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
1.1.1 ExampleExample1.1A very simple RLC network shown in Figure 1.1 isRTconsidered.i(t)tutinputtothe RLCSuppose that the voltage u(t) is thenetwork. +theThis circuit contains twoelements:energy-storageinductor and the capacitor
1.1.1 Example
Kirchhoff's laws,the voltage u,(t) across theApplyingcapacitor C and the current i,(t) through the inductor Lsatisfy the following differential equations.LRdu.(t)YYi(t)dtu(tu.tdi, (t)R·it(t)+u.(t)=u(t)业dtThe voltage u,(t) across the capacitor C is considered tobe the output y(t) y(t) = u (t)
we get a second-order differential equationRLd'u.(t)du.(t)SYYLC+ RC+u.(t) = u(t)dr?dt.(tw(t)TLdifferentialequationmodelTaking the Laplace transform and assuming the zero initialconditionsholdtrue.+1U.(s)G(s)transferfunctionmodelU(s)LCs?+ RCs +1the differential eguation model and the transfer functionmodel are all externalmodels, sometimes called I/O model
make the definitions, x,(t) =u,(t) and x, (t) =i,(t)RLYYYx, (t)CR-.1t7TTand are thex;(t) and x,(t) are calledstatevariablescomponents ofthe state vector X(t) =[x() x;(t)]f. It can be expressed in matrix form notation as100x(t)x (t)c[u(t)]1stateequationRx(t)[x(t)TLL