if all poles lie in the left half of the s-plane (negative real parts):

$$\text{Re}\{p_i\} < 0 \quad \text{for all poles } p_i$$

Discrete-Time (z-domain)

A causal LTI system with transfer function $H(z)$ is stable if and only if all poles lie inside the unit circle:

$$|p_i| < 1 \quad \text{for all poles } p_i$$

These geometric criteria make stability checking straightforward: find the poles and verify their locations.

Examples

Stable Systems

First-order decay: $h(t) = e^{-2t}u(t)$ — decays exponentially, $\int_0^\infty e^{-2t}dt = 0.5 < \infty$ ✓

Moving average: $h[n] = 1/3$ for $n = 0,1,2$ — finite duration, $\sum|h[n]| = 1 < \infty$ ✓

Damped oscillation: $h(t) = e^{-t}\cos(10t)u(t)$ — amplitude decays, integral converges ✓

Unstable Systems

Pure integrator: $h(t) = u(t)$ — $\int_0^\infty 1\,dt = \infty$ ✗. A constant input produces linearly growing output.

Growing exponential: $h(t) = e^{2t}u(t)$ — grows without bound ✗. Pole at $s = +2$ (right half-plane).

Discrete accumulator: $h[n] = u[n]$ — $\sum_{n=0}^\infty 1 = \infty$ ✗. A constant input causes output to grow indefinitely.

Marginally Stable (Boundary Case)

Undamped oscillator: $h(t) = \cos(\omega_0 t)u(t)$ — does not decay but does not grow. $\int|\cos|dt = \infty$, so technically BIBO unstable. Poles on the imaginary axis ($s = \pm j\omega_0$) indicate marginal stability.

Pure delay: $h[n] = \delta[n-k]$ — finite (just a shifted impulse), so $\sum|h| = 1 < \infty$ ✓ Stable.

Physical Consequences of Instability

Electronic oscillation: An unstable amplifier circuit oscillates, producing unwanted tones or saturating at power supply rails.

Control system runaway: An unstable controller causes the controlled variable (temperature, position, speed) to diverge from the setpoint, potentially causing physical damage.

Numerical overflow: In digital systems, unstable recursive filters produce output values that exceed the representable range, causing overflow and signal corruption.

Positive feedback: When a system\'s output reinforces its input through a feedback loop with gain > 1, the signal grows exponentially until physical limits are reached.

Stability Analysis Methods

Pole location: The most direct method — find all poles and check their locations.

Routh-Hurwitz criterion: For continuous-time systems, determines whether any roots of the characteristic polynomial have positive real parts without explicitly computing the roots.

Jury stability test: The discrete-time equivalent of Routh-Hurwitz.

Nyquist criterion: Uses the frequency response to determine closed-loop stability.

Lyapunov methods: For nonlinear systems, constructs an energy-like function to prove stability without solving the differential equations.

Key Takeaways

• BIBO stable: every bounded input produces a bounded output
• LTI stability criterion: impulse response must be absolutely integrable (or summable)
• Continuous-time: stable ⟺ all poles in left half-plane ($\text{Re}\{p_i\} < 0$)
• Discrete-time: stable ⟺ all poles inside unit circle ($|p_i| < 1$)
• Marginally stable systems (poles on boundary) are technically BIBO unstable
• Instability causes oscillation, divergence, overflow, or physical system failure
• All practical systems must be designed with adequate stability margins