The State Transition Matrix
The matrix exponential propagates a linear system's state forward in time and forms the core of its solution.
Solving x' = A x
For the unforced linear system x' = A*x with initial state x(0), the solution is x(t) = e^(A*t)*x(0). The quantity e^(A*t) is the state transition matrix, often written Phi(t). It maps the state at time zero to the state at time t, and it fully describes how the system evolves on its own.
The matrix exponential
The matrix exponential is defined by the same power series as the scalar one: e^(A*t) = I + A*t + (A*t)^2/2! + (A*t)^3/3! and so on. It is not computed term by term in practice; instead it is found by diagonalizing A, using its Jordan form, or via numerical scaling-and-squaring algorithms.
Properties
- Phi(0) = I, the identity: no time, no change.
- Phi(t1 + t2) = Phi(t1)*Phi(t2): transitions compose.
- Phi(t) is always invertible, with inverse Phi(-t).
- The derivative of Phi(t) is A*Phi(t).
Forced response
With an input u(t), the full solution adds a convolution: x(t) = Phi(t)*x(0) plus the integral from 0 to t of Phi(t - tau)*B*u(tau) d tau. The first term is the free response set by initial conditions, the second is the forced response set by the input history. This variation-of-parameters formula is the state-space analogue of solving with the Laplace transform.
Eigenvalues and modes
If A has eigenvalues lambda with eigenvectors v, the transition matrix decomposes the response into modes e^(lambda*t) along each eigenvector. Eigenvalues with negative real parts give decaying modes (stability); positive real parts give growing ones. This is the state-space mirror of the pole-location stability test.
Discrete-time systems use the analogous power A^k, and the transition matrix underlies the discretization formulas that convert a continuous model into the difference equation a digital controller actually runs.