西安理工大学自动控制理论双语chap .pptx

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1、Chapter 2 mathematical models of systems2.2.1 Examples2.1.4 types 1)Differential equations 2)Transfer function 3)Block diagram、signal flow graph 4)State variables2.2 The input-output description of the physical systems differential equations The input-output descriptiondescription of the mathematica

2、l relationship between the output variable and the input variable of physical systems.第1页/共38页Chapter 2 mathematical models of systems define:input ur output uc。we have：Example 2.1:A passive circuit 第2页/共38页Chapter 2 mathematical models of systemsExample 2.2:A mechanismDefine:input F，output y.We hav

3、e:Compare with example 2.1:ucy,urF-analogous systems第3页/共38页Chapter 2 mathematical models of systems Example 2.3:An operational amplifier(Op-amp)circuitInput ur output uc(2)(3);(2)(1);(3)(1)：第4页/共38页Chapter 2 mathematical models of systems Example 2.4:A DC motorInput ua，output 1(4)(2)(1)and(3)(1):(4

4、)(2)(1)and(3)(1):第5页/共38页Chapter 2 mathematical models of systemsMake:第6页/共38页Chapter 2 mathematical models of systems Assume the motor idle:Mf=0,and neglect the friction:f=0,we have:the differential equation description of the DC motor is:Compare with example 2.1 and example 2.2:-Analogous systems第

5、7页/共38页Chapter 2 mathematical models of systemsExample 2.5:A DC-Motor control systemInput ur,Output;neglect the friction:第8页/共38页Chapter 2 mathematical models of systems（2）（1）（3）（4），we have：2.2.2 steps to obtain the input-output description (differential equation)of control systems1)Identify the out

6、put and input variables of the control systems.2)Write the differential equations of each systems component in terms of the physical laws of the components.*necessary assumption and neglect.*proper approximation.3)dispel the intermediate(across)variables to get the input-output description which onl

7、y contains the output and input variables.第9页/共38页Chapter 2 mathematical models of systems4)Formalize the input-output equation to be the“standard”form:Input variable on the right of the input-output equation.Output variable on the left of the input-output equation.Writing the polynomialaccording to

8、 the falling-power order.2.2.3 General form of the input-output equation of the linear control systems A nth-order differential equation:Suppose:input r，output yE2.14,P2.2,P2.3,P2.7第10页/共38页Chapter 2 mathematical models of systems2.3 Linearization of the nonlinear components2.3.1 what is nonlinearit

9、y？The output of system is not linearly vary with the linear variation of the systems (or components)input nonlinear systems(or components).2.3.2 How do the linearization？Suppose:y=f(r)The Taylor series expansion about the operating point r0 is:第11页/共38页Chapter 2 mathematical models of systemsExample

10、s:Example 2.6:Elasticity equation Example 2.7:Fluxograph equationQ Flux;p pressure difference第12页/共38页Chapter 2 mathematical models of systems2.4 Transfer function Another form of the input-output(external)description of control systems,different from the differential equations.2.4.1 definition Tran

11、sfer function:The ratio of the Laplace transform of the output variable to the Laplace transform of the input variable with all initial condition assumed to be zero and for the linear systems,that is:第13页/共38页Chapter 2 mathematical models of systemsC(s)Laplace transform of the output variable R(s)La

12、place transform of the input variable G(s)transfer function*Only for the linear and stationary(constant parameter)systems.*Zero initial conditions.*Dependent on the configuration and coefficients of the systems,independent on the input and output variables.2.4.2 How to obtain the transfer function o

13、f a system1)If the impulse response g(t)is knownNotes:第14页/共38页Chapter 2 mathematical models of systems Example 2.8:2)If the output response c(t)and the input r(t)are knownWe have:Because:We have:Then:第15页/共38页Chapter 2 mathematical models of systems Example 2.9:Then:3)If the input-output differenti

