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Transcript of 581_Lec_Sec 1n
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California State University Fullerton Shahin Ghazanshahi
EGEE 581
Theory of Linear Systems
-
Introduction
Previous Courses In System Theory Involved:
1) Linear Systems (Superposition holds)
2) Time Invariant Systems
3) Single Input Single Output (SISO)
4) Ordinary Differential Equations
5) Transfer Functions
6) Poles and Zeros
7) Hand Calculations
Previous Courses Are Inadequate For Modern, Complex Systems:
State Space Theory Addresses:
1) Linear or Non-Linear Systems
2) Time Invariant Or Time Variant Systems
3) Multiple Input Multiple Output
4) Vector Matrix Differential Equations
5) Time Domain Solutions
6) Transfer Functions Must Be Converted To Vector Matrix Differential
Equations
7) Digital Computation Can Be Exploited
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Limitations:
Limitations Of Previous Methods:
1) 3rd
Order Systems; Single Input Single Output
Limitations of New Methods:
1) 50 -200th
Order Systems
2) Limitation Is Based On Adequacy Of Model And Power Of Digital
Computer
Paradox N0. 1:
Major Benefit Of State Space Method Is The Ability To Use Very
Powerful Digital Computer Capacity
Yet Courses And Texts Put Little Emphasis On Use Of Digital
Computation:
Paradox N0. 2:
The Critical Element In Successful Application Of Any Theory
( Including State Space) Is Adequacy Of Mathematical Models.
Modeling Success Is Gainted Through Experience; Its An Art, Requiring Detailed Knowledge Of A Physical System.
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Chapter 1
Physical System: Collection of physical devices in real world.
system : is a model of physical system.
Two methods to study and design physical system:
1) Imperical Method:
Apply various Signals Measure Responses
Then adjust parameters or connect to a compensator to improve
performance. This approach relies on past experience and it is carried out
by trial and error and it has succeeded in designing many physical systems.
Not good for complex systems
Expensive
Too Dangerous
2) Analytical Method:
Modeling
Developing Mathematical Descriptions
Analysis
Design
1) Modeling
A physical system may have different model based on question we are
looking and its experimental range.
1) Circuit or control systems are models.
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2) R, L, C1 are model of physical elements
3) Electronic amplifier is modeled differently at high or low
frequencies.
Choosing model that is close to a physical system and yet simple enough
to be studied analytically is the most difficult and the most important in
system design.
In this class we shall assume that models of physical systems are
available to us.
2) Mathematical Model:
Once a system (model) is selected, physical laws (i.e, KVL, KCL.
Newtons Law) will be apply to get mathematical model of physical
system. The mathematical model can be Linear/Non-linear;
Differential/Difference equations; Integral Equations; and etc.
3) Analysis:
Quantitative Analysis Interested in response of the system
Qualitative Analysis Interested in general properties of the system
such as Stability, Controllability and Observability
1 Resistor, Inductor, Capacitor
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Systems:
Linear ( L )
Non-Linear ( NL )
Time-Invariant ( TI )
Time-Variant ( TV )
Single Input-Single Output( SISO )
Multiple Input-Multiple Output ( MIMO )
Continues Time ( CT )
Discrete Time ( DT )
Time Domain Description:
Differential Equstion
() + () + () = () + 2 ()
Convolution
() = ( )()
= 0
() := impulse response
State Space Equation:
() = () + ()
() = () + ()
() := State and internal variable
Transfer Domain Description:
() = ()()
() = Transfer function
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Chapter 3-Review Of Matrix:
I assume you have basic knowledge of matrix theory. You should know:
A = {} = {}
C = AB = {} = =1
KA = {}
(A+B)C = AC + BC
AB BA (in general)
(AB)T = BTAT
det A = || = Scalar
det (AB) = det A det B (A , B n by n)
det A 0 A is non-singular; A-1 exists
det A = 0 A is singular, A-1 doesnt exist.
(AB)-1 = B-1A-1
A-1 =
|| used in hand calculations limit 3 *3
A-1 numerical methods (See EE 403 text)
Rank A n*m = Number of linearly independent rows
= Number of linearly independent columns
(A-1)T = (AT)-1
Trace A =
() =
+
(1) = 1
1
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() = { ()}
Math Sinology:
Common short hand symbols used
= There exist
= Such that
= For all
|| = Determinant Of A
{} = Set of s
= Implied by (Only If)
= Implies (If)
= Iff = If and Only If
= Is included in (member of a set)
= Is included in (subset of a set)
One= Exactly One = One and Only One
= One Or More (at least one)
Necessity And Sufficiency:
= A implies B
= A is sufficient condition for B but it may not be necessary.
