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Is the system stable, marginally stable, or unstable

Is the system stable, marginally stable, or unstable

Introductory Circuit Analysis (13th Edition)
Introductory Circuit Analysis (13th Edition)
13th Edition
ISBN:9780133923605
Author:Robert L. Boylestad
Publisher:Robert L. Boylestad
Chapter1: Introduction
Section: Chapter Questions
Problem 1P: Visit your local library (at school or home) and describe the extent to which it provides literature...
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**Problem 5: System Stability Analysis** A system has the transfer function: \[ H(z) = \frac{z^2 - 3z + 1}{z^3 + z^2 - 0.5z + 0.5} \] **Question:** Is the system stable, marginally stable, or unstable? **Analysis Guide:** To determine the stability of a system based on its transfer function, consider the following general guidelines: 1. **Stability Criterion:** - For a discrete system, evaluate the locations of the poles of the transfer function \( H(z) \). - The system is stable if all poles lie inside the unit circle in the complex plane. 2. **Poles and Zeros:** - Poles: The values of \( z \) that make the denominator zero. - Zeros: The values of \( z \) that make the numerator zero. 3. **Steps to Assess Stability:** - Factor the denominator to find the poles. - Check the magnitude of each pole. All poles must have a magnitude less than 1 for the system to be stable. By following these procedures, you can determine if the system is stable, marginally stable, or unstable.
Transcribed Image Text:**Problem 5: System Stability Analysis** A system has the transfer function: \[ H(z) = \frac{z^2 - 3z + 1}{z^3 + z^2 - 0.5z + 0.5} \] **Question:** Is the system stable, marginally stable, or unstable? **Analysis Guide:** To determine the stability of a system based on its transfer function, consider the following general guidelines: 1. **Stability Criterion:** - For a discrete system, evaluate the locations of the poles of the transfer function \( H(z) \). - The system is stable if all poles lie inside the unit circle in the complex plane. 2. **Poles and Zeros:** - Poles: The values of \( z \) that make the denominator zero. - Zeros: The values of \( z \) that make the numerator zero. 3. **Steps to Assess Stability:** - Factor the denominator to find the poles. - Check the magnitude of each pole. All poles must have a magnitude less than 1 for the system to be stable. By following these procedures, you can determine if the system is stable, marginally stable, or unstable.
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