1. سیستمهای کنترل خطی پاییز 1389 بسم ا... الرحمن الرحيم دکتر حسين بلندي- دکتر سید مجید اسما عیل زاده

2. مرور 2

3. 1) استخراج معادلات ديفرانسيل از مدل فيزيكي سيستم. 2) استخراج مدل رياضي سيستم و خلاصه کردن نتيجه بصورت يك بلوك دياگرام. 3) نتيجه خلاصه شدن يك سيگنال فلوگراف. مثال : اعمال وروديهای تست تحليل پاسخ سيستم پايداري مطلق و نسبي طراحی : تنظیم پارامترها جبران سازها

4. 1) تك ورودي - تك خروجي 2) تابع تبديل در حوزة s 3) روشهاي فركانسي 4) رسم مكان هندسي ريشه‌ها حصول اهداف کنترلی طراحی جبرانسازها

5. OBJECTIVES On completion of this course, the student will be able to do the following: Define the basic terminologies used in controls systems Explain advantages and drawbacks of open-loop and closed loop control systems Obtain models of simple dynamic systems in ordinary differential equation, transfer function, state space, or block diagram form Obtain overall transfer function of a system using either block diagram algebra, or signal flow graphs, or Matlab tools Compute and present in graphical form the output response of control systems to typical test input signals 5

6. Explain the relationship between system output response and transfer function characteristics or pole/zero locations Determine the stability of a closed-loop control systems using the Routh- Hurwitz criteria Analyze the closed loop stability and performance of control systems based on open-loop transfer functions using the Root Locus technique Design PID or lead-lag compensator to improve the closed loop system stability and performance using the Root Locus technique Analyze the closed loop stability and performance of control systems based on open-loop transfer functions using the frequency response techniques Design PID or lead-lag compensator to improve 6

7. Topics Covered Modeling of control systems using ode, block diagrams, and transfer functions Block diagrams and signal flow graphs Modeling and analysis of control systems using state space methods Analysis of dynamic response of control systems, including transient response, steady state response, and tracking performance. Closed-loop stability analysis using the Routh-Hurwitz criteria Stability and performance analysis using the Root Locus techniques Control system design using the Root Locus techniques Stability and performance analysis using the frequency response techniques Control system design using the frequency response techniques 7

8. References for reading 1.R.C. Dorf and R.H. Bishop, Modern Control Systems, 10th Edition, Prentice Hall, 2008, 2. Golnaraghi and Kuo, Automatic Control Systems,, ninth edition, Wiley, 2009 8

9. Grading Midterm 40% Final 40% Quiz 10% H.W. 10% 9

10. Mathematical Models of Systems

11. Objectives We use quantitative mathematical models of physical systems to design and analyze control systems. The dynamic behavior is generally described by ordinary differential equations. 11

12. A wide range of systems, including mechanical, hydraulic, and electrical could be considered. Since most physical systems are nonlinear, we will discuss linearization approximations, which allow us to use Laplace transform methods. 12

13. A wide range of physical Systems, including: mechanical, hydraulic, and electrical could be considered. 13

14. The transfer function blocks can be organized into: block diagrams or signal-flow graphs to graphically depict the interconnections. We will then proceed to obtain the input– output relationship for components and subsystems in the form of transfer functions. 14

15. Mathematical Models of Systems Review the Laplace transform, Learn how to find a mathematical model, called a transfer function, for LTI electrical, mechanical, and electromechanical systems, 15

16. 16 Introduction To understand and control complex systems we must obtain quantitative mathematical models of system.

17. 17 A model is a representation of the process or a system existing in reality or planned for realization which expresses the essential attributes of a process or a system in a useful form. Norbert Wiener, 1945

18. 18

19. 19 Approach to dynamic systems 1.Define the system and its components. 2.Formulate the MM and list the necessary assumptions. 3.Write the differential equations describing the model. 4.Solve the equations for the desired output variables. 5.Examine the solutions and the assumptions. 6.If necessary, reanalyze or redesign the system.

20. 20 Differential Equations of Physical Systems The differential equations describing the dynamic performance of a physical system are obtained by utilizing the physical laws of the process. A differential equation is any algebraic equality which involves either differentials or derivatives.

21. 21 This approach applies equally well to; Mechanical, Electrical, Fluid, Thermodynamic systems.

22. 22 Physical laws The physical laws define relationships between fundamental quantities and are usually represented by equations.

