unit1 pg math model
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Mathematical Modeling & Analysis
PG Mechanical Engineering Design ModuleLecture Slides: July-December 2013Prof. (Dr.) Gaurang V. Shah Doctor of EngineeringUniversity of Massachusetts Lowell, USAMechanical Engineering Department, PVPIT, Bavdhan Pune
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What is Mathematical Model?A set of mathematical equations (e.g., differential eqs.) that describes the input-output behavior of a system.
What is a model used for?
• Simulation
• Prediction/Forecasting
• Prognostics/Diagnostics
• Design/Performance Evaluation
• Control System Design
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Definition of SystemSystem: An aggregation or assemblage of things so combined by man or nature to form an integral and complex whole.
From engineering point of view, a system is defined as an interconnection of many components or functional units act together to perform a certain objective, e.g., automobile, machine tool, robot, aircraft, etc.
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To every system there corresponds three sets of variables:
Input variables originate outside the system and are not affected by what happens in the system
Output variables are the internal variables that are used to monitor or regulate the system. They result from the interaction of the system with its environment and are influenced by the input variables
Systemu y
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Dynamic SystemsA system is said to be dynamic if its current output may depend on the past history as well as the present values of the input variables. Mathematically,
Time : Input, :
]0),([)(
tu
tuty ≤τ≤τϕ=
Example: A moving mass
M
yu
Model: Force=Mass x Acceleration
uyM =
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Velocity-Force: ∫ ττ+==t
duM
ytytv0
)(1
)0()()(
Therefore, this is a dynamic system. If the drag force (bdx/dt) is included, then
uybyM =+
2nd order ordinary differential equation (ODE)
Position-Force:
dsduM
ytytyt s
∫ ∫ ττ++=0 0
)(1
)0()0()(
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Mathematical model of a real world system is derived using a combination of physical laws (1st principles) and/or experimental means
• Physical laws are used to determine the model structure (linear or nonlinear) and order.
• The parameters of the model are often estimated and/or validated experimentally.
• Mathematical model of a dynamic system can often be expressed as a system of differential (difference in the case of discrete-time systems) equations
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Input-output differential or difference equation
State equations (system of 1st order eqs.)
Transfer function
Nonlinear
Linear
Linear Time Invariant
System Type Model Type
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• A nonlinear model is often linearized about a certain operating point
• Model reduction (or approximation) may be needed to get a lumped-parameter (finite dimensional) model
• Numerical values of the model parameters are often approximated from experimental data by curve fitting.
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Differential Equations (Continuous-Time Systems)
ubububyayayay nnn
nnnn +++=++++ −
−−
− 1)1(
11)1(
1)(
)()1()()1()( 11 nkubkubnkyakyaky nn −++−+−++−=
Difference Equations (Discrete-Time Systems)
DiscretizationInverse Discretization
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Consider the mass-spring-damper (may be used as accelerometer or seismograph) system shown below:
Free-Body-Diagram
M
fs
fd
fs
fd
x
fs(y): position dependent spring force, y=x-ufd(y): velocity dependent spring force
Newton’s 2nd law ( ) )()( yfyfuyMxM sd −−=+=
Linearizaed model: uMkyybyM =++
M
ux
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Delayz -1
u y
Consider the digital system shown below:
Input-Output Eq.: )1()1()( −+−= kukyky
Equivalent to an integrator:
∑−
=
=1
0
)()(k
j
juky
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Transfer Function is the algebraic input-output relationship of a linear time-invariant system in the s (or z) domain
GU Y
dt
ds
kbsms
ms
sU
sYsGukyybym ≡
++==⇔=++ ,
)(
)()(
2
2
Example: Accelerometer System
Example: Digital Integrator
≡−
==⇔−+−= −
−
zz
z
zu
zYGkukyky ,
1)(
)()1()1()(
1
1 Forward shift
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• Transfer function is a property of the system independent from input-output signal
• It is an algebraic representation of differential equations
• Systems from different disciplines (e.g., mechanical and electrical) may have the same transfer function
• Chemical system of reacting species can be represented by TF
• Electromagnetic wave like LASER rep by TF(?).
