2. Chapter 2 Modeling of Control Systems NUAA-Control System Engineering

3. Control System Engineering-2008 How to analyze and design a control system 2 ControllerActuatorPlant Sensor - r Expected value e Error Disturbance Controlled variable n y u The first thing is to establish system model (mathematical model)

4. Control System Engineering-2008 3 Content in Chapter 2 2-1 Introduction 2-2 Establishment of differential equation and linearization 2-3 Transfer function 2-4 Structure diagram 2-5 Signal-flow graphs

5. Control System Engineering-2008 4 2-1 Introduction on mathematical models

6. Control System Engineering-2008 5 System model Definition ： Mathematical expression of dynamic relationship between input and output of a control system. Mathematical model is foundation to analyze and design automatic control systems No mathematical model of a physical system is exact. We generally strive to develop a model that is adequate for the problem at hand but without making the model overly complex.

7. Control System Engineering-2008 6 Laplace transform Fourier transform Three models Differential equation Transfer function Frequency characteristic Transfer function Differential equation Frequency characteristic Linear system Study time-domain response study frequency-domain response

8. Control System Engineering-2008 7 Modeling methods Analytic method According to A. Newton’s Law of Motion B. Law of Kirchhoff C. System structure and parameters the mathematical expression of system input and output can be derived. Thus, we build the mathematical model ( suitable for simple systems).

9. Control System Engineering-2008 Modeling methods 8 System identification method Building the system model based on the system input—output signal This method is usually applied when there are little information available for the system. Black box Input Output Black box: the system is totally unknown. Grey box: the system is partially known. Neural Networks, Fuzzy Systems

10. Control System Engineering-2008 9 Why Focus on Linear Time-Invariant (LTI) System What is linear system? system Is y(t)=u(t)+2 a linear system ？ - A system is called linear if the principle of superposition applies

11. Control System Engineering-2008 Advantages of linear systems The overall response of a linear system can be obtained by 10 -- decomposing the input into a sum of elementary signals -- figuring out each response to the respective elementary signal -- adding all these responses together. Thus, we can use typical elementary signal (e.g. unit step,unit impulse, unit ramp) to analyze system for the sake of simplicity.

12. Control System Engineering-2008 11 What is time-invariant system? –A system is called time-invariant if the parameters are stationary with respect to time during system operation –Examples ？

13. Control System Engineering-2008 12 2.2 Establishment of differential equation and linearization

14. Control System Engineering-2008 13 Differential equation Linear ordinary differential equations --- A wide range of systems in engineering are modeled mathematically by differential equations --- In general, the differential equation of an n- th order system is written Time-domain model

15. Control System Engineering-2008 14 How to establish ODE of a control system --- list differential equations according to the physical rules of each component; --- obtain the differential equation sets by eliminating intermediate variables; --- get the overall input-output differential equation of control system.

16. Control System Engineering-2008 15 Examples-1 RLC circuit RL C u(t) u c (t) i(t) Input u(t) system Output u c (t) Defining the input and output according to which cause-effect relationship you are interested in.

17. Control System Engineering-2008 16 According to Law of Kirchhoff in electricity RL C u(t) u c (t) i(t)

18. Control System Engineering-2008 It is re-written as in standard form 17 Generally ， we set the output on the left side of the equation the input on the right side the input is arranged from the highest order to the lowest order

19. Control System Engineering-2008 18 Examples-2 Mass-spring-friction system m k F(t) Displacement x(t) f friction Spring We are interested in the relationship between external force F(t) and mass displacement x(t) Define: input—F(t); output---x(t) Gravity is neglected.

20. Control System Engineering-2008 19 By eliminating intermediate variables, we obtain the overall input-output differential equation of the mass-spring-friction system. Recall the RLC circuit system These formulas are similar, that is, we can use the same mathematical model to describe a class of systems that are physically absolutely different but share the same Motion Law.

