EE 580 Linear Control Systems
Exam 1
Due on Thursday, September 30 by 2:30 PM
Department of Electrical Engineering
Pennsylvania State University
Fall 2010
1. Closed-Loop Descriptions. A linear time-invariant plant is described by
x(t) = Ax(t) + Bu(t),
t R
EE 580 Linear Control Systems
III. Input-Output Description of Linear Systems
Department of Electrical Engineering
Pennsylvania State University
Fall 2010
3.1
Denitions and Facts About Matrix Inversion
A function f : Rn Rm is said to be one-to-one (or inj
EE 580 Linear Control Systems
IV. Finite-Dimensional Vector Spaces
Department of Electrical Engineering
Pennsylvania State University
Fall 2010
4.1
Introduction
The space of complex numbers is denoted by C. Depending on the context, we will use the n-tupl
EE 580 Linear Control Systems
I. Models of Dynamical Systems
Department of Electrical Engineering
Pennsylvania State University
Fall 2010
c 2010 Ji-Woong Lee
1.1
Introduction
We are concerned with the analysis and synthesis of dynamical systems. Analysis
EE 580 Linear Control Systems
IX. Observability of Linear Systems
Department of Electrical Engineering
Pennsylvania State University
Fall 2010
9.1
Introduction
Consider the linear time-invariant system
x(t) = Ax(t) + Bu(t),
(9.1)
y(t) = Cx(t) + Du(t),
whe
EE 580 Linear Control Systems
II. Solution and Linearization of State-Space Models
Department of Electrical Engineering
Pennsylvania State University
Fall 2010
2.1
Notation and Denitions
n
2 1/2 . The
The Euclidean (vector) norm x of x Rn is dened by x =
EE 580 Linear Control Systems
XIII. State-Space Realization of Linear Systems
Department of Electrical Engineering
Pennsylvania State University
Fall 2010
13.1
Introduction
Let A Rnn , B Rnm , C Rln , and D Rlm . We know that the external description (i.e
EE 580 Linear Control Systems
X. Structural Properties of Linear Systems (Part 1)
Department of Electrical Engineering
Pennsylvania State University
Fall 2010
10.1
Introduction
Recall that a linear map, or its matrix representation, between nite-dimension
EE 580 Linear Control Systems
VI. State Transition Matrix
Department of Electrical Engineering
Pennsylvania State University
Fall 2010
6.1
Introduction
Typical signal spaces are (innite-dimensional) vector spaces. Consider the space C(J, Rn ) of Rn valued
EE 580 Linear Control Systems
VII. Stability of Linear Systems
Department of Electrical Engineering
Pennsylvania State University
Fall 2010
7.1
Denitions and Facts
We will see that internal stability of linear systems depends on the structural properties
EE 580 Linear Control Systems
XI. Structural Properties of Linear Systems (Part 2)
Department of Electrical Engineering
Pennsylvania State University
Fall 2010
11.1
Introduction
In this section, we continue to study the structural properties of LTI system
EE 580 Linear Control Systems
V. Structural Properties of Matrices
Department of Electrical Engineering
Pennsylvania State University
Fall 2010
5.1
5.1.1
Denitions and Facts
Determinants
Given a square matrix A Cnn , the determinant of A, denoted by det A
EE 580 Linear Control Systems
Exam 1
Due on Thursday, September 30 by 2:30 PM
Department of Electrical Engineering
Pennsylvania State University
Fall 2010
1. Closed-Loop Descriptions. A linear time-invariant plant is described by
x(t) = Ax(t) + Bu(t),
t R
EE 580 Linear Control Systems
XII. Eigenvalue Assignment and Stabilization
Department of Electrical Engineering
Pennsylvania State University
Fall 2010
12.1
Introduction
Recall that a control system is a feedback interconnection of two dynamical systems,
