1 Linear systems and solution representations
Linear systems describe how a state evolves under linear dynamics, typically expressed through differential equations (continuous time) or difference equations (discrete time). Stability questions ask whether trajectories approach an equilibrium, stay bounded, or grow without bound as time progresses.
1.1 Continuous-time linear systems
Continuous-time linear systems are modeled by linear ordinary differential equations with constant coefficients in their most common form.
1.1.1 State-space form and state transition
A standard continuous-time state-space model is \[ \dot{x}(t)=Ax(t), \] where \(x(t)\in\mathbb{R}^n\) and \(A\in\mathbb{R}^{n\times n}\). The system’s evolution is captured by the state transition map \(x(t)=\Phi(t)x(0)\), where \(\Phi(t)\) depends only on \(A\) when the system is time-invariant.
When inputs are present, one typically writes \[ \dot{x}(t)=Ax(t)+Bu(t), \] and the stability of the homogeneous part largely determines whether perturbations in initial conditions decay or persist.
1.1.2 Matrix exponential and modes
For \(\dot{x}=Ax\), the state transition is given by the matrix exponential: \[ x(t)=e^{At}x(0). \] The eigenstructure of \(A\) governs the long-term behavior. If eigenvalues have negative real parts, the exponential terms decay; if any eigenvalue has positive real part, solutions grow. In defective cases, Jordan blocks introduce polynomial factors multiplying the exponential, influencing transient growth even when asymptotic rates are set by eigenvalues.
1.2 Discrete-time linear systems
Discrete-time linear systems update states in steps, typically described by recurrence relations.
1.2.1 State update and iterated dynamics
A basic discrete-time model is \[ x_{k+1}=Ax_k, \] leading to iterated dynamics \(x_k=A^k x_0\). Stability then depends on the behavior of powers of \(A\) as \(k\to\infty\). If trajectories contract to the origin, the system is stable in the usual discrete-time sense; if powers grow, instability follows.
With inputs, the model \[ x_{k+1}=Ax_k+Bu_k \] splits into homogeneous evolution and a forced response determined by sums involving \(A^k\).
1.2.2 Powers of system matrices
| Eigenvalues of \(A\) determine whether \(A^k\) decays or grows. In diagonalizable cases, each mode contributes a factor \(\lambda^k\). More generally, Jordan blocks yield terms like \(k^m\lambda^k\), where the polynomial factor can create significant transient effects even when the limiting behavior is controlled by \( | \lambda | \). Stability is often characterized by whether all eigenvalues lie strictly inside a unit disk (for asymptotic decay). |
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1.3 Homogeneous vs. forced systems
Many stability notions are first defined for homogeneous systems, then extended to forced dynamics.
1.3.1 Steady-state behavior with inputs
For linear systems with inputs, stability is frequently assessed by how the state responds over time. If the homogeneous dynamics are stable, the system’s forced response can converge to a steady regime or remain bounded depending on the input class. Conversely, if the homogeneous part is unstable, bounded inputs can still produce unbounded state trajectories.
1.3.2 Superposition and decomposition
Linearity allows decomposition of solutions into a homogeneous term (carrying initial-condition effects) and a particular term (carrying input effects). This superposition property is central to many stability tests, including frequency-domain interpretations for time-invariant systems and norm-based bounds derived from induced system operators.
2 Notions of stability
Stability is not a single property but a family of related concepts. The core idea is to specify how trajectories behave under perturbations and how quickly they settle.
2.1 Equilibrium points and stability under perturbations
For linear time-invariant systems, an equilibrium often corresponds to the origin when there is no constant forcing. Stability describes how solutions behave when initial conditions are slightly changed.
2.1.1 Lyapunov stability (definition and intuition)
Lyapunov (or “stable”) behavior means that trajectories starting close to an equilibrium remain close for all future time. Formally, for every desired bound on the deviation, there exists a proximity level such that initial perturbations within that level guarantee the solution never leaves the prescribed neighborhood.
Intuitively, Lyapunov stability rules out runaway divergence, but it does not require convergence back to the equilibrium.
2.1.2 Asymptotic stability (convergence criteria)
Asymptotic stability requires not only that trajectories stay near the equilibrium, but also that they converge to it as time grows. In linear systems, asymptotic stability often corresponds to a spectral condition (e.g., negative real parts in continuous time, eigenvalues strictly inside the unit circle in discrete time).
