lecture-slides chapter 03

7/25/2019 Lecture-Slides CHAPTER 03

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ME464-Sys Dyn & Ctrl Spring-2013 Dr. Shaukat Ali


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State Variable Models

Chapter 3


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Introduction

The state variables

The state differential equations

Signal flow graphs

Outline


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So far:

Dynamic models of systems In time domain (t-domain)

In Laplace domain (s-domain)

In s-domain (for linear time-invariant ODEs/systems)

Transfer functions Block diagram models

Signal flow graphs

In this chapter

Write time domain models in state variable models

Introduction


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State variable model

Model of a system in the time domain with state variables

Usage:

Modern control systems

Analysis and design of control system through computer

Can be used for nonlinear, and time-varying systems

State Variable Models


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The state variables of a dynamic system

Those variables that determine the future behavior of asystem WHEN

the present state of the system and the excitation signals are known

State of a system

The condition of a system that is described in terms of a setof state variables

State Variables


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Two rolling carts attached with spring and dampers

p, q: position of carts

u: external force

Write down the state variable model

State Variable ModelsExample


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The dynamic model in time domain

Free body diagrams:

Newtons second law of motion:


uM

qM

bp

M

bq

M

kp

M

kp

11

1

1

1

1

1

1

1

1

qM

bbp

M

bq

M

kkp

M

kq

2

21

2

1

2

21

2

1


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State variables:

Equations of motion in terms of state variables:


q

p

q

p

x

x

x

x

x

4

3

2

1

u

M

x

M

bx

M

bx

M

kx

M

kx

1

4

1

1

3

1

1

2

1

1

1

1

1

3

1

4

2

21

3

2

1

2

2

21

1

2

1

4 x

M

bbx

M

bx

M

kkx

M

kx


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Transfer Function of state variable model:

Take the Laplace Transform:

Transfer function:

State Variable ModelsTransfer Function

sDUsBUCsY

sBUsXAsI

tDutCxty

tButAxtx

DBC

sU

sYsG


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Example: Transfer function of an RLC circuit

State Variable Model

xRy 0

uCx

L

R

L

Cx

0

1

1

10

LCs

L

Rs

LCRsUsYsG

12


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E3.2:

A robot arm drive system for one joint can be represented bythe differential equation:

Wherev(t) = velocity

y(t) = position

i(t) = control motor current (input)

Required:1. Put the equation in the state variable form

2. Setup the matrix form for k1=k2=k3=1 and find the statetransition matrix

ExercisesChapter 03

tiktyktvk

dt

tdv321


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E3.13

A system is described by the following two differentialequations:

Where y and are functions of time, and u is an input.

Required:

a) Select a set of state variables

b) Write the state differential equation

ExercisesChapter 03

04

02

ubydt

d

auydt

dy


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E3.19

A system has the following state variable model:

Determine the transfer function:

ExercisesChapter 03

xy

uxx

010

1

0

43

10

sU

sYsG

lecture-slides chapter 03

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