EE 580 Linear Control Systems
VIII. Controllability of Linear Systems
Department of Electrical Engineering
Pennsylvania State University
Fall 2010
8.1
Introduction
Analogous to the fact that matrices are linear maps between nite-dimensional vector spaces,
EE 580 Linear Control Systems
Exam 2
Due on Thursday, November 11 by 2:30 PM
Department of Electrical Engineering
Pennsylvania State University
Fall 2010
1. Piecewise-Linear Periodic Systems. Let T > 0. Let Ai Rnn , Bi Rnm , and Ci Rln
for i = 1, 2. Consi
EE 580 Linear Control Systems
Exam 3
Due on Thursday, December 16 by 2:30 PM
Department of Electrical Engineering
Pennsylvania State University
Fall 2010
1. Output Regulation. A linear plant is given by
x(t) = Ax(t) + B1 w(t) + B2 u(t),
y(t) = Cx(t) + Dw(
EE 580 Linear Control Systems
Homework 1
Due on Tuesday, September 14 by 2:30 PM
Department of Electrical Engineering
Pennsylvania State University
Fall 2010
1. An Initial-Value Problem. Consider the state equation
x(t) = ax(t) + t,
t 0,
with the initial
EE 580 Linear Control Systems
Homework 2
Due on Tuesday, September 21 by 2:30 PM
Department of Electrical Engineering
Pennsylvania State University
Fall 2010
1. Simulation of Simple Pendulum Models. [Antsaklis & Michel (2007),
Problem 1.14] Consider a sim
EE 580 Linear Control Systems
Homework 3
Due on Tuesday, October 26 by 2:30 PM
Department of Electrical Engineering
Pennsylvania State University
Fall 2010
1. Companion-Form Systems (Revisited). Consider the n-th order SISO system of the form
x(t) = Ax(t)
EE 580 Linear Control Systems
Homework 4
Due on Tuesday, November 2 by 2:30 PM
Department of Electrical Engineering
Pennsylvania State University
Fall 2010
1. Equivalent Systems. Consider two equivalent linear time-invariant systems
x1 (t) = A1 x1 (t) + B
EE 580 Linear Control Systems
Homework 5
Due on Tuesday, November 30 by 2:30 PM
Department of Electrical Engineering
Pennsylvania State University
Fall 2010
1. Review of Facts (that have been useful). [Expanded on Antsaklis & Michel (2007), Problem
5.1] D
EE 580 Linear Control Systems
Homework 6
Due on Tuesday, December 7 by 2:30 PM
Department of Electrical Engineering
Pennsylvania State University
Fall 2010
1. Servo Motor Control. [Antsaklis & Michel (2007), Problem 9.9] A servomotor that drives
a load is
EE 580 Linear Control Systems
I. Models of Dynamical Systems
Department of Electrical Engineering
Pennsylvania State University
Fall 2010
c 2010 Ji-Woong Lee
1.1
Introduction
We are concerned with the analysis and synthesis of dynamical systems. Analysis
EE 580 Linear Control Systems
II. Solution and Linearization of State-Space Models
Department of Electrical Engineering
Pennsylvania State University
Fall 2010
2.1
Notation and Denitions
n
2 1/2 . The
The Euclidean (vector) norm x of x Rn is dened by x =
EE 580 Linear Control Systems
III. Input-Output Description of Linear Systems
Department of Electrical Engineering
Pennsylvania State University
Fall 2010
3.1
Denitions and Facts About Matrix Inversion
A function f : Rn Rm is said to be one-to-one (or inj
Vector Space
A vector space is a set of elements called vectors such that it is closed under finite vector
addition and scalar multiplication. In order for to be a vector space, the following
conditions must hold for all vectors and scalars and :
1. Closu
EE 580 Linear Control Systems
Exam 3
Due on Thursday, December 16 by 2:30 PM
Department of Electrical Engineering
Pennsylvania State University
Fall 2010
1. Output Regulation. A linear plant is given by
x(t) = Ax(t) + B1 w(t) + B2 u(t),
y(t) = Cx(t) + Dw(