introduction to system modeling and control n.
Download
Skip this Video
Loading SlideShow in 5 Seconds..
Introduction to System Modeling and Control PowerPoint Presentation
Download Presentation
Introduction to System Modeling and Control

Loading in 2 Seconds...

play fullscreen
1 / 52

Introduction to System Modeling and Control - PowerPoint PPT Presentation


  • 140 Views
  • Uploaded on

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

loader
I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.
capcha
Download Presentation

PowerPoint Slideshow about 'Introduction to System Modeling and Control' - hollie


An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript
introduction to system modeling and control
Introduction to System Modeling and Control
  • Introduction
  • Basic Definitions
  • Different Model Types
  • System Identification
  • Neural Network Modeling
mathematical modeling mm
Mathematical Modeling (MM)
  • A mathematical modelrepresent 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

Numerical Solution

Theory

Solution Data

Data

Engineering System

Math. Model

Model Reduction

Control Design

Graphical Visualization/Animation

Engineering Modeling Process
  • Example: Automobile
  • Engine Design and Control
  • Heat & Vibration Analysis
  • Structural Analysis
definition of system
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 variables
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.

y

u

System

x

mathematical model types
Mathematical Model Types

discrete-event

distributed

Lumped-parameter

Most General

Input-Output Model

Linear-Time invariant (LTI)

LTI Input-Output Model

Discrete-time model:

Transfer Function Model

example accelerometer text 6 6 1

u

x

x

fs

fs

M

M

fd

fd

Example: Accelerometer (Text 6.6.1)

Consider the mass-spring-damper (may be used as accelerometer or seismograph) system shown below:

Free-Body-Diagram

fs(y): position dependent spring force, y=u-x

fd(y): velocity dependent spring force

Newton’s 2nd law

Linearizaed model:

example ii delay feedback
Example II: Delay Feedback

Consider the digital system shown below:

Input-Output Eq.:

Equivalent to an integrator:

transfer function

U

G

Y

Transfer Function

Transfer Function is the algebraic input-output relationship of a linear time-invariant system in the s (or z) domain

Example: Accelerometer System

Example: Digital Integrator

Forward shift

comments on tf
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
Acceleromter Transfer Function
  • Accelerometer Model:
  • Transfer Function: Y/A=1/(s2+2ns+n2)
    • 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
mixed systems
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

Ra

La

B

ia

dc

u

J

Electro-Mechanical Example

Input: voltage u

Output: Angular velocity 

Elecrical Subsystem (loop method):

Mechanical Subsystem

electro mechanical example1

Ra

La

B

ia

dc

u

Electro-Mechanical Example

Power Transformation:

Torque-Current:

Voltage-Speed:

where Kt: torque constant, Kb: velocity constant For an ideal motor

Combing previous equations results in the following mathematical model:

brushless d c motor
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.
dq coordinates
dq-Coordinates

b

q

d

e

a

e=p + 0

c

offset

Electrical angle

Number of poles/2

mathematical model
Mathematical Model

Where p=number of poles/2, Ke=back emf constant

system identification
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 example2

Ra

La

B

ia

Kt

u

12

10

u

8

t

6

Amplitude

4

T

2

0

0

0.1

0.2

0.3

0.4

0.5

Time (secs)

Electro-Mechanical Example

Transfer Function, La=0:

k=10, T=0.1

comments on first order identification
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
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.
frequency domain identification

Method I (Sweeping Sinusoidal):

f

Ao

Ai

system

t>>0

Method II (Random Input):

system

Transfer function is determined by analyzing the spectrum of the input and output

Frequency-Domain Identification
system models
System Models

high order

low order

nonlinear system modeling control

Nonlinear System Modeling& Control

Neural Network Approach

introduction
Introduction
  • Real world nonlinear systems often difficult to characterize by first principle modeling
  • First principle models are oftensuitable 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
what is a neural network
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 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
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
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

x1

y

x2

weights

activation function

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)
multilayer netwoks

y

x

W0

Wp

Multilayer Netwoks

Wk,ij: Weight from node i in layer k-1 to node j in layer k

universal approximation theorem uat
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
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.
neural network approximation of narma model
Neural Network Approximation of NARMA Model

y

u[k-1]

y[k-m]

Question: Is an arbitrary neural network model consistent with a physical system (i.e., one that has an internal realization)?

state space model
State-Space Model

u

y

system

States: x1,…,xn

a class of observable state space realizable models
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
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
slide45

INTRODUCTION

APPLICATIONS:

  • Robotics
  • Manufacturing
  • Automobile industry
  • Hydraulics

EXAMPLE:

EHPV control

(electro-hydraulic poppet valve)

  • Highly nonlinear
  • Time varying characteristics
  • Control schemes needed to open two or more valves simultaneously
motivation
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
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
slide48

INTRODUCTION

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
control design
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