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MAE 280A Linear Dynamic Systems

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

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MAE 280A Linear Dynamic Systems

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  1. MAE 280A Linear Dynamic Systems Robert E Skelton, bobskelton@ucsd.edu, 858 822 1054office 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

  2. MAE280A Syllabus • 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 • How to modify performance with control

  3. MAE280A Syllabus • How to get Models of Dynamics • Dynamics • State space models • Linearization

  4. 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!

  5. How to get Dynamic Models • Particle dynamics • Put model in state form y y mg x u x

  6. How to get Dynamic Models Rigid body dynamics Linearize about u = 90, ux = 0Put model in state form y y r u mg x u x

  7. What is a Linear System? • A linear algebraic system • A linear dynamic system

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

  9. 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

  10. 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.

  11. Taylor’s series

  12. Nonlinear Systems/Taylor’s Series

  13. 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

  14. Coordinate Transformations

  15. LTI Systems State: enough IC required to SOLVE the ODE (together with u(t))

  16. LTI Solutions

  17. 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

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