2.1.3 Uniform vs. non-uniform variants
Uniform versions refine how “closeness” and “eventual behavior” are quantified over time intervals. In continuous time, uniformity can matter when system dynamics or time scaling create differences in convergence speed. For time-invariant linear systems, the distinction is typically less subtle, but the terminology is still relevant for theoretical completeness.
2.2 Boundedness properties
Boundedness concepts focus on ensuring trajectories or outputs do not grow beyond limits, sometimes without requiring convergence to a specific equilibrium.
2.2.1 BIBO stability for linear time-invariant systems
For input-output models with output \(y(t)\) and input \(u(t)\), bounded-input bounded-output (BIBO) stability requires that any bounded input produces a bounded output. In linear time-invariant systems, this is closely related to whether the impulse response is absolutely integrable and whether the transfer function has appropriate pole placement relative to the region of convergence.
While BIBO stability is an input-output property, it indirectly encodes information about internal stability for certain well-posed realizations.
2.2.2 Mean-square and norm-bounded perspectives
In stochastic settings or systems with uncertainties, stability may be described via expected squared norms (mean-square stability) or via boundedness under norm constraints on trajectories or signals. Even in deterministic linear analysis, norm-bounded perspectives motivate robust variants of stability tests based on operator norms.
2.3 Stronger stability concepts
Stronger notions demand quantitative decay rates or additional performance guarantees.
2.3.1 Exponential stability
| Exponential stability requires that the magnitude of the state decays at least as fast as an exponential function. In continuous time, one expects bounds of the form \(\|x(t)\|\le Me^{-\alpha t}\|x(0)\|\); in discrete time, bounds resemble \(\|x_k\|\le M\rho^k\|x_0\|\) with \(\rho<1\). For linear time-invariant systems, exponential stability is closely tied to strict spectral inequalities. |
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2.3.2 Input-to-state viewpoints (linear case focus)
Input-to-state stability is commonly used to capture how the size of the state depends on the size of disturbances or inputs. For linear systems, these ideas simplify: bounded inputs yield bounded states, and the state magnitude can be bounded by a combination of decay from initial conditions plus a gain applied to the input. This viewpoint bridges internal stability and input-output response.
3 Eigenvalue-based criteria
Eigenvalues provide a direct route to stability characterization because they determine the asymptotic behavior of matrix exponentials and matrix powers.
3.1 Spectral conditions for continuous time
For continuous-time linear systems \(\dot{x}=Ax\), stability is governed by the location of eigenvalues in the complex plane.
3.1.1 Jordan form and decay rates
Using Jordan canonical form, one decomposes \(A\) into modal blocks. Each block contributes terms involving \(e^{\lambda t}\) multiplied by polynomials in \(t\) whose degrees depend on Jordan block sizes. If every eigenvalue satisfies \(\Re(\lambda)<0\), the exponential decay dominates any polynomial growth, producing asymptotic stability. If some eigenvalues satisfy \(\Re(\lambda)>0\), growth dominates and solutions become unbounded.
3.1.2 Complex eigenvalues and oscillatory modes
Complex conjugate eigenvalues introduce oscillations through sinusoidal factors in the real coordinate representation. Stability then depends on whether those oscillations are exponentially damped or amplified. Eigenvalues with negative real parts yield decaying oscillatory behavior; zero real parts can lead to persistent or marginal oscillations depending on Jordan structure.
3.2 Spectral conditions for discrete time
For discrete-time linear systems \(x_{k+1}=Ax_k\), eigenvalues relative to the unit circle determine growth or decay.
3.2.1 Unit-circle criteria and growth/decay
| If all eigenvalues satisfy \( | \lambda | <1\), then \(A^k\to 0\) and trajectories converge to the origin. If at least one eigenvalue lies outside the unit circle, corresponding modes grow geometrically and instability results. Eigenvalues on the unit circle require careful handling because marginal cases can involve persistent oscillations or polynomially growing terms. |
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3.2.2 Geometric multiplicity vs. stability behavior
| When eigenvalues are repeated, the Jordan form matters. Even if \( | \lambda | =1\), defective Jordan blocks can produce growth via polynomial factors \(k^m\). Geometric multiplicity influences whether such blocks are present. As a result, stability on the boundary of the unit disk is not determined by eigenvalues alone but also by their Jordan structure. |
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3.3 Special cases and edge conditions
Boundary cases define the boundary between “stable but not convergent” and “unstable.”
