Questions: State Transition Matrix and Solution Computation

Question 1

A linear system has state matrix A with eigenvalues {-2, -3+4j, -3-4j}. What can you conclude about the state transition matrix Φ(t) = eAt as t → ∞?

• A. Φ(t) grows without bound because complex eigenvalues produce oscillation that amplifies over time
• B. Φ(t) → 0 because all eigenvalues have strictly negative real parts, so every mode decays
• C. Φ(t) oscillates permanently at finite amplitude because of the complex eigenvalue pair
• D. Nothing can be concluded without explicitly computing eAt from the matrix series

Answer: (B)

The eigenvalues of A determine the character of Φ(t) directly: each eigenvalue λ = σ + jω contributes a mode of the form eσt(cos ωt + ...). The real part σ determines growth or decay. Here all eigenvalues have σ < 0 (−2, −3, −3), so all modes decay exponentially. The complex pair produces oscillation, but a decaying oscillation — eAt → 0 as t → ∞. Stability is determined entirely by the real parts of eigenvalues; complex parts add oscillation but not growth.

Question 2

In the complete solution x(t) = Φ(t)x(0) + ∫₀ᵗ Φ(t−τ)Bu(τ)dτ, what does the factor Φ(t−τ) inside the integral represent?

• A. The input applied at time τ, scaled by the system's DC gain
• B. The inverse of the transition matrix, used to back-propagate the current state
• C. The evolution of an impulse input applied at time τ through the remaining t−τ seconds of system dynamics
• D. A weighting function that normalizes the input magnitude across the integration interval

Answer: (C)

The convolution integral accumulates the effect of all inputs over time. At each instant τ, the input Bu(τ) enters the system as an effective impulse. That impulse then evolves through the system dynamics for the remaining t−τ seconds before the observation time t — and Φ(t−τ) is exactly the transition matrix that describes how a state vector is propagated forward by t−τ seconds. Summing (integrating) these 'aged' input contributions from τ = 0 to τ = t gives the total zero-state response. Older inputs (small τ, large t−τ) have been propagated longer and carry the imprint of the system's dynamics more heavily.

Question 3

The eigenvalues of the state matrix A completely determine whether the zero-input response of a linear system is stable, oscillatory, or divergent — without requiring explicit computation of eAt.

• T. True
• F. False

Answer: True

Each eigenvalue λᵢ of A contributes a mode eλᵢt to Φ(t). Real negative eigenvalues → exponential decay. Real positive eigenvalues → exponential growth (unstable). Complex pairs σ ± jω → oscillation at frequency ω with growth/decay rate σ. Purely imaginary pairs → undamped oscillation. So reading the eigenvalues directly tells you the qualitative behavior of all modes without computing the full matrix exponential. This is why stability analysis focuses on eigenvalue locations in the complex plane.

Question 4

A linear system whose state transition matrix Φ(t) does not decay to zero is typically unstable and will produce unbounded output for any input.

• T. True
• F. False

Answer: False

A system with purely imaginary eigenvalues (e.g., λ = ±jω) has a transition matrix that oscillates at constant amplitude — Φ(t) does not decay to zero, but the system is marginally stable, not unstable. For bounded inputs, outputs remain bounded (BIBO stable in some formulations). Only when eigenvalues have strictly positive real parts does Φ(t) grow without bound and produce unbounded outputs. The distinction between asymptotic stability (all eigenvalues strictly in left half-plane), marginal stability (eigenvalues on imaginary axis), and instability (any eigenvalue in right half-plane) is critical for control design.

Question 5

Why does computing eAt by diagonalization require A to have distinct eigenvalues, and what method must be used when eigenvalues repeat?

Model answer: Diagonalization requires that A = PΛP⁻¹, where P is the matrix of eigenvectors and Λ is diagonal. This is only possible when A has n linearly independent eigenvectors — which is guaranteed when eigenvalues are distinct but not when they repeat. For a repeated eigenvalue, the eigenvector set may be deficient (fewer independent eigenvectors than the eigenvalue's algebraic multiplicity). In this case, A must be reduced to Jordan canonical form J = P⁻¹AP, where J has the eigenvalues on the diagonal and 1s on the superdiagonal of each Jordan block. The matrix exponential then involves terms of the form t^k e^λt corresponding to the Jordan block structure.

The physical meaning: a Jordan block of size 2 for eigenvalue λ produces modes eλt and t·eλt. The polynomial factor t multiplying the exponential is the hallmark of a defective matrix. For stable systems (Re(λ) < 0), these modes still decay to zero (exponential beats polynomial), but they can cause large transient responses before doing so.