Time-Domain Solution of LTI State Equations

that is, as a set of coupled, first-order differential equations. The solution proceeds in two steps; first the state-variable response x(t) is found by solving the set of first-order state equations, Eq. (1), and then the state response is substituted into the algebraic output equations, Eq. (2) in order to compute y(t). As in the classical solution method for ordinary differential equations with constant coefficients, the total system state response x(t) is considered in two parts: a homogeneous solution xh(t) that describes the response to an arbitrary set of initial conditions x(0), and a particular solution xp(t) that satisfies the state equations for the given input u(t). The two components are then combined to form the total response. The solution methods used in this note rely heavily on matrix algebra. In order to keep the treatment simple we attempt wherever possible to introduce concepts using a first-order system, in which the A, B, C, and D matrices reduce to scalar values, and then to generalize results by replacing the scalars with the appropriate matrices.