System stability

Definition

System stability refers to the ability of a dynamic system to return to a state of equilibrium after being disturbed. A stable system will naturally settle back to its original position, while an unstable system may diverge away from equilibrium, leading to uncontrolled behavior. In engineering, assessing stability is crucial for ensuring that systems respond predictably under various conditions, which is evaluated using criteria like the Routh-Hurwitz Stability Criterion and concepts related to discrete-time transfer functions.

5 Must Know Facts For Your Next Test

1. In the context of the Routh-Hurwitz Stability Criterion, a system is stable if all roots of its characteristic equation have negative real parts.
2. Discrete-time systems can be analyzed using their transfer functions, where stability is determined by the location of poles in the z-plane; they must lie inside the unit circle for stability.
3. A stable system exhibits bounded input-output behavior, meaning that for any bounded input, the output will also remain bounded.
4. Routh's array is a systematic way to determine stability without calculating roots directly, simplifying the assessment process for complex systems.
5. Stability can be influenced by feedback mechanisms; positive feedback can lead to instability while negative feedback tends to promote stability.

Review Questions

• How does the Routh-Hurwitz Stability Criterion help in determining system stability?
  • The Routh-Hurwitz Stability Criterion provides a systematic method for assessing whether a linear time-invariant system is stable without needing to compute the roots of its characteristic polynomial. By constructing a Routh array from the coefficients of this polynomial, one can determine how many roots have positive real parts. If all entries in the first column of the array are positive, it indicates that all roots lie in the left half-plane, confirming that the system is stable.
• Discuss how discrete-time transfer functions relate to system stability and what role poles play in this context.
  • In discrete-time systems, stability is closely tied to the location of poles in the z-plane derived from their transfer functions. For a discrete-time system to be considered stable, all poles must reside within the unit circle; if any pole lies on or outside this circle, the system will exhibit unbounded output responses over time. This relationship emphasizes how analyzing pole locations is critical for predicting and ensuring stable behavior in digital control systems.
• Evaluate how feedback can impact system stability and provide an example illustrating both positive and negative feedback effects.
  • Feedback significantly influences system stability by either promoting or destabilizing it. For example, negative feedback reduces deviations from desired outputs by adjusting inputs based on output measurements, thereby enhancing stability. Conversely, positive feedback amplifies changes, which can lead to runaway conditions or oscillations. An example of negative feedback is an automatic thermostat that maintains temperature by adjusting heating or cooling based on current conditions, while an example of positive feedback might be a microphone placed too close to a speaker, causing a loud screeching sound as sound is continuously amplified.

Related terms

Equilibrium Point: A state where the system experiences no net change, and all forces acting on it are balanced.

Transient Response: The temporary behavior of a system as it reacts to changes or disturbances before settling into a steady-state condition.

Pole-Zero Analysis: A method used in control theory to analyze the stability and behavior of linear systems by examining the locations of poles and zeros in the complex plane.