3.3.1 Zero and repeated eigenvalues
Zero eigenvalues in continuous time correspond to modes with \(\Re(\lambda)=0\), which can lead to non-decaying components. Repeated eigenvalues exacerbate transient behavior because Jordan blocks can yield polynomially weighted terms. Stability characterization then depends on whether these polynomial terms vanish asymptotically or persist.
3.3.2 Marginal stability and its characterization
Marginal stability refers to cases where trajectories remain bounded but do not converge to the equilibrium in general. In continuous time, eigenvalues on the imaginary axis can produce bounded oscillations when the Jordan blocks are trivial (no defective blocks) and when the system is structurally such that those modes do not generate polynomial growth. In discrete time, analogous conditions apply to eigenvalues on the unit circle with restriction against defective Jordan blocks.
4 Lyapunov methods for linear systems
Lyapunov methods provide stability certificates without explicitly solving the system. For linear systems, they often reduce to checking matrix inequalities.
4.1 Quadratic Lyapunov functions
Quadratic forms \(V(x)=x^\top P x\) are common because they lead to tractable conditions.
4.1.1 Existence of P for asymptotic stability
For stable linear time-invariant systems, one can often find a symmetric positive definite matrix \(P\) such that the Lyapunov function decreases along trajectories. For asymptotic stability, the derivative (or difference) of \(V\) must be negative in an appropriate sense. Existence of such a \(P\) provides a constructive stability proof.
4.1.2 Connections to matrix inequalities
The requirement that \(V\) decreases translates into inequalities involving \(P\) and the system matrix. These inequalities can be expressed as linear matrix inequalities (LMIs) in \(P\) for both continuous and discrete settings, enabling computational verification.
4.2 Continuous-time Lyapunov equation/inequality
For \(\dot{x}=Ax\), differentiating \(V(x)=x^\top P x\) links stability to a condition on \(PA + A^\top P\).
4.2.1 Derivation using V(x)=xᵀPx
Compute \[ \dot{V}(x)=\frac{d}{dt}(x^\top P x)=\dot{x}^\top P x + x^\top P \dot{x}. \] Substituting \(\dot{x}=Ax\) yields \[ \dot{V}(x)=x^\top (A^\top P + P A)x. \] If \(P\) is positive definite and \(A^\top P + P A\) is negative definite, then \(\dot{V}(x)\) is strictly negative away from the origin, ensuring asymptotic stability.
4.2.2 Strict vs. non-strict inequalities
Negative definite conditions yield exponential-type decay estimates, while negative semidefinite conditions yield weaker results: trajectories may be stable but may not converge to the equilibrium. In marginal scenarios, the derivative of \(V\) can vanish on invariant subspaces, which limits the strength of the conclusion.
4.3 Discrete-time Lyapunov inequality
For \(x_{k+1}=Ax_k\), the Lyapunov condition uses a difference \(V(x_{k+1})-V(x_k)\).
4.3.1 Conditions using V(x_{k}) forms
Take \(V(x_k)=x_k^\top P x_k\) with \(P\succ 0\). Then \[ V(x_{k+1})-V(x_k)=x_k^\top(A^\top P A - P)x_k. \] If \(A^\top P A - P\) is negative definite, the function decreases at each step, implying convergence. Negative semidefinite conditions again correspond to weaker or potentially marginal outcomes.
4.3.2 Practical verification strategies
In computations, one typically searches for \(P\) satisfying a strict inequality with a small margin to reduce numerical sensitivity. This can be achieved by formulating an LMI feasibility or optimization problem and then verifying that the obtained matrices satisfy the chosen tolerances.
5 Stability via matrix inequalities and norms
While eigenvalues give exact boundary conditions, matrix inequalities and induced norms provide robust, certificate-based tools.
5.1 Operator norms and induced growth bounds
Norms measure system amplification in a way that can yield sufficient stability criteria.
