MAE 280A Linear Dynamic Systems

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MAE 280A Linear Dynamic Systems

• Robert E Skelton,
  bobskelton_at_ucsd.edu, 858 822 1054
  office hours (help session) 400-500 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

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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

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MAE280A Syllabus

• How to get Models of Dynamics
• Dynamics
• State space models
• Linearization

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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!

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How to get Dynamic Models

• Particle dynamics
• Put model in state form

y
y
mg
x
u
x
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How to get Dynamic Models

• Rigid body dynamics
• Linearize about u 90, ux 0 Put model in
  state form

y
y
r
u
mg
x
u
x
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What is a Linear System?

• A linear algebraic system
• A linear dynamic system

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State Space form of Dynamic Models
Nonlinear Models
LTV (Linear Time-Varying) Models
LTI (Linear Time-Invariant) Models
LaPlace Transform of LTI Model
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State form of Dynamic Models, Discrete
Nonlinear Models
LTV (Linear Time-Varying) Models, Discrete
LTI (Linear Time-Invariant) Models, Discrete
z Transform of LTI Model, Discrete
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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.

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Taylors series
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Nonlinear Systems/Taylors Series
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MAE280A Syllabus

• How to get Models of Dynamics
• How to get linear Models of Dynamics
• How to get solution of linear models
• Coordinate Transformations
• The Liapunov Transformation
• The State Transition Matrix
• HW2 chapter 4, exercises 4.11, 4.13, 4.14, 4.23,
  4.25, 4.28

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Coordinate Transformations
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LTI Systems
State enough IC required to SOLVE the ODE
(together with u(t))
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LTI Solutions
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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 odes
• controllability
• pole assignment
• observability
• state estimation
• stability
• trackability
• optimality
• model reduction