Mathematical Models and Block Diagrams of Systems Regulation And Control Engineering

1. Mathematical Models and Block Diagrams of Systems Regulation And Control Engineering

2. Introduction Differential Equations of Physical Systems The Laplace Transform Transfer Function of Linear Systems Block Diagram Contents

3. Define the system and its components Formulate the mathematical model and list the necessary assumptions Write the differential equations describing the model Solve the equations for the desired output variables Examine the solutions and the assumptions If necessary, reanalyze or redesign the system Step and Procedure

4. A mathematical model is a set of equations (usually differential equations) that represents the dynamics of systems. In practice, the complexity of the system requires some assumptions in the determination model. The equations of the mathematical model may be solved using mathematical tools such as the Laplace Transform. Before solving the equations, we usually need to linearize them. Introductions

5. Differential Equations • Examples: How do we obtain the equations? Physical law of the process  Differential Equation Mechanical system (Newton’s laws) Electrical system (Kirchhoff’s laws)

6. Example: Springer-mass-damper system Assumption: Wall friction is a viscous force. Differential Equations The time function of r(t) sometimes called forcing function Linearly proportional to the velocity

7. Example: Springer-mass-damper system Newton’s 2nd Law: Differential Equations

8. Example: RLC Circuit Differential Equations

9. The differential equations are transformed into algebraic equations, which are easier to solve. The Laplace transformation for a function of time, f(t) is: If, , then, Similarly, Thus, The Laplace Transform

10. Example: Spring-mass-damper dynamic equation The Laplace Transform Laplace Transform for the equation above: When r(t)=0, y(0)= y0 and (0)=0:

11. Example: Spring-mass-damper dynamic equation The Laplace Transform Some Definitions q(s) = 0 is called characteristic equation (C.E.) because the roots of this equation determine the character of the time response. The roots of C.E are also called the poles of the system. The roots of numerator polynomial p(s) are called the zeros of the system.

12. Transform table: The Laplace Transform Impulse function Step function Ramp function

13. Transform Properties The Laplace Transform

14. Example: Find the Laplace Transform for the following. • Unit function: • Ramp function: • Step function: The Laplace Transform

15. Transform Theorem • Differentiation Theorem • Integration Theorem: • Initial Value Theorem: • Final Value Theorem: The Laplace Transform

16. The inverse Laplace Transform can be obtained using: Partial fraction method can be used 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 inverse Laplace Transform. The Laplace Transform

17. We will consider three cases and show that F(s) can be expanded into partial fraction: • Case 1: Roots of denominator A(s) are real and distinct. • Case 2: Roots of denominator A(s) are real and repeated. • Case 3: Roots of denominator A(s) are complex conjugate. The Laplace Transform

18. Case 1: Roots of denominator A(s) are real and distinct. Example: Solution: The Laplace Transform It is found that: A = 2 and B = -2

19. Case 1: Roots of denominator A(s) are real and distinct. Problem: Find the Inverse Laplace Transform for the following. The Laplace Transform

20. Case 2: Roots of denominator A(s) are real and repeated. Example: Solution: The Laplace Transform It is found that: A = 2, B = -2 and C = -2

21. Case 3: Roots of denominator A(s) are complex conjugate. Example: Solution: The Laplace Transform It is found that: A = 3/5, B = -3/5 and C = -6/5

22. Case 3: Roots of denominator A(s) are complex conjugate. Example: Solution: The Laplace Transform

23. Problem: Find the solution x(t) for the following differential equations. The Laplace Transform

24. The transfer function of a linear system is the ratio of the Laplace Transform of the output to the Laplace Transform of the input variable. Consider a spring-mass-damper dynamic equation with initial zero condition. The Transfer Function

25. The transfer function is given by the following. The Transfer Function R(s) Y(s)

26. Electrical Network Transfer Function The Transfer Function V-I I-V V-Q Impedance Admittance Component

27. Problem: Obtain the transfer function for the following RC network. The Transfer Function

28. Problem: Obtain the transfer function for the following RLC network. Answer: The Transfer Function

29. Mechanical System Transfer Function Problem: Find the transfer function for the mechanical system below. The Transfer Function • The external force u(t) is the input to the system, and the displacement y(t) of the mass is the output. • The displacement y(t) is measured from the equilibrium position. • The transfer function of the system.

30. A block diagram of a system is a practical representation of the functions performed by each component and of the flow of signals. Cascaded sub-systems: Block Diagram Transfer Function G(s) Input Output

31. Feedback Control System Block Diagram

32. Feedback Control System Therefore, Block Diagram The negative feedback of the control system is given by: Ea(s) = R(s) – H(s)Y(s) Y(s) = G(s)Ea(s)

33. Reduction Rules Block Diagram

34. Reduction Rules Block Diagram

35. Problem: Block Diagram

36. Problem: Block Diagram

37. Chapter 2 • Dorf R.C., Bishop R.H. (2001). Modern Control Systems (9th Ed), Prentice Hall. • Nise N.S. (2004). Control System Engineering (4th Ed), John Wiley & Sons. Further Reading…

38. “The whole of science is nothing more than a refinement of everyday thinking…” The End…