Stability Theory and Analysis

Introduction and Context

Stability is the central property that determines whether a control system performs predictably after disturbances, command changes, measurement noise, or parameter variations. A system may track a reference accurately yet be unacceptable if its response grows without bound. Conversely, a stable system may be robust but too slow or poorly damped. Stability analysis therefore precedes most control-system design decisions.

For linear time-invariant (LTI) systems, stability can often be assessed from the locations of poles in the complex plane. The Routh–Hurwitz criterion determines stability without explicitly calculating roots, while Lyapunov methods provide a broader energy-based framework applicable to nonlinear and time-varying systems. Stability margins quantify how close a system is to instability and are especially important when modelling uncertainty is present.

This chapter develops internal, external, bounded-input bounded-output (BIBO), asymptotic, marginal, and relative stability. It also explains pole-zero considerations, Routh tables, Lyapunov stability, gain and phase margins, and an applied servo-system case.

Learning Objectives

After studying this chapter, the learner should be able to:

1. Distinguish internal, external, BIBO, asymptotic, marginal, and relative stability.

2. Relate pole locations to the response of continuous- and discrete-time LTI systems.

3. Apply the Routh–Hurwitz criterion, including special cases.

4. Formulate Lyapunov functions and use direct and indirect methods.

5. Interpret gain margin, phase margin, damping, and dominant poles.

6. Analyse the influence of uncertainty, delays, nonlinearities, and actuator limitations.

7. Select suitable stability tests for practical control problems.

1. Fundamental Definitions of Stability

1.1 Equilibrium and perturbation concepts

Consider a state-space model

x_dot = f(x,u,t), y = h(x,u,t),

where x is the state vector, u is the input, and y is the output. An equilibrium x_e under constant input u_e satisfies

f(x_e,u_e,t) = 0.

For an autonomous system, u is absent and the equilibrium is commonly shifted to the origin by defining z = x − x_e. Stability concerns the behaviour of z when the initial state is close to zero.

1.2 Lyapunov stability

An equilibrium is stable in the sense of Lyapunov if, for every ε > 0, there exists δ > 0 such that

||x(0) − x_e|| < δ implies ||x(t) − x_e|| < ε for all t ≥ 0.

Thus, sufficiently small initial deviations remain small. This definition does not require the deviation to decay.

The equilibrium is asymptotically stable if it is Lyapunov stable and

x(t) → x_e as t → ∞

for all initial states sufficiently close to x_e. It is globally asymptotically stable if this convergence holds for every initial state in the state space.

Exponential stability is stronger: there exist positive constants c and α such that

||x(t) − x_e|| ≤ c exp(−αt)||x(0) − x_e||.

Exponential stability provides a quantitative rate of convergence and is particularly useful in robustness analysis.

1.3 Internal and external stability

Internal stability concerns every state trajectory generated by initial conditions and inputs. A system is internally asymptotically stable if all unforced state trajectories decay to zero.

External stability concerns the input-output relationship. A continuous-time system is BIBO stable if every bounded input produces a bounded output. Formally, if |u(t)| ≤ M_u < ∞, then there must exist M_y < ∞ such that |y(t)| ≤ M_y for all t ≥ 0.

For a finite-dimensional, causal, proper LTI transfer function with no unstable hidden modes, internal asymptotic stability and BIBO stability are equivalent. However, the qualifications are important. Pole-zero cancellation can conceal an unstable internal mode in a transfer function, so an apparently stable input-output model may not represent a physically internally stable realization.

1.4 Marginal and unstable behaviour

A continuous-time LTI system is marginally stable when its response remains bounded but does not necessarily decay. Simple poles on the imaginary axis are permitted, provided no repeated poles occur there and all remaining poles lie in the open left half-plane. A repeated imaginary-axis pole produces terms such as t sin(ωt), which grow without bound and therefore cause instability.

A system is unstable if at least one mode grows without bound. This occurs when a pole lies in the open right half-plane, or when a repeated pole lies on the imaginary axis. In discrete time, the corresponding stability region is the open unit circle: poles must satisfy |z_i| < 1 for asymptotic stability.

2. Pole Locations and Time Response

For a continuous-time LTI system, the natural response is determined by the roots of the characteristic polynomial. A pole p = σ + jω contributes a term proportional to exp(σt)cos(ωt) or exp(σt)sin(ωt). Therefore:

• σ < 0: the mode decays.
• σ = 0: the mode persists without decay.
• σ > 0: the mode grows.

