Control Engineering Review Slides: Modeling Electrical and Mechanical Systems

1. Lecture #9Control EngineeringREVIEW SLIDESReference: Textbook by Phillips and Habor

2. Mathematical Modeling

3. Models of Electrical Systems • R-L-C series circuit, impulse voltage source:

4. Model of an RLC parallel circuit:

5. Kirchhoff’ s voltage law: The algebraic sum of voltages around any closed loop in an electrical circuit is zero. • Kirchhoff’ s current law: The algebraic sum of currents into any junction in an electrical circuit is zero.

6. Models of Mechanical Systems Mechanical translational systems. • Newton’s second law: • Device with friction (shock absorber): B is damping coefficient. • Translational system to be defined is a spring (Hooke’s law): K is spring coefficient

7. Model of a mass-spring-damper system: • Note that linear physical systems are modeled by linear differential equations for which linear components can be added together. See example of a mass-spring-damper system.

8. Simplified automobile suspension system:

9. Mechanical rotational systems. • Moment of inertia: • Viscous friction: • Torsion:

10. Model of a torsional pendulum (pendulum in clocks inside glass dome); Moment of inertia of pendulum bob denoted by J Friction between the bob and air by B Elastance of the brass suspension strip by K

11. Differential equations as mathematical models of physical systems: similarity between mathematical models of electrical circuits and models of simple mechanical systems (see model of an RCL circuit and model of the mass-spring-damper system).

12. Laplace Transform

14. Find the inverse Laplace transform ofF(s)=5/(s2+3s+2). Solution:

15. Find inverse Laplace Transform of

16. Find the inverse Laplace transform of F(s)=(2s+3)/(s3+2s2+s). • Solution:

17. Laplace Transform Theorems

18. Transfer Function

19. Transfer Function • After Laplace transform we have X(s)=G(s)F(s) • We call G(s) the transfer function.

20. System interconnections • Series interconnection Y(s)=H(s)U(s) where H(s)=H1(s)H2(s). • Parallel interconnection Y(s)=H(s)U(s) where H(s)=H1(s)+H2(s).

21. Feedback interconnection

22. Transfer function of a servo motor:

23. Mason’s Gain Formula • This gives a procedure that allows us to find the transfer function, by inspection of either a block diagram or a signal flow graph. • Source Node: signals flow away from the node. • Sink node: signals flow only toward the node. • Path: continuous connection of branches from one node to another with all arrows in the same direction.

24. Loop: a closed path in which no node is encountered more than once. Source node cannot be part of a loop. • Path gain: product of the transfer functions of all branches that form the loop. • Loop gain: products of the transfer functions of all branches that form the loop. • Nontouching: two loops are non-touching if these loops have no nodes in common.

26. An Example • Loop 1 (-G2H1) and loop 2 (-G4H2) are not touching. • Two forward paths:

28. State Variable System:

29. Solutions of state equations:

30. Responses

31. System Responses (Time Domain) • First order systems: Transient response Steady state response Step response Ramp response Impulse response • Second order systems Transient response Steady state response Step response Ramp response Impulse response

32. Time Responses of first order systems The T.F. for first order system:

34. is called the time constant • Ex. Position control of the pen of a plotter fora digital computer: is too slow, is faster.

35. System DC Gain • In general: