State-transition matrix

In control theory, the state-transition matrix is a matrix whose product with the state vector x {displaystyle x} at an initial time t 0 {displaystyle t_{0}} gives x {displaystyle x} at a later time t {displaystyle t} . The state-transition matrix can be used to obtain the general solution of linear dynamical systems. In control theory, the state-transition matrix is a matrix whose product with the state vector x {displaystyle x} at an initial time t 0 {displaystyle t_{0}} gives x {displaystyle x} at a later time t {displaystyle t} . The state-transition matrix can be used to obtain the general solution of linear dynamical systems. The state-transition matrix is used to find the solution to a general state-space representation of a linear system in the following form where x ( t ) {displaystyle mathbf {x} (t)} are the states of the system, u ( t ) {displaystyle mathbf {u} (t)} is the input signal, and x 0 {displaystyle mathbf {x} _{0}} is the initial condition at t 0 {displaystyle t_{0}} . Using the state-transition matrix Φ ( t , τ ) {displaystyle mathbf {Phi } (t, au )} , the solution is given by: The first term is known as the zero-input response and the second term is known as the zero-state response.