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