ECE 801
Homework #11 Solutions
Fall 2011
Homework #11 Solutions
ECE801, R. Groff
Determine whether each of the following types of stability hold for each of the systems below: (i)
Lyapunov Stable (ii) Asymptotically Stable (iii) BIBS stable (iv) BIBO stab
The Fundamental and State Transition Matrices
Consider the zero-input linear ODE
x = A(t )x . (*)
where A(t ) has continuous entries.
Definition: () : nn (an n n matrix-valued function of time) is the fundamental matrix of
x = A(t )x if and only if the n
Properties of Linear State Space Differential Equations with Zero Input
(*)
x = A(t )x
x (t ) n
A(t ) nn
FACT: Let the entries of A(t ) be continuous in t . Let the initial time t0 be given. Then for any
x 0 n , there exists a unique solution x () F (, n
Linearization Example
ECE801, R. Groff
10/05/2011
1/4
The Matlab code plots the solution (the trajectory of the state through state space) that the system
takes for two different conditions.
1)
1.3
x0 = 0.5 , u (t ) = 0.20sin(t )
1.4
0.2
2) x0 = 0
Checking Linear Independence with the Grammian
The Grammian can used to check linear independence of vectors, since
FACT: y1 , y2 , , yk are linearly independent if and only if the Grammian satisfies det(G ) 0 .
Example:
( n , )
A = [ y1
y2
yk ] with yi n
Projection Theorem Examples:
Example 1:
3
T
Vector space ( , ) with x, y = x y .
M = spancfw_ y1 , y2 ,
1
y1 = 2 ,
0
1
y2 = 0 .
1
7
Find the point m0 M that is closest to x = 1 in terms of the norm generated by the inner product.
2
Norm
Pythagorean Theorem
Consider the inner product space ( X , F , , ) and the norm x = ( x, x
product. Then the Pythagorean theorem holds in this space. i.e.
2
2
2
If x, y X are orthogonal, then x + y = x + y .
)
1
2
generated by the inner
Classical Projecti
Inner Product Spaces
The inner product (also called the scalar product) is a generalization of the concept of dot product. The
inner product allows us to introduce the important concepts of projection and orthogonality.
Let ( X , F ) be a vector space ove
Induced Norms
Let ( X , F , X ) and (Y , F, Y ) be normed vector spaces.
The set of all linear transformation A : X Y forms a vector space over F . An altogether new norm on
this vector space of linear transformations could be defined, but it is more conv
Systems as Linear Transformations
Consider a system described by a linear state space ODE and output equation
x = A(t ) x + B (t )u . (*)
y = C (t ) x + D (t )u
where A(t ) has entries that are continuous in t , x n is the state, u r is the input, and y m
Mathematical Induction
Mathematical induction is a technique for formally proving that a statement is true for all
integers greater than a certain value. Consider a mathematical statement P (k ) that depends on
an integer index k . A proof by mathematical
Kalman Canonical Forms
Any linear time invariant system can be transformed to any of the following canonical forms using an
appropriate change of basis:
Kalman Controllable Canonical Form (KCCF)
Kalman Oberservable Canonical Form (KOCF)
Kalman Canonica
Singular Value Decomposition
Any matrix A mn can be decomposed as A = U V T , called the singular value decomposition or
SVD, where
U = [u1
um ] mm is an orthogonal matrix1. The columns
of U form an orthonormal basis for the codomain.
V = [u1
um ] nn is
Observability
Consider the state space system
x = A(t )x + B(t )u
y = C (t )x + D(t )u
(*)
D
u
x
x = Ax + Bu
x0
C
+ +
y
Definition: The linear state space system (*) is said to be observable at time t0 if there exists a finite
time t1 > t0 such that for a
Controllability Tests for LTI Systems
Consider the LTI state space system
x = Ax + Bu
y = Cx + Du
with A nn , B nr ,C mn , D mr .
Definition: The controllability matrix for the system is given by
Q = B AB A2B
An 1B nnr .
Definition: The range of the contr
Controllability of Time-Varying and Time-Invariant Linear State Space Systems
[Brogan Ch 11, Chen 84 Ch 5, Chen 99 Ch 6]
Consider a linear state space system, possibly time-varying,
x = A(t )x + B(t )u
y = C (t )x + D(t )u
(*)
x n (or n ) , u r , y m
D(t
Stability
(Brogan Ch. 10, Chen Ch 5.)
Conceptually, stability indicates that a small change (i.e. a perturbation) in the conditions of a system
does not lead to much larger changes in (possibly other) conditions of the system.
The classical illustration c
Distinct Eigenvalues
Definition: A matrix A nn with eigenvalues 1 , 2 , , n is said to have distinct eigenvalues if and
only if i j whenever i j .
Theorem: Let A nn have distinct eigenvalues 1 , 2 , , n , and let v1 , v2 , vn n be
corresponding eigenvecto
Normed vector spaces
The norm is a generalization of the concept of Euclidean distance for measuring vector length.
Let ( X , F ) be a vector space over either F = or F = . (The following discussion is restricted to the
fields or .)
Definition: A norm is
Change of Basis
Let ( X , F ) be a vector space.
Let
U = cfw_u1 , u2 , , un
U = cfw_u1 , u2 , un
be two different bases for X .
Definition: The identity map for X is the transformation I : X X that defined by x X , I ( x) = x .
Note: The identity map is
Representing Linear Transformation using Coordinates
Let ( X , F ) be a vector space with basis U = cfw_u1 , u2 , , u n .
Let (Y , F ) be a vector space with basis V = cfw_v1 , v2 , , vm .
Let A : X Y be a linear transformation (but not necessarily inve
ECE 801
Homework #12 Solutions
Fall 2011
Homework #12 Solutions
ECE801, R. Groff
Consider a cart and double pendulum system, shown in the
figure, with M = 10 and m = 1 . For small angles, the
dynamics are given by
Mv = mg1 mg 2 + u
mli i = mg i + ( 1)i +1
ECE 801
Homework #6 Solutions
Fall 2011
Homework #6 Solutions
ECE801, R. Groff
Fall 2011
Consider L2 ([0, ]) with inner product u (), v() = 0 u (t )v (t )dt . Let y1 (t ) = t , y2 (t ) = t 3 . Let x(t ) = sin(t ) .
2
/2
a.
Find xa (t ) , the linear combi
ECE 801
Homework #5 Solutions
1.
4 3 1
16 10 8
x .
Consider the linear transformation A : 3 4 given by A( x ) =
12 6 9
12 8 5
Find a basis for R( A) .
a.
Find a basis for N ( A) .
b.
2.
Consider the normed vector space ( 2 , , p ) for the following n
ECE 801
Homework #4 Solutions
1. (Exercise) Consider the vector space ( 2 , ) . Let A : 2 2 be the linear transformation that
rotates a vector counterclockwise by radians about the origin. Find the matrix representation A of
A using the standard basis for