= not A not B
= If A, then B
not B not A
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= A implied by B
A is a necessary condition for B but it may not be sufficient.
is equivalent to NOT A NOT B
Definition:
Dimension: The dimension of a vector space V = maximum number of
the linearly independent vectors V
:
(R3, R) 3 dimensional real space
Note: Any 3 of {} are linearly independent
All 4 are Not
Basis: coordinate system {} are a basis of V iff :
1) {} are linearly independent
2) Every X V is a Unique linearly combination of {}
X = 1 1 + 2 2 + 3 3
[1 , 2 , 3 ] Representation of vector X w.r.t. the basis
[1 , 2 , 3 ]
X3 X4
X2
X1
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:
In the example above:
1) 1 , 2 are NOT a basis
2) 1 , 2 , 3 , 4 are NOT a basis
3) 1 , 2 , 3 is a basis
4) 1 , 3 , 4 is a basis
Theorem:
If the dimension of V = n, then any set of n linearly independent {} is a
basis of V
Proof:
1) {} are linearly independent
2) Choose arbitrary X V, then ; X, 1 , 2 , , 3 are linearly
dependent, since maximum number of linearly independent is n
0 + 1 1 + + = ; with {} not all zero
0 0, otherwise 0 = 0 and some other 0 , j0
1 1 + + + + = 0 linearly independent
(contradiction)
0 0 and;
X=- 1
0 (1 1 + + )
X= (1 1 + + )
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:
An arbitrary X V can be written as a linearly combination of {}i=1
show linearly combination is Unique
Let;
X = 1 1 + + Be another linear combination
Then;
X = 1 1 + + = X = 1 1 + +
(1 1)1 + + ( ) = ;
{} linearly independent
( ) = ; i
= linearly combination is Unique
key Result:
Let {} be a Known Basis of V and X V an arbitrary vector
Then;
X = 1 1 + + = [1, , ] [
1 .
]
And X V is completely Represented by
= [
1 .
] ; a column vector of n real numb
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Important Notes:
1) A basis is Not Unique; i.e, different Bases for some V can be chosen
(equivalent to changing coordinate systems)
2) But for Each basis, X V has Unique representation with respect to
That basis
3) The same X V has different representations (one for each basis)
4) Some basis are better (more convenient to use). Orthonormal basis
often used.
:
(1)
(2) (, R) = n dimension real vector space of n-tuples column vectors)
1 = [
1 .0
] = [
0 .1
]
{} is a basis of (, R)
Then X , is written as:
n 1
n 2
n3
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X = 1| | = [
1 0 00 1 0 0 0 1
] [
12
]
Changing Bases:
Let X () , X arbitrary
Let V have two bases {} and {}
Then;
(*) X = [1 . ] = [1 ] ; = [
1 .
] , = [
1 .
]
How are and related.
Note:
can be written as a linearly combination of {}
[1 ] = [1 ] [11 1
1
]
= [1 ] P
Substitute into (*)
[1 ] P = [1 ] ; but linear combination is Unique
Coefficients are unique
P =
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Similarly each can be written as a linear combination of {}
[1 ] = [1 ] [11 1
1
]
Substitute into (*)
[1 ] = [1 ] Q = Q
Previously P = = QP (**)
But XV was arbitrary chosen
is arbitrary, i.e, (**) holds
= P = 1
Summary:
Change of basis = P
= P1
:
X = [13] 2 and consider 1 = [
31] and 2 = [
22] as new basis. Note
X = [13] is representation of X at orthonormal basis. Draw from X a line, in
parallel with 2 line which interacts the 1line at 1. Draw from X a line
in
P
1 =Q
and represent X V
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parallel with 1 which interacts the 2 line at 22. So representation of X
respect to 1 and 2 is [12
].
Check
X= [13] = [1 2] [
12
] = [3 21 2
] [12
]
= Q
= Q1
Norm of vectors:
Norm is the length or magnitude of the vector shown as when;
1) 0
2) = || real
3) 1 + 2 1 + 2
let;
X= [1, 2, , ]t , then norm of X are any of the follow.
