State Feedback and State Estimators
Introduction
Two types of control:
Open-loop control: the actuating signal depends
only on the reference signal and is independent
of the plant output
Closed-loop control: the actuating signal depends on
both the refe
Controllability Indices
Let A and B be n n and n p constant matrices. Suppose the pair cfw_A, B is
controllable then the controllability matrix
U = [B AB
A2B
An-1B]
has rank n which mean that there are n linearly independent columns in U.
It may be that w
State Feedback and Zeros of the Transfer Function
State feedback does not change the zeros of a
realization:
This simple to see in the controllable canonical form:
cfw_Ac, bc, Cc
cfw_Ac +bck, bc, Cc
State feedback with
gain kc
Hence
gcl(s) = Cc(sI Ac bc k
In this Lecture:
Canonical Decomposition of Linear Time-Invariant System
1
Canonical Decomposition of Linear Time-Invariant System
Consider the linear time-invariant dynamical equation
(*)
:=
x = Ax + Bu
y = Cx + Du
where A, B, C and D are n n, n p, q n,
Irreducible realization of proper rational transfer functions
Realization from the Hankel matrix
0 s n 1s n1 . n
g ( s) n
s 1s n1 . n
g(s) h(0) h(1)s 1 h(2)s 2 h(3)s 3 .
The coefficients h(i) will be called Markov parameters.
h(2)
h(3)
h(1)
h(2)
h(3)
h(
Minimal Realizations
A transfer matrix G(s) is said to be realizable if there exist a
state space equation
x = Ax + Bu
y = Cx + Du
That has G(s) as its transfer matrix.
Why do we study realizations?
Because
1. In order to simulate systems you need dynamic
In this Lecture:
Controllability of Linear Dynamical Equations
Observability of Linear Dynamical Equations
Canonical Decomposition of a Linear Time-invariant
Dynamical Equation
1
Controllability and Observability of Linear Systems
System analysis consis
Observability of Linear Time Invariant Systems
Consider the n-dimensional p-input state equation
x = Ax + Bu
y = Cx + Du
*
Definition: The state equation (*) is said to be observable
if for any unknown initial state x(0) = x0, there exists a
finite t1 > 0
In this Lecture:
Stability
Input-Output stability of LTI systems
Internal stability
Stability of LTV systems
1
Stability
Unstable System tends to burn out, disintegrate, or saturate
when a signal is applied.
Stability is a basic requirement for all
What can be achieved by using State Feedback?
Pole Placement
Example
Consider the system
1 1 1
x=
x + u
1
2 0
Determine the stability of the above system. If the
system is unstable use State Feedback to stabilize the
system.
Solution
a(s) = det(sI A
Controllability and Observability of Jordan Form
Dynamical System
Consider the n-dimensional LTI Jordan form system
x = Ax + Bu
JFE
y = Cx + Du
Where A, B, C are assumed to be of the form
A1
A =
( n n )
Am
A2
.
.
B1
B
2
B = .
( n p )
.
Bm
C
In this Lecture:
Equivalent Dynamical Equations
Realizations
1
Equivalent Dynamical Equations
Consider the linear time-invariant dynamical equation
:
x Ax Bu
x(0)=x0
y Cx Du
Define
x Px where P is any n n nonsingular matrix
Then
x Px
PAx PBu PAP 1x PBu
Normed spaces
Want to add more structures to a vector space
Norms: Generalization of the idea of length
In order to talk about the length of vector in a vector space or the distance
between two vectors, we must define the concept of a norm.
A norm of a v
Methods for Computing eAt
1. Taylor Series
e = t k Ak / k!
At
k =0
This method does not give closed form solution.
2.
e At = 1cfw_( sI A) 1
To show this
It is well known that the infinite series
F() = (1 )-1 = 1 + + 2 +
Converges for | | < 1.
1
Now
( sI
State-Space Solutions and Realizations
Solution of LTV State Equations
Consider a linear time varying system:
x(t ) = A(t ) x(t ) + B (t )u (t )
y (t ) = C (t ) x(t ) + D(t )u (t )
(*)
x(t o ) = xo
Given x(t0) = x0 and u() A unique solution x(), y()
What
Some Direct Realizations of Multivariable Transfer
Functions
Consider systems in which the denominator polynomial has
distinct roots
r
d ( s) ( s i )
i 1
i j
d(s) = the least common multiple of the denominators of all
the entries of the transfer matrix H(
In this Lecture:
Function of a Square Matrix
Cayley-Hamilton Theorem
1
Function of a Square Matrix
Examples of square matrices
AK , eAt
Why those functions?
AK raised from the solution to the difference equation
x(k + 1) = Ax(k)
given x(0)
which is
x(k)
Investigation and Control of Linearizable, Dynamical systems
EE550 Course Project
Dr. Ahmad A. Masoud, Electrical Engineering, KFUPM, 2012-1
The objective of this project is to explore the use of linear systems in understanding and controlling practical s
X 1
Consider the under-constrained linear system: [1 2 3] X 2 = [6]
X 3
0
A particular solution of the system is: Xp = 0
2
Lets now construct the general solution using vectors that are only independent of [1 2 3]T
X = Xp + 1 N 1 + 2 N 2
EE 550-1 Take home Quiz-3
Dr. Ahmad A. Masoud, Due Sunday October 30, 2013
All of you know by now that system stability is important it is the minimum requirement a system must satisfy
for it to be useful. Also all of you know that one famous test of asym
EE 550-1 Take home Quiz-2
Dr. Ahmad A. Masoud, Due Sunday October 27, 2013
Name:
Number:
Consider a SISO system with transfer function shown below
1
H ( s) = n
s + a n 1 s n 1 +.+ a1 s + a 0
Consider the A matrix in the controller canonical form realizati
EE 550-1 Take home Quiz-1
Dr. Ahmad A. Masoud, Due Tuesday October 8, 2013
Name:
Number:
This is the first take home quiz. It is optional and its mark is a bonus ( 5 marks = half the mark of a regular quiz). It is due next Tuesday on which the
second regu
Copyrighted Material
LECTURE 1
State-Space Linear Systems
CONTENTS
This lecture introduces state-space linear systems, which are the main focus of this course.
1. State-Space Linear Systems
2. Block Diagrams
3. Exercises
1.1 STATE-SPACE LINEAR SYSTEMS
A c
2.14 Analysis and Design of Feedback Control Systems
State-Space Representation of LTI Systems
Derek Rowell October 2002
1
Introduction
The classical control theory and methods (such as root locus) that we have been using in class to date are based on a s
1
State-Space Canonical Forms
For any given system, there are essentially an innite number of possible state
space models that will give the identical input/output dynamics. Thus, it
is desirable to have certain standardized state space model structures:
ME2025 Digital Control
Z Transform in State Space
Jee-Hwan Ryu
School of Mechanical Engineering
Korea University of Technology and Education
State-Space Representations of
Discrete-time Systems
Many ways to realize state-space
representation
Controllable