14、al equation is known Assume:zero initial conditions;Make:Laplace transform of the differential equation;Deduce:G(s)=C(s)/R(s).第16页/共38页Chapter 2 mathematical models of systemsExample 2.10:4)For a circuit*Transform a circuit into a operator circuit.*Deduce the C(s)/R(s)in terms of the circuits theory

15、.第17页/共38页Chapter 2 mathematical models of systems Example 2.11:For a electric circuit:第18页/共38页Chapter 2 mathematical models of systemsExample 2.12:For a op-amp circuit第19页/共38页Chapter 2 mathematical models of systems5)For a control system Write the differential equations of the control system;Make

16、 Laplace transformation,assume zero initial conditions,transform the differential equations into the relevant algebraic equations;Deduce:G(s)=C(s)/R(s).Example 2.13the DC-Motor control system in Example 2.5第20页/共38页Chapter 2 mathematical models of systems In Example 2.5,we have written down the diff

17、erential equations as:Make Laplace transformation,we have:(2)(1)(3)(4),we have:第21页/共38页Chapter 2 mathematical models of systemsE2.2,E2.6,E2.15,E2.19,E2.20,E2.27,P2.7,P2.8第22页/共38页Chapter 2 mathematical models of systems2.5.1 Proportioning elementRelationship between the input and output variables:T

18、ransfer function:Block diagram representation and unit step response:Examples:amplifier,gear train,tachometer2.5 Transfer function of the typical elements of linear systems A linear system can be regarded as the composing of several typical elements,which are:第23页/共38页Chapter 2 mathematical models o

19、f systems2.5.2 Integrating elementRelationship between the input and output variables:Transfer function:Block diagram representation and unit step response:Examples:Integrating circuit,integrating motor,integrating wheel第24页/共38页Chapter 2 mathematical models of systems2.5.3 Differentiating elementRe

20、lationship between the input and output variables:Transfer function:Block diagram representation and unit step response:Examples:differentiating amplifier,differential valve,differential condenser第25页/共38页2.5.4 Inertial elementChapter 2 mathematical models of systemsRelationship between the input an

21、d output variables:Transfer function:Block diagram representation and unit step response:Examples:inertia wheel,inertial load(such as temperature system)第26页/共38页Chapter 2 mathematical models of systems2.5.5 Oscillating elementRelationship between the input and output variables:Transfer function:Blo

22、ck diagram representation and unit step response:Examples:oscillator,oscillating table,oscillating circuit第27页/共38页2.5.6 Delay elementChapter 2 mathematical models of systemsRelationship between the input and output variables:Transfer function:Block diagram representation and unit step response:Exam

23、ples:gap effect of gear mechanism,threshold voltage of transistors第28页/共38页.2.6.1 Block diagram representation of the control systemsChapter 2 mathematical models of systemsExamples:2.6 block diagram models(dynamic)Portray the control systems by the block diagram models more intuitively than the tra

24、nsfer function or differential equation models第29页/共38页Example 2.14 Chapter 2 mathematical models of systemsFor the DC motor in Example 2.4 In Example 2.4,we have written down the differential equations as:Make Laplace transformation,we have:第30页/共38页Chapter 2 mathematical models of systemsDraw bloc

25、k diagram in terms of the equations(5)(8):Consider the Motor as a whole:1)(12+ffemmeeTsTTTsTTC1)()(12+ffemmemmeTsTTTsTTTsTTJUa(s)(sW)(sM-第31页/共38页Chapter 2 mathematical models of systemsExample 2.15The water level control system in Fig 1.8:第32页/共38页Chapter 2 mathematical models of systemsThe block d

26、iagram model is:第33页/共38页Chapter 2 mathematical models of systemsExample 2.16The DC motor control system in Fig 1.9第34页/共38页Chapter 2 mathematical models of systemsThe block diagram model is:第35页/共38页Chapter 2 mathematical models of systems2.6.2 Block diagram reduction purpose:reduce a complicated block diagram to a simple one.2.6.2.1 Basic forms of the block diagrams of control systemsChapter 2-2.ppt第36页/共38页Chapter 2 mathematical models of systems第37页/共38页感谢您的观看！第38页/共38页