23. Modeling Guid lines Focus on important variables Use reasonable approximations Write mathematical equations from physical laws, don’t invent your own Eliminate intermediate variables Obtain o.d.e. involving input/output variables  I/O model Or obtain 1 st order o.d.e.  state space Get I/O transfer function

24. بطور كلي دو ديدگاه جهت مدلسازي وجود دارد : الف: تقسيم نمودن سيستم به اجزاء تشكيل دهنده و مدلسازي آن توسط روابط رياضي. ب : شناسايي پارامتري سيستم : در اين حالت آزمايشهايي سيستم انجام مي‌پذيرد و با بررسي نتايج حاصله يك مدل رياضي براي سيستم تعيين مي‌‌شود.  در راستاي پايه‌گذاري و تبيين سيستم، مدل بدست آمده بايد مبين پارامترهای زير باشد: ـ ارتباط ديناميكي بين پارامترهاي دستگاه ـ ورودي كارانداز ـ خروجي قابل اندازه‌گيري باشد. جمعبندی اولیه:

25. نکاتی که در مدلسازی سيستمها بايد در نظر داشت  مدلسازي دربرگيرنده اطلاعات دروني سيستم بوده و همچنين ارتباط بين effect, cause متغيرهاي سيستم مي‌باشد.  پايه و اساس اصلي جهت انجام كار استفاده از قوانين فيزيكي حاكم بر سيستم مي ‌ باشد.  انتخاب متغيرهاي حالت در روش متغيرهاي فيزيكي براساس عناصر موجود نگهدارنده انرژي سيستم بنا مي‌شود.  متغير فيزيكي در معادلة انرژي براي هر عنصر نگهدارنده انرژي مي‌تواند بعنوان متغير حالت سيستم انتخاب شود. لازم به يادآوری است که متغيرهاي فيزيكي بايد بگونه ای انتخاب شوند كه ناوابسته باشند.

26. عناصر نگهدارندة انرژي

27. Circuit: KCL:  ( i into a node) = 0 KVL:  ( v along a loop) = 0 RLC: v=Ri, i=Cdv/dt, v=Ldi/dt Linear motion: Newton: ma =  F Hooke’s law: F s = K  x damping:Fd = C  x_dot Angular motion: Euler: J  =   K   C  dot In GeneraL: Common Physical Laws

28. 28 Symbols and units

29. Voltage-current, voltage-charge, and impedance relationships for capacitors, resistors, and inductors

30. Ex.1; Find ODE eqn. RLC network

31. مثال 2 : معادلات ode را بنویسید.

32. سيستمهای مکانيکی (1) انتقالي : مجموعة نيروها برابر است با حاصلضرب شتاب در جرم (N) (2) دوراني : مجموعة گشتاورها برابر است با حاصلضرب ممان اينرسي در شتاب زاويه‌اي

33. اجزای اصلی سيستمهای مکانيکی

34. مثال 3 : :

35. مثال 4 :

36. 36 MECHANICAL ROTATIONAL SYSTEMS

37. 37 Transforms The term transform refers to a mathematical operation that takes a given function and returns a new function. The transformation is often done by means of an integral formula. Commonly used transforms are named after Laplace and Fourier.

38. 38 Transforms are frequently used to change a complicated problem into a simpler one. The simpler problem is then solved, usually using elementary algebraic means. The solution to the simpler problem is taken over to the original problem using the inverse transform.

39. 39

40. 40 Laplace transform Laplace transform can significantly reduce the effort required to solve linear differential equations. A major benefit is that this transformation convert differential equations to algebraic equations, which can simplify the mathematical manipulations required to obtain a solution.

41. 41

42. 42 Step 1. Take the Laplace transform of both sides of the differential equation. Step 2. Solve for Y(s) If the expression for Y(s) does not appear in Laplace Transform Table Step 3a. Factor the characteristic equation polynomial. Step 3b. Perform the partial fraction expansion. Step 4. Use the inverse Laplace transform relations to find y(t). General solution procedure:

43. 43 Example 5

44. 44 Example 5

45. 45 Example 5

46. 46 Disadvantage: The solution of the differential equation involves use of Laplace transforms as an intermediate step. Any change in the initial conditions or in the forcing function requires that the complete solution be redeliver.

47. 47 The transfer function - a modified approach. The transfer function is an algebraic expression for the dynamic relation between input and output of the process model. It is defined so as to be independent of initial conditions and of the particular choice of forcing function.

48. 48 G(s) To obtain the transfer function G(s) of the LTI system, we take the Laplace transform on both sides of the equation, and assume zero initial conditions.

