Basic ModelingMethods
Basic Modeling Methods
References: Automatic Control Systems, 8th Edition, B.C.KuoF. Golnaraghi, John Wiley & Sons, 2002Matlab/Simulink
References • Automatic Control Systems, 8 th Edition, B.C. Kuo, F. Golnaraghi, John Wiley & Sons, 2002 • Matlab/Simulink
What are mathematics models forphysical systems?They are empirical representations of a physical system'sinput/output relationships and internal behavior by usingmathematics expressions.Themodels can be in differentforms:FunctionsDifferential equationsState space modelsTransfer functionsBlock diagramSimulation modulesThe models can be obtained by different methodsDerivations based on basic principles Experimental data and model fitting Real-time updatinglearning
What are mathematics models for physical systems? • They are empirical representations of a physical system’s input/output relationships and internal behavior by using mathematics expressions. • The models can be in different forms: • Functions • Differential equations • State space models • Transfer functions • Block diagram • Simulation modules • The models can be obtained by different methods: • Derivations based on basic principles • Experimental data and model fitting • Real-time updating • learning
Why do we need mathematics models?:They are cost effective in studying the main features ofphysical, environment, social systemsExamples: Battery SOC estimation, vehicle fuel economy andemission, power system security and reliability, ...They can be used to predict the future behavior of thephysical, environmental, and social systemExamples: Covid infection prediction, trafficpatterns, ... They can be used to evaluate different controls, designs andimpact of decisions.Examples: battery management systems, control systems formotors, autonomous vehicles, ... They can be used to coordinate component designs fromdifferent teams and companies.Examples: autonomous vehicles, battery managementsystems,
Why do we need mathematics models? • They are cost effective in studying the main features of physical, environment, social systems. Examples: Battery SOC estimation, vehicle fuel economy and emission, power system security and reliability, . • They can be used to predict the future behavior of the physical, environmental, and social system. Examples: Covid infection prediction, traffic patterns,. • They can be used to evaluate different controls, designs and impact of decisions. Examples: battery management systems, control systems for motors, autonomous vehicles, . • They can be used to coordinate component designs from different teams and companies. Examples: autonomous vehicles, battery management systems,
Derivation ofDifferential Equation Models
Derivation of Differential Equation Models
Derivation of Models for Electrical Systems(1) Start from the basic circuit component principle:BasicVoltage-CurrentRelationsv(t) = Ri(t)Resistor of R (ohms):V(s) = R I(s)cdv() =i(t)= C(sV(s) - v(0) = I(s)Capacitor of C (farads):dtI(s) + v(0) -[1(s) +Cv(0)]V(s) =SCsI di(t)Inductor ofL (henries):v(t)dt(0)V(s) = LsI(s) - Li(O) = Ls I(s)1S
Derivation of Models for Electrical Systems dt C dv(t) = i(t) C(sV (s) − v(0)) = I (s) Resistor of R (ohms): Capacitor of C (farads): Cs s Cs Inductor of L(henries): V(s) = 1 I(s) + v(0) = 1 I(s) +Cv(0) s V (s) = LsI(s) − Li(0) = Ls I (s) − i(0) v(t) = Ri(t) V(s) = R I(s) dt v(t) = L di(t) (1) Start from the basic circuit component principle: Basic Voltage-CurrentRelations:
V=-RIVRV= RIVR1dtdtdlVdtdt
+V I -+VI -+V I R V = RI CL dt I = C dV dt V = L dI +V I -+VI -+V I R V = −RI CL dt I = − C dV dt V = − L dI
V=RIVRV=-RI1dvdtdtdlVdlVdtdt
-V I +-V+ I +-V I R V = −RI CL dV I = −C dt dt V = − L dI I -V+-V+ I +-V I R V = RI CL dt I = C dV dt V = L dI
(2)BuildUp Circuit InterconnectionsExample: The RC-Branch Model in a BatteryRpVocvVpR,dtpdvdv2PRdtdtRD
Rp R v i vocv vp + - Cp ip - + p p p p p p p p p dvp , vp R dvp dvp vp C dt R dt R C C i =C = i −i dt = i − vp = − + 1 i (2) Build Up Circuit Interconnections Example: The RC-Branch Model in a Battery
Initial Condition Response (Zero-Input Response)i = O, the initial conditionis v, (O)dyVRpCp v(t)=v(0)e二dtR,C.pDT,= R,C,= Time ConstantIf C, is small, then the time constant is small= The initial consdition response will go down to zero relativelyfast= The RC branch will reach the steady state fastSteady State of theRC Branch(after the initial condition response diminishes)dyP= O and v(o) is now a constant.The steady state meansdtdyvFromwehave i, =0dt(8)VFromi-i, =i, we have v,(o)= R, iR,p
p p RpCp p p p p dv v dt R C i = 0, the initial condition is vp (0) − t = − v (t) = v (0)e Initial Condition Response (Zero-Input Response) Tp = RpCp = Time Constant If Cp is small, then the time constant is small. The initial consdition response will go down to zero relativelyfast The RC branch will reach the steady state fast. Steady State of the RC Branch (after the initial condition response diminishes) p p p p p p p p dt dt R dv The steady state means p = 0 and v () is now a constant. Fromi = C dvp , wehave i = 0 v () From p = i −i = i, we have v () = R i