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• Most systems in mechatronics are of the mixed type, e.g., electromechanical, hydromechanical, etc
• Each subsystem within a mixed system can be modeled as single discipline system first
• Power transformation among various subsystems are used to integrate them into the entire system
• Overall mathematical model may be assembled into a system of equations, or a transfer function
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voltage emf-backe,edt
diLiRu bb
aaaa =++=
Mechanical Subsystem BωωJTmotor +=
uia dc
Ra La
Jω
BInput: voltage uOutput: Angular velocity ω
Elecrical Subsystem (loop method):
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uia dc
Ra La
ω
Torque-Current:
Voltage-Speed:
atmotor iKT =
Combing previous equations results in the following mathematical model:
B
Power Transformation:
ωKe bb =
=+
=ω++
0at
baaa
a
iK-BωωJ
uKiRdt
diL
where Kt: torque constant, Kb: velocity constant For an ideal motor bt KK =
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Taking Laplace transform of the system’s differential equations with zero initial conditions gives:
Eliminating Ia yields the input-output transfer function
( ) btaaa2
a
t
KKBRBLJRJsL
K
U(s)
Ω(s)
++++=
uia Kt
Ra La
ω
B( )( )
=Ω+=Ω++
0)(
)()()(
sIK-(s)BJs
sUsKsIRsL
at
baaa
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Assuming small inductance, La ≈0
( )( )abt
at
RKKBJs
RK
U(s)
Ω(s)
++=
which is equivalent to
ωat RuK
Babt RKK
• The D.C. motor provides an input torque and an additional damping effect known as back-emf damping
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• A brushless PMSM has a wound stator, a PM rotor assembly and a position sensor.
• The combination of inner PM rotor and outer windings offers the advantages of– low rotor inertia– efficient heat dissipation, and – reduction of the motor size.
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a
qb
c
d
θe
θe=p θ+ θ0
Electrical angleNumber of poles/2
offset
θ
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qme
dmqq
dqmdd
vLL
Kipi
L
R
dt
di
vL
ipiL
R
dt
di
1
1
+ω−ω−−=
+ω+−=
Where p=number of poles/2, Ke=back emf constant
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• Parametric Identification: The input-output model coefficients are estimated to “fit” the input-output data.
• Frequency-Domain (non-parametric): The Bode diagram [G(jw) vs. w in log-log scale] is estimated directly form the input-output data. The input can either be a sweeping sinusoidal or random signal.
Experimental determination of system model. There are two methods of system identification:
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uia Kt
Ra La
ω
B
( )( ) 1+
=++
=Ts
k
RKKBJs
RK
U(s)
Ω(s)
abt
at
Transfer Function, La=0:
0 0.1 0.2 0.3 0.4 0.50
2
4
6
8
10
12
Time (secs)
Am
plitu
de
ku
T
u
t
k=10, T=0.1
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Graphical method is
• difficult to optimize with noisy data and multiple data sets
• only applicable to low order systems
• difficult to automate
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Given a linear system with uniformly sampled input output data, (u(k),y(k)), then
noisenkubkubnkyakyaky nn +−++−+−++−= )()1()()1()( 11
Least squares curve-fitting technique may be used to estimate the coefficients of the above model called ARMA (Auto Regressive Moving Average) model.
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Input: Random or deterministic
Random Noise
u
Output
n
plant
Noise model
• persistently exciting with as much power as possible;• uncorrelated with the disturbance • as long as possible
y
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• Analytical
• Experimental
– Time response analysis (e.g., step, impulse)
– Parametric
* ARX, ARMAX
* Box-Jenkins
* State-Space
– Nonparametric or Frequency based
* Spectral Analysis (SPA)
* Emperical Transfer Function Analysis (ETFE)
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Bode Diagram of
10-1
100
101
102
-10
0
10
20
Frequency (rad/sec)
Ga
in d
B
10-1
100
101
102
-30
-60
-90
0
Frequency (rad/sec)
Ph
ase
de
g
1/T
20log( )k
1)(
+=
Ts
ksG
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Method I (Sweeping Sinusoidal):
systemAiAo
f
t>>0
Magnitude Phasedb
=
=
A
Ai
0 , φ
Method II (Random Input):
system
Transfer function is determined by analyzing the spectrum of the input and output
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• Pointwise Estimation:)(
)()(
ωω=ω
U
YjG
This often results in a very nonsmooth frequency response because of data truncation and noise.
