State Space Analysis (GPS) Part 2

State Transition Matrix Properties

The homogeneous part of eqn. (3.43) is

Definition 3.2 A continuous and differentiable matrix function

is the fundamental solution of eqn. (3.53) onif and only iffor all

The fundamental solutionis important since it will serve as the basis for finding the solution to both eqns. (3.43) and (3.53).

Ifthen x(t) satisfies the initial value problem corresponding to eqn. (3.53) with initial condition x(0):

Ifis nonsingular, thenand

whereis called the state transition matrix from time t to time t . The state transition matrix transforms the solution of the initial value problem corresponding to eqn. (3.53) at time t to the solution at time t. The state transition matrix has the following properties:

The general solution to eqn. (3.43) is

which can be verified as follows:

where Leibnitz rule

has been used to move derive the second equation from the first.

Linear Time-Invariant Systems

In the case that F is a constant matrix, it can be shown by direct differentiation that

The matrix exponential, its properties, and its computation are discussed in Section B.12. The Laplace transform of the impulse response is the transfer functionderived in eqn. (3.51).

An approximate method to computefor smallwhen F(t) is slowly time-varying is discussed in Section 7.2.5.2.

Discrete-Time Equivalent Models

It is often the case that a system of interest is naturally described by continuous-time differential equations, but that the system implementation is more convenient in discrete-time. In these circumstances, it is of interest to determine a discrete-time model that is equivalent to the continuous-time model at the discrete-time instants‘ for some fixed valueand k = 0,1, 2,…. Equivalence meaning that the discrete and continuous-time models predict the same system state at the specified discrete-time instants.

If F is a constant matrix, then from eqn. (3.62)

whereSimplification of the second term on the right hand side is possible under various assumptions. The most common assumption is thatG(t) is a constant vector and that u(t) has the constant valueWith this assumption, eqn. (3.64) reduces to

where

Example 3.13 For the continuous-time system described in eqn. (3.16),. Denoting the sampling time by T,can be found in closed form, via Taylor series approach described in Section B.12.1. Sinceforthe Taylor series terminates after the second term

If, in addition, the applied force is held constant at

for all k = 0,1, 2,…, then using the change of variables we obtain

Thus the discrete-time equivalent model to eqn. (3.16) is