For a second-order denominator

s² + 2ζω_n s + ω_n²,

ω_n is the natural frequency and ζ is the damping ratio. Its poles are

s = −ζω_n ± jω_n√(1 − ζ²),

for 0 < ζ < 1. The real part is −ζω_n, and the damped frequency is ω_d = ω_n√(1 − ζ²). Increasing ζ generally reduces oscillation but may increase rise time when the natural frequency is fixed.

For a dominant second-order approximation, commonly used performance estimates include

T_s ≈ 4/(ζω_n)

for a 2% settling-time criterion, and

M_p ≈ exp[−πζ/√(1−ζ²)] × 100%,

where M_p is the percentage overshoot. These expressions assume a standard underdamped second-order response and may be inaccurate when additional poles or zeros significantly affect the transient.

Pole-zero cancellation and hidden modes

Suppose a transfer function is written as

G(s) = (s − a)/[(s − a)(s + 2)] = 1/(s + 2).

The reduced transfer function appears stable, but a physical realization containing the mode at s = a still has that internal mode. If a > 0, the realization is internally unstable even though the reduced input-output transfer function is stable. Exact cancellation is also non-robust: small modelling errors prevent exact cancellation and expose the unstable pole.

3. Characteristic Equations and Feedback Stability

For a negative-feedback loop with forward transfer function G(s) and feedback transfer function H(s), the closed-loop transfer function is

T(s) = G(s)/[1 + G(s)H(s)],

assuming scalar transfer functions and compatible signal definitions. The characteristic equation is

1 + G(s)H(s) = 0.

The roots of this equation are the closed-loop poles. Stability must be assessed using the complete characteristic equation, not merely the poles of G(s) or H(s) separately.

In unity feedback, H(s) = 1, and the equation reduces to 1 + G(s) = 0. A sign error in the feedback convention changes the characteristic equation and can reverse the stability conclusion.

4. The Routh–Hurwitz Criterion

4.1 Purpose and theorem

Let the characteristic polynomial be

P(s) = a_n s^n + a_{n−1}s^{n−1} + ... + a_1s + a_0,

where a_n > 0 and all coefficients are real. The Routh–Hurwitz criterion states that all roots of P(s) lie in the open left half-plane if and only if all elements in the first column of the Routh array are positive, assuming no zero or singular special case requires separate treatment.

The criterion avoids explicit root calculation and is valuable for symbolic parameter studies.

4.2 Construction of the Routh array

The first two rows are formed from alternating coefficients:

s^n: a_n a_{n−2} a_{n−4} ...

s^{n−1}: a_{n−1} a_{n−3} a_{n−5} ...

Each subsequent element is calculated from the preceding two rows. For example, the first element in the s^{n−2} row is

b_1 = (a_{n−1}a_{n−2} − a_n a_{n−3})/a_{n−1}.

The next element is

b_2 = (a_{n−1}a_{n−4} − a_n a_{n−5})/a_{n−1}.

Equivalent sign conventions may be used, but the complete first-column convention must be applied consistently.

The number of sign changes in the first column equals the number of roots in the open right half-plane, provided the array is constructed correctly and special cases are handled.

4.3 Worked example: stability range of a parameter

Consider

P(s) = s³ + 2s² + (K + 1)s + 4,

where K is a real controller parameter. The Routh array is

s³: 1 K + 1

s²: 2 4

s¹: [(2(K+1) − 4)/2] 0 = K − 1

s⁰: 4

For all first-column entries to be positive:

1 > 0, 2 > 0, K − 1 > 0, 4 > 0.

Thus the system is asymptotically stable when

K > 1.

At K = 1, the first-column element becomes zero and the system is on a stability boundary; the ordinary positivity test no longer establishes asymptotic stability. For K < 1, there is at least one right-half-plane root.

4.4 Special Routh cases

A zero in the first column. If a single first-column element is zero while other entries in its row are nonzero, replace the zero by a small positive number ε, complete the table, and examine the limit as ε approaches zero from the positive side. The resulting sign changes identify right-half-plane roots.

A complete row of zeros. A row of zeros indicates symmetric roots, often including roots on the imaginary axis. Form the auxiliary polynomial from the row immediately above the zero row, differentiate it, and use the derivative coefficients to replace the zero row. The auxiliary polynomial also helps identify the symmetric factors.

