ME 451 Control Systems Jongeun Choi Ph D

ME 451: Control Systems Jongeun Choi, Ph. D. Assistant Professor Department of Mechanical Engineering, Michigan State University http: //www. egr. msu. edu/classes/me 451/jchoi/ http: //www. egr. msu. edu/jchoi@egr. msu. edu

Course Information (Syllabus) § Lecture: 2205 EB, Sections: 5, 6, 7, 8, MWF 12: 40 -1: 30 pm § Class website: http: //www. egr. msu. edu/classes/me 451/jchoi/ § Laboratory website: http: //www. egr. msu. edu/classes/me 451/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: 30 pm, 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, 4 th edition, 2000, ISBN 0 -13 -949090 -6 § Grading: Homework (15%), Exam 1 (15%), Exam 2 (15%), Final § Note Exam(comprehensive) (30%), Laboratory work (25%) § 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. ME 451 S 07 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/ ME 451 S 07 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, ME 451 S 07 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 ME 451 S 07 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 ME 451 S 07 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 + - Algorithm Actuator Plant Sensor ME 451 S 07 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 ME 451 S 07 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 ME 451 S 07 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 ME 451 S 07 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 ME 451 S 07 11

Why Linear Systems? § Easier to understand 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 ME 451 S 07 12

Convolution Integral with Impulse § Input signal u(t) ME 451 S 07 13

Output Signal of a Linear System § Input signal § Output signal Superposition! def: impulse response def: convolution def: causality ME 451 S 07 14

Impulse Response ME 451 S 07 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 ME 451 S 07 16

Causal System (Physically Realizable) past future current ME 451 S 07 System past future current 17

Causal System? § Derivative operator (input: position, output: velocity) § Integral operator (input: velocity, output: position) ME 451 S 07 18

Complex Numbers § Ordered pair of two real numbers § Conjugate § Addition § Multiplication ME 451 S 07 19

Complex Numbers § Euler’s identity § Polar form § Magnitude § Phase ME 451 S 07 20

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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 ME 451 S 07 22

Frequency Response § Input § We know § Complex numbers Magnitude Phase shift ME 451 S 07 23

Frequency Response ME 451 S 07 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 ME 451 S 07 25

The Laplace Transform (Appendix B) § Laplace Transform of a function f(t) § Convolution integral ME 451 S 07 26

Properties of Laplace Transforms (page 641 -643) § Linearity § Time Delay Non-rational function ME 451 S 07 27

Properties of Laplace Transforms § Shift in Frequency § Differentiation ME 451 S 07 28

Properties of Laplace Transforms § Differentiation ( in time domain , s in Laplace domain) § Integration ( in time domain , 1/s in Laplace domain) ME 451 S 07 29

Laplace Transform of Impulse and Unit Step § Impulse § Unit Step ME 451 S 07 30

Unit Ramp ME 451 S 07 31

Exponential Function ME 451 S 07 32

Sinusoidal Functions ME 451 S 07 33

Partial-fraction Expansion (Text, page 637 -641) § F(s) is rational, realizable) realizable condition (d/dt is not zeros poles ME 451 S 07 34

Cover-up Method § Check the repeated root for the partial-fraction expansion (page 638) ME 451 S 07 35

Example § Obtain y(t)? ME 451 S 07 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 ME 451 S 07 37

Mass-Spring-Damper System § ODE § Assume all initial conditions are zero. Then take Laplace transform, Output Transfer function Input ME 451 S 07 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: ME 451 S 07 39

Kirchhoff’s Voltage Law § The algebraic sum of voltages around any closed loop in an electrical circuit is zero. ME 451 S 07 40

Kirchhoff’s Current Law § The algebraic sum of currents into any junction in an electrical circuit is zero. ME 451 S 07 41

Theorems § Initial Value Theorem § Final Value Theorem § If all poles of s. F(s) are in the left half plane, then ME 451 S 07 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 ME 451 S 07 43

Pure Integrator § Impulse response § Step response ME 451 S 07 44

First Order System § Impulse response § Step response § DC gain: (Use final value theorem) ME 451 S 07 45

Matlab Simulation § G=tf([0 5], [1 2]); § impulse(G) § step(G) ME 451 S 07 46

Second Order Systems with Complex Poles § Assume § Poles: ME 451 S 07 47

Second Order Systems with Complex Poles ME 451 S 07 48

Impulse Response of the 2 nd Order System ME 451 S 07 49

Matlab Simulation § zeta = 0. 3; wn=1; § G=tf([wn], [1 2*zeta*wn wn^2]); § impulse(G) ME 451 S 07 50

Unit Step Response of the 2 nd Order System § DC gain ME 451 S 07 51

Unit Step Response (page 122) ME 451 S 07 52

Matlab Simulation § zeta = 0. 3; wn=1; G=tf([wn], [1 2*zeta*wn wn^2]); § step(G) ME 451 S 07 53

Laplace Transform Table ME 451 S 07 54

Laplace Transform Table ME 451 S 07 55

Laplace Transform Table ME 451 S 07 56

Laplace Transform Table ME 451 S 07 57

Resistance § Voltage Source § Kirchhoff’s voltage law: § Current Source ME 451 S 07 58

Linearization of nonlinear systems § Identify an operating point § Perform Taylor series expansion and keep only constant and 1 st derivative terms § For a nonlinear function ME 451 S 07 linearized around 59

Linearization § Define § Linearize ME 451 S 07 at 60