Finding & Using Reciprocal
Basis Vectors
Text Chapter 5.8
Vector Expansions
Each vector in the space has a unique expansion
n
x = ai vi = a1v1 + a2v2 + . + an vn
i =1
Where the vis are the basis vectors
To get the ais consider
<vj,x> = <vj,
i=1,n
aiv
Matrix Algebra
Text Chapter 4
Matrix Fundamentals
Amxn => m rows & n columns
Square matrix?
Algebra
A = B iff aij = bij
A B = C => aij bij = cij
Multiplication
By a scalar A = [ ]
AnxmBmxr = Cnxr matrices must be conformable
M
n
c = a bj
k k
i
j
i
Matrix Algebra
Text Chapter 4
Matrix Fundamentals
Amxn => m rows & n columns
Square matrix?
Algebra
A = B iff aij = bij
A B = C => aij bij = cij
Multiplication
By a scalar A = [ ]
AnxmBmxr = Cnxr matrices must be conformable
M
n
c = a bj
k k
i
j
i
Square Matrices
Text Chapter 8
Building Polynomials
Exponents (Am)n = Amn
(Am)n = (Am)(Am) Am
(AAAA)(AAAA)(AAAA)
A0 = I
for A nonsingular
Matrix Polynomials
If p(x) = cm xm + cm-1 xm-1 + +c1 x + c0
p(A) = cm Am + cm-1 Am-1 + +c1 A + c0
Factored for
Matrix Determinants & Inversion
Text Chapter 4b
Determinants
The determinant of A = |A| is a scalar
Several methods for finding determinants
Laplace expansion using Cofactors
Pivotal condensation (see book)
If |A| = 0 the matrix is singular & we cant
Lyapunovs Methods
Text Chapter 10.6
Lyapunovs Two
Methods
First = Indirect
Needs knowledge of solutions
The eigenvalue stuff we just did
Second = Direct
No need for solution information
Uses the idea of generalized energy
Stable if total energy (li
Linear Independence
Text Chapter 5.4
Definition
For cfw_x = cfw_x1, x2, xn finite set of vectors
cfw_xi are linearly dependent if there exists a set of
ais not all zero, such that
a1x1 + a2x2 + a3x3 + anxn = 0
Lemma: If a set of vectors is linearly de
Stability & Eigenvalues
Text Chapter 10.5
State Transition Matrix
Stable requires lim (t->infinity) | | = 0
For constant A
Continuous => eAt
Discrete =>Ak
Look at Jordan Form
Continuous Time
eAt = M eJt M-1
If one element -> infinity, whole thing do
Transfer Function Methods
Text Chapter 3d
State Equations from Transfer
Functions
Start with continuous time first
SISO systems
yn + an-1 yn-1 + + a2 y + a1 y + a0 y =
b0 u + b1 u + b2 u + + bm um
Laplace Form (derivative = s, zero ICs)
snY + an-1 sn
Simulation Diagram Method
Text Chapter 3c
Simulation Diagrams
Steps lead to observable cannonical form
1. Rearrange equation by grouping same order
derivatives of y & u
2. Integrate as many times as highest order derivative
3. Bring inner integrals in
Input-Output Method for System
Identification
Text Chapter 3b
Single Input Single
Output (SISO) Cases
SISO with no input derivatives
Typical equation
y + a3 y+ a2 y + a1 y + a0 y = u(t)
How many state variables do we need?
Number of states = order of
State Space Definitions and
Notation
Text Chapter 3
State Variable Approach
Determine a system description
Internal status of the system
Input-Output relationships
Functions
Function
Generally: rule giving the relationship between
the variables in 2 s
Conclusion of State Space Forms
Text End of Chapter 3
Interconnection of
Systems
u1
y1= u2
y2
System 1
System 2
x1 = A1 x1 + B1u1
y1 = C1 x1 + D1u1
x2 = A2 x2 + B2u2
y2 = C2 x2 + D2u2
Substitute in for u2 = y1 = C1x1 + D1u1 to get
x2 = A2 x2 + B2 ( C1 x1
Solving Unforced Equations
Text Chapter 8
Solving for x(t) When
u(t) = 0
x = Ax
State equation
Assume solution of the form x(t) = eAt x(0)
cfw_
d
d At
At
x = cfw_ x (t ) = e x (0) = Ae x (0) = Ax (t )
dt
dt
Finding eAt
Cayley-Hamilton
Laplace
Jorda
Stability Definitions
Text Chapter 10
Stability
Poles
Nyquist
Root Locus
Eigenvalues!
Equilibrium Points
x = 0
Steady state
No motion
Stays there if no input or disturbance
SEP or UEP?
Stable Equilibrium Points (SEPs)
EP such that if the system is distu