Introduction to System Modeling and Control

1 / 52

# Introduction to System Modeling and Control - PowerPoint PPT Presentation

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

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

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

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
• Basic Definitions
• Different Model Types
• System Identification
• Neural Network Modeling
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.

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
• 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

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

discrete-event

distributed

Lumped-parameter

Most General

Input-Output Model

Linear-Time invariant (LTI)

LTI Input-Output Model

Discrete-time model:

Transfer Function Model

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

Consider the digital system shown below:

Input-Output Eq.:

Equivalent to an integrator:

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

• 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+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
• 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

Ra

La

B

ia

dc

u

J

Electro-Mechanical Example

Input: voltage u

Output: Angular velocity 

Elecrical Subsystem (loop method):

Mechanical Subsystem

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
• 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

b

q

d

e

a

e=p + 0

c

offset

Electrical angle

Number of poles/2

Mathematical Model

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

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.

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

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.

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

high order

low order

### Nonlinear System Modeling& Control

Neural Network Approach

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?
• 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
• Learning from
• Parallel architecture
• Fault tolerance and redundancy
• 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
• NLPN (Sadegh et al 1998,2010)

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)

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)

• 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.
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

u

y

system

States: x1,…,xn

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

### Fluid Power Application

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
• 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

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
• 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