CHAPTER1 STATE SPACE MODELCONTENT> 1.1 Definition of State Space> 1.2 Obtaining State Space Model from I/O Model> 1.3ObtainingTransferFunctionMatrixfromStateSpace Model>1.4 ModelofCompositeSystems> 1.5 State Transformation of the LTI system>1.6Obtaininga Jordan CanonicalFormby StateTransformation
CHAPTER1 STATE SPACE MODEL • CONTENT 1.1 Definition of State Space 1.2 Obtaining State Space Model from I/O Model 1.3 Obtaining Transfer Function Matrix from State Space Model 1.4 Model of Composite Systems 1.5 State Transformation of the LTI system 1.6 Obtaining a Jordan Canonical Form by State Transformation
1.2 Obtaining State Space Model from I/O ModelState space Model can be developed directly from a physicalsystem, such as electrical network and mechanical system.we can also build up the state space model from differentialequation and transfer function, which are two kinds ofexternal models of a system. In other words, we can build upthe internal model of system from its external model. It is, infact,aRealizationproblem
1.2 Obtaining State Space Model from I/O Model
1.2.1 Obtaining State Space Model fromDifferential EquationThe general differential equation model of an n-order SISOLTI system is shown as +y(n) + ay(n-1) + .. + an-ij+ any= b,u(n) +b,u(o-1) + ..+ bn-iu+b,uFirstly, several parametersβ,（i=0,,n ) may beconstructed with the coefficients, +[β。= boβ, = b1-atβoβ, = b, -a,β -αzββn-- =br-1 -a,βr-2 -α, βn-3 -..-an-2β, -an--βoβ, =bn -a,βn-1 -a,βn-2 -..-an-2β, -an--β -a,β
1.2.1 Obtaining State Space Model from Differential Equation
1.2.1 Obtaining State Space Model fromDifferential Equation[β。= boβ, = bi -αrββ, =b, -aβ-azββn-1 = bn-1 -aiβn-2 -α,βr-3 -..-an-2β, -an-1βoβ, = bn -a,βn-1 -a,βn-2 -...-an-2β, -an--β -anβ[b。= β。bi = β +arβob, = β, +aβ +a2βbn-1 =βn-1 +a,βn-2 +a2βn-3 +..-+an-2β +an-1βb,=β, +a,β-1 +a, βn-2 +..-+an-2β, +an-1β,+anβ
1.2.1 Obtaining State Space Model from Differential Equation
[b。= β。b, = β +a,βb2 =β, +aβ +a,βbn-1 = βn-1 +aiβn-2 +a2βn-3 +..-+an-2 P +an-1βob, =β, +a,βn-1 +a,βn-2 +..-+an-2β, +an-1β +anβSo, the state variables may be defined byXi= y-βux2 =j-βu-βux, =j-βu-β,u-β,uXn-1 = y(n-2) - β,u (n-) -. - β,n-3t -β,-2ul[, = y(-1) - βeou(-1) -..- βn-2 i - Bn-u
xi= y-βux, =j-βu-βux=j-βi-βu-βun-1 = y(n-2) - β,u (n-) -.- β,-,t -β,-_ [xn = y(o-1) - βou(r-1) ....-βn-2i-βn-1useveral phase variables y, j, ., ,(r-1)can be developedy=x + βuj=x +βu+βuj=x, +βi+βu+β,un(n-2)1+ β,uor-2) + ..+ βμsu + βe,-2-uEx.1(n-1)) = , + β,ur-1) +..+ β,--u+ β,-u(n) +ay(n-1)+..+an-1j+any=bum) +b,u(n-1) +...+bn-ju+b,u
() =-{ax, +axn- +.+an--+ +a,x)+ βBu) + βu(n-1) ++ β.-u+β,uy=x +βuj=x, +βu+ βuj=x, +βu+Bu+βuyon-2) = X- + Bou-2) +.+ βu+ β2u,(r-1) = , + β,u-1) +.-+ βn-i + β,uxi= y-βouxz =j-βu-βux=j-βi-βu-β,uDifferentiating x- = (-2) -βB,u-2) -.-- β,ju- β-u[, = ,(-1) - eou (-1) .. - β,- i - β,-
x=j-βu=(x+u+Bu)-βi=x +Bux=j-βi-βu=x+βux =x+βu:美xn-1 = xn +βn-iux, = J() - βpu() _..- β,2 - β,-ri=-(ax, +a2Xm-1 + .-+an-ix + a,++βou( + βu(r-)+..+ βu + β,u - βoun ...- β-ii - βui=(-a, -ar-r, -..-a,xn-1 -axn)+β,uy= xi + βuwe can obtain the state space model of the system in vectornotation
(n) +a1y(on-1)+...+an-1ij+any= b,u() +bu(n-1) +.+bn-u+ b,ucompanion matrix010β,00x×200β,001In-)00000βsX3u+-.-100001βr-1XmXn-1专B.an小→ar-lan-2aastateequationXiX2X30+[β. ] u００...0output equationXn-lXn
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The state spacemodel and can be illustrated asoA71...十了妆a1