1、Chapter 2 mathematical models of systems,2.1 Introduction 2.1.1 Why？1) Easy to discuss the full possible types of the control systems only in terms of the systems “mathematical characteristics”.2) The basis of analyzing or designing the control systems.,2.1.3 How get？1) theoretical approaches 2) exp
2、erimental approaches 3) discrimination learning,2.1.2 What is ？Mathematical models of systems the mathematical relation- ships between the systems variables.,Chapter 2 mathematical models of systems,2.2.1 Examples,2.1.4 types1) Differential equations2) Transfer function3) Block diagram、signal flow g
3、raph4) State variables,2.2 The input-output description of the physical systems differential equations The input-output descriptiondescription of the mathematical relationship between the output variable and the input variable of physical systems.,Chapter 2 mathematical models of systems,define: inp
4、ut ur output uc。 we have：,Example 2.1 : A passive circuit,Chapter 2 mathematical models of systems,Example 2.2 : A mechanism,Define: input F ，output y. We have:,Compare with example 2.1: ucy, urF-analogous systems,Chapter 2 mathematical models of systems,Example 2.3 : An operational amplifier (Op-am
5、p) circuit,Input ur output uc,(2)(3); (2)(1); (3)(1)：,Chapter 2 mathematical models of systems,Example 2.4 : A DC motor,Input ua， output 1,(4)(2)(1) and (3)(1):,Chapter 2 mathematical models of systems,Make:,Chapter 2 mathematical models of systems,Assume the motor idle: Mf = 0, and neglect the fric
6、tion: f = 0, we have:,the differential equation description of the DC motor is:,Compare with example 2.1 and example 2.2:,-Analogous systems,Chapter 2 mathematical models of systems,Example 2.5 : A DC-Motor control system,Input ur, Output ; neglect the friction:,Chapter 2 mathematical models of syst
7、ems,（2）（1）（3）（4），we have：,2.2.2 steps to obtain the input-output description (differential equation) of control systems,1) Identify the output 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.* n
8、ecessary assumption and neglect.* proper approximation.,3) dispel the intermediate(across) variables to get the input-output description which only contains the output and input variables.,Chapter 2 mathematical models of systems,4) Formalize the input-output equation to be the “standard” form:Input
9、 variable on the right of the input-output equation .Output variable on the left of the input-output equation.Writing the polynomialaccording to the falling-power order.,2.2.3 General form of the input-output equation of the linear control systemsA nth-order differential equation:,Suppose: input r ，
10、output y,Chapter 2 mathematical models of systems,2.3 Linearization of the nonlinear components 2.3.1 what is nonlinearity？,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？Suppo
11、se: y = f(r)The Taylor series expansion about the operating point r0 is:,Chapter 2 mathematical models of systems,Examples:,Example 2.6 : Elasticity equation,Example 2.7 : Fluxograph equation,Q Flux; p pressure difference,Chapter 2 mathematical models of systems,2.4 Transfer function,Another form of
12、 the input-output(external) description of control systems, different from the differential equations.,2.4.1 definition,Transfer 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 th
13、e linear systems, that is:,Chapter 2 mathematical models of systems,C(s) Laplace transform of the output variable R(s) Laplace transform of the input variable G(s) transfer function,2.4.2 How to obtain the transfer function of a system,1) If the impulse response g(t) is known,Notes:,Chapter 2 mathem
14、atical models of systems,Example 2.8 :,2) If the output response c(t) and the input r(t) are known,We have:,Chapter 2 mathematical models of systems,Example 2.9:,Then:,3) If the input-output differential equation is known,Assume: zero initial conditions; Make: Laplace transform of the differential e
15、quation; Deduce: G(s)=C(s)/R(s).,Chapter 2 mathematical models of systems,Example 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.,Chapter 2 mathematical models of systems,Example 2.11: For a electric circuit:,Chapter 2 mat
16、hematical models of systems,Example 2.12: For a op-amp circuit,Chapter 2 mathematical models of systems,5) For a control system,Write the differential equations of the control system;Make Laplace transformation, assume zero initial conditions,transform the differential equations into the relevant al
17、gebraicequations; Deduce: G(s)=C(s)/R(s).,Example 2.13,the DC-Motor control system in Example 2.5,Chapter 2 mathematical models of systems,In Example 2.5, we have written down the differential equations as:,Make Laplace transformation, we have:,(2)(1)(3)(4), we have:,Chapter 2 mathematical models of
18、 systems,Chapter 2 mathematical models of systems,2.5.1 Proportioning element,Relationship between the input and output variables:,Transfer function:,Block diagram representation and unit step response:,Examples:,amplifier, gear train, tachometer,2.5 Transfer function of the typical elements of line
19、ar systemsA linear system can be regarded as the composing of several typical elements, which are:,Chapter 2 mathematical models of systems,2.5.2 Integrating element,Relationship between the input and output variables:,Transfer function:,Block diagram representation and unit step response:,Examples:
20、,Integrating circuit, integrating motor, integrating wheel,Chapter 2 mathematical models of systems,2.5.3 Differentiating element,Relationship between the input and output variables:,Transfer function:,Block diagram representation and unit step response:,Examples:,differentiating amplifier, differen
21、tial valve, differential condenser,2.5.4 Inertial element,Chapter 2 mathematical models of systems,Relationship between the input and output variables:,Transfer function:,Block diagram representation and unit step response:,Examples:,inertia wheel, inertial load (such as temperature system),Chapter
22、2 mathematical models of systems,2.5.5 Oscillating element,Relationship between the input and output variables:,Transfer function:,Block diagram representation and unit step response:,Examples:,oscillator, oscillating table, oscillating circuit,2.5.6 Delay element,Chapter 2 mathematical models of sy
23、stems,Relationship between the input and output variables:,Transfer function:,Block diagram representation and unit step response:,Examples:,gap effect of gear mechanism, threshold voltage of transistors,.2.6.1 Block diagram representation of the control systems,Chapter 2 mathematical models of syst
24、ems,Examples:,2.6 block diagram models (dynamic)Portray the control systems by the block diagram models more intuitively than the transfer function or differential equation models,Example 2.14,Chapter 2 mathematical models of systems,For the DC motor in Example 2.4,In Example 2.4, we have written do
25、wn the differential equations as:,Make Laplace transformation, we have:,Chapter 2 mathematical models of systems,Draw block diagram in terms of the equations (5)(8):,Consider the Motor as a whole:,Chapter 2 mathematical models of systems,Example 2.15,The water level control system in Fig 1.8:,Chapte
26、r 2 mathematical models of systems,The block diagram model is:,Chapter 2 mathematical models of systems,Example 2.16,The DC motor control system in Fig 1.9,Chapter 2 mathematical models of systems,The block diagram model is:,Chapter 2 mathematical models of systems,2.6.2 Block diagram reductionpurpose: reduce a complicated block diagram to a simple one.,2.6.2.1 Basic forms of the block diagrams of control systems Chapter 2-2.ppt,Chapter 2 mathematical models of systems,
