Introduction to System Modeling and Control
Introduction Basic Definitions Different Model Types System Identification Neural Network Modeling
Mathematical Modeling (MM)
A mathematical model represent a physical system in terms of mathematical equations
It is derived based on physical laws (e.g.,Newton’s law, Hooke’s, circuit laws, etc.) in combination with experimental data.
It quantifies the essential features and behavior of a physical system or process.
It may be used for prediction, design modification and control.
Engineering Modeling Process
Graphical Visualization/Animation
Engineering System
Theory
Data
Example: Automobile• Engine Design and Control• Heat & Vibration Analysis• Structural Analysis
Solution Data
Math. Model
fx
v
T
vc
bvdt
dvmf
Numerical Solution
Control Design
Model Reduction
Definition of System
System: 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.
System VariablesEvery system is associated with 3 variables:
Input variables (u) originate outside the system and are not affected by what happens in the system
State variables (x) constitute a minimum set of system variables necessary to describe completely the state of the system at any given time.
Output variables (y) are a subset or a functional combination of state variables, which one is interested to monitor or regulate.
Systemu y
x
Mathematical Model Types
Lumped-parameter
discrete-event
Most General ),,( tuxfx
),,( tuxhy
Linear-Time invariant (LTI)
DuCxy BuAxx
Input-Output Model ),,,,,,,,( )()1()( tuuuyyyfy nnn
distributed
LTI Input-Output Model
ubububyayayay nnn
nnnn 1
)(01
)1(1
)(
Transfer Function Model)()()( sUsGsY
Discrete-time model:
)()()( ityty i )1()( txtx
Example: Accelerometer (Text 6.6.1)
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=u-xfd(y): velocity dependent spring force
Newton’s 2nd law )()( yfyfyuMxM sd
Linearizaed model: uMkyybyM
M
ux
Example II: Delay Feedback
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
Transfer Function
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
1Forward shift
Comments on TF
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
Acceleromter Transfer Function
Accelerometer Model: Transfer Function:
Y/A=1/(s2+2ns+n2) n=(k/m)1/2, =b/2n
Natural Frequency n, damping factor
Model can be used to evaluate the sensitivity of the accelerometer Impulse Response Frequency Response
uMkyybyM
Frequency Response
Frequency (rad/sec)
Ph
as
e (
de
g);
Ma
gn
itu
de
(d
B)
Bode Diagrams
-60
-40
-20
0
20
40From: U(1)
10-1 100 101-200
-150
-100
-50
0
To
: Y
(1)
/n
Mixed Systems
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
Electro-Mechanical Example
voltage emf-back bba
aaa e,edt
diLiRu
Mechanical Subsystem BJTmotor
uia dc
Ra La
J
BInput: voltage uOutput: Angular velocity
Elecrical Subsystem (loop method):
Electro-Mechanical Example
Torque-Current:
Voltage-Speed:
at iKT motor
Combing previous equations results in the following mathematical model:
uia dc
Ra La
B
Power Transformation:
bb Ke
0at
baaa
a
iKBJ
uKiRdt
diL
where Kt: torque constant, Kb: velocity constant For an ideal motor bt KK
Brushless D.C. Motor 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.
Mathematical Model
qme
dmqq
dqmdd
vLL
Kipi
L
R
dt
di
vL
ipiL
R
dt
di
1
1
Where p=number of poles/2, Ke=back emf constant
qtem iKTJ
System identification
Experimental determination of system model. There are two methods of system identification:
Parametric Identification: The input-output model coefficients are estimated to “fit” the input-output data.
Frequency-Domain (non-parametric): The Bode diagram [G(j) vs. in log-log scale] is estimated directly form the input-output data. The input can either be a sweeping sinusoidal or random signal.
