의료기기를 위한 공학이론
Linear Systems & Control
Professor Sungwan Kim
Biomedical Control & Modeling Laboratory (BMC Lab)
Dept. of Biomedical Engineering, College of Medicine
Seoul National University
Seoul National University Hospital
2012년 9월25일
SNU College of Medicine (의예과 2학년)
Outline
• Introduction
• Linear System
• Linear System Theory
• Linear System: Control & Modeling
• Examples #1
– PID Control for Mass-Damper-Spring System
• Examples #2
– Automobile Platooning System
• Examples #3
– Bio-modeling & Simulation for Oculomotor System
• Concluding Remarks
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Linear System (1 of 4)
• System, Sub-system, Sub-sub-system, etc.
• System is defined as a collection of objects linked in some
form of interaction or interdependence
• Static System vs. Dynamic System (Mass-Spring-Damper)
• Communication System
• Transmitter, Antenna, Speaker, Microphone, etc.
• Navigation System
• GPS Satellite, GPS Receiver, Nav Computer, etc.
• Circulation System
• Heart, Atrium, Ventricle, Valve, etc.
• Human System: Circulation Sub-system
• Control System, Power System, etc.
• Aircraft System: Control Sub-system, Power Sub-system
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Mass-Spring-Damper Model or RLC Circuit
 2nd Order System
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Linear System (2 of 4)
• Linear System vs. Non-linear System
• Time-invariant System vs. Time-varying System
Linear Time-Invariant (LTI) System
• Laplace Transformation, Fourier Transformation, etc.
• Why?
• Algebraic Equation, Easy-to-solve, Frequency Domain Analysis
• Mathematical Formulation & Proofs
• Analytic Solution
• Mainly for LTI System
• http://en.wikipedia.org/wiki/Linear_system
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http://en.wikipedia.org/wiki/Linear_system
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http://en.wikipedia.org/wiki/Linear_system
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Linear System (3 of 4)
• Linear System should satisfy the below two characteristics:
• Additivity: the response to {x1(t) + x2(t)} is {y1(t) + y2(t)}
• Scalibility: the response to {ax1(t)} is a{y1(t)}
where y1(t) & y2(t) are responses of the x1(t) & x2(t), respectively
 Therefore, the response to {ax1(t)+bx2(t)} is a{y1(t)} + b{y2(t)}
where a & b are constants
• Quiz: 2* x(t) vs. 4*x2(t)?
• The Linear System satisfies the Principle of Superposition
x[n]  k ak xk [n]  a1 x1[n]  a2 x2 [n]  a3 x3[n]  
 y[n]  k ak yk [n]  a1 y1[n]  a2 y2 [n]  a3 y3[n]  
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Linear System (4 of 4)
• Linear System vs. Non-linear System
• Modeling: Non-linear System  Linearization
• Perturbation Theory
• Piecewise Linearization
• w.r.t. Normal Operating Points
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Linear System Theory
• Linearization
• Modeling & Uncertainty & Noise
• Input-Output System & Sensors & Actuators
• Transfer Function
• State-space Model
• Stability
• Causality
• Controllability
• Observability
• Reachability
• Identifiability
• Invertability
......
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Input / System (or Plant) / Output
Input
Plant
Output
• Control: Find Input for Given Plant Info and Output
• System ID: Find Plant Info for Given Input and Output
• Analysis: Find Output for Given Input and Plant Info
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Conventional Block Diagram
Dynamics
Uncertainty
Command
Measurement
Noise
Input
Regulator
System
Sensor
Measured
Output
Gain
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Example #1: Mass-Spring-Damper Model
(or RLC Circuit)  2nd Order System
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Purpose
• The purpose of this project is to achieve the quickest and
most efficient time response by designing a proportionalintegral-derivative (PID) controller.
• The PID Controller is a feedback control system and a
mass-spring-damper (MSD) model is used for this
experiment.
• A formula for the PID controller is shown below where
KP, KI, and KD denote proportional gain, integral gain,
and derivative gain.
Mathematical and Engineering Goal
• To achieve the engineering goal of finding the quickest
and most efficient time response for a given dynamic
system, a PID controller should be designed.
• A MSD (Mass Spring Damper) Model is used to
represent a dynamic system.
• The first method to achieve the goal is to derive a
Mathematical formula for the MSD Model and PID
controller, then solve the formula analytically with the
Laplace transformation technique.
• The second method to achieve the goal is to solve the
problem with a Simulation technique by conducting
experiment with various sets of proportional gain,
integral gain, and derivative gain. KP, KI, and KD need to
be optimally designed to achieve the engineering goal.
A commercial-off-the-shelf (COTS) simulation tool will be
used to conduct the experiment.
