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1、FUNDAMENTALS OF CONTROL ENGINEERING Lecture 3 Feiyun Xu Email: http:/ 2: Mathematical Models of Systems,2.4 The Laplace transform and its inverse transform,Laplace transform and its inverse transform,Chapter 2: Mathematical Models of Systems,The Inverse Laplace transform for the function F(s) is :,
2、Chapter 2: Mathematical Models of Systems,Laplace transform of some typical functions,the unit step function,Chapter 2: Mathematical Models of Systems,the unit ramp function,Chapter 2: Mathematical Models of Systems,the unit parabolic function,Chapter 2: Mathematical Models of Systems,the unit impul
3、se function,Chapter 2: Mathematical Models of Systems,the damping exponential function,(a is constant),Chapter 2: Mathematical Models of Systems,the sine and cosine function,By Euler fomula:,Chapter 2: Mathematical Models of Systems,Therefore:,Similarly：,Chapter 2: Mathematical Models of Systems,Pro
4、perties of Laplace transform,Linearity,Chapter 2: Mathematical Models of Systems,Real differential theorem,Chapter 2: Mathematical Models of Systems,Chapter 2: Mathematical Models of Systems,This means the Laplace variable s can be considered as a differential operator.,Chapter 2: Mathematical Model
5、s of Systems,Integral theorem,If,integral operator,Chapter 2: Mathematical Models of Systems,Delay theorem,Provided that f(t) = 0 while t0, exists,Chapter 2: Mathematical Models of Systems,Translational theorem,Initial value theorem,Chapter 2: Mathematical Models of Systems,Final value theorem,Chapt
6、er 2: Mathematical Models of Systems,Chapter 2: Mathematical Models of Systems,Convolution theorem,Chapter 2: Mathematical Models of Systems,Chapter 2: Mathematical Models of Systems,Scale transform,Example：,Chapter 2: Mathematical Models of Systems,Find the inverse Laplace transform with partial fr
7、action expansion,Partial fraction expansion,If F(s)=F1(s)+F2(s)+Fn(s),Chapter 2: Mathematical Models of Systems,In control engineering, F(s) can be written as:,Where -p1, -p2, , -pn are the roots of the characteristic equation A(s) = 0, i.e. the poles of F(s). ci=bi /a0 (i = 0,1,m),Chapter 2: Mathem
8、atical Models of Systems,Partial fraction expansion for F(s) with different real poles,Where the constant coefficients Ai are called residues at the pole s = -pi.,Therefore：,How to find the coefficients Ai ?,Chapter 2: Mathematical Models of Systems,Example1: Find the inverse Laplace transform,Chapt
9、er 2: Mathematical Models of Systems,i.e.,Chapter 2: Mathematical Models of Systems,Partial fraction expansion for F(s) with complex poles,Supposing F(s) only has one pair of conjugated complex poles -p1 and -p2, and the other poles are different real poles. Then,Where,Chapter 2: Mathematical Models
10、 of Systems,Or:,Where A1 and A2 can be calculated with the following equation.,Chapter 2: Mathematical Models of Systems,Example 2: Find the inverse Laplace transform,Given,Chapter 2: Mathematical Models of Systems,i.e.,Chapter 2: Mathematical Models of Systems,Therefore,Chapter 2: Mathematical Mode
11、ls of Systems,Finally,The inverse Laplace transform will be:,Chapter 2: Mathematical Models of Systems,Partial fraction expansion for F(s) with repeated poles,Supposing F(s) only has a r-order repeated pole -p0,Where the coefficients Ar+1,An can be found with the forenamed single pole residue method
12、.,Chapter 2: Mathematical Models of Systems,Chapter 2: Mathematical Models of Systems,Therefore,From the Laplace transform table, we obtain,Chapter 2: Mathematical Models of Systems,Example 3: Find the inverse Laplace transform,Chapter 2: Mathematical Models of Systems,Chapter 2: Mathematical Models
13、 of Systems,Using Laplace transform to solve the differential equations,Image function of output in s-domain,Algebraic equation in s-domain,Chapter 2: Mathematical Models of Systems,Example 5: Solving a differential equation with Laplace transform,Chapter 2: Mathematical Models of Systems,Do the Lap
14、lace transform to the left-hand of the differential equation, we have,i.e.,Chapter 2: Mathematical Models of Systems,Since we obtain,Chapter 2: Mathematical Models of Systems,Chapter 2: Mathematical Models of Systems,Therefore:,From the Laplace transform table, we obtain,Chapter 2: Mathematical Mode
