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 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

MAE280A Syllabus 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 Put model in state form y y mg u x x

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

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

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

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

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.

Taylor’s series

Nonlinear Systems/Taylor’s Series

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

Coordinate Transformations

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

LTI Solutions

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