• Spectral estimation: uses “smoothed” sample estimators based on input-output covariance and crosscovariance.
The smoothing process reduces variability at the expense of adding bias to the estimate
)(ˆ)(ˆ
)(ˆωΦωΦ
=ω
u
yujeG
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0.1 1 10 100 1 10375
50
25
0
25
Frequency (Hz)
Ma
gn
itud
e (
dB
)
0.1 1 10 100 1 103180
90
0
90
180
Frequency (Hz)
Ph
ase (
Deg
)
high order
low order
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Neural Network Approach
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Prof. (Dr.) Gaurang ShahPVPIT, Pune
July-December 2013Lecture Notes: Mathematical Modeling Analysis PG ME (Design)
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IntroductionReal world nonlinear systems often difficult to
characterize by first principle modelingFirst principle models are often
suitable for control designModeling often accomplished with input-output maps
of experimental data from the systemNeural networks provide a powerful tool for data-
driven modeling of nonlinear systems
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])1[],...,[],1[],...,[(][ −−−−= kumkukymkygky
g
z -1 z -1 z -1
z -1 z -1 z -1
u
y
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• Artificial Neural Networks (ANN) are massively parallel computational machines (program or hardware) patterned after biological neural nets.
• ANN’s are used in a wide array of applications requiring reasoning/information processing including–pattern recognition/classification–monitoring/diagnostics–system identification & control–forecasting–optimization
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• Learning from examples rather than “hard” programming
• Ability to deal with unknown or uncertain situations
• Parallel architecture; fast processing if implemented in hardware
• Adaptability
• Fault tolerance and redundancy
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• Hard to design
• Unpredictable behavior
• Slow Training
• “Curse” of dimensionality
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• A neuron is a building block of biological networks
• A single cell neuron consists of the cell body (soma), dendrites, and axon.
• The dendrites receive signals from axons of other neurons.
• The pathway between neurons is synapse with variable strength
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Artificial Neural NetworksThey are used to learn a given input-output
relationship from input-output data (exemplars).Most popular ANNs:
Multilayer perceptronRadial basis functionCMAC
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Objective: Find a finite-dimensional representation of a function with compact domain
• Classical Techniques
-Polynomial, Trigonometric, Splines
• Modern Techniques
-Neural Nets, Fuzzy-Logic, Wavelets, etc.
mn RRAf →⊂:
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x1
x2
y
• MLP is used to learn, store, and produce input output relationships
Training: network are adjusted to match a set of known input-output (x,y) training data
Recall: produces an output according to the learned weights
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x y
Wk,ij: Weight from node i in layer k-1 to node j in layer k
( )( )( )( )xWσWσσWσWy TTTp
Tp 011 −=
σ: Activation function, e.g.,
+
+=
−
−
nx
x
e
e
1
1
1
1
)(1
xσ
W0 Wp
p: number of hidden layers
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A single hidden layer perceptron network with a sufficiently large number of neurons can approximate any continuous function arbitrarily close.
Comments:
• The UAT does not say how large the network should be
• Optimal design and training may be difficult
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Objective: Given a set of training input-output data (x,yt) FIND the network weights that minimize the expected error )yy(
2
tEL −=
Steepest Descent Method: Adjust weights in the direction of steepest descent of L to make dL as negative as possible.