15、ls of Systems,Comments:,The final solution of a differential equation is obtained directly with Laplace transform method. No need for finding the general and the particular solution of the differential equation.,If the initial conditions is zero, the transformed algebraic equation in s-domain can be
16、 gotten simply with replacing the dn/dtn operator with variable sn.,Chapter 2: Mathematical Models of Systems,Note that the output response X0(s) includes two parts: the forced response determined by the input and the natural response determined by the initial conditions.,Chapter 2: Mathematical Mod
17、els of Systems,Obviously, the transient response of the system will be decreased to zero with time t.,Chapter 2: Mathematical Models of Systems,2.5 The transfer function of linear systems,Transfer function,The transfer function of a linear system is defined as the ratio of the Laplace transform of t
18、he output variable to the Laplace transform of the input variable, with all initial conditions assumed to be zero.,The system is in steady-state, i.e. output variable and its derivative of all order are equal to zero while t0.,Chapter 2: Mathematical Models of Systems,Example 1: Finding the transfer
19、 function of the spring-mass-damper system,Chapter 2: Mathematical Models of Systems,Example 2: Finding the transfer function of an op-amp circuit,i.e.,Chapter 2: Mathematical Models of Systems,Example 3: Finding the transfer function of a two-mass mechanical system,Chapter 2: Mathematical Models of
20、 Systems,If the transfer function in terms of the position x1(t) of mass M1 is desired, then we have,Mini-Test: Please write the differential equation of the two-mass mechanical system.,Chapter 2: Mathematical Models of Systems,Example 4: Transfer function of DC motor,Chapter 2: Mathematical Models
21、of Systems,The transfer function of the dc motor will be developed for a linear approximation to an actual motor, and second-order effects, such as hysteresis and the voltage drop across the brushes, will be neglected.,The air-gap flux of the motor is proportional to the field current, provided the
22、field is unsaturated, so that,The torque developed by the motor is assumed to be related linearly to and the armature current as follows:,Chapter 2: Mathematical Models of Systems,Field current controlled dc motor (ia=Ia is constant),where Km is defined as the motor constant.,The field current is re
23、lated to the field voltage as,Chapter 2: Mathematical Models of Systems,The load torque for rotating inertia as shown in the Figure is written as,Therefore the transfer function of the motorload combination, with Td(s) = 0, is,Chapter 2: Mathematical Models of Systems,Armature current controlled dc
24、motor (if=If is constant),The armature current is related to the input voltage applied to the armature as,where Vb(s) is the back electromotive-force voltage proportional to the motor speed. Therefore we have：,Chapter 2: Mathematical Models of Systems,The armature current is,Therefore the transfer f
25、unction of the motorload combination, with Td(s) = 0, is,Chapter 2: Mathematical Models of Systems,Range of control response time and power to load for electro-mechanical and electrohydraulic devices.,Chapter 2: Mathematical Models of Systems,General form of transfer function,Chapter 2: Mathematical
26、 Models of Systems,Remarks,The transfer function of a system (or element) represents the relationship describing the dynamics of the linear system under consideration.,A transfer function is an inputoutput description of the behavior of a system. Thus the transfer function description does not inclu
27、de any information concerning the internal structure of the system and its behavior.,Transfer function is defined under the zero state of a system. Thus the response of the system under non- zero state cannot be obtained with the transfer function.,Chapter 2: Mathematical Models of Systems,Zeros and
28、 poles,The transfer function can be rewritten as follows,Where the roots of numerator polynomial M(s) = 0, i.e. s=zi (i=1, 2, , m) are called zeros of the system; the roots of denominator polynomial N(s) = 0 (characteristic equation), i.e. s=pi (i=1, 2, , n) are called poles of the system.,Chapter 2: Mathematical Models of Systems,Plot of the zeros and poles,