Q6Control System
Question
Q.6. Obtain STM (State Transition Matrix) for the state model whose matrix A is given by :
Answer
For A = [[0,1],[-2,-3]], eigenvalues are -1 and -2; using the Laplace method φ(t) = L⁻¹[(sI-A)⁻¹] gives STM φ(t) with entries built from e^(-t) and e^(-2t) terms.
The state transition matrix (STM), φ(t), is the fundamental matrix that propagates the state of a linear time-invariant system ẋ = Ax + Bu forward in time: given any initial state x(t0), the state at a later time is x(t) = φ(t-t0)x(t0) + [convolution term with the input]. For the unforced (zero-input) system ẋ = Ax, the complete solution is x(t) = φ(t)x(0), so φ(t) fully characterizes the system's natural (homogeneous) response modes, independent of any particular input or initial condition. There are two standard methods to compute φ(t): the infinite matrix-exponential series φ(t) = I + At + A²t²/2! + A³t³/3! + ⋯, which converges for any square matrix A but is rarely convenient for hand calculation beyond low powers, and the Laplace-transform method used here, φ(t) = L⁻¹[(sI-A)⁻¹], which is systematic and exact for any finite-dimensional A.
The state transition matrix (STM) is φ(t) = e^(At) = L⁻¹[(sI - A)⁻¹]. First form sI - A:
Determinant: |sI-A| = s(s+3) + 2 = s² + 3s + 2 = (s+1)(s+2). The characteristic roots (eigenvalues) are λ1 = -1, λ2 = -2, confirming the system is asymptotically stable. The inverse is adj(sI-A)/|sI-A|:
Performing partial fraction expansion element-by-element: for (s+3)/[(s+1)(s+2)] = A/(s+1) + B/(s+2), A = 2, B = -1, so this term is 2/(s+1) - 1/(s+2). For 1/[(s+1)(s+2)] = 1/(s+1) - 1/(s+2). For -2/[(s+1)(s+2)] = -2/(s+1) + 2/(s+2). For s/[(s+1)(s+2)] = -1/(s+1) + 2/(s+2). Taking the inverse Laplace transform of each element (L⁻¹[1/(s+a)] = e^(-at)):
This φ(t) satisfies φ(0) = I (identity) and dφ/dt = Aφ(t), and is used to compute the free (zero-input) response x(t) = φ(t)x(0) of the state-space system.