1. page 1 241-482 : PID Controller Chapter 3 Design of Discrete- Time control systems PID C ontroller

2. page 2 241-482 : PID Controller Introduction 1 P Controller 2 PD Controller 3 PID controller

3. page 3 241-482 : PID Controller Introduction Design Specifications : 1 Relative Stability 2 Steady-State Error 3 Transient Response 4 Frequency Response Characteristics

4. page 4 241-482 : PID Controller Time Domain Specifications : - Steady-State Error - Maximum Overshoot - Rise Time - Setting Time Frequency Domain Specification : - Gain Margin - Phase Margin

5. page 5 241-482 : PID Controller Controller Configuration One degree of freedom : 1 Cascade (Series) Compensation 2 Feedback Compensation 3 State Feedback Compensation Two degree of freedom : 1 Series-Feedback Compensation 2 Feedforward Compensation

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8. page 8 241-482 : PID Controller Fundamental Principle of design : 1 Choose a controller configuration 2 Choose a controller type that will be satify all design specification 3 Choose the controller parameters

9. page 9 241-482 : PID Controller Useful Time- and Frequency- Domain Characteristics 1 Complex-conjugate poles of the closed-loop transfer function lead to a step response that is underdamped. If all system poles are real, the step response are overdamped. However, zeros of the closed-loop transfer function may cause overshoot even if the system is overdamped. 2 The response of the system is dominated by those poles closest to the origin in the s-plane. Transient due to those poles farther to the left decay faster.

10. page 10 241-482 : PID Controller 3 The farther to the left the in the s-plane the system's dominant poles are, the faster the system will response and the greater its bandwidth will be. 4 The farther to the left in the s- plane the system's dominant poles are, the more expensive it will be and the larger its internal signals will be.

11. page 11 241-482 : PID Controller 5 When a pole and zero of a system transfer function nearly cancel each other, the portion of the system response associated with the pole will have a small magnitude. 6 Time- and frequency-domain specifications are loosely associated with each other. Rise time and bandwidth are inversely proportional. Phase margin, gain margin and damping are inversely proportional.

12. page 12 241-482 : PID Controller Bode plots showing gain adjustment for a desired phase margin Transient Response via Gain Adjustment

13. page 13 241-482 : PID Controller Proportional-Derivative Controller Transfer function

14. page 14 241-482 : PID Controller Time-domain interpretation of PD Controller

15. page 15 241-482 : PID Controller Frequency-domain interpretation of PD Controller

16. page 16 241-482 : PID Controller Effects of PD Controller 1 Improves damping and reduces maximum overshoot. 2 Reduces rise time and settimg time. 3 Increases bandwidth. 4 Improves GM, PM. 5 May accentuate noise at higher frequency. 6 Not effective for highly damped or initially unstable systems. 7 May require a relatively large capacitor in circuit implementation.

17. page 17 241-482 : PID Controller Example 8.1 Consider the second-order model of the aircraft attitude control system. The forward-path transfer function of the system is given by, K = 181.17 From Example 10.1 B. KUO

18. page 18 241-482 : PID Controller The performance specifications is follows : - Steady-state error due to unit- ramp function <= 0.000443 - Maximum overshoot <= 5% - Rise time t r <= 0.005 sec - Settling time t s <= 0.005 sec

19. page 19 241-482 : PID Controller Time-domain design With PD controller, the forward path thansfer function of the system becomes

20. page 20 241-482 : PID Controller Closed-loop transfer function is Ramp-error constant is

21. page 21 241-482 : PID Controller Steady-state error due to a unit- ramp function input is From the closed-loop transfer function show that the effect of PD controller are to : 1 Add a zero at s=-K p /K D to the closed-loop transfer function. 2 Increase the “damping term”

22. page 22 241-482 : PID Controller Characteristic equation is We can arbitrarily set K p =1 which is acceptable from the steady- state error requirement. The damping ratio is

23. page 23 241-482 : PID Controller By setting K D to zero the characteristic equation becomes The root locus of this eq. As K p varies from 0 to infinity are

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25. page 25 241-482 : PID Controller The root locus with K p = constant and KD varies are shown for K p =1 and K p =0.25.

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27. page 27 241-482 : PID Controller Attributes of the Unit- step Response of the system with PD Control

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30. page 30 241-482 : PID Controller Frequency domain Characteristics of the system with PD Control

31. page 31 241-482 : PID Controller PID Controller Transfer function

32. page 32 241-482 : PID Controller The PD controller could add damping to a system, but steady- state response is not affected. The PI controller could improve the relative stability and improve the steady-state error at the same time, but the rise time is increased. The PID controller utilize the best feature of each of the PI and PD controller.

33. page 33 241-482 : PID Controller Design procedure of the PID Controller 1 Consider that the PID controller consists of a PI portion connected in cascade with a PD portion. The transfer function of the PID controller is written as

34. page 34 241-482 : PID Controller The proportional constant of the PID portion is set to unity, since we need only three parameters in PID controller. Equating both side, we have Design procedure of the PID Controller (cont.)

35. page 35 241-482 : PID Controller 3 Select the parameter K I2 and K P2 so that the total requirement on relative stability is satisfied. Design procedure of the PID Controller (cont.) 2 Consider that the PD portion is in effect only. Select the value of K D1 so that the portion of the desired relative stability is achieved. In time domain, this relative stability may be measured by the maxinum overshoot, and in the frequency domain it is the phase margin.

36. page 36 241-482 : PID Controller Example 8.3 Consider the third-order model of the aircraft attitude control system. The forward-path transfer function of the system is given by From Example 10.5 B. KUO

37. page 37 241-482 : PID Controller The time-domain performance specifications is follows : - Steady-state error due to parabolic function t 2 /2 <= 0.2 - Maximum overshoot <= 5% - Rise time t r <= 0.005 sec - Settling time t s <= 0.005 sec

38. page 38 241-482 : PID Controller Apply the PD control with transfer function (1+K D1 s). The forward-path transfer function becomes The best PD controller is K D1 =0.002 and the maximum overshoot is 11.37 %. The rise and settling times are within the required values. Characteristic equation is Time-domain design

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41. page 41 241-482 : PID Controller We add the PI controller, the forward-path transfer function becomes Let K I2 /K P2 =15

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43. page 43 241-482 : PID Controller Select K P2 = 0.3, K D1 =0.002 and K I2 =15K P2 = 4.5, the parameters of the PID controller are :

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45. page 45 241-482 : PID Controller The forward-path transfer function with PD controller is Frequency-domain design Specifications Phase margin >= 70 degree |G(j  )| = 7 dB,  g = 811 rad/sec. K P2 = 10 - 7/20 = 0.45

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48. page 48 241-482 : PID Controller Bilinear Transformation we introduce the bilinear transformation, which use an algebraic transform between the variables s and z. This transform is

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