Method
• To conduct the experiment, a MSD Model is defined first. The
MSD Model is chosen in this project because almost all
dynamic systems can be mathematically represented as a MSD
model.
• In the picture of a MSD model, m, k, B, F, and x denote Mass,
Spring Constant, Dampening coefficient, Applied Force, and
Displacement.
• A feedback control system is used to design the PID controller.
A feedback control system was chosen to stabilize the dynamic
system and improve the system performance. A block diagram
of the feedback control system is shown in the picture.
• For an analytical solution, the Laplace Transformation is used.
The Laplace Transformation converts differential equations to
algebraic equations and a Laplace Transformation Table is
used for conversation.
• To conduct an experiment with a Simulation Tool, various gain
sets are used to find the optimal one. Eleven gain sets are
investigated for this experiment.
P, I, D, Denotations
• Proportional gain (KP) has the effect of reducing the rise time
and increasing overshoot and settling time.
• Integral gain (KI) has the effect of eliminating the steady-state
error while increasing settling time.
• Derivative gain (KD) has the effect of decreasing overshoot and
settling time while increasing rise time and steady state error.
CL RESPONSE
RISE TIME
OVERSHOOT
SETTLING TIME
S-S ERROR
KP
Decrease
Increase
Increase
-
KI
-
-
Increase
Eliminate
KD
Increase
Decrease
Decrease
Increase
Time Responses for Three Different Integral Gain Sets
Mathematical Solution
• The first method was to derive a mathematical formula for the MSD Model
and PID controller and then using the formula to find the analytical solution.
Newton’s Law of Motion is applied to the MSD model shown above and the
force equations of the MSD system is written as a standard second order
differential equation:
F = m*a
F(t) – B*(d/dt)*x(t) – k*x(t) = m*(d/dt)2*x(t)
F(t) = m*(d/dt)2*x(t) + B*(d/dt)*x(t) + k*x(t)
where
m = Mass
k = Spring constant
B = Dampening coefficient
F = Applied Force
x = Displacement
• The Laplace transformation is applied to solve the above differential
equation. From the Laplace Transformation Table, the above differential
equation is written below:
F(s) = m*s2*X(s) + B*s*X(s) + k*X(s)
= (m*s 2 + B*s + k) * X(s)
• The above algebraic equation can be solved for the given MSD model (given
m, k, B, and F) below:
X(s) = F(s) / (m*s2 + B*s + k)
• By applying the Inverse Laplace Transformation, time response, x(t), can be
found analytically.
• A PID controller can also be written with the equation below by applying the
Laplace Transformation.
KP + KI/s + KD*s = (s2*KD + s*KP + KI) / s
• From the two equations above and Figure 1, a formula for the MSD model
with a PID controller is written below:
X(s) = F(s) * (KD*s2 + KP*s + KI) / (m*s3 + (B + KD)*s2 + (k + KP)*s + KI)
• By applying the Inverse Laplace Transformation, time response with the PID
controller, x(t), can be found analytically.
Engineering Solution
• For the second method, a commercial-off-the-shelf (COTS) simulation tool
was used and ten gain sets were found to investigate the MSD system
performance. The simulation tool being used in this project the MathWorks’
Matlab Simulation Tool. The eleven cases are:
1 open loop case: without PID controller
3 KP gains (low, nominal, high)
3 KD gains (low, nominal, high)
3 KI gains (low, nominal, high)
1 optimal case (the quickest and most efficient time response)
• After examining the open loop case, a proportional gain (KP) is applied to
reduce a rise time and 3 KPs are investigated to see how the KP changes the
system performance. With a nominal value of KP, a differential gain (KD) is
adjusted to reduce an overshoot and 3 KDs are also investigated to see how
the KD changes the system performance. Then, with a nominal values of KP
and KD, an integral gain (KI) is adjusted to reduce the steady state error.
Then, an optimal gain set is selected based on the experiment.
Engineering Solution
KP: Rise Time
KD: Overshoot
KI: S-S
Error
The Quickest and Most Efficient Time Response
Rise Time: 1.37 sec
Overshoot: N/A
Settling Time: N/A
Steady-state Error: 0.7
Rise Time: 0.13 sec
Overshoot: 16% at 0.35 sec
Settling Time: 0.764 sec
Steady-state Error: 0
With a PID Controller
(KP=200, KI=100, KD=50)
Conclusion
• The purpose of this experiment was to achieve the quickest and
most efficient time response in a mass spring damper model
using two methods. The first method was to derive a
mathematical formula and solve it analytically with the Laplace
transformation technique to find the optimal PID gain set. The
second method was to design a PID controller and simulate to
find the optimal PID gain set. The engineering goal of
achieving the quickest and most efficient time response in a
mass spring damper model using a PID Controller was
successful for both methods reached the optimal PID gain set.
