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- 1. 1 Modeling in the Frequency Domain• Review of the Laplace transform• Learn how to find a mathematical model, called a transfer function, for linear, time-invariant electrical, mechanical, and electromechanical systems• Learn how to linearize a nonlinear system in order to find the transfer function Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
- 2. 2 Introduction to Modeling• we look for a mathematical representation where the input, output, and system are distinct and separate parts • also, we would like to represent conveniently the interconnection of several subsystems. For example, we would like to represent cascaded interconnections, where a mathematical function, called a transfer function, is inside each block Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
- 3. 3 Laplace Transform Review• a system represented by a differential equation is difficult to model as a block diagram• on the other hand, a system represented by a Laplace transformed differential equation is easier to model as a block diagram The Laplace transform is defined as where s = σ + jω, is a complex variable. Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
- 4. 4 Laplace Transform ReviewThe inverse Laplace transform, which allows usto find f (t) given F(s), is where Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
- 5. 5Laplace Transform Review Laplace transform table Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
- 6. 6Problem: Find the Laplace transform ofSolution: Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
- 7. 7Laplace Transform Review Laplace transform theorems Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
- 8. 8Laplace Transform Review Laplace transform theorems Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
- 9. 9 Partial-Fraction Expansion• To find the inverse Laplace transform of a complicated function, we can convert the function to a sum of simpler terms for which we know the Laplace transform of each term. N ( s) F ( s) = D( s) Case 1: Roots of the Denominator of F(s) Are Real and Distinct Case 2: Roots of the Denominator of F(s) Are Real and Repeated Case 3: Roots of the Denominator of F(s) Are Complex or Imaginary Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
- 10. 10Case 1: Roots of the Denominator of F(s) Are Real and Distinct If the order of N(s) is less than the order of D(s), then Thus, if we want to find Km, we multiply above equation by ( s + pm ) Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
- 11. 11If we substitute s = − pm in the above equation, then we can find Km Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
- 12. 12Problem Given the following differential equation, solve for y(t) ifall initial conditions are zero. Use the Laplace transform. Solution Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
- 13. 13Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
- 14. 14Case 2: Roots of the Denominator of F(s) Are Real and Repeated First, we multiply by (s+ p )r and we can solve immediately for K1 if s= - p1 1 Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
- 15. 15• we can solve for K2 if we differentiate F1(s) with respect to s and then let s approach –p1• subsequent differentiation allows to find K3 through Kr• the general expression for K1 through Kr for the multiple roots is Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
- 16. 16Case 3: Roots of the Denominator of F(s) Are Complex or Imaginary has complex or pure imaginary roots • the coefficients K2, K3 are found through balancing the coefficients of the equation after clearing fractions (K 2s + K3 ) is put in the form of by completing ( s + as + b) 2 the squares on ( s 2 + as + b) and adjusting the numerator Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
- 17. 17Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
- 18. 18 The Transfer Function (1)Let us consider a general nth-order, linear, time-invariant differentialequation is given as:where c(t) is the output, r(t) is the input, and ai, bi are coefficients.Taking the Laplace transform of both sides (assuming all initialconditions are zero) we obtain Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
- 19. 19 The Transfer Function (2)Transfer function is the ratio G(s) of the output transform, C(s),divided by the input transform, R(s) C ( s) (bm s m + bm −1s m −1 + ... + b0 ) = G ( s) = R( s) (an s n + an −1s n −1 + ... + a0 ) Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
- 20. 20 Electric Network Transfer FunctionsVoltage-current, voltage-charge, and impedance relationships for capacitors,resistors, and inductors Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
- 21. 21Problem Find the transfer function relating the capacitor voltage,Vc(s), to the input voltage, V(s) Solution: Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
- 22. 22Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
- 23. 23Transfer function-single loop via transform methods Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
- 24. 24Complex Electric Circuits via Mesh Analysis 1. Replace passive element values with their impedances. 2. Replace all sources and time variables with their Laplace transform. 3. Assume a transform current and a current direction in each mesh. 4. Write Kirchhoffs voltage law around each mesh. 5. Solve the simultaneous equations for the output. 6. Form the transfer function Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
- 25. 25Problem: find the transfer function, I2(s)/V(s) Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
- 26. 26Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
- 27. 27Complex Electric Circuits via Nodal Analysis1. Replace passive element values with their admittances2. Replace all sources and time variables with their Laplace transform.3. Replace transformed voltage sources with transformed current sources.4. Write Kirchhoffs current law at each node.5. Solve the simultaneous equations for the output.6. Form the transfer function. Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
- 28. 28 Operational AmplifiersInverting OP AMP Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
- 29. 29Non-inverting OP AMP Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
- 30. 30Example Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