These procedures are not merely algebraic tricks: they signal a structural stability boundary or a pair of roots symmetric about the origin.

4.5 Limitations of Routh–Hurwitz analysis

Routh–Hurwitz applies directly to real-coefficient polynomials and continuous-time LTI characteristic equations. It determines the number of roots in half-planes but does not directly provide damping ratios, settling times, or robustness to unmodelled dynamics. It also becomes cumbersome for high-order uncertain systems unless symbolic or computational tools are used. Numerical root calculation, Nyquist analysis, or structured singular-value methods may be more informative for complex uncertainty.

5. Lyapunov Stability Theory

5.1 Energy interpretation

Lyapunov’s direct method does not require solving the system trajectories. For an autonomous system

x_dot = f(x),

choose a scalar function V(x), called a Lyapunov function, with V(0) = 0. A typical set of sufficient conditions is:

• V(x) is positive definite: V(x) > 0 for x ≠ 0.
• Its derivative along trajectories, V_dot = ∇V(x)^T f(x), is negative semidefinite or negative definite.

If V is positive definite and V_dot is negative semidefinite, the origin is stable under suitable regularity and invariance conditions. If V_dot is negative definite, the origin is asymptotically stable locally. If V is radially unbounded and the conditions hold globally, global asymptotic stability may be established.

Positive definite means V(0) = 0 and V(x) > 0 elsewhere. Radially unbounded means V(x) tends to infinity as ||x|| tends to infinity.

5.2 Linear systems and the Lyapunov equation

For the continuous-time linear system

x_dot = Ax,

if A is Hurwitz—that is, all eigenvalues of A have negative real parts—then for every positive-definite matrix Q, the equation

A^T P + PA = −Q

has a unique positive-definite solution P. The quadratic function

V(x) = x^T P x

proves asymptotic stability because

V_dot = x^T(A^TP + PA)x = −x^TQx < 0

for every nonzero x.

Conversely, existence of a positive-definite P for a positive-definite Q is equivalent to A being Hurwitz. This result links eigenvalue analysis and an energy-like certificate.

5.3 Indirect method for nonlinear systems

For a nonlinear system x_dot = f(x), linearize about an equilibrium:

x_dot ≈ A x, A = [∂f/∂x] evaluated at the equilibrium.

If A is Hurwitz, the equilibrium is locally asymptotically stable. If A has an eigenvalue with positive real part, the equilibrium is unstable. If eigenvalues lie on the imaginary axis with none in the right half-plane, linearization is inconclusive; nonlinear terms must then be examined using a direct Lyapunov function, centre-manifold analysis, or another nonlinear method.

5.4 LaSalle’s invariance principle

If V_dot is negative semidefinite rather than negative definite, trajectories may still converge. LaSalle’s principle states, informally, that solutions approach the largest invariant set contained in

{x : V_dot(x) = 0}.

For example, mechanical systems with damping may have V_dot = −b v², which is zero whenever velocity v is zero. If the only invariant trajectory in that set is the equilibrium, asymptotic stability follows even though V_dot is not negative definite everywhere.

6. Relative Stability and Frequency-Domain Margins

Absolute stability answers whether a system is stable or unstable. Relative stability indicates how close it is to instability.

6.1 Dominant-pole measures

For a pole p = σ + jω with σ < 0, the distance from the imaginary axis is −σ. A pole close to the imaginary axis produces a slowly decaying mode and usually dominates the transient response. For a second-order pair, the damping ratio

ζ = −σ/√(σ² + ω²)

measures angular damping, while the natural frequency is

ω_n = √(σ² + ω²).

A system can be asymptotically stable but poorly relatively stable if its dominant poles are close to the imaginary axis or if its damping ratio is small.

6.2 Gain and phase margins

Let L(s) = G(s)H(s) be the loop transfer function. The gain crossover frequency ω_gc satisfies

|L(jω_gc)| = 1.

The phase margin is

PM = 180° + angle L(jω_gc),

when the phase is expressed in degrees and the conventional negative-feedback interpretation is used.

The phase crossover frequency ω_pc satisfies

angle L(jω_pc) = −180°.

The gain margin is

GM = 1/|L(jω_pc)|,

or in decibels,

GM_dB = 20 log10(GM).

