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 [e-mail]

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ME451 Control SystemsJongeun Choi,
Ph.D.Assistant Professor Department of
Mechanical Engineering, Michigan State
Universityhttp//www.egr.msu.edu/classes/me451/j
choi/http//www.egr.msu.edu/jchoijchoi_at_egr.msu
.edu
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Course Information (Syllabus)

• Lecture 2205 EB, Sections 5, 6, 7, 8, MWF
  1240-130pm
• Class website http//www.egr.msu.edu/classes/me45
  1/jchoi/
• Laboratory website http//www.egr.msu.edu/classes
  /me451/radcliff/lab
• Class Instructor Jongeun Choi, Assisntant
  Professor, 2459 EB, Email jchoi_at_egr.msu.edu
• Office Hours of Dr. Choi 2459 EB, MW
  0140-230pm, Extra hours by appointment only
  (via email)
• Laboratory Instructor Professor C. J. Radcliffe,
  2445 EB, Phone [phone]
• 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.

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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 [phone], Email
  jchoi_at_egr.msu.edu, Website http//www.egr.msu.edu
  /jchoi/

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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,

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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
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Open-Loop vs. Closed-Loop Control

• Closed-loop control system
• Driving, cruise control, home heating, guided
  missile

Manipulated variable
Signal Input
output
Error
Controller (Actuator)
Plant

-
Sensor
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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
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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

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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

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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
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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

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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

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Convolution Integral with Impulse

• Input signal u(t)

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Output Signal of a Linear System

• Input signal
• Output signal

Superposition!
def impulse response
def convolution
def causality
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Impulse Response
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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

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Causal System (Physically Realizable)
future
past
past
future
System
current
current
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Causal System?

• Derivative operator (input position, output
  velocity)
• Integral operator (input velocity, output
  position)

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Complex Numbers

• Ordered pair of two real numbers
• Conjugate
• Addition
• Multiplication

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Complex Numbers

• Eulers identity
• Polar form
• Magnitude
• Phase

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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
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Frequency Response

• Input
• We know
• Complex numbers

Magnitude
Phase shift
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Frequency Response
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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
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The Laplace Transform (Appendix B)

• Laplace Transform of a function f(t)
• Convolution integral

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Properties of Laplace Transforms (page 641-643)

• Linearity
• Time Delay

Non-rational function
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Properties of Laplace Transforms

• Shift in Frequency
• Differentiation

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Properties of Laplace Transforms

• Differentiation ( in time domain , s in
  Laplace domain)
• Integration ( in time domain , 1/s in
  Laplace domain)

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Laplace Transform of Impulse and Unit Step

• Impulse
• Unit Step

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Unit Ramp
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Exponential Function
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Sinusoidal Functions
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Partial-fraction Expansion (Text, page 637-641)

• F(s) is rational, realizable condition (d/dt
  is not realizable)

zeros
poles
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Cover-up Method

• Check the repeated root for the partial-fraction
  expansion (page 638)

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Example

• Obtain y(t)?

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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
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Mass-Spring-Damper System

• ODE
• Assume all initial conditions are zero. Then take
  Laplace transform,

Output
Transfer function
Input
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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

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Kirchhoffs Voltage Law

• The algebraic sum of voltages around any closed
  loop in an electrical circuit is zero.

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Kirchhoffs Current Law

• The algebraic sum of currents into any junction
  in an electrical circuit is zero.

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Theorems

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

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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

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Pure Integrator

• Impulse response
• Step response

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First Order System

• Impulse response
• Step response
• DC gain (Use final value theorem)

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Matlab Simulation

• Gtf(0 5,1 2)
• impulse(G)
• step(G)

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Second Order Systems with Complex Poles

• Assume
• Poles

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Second Order Systems with Complex Poles
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Impulse Response of the 2nd Order System
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Matlab Simulation

• zeta 0.3 wn1
• Gtf(wn,1 2zetawn wn2)
• impulse(G)

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Unit Step Response of the 2nd Order System

• DC gain

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Unit Step Response (page 122)
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Matlab Simulation

• zeta 0.3 wn1 Gtf(wn,1 2zetawn wn2)
• step(G)

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Laplace Transform Table
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Laplace Transform Table
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Laplace Transform Table
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Laplace Transform Table
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Resistance

• Voltage Source
• Kirchhoffs voltage law
• Current Source

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Linearization of nonlinear systems

• Identify an operating point
• Perform Taylor series expansion and keep only
  constant and 1st derivative terms
• For a nonlinear function linearized
  around

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Linearization

• Define
• Linearize at