Introduction to System Modeling and Control

guineanhillAI and Robotics

Oct 20, 2013 (3 years and 5 months ago)

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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