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
Numerical
Solution
Control
Design
Model
Reduction
System Variables
Every 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.
System
u
y
x
Mathematical Model Types
Lumped

parameter
discrete

event
Most General
Linear

Time invariant (LTI)
Input

Output Model
distributed
LTI Input

Output Model
Transfer Function Model
Discrete

time model:
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
f
s
f
d
f
s
f
d
x
f
s
(
y
): position dependent spring force, y=u

x
f
d
(
y
): velocity dependent spring force
Newton’s 2nd law
Linearizaed model:
M
u
x
Acceleromter Transfer Function
Accelerometer
Model:
Transfer Function: Y/A=1/(s
2
+2
n
s
+
n
2
)
n
=(k/m)
1/2
,
=b/2
n
Natural Frequency
n
, damping factor
Model can be used to evaluate the
sensitivity of the accelerometer
Impulse Response
Frequency Response
Impulse Response
Frequency Response
/
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
Mechanical Subsystem
u
i
a
dc
R
a
L
a
J
B
Input
: voltage
u
Output
: Angular velocity
Elecrical Subsystem
(loop method):
Electro

Mechanical Example
Torque

Current:
Voltage

Speed:
Combing previous equations results in the following
mathematical model:
u
i
a
dc
R
a
L
a
B
Power Transformation
:
where K
t
: torque constant, K
b
: velocity constant For an
ideal motor
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
u
i
a
K
t
R
a
L
a
B
Transfer Function, L
a
=0:
0
0.1
0.2
0.3
0.4
0.5
0
2
4
6
8
10
12
Time (secs)
Amplitude
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.
Nonlinear System Modeling
& Control
Neural Network Approach
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
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)
x
1
x
2
y
weights
activation
function
Sigmoidal
and Gaussian
Activation Functions
Multilayer
Netwoks
W
k,ij
: Weight from node i in layer k

1 to node j in layer k
x
y
W
0
W
p
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,y
t
) 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.
Neural Networks with Local Basis
Functions
These networks employ basis (or
activation) functions that exist
locally
, i.e.,
they are activated only by a certain type of
stimuli
Examples:
Cerebellar Model Articulation Controller (CMAC,
Albus
)
B

Spline CMAC
Radial Basis Functions
Nodal Link Perceptron Network (NLPN, Sadegh)
Biological Underpinnings
Cerebellum: Responsible for complex
voluntary movement and balance in
umans
Purkinje cells in cerebellar cortex is believed
to have CMAC like architecture
Nodal Link Perceptron Network
(NLPN) [Sadegh, 95,98]
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
NLPN Architecture
Input

Output Equation
Basis Function:
Each
ij
is a
1

dimensional
triangular
basis function over a finite interval
x
y
w
i
NLPN
Approximation: 1

D Functions
Consider a scalar function f(x
)
f(x) on interval [a
i
,a
i+1
] can be approximated
by a line
a
i+1
w
i
a
i
w
i+1
Basis Function Approximation
Defining the
activation/
basis
functions
Function f can expressed as
This is also similar to fuzzy

logic approximation
with “triangular” membership functions.
(1st order B

spline CMAC)
a
i
a
i

1
a
i+1
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)?
State

Space Model
u
y
States: x
1
,…,x
n
system
A Class of Observable State
Space Realizable Models
Consider the input

output model:
When does the input

output model have a
state

space realization?
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
The Model Form
The following Input

Output Model
always admits a minimal state
realization:
State Space Realization
The
state

model of the input

output model is
as follows with y=x
1
:
Neural Networks
Reduced coupling results in sub

networks:
Can’t
use prepackaged software, but
standard training methods are the same
Nodal Link Perceptron Networks
Local basis functions, similar to CMAC
networks. Reduced Coupling also results
in sub

networks:
Simulation
Example
Nonlinear mass spring
damper
Data
sampled at 0.01s, output is the velocity
of the 2
nd
mass
Simulation Results
I: Linear model. mse=0.0281. training(static) mse=0.0059.
II: NARMA model. mse=0.0082. training(static) mse=0.0021.
III. Neural network. mse=3.6034e

4. training(static) mse=0.0016.N
IV. NLPN. mse=7.2765e

4. training(static) mse=2.6622e

4.
Simulation Results
I: Linear model. mse=0.0271.
II: NARMA model. mse=0.0067.
III. Neural network. mse=5.3790e

4.
IV. NLPN. mse=7.1835e

4.
Conclusions
A number of data driven modeling techniques are
suitable for an observable state space
transformation
Rough
guidelines were given for when and how to
use NARMA, neural network and NLPN models
NLPN modifications make it an easily trainable
option with excellent capabilities
Substantial training & design issues include data
sampling rate and input repetition due to the
reduced coupling restriction
Fluid Power Application
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
Learning Control Block Diagram
Experimental Results
Experimental Results
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