Course Information (Syllabus)

1. ME451: Control SystemsJongeun Choi, Ph.D.Assistant ProfessorDepartment of Mechanical Engineering, Michigan State Universityhttp://www.egr.msu.edu/classes/me451/jchoi/http://www.egr.msu.edu/jchoijchoi@egr.msu.edu

2. Course Information (Syllabus) • Lecture: 2205 EB, Sections: 5, 6, 7, 8, MWF 12:40-1:30pm • Class website: http://www.egr.msu.edu/classes/me451/jchoi/ • Laboratory website: http://www.egr.msu.edu/classes/me451/radcliff/lab • Class Instructor:Jongeun Choi, Assisntant Professor, 2459 EB, Email: jchoi@egr.msu.edu • Office Hours of Dr. Choi:2459 EB, MW 01:40-2:30pm, Extra hours by appointment only (via email) • Laboratory Instructor: Professor C. J. Radcliffe, 2445 EB, Phone: (517)-355-5198 • Required Text: Feedback Control Systems,C. L. Phillips and R. D. Harbor, Prentice Hall, 4th edition, 2000, ISBN 0-13-949090-6 • Grading: Homework (15%), Exam 1 (15%), Exam 2 (15%), Final Exam(comprehensive) (30%), Laboratory work (25%) • Note • Homework will be done in one week from the day it is assigned. • 100% laboratory attendance and 75% marks in the laboratory reports will be required to pass the course. • Laboratory groups for all sections will be posted on the door of 1532 EB. ME451 S07

3. About Your Instructor • Ph.D. (‘06) in Mechanical Engineering, UC Berkeley • Major field: Controls, Minor fields: Dynamics, Statistics • M.S. (‘02) in Mechanical Engineering, UC Berkeley • B.S. (‘98) in Mechanical Design and Production Engineering, Yonsei University at Seoul, Korea • Research Interests: Adaptive, learning, distributed and robust control, with applications to unsupervised competitive algorithms, self-organizing systems, distributed learning coordination algorithms for autonomous vehicles, multiple robust controllers, and micro-electromechanical systems (MEMS) • 2459 EB, Phone: (517)-432-3164, Email: jchoi@egr.msu.edu, Website: http://www.egr.msu.edu/~jchoi/ ME451 S07

4. Motivation • A control system is an interconnected system to manage, command, direct or regulate some quantity of devices or systems. • Some quantity: temperature, speed, distance, altitude, force • Applications • Heater, hard disk drives, CD players • Automobiles, airplane, space shuttle • Robots, unmanned vehicles, ME451 S07

5. Open-Loop vs. Closed-Loop Control • Open-loop Control System • Toaster, microwave oven, shoot a basketball • Calibration is the key! • Can be sensitive to disturbances Manipulated variable Signal Input output Controller (Actuator) Plant ME451 S07

6. Open-Loop vs. Closed-Loop Control • Closed-loop control system • Driving, cruise control, home heating, guided missile Manipulated variable output Signal Input Error Controller (Actuator) Plant + - Sensor ME451 S07

7. Feedback Control • Compare actual behavior with desired behavior • Make corrections based on the error difference • The sensor and the actuator are key elements of a feedback loop • Design control algorithm Signal Input Error output Control Algorithm Plant Actuator + - Sensor ME451 S07

8. Common Control Objectives • Regulation (regulator): maintain controlled output at constant setpoint despite disturbances • Room temperature control, • Cruise control • Tracking (servomechanism): controlled output follows a desired time-varying trajectory despite disturbances • Automatic landing aircraft, • Hard disk drive data track following control ME451 S07

9. Control Problem • Design Control Algorithm • such that the closed-loop system meets certain performance measures, and specifications • Performance measures in terms of • Disturbance rejection • Steady-state errors • Transient response • Sensitivity to parameter changes in the plant • Stability of the closed-loop system ME451 S07

10. Why the Stability of the Dynamical System? • Engineers are not artists: • Code of ethics, Responsibility • Otherwise, Tacoma Narrows Bridge: Nov. 7, 1940 Wind-induced vibrations Catastrophe ME451 S07

11. Linear (Dynamical) Systems • H is a linear system if it satisfies the properties of superposition and scaling: • Inputs: • Outputs: • Superposition: • Scaling: • Otherwise, it is a nonlinear system ME451 S07

12. Why Linear Systems? • Easier to understand and obtain solutions • Linear ordinary differential equations (ODEs), • Homogeneous solution and particular solution • Transient solution and steady state solution • Solution caused by initial values, and forced solution • Add many simple solutions to get more complex ones (Utilize superposition and scaling!) • Easy to check the Stability of stationary states (Laplace Transform) • Even nonlinear systems can be approximated by linear systems for small deviations around an operating point ME451 S07

