Stability

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Questions and Answers

What are the three requirements that enter into the design of a control system?

Transient response, frequency response, and error analysis

Steady-state response, stability, and error analysis

Stability, error analysis, and natural response

Transient response, stability, and steady-state errors (correct)

In what section of the text are definitions of stability, instability, and marginal stability presented?

Section 1.3

Section 1.2

Section 1.8

Section 1.5 (correct)

What is the definition of stability for a linear, time-invariant system?

The forced response grows without bound as time approaches infinity

The natural response approaches zero as time approaches infinity (correct)

The forced response approaches zero as time approaches infinity

The natural response grows without bound as time approaches infinity

What happens if a system is unstable?

Transient response and steady-state errors are moot points Signup and view all the answers

What is the sum of the forced and natural responses in a system's total response?

$c(t) = c_{forced}(t) + c_{natural}(t)$ Signup and view all the answers

What does the Routh-Hurwitz criterion help determine?

The number of poles in each section of the s-plane Signup and view all the answers

In a linear, time-invariant system, what indicates system instability?

Presence of any poles in the right half-plane Signup and view all the answers

What is the relationship between sign changes in the Routh table and the roots of an even polynomial?

The number of sign changes equals the number of right-half-plane roots Signup and view all the answers

What does the state-space representation of a system help determine?

The eigenvalues of the system matrix Signup and view all the answers

When is a system marginally stable based on the location of its poles?

When the poles are on the jω-axis and in the left half-plane Signup and view all the answers

According to the Routh-Hurwitz Criterion, what does an entire row of zeros in the Routh table indicate?

The presence of a purely even polynomial factor with symmetrical roots about the origin Signup and view all the answers

What does the number of sign changes in the first column of the Routh table indicate?

The number of roots in the right half-plane Signup and view all the answers

What does the Routh-Hurwitz Criterion provide without solving for closed-loop system poles?

Stability information Signup and view all the answers

How are Routh table entries calculated?

Using negative determinants of previous rows Signup and view all the answers

What does an epsilon (ε) replace in the first column of the Routh table?

Zero Signup and view all the answers

What is the BIBO definition of stability for a linear, time-invariant system?

A system is stable if every bounded input yields a bounded output. Signup and view all the answers

What is the alternate definition for instability based on the total response of a linear, time-invariant system?

A system is unstable if any bounded input yields an unbounded output. Signup and view all the answers

What is the condition for a linear, time-invariant system to be marginally stable based on the natural response?

The natural response neither decays nor grows but remains constant or oscillates as time approaches infinity. Signup and view all the answers

How are stable systems characterized in terms of the location of their poles?

Stable systems have closed-loop transfer functions with poles only in the left half-plane. Signup and view all the answers

What makes it difficult to determine if a feedback control system is stable?

The complexity of locating the poles of the equivalent closed-loop system without factoring or solving for the roots. Signup and view all the answers

According to the natural response definitions of stability, a system is unstable if:

the natural response approaches infinity as time approaches infinity Signup and view all the answers

What characterizes an unstable system according to the BIBO definitions of stability?

Any bounded input yields an unbounded output Signup and view all the answers

What type of natural responses yield stable systems?

Pure exponential decay or damped sinusoidal natural responses Signup and view all the answers

What is the alternate definition of instability based on the total response?

A system is unstable if any bounded input yields an unbounded output Signup and view all the answers

What is the implication of poles in the right half-plane of the s-plane for system stability?

The system is unstable Signup and view all the answers

Study Notes

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.

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Description

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.