ME451: Control Systems Jongeun Choi, Ph.D. Assistant Professor Department of Mechanical Engineering, Michigan State University http://www.egr.msu.edu/classes/me451/jchoi/ http://www.egr.msu.edu/jchoi jchoi@egr.msu.edu 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 2 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 3 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 4 Open-Loop vs. Closed-Loop Control Open-loop Control System Toaster, microwave oven, shoot a basketball Manipulated variable Signal Input Controller output Plant (Actuator) Calibration is the key! Can be sensitive to disturbances ME451 S07 5 Open-Loop vs. Closed-Loop Control Closed-loop control system Driving, cruise control, home heating, guided missile Signal Input + Manipulated variable Error Controller - output Plant (Actuator) Sensor ME451 S07 6 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 + - Actuator Plant Algorithm Sensor ME451 S07 7 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 8 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 9 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 ME451 S07 Catastrophe 10 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 11 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 12 Convolution Integral with Impulse Input signal u(t) ME451 S07 13 Output Signal of a Linear System Input signal Output signal Superposition! def: impulse response def: convolution def: causality ME451 S07 14 Impulse Response ME451 S07 15 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 16 Causal System (Physically Realizable) past future current ME451 S07 System past future current 17 Causal System? Derivative operator (input: position, output: velocity) Integral operator (input: velocity, output: position) ME451 S07 18 Complex Numbers Ordered pair of two real numbers Conjugate Addition Multiplication ME451 S07 19 Complex Numbers Euler’s identity Polar form Magnitude Phase ME451 S07 20 ME451 S07 21 Transfer Function: Laplace Transform of Unit Impulse Response of the System Input signal: Output signal: def: Transfer Function Take Laplace transform of the impulse response ME451 S07 22 Frequency Response Input We know Complex numbers Magnitude Phase shift ME451 S07 23 Frequency Response ME451 S07 24 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 25 The Laplace Transform (Appendix B) Laplace Transform of a function f(t) Convolution integral ME451 S07 26 Properties of Laplace Transforms (page 641-643) Linearity Time Delay Non-rational function ME451 S07 27 Properties of Laplace Transforms Shift in Frequency Differentiation ME451 S07 28 Properties of Laplace Transforms Differentiation ( Integration ( ME451 S07 in time domain , s in Laplace domain) in time domain , 1/s in Laplace domain) 29 Laplace Transform of Impulse and Unit Step Impulse Unit Step ME451 S07 30 Unit Ramp ME451 S07 31 Exponential Function ME451 S07 32 Sinusoidal Functions ME451 S07 33 Partial-fraction Expansion (Text, page 637-641) F(s) is rational, realizable) realizable condition (d/dt is not zeros poles ME451 S07 34 Cover-up Method Check the repeated root for the partial-fraction expansion (page 638) ME451 S07 35 Example Obtain y(t)? ME451 S07 36 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 Laplace transform Inverse Laplace transform Frequency domain ME451 S07 37 Mass-Spring-Damper System ODE Assume all initial conditions are zero. Then take Laplace transform, Output Transfer function Input ME451 S07 38 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 39 Kirchhoff’s Voltage Law The algebraic sum of voltages around any closed loop in an electrical circuit is zero. ME451 S07 40 Kirchhoff’s Current Law The algebraic sum of currents into any junction in an electrical circuit is zero. ME451 S07 41 Theorems Initial Value Theorem Final Value Theorem If all poles of sF(s) are in the left half plane, then ME451 S07 42 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 43 Pure Integrator Impulse response Step response ME451 S07 44 First Order System Impulse response Step response DC gain: (Use final value theorem) ME451 S07 45 Matlab Simulation impulse(G) Impulse Response Amplitude G=tf([0 5],[1 2]); 5 4.5 4 3.5 3 2.5 2 1.5 1 0.5 0 0 0.5 2.5 3 2.5 3 Step Response 2.5 2 Amplitude step(G) 1 1.5 2 Time (sec) 1.5 1 0.5 0 0 ME451 S07 0.5 1 1.5 2 Time (sec) 46 Second Order Systems with Complex Poles Assume Poles: ME451 S07 47 Second Order Systems with Complex Poles ME451 S07 48 Impulse Response of the 2nd Order System ME451 S07 49 Matlab Simulation zeta = 0.3; wn=1; G=tf([wn],[1 2*zeta*wn wn^2]); impulse(G) Amplitude 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 -0.1 -0.2 -0.3 0 2 Impulse Response ME451 S07 4 6 8 10 12 14 16 18 20 Time (sec) 50 Unit Step Response of the 2nd Order System DC gain ME451 S07 51 Unit Step Response (page 122) ME451 S07 52 Matlab Simulation zeta = 0.3; wn=1; G=tf([wn],[1 2*zeta*wn wn^2]); step(G) 1.4 Step Response Amplitude 1.2 1 0.8 0.6 0.4 0.2 0 0 2 4 6 8 10 12 14 16 18 20 Time (sec) ME451 S07 53 Laplace Transform Table ME451 S07 54 Laplace Transform Table ME451 S07 55 Laplace Transform Table ME451 S07 56 Laplace Transform Table ME451 S07 57 Resistance Voltage Source Kirchhoff’s voltage law: Current Source ME451 S07 58 Linearization of nonlinear systems Identify an operating point Perform Taylor series expansion and keep only constant and 1st derivative terms For a nonlinear function ME451 S07 linearized around 59 Linearization Define Linearize ME451 S07 at 60