MAE 280A Linear Dynamic SystemsMAE 280A Linear Dynamic Systems

Robert E Skelton, bobskelton@ucsd.edu, 858 822 1054 office hours (help session): 4:00-5:00 TU, 1804 EBU-1

text 1: skelton, dynamic systems control, Wiley 1988 ISBN 0-471-83779-2

text 2: skelton, iwasaki and grigoriadis, a unified algebraic approach to control design, Taylor and Francis 1998 ISBN 0-7484-0592-5

prerequisites: linear algebra, differential eqs

homework turned in on every Monday, late homework cannot be accepted (solutions will be posted on web)

hwnotebook = corrected homework solutions, bound in a notebook (due last week of class)

grade=.25exam + .25exam + .3final + .2hwnotebook

read assignment before lecture (come to class with questions in your head)

Homework 1: read chapter 3, text 1. Do exercises 3.2, 3.7, and 3.8

MAE280A Syllabus

1. How to get Models of Dynamics

2. How to get linear Models of Dynamics

3. How to get solution of linear models

4. How to measure performance of dynamic system

5. How to compute performance without solving the ODEs

6. How to modify performance with control

MAE280A Syllabus

1. How to get Models of Dynamics– Dynamics– State space models– Linearization

Modeling, the Most Difficult partModeling, the Most Difficult part

• How should we model a pendulum?• Should we model:

– Flexibility of rod?

– Bearing dynamics?

– Friction?

– Aerodynamic disturbances?

• Depends on control accuracy required of y• Control accuracy will depend on model, hence,• Modeling and Control Problem not independent• How do we get a model suitable for control design?• An ongoing research topic!

• Particle dynamics

• Put model in state form

How to get Dynamic Models

mg

x

y

x

y

m x u

um y mg

u x

y

x Ax Bu

Rigid body dynamics

Linearize about = 90, ux = 0

Put model in state form

How to get Dynamic Models

mg

x

y

( )x y

x

y

u u

u

J Cos Sin

m x

y

r

m mg u

u x

y

x Ax Bu

r

What is a Linear System?What is a Linear System?• A linear algebraic system

• A linear dynamic system

yAx

y

y

x

x

x

aaa

aaa

yxaxaxa

yxaxaxa

2

1

3

2

1

232221

131211

2323222121

1313212111

2

1

2

2

1

1

1

2221

1211

2222121

1212111

000

000

u

u

x

x

x

x

x

aa

aa

uxaxa

uxaxa

State Space form of Dynamic Models

( ) ( ) ( ), ( ) (( ) ( ) ( ) ( )) ( )x t x t u t y tA t B t C t Dx t u tt

( ) ( ) ( ), ( ) ( ) ( )x t x t u t y t x tA B C uD t

1( ) , (( ) ( ) ( ))G s G s C sIy s u s A B D

Nonlinear Models

LTI (Linear Time-Invariant) Models

LaPlace Transform of LTI Model

( ) ( ( ), ( )), ( ) ( ( ), ( ))x t f x t u t y t g x t u t

LTV (Linear Time-Varying) Models

State form of Dynamic Models, Discrete

1( ) ( ) ( )( ) ( ) , ( ) ( ) ( )( ) ( )k k k k k kk kk kx t x t u t y tA t B t x t tt D t uC

1( ) ( ) ( ), ( ) ( ) ( )k k k k k kx t x t u t y tA B C Dx t u t

1( ) , (( ) ( ) ( ))G z G z C zIy z u z A B D

Nonlinear Models

LTI (Linear Time-Invariant) Models, Discrete

z Transform of LTI Model, Discrete

( ) ( ( ), ( )), ( ) ( ( ), ( ))x t f x t u t y t g x t u t

LTV (Linear Time-Varying) Models, Discrete

What is a Linear System?What is a Linear System?

• The math model is an abstraction (always erroneous) of the Real System

• Are there any Real Systems that are linear?

Yes. Annually compounded interest at the bank.

0)1(1

System Time-cretelinear Dis A1

02

1112

0001

!