5.1.1 Norm-based sufficient conditions
| For discrete-time systems, if \(\|A\|<1\) under an induced matrix norm, then \(\|A^k\|\le \|A\|^k\) and the system is asymptotically stable. For continuous time, similar reasoning can be applied using bounds on \(\|e^{At}\|\) derived from norm estimates or by relating the logarithmic norm to a decay rate. |
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These conditions are typically conservative but easy to compute and useful for robustness analysis.
5.1.2 Spectral abscissa vs. norm growth
The spectral abscissa (maximum real part of eigenvalues) determines the best possible exponential rate, but induced norms can overestimate growth. There is a gap between “true” asymptotic decay and “certified” decay when relying solely on a fixed norm. Tightness can sometimes be improved by choosing suitable norms or by scaling transformations.
5.2 Small-gain style stability checks (linear setting)
Small-gain reasoning interprets stability as contraction of a feedback loop.
5.2.1 Loop gain interpretation
In linear interconnections, stability can be related to whether a loop gain operator has norm less than unity. In simpler linear settings, this can translate to conditions on products of transfer functions or on algebraic relationships among system matrices, depending on the modeling framework.
5.2.2 Robustness-flavored variants (linear constraints)
Small-gain logic naturally extends to robustness: if uncertainty perturbs a system, stability can be preserved when the perturbation is small enough that the overall gain remains below the critical threshold. In linear analysis, this often leads to inequalities that include uncertainty bounds and can be checked via conservative norm estimates.
5.3 Scaling and coordinate transforms
Stability properties are invariant under nonsingular changes of coordinates, but numerical tests are not.
5.3.1 Similarity transforms and invariance
If \(x=Tz\) with invertible \(T\), then the transformed system has matrix \(T^{-1}AT\). Eigenvalues and true stability are preserved. However, Lyapunov matrices and norm bounds change with the transformation, affecting how easily conditions are verified computationally.
5.3.2 Conditioning effects on numerical tests
When matrices are ill-conditioned, computed eigenvalues and derived inequalities can be sensitive to rounding errors. Scaling strategies can reduce such effects by improving numerical conditioning, thereby making stability certificates more reliable in practice.
6 Stability of interconnections (linear systems)
Interconnected systems can exhibit stability properties that differ from those of individual subsystems, especially when coupling is strong.
6.1 Series, parallel, and feedback structures
System interconnections are commonly organized into standard configurations that simplify analysis.
6.1.1 Closed-loop dynamics from block models
In block state-space form, interconnections produce an aggregated state matrix that determines closed-loop dynamics. Stability then depends on the eigenvalues of this composite matrix, or on Lyapunov conditions tailored to the interconnection structure.
6.1.2 Relationship to poles/eigenvalues
In frequency-domain terms, closed-loop poles correspond to eigenvalues of the closed-loop state matrix (for realizations under standard assumptions). Stability is therefore linked to whether those poles lie in the stable region for the time domain under consideration.
6.2 State-space interconnection stability
State-space approaches often require constructing certificates for the combined system.
6.2.1 Aggregate system matrix properties
When two subsystems with states \(x_1\) and \(x_2\) are coupled, the combined state vector \(x=[x_1^\top\ x_2^\top]^\top\) evolves according to a block matrix. Stability can be studied by analyzing properties of that block matrix, including spectral location and whether coupling introduces destabilizing eigenvalues.
6.2.2 Common Lyapunov functions for coupled systems
For families of coupled dynamics, one may look for a single Lyapunov function valid across the interconnection. Existence of a common quadratic Lyapunov function can imply stability for all admissible couplings within a structured class, providing a unifying certificate.
6.3 Separation of dynamics (when applicable)
In some cases, interconnections decouple or exhibit mode independence.
6.3.1 Decoupled subsystems analysis
If the combined system matrix can be arranged into invariant blocks (e.g., via a change of basis), each block can be analyzed separately. This can reduce computational effort and clarify how each subsystem contributes to stability.
6.3.2 Invariant subspaces and modal independence
Invariant subspaces correspond to sets of states that evolve under the dynamics without mixing with the complementary subspace. When such subspaces exist, eigenmodes associated with different subspaces do not interfere, and stability can be determined from the behavior within each invariant component.
7 Stability in control design context (linear analysis)
Control design uses stability concepts to ensure that controllers and observers achieve desired closed-loop behavior.
