MAE 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.
2.
3.
4.
5.
How to get Models of Dynamics
How to get linear Models of Dynamics
How to get solution of linear models
How to measure performance of dynamic system
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 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!
How to get Dynamic Models
• Particle dynamics
y
y
m x  ux
m y  mg  u y
u
mg
x
x
• Put model in state
form
x
Ax  Bu
How to get Dynamic Models
Rigid body dynamics
y
J  r (u xCos  u y Sin )
m x  ux
y
r
u
u
mg
m y  mg  u y
Linearize about u = 90, ux = 0
Put model in state form x 
x
x
Ax  Bu
What is a Linear System?
• A linear algebraic system
a11x1  a12 x2  a13 x3  y1
a21x1  a22 x2  a23 x3  y2
 a11 a12
a
 21 a22
 x1 
a13     y1 
x2   


a23    y2 
 x3 
Ax  y
• A linear dynamic system


a11 x1  a12 x2  u1
a21 x1  a22 x2  u2
a11
0

0
0
0
a21
a12
0
 x 
 1 
x 
0  1 

x



1
a22  
 
 x2 
 x2 
 u1 
u 
 2
State Space form of Dynamic Models
Nonlinear Models

x(t )  f ( x(t ), u (t )),
y(t )  g ( x(t ), u(t ))
LTV (Linear Time-Varying) Models

x(t )  A(t ) x(t )  B(t )u (t ),
y(t )  C (t ) x(t )  D(t )u (t )
LTI (Linear Time-Invariant) Models

x(t )  Ax(t )  Bu (t ),
y(t )  Cx(t )  Du(t )
LaPlace Transform of LTI Model
y( s)  G(s)u( s),
1
G(s)  C (sI  A) B  D
State form of Dynamic Models, Discrete
Nonlinear Models

x(t )  f ( x(t ), u (t )),
y(t )  g ( x(t ), u(t ))
LTV (Linear Time-Varying) Models, Discrete
x(tk 1 )  A(tk ) x(tk )  B(tk )u(tk ),
y(tk )  C (tk ) x(tk )  D(tk )u(tk )
LTI (Linear Time-Invariant) Models, Discrete
x(tk 1 )  Ax(tk )  Bu (tk ),
y (tk )  Cx(tk )  Du(tk )
z Transform of LTI Model, Discrete
y( z )  G( z )u( z ),
1
G( z )  C ( zI  A) B  D
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.
r  .07 for interest rate of 7%
Pk  principal at beginning of year k
P1  P0  rP0  (1  r ) P0
P2  P1  rP1  (1  r ) P1  (1  r ) 2 P0
Pk 1  (1  r ) Pk :
P 
1 (1 r )  k  !   0
 Pk 


A linear Discrete - Time System
Taylor’s series

_
_
_
f ( x)
1  2 f ( x)
1 i f
2
f , x  R : f ( x)  f ( x) 
( x  x) 
(
x

x
)

...

(
x

x
)i  aT z

2
i
x
2 x
i  0 i! x
_
1
1 i f
a  (...
...) ,
i! x i
f  R1 , x  R 2 :
T
_
f ( x)  f ( x) 
_


_
f ( x)   x1  x1  1 

x

x
1
1

_
x2   x  x  2 
2
 2
2
_
_
_
f
1
T  f
 f ( x)  ( x  x)  ( x  x)
(
x

x
)
x
2
x 2
_
Homework :
z  Az , find
A
_
_
_
f ( x)
f ( x)
1  2 f ( x)
 2 f ( x)
1  2 f ( x)
2
( x1  x1 ) 
( x2  x2 ) 
(
x

x
)

(
x

x
)(
x

x
)

( x2  x 2 ) 2
1
1
1
1
2
2
2
2
x1
x2
2 x1
x1x2
2 x2
 f ( x)
 f ( x)  
 x1
_

_
z T  (...( x  x) i ...) :
  2 f ( x)
2
_ 
  x1
x2  x2  2
   f ( x)
 x x
 2 1
 2 f ( x) 
_


x1x2   x1  x1 

 2 f ( x)   x  x_ 
2
 2
 2 x2 
Nonlinear Systems/Taylor’s Series
2
_
_
_
f
1
T  f
x  f ( x)  f ( x) 
( x  x)  ( x  x)
( x  x) 
2
x
2
x

_
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
1. Coordinate Transformations
2. The Liapunov Transformation
3. The State Transition Matrix
HW2: chapter 4, exercises 4.11, 4.13, 4.14, 4.23, 4.25, 4.28
Coordinate Transformations
x  Ax  Bu ,
x  Tz ,
T  AT
Tz  Tz  ATz  Bu ,  z  T 1[( AT  T ) z  Bu ]  T 1Bu
integrating 
t
z (t )  z (t0 )   T 1 ( ) B( )u ( ) d . Hence,
t0
t
x(t )  T (t )T (t0 ) x(t0 )   T (t )T 1 ( ) B( )u ( ) d
1
t0
t
  (t , t0 ) x(t0 )    (t , ) B( )u ( )d ,
t0
 (t , t0 )  A(t )  (t , t0 ),
 (t0 , t0 )  I
LTI Systems
 (t , t0 )  e
A ( t  t0 )

  [ A(t  t0 )]i (ifactorial)-1  I  A(t  t0 ) 
i 0
1 2
A (t  t0 ) 2  ......
2
proof :
d A ( t  t0 )
e
 Ae A(t t0 ) ,
dt
x(t )  e
A ( t  t0 )
e A ( t0  t0 )  I ,
Hence
t
x(t0 )   e A(t  ) Bu ( )
t0
State: enough IC required to SOLVE the ODE (together with u(t))
ZIR:
LTI Solutions
x  e A(t t0 ) x(t0 )
t
ZSR: x   e A( t  ) Bu ( )d
t0
u (t )  u  (t )
x  e At Bu ,
Impulse Response:
OAT Impulse Response: x(t , i )  e At bi ui
r
B  [b1 b2 ......bnu ]

X    x(t , i ) xT (t , i ) dt ,
Deterministic Covariance:
i=1
r
0

   e At bi ui ui biT e A t dt ,
i=1
T
0

  e At BUBT e A t dt ,
T
0
r  nu
r  nu
 u12

U  0
 0

0
u2 2
0
0 

0 
u32 
Theorem 4.12 (text )
Z  AZ  ZAT  W
has solution
t
Z (t )  e At Z (0)e A t   e A(t  )We A
T
T
0
t
Z ()  Z  lim  e A(t  )We A
t  0
0  AZ  ZA  W .
T
( t  )
( t  )
d ,

Hence, if
lim e At Z (0)e A t  0,
T
t 
d   e A We A  d , Hence, if lim Z  0
0
T
t 
MAE 280A Outline
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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