: )1(

)1()1(

)1(

year of beginningat principal

7% of rateinterest for 07.

k

k

P

Pr

kk

k

PrP

PrPrrPPP

PrrPPP

kP

r

Taylor’s seriesTaylor’s series

)()(2

1)()(

)()(

)()(

2

1)()()(

)()(

2

1))((

)()(

)(

2

1)(

)()(

)()()(

:,

,::...))(...(,...)!

1(...

)(!

1...)(

)(

2

1)(

)()()(:,

_

2

2___

_

22

_

11

22

2

12

221

2

21

2

_

22

_

11_

22

_

11

21

_

2222

2

2

221121

22

_

1121

2_

222

_

111

_

21

_

_

0

2_

2

2__1

xxx

fxxxx

x

fxf

xx

xx

x

xf

xx

xfxx

xf

x

xf

xxxxxx

xxx

xf

x

xfxf

xxx

xfxxxx

xx

xfxx

x

xfxx

x

xfxx

x

xfxfxf

RxRf

AfindAzzHomeworkxxzx

f

ia

zaxxx

f

ixx

x

xfxx

x

xfxfxfRxf

T

iTi

iT

Ti

ii

i

Nonlinear Systems/Taylor’s Series

)()(2

1)()()(

_

2

2___

xxx

fxxxx

x

fxfxfx T

MAE280A Syllabus

1. How to get Models of Dynamics2. How to get linear Models of Dynamics3. How to get solution of linear models

1. Coordinate Transformations2. The Liapunov Transformation3. The State Transition Matrix

HW2: chapter 4, exercises 4.11, 4.13, 4.14, 4.23, 4.25, 4.28

Coordinate Transformations

0

0

0

1 1

10

1 10 0

0 0 0

, ,

, [( ) ]

( ) ( ) ( ) ( ) ( ) .

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

( , ) ( ) ( , ) ( ) ( ) , ( , )

integrating Hence,t

t

t

t

t

t

x Ax Bu x Tz T AT

Tz Tz ATz Bu z T AT T z Bu T Bu

z t z t T B u d

x t T t T t x t T t T B u d

t t x t t B u d t t

0

0 0

( ) ( , ),

( , )

A t t t

t t I

LTI Systems

0

0 0 0 0

0

0

( ) 2 20 0 0 0

0

( ) ( ) ( )

( ) ( )0

1( , ) [ ( )] ( ( ) ( ) ......

2

:

, ,

( ) ( ) ( )

-1factorial)

Hence

A t t i

i

A t t A t t A t t

tA t t A t

t

t t e A t t i I A t t A t t

proof

de Ae e I

dt

x t e x t e Bu

State: enough IC required to SOLVE the ODE (together with u(t))

LTI Solutions0

0

( )0

( )

1 2

0

( )

( )

, ( ) ( )

( , ) [ ...... ]

( , ) ( , ) ,r

i=1

ZIR:

ZSR:

Impulse Response:

OAT Impulse Response:

Deterministic Covariance:

u

A t t

t A t

t

At

Ati i n

Tu

At Ti i i i

x e x t

x e Bu d

x e Bu u t u t

x t i e bu B b b b

X x t i x t i dt r n

e bu u b

0

21

220

23

( ) ( )

0

( ) (

0

,

0 0

, 0 0

0 0

4.12 ( )

( ) (0) , lim (0) 0,

( ) lim

r

i=1

has solution

Hence, if

T

T

T T T

T

A tu

At T A t

T

tAt A t A t A t At A t

t

t A t A

t

e dt r n

u

e BUB e dt U u

u

Theorem text

Z AZ ZA W

Z t e Z e e We d e Z e

Z Z e We

)

0, lim 0

0 .

Hence, ifTt A A

td e We d Z

AZ ZA W

MAE 280A Outline• Modeling, introduction to state space models• linearization• vectors, inner products, linear independence• Linear algebra problems, matrices, matrix calculus• least squares• Spectral decomposition of matrices: Eigenvalues/eingenvectors• coordinate transformations• solutions of linear ode’s• controllability• pole assignment• observability• state estimation• stability• trackability• optimality• model reduction