L INEAR S YSTEMS T HEORY
Joao P. Hespanha
October 21, 2010 1
Disclaimer: This is a draft and probably contains a few typos.
Comments and information about typos are welcome.
Please contact the author at hespanha@ece.ucsb.edu.
c Copyright to Joao Hespanha.
Controllability
Consider the continuous-time LTV state equation
x = A(t)x + B(t)u
!
with initial condition at t0 x(t0 ) = x0 and
Z solution at time t1
t1
!
x(t1 ) =
(t1 , t0 )x0 +
(t1 , )B( )u( ) d
t0
Definition: Given two times t1>t0 the reachable subspa
Observability
We have seen observability twice already
It was the property which permitted us to retrieve the initial state from the
initial data cfw_u(0), y(0), u(1), y(1), . . . , u(n 1), y(n 1)
It was the property in Lyapunov stability which allowed us
Solution of Linear State-space Systems
Homogeneous (u=0) LTV systems first
Theorem (Peano-Baker series)
= A(t)x(t), x(t0 ) = x0 2 Rn is given by
The unique solution to x(t)
! x(t) =
(t, t0 )x0
!
Z
!
+
where
(t, t0 ) = I +
!
t
A(s1 ) ds1 +
t0
Z
t
A(s1 )
t0
State Space Structure
Standard LTI system, D=0 for convenience only
x(t)
= Ax(t) + Bu(t)
D plays no role in state-space structure since its
y(t) = Cx(t)
effect is easily accommodated in y(t)
We identify these two subspaces of Rn based on the range of C th
Homework 2 for MAE280A Linear Systems Theory, Fall 2016:
due Thursday October 13 in class
Question 1
Let A Cmn , B C`p , C Cmp .
(i) Show that the matrix equation AX = C is soluble if and only if R(C) R(A). [Note that this consists of two
parts: the if, a
MAE280A Linear Systems Theory
http:/numbat.ucsd.edu/~bob/linearsystems
Regular Professor Bob Bitmead
rbitmead@ucsd.edu, 858 822 3477, Jacobs Hall 1609
email policy: If you send me an email I will read it.
If it takes longer to deal with it than it took yo
Linear Systems
Linear systems?!?
(Roughly) Systems which obey properties of superposition
Input u(t) output y(t) = L(u(t)
L(u1 (t) + u2 (t) = L(u1 (t) + L(u2 (t)
Our interest is in dynamic systems
Dynamic system means a system with memory
of course includ
function [U,M,V,W] = myGaussGram(A)
[row,col] = size(A);
if nargout >= 2
% If there are 2 outputs or more
A=[A eye(row)];
% Add an I matrix in operation
col_eye=size(eye(row),2);
end
[row,col] = size(A);
tol = 0.00000000001; %tolerance for the matrix
i =
Gram-Schmidt Orthogonalization
If an orthogonal basis for a vector space V is a
desirable thing for the simple representation of
objects in V, then how does one find an orthogonal
basis for V? Given a basis B = cfw_x1, x 2 , x n for V,
there is a straigh
Linear Algebra and Applications:
Numerical Linear Algebra
David S. Watkins
watkins@math.wsu.edu
Department of Mathematics
Washington State University
IMA Summer Program, 2008 p.
My Pledge to You
IMA Summer Program, 2008 p.
My Pledge to You
I promise not t
Japan Patent Office
Digitization of Business Processes and
Information Systems
Isao HONZAWA
Japan Patent Office (JPO)
7 Dec, 2010
Contents
Japan Patent Office
Outline of JPO Paperless System
Effects of Implementing Paperless System
Key issues in syste
function [Q,R]=gschmidt(V)
% Input: V is an m by n matrix of full rank m<=n % Output: an m-by-n upper
triangular matrix R
% and an m-by-m unitary matrix Q so that A = Q*R.
[m,n]=size(V)
R=zeros(n)
R(1,1)=norm(V(:,1)
%1-st column's unit length
Q(:,1)=V(:,1