1) 1-norm: 1 ||=1
2) 2-norm, Euclidian : 2 ||2=1
3) Infinite norm: = max ||
:
X = [2 4]
1 = 2 + |4| = 6
2 = = 22 + (4)2 = 4.47
= 4
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Orthogonal vectors:
Two vectors 1 and 2 are said to be orthogonal if;
12 = 21
=0
Note:
= scalar and is n
Orthonormal vectors:
; = 1, 2, , n is said to be orthonormal if;
= {
01
Orthogonal Matrix:
A square matrix is an orthogonal matrix if;
= = I
Note:
If A is not square such as 32 then;
23 32 = 2 but 32
23 3
Note:
For square orthogonal matrix A , 1 is
= A = I
A1 = I 1 = AT
i i=j
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Schmidt Orthonormalization:
Consider a set of linearly independent vectors 1, 2, , , we can
normalize them as:
1 1 1 1
1
2 2 - (12 )1 2
22
.
(
)1=1
Note:
1 2 1 is projection of the vector 2 along 1. Its subtraction from 2
yields the vertical part 2. And 2
2 normalized 2.
Finding the rank of a matrix:
Apply row operations on A until A becomes upper triangular matrix as in
Gaussian elimination.
Rank of A will not change after pre or post- multiplication of A by a non
singular matrix.
Rank (AC) = Rank (A)
Rank (DA) = Rank (A)
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Row Operations:
1) Interchange rows
2) Multiply a row by a non zero constant
3) Add a multiple of one row to another row
:
Find rank A
A = [0 1 1 21 2 3 42 0 2 0
] [1 2 3 42 0 2 00 1 1 2
] [1 2 3 40 4 4 80 1 1 2
]
[1 2 3 40 1 1 20 1 1 2
] [1 2 3 40 1 1 20 0 0 0
] Rank =2
1and 2 are linearly independent
Linear Algebraic Equations:
1 = 1
Note:
If AX = 0 then X null vector
Definition:
1) Nullity Maximum number of linearly independent null vectors of A
2) Nullity Number of columns of A Rank A= n- ()
3) Rank A min (m,n)
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:
A= [
0 1 1 21 2 3 42 0 2 01 2 3 4
]
1 and 2 are linearly independent and can be used as basis for range
space of A.
Range space of A:
All possible linear combinations of all columns of A.
Back to the example above!
Nullity = 4 () = 4-2=2
1 = [
11
10
] and 2 = [
020
1
] are null vectors and are basis for null space;
1 = 2 = 0 ; 1 and 2 are linearly independent
Theorem:
1 = 1
consider:
K = n - ()
= solution of AX = Y
1) If () = ; then k=0 then = unique solution
2) If K >0; then for every real ; i=1, 2, ,k; the vector
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X = + 11 + . +
Is the solution of AX=Y where; {1, 2. , } is null basis.
Back to the previous example:
AX = [0 1 1 21 2 3 42 0 2 0
] X = [480
] = Y
Let = [0 4 0 0] be the solution, so the general solution is;
X= + 11 + 22 = [
0400
] + 1 [
11
10
] + 2 [
020
1
] 1 and 2 are real
n- ()= number of components in X which can be arbitrarily chosen
(free parameters)
Theorem:
Consider square matrix as:
1 = 1
1) If A is a non singular or ()=n, then X = 1 Y is the unique
solution
2) Homogenous equation AX = 0 has non-zero solution iff A is a
singular matrix or () < where the number of linearly
independent solution is n-(), the nullity of A.
This theorem is important and we will use it later!
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Similarity Transformation:
Consider = . Square matrix A maps X to Y , change
basis to new one as {1 2 } = Q
The equation become;
* =
where , are representation of X and Y with respect to the basis
{1 } where as discussed before:
X=Q
Y=Q
Substituting these in AX=Y
AQ = Q
Or:
1AQ = 1Q =
So:
1AQ = comparing with *
** = A Q or A= Q
This is called similarity transformation and A and are said to be similar.