• Throughout the project, many things have been learned and the
acquired knowledge will be a great addition while pursuing
career in Mathematics and Engineering. Concepts utilized in
this project include Dynamics and Control Theory, Feedback
Control System, PID Controller, Time Response, Laplace
Transformation, and simulation tools.
Example #2: The effect of a PID Controller on variable-speed
automobile cruise control to increase safety while driving
100 Yards
Objective: Maintain 100 Yards
Measurement: Distance between two vehicles
V1 = VC + dVPID
VC = 60 MPH
dVDisturb. = [-15, -10, -5, 0, 5, 10, 15]
dVPID = f (Distance, PID Gain Set)
V2 = VC + dVDisturb.
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Background on Cruise Control
• Cruise control is commonly used on highways to maintain a
constant speed and to have more efficient mileage
• Problem 1: However, cruise control leads to laziness on
highways for one does not have to focus on maintaining a
constant speed
• Problem 2: Another problem with cruise control is that it does
not adjust to its environment
• By implementing a feedback control system, in this case a
proportional-integral-differential controller, a variable-speed
automobile cruise control system will be created
• In other words, the Proportional-Integral-Differential controller
will measure variables such as distance and front car velocity,
and adjust the automobile’s velocity and delta speed, rate of
change
Objective and Hypothesis
• Objective - Observe the effects of the implementation
of a Proportional-Integral Differential (PID) controller in
an automobile
• Hypothesis - Automobile velocity and delta speed can
be adjusted using a Proportional-Integral-Differential
controller to maintain the desired distance between
the two vehicles
Variables
• Independent variable – Difference in distance between the front
car and the rear car
• Dependent variables – rear car velocity and delta speed
• Constants – initial conditions of 100 yards between the two
vehicles and initial speeds of 60 miles per hour.
• Extraneous variables – wind velocity and curvature of the road
is assumed to be zero to simply test if a PID controller can be
applied for variable speed cruise control
Method & Matlab Code
• A simulation that incorporates the previously mentioned initial
conditions, randomized front car velocity, distance between the
two vehicles, and the implemented PID controller will be used
to collect the data
• The front car’s change in position will be randomized between
values of -15 and 15
• The front car’s position will be read at one second intervals,
while the rear car itself will adjust its speed according to the PID
controller at one tenth second intervals
• The control that will be used is a standard cruise control setting
with no feedback system to intelligently adjust the car’s own
speed
• Five trials will be performed for the control and five trials will be
performed with the implementation of the PID controller
• Matlab Code: M-file
No Control Case
Collision!
With a PID Controller (KP=0.1, KI=1.0, KD=0.01)
Collision!
With a PID Controller (KP=0.1, KI=1.0, KD=0.01)
Collision!
Optimal vs. Bad PID Gain Case #1 (Slow Response)
KP=0.005, KI=0.5, KD=0.01
KP=0.1, KI=1.0, KD=0.01
Optimal vs. Bad PID Gain Case #2 (Overshoot)
KP=0.5, KI=1.0, KD=0.1
KP=0.1, KI=1.0, KD=0.01
Optimal vs. Bad PID Gain Case #3 (Diverge)
KP=1.0, KI=1.0, KD=0.1
KP=0.1, KI=1.0, KD=0.01
With a PID Controller (KP=0.1, KI=1.0, KD=0.01): Random Case #1
With a PID Controller (KP=0.1, KI=1.0, KD=0.01): Random Case #10
Control System Designer’s Wish vs. Reality
• Model: Perfect
• Sensor: Measure All the Info
• Noise: None
• Actuator: Enough Control Authority
None is Achievable!
 What we Need!
• Understand Physics & Limitations
• Modeling Technique (+ Sys ID) for a Good Model
• Multiple Sensor Suites for Better Measurements
• Design Actuators for a System with Enough Margin ( X 3)
• Simulation & Tests for V&V
39
Modeling Technique
• Modeling
• Mathematical Model
– Unmodelled Dynamics
– Uncertainty  Uncertainty Analysis
• TOC
– Modeling Examples including Oculomotor
– Why need a model?
– Why need a good model?
– How to build a good model?
– How do we know a good model?
– How to use a good model?
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Bio-modeling & Sim: Oculomotor
(Introduction for Biomedical Engineering by John D. Enderle et. al)
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Bio-modeling & Sim: Oculomotor - EoM
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Bio-modeling & Sim: Oculomotor
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Bio-modeling & Sim: Oculomotor
.
.
.
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Bio-modeling & Sim: Oculomotor
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Bio-modeling & Sim: Oculomotor
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Modeling Examples
• Curve Fitting
– Linear Regression: Line (First-order), Y=AX+B, Least Square Technique
– Parabola (Second-order), Third-order, etc.
