Example 1.4 Find a state space model for the systemdescribedby the following differential equation+16j+194j+640y=4u+160u+720uSolutionFor the given system,n=3, and thosecoefficientsinvolvedareα =16,αz =194，a,=640 ；b。=4,b, =0,b,=160，b,=720Theparameters β, (i= 0,..-,3)canbe constructedasβ。 =b。= 4β = b - αiβ。= 0-16 ×4 =-64β, = b, -α,β -α,β。=160 -16 ×(-64)-194 × 4 = 408β, =b, -a,β, -αzβ -a, β。= 720 -16×408 -194 ×(-64)-640 × 4= 4048
Example 1.4 Find a state space model for the systemdescribedby the following differential equationj+16j+194j+640y=4u+160u+720uSolution[β。= b。= 4β = b -αβ。= 0 -16 × 4 = -64β, = b, -a,β -a,β。=160 -16 ×(-64) -194 × 4 = 408β, =b, -a,βz -aαzβ -a,β。=720 -16×408 -194 ×(-64)-640 × 4 = 4048Thestatevariablescanbedefinedbyxi=y-βoux=j-βou-βux=j-βoui-βu-βzu
Example 1.4 Find a state space model for the systemdescribed by the following differential equationj+16j+194j+640y=4+160u+720uSo, the state space model of system can be obtained asSolution100[B.]0X=01β2X+uLB.-a3-a2-a010-64001X+408uV-164048-640-194y=[1 0 o]X+[β.]u=[1 0 o]+4uWhere X=[x x, x,]
010-640X=01X +408y=[1 0 ]X+4u1-164048-640-19464408uX28"404816194640
1.2.2 Obtaining State Space Model fromTransfer FunctionThe general transfer function model of a sISO LTI n-ordersystem is shown asb,s" + b,s-1 + b,sm-2 +...+b-s + b.Y(s)G(s) :U(s)s" +as"-I +.+an-is+anThe system, is called proper rational system if m ≤n . Bythe way, it is called strictly proper rational system ifm<n.In this section, two methods for finding the state space modelfrom transfer function of a system will be discussed
1.2.2 Obtaining State Space Model from Transfer Function