7.1 Stabilizability and detectability (overview)
These properties determine whether feedback can assign stable dynamics in the presence of uncontrollable or unobservable modes.
7.1.1 Intuitive meaning for linear systems
Stabilizability means that unstable modes are controllable through the available input channels. Detectability means that unstable (or non-decaying) modes are observable through the output measurements. Together, these properties dictate whether one can design a controller (and separately an observer) to achieve stability.
7.1.2 Implications for controller/observer stability
If the system is stabilizable, state feedback can be designed so that the closed-loop system is stable. If the system is detectable, an observer can be designed so that estimation errors decay. In combined observer-based control, stability of the overall loop depends on both aspects and on the chosen gains.
7.2 Pole placement and stability targets
Pole placement aims to position eigenvalues of the closed-loop dynamics in desired locations.
7.2.1 Assignable regions for eigenvalues
For a controllable pair, eigenvalues can often be assigned arbitrarily through appropriate feedback. When only stabilizability holds, one can still typically place the unstable eigenvalues inside the stable region, while leaving stable uncontrollable modes unaffected.
7.2.2 Effects of constraints and multiplicities
In practical scenarios, constraints on gains, limited actuator authority, or desired eigenvalue multiplicities can affect achievable placement. Repeated eigenvalues or complex conjugate pairs influence transient responses and may require careful handling because Jordan structure affects time-domain behavior beyond eigenvalue locations.
7.3 Observer-based and state-feedback loops
Observer-based designs replace full state information with estimated states.
7.3.1 Closed-loop combined dynamics
When using a state observer with gain \(L\) and a state-feedback controller with gain \(K\), the combined system (plant plus estimator) has dynamics that depend on both \(A-BK\) and \(A-LC\). Stability of the combined system is typically ensured when both components are stable and when the coupling structure does not introduce unexpected instabilities.
7.3.2 Robustness to small modeling errors (linear view)
Linear analysis often treats modeling errors as perturbations in system matrices. Stability margins, norm bounds, and sensitivity of eigenvalues to perturbations provide a basis for assessing whether the designed gains remain stabilizing under small changes. Although exact robustness guarantees may require more advanced tools, the eigenvalue and Lyapunov viewpoints remain central.
8 Numerical aspects and computation of stability
Computational verification introduces practical issues that may obscure theoretical properties.
8.1 Eigenvalue computations and pitfalls
Eigenvalue-based methods can be sensitive to numerical accuracy, especially near stability boundaries.
8.1.1 Conditioning of eigenvalues
Eigenvalues associated with non-normal matrices can be highly sensitive to perturbations. In such cases, computed eigenvalues may differ significantly from the true values, and transient growth may occur even when asymptotic decay is present. This means eigenvalue location alone can be insufficient for understanding time-domain behavior.
8.1.2 Handling nearly marginal systems
When eigenvalues are close to the imaginary axis (continuous time) or close to the unit circle (discrete time), numerical tolerances determine whether a system is declared stable. Small rounding errors can switch the classification, so practical workflows often incorporate margins or additional checks beyond raw eigenvalue tests.
8.2 Approximating the matrix exponential
For continuous-time verification, discretization and approximation can alter stability assessment.
8.2.1 Time discretization impact on stability
When simulating \(\dot{x}=Ax\) using a discrete time step, the numerical method effectively applies a transformation to the dynamics. Some integrators preserve stability properties (under conditions), while others can introduce artificial instability. The discrete-time interpretation depends on the chosen step size and method.
8.2.2 Stability of numerical integrators (linear case)
For linear systems, the stability of common numerical integrators can be analyzed by studying their stability functions. Matching continuous-time stability with discrete-time decay often requires ensuring that the step size is sufficiently small or that the method is A-stable or L-stable in relevant contexts.
8.3 Verifying Lyapunov inequalities computationally
Matrix inequality checks can be performed using convex optimization tools, but strictness and tolerances must be managed.
8.3.1 Using semidefinite programming (conceptual)
Quadratic Lyapunov conditions yield semidefinite programs when framed as LMI feasibility or minimization. Solvers can search for \(P\succ 0\) satisfying continuous or discrete Lyapunov inequalities, producing a certificate of stability when successful.
