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1、1,Ch2 Mathematical models of systems,IntroductionModels in time-domain(Differential equations)Models in frequency-domain(Transfer function)Block diagram modelsSignal-flow graph modelSimulation and Examples,2,2.1 Introduction,Mathematical models can help us to understand and control complex system ef
2、fectively.So it is necessary to analyze the relationships between the system variables and to obtain a mathematical model.Because the systems are dynamic in nature,the descriptive equations are usually differential equations,if they can be linearized,then Laplace transform can be used to simplify th
3、e method of solution.,3,Two Tasks in Control Theory,Analysis Problems:Synthesis Problem:,Given a Controller,determine if the controlled signals(including outputs,tracking error&control signal,etc.)satisfy the desired performance for all admissible external disturbance and model uncertainties.,Design
4、 a controller such that the controlled signals satisfy the desired properties for all external and inner uncertainties(such as noise,extraneous disturbance&modeling errors).,4,Modeling method,Analytical methodExperimental method,Goals of modeling,AnalysisDesign or Control,5,1 Analytical method(解析法)：
5、,Routh判据,英国建立（Routh-Hurwitz Stability Criteria）（1875年）,J.C.Maxwell formulates a mathematical model for a flyball governor,6,2 Experimental method(实验法)：,实际系统,7,Hypothesis(猜想法)：,宇宙大爆炸理论,8,Models under Consideration,Linear Time-invariant(constant)Parameter-lumped,9,2.2 Differential equations of physica
6、l systems,Through-variable and across-variableAnalogous variables,Voltage-Velocity analogyForce-current analogy,Refer to(P39-40),10,How to obtain DE?,Newtons laws for Mechanical systemKirchhoffs laws for electrical systemsTwo examples,(Refer to script 2-2),11,弹簧阻尼系统如图示：其中，f 为阻尼系数 k 为弹簧系数试确定该系统的模型。,例
7、1 机械力学系统,12,解：系统的微分方程如下 拉氏变换后（零初始条件下）,13,例2 电学系统,RLC无源网络如图所示，其中：电阻为R，电感为L，电容为C，确定其输入输出模型。,R,L,C,14,解：系统的微分方程如下 拉氏变换后（零初始条件下）,15,Solving the equation,Analytical or classical methodLaplace transform methodExample of solving differential equation,(Refer to script 2-3,4),16,Modes of dynamic system,Modes
8、 is determined by Characteristic roots:real and distinct roots real and repeated roots complex conjugate roots,17,2.3 Linear approximations of physical systems,Non-linearity is essential and pervasiveA system is defined as linear in terms of the system excitation and response.Principle of superposit
9、ionHomogeneityA linear system satisfies the properties of superposition and homogeneity,18,Linear approximations method,Taylor series expansionExample:pendulum oscillator model,Refer to next page or P45,19,Pendulum oscillator,20,2.4 The Laplace transform,Review complex algebraic(refer to P46-52):,La
10、place transform Inverse Laplace transform residue evaluation Final value theorem,21,Time response by Laplace transform,Time response solution is obtained by:1.obtain the differential equations 2.obtain the Laplace transformation of differential equations 3.solve the algebraic equation of the variabl
11、e of interest 4.obtain the response by the inverse Laplace transform,22,2.5 The transfer function of linear system,Transfer function is defined as the ratio of the Laplace transform of the output variable to the laplace transform of the input variable,with all initial conditions assumed to be zero.E
12、xamples:(refer to P53),23,Characteristics of TF（传递函数的特性）,Black-box modelrational real fractional function transition between TF and DETF and impulse response,Transfer function is the Laplace transform of the impulse response and vice versa.,24,Implication of TF（传递函数的含义）,Zero initial conditionReprese
13、ntation of TF,Inputs and outputs of system are zero at t=0-,25,TF of typical elements and dynamic systems,Gain：Zeros at Origin：Poles at Origin：Zeros：Poles：,26,Complex Conjugate Zeros：Complex Conjugate Poles：Time-Delay：,Refer to P55-63 and Table 2.5(P64-67),Examples of TF,27,Examples of TF,1.Potentio
14、meter：(线位移或角位移变换为电压量的装置)（a）,28,（b）where EVoltage of Source；maxMaximum angle of potentiometer,29,(c)Potentiometer error detector,2.DC Motor:,M,+,-,R,a,L,a,E,b,i,a,u,a,(),31,3.Tachometer：,32,4.RC passive network：,33,5.Series of RC passive network(无源网络级联),34,Series of RC passive network(without load ef
15、fect),35,6.Integrating Circuit(有源网络),36,2.6 Block diagram models（结构图或方框图模型）,Block diagrams representation is prevalent in control engineering,and consist of unidirectional operational blocks that represent the transfer function of the variables of interests.Representation of block diagram,Refer to P
16、68,37,Transformation of Block Diagram,BD=Scheme+EquationFour Components of BD(refer to 2-14,15),Signal linePickoff pointSumming pointBlock,38,Transition between TF and BD,Refer to 2-15,16,17An example of RC passive network,i,Ur,39,Block Diagram of RC passive network,40,41,42,The Block Diagram of the
17、 system,43,Characteristics of BD,Unidirectional Principle Without load effect Connection ways:,Series parallel and feedback connection,44,Block diagram reduction,Combination of blocks in cascadeCombination of blocks in parallelElimination of feedback loopMoving pickoff points and summing points,Prin
18、ciple:Keep every signal to be equivalent,45,1.Combination of blocks in cascade,46,2.Combination of blocks in parallel,47,3.Elimination of feedback loop,48,4.Moving pickoff points and summing points,a)Moving a summing point behind a block,49,b)Moving a summing point ahead of a block,50,c)Combination
19、of summing points,51,d)Moving a pickoff point ahead of a block,52,e)Moving a pickoff point behind a block,53,Examples of BD reduction,Refer to textbook(P70-72)Other examples:refer to(2-19,20,21),54,Example 1,55,-,-,-,56,Example 2,57,U,r,(s),-,-,-,G,1,G,2,G,3,H,3,H,2,H,1,-,58,U,r,(s),-,-,-,G,1,G,2,G,3,H,3,H,2,H,1,-,59,U,r,(s),-,-,-,G,1,G,2,G,3,H,3,H,2,-,60,U,r,(s),-,-,-,G,1,G,2,G,3,H,3,H,2,-,61,U,r,(s),-,-,-,G,1,G,2,G,3,H,3,H,2,-,62,Example 3,63,64,Assignment(课后作业）,Review Ch2(P32-)E2.4 E2.6E2.8E2.18,
