Stable/unstable systems; test stability

Definitions

• Stable system (BIBO): every bounded input |x(t)| ≤ M₁ produces a bounded output |y(t)| ≤ M₂. For an LTI system, this holds if and only if ∫|h(t)| dt < ∞.
• Unstable system: some bounded input produces an unbounded output (∫|h| dt = ∞). Example: an ideal integrator, h(t) = u(t).

Tests

The exponent printed as ψ is unclear in the scan; I treat it as a real constant ψ > 0 (for example 2) and give the general condition.

(i) h(t) = e^(ψt) u(t)

∫₀^∞ e^(ψt) dt = [e^(ψt)/ψ]₀^∞ = ∞ for ψ > 0

The integral diverges, so the system is unstable. (It would be stable only if ψ were negative, ψ < 0, giving 1/|ψ|.)

(ii) h(t) = e^(-ψt) u(t)

∫₀^∞ e^(-ψt) dt = [-e^(-ψt)/ψ]₀^∞ = 1/ψ < ∞ for ψ > 0

The system is stable. Its pole s = -ψ lies in the left half-plane.

Answer: for ψ > 0, (i) is unstable and (ii) is stable with ∫|h| dt = 1/ψ.