We can write ** as:
AQ = Q
A[1 ] = [1 ] = [1 ]
So the ith column of is the representation of A with respect to the
basis [1 ]
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:
Let A=[3 2 1
2 1 04 3 1
]
Let b=[0 0 1] then:
Ab = [101
] , 2b = A(Ab) = [42
3] , 3b = A(2) = [
510
13], Also, we can
verify that:
3 = 17b 15Ab + 52
since b, Ab and 2b are linearly independent, they can be used as new
basis. Lets find the representation of A with respect to this basis. =?
Q = [0 1 40 0 21 1 3
] = [ 2]
= 1AQ = [0 0 171 0 150 1 5
]
Note:
A(b) = [ 2] [010]
A(Ab) = [ 2] [001] As we said: AQ=Q
A(2) = [ 2] [17
155
]
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In General:
Let A be n matrix. If there exists a n 1 vector b such that b, Ab, ,
1b are linearly independent and if
b = 1b + 2(Ab) + + (1)
then which is representation of A with respect to basis
[ . . 1] is
=
[ 0 0 . 0 11 0 . 0 20 1 . 0 3 . 0 40 0 . 0 0 0 . 1 1]
companion form
Eigenvalues and Eigenvectors:
Let A be an n matrix. Then a nonzero vector X in is called an
eigenvector of A if AX is a scalar multiple of X; that is:
AX= ; for some scalar
The scalar is called an Eigenvalue of A and X is the Eigenvector of A
corresponding to eigenvalue .
Find :
AX = X, X0 ; is a scalar
AX- X = (A- )X = 0 ; X0
i.e; (A- ) is singular
() = det (A- ) = |A | = 0 = | A|
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Note:
1) , |A | are defined iff A is square ()
2) () = 0 is a polynomial of degree n in and is called the
characteristic polynomial of A.
() = + 11+ +1+ =0
3) () has n roots {} and can be written as:
() = |A | = ( 1) ( 2) ( )=0
where {}are not necessarily distinct
4) why do we care? A matrix A will replace the transfer function in
the role of determining system stability. Later, well show that:
= Poles of the transfer function
:
A=[1 10 1
] ; | | = |1 1
0 (1 + )| = ( 1)( + 1)
() = 2-1 = ( 1)( + 1) = 0 = 1
( ) = 0 , X0
If ; = 1 [0 10 2
] [12
] = 0 = [10
] Eigenvector
Where 1 is arbitrary
Note: | | =0 Eigenvector is not unique
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If ; = -1 [2 10 0
] [12
] = 0 = [1/22
] Eigenvector
Where 2 is arbitrary
Note: | | =0
Theorem: {} distinct:
Let {}i=1 be distinct eigenvalues of A and be the eigenvector associated
with . Then {}i=1 is a linearly independent set { is a basis}
Corollary:
Let {}i=1 be distinct eigenvalues of A. Then A is similar to a diagonal
matrix D
D = [1 0 0
]
i.e; = 1 , where Q = [1 ]
is the eigenvector associated with
Proof:
1) Q is nonsingular since the columns are linearly independent.
2) Must merely show that QD = AQ (equivalent to D = 1AQ)
QD = [1 ] [1 0 0
] = [11 ]
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AQ = A[1 ] = [1 ]
= [11 ]
:
A=[0 0 21 0 20 1 1
] ; () = =|A | = | 0 0
1 20 1 1
|=( 2)( + 1)
1,2,3= 2, -1, 0
(A-2I) 1 = [2 0 01 2 20 1 1
] 1 = 0 1 = [011]
Note: 1 is not unique; 1 = [0 , , ]
2 = [0
21
] 3 = [21
1]
Thus D = = [2 0 00 1 00 0 0
] which is diagonal with eigenvalues at
diagonal
= 1 where Q=[0 0 21 2 11 1 1
]
Check: Q =
This Q is said to be diagonalization matrix
1, 2, 3 are eigenvectors
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Theorem: {} not distinct = Repeated eigenvalues:
1) If {} are not distinct, then A is similar to D Diagonal Matrix; iff
the eigenvectors are linearly independent and D = 1
2) Otherwise, A is similar to J (of Jordan Canonical Form),
i.e, = 1 =
:
A = [1 0 10 1 00 0 2
] = 1, 1, 2
If; = 1 1 = [100] 2 = [
010]
If; = 2 3 = [101
]
D = [1 0 00 1 00 0 2
] = 1
:
A = [1 1 20 1 30 0 2
] = 1, 1, 2
Three
linearly
independent
{}
But only two linearly
independent {}
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Definition:
If is repeated times, it has multiplicity . Jordan Canonical Form:
J = [
1
] Block Diagonal
Where block i (corresponding to ) is of the form:
= [
0 00 00 0 0 0 0
] 1s on first super diagonal
Note:
A = Q 1
det A = det Q det det 1 = det .