21. Control System Engineering-2008 20 Examples-3 nonlinear system In reality, most systems are indeed nonlinear, e.g. then pendulum system, which is described by nonlinear differential equations. It is difficult to analyze nonlinear systems, however, we can linearize the nonlinear system near its equilibrium point under certain conditions.

22. Control System Engineering-2008 Linearization of nonlinear differential equations Several typical nonlinear characteristics in control system. 21 input output 0 Saturation (Amplifier) input output 0 Dead-zone (Motor)

23. Control System Engineering-2008 22 Methods of linearization （ 1 ） Weak nonlinearity neglected （ 2 ） Small perturbation/error method Assumption: In the system control process, there are just small changes around the equilibrium point in the input and output of each component. If the nonlinearity of the component is not within its linear working region, its effect on the system is weak and can be neglected. This assumption is reasonable in many practical control system: in closed-loop control system, once the deviation occurs, the control mechanism will reduce or eliminate it. Consequently, all the components can work around the equilibrium point.

24. Control System Engineering-2008 23 The input and output only have small variance around the equilibrium point. This is linearized model of the nonlinear component. Example A(x0,y0) is equilibrium point. Expanding the nonlinear function y=f(x) into a Taylor series about A(x0,y0) yields Saturation (Amplifier)

25. Control System Engineering-2008 24 Note ： this method is only suitable for systems with weak nonlinearity. RelaySaturation For systems with strong nonlinearity, we cannot use such linearization method.

26. Control System Engineering-2008 25 Flash toilet Example-4 modeling a nonlinear system Problem: Derive the differential equation of water tank (the cross-sectional area of the water tank is C). Q 1 : inflow per unit time Q 2 : outflow per unit time Initial water level : H 0 Q 10 =Q 20 =0 Define: Input—Q 1,Output—H

27. Control System Engineering-2008 26 Solution: Outflow or inflow within dt time should be equal to the total amount of water change （ Q1-Q2 ） dt, that is: According to ‘Torricelli Theorem’, the water yield is in direct proportion to the square root of the height of water level, thus: It is obvious that this formula is nonlinear, On the basis of the Taylor Series expansion of functions around operation points (Q 10,H 0 ) ， We have

28. Control System Engineering-2008 Therefore, the Linearized Differential Equations of the water tank is: 27

29. Control System Engineering-2008 Exercise E1. Please build the differential equations of the following two systems. 28 Output Input Output Input x A B

30. Control System Engineering-2008 29 Solutions. (1) RC circuit (2) Mass-spring system

31. Control System Engineering-2008 30 2-3 Transfer function

32. Control System Engineering-2008 31 Solving Differential Equations Example Solving linear differential equations with constant coefficients: To find the general solution (involving solving the characteristic equation) To find a particular solution of the complete equation (involving constructing the family of a function)

33. Control System Engineering-2008 32 WHY need LAPLACE transform? s-domain algebra problems Solutions of algebra problems Time-domain ODE problems Solutions of time- domain problems Laplace Transform (LT) Inverse LT DifficultEasy

34. Control System Engineering-2008 33 Laplace Transform Laplace, Pierre-Simon 1749-1827 The Laplace transform of a function f(t) is defined as where is a complex variable.

35. Control System Engineering-2008 34 Examples Step signal: f(t)=A Exponential signal f(t)=

36. Control System Engineering-2008 35 Laplace transform table f(t) F(s) f(t) F(s) δ(t) 1 1(t) t

37. Control System Engineering-2008 36 Properties of Laplace Transform (1) Linearity (2) Differentiation where f(0) is the initial value of f(t). Using Integration By Parts method to prove

38. Control System Engineering-2008 37 (3) Integration Using Integration By Parts method to prove ）

39. Control System Engineering-2008 (5) Initial-value Theorem 38 （ 4 ） Final-value Theorem The final-value theorem relates the steady-state behavior of f(t) to the behavior of sF(s) in the neighborhood of s=0

40. Control System Engineering-2008 39 (6)Shifting Theorem ： a. shift in time (real domain) b. shift in complex domain (7) Real convolution (Complex multiplication) Theorem

41. Control System Engineering-2008 40 Inverse Laplace transform Definition ： Inverse Laplace transform, denoted by is given by where C is a real constant 。 Note: The inverse Laplace transform operation involving rational functions can be carried out using Laplace transform table and partial-fraction expansion.

42. Control System Engineering-2008 41 Partial-Fraction Expansion method for finding Inverse Laplace Transform If F(s) is broken up into components If the inverse Laplace transforms of components are readily available, then

43. Control System Engineering-2008 Poles and zeros Poles A complex number s0 is said to be a pole of a complex variable function F(s) if F(s0)=∞. 42 Examples: zeros: 1, -2 poles: -3, -4; poles: -1+j, -1-j; zeros: -1 Zeros –A–A complex number s 0 is said to be a zero of a complex variable function F(s) if F(s 0 )=0.

44. Control System Engineering-2008 43 Case 1: F(s) has simple real poles Parameters p k give shape and numbers c k give magnitudes. Partial-Fraction Expansion Inverse LT

45. Control System Engineering-2008 44 Example 1 Partial-Fraction Expansion

46. Control System Engineering-2008 45 Case 2: F(s) has simple complex-conjugate poles Example 2 Laplace transform Partial-Fraction Expansion Inverse Laplace transform Applying initial conditions

47. Control System Engineering-2008 46 Case 3: F(s) has multiple-order poles The coefficients corresponding to the multi-order poles are determined as Simple polesMulti-order poles

48. Control System Engineering-2008 47 Example 3 Laplace transform: Applying initial conditions: Partial-Fraction Expansion s= -1 is a 3- order pole

49. Control System Engineering-2008 48 Determining coefficients: Inverse Laplace transform:

50. Control System Engineering-2008 49 With aid of MATLAB 1. Laplace Transform L=laplace(f) 2. Inverse Laplace Transform F=ilaplace(L) >> syms t >> L=laplace(t) L= 1/s^2 >> L=laplace(sin(t)) L= 1/(s^2+1) >> F=ilaplace(L) F= sin(t)

51. Control System Engineering-2008 50 Transfer function LTI system Input u(t) Output y(t) Consider a linear system described by differential equation Assume all initial conditions are zero, we get the transfer function(TF) of the system as

52. Control System Engineering-2008 51 Example 1. Find the transfer function of the RLC 1) Writing the differential equation of the system according to physical law: RL C u(t) u c (t) i(t) Input Output 2) Assuming all initial conditions are zero and applying Laplace transform 3) Calculating the transfer function as Solution:

53. Control System Engineering-2008 Exercise Find the transfer function of the following system : 52

54. Control System Engineering-2008 53 Transfer function of typical components ComponentODETF

55. Control System Engineering-2008 Properties of transfer function The transfer function is defined only for a linear time-invariant system, not for nonlinear system. All initial conditions of the system are set to zero. The transfer function is independent of the input of the system. The transfer function G(s) is the Laplace transform of the unit impulse response g(t). 54

56. Control System Engineering-2008 55 How poles and zeros relate to system response Why we strive to obtain TF models? Why control engineers prefer to use TF model? How to use TF model to analyze and design control systems? we start from the relationship between the locations of zeros and poles of TF and the output responses of a system

57. Control System Engineering-2008 56 Position of poles and zeros -a j i 0 Transfer function Time-domain impulse response 0

58. Control System Engineering-2008 57 Transfer function Time-domain impulse response Position of poles and zeros -a j i b 0 0

59. Control System Engineering-2008 58 Transfer function Time-domain impulse response Position of poles and zeros j i b 0 0

60. Control System Engineering-2008 59 Position of poles and zeros -a j i 0 Transfer function Time-domain impulse response