Positive margins generally indicate a useful separation from the nominal instability condition, but their interpretation depends on loop shape, delay, uncertainty structure, and crossover multiplicity. A rule such as “45° phase margin is always adequate” is not universal; performance, noise, actuator limits, and plant uncertainty must also be considered.

6.3 Delay and phase margin

A pure delay e^(−sT) has unit magnitude and phase −ωT radians. It therefore reduces phase margin without changing the magnitude plot. Even a stable nominal system can become unstable when computation, sensing, communication, or actuation delays are increased. Delay robustness should be evaluated at the relevant crossover frequencies, not only at low frequency.

6.4 Nyquist perspective

The Nyquist criterion assesses closed-loop stability from the contour of L(s) around the critical point −1 + j0. The required encirclements depend on the number of open-loop right-half-plane poles. Unlike a simple margin calculation, Nyquist analysis remains applicable when the open-loop plant is unstable, provided the encirclement convention is used correctly.

7. Discrete-Time Stability

For a discrete-time state model

x[k+1] = A_d x[k],

asymptotic stability requires every eigenvalue λ_i of A_d to satisfy

|λ_i| < 1.

The unit circle replaces the imaginary axis as the stability boundary. A pole at z = 1 or z = −1 produces a persistent mode; repeated poles on the unit circle produce unbounded polynomial factors in k. Mapping a continuous-time pole through zero-order-hold sampling gives z = e^(sT), so a continuous pole with negative real part maps inside the unit circle for any positive sampling period T. Nevertheless, sampling, delay, quantization, and computational effects can significantly alter relative stability.

The discrete-time Lyapunov equation is

A_d^T P A_d − P = −Q,

where Q is positive definite. Existence of a positive-definite P is equivalent to Schur stability, meaning all eigenvalues of A_d lie inside the unit circle.

8. Applied Case: Stability Assessment of a Servo Position Loop

The following is an illustrative engineering case, not a report of a specific commercial product.

Background

A laboratory servo positions a motor shaft using a proportional controller. The simplified plant is approximated by

G_p(s) = 10/[s(s + 2)],

and unity feedback is used. The controller is K, so the loop transfer function is

L(s) = 10K/[s(s + 2)].

Problem

The design team wants to determine the allowable controller gain and understand why a gain that improves speed may create excessive oscillation.

Analysis

The closed-loop characteristic equation is

1 + 10K/[s(s+2)] = 0,

which gives

s² + 2s + 10K = 0.

For a second-order polynomial s² + a_1s + a_0, asymptotic stability requires a_1 > 0 and a_0 > 0. Therefore K > 0 gives asymptotic stability. However, stability alone does not guarantee acceptable performance.

Comparing with s² + 2ζω_n s + ω_n² gives

ω_n = √(10K),

ζ = 1/√(10K).

At K = 0.4, ω_n = 2 rad/s and ζ = 0.5. The estimated 2% settling time is

T_s ≈ 4/(ζω_n) = 4/(1) = 4 s.

The overshoot estimate is

M_p ≈ exp[−π(0.5)/√(1−0.25)] × 100%

≈ exp(−1.814) × 100%

≈ 16.3%.

If K is increased to 2.5, then ω_n = 5 rad/s and ζ = 1/5 = 0.2. The approximate settling time remains 4 s because ζω_n = 1, but the overshoot becomes approximately

exp[−π(0.2)/√(1−0.04)] × 100% ≈ 52.7%.

Thus, increasing K raises the oscillation frequency but reduces damping. A design based only on “faster natural frequency” would miss the deterioration in transient quality.

Approach and solution

A practical design would specify both stability and performance constraints, such as a minimum damping ratio, an overshoot limit, and adequate phase margin. If ζ ≥ 0.5 is required, then

1/√(10K) ≥ 0.5,

which implies

√(10K) ≤ 2,

and hence K ≤ 0.4. Under this simplified model, the region 0 < K ≤ 0.4 satisfies the damping requirement. A lead compensator could then be considered if faster response and adequate phase margin are both required. Real implementation would additionally include motor saturation, sensor noise, friction, sampling, computational delay, and unmodelled flexible modes.

Outcome and learning points

The loop is asymptotically stable for every positive K in the ideal second-order model, but relative stability changes substantially with K. The case demonstrates that:

• Stability is a necessary, not sufficient, performance requirement.
• Pole damping must be considered along with pole location and speed.
• Simplified models can conceal actuator and delay limitations.
• A compensator should be evaluated using both time-domain specifications and frequency-domain margins.

