From Modern Control Systems: An Introduction
1.9 SOLUTION OF STATE EQUATIONS FOR LINEAR TIMEINVARIANT SYSTEMS
1.9.1 State Transition Matrix (STM)
The state equation of a linear timeinvariant system is given by
For a homogeneous (unforced) system
we have
Take the Laplace Transform on both sides
Equation (1.20) may also be written as
Take the inverse Laplace
Equation (1.21) gives the solution of the LTI homogeneous state Equation (1.19). From Equation (1.21) it is observed that the initial state x(0) at t = 0, is driven to a state x( t) at time t. This transition in state is carried out by the matrix exponential e ^{ A t}. Because of this property, e A ^{t} is termed as the State Transition Matrix and is denoted by ( t).
Thus,
where, is called the Resolvent Matrix.
As e ^{ A t} represents a power series of the matrix A t, thus,
1.9.1.1 Properties of the State Transition Matrix

?(0) = I.
Proof. ?(t) = e ^{ At}
Put t = 0
We have ?(0) = I.

? ^{1}(t) = ?(t)
Proof. ?(t) = e ^{ At}
Postmultiply both sides by e ^{At}
We have ?(t)e ^{ At} = e ^{ At}.e ^{ At}
or, ?(t) ?(t) = I
Premultiply both sides by ? ^{1}( t).
We have ? ^{1}( t) ?(t) ?(t) = ? ^{1}( t)
or, ?
Products & Services
Loop powered devices are electronic devices that can be connected in a transmitter loop, normally a current loop, without the need to have a separate or independent power source. Typical loop powered devices include sensors, transducers, transmitters, isolators, monitors, PLCs, and many field instruments.
Topics of Interest
1.10 SOLUTION OF STATE EQUATIONS FOR LINEAR TIMEVARYING SYSTEMS 1.10.1 State Transition Matrix The state equation of a linear timevarying system is given by For a homogeneous (unforced) system...
5.9 The State Transition Matrix Let us again consider the state equations pair where for two or more simultaneous differential equations A and C are 2 2 or higher order matrices, and b and d are...
5.9 The State Transition Matrix Let us again consider the state equations pair where for two or more simultaneous differential equations A and C are 2 2 or higher order matrices, and b and d are...
5.3 Solutions of Ordinary Differential Equations (ODE) A function y = f (x) is a solution of a differential equation if the latter is satisfied when y and its derivatives are replaced throughout by...
5.3 Solutions of Ordinary Differential Equations (ODE) A function y = f( x) is a solution of a differential equation if the latter is satisfied when y and its derivatives are replaced throughout by f...