49. 49 Properties of the G(s) The G(s) is defined only for a LTI system. All initial conditions of the system are set to zero. The G(s) is independent of the input of the system. The G(s) of a continuous-data system is expressed only as a function of the complex variable s. For discrete-data systems modeled by difference equations, the transfer function is a function of z when the z-transform is used.

50. 50 A transfer function can be derived only for a LTI differential equation model.

51. 51 A transfer function A transfer function of the LTI system is defined as a 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.

52. 52 EX. 6: An automobile shock absorber Spring-mass-damperFree-body diagram

53. 53 The automobile shock absorber

54. 54 EX.7: Transfer function of the RC network V 1 (s) = (R + 1/Cs) I(s) V 2 (s) = I(s) 1/Cs G(s) = V 2 (s)/V 1 (s) = 1/(RC s + 1) = 1/T/(s + 1/T)

55. 55 The transfer function of the RC network is obtained by writing the Kirchhoff voltage equation. The circuit is a voltage divider, where V 2 (s)/V 1 (s) = Z 2 (s)/(Z 1 (s) + Z 2 (s)), where Z 1 (s)= R and Z 2 = 1/Cs The single pole s = -1/T

56. Ex.8; Find G(S)? RLC network

58. Ex. 9: Mesh analysis Mesh 1 Mesh 2

59. Sum of impedance around mesh 1 Sum of impedance around mesh 2 Sum of impedance common to two meshes Sum of applied voltages around the mesh Write equations around the meshes

60. Determinant

62. Kirchhoff current law at these two nodes i1i1 i3i3 i2i2 i 1 + i 2 +i 3 =0 i4i4 i 3 + i 4 =0 Ex. 10: Nodal analysis

63. conductance Kirchhoff current law

64. Sum of admittance at each node Admittance between node I and node j Sum of injected current into each node

65. 65 A transfer function of LTI system is defined as the Laplace transform of the impulse response, with initial conditions set to zero.

66. 66 Input-Output description A transfer function is an input-output description of the behavior of a system. Thus the transfer function description does not include any information concerning the internal structure of the system.

67. 67 Summary 1. The differential equations describing the dynamic performance of physical systems were utilized to construct a mathematical model. The physical systems included mechanical, electrical, fluid, and thermodynamic systems. 2. For linear systems we apply the Laplace transformation and its related input-output relationship given by the transfer function.

68. 68 Summary 3. The transfer function allows to determine the response of the system to various input signals. Y(s) = X(s) G(s)

69. 69

70. 70

71. 71

72. روش مدرن از معادلات ديفرانسيل معادلات فضاي حالت پيش گفتار: يادآوری :  اكثر روشهاي طراحي سيستم‌هاي كنترل مبتني بر نوعي مدل رياضي از سيستم فيزيكي مي‌باشد.  طراحي‌هاي كلاسيك سيستم‌هاي كنترل از روشهايي مانند مكان، پاسخ فركانسي جهت تحليل و طراحي سيستم‌ها استفاده مي‌کنیم.  شايان توجه است كه در اين ديدگاه، فعاليت متمركز بر استفاده از تابع تبديل است.

73. 1) اين روش براي سيستمهاي صنعتي SISO قابل بهره‌وري میباشد و مي‌تواند نتايج مطلوبي را بدنبال داشته باشد 2) تحليل دقيق سيستمهاي صنعتي پيشرفته مدلهاي كاملتري را طلب مي‌كند. 3) سيستم‌هاي صنعتي پيچيده براي دقت، سرعت عمل و كارايي بيشتر نيازمند به طراحي‌هاي مدرن سيستم‌هاي كنترل مي‌باشند. معايب روشهای کلاسيک

74.  مدلسازي سيستم‌هاي كنترل با استفاده از متغيرهاي حالت در راستاي تحقق اهدافي است كه به آن اشاره كرده‌ايم.  متغيرهاي حالت در واقع مي‌توانند ديناميكي از سيستم را شامل شوند كه در مدل خروجي ـ ورودي ظاهر نمی شوند. از اين جهت مدل متغيرهاي حالت را مدل داخلي نيز مي‌گويند.  توصيف فضاي حالت، تصوير كاملي را از ساختار داخلي سيستم فراهم مي‌كند. اين مدل نشان مي‌دهد كه متغيرهاي حالت چگونه با يكديگر تداخل نموده، ورودي سيستم چگونه بر متغيرهاي حالت تأثير مي‌گذارد و چگونه با تركيبهاي متفاوت مي‌توان يك سيستم خاص را نشان داد. نتيجه