Since det Q det 1 = det I =1
= D ; det D = det = 1 2
det A = product of all eigenvalues
This implies that A is non-singular iff it has non-zero eigenvalue
Companion form matrices:
[
0 0 0 1 0 0 1 0 0 1 1
] Or [
1 2 0 1 0 0 0 0 0 1 0
]
or their transpose such as
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[
0 1 0 00 0 1 0 1
1 2 1
]
All have same characteristic equation:
() = + 11+ +1+ = 0
Jordan form representation:
If has repeated eigenvalues with not n independent eigenvectors then
A has Jordan-form representation;
J = [
1 1 0 00 1 1 00 0 1 10 0 0 1
]
This matrix which has its eigenvalues at diagonal and 1 on the super
diagonal is called a Jordan block of order 4. The characteristic polynomial
of J is: () = | | = ( 1)4.
We can have Jordan block of various orders. Consider 55
With repeated value 1with multiplicity of 4 and single eigenvalue 2 .
There exists a non-singular matrix Q such that the similarity matrix .
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Assume one of the following form:
= 1
1 =
[ 1 1 0 0 00 1 1 0 00 0 1 1 00 0 0 1 00 0 0 0 2]
2 =
[ 1 1 0 0 00 1 1 0 00 0 1 1 00 0 0 1 00 0 0 0 2]
Jordan block of order 4 Jordan block of order 3 and 1
3 =
[ 1 1 0 0 00 1 0 0 00 0 1 1 00 0 0 1 00 0 0 0 2]
4 =
[ 1 0 0 0 00 1 0 0 00 0 1 0 00 0 0 1 00 0 0 0 2]
Jordan block of order 2 and2 3 Jordan block of order 2 and 1 and 1
5 =
[ 1 0 0 0 00 1 0 0 00 0 1 0 00 0 0 1 00 0 0 0 2]
4 Jordan block of orders 1
All these matrices are said to be Jordan form! If nullity of (A -1) is four,
then 4 linearly independent associated with 1.
And A is said to be diagonalizable. So 5 eigenvectors can be used as basis.
If nullity of (A -1) = 1, then one eigenvector associated with 1.
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Conclusion:
Computing Jordan form matrices is complicated and will not discuss
future. But we discuss a useful property of it as for Jordan block of order
4:
(J -) = [
0 1 0 00 0 1 00 0 0 10 0 0 0
]
(J ) 2 = [
0 0 1 00 0 0 10 0 0 00 0 0 0
]
(J ) 3 = [
0 0 0 10 0 0 00 0 0 00 0 0 0
]
(J ) = 0 K 4
This is called Nilpotent
Definition:
A monic polynomial is a polynomial with 1 as its leading coefficient. The
minimal polynomial of matrix A is the polynomial () with load
coefficient = 1 and the least degree () = 0
From a practical viewpoint, finding () is NOT easy and is not used,
except in theoretical derivations.
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Caley-Hamilton Theorem:
Every finite matrix A satisfies its own characteristic equation.
() = |A | = + 11+ +1+ = 0
Then;
() = + 11+ +1+ = 0
Applications:
1) Computing ;
pre-multiply () with 1;
1+ 12+ +1+
1 = 0
1 = -1
[1 + 1
2 + + 1]
i.e; given non-singular , then 1 is a linear combination of 1 and
lower power of A ( n- terms total)
2) Reducing the degree of Polynomial;
() = + 11+ +1+ = 0
= - [ 1 + + ]
Which implies +1 can be written as a linear combination of
[1, 2 , ] which in turn can be written as linear combination of
[, 1 , 1].
Proceeding forward, we conclude that, for any f(), no matter how large
its degree is, f() can always be expressed as:
f(A) = 0 + 1 + + 11 =
1=1
One way to get f(A) is to use long division to express f().
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f() = q() () + h() = h() = 0 + 1 + + 11
0
() is the quotient and h() is the reminder with degree less than n. then
we have:
() = q() () + h() = h() = 0 + 1 + + 11
0