13. Convolution Integral with Impulse • Input signal u(t) ME451 S07

14. Output Signal of a Linear System • Input signal • Output signal Superposition! def: impulse response def: convolution def: causality ME451 S07

15. Impulse Response ME451 S07

16. Causal Linear Time Invariant (LTI) System • A causal system (a physical or nonanticipative system) is a system where the output only depends on the input values • Thus, the current output can be generated by the causal system with the current and past input values • Causal LTI impulse response • Thus, we have ME451 S07

17. Causal System (Physically Realizable) past future past future System current current ME451 S07

18. Causal System? • Derivative operator (input: position, output: velocity) • Integral operator (input: velocity, output: position) ME451 S07

19. Complex Numbers • Ordered pair of two real numbers • Conjugate • Addition • Multiplication ME451 S07

20. Complex Numbers • Euler’s identity • Polar form • Magnitude • Phase ME451 S07

21. ME451 S07

22. Transfer Function: Laplace Transform of Unit Impulse Response of the System • Input signal: • Output signal: • Take def: Transfer Function Laplace transform of the impulse response ME451 S07

23. Frequency Response • Input • We know • Complex numbers Magnitude Phase shift ME451 S07

24. Frequency Response ME451 S07

25. The Laplace Transform (Appendix B) • Laplace transform converts a calculus problem (the linear differential equation) to an algebra problem • How to Use it: • Take the Laplace transform of a linear differential equation • Solve the algebra problem • Take the Inverse Laplace transform to obtain the solution to the original differential equation def: Laplace transform def: Inverse Laplace transform ME451 S07

26. The Laplace Transform (Appendix B) • Laplace Transform of a function f(t) • Convolution integral ME451 S07

27. Properties of Laplace Transforms (page 641-643) • Linearity • Time Delay Non-rational function ME451 S07

28. Properties of Laplace Transforms • Shift in Frequency • Differentiation ME451 S07

29. Properties of Laplace Transforms • Differentiation ( in time domain , s in Laplace domain) • Integration ( in time domain , 1/s in Laplace domain) ME451 S07

30. Laplace Transform of Impulse and Unit Step • Impulse • Unit Step ME451 S07

31. Unit Ramp ME451 S07

32. Exponential Function ME451 S07

33. Sinusoidal Functions ME451 S07

34. Partial-fraction Expansion (Text, page 637-641) • F(s) is rational, realizable condition (d/dt is not realizable) zeros poles ME451 S07

35. Cover-up Method • Check the repeated root for the partial-fraction expansion (page 638) ME451 S07

36. Example • Obtain y(t)? ME451 S07

37. Transfer Function • Defined as the ratio of the Laplace transform of the output signal to that of the input signal (think of it as a gain factor!) • Contains information about dynamics of a Linear Time Invariant system • Time domain • Frequency domain Laplace transform Inverse Laplace transform ME451 S07

38. Mass-Spring-Damper System • ODE • Assume all initial conditions are zero. Then take Laplace transform, Output Transfer function Input ME451 S07

39. Transfer Function • Differential equation replaced by algebraic relation Y(s)=H(s)U(s) • If U(s)=1 then Y(s)=H(s) is the impulse response of the system • If U(s)=1/s, the unit step input function, then Y(s)=H(s)/s is the step response • The magnitude and phase shift of the response to a sinusoid at frequency is given by the magnitude and phase of the complex number • Impulse: • Unit step: ME451 S07

40. Kirchhoff’s Voltage Law • The algebraic sum of voltages around any closed loop in an electrical circuit is zero. ME451 S07

41. Kirchhoff’s Current Law • The algebraic sum of currents into any junction in an electrical circuit is zero. ME451 S07

42. Theorems • Initial Value Theorem • Final Value Theorem • If all poles of sF(s) are in the left half plane, then ME451 S07

43. DC Gain of a System • DC gain: the ratio of the steady state output of a system to its constant input (1/s) • For a stable transfer function • Use final value theorem to compute the steady state of the output ME451 S07

44. Pure Integrator • Impulse response • Step response ME451 S07

45. First Order System • Impulse response • Step response • DC gain: (Use final value theorem) ME451 S07

46. Matlab Simulation • G=tf([0 5],[1 2]); • impulse(G) • step(G) ME451 S07

47. Second Order Systems with Complex Poles • Assume • Poles: ME451 S07

48. Second Order Systems with Complex Poles ME451 S07

49. Impulse Response of the 2nd Order System ME451 S07

50. Matlab Simulation • zeta = 0.3; wn=1; • G=tf([wn],[1 2*zeta*wn wn^2]); • impulse(G) ME451 S07