- 31. 31 Translational Mechanical System Transfer FunctionsNewtons law: The sum of forces on a body equals zero; the sum ofmoments on a body equals zero. K… spring constant fv … coefficient of viscious friction M … mass Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
- 32. 32Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
- 33. 33Problem Find the transfer function, X(s)/F(s), for the systemSolution: Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
- 34. 34 Translational Mechanical System Transfer Functions• The required number of equations of motion is equal to the number of linearly independent motions. Linear independence implies that a point of motion in a system can still move if all other points of motion are held still.• Another name for the number of linearly independent motions is the number of degrees of freedom. Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
- 35. 35 Problem: Find the transfer function, X2(s)/F(s), of the systemSolution• we draw the free-body diagram for each point of motion and then use superposition• for each free-body diagram we begin by holding all other points of motion still and finding the forces acting on the body due only to its own motion.• then we hold the body still and activate the other points of motion one at a time, placing on the original body the forces created by the adjacent motion Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
- 36. 36a. Forces on M1 due onlyto motion of M1b. forces on M1 due only tomotion of M2 c. all forces on M1 Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
- 37. 37a. Forces on M2 due onlyto motion of M2; b. forces on M2 due only to motion c. all forces on M2 Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
- 38. 38Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
- 39. 39Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
- 40. 40 Rotational Mechanical System Transfer FunctionsTorque-angular velocity, torque-angular displacement, andimpedance rotational relationships for springs, viscousdampers, and inertia • the mass is replaced by inertia • the values of K, D, and J are called spring constant, coefficient of viscous friction, and moment of inertia, respectively • the concept of degrees of freedom is the same as for translational movement Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
- 41. 41Problem Find the transfer functionSolution a. Torques on J1 due only to the motion of J1 Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
- 42. 42b. torques on J1due only to themotion of J2c. final free-bodydiagram for J1 Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
- 43. 43a. Torques on J1due only to themotion of J1b. torques on J1due only to themotion of J2c. final free-bodydiagram for J1 Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
- 44. 44a. Torques on J2due only to themotion of J2b. torques on J2due only to themotion of J1c. final free-bodydiagram for J2 Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
- 45. 45Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
- 46. 46Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
- 47. 47Transfer Functions for Systems with Gears- the distance travelled along each gears circumference is the same θ 2 r1 N1 = = θ1 r2 N 2- the energy into Gear 1 equals the energy out of Gear 2 T2 θ1 N 2 = = T1 θ 2 N1 Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
- 48. 48T2 θ1 N 2 = =T1 θ 2 N1 Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
- 49. 49Rotational mechanical impedances can be reflected through geartrains by multiplying the mechanical impedance by the ratio Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
- 50. 50A gear train is used to implement large gear ratios by cascadingsmaller gear ratios. Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
- 51. 51Problem Find the transfer function, Solution: Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
- 52. 52Electromechanical System Transfer FunctionsElectromechanical systems: robots, hard disk drives, …Motor is an electromechanical component that yields a displacement outputfor a voltage input, that is, a mechanical output generated by an electricalinput. Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
- 53. 53 Derivation of the Transfer Function of MotorBack electromotive force (back emf): where Kb is a constant of proportionality called the back emf constant; dθ m / dt = ωm (t ) is the angular velocity of the motorTaking the Laplace transform, we get: Torque: Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
- 54. 54The relationship between the armature current, ia(t), the applied armature voltage,ea(t), and the back emf, vb(t), Tm ( s ) I a ( s) = Kt Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
- 55. 55We assume that the armature inductance, La, is small compared to the armatureresistance, Ra, Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
- 56. 56 The mechanical constants Jm and Dm can be found as:Electrical constants Kt/Ra and Kb can be found through a dynamometer test ofthe motor. Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
- 57. 57K t Tstall =Ra ea ea Kb = ωno −load Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
- 58. 58 NonlinearitiesA linear system possesses two properties: superposition and homogeneity. Superposition - the output response of a system to the sum of inputs is the sum of the responses to the individual inputs. Homogeneity - describes the response of the system to a multiplication of the input by a scalar. Multiplication of an input by a scalar yields a response that is multiplied by the same scalar. Linear system Nonlinear system Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
- 59. 59Nonlinearities Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
- 60. 60 Linearization• if any nonlinear components are present, we must linearize the system before we can find the transfer function• we linearize nonlinear differential equation for small-signal inputs about the steady-state solution when the small signal input is equal to zero ma is the slope of the curve at point A Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
- 61. 61Solution: At π/2 df/dx = - 5 Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.

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