8.3.2 Practical tolerance and strictness checks
Because floating-point arithmetic cannot guarantee exact inequalities, one typically checks whether the smallest eigenvalue margins exceed a preset tolerance. For strict inequalities, a small buffer is often introduced to reduce false positives due to numerical noise.
9 Examples and worked case studies
Examples illustrate how definitions, spectral criteria, and Lyapunov certificates relate in concrete systems.
9.1 Single-mode systems (scalar)
Scalar systems provide the simplest mapping between eigenvalues and stability.
9.1.1 Explicit stability classification
| For \(\dot{x}=ax\), solutions are \(x(t)=e^{at}x(0)\). Stability holds when \(a<0\) and marginal behavior occurs when \(a=0\). For discrete time \(x_{k+1}=ax_k\), solutions are \(x_k=a^k x_0\), giving stability when \( | a | <1\). |
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9.1.2 Behavior under parameter changes
As \(a\) crosses the boundary (\(0\) in continuous time or \(\pm 1\) in discrete time), the qualitative behavior switches from decay to growth or from convergence to persistence. This makes scalar cases useful for building intuition about boundary conditions.
9.2 2×2 systems with complex eigenpairs
Two-dimensional systems demonstrate oscillations and how decay rates emerge from eigenvalues.
9.2.1 Oscillation with decay vs. growth
| Consider a system with complex eigenvalues \(\lambda=\alpha\pm i\beta\). The real dynamics exhibit oscillatory motion with an amplitude scaling like \(e^{\alpha t}\) (continuous time) or \( | \lambda | ^k\) (discrete time). If \(\alpha<0\), oscillations damp out; if \(\alpha>0\), they grow. |
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9.3 Repeated-eigenvalue examples
Repeated eigenvalues highlight the role of Jordan blocks in transient and marginal behavior.
9.3.1 Jordan blocks and polynomial terms
When a matrix has a defective eigenvalue, the matrix exponential includes terms with powers of \(t\) multiplying \(e^{\lambda t}\). Even if \(\Re(\lambda)<0\), these polynomial factors affect the transient amplitude. If \(\Re(\lambda)=0\), polynomial factors can lead to unbounded growth even though eigenvalues are not in the unstable half-plane.
9.4 System with input and steady-state response
Forced linear systems connect internal stability with long-run behavior.
9.4.1 Stability and long-run behavior under forcing
For \(\dot{x}=Ax+Bu(t)\), if \(A\) is asymptotically stable and the input is bounded and well-behaved (e.g., convergent or square-integrable under suitable assumptions), the state typically approaches a steady response or remains bounded. If \(A\) is unstable, inputs that are bounded may still cause unbounded state growth because unstable modes are excited by the dynamics.
10 Summary of key results and takeaways
Stability in linear systems is fundamentally determined by how trajectories evolve under repeated application of linear dynamics, and it can be certified using spectral, Lyapunov, or inequality-based methods.
10.1 Checklist by system type (continuous vs. discrete)
For continuous-time \(\dot{x}=Ax\), asymptotic stability aligns with eigenvalues having strictly negative real parts; marginal and unstable behavior depend on eigenvalues on the imaginary axis and corresponding Jordan structure. For discrete-time \(x_{k+1}=Ax_k\), asymptotic stability corresponds to eigenvalues strictly inside the unit circle, while eigenvalues on the unit circle require checking for defective Jordan blocks.
10.2 Eigenvalue vs. Lyapunov equivalences (linear case)
For linear time-invariant systems, eigenvalue-based criteria and Lyapunov methods are closely connected: asymptotic stability implies existence of a quadratic Lyapunov function satisfying a strict inequality, and such Lyapunov certificates in turn imply stability properties. This equivalence underlies both theoretical results and practical computation via LMIs.
10.3 Common mistakes and quick diagnostics
A frequent mistake is to treat eigenvalues alone as sufficient in marginal cases; Jordan structure can change conclusions. Another pitfall is confusing stability of the homogeneous system with input-output boundedness without checking assumptions. Quick diagnostics include: (i) checking eigenvalue locations with boundary tolerance, (ii) confirming strict Lyapunov inequalities when eigenvalues are near the boundary, and (iii) accounting for numerical discretization effects when simulating continuous-time systems.