61. Control System Engineering-2008 60 Transfer function ： Time-domain dynamic response Position of poles and zeros -a j i b 0 0

62. Control System Engineering-2008 61 Summary of pole position & system dynamics

63. Control System Engineering-2008 62 Note: stability of linear single-input, single-output systems is completely governed by the roots of the characteristics equation. Characteristic equation -obtained by setting the denominator polynomial of the transfer function to zero

64. Control System Engineering-2008 63 Transfer function(TF) models in MATLAB Suppose a linear SISO system with input u(t), output y(t), the transfer function of the system is Descending power of s TF in polynomial form >> Sys = tf （ num ， den ） >> [num, den] = tfdata (sys)

65. Control System Engineering-2008 64 TF in zero-pole form >> sys = zpk （ z, p, k ） >> [z, p,k] = tfdata (sys) Transform TS from zero-pole form into polynomial form >> [z, p, k] = tf2zp(num, den)

66. Control System Engineering-2008 Review questions What is the definition of “transfer function”? When defining the transfer function, what happens to initial conditions of the system? Does a nonlinear system have a transfer function? How does a transfer function of a LTI system relate to its impulse response? Define the characteristic equation of a linear system in terms of the transfer function. 65

67. Control System Engineering-2008 66 2-4 Block diagram and Signal- flow graph (SFG)

68. Control System Engineering-2008 Block diagram The transfer function relationship 67 can be graphically denoted through a block diagram. G(s) U(s)Y(s)

69. Control System Engineering-2008 68 Equivalent transform of block diagram 1 Connection in series G(s) U(s) Y(s) X(s) G 1 (s) G 2 (s) U(s) Y(s)

70. Control System Engineering-2008 69 2.Connection in parallel G(s) U(s) Y(s) U(s) G 2 (s) G 1 (s) Y 1 (s) Y 2 (s) Y(S)

71. Control System Engineering-2008 70 3. Negative feedback M(s) R(s) Y(s) G(s) H(s) U(s) R(s) _ Transfer function of a negative feedback system:

72. Control System Engineering-2008 71 Signal flow graph (SFG) SFG was introduced by S.J. Mason for the cause- and-effect representation of linear systems. 1.Each signal is represented by a node. 2.Each transfer function is represented by a branch. G(s) U(s)Y(s) G(s) H(s) U(s) R(s) _ Y(s) G(s) U(s) Y(s) G(s) U(s) Y(s)R(s) -H(s) 1

73. Control System Engineering-2008 Block diagram and its equivalent signal-flow graph 72 － － － －1－1 －1－1 －1－1 1 1 1

74. Control System Engineering-2008 Note 73 A signal flow graph and a block diagram contain exactly the same information (there is no advantage to one over the other; there is only personal preference)

75. Control System Engineering-2008 Mason’s rule 74 path gain of the kth forward path. value of ∆ for that part of the block diagram that does not touch the kth forward path. N total number of forward paths between output Y(s) and input U(s)

76. Control System Engineering-2008 75 Example 1 Find the transfer function for the following block diagram Solution. 1236 12346 123456 b1b1 1/s a3a3 b2b2 b3b3 a2a2 a1a1 + _ _ _ + + + Y(s) U(s) ① ② ③ ④⑤ ⑥

77. Control System Engineering-2008 76 Example 1 b1b1 1/s a3a3 b2b2 b3b3 a2a2 a1a1 + _ _ _ + + + Y(s) U(s) ① ② ③ ④⑤ ⑥ Find the transfer function for the following block diagram Solution. Applying Mason’s rule, we find the transfer function to be

78. Control System Engineering-2008 77 Example 2 Find the transfer function for the following SFG Solution. 1 1 ① ② ③ ④ ⑤ ⑥

79. Control System Engineering-2008 78 Solution. Applying Mason’s rule, we find the transfer function to be Example 2 Find the transfer function for the following SFG 1 1 ① ② ③ ④ ⑤ ⑥