Chapter 2 Mathematical Models of Systems重点掌握：微分方程传递函数 系统结构图及信号流图 梅逊公式1Main contentsnDifferential Equations of Physical Systems.nThe Laplace Transform and Inverse Transform.nThe Transfer function of Linear Systems.nBlock Diagram and Block Diagram Reduction.nSignal-flow Graph and Masons gain formula2Definition of Mathematical model of system Mathematical model:Descriptions of the behavior of a system usingmathematics.描述系统的输入、输出变量以及系统内 部各个变量之间的数学表达式。3Types of mathematical models1、Differential Equation2、Transfer Function3、Frequency Response4、State Equation5、Difference Equation4Differential Equations of Physical SystemsThe differential equations describing the dynamic performance of a physical system are obtained by utilizingthe physical laws of the process. How to get the differential equations of physical systems?Step1:确定系统中各元件的输入、输出变量。Step2:按信号传递顺序列写微分方程。Step3:化简（线性化、消去中间变量），写出输入、输出变 量间的数学表达式。5Differential Equations for Ideal Elements(1) Electrical Resistance UiR 6(2) Electrical Capacitance CUi(3)Electrical InductanceLi(4) Mass blockFvMU7(5) Springkx1x2F(6) Damperbv1v2F8examplesExample1 :RLC circuit RLCr(t)c(t)i(t)9Example 2:mass-spring-damperr(t)ykb MMyr(t)ky10Linear Approximations of Physical SystemsA linear system satisfies the properties of superposition and Homogeneity: (Principle of Superposition).What is the linear system?满足叠加原理的系统称为线性系统。叠加 原理又可分为可加性和齐次性。11Principle of superpositionSuperposition PropertysystemsystemsystemHomogeneity Propertysystemsystem12Example (1)(2)(3)Does not satisfy the homogeneityproperty Does not satisfy the superpositionpropertyWhen and Equation (2) can be rewrittenasWe haveor 13Linearization of Weak Nonlinear Characteristic14The output-input nonlinear characteristic of y=f(x) is illustrated in the following figure: Linearization using Taylor series expansion about the operating point( Equilibrium Position)15So we get:Set y=f(x)-f(x0), so we haveSet We get y=kxOr y=kx16The Laplace TransformDefinitionIf a function of time,f(t),satisfyWe have the Laplace transformation for function f(t),is17The Laplace variable s can be considered to be differential operatorso that,we have18Important Laplace Transform Pairs19Inverse Laplace TransformationInverse Laplace transformation can be denotedor20Important Theorems of Laplace Transform(1) linearity(2) differentiation21(3) Shife in Time(4) Complex Shifting(5)Initial-Value Theorem(6) Final-Value TheoremIF sF(s) does not have poles on or to the right of the imaginary axis in the s-plane. 22Solve the differential equations using the Laplace transformExample 1The Laplace transform of the equation iswhen ,23We can getwhen,Then Y(s) becomes(1)The inverse Laplace transform of Eq.(1)is24Example 2When the initial conditions are,andThe Laplace transform,we obtainso25Example 3: Consider the functionCalculate g(t).answer26The transfer function of linear systemsTransfer function: 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. definition零初始条件下，输出变量的拉氏变换与输入变量 的拉氏变换之比。27RCExample: RC electrical network28Consider the dynamic system represented by the differential equationIf the initial conditions are all zero,then the Laplace transform of the Eq.yields The transfer function is29Some Comments about the Transfer Function 1. The concept of transfer function only applies to the LTI system.2. Transfer function is only determined by the structure andparameter of system.3. Transfer function is a rational proper fraction, and there relationship of the orders of the numerator and denominator is nthe order of the denominator mthe order of the numerator nm4. The inverse Laplace transform of transfer function is the impulse response function of the system.306. The method of the transfer function has some limitation. (1). It only applies to the SISO system. (2). It only can reflect the relationship of input and output.(3). It only can analyze the motion characteristic of zero initialconditions.5. A certain transfer function correspond a certain portrait of zero- poles distribution.setPoles: The roots of the denominator polynomial D(s).Zeros:The roots of the numerator polynomial N(s).31The transfer functions of some components1. Potentiometer. E UK 322. Potentiometer Bridge.333.Gear train344.armature-controlled dc motor Inertial=J Friction=b(1)35Motor torque:load torque:disturbance torquewhere(2)(3)(4)We can obtain the transfer function(with )36The block diagram of dc-motor37Components of block diagram1. Signal line: A line with arrow that indicate the direction of signal transform. 2. Block: It expresses the transfer function. 3. Derivation point (measuring point). 384. Synthesis Point (Comparing point).Example:Position slave system391/iK140Block Diagram Transformations (1) Combining blocks in cascade(2) Parallel Connection of Blocks 41(3)Eliminating a