Discretization of state equations for continuous time-varying systems

Discretization of state equations for continuous time-varying systems

To solve the state equation of continuous-time system by computer, it is necessary to turn its state equation into discrete equation.

Assumptions: (1) t=kt,t is sampling period, and very small,k=0,1,2... to a positive integer.

(2) U (t) is only discretized at sample time, i.e. at kt≤t≤(k+1) t,u (t) =u (KT) = Constants

This paper is mainly aimed at the discretization of linear time-varying systems, in a certain degree, the linear system is only one of the most special forms of linear time-varying systems, and nonlinear systems are usually linearized when solved, so the main problem is to generalize the solution of linear time-varying systems, which has high generality and practicability.

First, introduce the main formula of linear time-varying system.

Induction: discretization of continuous State equations

Three Add an example of a linear stationary system

Four Finally, to add a little bit

the above method is to book a sampling time first T, and then solve the equation in the bring-in formula. That would be appropriate for a more general situation, as described at the outset. First, then the integration of the main problem is to find the state transfer matrix , the following describes several methods to solve the state transfer matrix.

1. Series Summation method

2. Jordanian norm-form law

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Discretization of state equations for continuous time-varying systems