discretization of continuous systems

Consider a continuous-time system with the following state space representation

P:  {x˙(t)=Ax(t)+Bu(t),y(t)=Cx(t)+Du(t), (1)

where x(t)n, u(t)r and y(t)m are the state vector, input vector and output vector of the system, respectively; An×n, Bn×r, Cm×n and Dm×r are the constant real or complex matrices.

Suppose that the sampling interval is τ. By using the step invariance transform or the zero-order hold (ZOH), i.e., u(t)=u(kτ),kτt<(k+1)τ, discretizing the system in (1) gives a discrete-time model,

Pτ:  {x(kτ+τ)=Gτx(kτ)+Fτu(kτ),y(kτ)=Cx(kτ)+Du(kτ),k=0,1,2, (2)

where x(kτ)=x(t)|t=kτ, y(kτ)=y(t)|t=kτ, and

Gτ:=eAτ,Fτ:=0τeAtdtB. (3)
Title discretization of continuous systems
Canonical name DiscretizationOfContinuousSystems
Date of creation 2013-03-22 15:50:45
Last modified on 2013-03-22 15:50:45
Owner mathcam (2727)
Last modified by mathcam (2727)
Numerical id 7
Author mathcam (2727)
Entry type Topic
Classification msc 93C55
Synonym Transform