tTdeEdL yye,0)y( −=≤=
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Examples:
• Cerebellar Model Articulation Controller (CMAC, Albus)
• B-Spline CMAC
• Radial Basis Functions
• Nodal Link Perceptron Network (NLPN, Sadegh)
These networks employ basis (or activation) functions that exist locally, i.e., they are activated only by a certain type of stimuli
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• Cerebellum: Responsible for complex voluntary movement and balance in humans
• Purkinje cells in cerebellar cortex is believed to have CMAC like architecture
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x y
)v,x(wy ii
ii∑ ϕ=
weightsbasis function
• One hidden layer only
• Local basis functions have adjustable parameters (vi’s)
• Each weight wi is directly related to the value of function at some x=xi
• similar to spline approximation
• Training algorithms similar to MLPs
wi
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Consider a function RRAf →⊂:
f(x) on interval [ai,ai+1] can be approximated by a lineai+1
wi
ai
wi+1
111
1)( +++
−
−+
−
−−= iii
ii
ii
i waa
axw
aa
axxf
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Defining the basis functions
)a,()( xwxfi
ii∑ ϕ=
∈−
−−
∈−−
=ϕ ++
−−
−
otherwise,0
],[,1
],[,
)( 11
11
1
iiii
i
iiii
i
i aaxaa
ax
aaxaa
ax
x
aiai-1 ai+1
Function f can expressed as
=
Na
a
1
a
This is also similar to fuzzy-logic approximation with “triangular” membership functions.
(1st order B-spline CMAC)
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Advantages of networks with local basis functions:
• Simpler to design and understand
• Direct Programmability
• Training is faster and localized
Main Disadvantage:
• Curse of dimensionality
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• Piecewise multilinear network (extension of 1-dimensional spline)
• Good approximation capability (2nd order)
• Convergent training algorithm
• Globally optimal training is possible
• Has been used in real world control applications
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x y
wi
)v,x(∑ ϕ=i
iiwy
)v,x()v,x()v,x()v,x( 21 21 niiii nϕϕϕ=ϕ
Input-Output Equation
Basis Function:
Each ϕ ij is a 1-dimensional triangular basis function over a finite interval
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y
y[k-m]
u[k-1]
Question: Is an arbitrary neural network model consistent with a physical system (i.e., one that has an internal realization)?
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])[x(][
])[],[x(f]1[x
khky
kukk
==+
u y
States: x1,…,xn
system
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A Class of Observable State Space Realizable Models
Consider the input-output model:
When does the input-output model have a state-space realization? :
])[x(][
])[],[x(f]1[x
khky
kukk
==+
])1[],...,[],1[],...,[(][ −−−−= kumkukymkygky
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• A Generic input-Output Model does not necessarily have a state-space realization (Sadegh 2001, IEEE Trans. On Auto. Control)
• There are necessary and sufficient conditions for realizability
• Once these conditions are satisfied the statespace model may be symbolically or computationally constructed
• A general class of input-Output Models may be constructed that is guaranteed to admit a state-space realization
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Prof. (Dr.) Gaurang ShahPVPIT, Pune
July-December 2013Lecture slideMathematical Modeling AnalysisPG
Mechanical Engineering (Design)
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APPLICATIONS:
Robotics Manufacturing Automobile industry Hydraulics
INTRODUCTION
EHPV control(electro-hydraulic poppet valve) Highly nonlinear Time varying characteristics Control schemes needed to
open two or more valves simultaneously
EXAMPLE:
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EXAMPLE:
Single EHPV learning control being investigated at Georgia Tech
Controller employs Neural Network in the feedforward loop with adaptive proportional feedback
Satisfactory results for single EHPV used for pressure control
INTRODUCTION
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IMPROVED CONTROL:
Linearized error dynamics - about (xd,k ,ud,k)
CONTROL DESIGN
( ) ( )kdkkkdkkdkkdnk o ,,,, , uueuuQeJe −+−+=+
kdkkkk ,1 ueJQu +−= −
Exact Control Law (deadbeat controller)
Nonlinear system (‘lifted’ to a square system)
( )kknk F uxx ,=+
Approximated Control Law
kkkkkk uueJQu ~~~1
11 ∆++−= −
−−
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IMPROVED CONTROL:
CONTROL DESIGN
kkkkkk uueJQu ~~~1
11 ∆++−= −
−−
Approximated Control Law
Estimation of Jacobian and controllability
Feedback correction
Nominal inverse mapping
inverse mapping correction
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