Question

A linear discrete-time system has the characteristic equation, $z^3 - 0.81z = 0$, the system is:

• A. stable
• B. marginally stable
• C. unstable
• D. stability cannot be assessed from the given information

Hint:

For a discrete-time system to be stable, all the roots of the characteristic equation must lie inside the unit circle (i.e., their magnitudes must be less than 1).

Answer 1

The Correct Option is A

The given characteristic equation is $z^3 - 0.81z = 0$. To analyze the stability of the system, we need to find the roots of this equation. We can factor it as:
\[ z(z^2 - 0.81) = 0 \]
Solving for $z$, we get: \[ z = 0 \text{or} z = \pm 0.9 \]
The roots are $z = 0$, $z = 0.9$, and $z = -0.9$. For the system to be stable, the magnitude of all roots must be less than 1.
- The root $z = 0$ has a magnitude of 0, which is less than 1.
- The roots $z = 0.9$ and $z = -0.9$ also have magnitudes less than 1.
Since all the roots have magnitudes less than 1, the system is stable.