1. Modern Control Systems 2-1 Lecture 02 Modeling (i) –Transfer function 2.1 Circuit Systems 2.2 Mechanical Systems 2.3 Transfer Function

2. Modern Control Systems 2-2 Resistor 應用的定律 ．克希荷夫電流定律 (Kirchhoff Current Law) ．克希荷夫電壓定律 (Kirchhoff Voltage Law) ．歐姆定律 (Ohm’s Law) InductorCapacitor 2.1 Circuit Systems

3. Modern Control Systems 2-3 Example 2.1 ： RC Series Circuit Example 2.2 ： By Kirchhoff Current Law (Node Analysis) By Kirchhoff and Ohm’s Law 2.1 Circuit Systems Fig. 2.1 Fig. 2.2

4. Modern Control Systems 2-4 Spring( 彈簧 ) ( 張力 ) : spring constant ( 彈簧常數 ) : Velocity ( 速率 ) : Viscous Friction Force ( 黏滯摩擦力 ) : viscous friction Constant ( 黏滯摩擦係數 ) Damper( 阻尼器 ) 2.2 Mechanical Systems (Translational Motion)

5. Modern Control Systems 2-5 Mass( 質量 ) : inertia force ( 慣性力 ) : Mass ( 質量 ) F ：所有外力之和 ： acceleration ( 加速度 ) ( 外力的方向與位移相同為正 ) 應用的定律 牛頓定律： 2.2 Mechanical Systems (Translational Motion)

6. Modern Control Systems 2-6 Form Newton’s Second Law Under ZIC, take Laplace transform both sides Example 2.3: Mass-Spring-Damper (Transfer Function) Note: ZIC=Zero Initial Condition 2.2 Mechanical Systems (Translational Motion) Fig. 2.3

7. Modern Control Systems 2-7 Example 2.4 ：懸吊系統 (Suspension system) 2.2 Mechanical Systems (Translational Motion) Mathematical Model: Fig. 2.4 Tire spring constant

8. Modern Control Systems 2-8 : angle 角度 T ： torque 轉矩 ： angular velocity ( 角速度 ) ： angular acceleration ( 角加速度 ) ：對轉動軸的慣量 2.2 Mechanical Systems (Rotational Motion) Spring( 彈簧 ) Damper( 阻尼器 ) Inertia( 慣量 )

9. Modern Control Systems 2-9 Gear train (1) (2)(3) (4) no energy loss

10. Modern Control Systems 2-10 應用的定律 = 物體的加速度 = 轉動慣量 = 所有外加轉矩之和 ( 與角位移方向相同者為正 ) 2.2 Mechanical Systems (Rotational Motion)

11. Modern Control Systems 2-11 Example2.5 ： Mass － Spring － Damper (Rotational System) Form Newton’s Second Law Under ZIC, take Laplace transform both sides 2.2 Mechanical Systems (Rotational Motion) Fig. 2.5

12. Modern Control Systems 2-12 Definition ZIC: Zero Initial Condition 2.3 Transfer Function

13. Modern Control Systems 2-13 Transfer Function: Gain that depends on the frequency of input signal Under ZIC, the steady state output where is also called the DC gain. When input Special Case: 2.3 Transfer Function (2.1) Conclusion: Under ZIC, for sinusoidal input, the steady state Output is also a sinusoidal wave.

14. Modern Control Systems 2-14 Example 2.6  A=0.707 <1, Attenuation! 2.3 Transfer Function With reference to (2.1), we know and Find the output y(t) ? Fig. 2.6

15. Modern Control Systems 2-15 Set I.C. =0 and Take L.T. both sides A Second-Order Example Derivation of T.F. from Differential Equation 2.3 Transfer Function (Transfer Function from r to y)

16. Modern Control Systems 2-16 Example 2.7: A second-order Circuit 2.3 Transfer Function From Kirchihoff Voltage Law, we obtain (Transfer Function from ) Fig. 2.7

17. Modern Control Systems 2-17 Form Newton’s Second Law Transfer Function Take Laplace Transfrom both sides Example 2.3: Mass-Spring-Damper 2.3 Transfer Function Fig. 2.8

18. Modern Control Systems 2-18 轉移函數的相關名詞 p(s)= 分子多項式， q(s)= 分母多項式 ( 特性多項式, characteristic polynomial) q(s)= 之階數稱為此系統之階數 (order) q(s) 之根稱為系統之極點 (pole) p(s) 之根稱為系統之零點 (zero) 方程式 q(s)=0 稱為 特性方程式 (characteristic equation) ◆ ◆ ◆ ◆ ◆ 2.3 Transfer Function Transfer Function

19. Modern Control Systems 2-19 Under ZIC, take L.T., we get the transfer function Poles ： Zeros ： Char. Equation: 2.3 Transfer Function Example 2.8 Fig. 2.9

20. Modern Control Systems 2-20 : 稱為系統的時間常數 (Time Constant) : 稱為穩態直流增益 2.3 Transfer Function Time Constant of a first-order system Consider

21. Modern Control Systems 2-21 is called step-function. When A=1, it is called unit step function. Example 2.9 The output voltage For RC=1 Fig. 2.10 2.3 Transfer Function

22. Modern Control Systems 2-22 A 1 2 3 45 gain-time constant form pole-zero form 0 Time Constant: Measure of response time of a first-order system Fig. 2.11 2.3 Transfer Function