A continuous-time Linear Time-Invariant (LTI) dynamic system has the state equation: ẋ(t) =...
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A continuous-time Linear Time-Invariant (LTI) dynamic system has the state equation: ẋ(t) = Ax(t) + Bu(t), where the system matrix A is given by: A = [[1, 1], [0, 1]]. Determine the state transition matrix Φ(t) = e^(At) of the system.
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The state transition matrix Φ(t) = e^(At) for a 2×2 matrix with eigenvalue λ=1 (repeated) and Jordan form structure is computed using the Jordan normal form decomposition. Since A is upper triangular with repeated eigenvalue 1, the exponential yields Φ(t) = e^t·I + t·e^t·N, where N is the nilpotent part. This gives the matrix with diagonal e^t entries and off-diagonal term te^t in position (1,2).
Step-by-step Derivation:
Step 1: Identify eigenvalues of A. det(A - λI) = det([[1-λ, 1], [0, 1-λ]]) = (1-λ)² = 0, so λ = 1 (repeated).
Step 2: Find Jordan form. A has one eigenvector [1, 0]ᵀ but geometric multiplicity 1 < algebraic multiplicity 2, so A has a Jordan block:
J = [[1, 1], [0, 1]] (already in Jordan form with P = I)
Step 3: Decompose A = λI + N where N = [[0, 1], [0, 0]] (nilpotent, N² = 0).
Step 4: Compute e^(At) = e^(λt)·e^(Nt) = e^t·e^(Nt).
Since N² = 0: e^(Nt) = I + Nt = [[1, t], [0, 1]]
Step 5: e^(At) = e^t·[[1, t], [0, 1]] = [[e^t, te^t], [0, e^t]]
Verification: d/dt[e^(At)] = A·e^(At) ✓