9. Challenges, Limitations, and Best Practices

Model uncertainty

Parameter errors shift poles and alter margins. Robust stability asks whether every plant in an uncertainty set remains stable. Independent gain and phase margins may not adequately describe structured multivariable uncertainty; robust-control tools such as structured singular-value analysis may be needed.

Nonlinearities

Saturation, dead zones, backlash, friction, hysteresis, and relay action invalidate unrestricted linear conclusions. Linear stability may describe local behaviour, while large-signal behaviour may include limit cycles or multiple equilibria.

Hidden unstable modes

Minimal transfer-function reduction can conceal uncontrollable, unobservable, or cancelled modes. Stability claims should therefore be based on a physically meaningful realization and a controllability-observability assessment where appropriate.

Numerical issues

High-order polynomial root calculations can be ill-conditioned. Routh tables can also suffer from numerical scaling problems. Polynomial coefficients should be scaled, independent computational checks should be used, and conclusions near a stability boundary should not rely on low-precision arithmetic.

Engineering best practices

1. Define the stability notion required by the application.

2. Derive the complete closed-loop characteristic equation.

3. Check internal modes, not only a reduced transfer function.

4. Use pole locations or Routh–Hurwitz for LTI screening.

5. Use Lyapunov analysis for nonlinear, time-varying, or energy-based arguments.

6. Evaluate gain and phase margins under realistic delays and uncertainty.

7. Verify analytical results through simulation and, when possible, hardware testing.

8. Test saturation, faults, sensor failure, communication loss, and worst-case disturbances.

9. Document assumptions, units, sign conventions, sampling rates, and numerical tolerances.

10. Contemporary and Professional Considerations

Modern control systems are increasingly networked and software-defined. Stability may be affected by variable communication delay, packet loss, asynchronous sampling, cyber-physical attacks, and adaptive or learning-based controllers. An AI-based controller should not be accepted merely because its nominal simulation is stable. Its policy or learned model requires verified operating bounds, runtime monitoring, fallback control, and safety constraints.

In safety-critical applications, stability evidence should be traceable to requirements and independently reviewed. A controller that remains mathematically stable but drives an actuator into thermal overload is not operationally safe. Governance therefore connects mathematical stability with hazard analysis, verification, validation, cybersecurity, and fail-safe design.

Chapter Summary

Stability describes whether system trajectories remain bounded and, in stronger forms, converge to an equilibrium. Internal stability concerns state behaviour, whereas BIBO stability concerns bounded inputs and outputs. For finite-dimensional LTI systems without hidden unstable modes, internal asymptotic and BIBO stability coincide.

Continuous-time asymptotic stability requires poles in the open left half-plane; discrete-time asymptotic stability requires poles inside the unit circle. Imaginary-axis or unit-circle poles require careful treatment because simple poles may yield bounded but non-decaying responses, while repeated poles cause instability. The Routh–Hurwitz criterion determines the number of right-half-plane roots from polynomial coefficients and provides a practical parameter-stability test.

Lyapunov’s direct method establishes stability through positive-definite functions and their derivatives. For linear systems, the Lyapunov equation gives an equivalent matrix-based certificate. LaSalle’s principle extends analysis when the Lyapunov derivative is only negative semidefinite. Relative stability is assessed using dominant-pole damping, gain margin, phase margin, Nyquist analysis, and delay sensitivity. In practice, uncertainty, nonlinearities, saturation, sampling, and hidden modes must be included in any credible stability assessment.

Key Terms

Lyapunov stability: The property that sufficiently small initial deviations remain small for all future time.

Asymptotic stability: Stability together with convergence of trajectories to the equilibrium.

Exponential stability: Convergence bounded by an exponentially decaying function.

BIBO stability: Bounded-input bounded-output behaviour.

Internal stability: Stability of all internal state trajectories.

Marginal stability: Bounded, non-decaying response under permitted simple boundary poles.

Characteristic equation: Equation whose roots are the closed-loop poles.

Hurwitz polynomial: A polynomial whose roots all lie in the open left half-plane.

Routh array: Tabular arrangement used to apply the Routh–Hurwitz criterion.

Imaginary-axis pole: A pole with zero real part in continuous time.

Dominant pole: A pole or pole pair chiefly determining the long-term transient response.

Damping ratio: A normalized measure of oscillatory decay in a second-order model.