• Fitting with Single, Quadratic, & Cubic Orders
– Right Order is required
– But, Over-fitting is also an Issue
• Dynamic Systems: Equation of Motion
– Newton’s Law: Mass-Spring-Damper, Gravitational Motion, etc.
• ODE, Time-derivative, dx/dt=f(t), dx/dt=A*x(t)+B*u(t)+F*N(t)
– Kepler's Laws of Planetary Motion
– Thermodynamics
– Complicated Systems: PDE
• Weather Forecasting: Computer Models for Hurricane
• Statistical Model
–
–
–
–
–
If Parents have some symptoms, the children are likely… (Family History)
Blood Type vs. Characteristics
Bio-rhythm vs. Luck/Fortune
Match-maker, Fortune-teller (점 based on 생년월일?), etc.
Sports (Baseball): If XXX, then we are expecting YYY…
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Computer Models for Hurricane
(http://www.wunderground.com/tropical/)
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Computer Models for Hurricane
(http://www.wunderground.com/tropical/)
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50
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Why Need a Model?
• Do Something
• Analysis / Control / System ID Concept Chart
• Physical System / Plant: Initially Completely Unknown (~Black box)
• For Analysis
• For Control
• Many Other Reasons including Fun: Name It!
• Sometimes, Another Model based on a Different Modeling Technique is
required to verify a Given Model
• Two Different Models for Same System / Plant
• Design Input Set for V&V
• Usually, Two Different Sets of Outputs for Same Input are Acquired
• How Much Different?
• Why?
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• Claim Two Math Models are Close Enough as Adequate Models?
Why Need a Good Model?
&
How to Build a Good Model?
• Straightforward!
• Very Difficult to Get!
•
•
•
•
•
Initially, Unavailable
Understanding Physics is Very Important
Start with a Preliminary Model with Many Assumptions
Assumptions  Build Model  Analysis  Verification
Eliminate Some Assumptions & Refine Model  Analysis 
Verification
• Repeat Steps, but until When?
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How Do We Know a Good Model?
• Compare with a Physical System / Plant
–
–
–
–
–
Measured Output vs. Computed Output
Residuals
Whiteness Test
Should Compare Various Operation Conditions
Comparison with a Physical System usually Expensive
since Various Tests are Required
• Sometimes Sub-scaled Tests Are Alternatives
– Automobile & Aerospace Vehicle: Wind Tunnel Test
– Rover & Robot: Tests with 1/10 Road / Env. Configuration
• Compare with a Different Model being Derived based
on the Different Modeling Technique
– Should Be Conducted with Care
54
How to Use a Good Model?
• For Better Analysis
– Weather Forecast, Prediction, etc.
– Trade-off Analysis
– Worst Case Scenario Analysis
• For Better Control System Design & Analysis
– Reduce Input Cost
• To Develop a Better Model
55
Personal Comments on Control Systems
• There has been progress
– Theory and Technology
– Development and Application
• Still promising Technical Area and will be Most Important &
Valuable Research Area in 21st Century
– Fully Autonomous System
– Fully Automated Environment
– Biomedical Engineering: Many More Things To Do!
• Recommendation
– Analytical Capability
– Mathematical Concept and Physics (see next chart)
– Science, Technology, Engineering, and Mathematics (STEM)
• Fast-growing Technology & Changes in Environments
–
–
–
–
Airplane & Automobile: ~100 Years
PC and Internet and e-mails: ~20 Years
Automobile Navigation: ~ 5 Years
Twitter, Facebook, Skype, Google: A Few Years Only
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Concluding Remarks
Control Systems
• There has been progress
– Theory and Technology
– Development and Application
• Still promising Technical Area and will be Most Important &
Valuable Research Area in 21st Century
– Fully Autonomous System
– Fully Automated Environment
– Biomedical Engineering: Many More Things To Do!
• Recommendation
– Analytical Capability
– Mathematical Concept and Physics (see next chart)
– Science, Technology, Engineering, and Mathematics (STEM)
57
IEEE Control Systems Magazine (Dec. 2007)
The following paradigm for research
domains that combine Models and Mathematics:
1) Get the physics right
2) The rest is mathematics
- By Rudy Kalman @ the 16th IFAC WC
in Prague on 7/4/2005
i) Did we, system theorists, get the physics right?
ii) Do our basic model structures adequately
translate physical reality?
iii) Does the way in which we view interconnections respect
the physics?
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Albert Einstein
It is impossible to get out of a
problem by using the same
kind of thinking that it took to
get into the problem
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THANK YOU!
&
Questions?
[email protected]
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