Stability

Routh-Hurwitz Criterion for Stability

• Closed-loop transfer function with only left-half-plane poles implies all coefficients of the denominator are positive
• Unstable system if all coefficients of the denominator have different signs
• Routh-Hurwitz criterion provides stability information without solving for closed-loop system poles
• The method generates a Routh table and interprets it to identify the number of closed-loop system poles in different sections of the s-plane
• The criterion's power lies in design rather than analysis, providing closed-form expressions for stability ranges
• Routh table entries calculated using negative determinants of previous rows
• Number of roots in the right half-plane equals the number of sign changes in the first column
• An epsilon (ε) replaces zero in the first column to avoid division by zero
• Entire row of zeros in the Routh table indicates the presence of a purely even polynomial factor with symmetrical roots about the origin
• The row previous to the row of zeros contains the even polynomial factor
• Polynomials generating entire rows of zeros have roots symmetrical about the origin and apply only to the even polynomial
• Routh table provides information about the existence and characteristics of even polynomials with symmetrical roots

Defining Stability and Instability in Linear, Time-Invariant Systems

• A linear, time-invariant system is marginally stable if the natural response neither decays nor grows but remains constant or oscillates as time approaches infinity.
• Stability implies that as the natural response approaches zero, only the forced response remains.
• The BIBO definition of stability states that a system is stable if every bounded input yields a bounded output.
• An alternate definition for instability based on the total response states that a system is unstable if any bounded input yields an unbounded output.
• Marginally stable systems, by the natural response definitions, are included as unstable systems under the BIBO definitions.
• Unstable systems, whose natural response grows without bound, can cause damage to the system, adjacent property, or human life.
• The natural response definitions of stability rely on the location of system poles in the left or right half-plane of the s-plane.
• Stable systems have closed-loop transfer functions with poles only in the left half-plane, while unstable systems have poles in the right half-plane or poles of multiplicity greater than 1 on the imaginary axis.
• Marginally stable systems have closed-loop transfer functions with only imaginary axis poles of multiplicity 1 and poles in the left half-plane.
• It is not always easy to determine if a feedback control system is stable due to the complexity of locating the poles of the equivalent closed-loop system without factoring or solving for the roots.
• The text provides insight into how to define, determine, and understand stability and instability in linear, time-invariant systems.
• The definitions presented are crucial for understanding the behavior and implications of stability and instability in systems.

Explore the Routh-Hurwitz Criterion for stability analysis in linear, time-invariant systems. Learn about closed-loop transfer functions, stability definitions, and interpreting Routh tables to determine the number of system poles. Understand the importance of stability for system behavior and design considerations.