Download ME451 - MSU College of Engineering

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
[email protected]
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:
[email protected]

Office Hours of Dr. Choi: 2459 EB, MW 01:40-2:30pm, Extra hours by appointment
only (via email)
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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.
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100% laboratory attendance and 75% marks in the laboratory reports will be
required to pass the course.
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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: (517)-432-3164, Email: [email protected],
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
Manipulated
variable
Signal Input
Controller
output
Plant
(Actuator)
 Calibration is the key!
 Can be sensitive to disturbances
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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
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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
+
-
Actuator
Plant
Algorithm
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
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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)
past
future
current
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System
past
future
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
 Euler’s identity
 Polar form
 Magnitude
 Phase
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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
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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 (
 Integration (
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in time domain , s in Laplace domain)
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)
realizable condition (d/dt is not
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
Laplace transform
Inverse Laplace transform
 Frequency domain
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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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Kirchhoff’s Voltage Law
 The algebraic sum of voltages around any closed loop in an
electrical circuit is zero.
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Kirchhoff’s 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
 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
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0.5
1
1.5
2
Time (sec)
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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; 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
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Time (sec)
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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; 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)
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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
 Kirchhoff’s 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
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linearized around
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Linearization
 Define
 Linearize
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at
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