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Linear System TheoryLecture 1IntroductionContentsnIntroduction to controlnDevelopment of control theorynSteps to design controlnMathematical model of systemnHow to build mathematical model of a systemnRelation of different modelsnBackground knowledgeMilestones in development of Control TheorynClassical Control Theory:nBefore 1950s.nCharacterized by Transfer Function analysis method.nNot very good for multi-variable or large-scale systemnRoot Locus,Frequency Response methods are extensively usednVery much dependent on design experienceMilestones in development of Control TheorynModern Control Theory:nAfter 1960s.nCharacterized by state variable analysis method.nVery good for multi-variable or large-scale system,and can be easily extended to time-variable system or even non-linear systemnSystematic methods are developed to analyze the controllability,observability and stabilitynPerformance can be clearly specified and analytical methods are developed to accomplish the designMilestones in development of Control TheorynLarge-scale System Theory and Intelligent ControlnAfter 1970snProposed for dealing with large-scale,complex,and hierarchical control of large system.nHighly intelligent,adaptive and robustControl ExamplesFlyingball control boilerManual control fluidControl ExamplesBoiler-generator ControlControl ExamplesPower Generation ControlModeling of physical systemVerificationSimulation/VisualizationValidationA control modelRoughly speaking,control system design deals with the problem of making a concrete physical system behave according to desired specificationsOpen-loop controlClosed-loop controlDefinition of“system”nDefinition:nLiterally:a group or combination of interrelated,interdependent,or interacting elements forming a collective entity;a methodical or coordinated assemblage of parts,facts,concepts,etc.nHere:Mathematical description of a relationship between externally supplied quantities(I.e.,those coming from outside of the system)and the dependent quantities that result from the action or effect on those external quantitiesSystem is described by“model”,which is a group of differential equations(partial or ordinary),or algebraic equations concerning relevant variablesWhy we need to study a system?nBecause we want to control it.What is control?nRoughly speaking,control is to make a system behave like we desire.Essential elementsSome termsnSISO:Single-Input-Single-OutputnMIMO:Multiple-Input-Multiple-OutputnSIMO:nMISO:Some terms-MemorynA system with memory is one whose output depends on itself from an earlier point in time;nE.g.,Capacitor C,Inductor LnA memoryless system is one whose output depends only on the current time and current input.E.g.,Resistor RSome terms-CausalitynA system is said to be causal if the value of the output at time t0 depends on the values of the input and output for all t0 up to t0,i.e.,t=t0nEvery practical system is causalnFor easy understanding,this means that we cannot use anyway to change what have already happened.Some termsstate variablenState Variable:a set of variables that define the system response.Once the initial conditions are given,the differential equations completely characterize any chosen output function from the initial time for any admissible input function.Vc,IL:state variableVR,IL:not state variableSome termsLumpednessnLumpedness:A system is said to be lumped if its number of state variable is finiteLimited state variablesInfinite state variablesSome terms-Time InvariancenTime Invariance:A time-invariant system is one whose output depends only on the difference between the initial time and the current time,y=y(t-t0).Otherwise,the system is time-variantnFor your easy understanding,TI means the response of a system is independent of the time when an excitation is applied.nTime invariance is a mathematical fiction.No man-made electronic system is time invariant in the strict sense.y(t)=(H(x)(t)y(t-)=(H(x)(t-)y(t-)=(H(x,)(t-)Some terms-LinearitynLinearity:A system is linear if it satisfies the superposition property,i.e.,The output of a system to the input f(t)is y(t)=Prove,or give an example as to whether the system is:A.LinearB.CausalC.Time invariantSome termsnZero-state responsenOutput excited exclusively by the inputnZero-input responsenOutput excited exclusively by the initial stateSuperposition property of linear systemnFor a linear system,nResponse=zero-input response+zero-state responsenTherefore,the overall response can be decomposed and studied separatelyLinearization of non-linear systemTaylor series linearizationFor multivariable case,Mathematical description of a systemMathematical description of a systemInput-output Description:For a causal systemMathematical descriptionnFor multivariable case,the G is a matrix,Where,Mathematical descriptiontransfer functionContinuous systemDiscrete systemZero-pole:Mathematical description-State space modelnEvery linear lumped system can be described by a set of state equation,nFor a Linear Time Invariant(LTI)system,the transfer function satisfies time shifting property,i.e.,Mathematical description-State space modelnHence the state equation has the form,(constant coefficients)Laplace TransformnDefinition:nAfter Laplace transformation,a system can be described,Expamplesntext example 2.2,2.3,2.4,2.5Properties of polynomialnLaplace transform transfers a transfer function(of a lumped system)into a fraction of two polynomialsn is said to be proper if nis said to be strictly proper ifnis biproper ifnis improper ifZeroes and polesnA rational transfer function can be decomposed into a form like,nzi and zj are not necessarily different and ,therefore zi is a zero of npi and pj are not necessarily different and ,therefore pi is a pole of Laplace transform of state space equationnFor an LTI system,nIts Laplace transform is,nSolve and ,Relationship between input-output and state space descriptionnHence,for a relaxed system(x(0)=0),nTherefore,the transfer function,How to build a model of a systemAll models are wrong,but some models are useful-G.E.BoxCase study:how to model a systemFree body motion and Newtons law:State variable modelPhysical modelMathematical modelState variableinputoutputTransfer function modelSimulation(step response)nM1=1 kg nM2=0.5 kg nk=1 N/sec nF=1 N nu=0.002 sec/m ng=9.8 m/s2 nM1=1;nM2=0.5;nk=1;nF=1;nu=0.002;ng=9.8;nnum=M2 M2*u*g 1;nden=M1*M2 2*M1*M2*u*g M1*k+M1*M2*u*u*g*g+M2*k M1*k*u*g+M2*k*u*g;nA=0 1 0 0;n -k/M1 -u*g k/M1 0;n 0 0 0 1;n k/M2 0 -k/M2 -u*g;n B=0;n 1/M1;n 0;n 0;n C=0 1 0 0;n D=0;nsys=ss(A,B,C,D);nstep(sys,0:0.1:200);Balance between external force and frictionGeostationary orbit satellitenAssume the satellite only move in equator plane.nControl goal:to limit the satellite in a fixed position with respect to the devices on earth.Desired geostationary orbit:Equilibrium:Linearization:Linear state space model:
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