Gain margin: Multiplicative gain increase permitted before the critical Nyquist condition is reached.

Phase margin: Additional phase lag permitted before instability at gain crossover.

Lyapunov function: A scalar measure used to certify stability through its sign and derivative.

Radially unbounded: A function that tends to infinity as the state norm tends to infinity.

LaSalle invariance principle: A result establishing convergence to the largest invariant set where a Lyapunov derivative is zero.

Schur stability: Discrete-time stability with all poles inside the unit circle.

Robust stability: Stability maintained for all plants within a specified uncertainty set.

Hidden mode: An internal dynamic mode concealed by pole-zero cancellation or nonminimal modelling.

Review Questions

Short Answer Questions

1. Distinguish internal stability from BIBO stability.

2. State the continuous-time pole condition for asymptotic stability.

3. What does a sign change in the first column of a Routh array represent?

4. Define a positive-definite Lyapunov function.

5. Explain the meaning of phase margin.

Long Answer Questions

1. Derive the relationship between the poles of a standard second-order system and its damping ratio and natural frequency.

2. Explain the Routh–Hurwitz procedure, including the zero first-column and complete-row-zero cases.

3. Discuss the direct and indirect Lyapunov methods and their applicability to nonlinear systems.

4. Compare absolute stability and relative stability using pole and frequency-domain measures.

5. Explain why pole-zero cancellation can produce a misleading stability conclusion.

Analytical / Application Questions

1. Construct the Routh array for P(s) = s⁴ + 3s³ + 5s² + 7s + 2 and determine the number of right-half-plane roots.

2. For the servo case with characteristic polynomial s² + 2s + 10K, determine the gain range that gives damping ratio ζ ≥ 0.7.

3. A controller has loop transfer function with a phase margin of 20° and a significant computational delay. Evaluate the risks and propose at least three design or verification actions.

Multiple Choice Questions with Answers

Question 1

For a continuous-time LTI system, which pole location is required for asymptotic stability?

A. All poles must lie in the open left half-plane

B. All poles must lie on the imaginary axis

C. At least one pole must lie in the right half-plane

D. All poles must lie outside the unit circle

Correct Answer: A. All poles must lie in the open left half-plane

Question 2

In a Routh array, the number of sign changes in the first column equals the number of:

A. Left-half-plane roots

B. Right-half-plane roots

C. Imaginary-axis roots only

D. Repeated roots only

Correct Answer: B. Right-half-plane roots

Question 3

For P(s) = s³ + 2s² + (K+1)s + 4, the asymptotically stable range is:

A. K < 1

B. K = 1 only

C. K > 1

D. K > −1

Correct Answer: C. K > 1

Question 4

A positive-definite Lyapunov function V(x) must satisfy which condition away from the equilibrium?

A. V(x) < 0

B. V(x) = 0

C. V(x) > 0

D. V_dot(x) > 0 necessarily

Correct Answer: C. V(x) > 0

Question 5

For a discrete-time LTI system, asymptotic stability requires all poles to be:

A. Inside the unit circle

B. In the open left half-plane

C. On the unit circle

D. Outside the unit circle

Correct Answer: A. Inside the unit circle

Question 6

The phase margin is measured at the frequency where:

A. The phase is zero degrees

B. The magnitude of the loop transfer function is unity

C. The magnitude is zero

D. The closed-loop output is zero

Correct Answer: B. The magnitude of the loop transfer function is unity

Question 7

For a standard second-order system, increasing the damping ratio generally:

A. Increases overshoot

B. Reduces oscillatory overshoot

C. Moves both poles into the right half-plane

D. Makes the natural frequency zero

Correct Answer: B. Reduces oscillatory overshoot

Question 8

For the servo case, when K = 0.4, the damping ratio is:

A. 0.2

B. 0.4

C. 0.5

D. 2.0

Correct Answer: C. 0.5

Question 9

A repeated pole on the continuous-time imaginary axis generally produces:

A. A decaying exponential response

B. An unbounded polynomially growing response

C. A finite-time settling response

D. A response that is always asymptotically stable

Correct Answer: B. An unbounded polynomially growing response

Question 10

The continuous-time Lyapunov equation A^T P + PA = −Q is used with Q positive definite to certify stability when:

A. P is positive definite

B. P is negative definite

C. A has a right-half-plane eigenvalue

D. Q is singular and indefinite

Correct Answer: A. P is positive definite