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
make the definitions, x,(t)=u,(t) and x,(t) =i,(t)RLYYY100x(t)cx (t)一11.u.(t)1R[x (t)x(t)LLLstateequationy(t) = u.(t)It can be expressed in matrix form notation asx()y(t)=[1 0output equationx(t)
100>x(t)x (t)c1[u(t)]state equation十1R[x (0)x,(t)LZLx(t)output equation0y(t) =[1x(t)x(t)x(t)X(t) =X(t) =x, (t)x,(t)It can also be expressed in a more compact form asX(t) = AX(t) + bu(t)state equationstate space modeloutput equationy(t) = cX(t) + du(t)The state space model is an internal model of system
1.1.2 Several DefinitionsDefinition 1.1 The state of a system is a minimum set ofvariables x,, called state variables, to describe the system'sbehavior.(1) The components of state, which are those state variablesare linearly independent at any instant, in other words, x,cannot be expressed linearlyby x,, j+ i,i, j =l,2,..,n
1.1.2 Several Definitions
1.1.2 Several DefinitionsDefinition 1.1 The state of a system is a minimum set ofvariables x,, called state variables, to describe the system'sbehavior. +(2) The initial values of this set and the inputs of system forte (to,co) are sufficient to describe uniquely the system'sresponse for all t≥t.. The state at any fixed time can berepresented by the values of a set of state variablesx,,i = 1,2, .., n
1.1.2 Several Definitions
1.1.2 Several DefinitionsDefinition 1.1 The state of a system is a minimum set ofvariables x,, called state variables, to describe the system'sbehavior. +(3) The state of a system can be represented by an n.x,()Tdimension state vector X(t) =x (t) x, (t)and the dimension of state vector is called the dimension ortheorderofthe system
1.1.2 Several Definitions