Lecture 9: Solution of Controlled Linear Systems
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Continuous-Time Controlled LTI Systems
A continuous-time LTI system with input:
x = Ax + Bu
y = Cx + Du,
where x Rn is the state, u Rm is the input, y Rp is the output.
A Rnn is the (state) dynamics
Lecture 10: Lumped Nonlinear Systems
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Lumped Nonlinear Systems
Lumped continuous-time nonlinear system:
d
x(t) = f (x, u, t),
dt
y (t) = g (x, u, t)
Lumped discrete-time nonlinear system:
x[k + 1] = f (x[k], u[k], k),
f : Rn Rm R Rn ,
y [k] = g (x[
Lecture 14: Observability I
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State Estimation of D-T LTI Systems
Discrete-time n-state m-input p-output LTI system
x[k + 1] = Ax[k] + Bu[k],
x[0] = x0
y [k] = Cx[k] + Du[k]
Matrices A Rnn , B Rnm , C Rpn , D Rpm are known
Input and output are know
Name: _
EE-602
Exam II
October 15, 2009
140 Point Exam
INSTRUCTIONS
This is a closed book, closed notes exam. No calculator is permitted. Work patiently,
efficiently, and in an organized manner clearly identifying the steps you have taken to solve each
pr
Name: _
EE-602
Exam II
October 13, 2011
140 Point Exam
INSTRUCTIONS
This is a closed book, closed notes exam. No calculator is permitted. Work patiently,
efficiently, and in an organized manner clearly identifying the steps you have taken to solve each
pr
Exam 2, ECE 602, 2015
Name: _
EE-602
Exam II
October 8, 2015
148 Point Exam
(1 hour and 22 minutes)
INSTRUCTIONS
This is a closed book, closed notes exam. No calculator is permitted. No additional scrap
paper is permitted. Work patiently, efficiently, and
Name: _
EE-602
Exam I
September 14, 2015
135 Point Exam
INSTRUCTIONS
Time limit 1 hour 20 minutes.
This is a closed book, closed notes exam. No calculators or scrap paper are permitted.
Work patiently, efficiently, and in an organized manner clearly ident
Name: _
EE-602
Exam II
October 9, 2008
140 Point Exam
INSTRUCTIONS
This is a closed book, closed notes exam. You are permitted only a calculator. Work
patiently, efficiently, and in an organized manner clearly identifying the steps you have taken to
solve
Name: _
EE-602
Exam II
October 16, 2012
140 Point Exam
(1 hour and 22 minutes)
INSTRUCTIONS
This is a closed book, closed notes exam. No calculator is permitted. No additional scrap
paper is permitted. Work patiently, efficiently, and in an organized mann
ECE 602 Midterm 1 Solution
Problem 1. (20 pts) Consider a system given in the above diagram, which has two inputs, u1 (t)
and u2 (t), and two outputs, y1 (t) and y2 (t). There are also two integrators in the diagram.
(a) (10 pts) Define a set of state var
Lecture 11: Quadratic Forms and SVD
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Symmetric and Skew Symmetric Matrices
A Rnn is symmetric if AT = A.
All eigenvalues are real and all eigenvectors are orthogonal
Diagonalized by an orthogonal matrix Q (Q 1 = Q T ):
Q 1 AQ = Q T AQ = diag (1 ,
Lecture 8: Stability of Linear Systems
1 / 14
Stability of C-T Autonomous Linear Systems
Continuous-time linear system has an equilibrium point at xe = 0
x(t)
= A(t)x(t)
(1)
Definition (Asymptotic Stability)
System (1) is called asymptotically stable at x
chapter3-1
Linear Algebra
Let A, B, C and D be respectively n m, m r, l n, and r p,
ai be the ith column of A, and b the jth row of B. Then
b
1
h
i
b2
= a1 b1 + a2 b2 + + am bm
AB= a1 a2 am
.
.