Electro-Mechanical Example
uia Kt
Ra La
B
1
Ts
k
b at
at
RKKBJs
RK
U(s)
Ω(s)
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
Comments on First Order Identification
Graphical method is difficult to optimize with noisy data
and multiple data sets only applicable to low order
systems difficult to automate
Least Squares Estimation
Given a linear system with uniformly sampled input output data, (u(k),y(k)), then
Least squares curve-fitting technique may be used to estimate the coefficients of the above model called ARMA (Auto Regressive Moving Average) model.
noisenkubkubnkyakyaky nn )()1()()1()( 11
Frequency-Domain Identification
Method I (Sweeping Sinusoidal):
systemAiAo
f
t>>0
Magnitude Phasedb
A
Ai
0 ,
Method II (Random Input):
system
Transfer function is determined by analyzing the spectrum of the input and output
System Models
0.1 1 10 100 1 10375
50
25
0
25
Frequency (Hz)
Ma
gn
itu
de
(d
B)
0.1 1 10 100 1 103180
90
0
90
180
Frequency (Hz)
Ph
ase
(D
eg
)
high order
low order
Introduction Real world nonlinear systems often difficult to
characterize by first principle modeling First principle models are often
suitable for control design Modeling often accomplished with input-
output maps of experimental data from the system
Neural networks provide a powerful tool for data-driven modeling of nonlinear systems
Input-Output (NARMA) Model
])1[],...,[],1[],...,[(][ kumkukymkygky
g
z -1 z -1 z -1
z -1 z -1 z -1
u
y
What is a Neural Network?
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
Advantages and Disadvantages of ANN’s
Advantages: Learning from Parallel architecture Adaptability Fault tolerance and redundancy
Disadvantages: Hard to design Unpredictable behavior Slow Training “Curse” of dimensionality
Biological Neural Nets 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
Artificial Neural Networks
They are used to learn a given input-output relationship from input-output data (exemplars).
The neural network type depends primarily on its activation function
Most popular ANNs: Sigmoidal Multilayer Networks Radial basis function NLPN (Sadegh et al 1998,2010)
Multilayer Perceptron MLP is used to learn, store, and produce
input output relationships
The activation function (x) is a suitable nonlinear function: Sigmidal: (x)=tanh(x) Gaussian: (x)=e-x2
Triangualr (to be described later)
x1
x2
y
)( ijj
ji
i vx wy
weightsactivation function
Sigmoidal and Gaussian Activation Functions
-5 -4 -3 -2 -1 0 1 2 3 4 50
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
x
sig(
x)
gaussian sigmoid
Multilayer Netwoks
Wk,ij: Weight from node i in layer k-1 to node j in layer k
xWσWσσWσWy TTTp
Tp 011
x y
W0 Wp
Universal Approximation Theorem (UAT)
Comments: The UAT does not say how large the
network should be Optimal design and training may be
difficult
A single hidden layer perceptron network with a sufficiently large number of neurons can approximate any continuous function arbitrarily close.
Training Objective: Given a set of training
input-output data (x,yt) FIND the network weights that minimize the expected error
Steepest Descent Method: Adjust weights in the direction of steepest descent of L to make dL as negative as possible.
)(2
tEL yy
tTdeEdL yyey ,0)(
Neural Network Approximation of NARMA Model
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)?
A Class of Observable State Space Realizable Models
Consider the input-output model:
When does the input-output model have a state-space realization?
])[(][
])[],[(]1[
khky
kukk
x
xfx
])1[],...,[],1[],...,[(][ kumkukymkygky
Comments on State Realization of Input-Output Model
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 state-space 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
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:
Motivation
The valve opening is controlled by means of the solenoid input current
The standard approach is to calibrate of the current-opening relationship for each valve
Manual calibration is time consuming and inefficient
Research Goals
Precisely control the conductivity of each valve using a nominal input-output relationship.
Auto-calibrate the input-output relationship
Use the auto-calibration for precise control without requiring the exact input-output relationship
EXAMPLE:
Several EHPV’s were used to control the hydraulic piston
Each EHPV is supplied with its own learning controller
Learning Controller employs a Neural Network (NLPN) in the feedback
Satisfactory results for single EHPV used for pressure control
INTRODUCTION
Control Design Nonlinear system (‘lifted’ to a square
system)
Feedback Control Law
is the neural network output The neural network controller is directly trained
based on the time history of the tracking error
kknk F uxx ,
)(),(ˆ
),(ˆ dd
ddpdd xx
x
xxKxx
u
),(ˆ dd xx