线性系统的根轨迹分析法 (4).pdf

1Chapter 2 Modeling of Automatic Control Systems2.1 Differential Equations of Dynamic Systems2.2 Transfer Functions of Control Systems2.3 Block Diagrams of Control Systems and TheirEquivalent Transformations2.4 Signal Flow Graphs of Control Systems21 12 2Definition of the transfer functionTransfer function of typical elements3 3Examples of deriving the transfer function2.2 Transfer Functions of Control Systems32.2 Transfer Functions of Control SystemsThe transfer function of a system is that when the linear system is with zeroinitial condition,the ratio of the output Laplace transform to the input Laplacetransform in the differential equation.()()()Y sG sU sIn other words,the transfer function of a linear time-invariant system can beexpressed as1 1Definition of the transfer function42.2 Transfer Functions of Control SystemsThe differential equation of a linear time-invariant system can always be expressed aswhere is the input of the system;is the output of the system;are constant coefficients.1110111101()()().()()()().()nnnnnnmmmmmmd y tdy tdy taaaa y tdtdtdtd u tdu tdu tbbbb u tdtdtdt)0,0(,mjnibaji()u t()y tAssume the initial condition is zero,the Laplace transform of the above formula iscarried out to obtain11110110(.)()(.)()nnmmnnmma sasa sa Y sb sbsb sb U s1 1Definition of the transfer function52.2 Transfer Functions of Control SystemsAccording to the definition of transfer functions of a linear time-invariant system,we haveWhen the transfer function and the input are known,according to,byderiving the inverse Laplace transform of the result,the time domain expression of theoutputcan be obtained.11101110.()()().mmmmnnnnb sbsb sbY sG sU sa sasa sa()()()Y sG s U s()y t1()()()y tG s U s 1 1Definition of the transfer function62.2 Transfer Functions of Control SystemsThe transfer function is the most important mathematical model in the classical control theory.Many problems can be solved by using the transfer function in system analysis and synthesis.Without solving differential equations,the dynamic process of a system with zero initialcondition under the action of input signal can be studied.It can also be applied to study the effect of the system parameter change or structural changeon the dynamic process of the system.Therefore,the analysis of the system is greatlysimplified.The requirements for system performance can be transformed into the requirements forsystem parameters or structural forms of system transfer functions.Thus,the complexproblems will be easy to solve.1 1Definition of the transfer function72.2 Transfer Functions of Control Systems1.The rational fraction form of linear time-invariant systems isRule:The order of the system is determined by the highest order of the denominator.The order of the system in the above formula is n.11101110.()()().mmmmnnnnb sbsb sbY sG sU sa sasa sawhere are real constants,generally .)0,0(,mjnibajinm1 1Forms of transfer functions 82.2 Transfer Functions of Control Systems2.The zero-pole form of linear time-invariant systems is11101110.()()().mmmmnnnnb sbsb sbY sG sU sa sasa sawhereis the zero point of the transfer function;is the pole of the transfer function;is the transfer function gain of the zero-pole form.isz 1110111101().()mmmimmignnnnnjjszbsbsbsbKasasa saspjsp mgnbKa1 1Forms of transfer functions 92.2 Transfer Functions of Control Systems3.The form of time constant of linear time-invariant systems is11101110.()()().mmmmnnnnb sbsb sbY sG sU sa sasa sawhere1011110111(1).1.1(1)mmmimminnnnnjjsbb sbsbsKaa sasa sT s0011,ijijbKTazpare time constants respectively,is the amplification coefficient.,ijTK1 1Forms of transfer functions 101 1Forms of transfer functions 2.2 Transfer Functions of Control SystemsThe transfer function of a linear time-invariant system(in the zero-pole form)is121222112211()(2)()()()()(2)mmikkkgiknnjllljlszssKY sG sU sssps where12122,2mmmnnnProportional elementFirst-order differentialSecond-order differentialIntegral elementInertial elementOscillation elementIt can be seen from the above formulas that transfer functions are the product of some basic factors.These basic factors are transfer functions of what we call typical elements.11Typical elementDifferential equation Transfer functionProportional ElementInertial ElementIntegration ElementDifferential ElementOscillation ElementLag Element2 2Transfer function of typical elements2.2 Transfer Functions of Control Systems()G sK()()y tKu t1()KG ssTs0()()ty tKu t dt()()()Ty ty tKu t()1KG sTs()()y tKu t()G sKs2100()()()()a y ta y ta y tb u t02210()bG sa sa sa()()y tu t()sG se12Example 1.Derive the transfer function of an armature controlled DC motor.22()cammuamacdMddT TTK uKTMdtdtdtThrough the Laplace transform on both sides of the above expression,we have2(1)()()(1)()ammuamacT T sT ssK UsKT sMs2.2 Transfer Functions of Control SystemsSolution:From the first section,we obtained the differential equation of the armature controlled DC motor as follows.3 3Examples of deriving the transfer function13Example 1.Derive the transfer function of an armature controlled DC motor.Solution:Assume：,the transfer function of the system is as follows with the armature voltage as the input,and the motor speed as the output.()0cM s 2()()()1uuaammKsGsUsT T sT sAssume ,the transfer function can be obtained with the load moment as the input,and motor speed as the output.()0aU s 2(1)()()()1mamcammKT ssGsMsT T sT sFinally,using the superposition principle,the transfer function form of motor speed is()()()()()uamcsGs UsGs Ms2.2 Transfer Functions of Control Systems3 3Examples of deriving the transfer function14Example 2.Derive the transfer function of the circuit as shown in the figure.Solution:1.If the loop current in the circuit is as shown in the figure,we can separately write circuit voltage equations of both and ,as well as output voltage equation.1iThe three equations are applied with Laplace transform under zero initial condition.212112()(1)()oiUsRR CsUsR R CsRR2.2 Transfer Functions of Control Systems2i11 11 210i dtR iR iC1 21 122iR iR iR iu22oR iu11121()()()0RIsR IsCs11122()()()()iR IsRRIsUs22()()oR IsUsEliminating these two intermediate variables ,12,II3 3Examples of deriving the transfer function1RC2iiu1i2ROu15Example 2.Derive the transfer function of the circuit as shown in the figure.Solution:2.If all the components in the original circuit arerepresented in the form of complex impedance,then the systemtransfer function can be obtained by using the principle of partialpressure.2.2 Transfer Functions of Control Systems221()1()1oiUsRUsRC sR21121212112(1)(1)(1)RR C sRRR C sRR C sR R C sRR3 3Examples of deriving the transfer function1RCs1iu2ROu16A few notes on transfer functionsThe concept of the transfer function is applicable to linear time-invariant systems.Its corresponding to the differentialequations with constant coefficients in linear time-invariant systems,and it corresponds to the system dynamiccharacteristics.Transfer functions cannot reflect the disciplinary and physical properties of a system or a component.The systems withdistinct physical properties and subject categories may also have exactly the same transfer functions.And the conclusionof studying some transfer functions can also be applied to various systems with such transfer functions.Transfer functions are only related to the structure and parameters of the system,and independent of the system input.Itreflects the relationship between the input and the output,but not the relationship between the intermediate variables.The concept of the transfer function is mainly applicable to the system with single input and single output.If a systemhas multiple input signals,when finding the transfer function,except for the related signals,all other inputs areconsidered zero.Transfer functions ignore the influence of initial conditions.Transfer functions are usually the rational fractions of s.For a practical system,the order of the denominator n isalways greater than or equal to the order of the numerator m,and the system is the n-th order system.2.2 Transfer Functions of Control Systems3 3Examples of deriving the transfer function