CA= C
h
a1
a2
am
bm
i h
= Ca1
Ca2
Cam
i
and
Fall 2014
chapter2-1
1
1.1
MATHEMATICAL DESCRIPTIONS
OF SYSTEMS
Introduction
xxxSingle-input single-output (SISO) system
Only one input terminal and only one output terminal.
xxxMulti-input multi-output (MIMO) system
Two or more input terminals and output terminals
System type - Internal description .- External description
Distributed. linear E y(t) = f G(!. r)u(r)dr
i (u
; j r
Lumped. linear x = A0: + Emu 5 W) = / GU. mum dr
; In
y = C(I)X + DU)
3 l
Distn'buted, linear, j i yo) = / GU r)u(r) dr
' : U
time-invariant
Lecture 13: Controllability II
1 / 12
Controllability of C-T LTI Systems
Continuous-time LTI system x = Ax + Bu,
x(0) = x0
Controllable at time tf > 0 if for any initial state x0 Rn and any
target state xf Rn , a control u(t) exists that can steer the sy
Lecture 12: Controllability I
1 / 23
Controllability of C-T LTI Systems
A continuous-time LTI system
x = Ax + Bu,
x(0) = x0
Given tf > 0. The control input u(t) over the time interval [0, tf ] steers
(or transfers) the state from x0 to
Z tf
Atf
xf := x(tf
ECE 602 Midterm 2 Solution
Problem 1. (20 pts) A nonlinear system is given by
x 1 = (2 x1 x2 )x1
x 2 = (x1 x2 )x2 .
Find all the equilibrium points of the system and determine their (local) stability, if possible.
Solution: The system has the following eq
ECE 602 Homework #2 Solution
Problem 1. For each of the following sets equiped with the natural addition and scalar multiplication operations, determine if the set is a vector space. If the answer is yes, find its dimension. If
the answer is no, state you
ECE 602 Homework #6 Solution
Problem 1. Consider the continuous-time
1
x = 1
1
y= 1
LTI system (A, B, C):
0 0
0
0 0 x + 1 u
1 1
1
1 0 x.
(a) Is the system controllable? What is its reachable subspace?
(b) Find a coordinate transform T1 R33 to transform
Lecture 16: Minimality, BIBO Stability, and
Canonical Forms
Nov. 11
1 / 17
Kalman Decomposition
For any continuous-time n-state m-input p-output LTI system
x = Ax + Bu,
Its Kalman Canonical Form is
Aco
0
c o
21 A
A
x =
0
0
y=
0
co
C
Eigenvalues of A
Lecture 17: State Feedback Control
Nov. 18, 2014
1 / 14
Controller Design
A continuous-time (or discrete-time) LTI system
x = Ax + Bu
or
x[k + 1] = Ax[k] + Bu[k]
Problem: Design input u so that behavior of system is better
Stabilize the system (if A is u
ECE 602 Homework #7 Solution
Problem 1. Consider a LTI system x = Ax + Bu, y = Cx, given specifically by
3 1
1
0
0
3 1
2
3
2
x +
u
x =
1
0
3 5
3 1
3
4 1
0 1 0 2 1
y=
x,
1 0 0 1 0
(1)
where the system matrices are dependent on the parameters , , , and
ECE 602 Homework #3 Solution
Problem 1. Given the matrix
3 1
1
A = 1 1 1 .
0
0 2
(a) Write the expression of the solution to the continuous-time LTI system x = Ax with arbitrary
initial condition x(0). What are the modes of the system?
(b) Write the expre
ECE 602 Homework #8 Solution
Problem 1. For the problem of finding the least-cost path from A to B in the above grid, moving
left to right, as discussed in Lecture 19, verify that the value funcitons for all the nodes are indeed
given by the quantities ma
Lecture 19: Linear Quadratic Regulation
Dec. 2, 2014
1 / 32
LQR Problem Formulation
A discrete-time LTI system
x[k + 1] = Ax[k] + Bu[k],
x[0] = x0
Problem: Given a time horizon k cfw_0, 1, . . . , N, find the optimal input
sequence U = cfw_u[0], . . . , u