University of California
Mechanical Engineering Department
ME 232 Advanced Control Systems I
Homework Set #3
Assigned: September 24 (Tu)
Due: October 3 (Th)
[1]
Consider a first order system described by
Fall 2013
dx(t )
= ax(t ) + bu (t ), x(0) = x 0
dt
UNIVERSITY OF CALIFORNIA, BERKELEY
Department of Mechanical Engineering
ME C232 / EE C220A Advanced Control Systems I
Fall 2014
Assigned:
Due:
Homework #2
Tuesday, September 9
Tuesday, September 16
1) A second order system is described by
dx1 (t)
= x2 (t)
ME232 Fall 2014
Solutions to Homework Set #9
1. (a) We have
e(k + 1) = y(k + 1) r
= Cx(k + 1) r
= C (x(k) + x(k + 1) r
= (y(k) r) + Cx(k + 1)
= e(k) + Cx(k + 1).
(b) First we need to nd the state dynamics for x(k)
x(k + 1) = x(k + 1) x(k)
= Ax(k) + Bu(k)
ME232 Fall 2014
Solutions to Homework Set #8
1) (a) We use the following Matlab command to compute Tb :
A =
B =
C =
sys
[0 2 0 0;0 0 4 0;0 0 0 8;-1.557 -4.803 -9.37 -14.6];
[0;0;0;2];
[0.2031 0.5625 0.6875 0.5];
= ss(A,B,C,0);
Wc = lyap(A,B*B);
Wo = lyap(
UNIVERSITY OF CALIFORNIA AT BERKELEY
Department of Mechanical Engineering
ME C232 / EE C220A Advanced Control Systems I
Fall 2014
Homework #1
Assigned:
Due:
Th., Aug. 28
Tu., Sept. 9
1) Obtain the Laplace transform F (s) for the time functions f (t) given
ME232 Fall 2014
Solutions to Homework Set #1
1. Computation of Laplace and Z-Transforms
(a) Let g(t) = e5t sin(3t), then
G(s) = L (g(t) =
t
0
With the property of L
3
(s + 5)2 + 9
g( )d = 1 G(s), we have
s
1
1
3
G(s) =
s
s (s + 5)2 + 9
F (s) = L (f (t) =
H WI sample solutions
1. [10 points] Denoting cfw_- as the operator for Laplace transform, we have
cfw_h (t)
* 12 (t)
= cfw_it h (t - T) 12 (T) dT
= le-st(lth(t-T)12(T)dT)dt
1
00
=
lte-sth(t-T)12(T)dTdt
The above is an integral over the region D = cfw_(t
UNIVERSITY OF CALIFORNIA AT BERKELEY
Department of Mechanical Engineering
ME C232 / EE C220A Advanced Control Systems I
Fall 2014
Assigned:
Due:
Homework #6
Tuesday, October 21
Thursday, October 30
1) Show in the z-plane how the imaginary axis of the s-pl
ME232 Fall 2014
Solutions to Homework Set #10
1. Axes Synchronization1
(a) Rewrite J = 0 (x2 + x2 + s(x1 x2 )2 + (u2 + u2 ) dt = 0 (X T QX + U T RU ) dt with Q =
1
2
1
2
1 + s s
and R = I. Then according to Theorem LQ-1, we have the optimal state feedback
UNIVERSITY OF CALIFORNIA, BERKELEY
Department of Mechanical Engineering
ME C232 / EE C220A Advanced Control Systems I
Fall 2014
Homework #3
Assigned:
Due:
Thursday, September 18
Thursday, September 25
1. (a) Let R(s) by the eld of rational functions of po
1
2
Continuous time forced response!
ME 232 Advanced Control Systems I
Lecture 8
Discrete Time Models from
Sampling Continuous Time Models"
Assume
and take Laplace transform
(ME232 Class Notes pp. 33-37)"
"
3
Continuous time forced response!
4
Continuous
1
2
Summary"
Solution of continuous time LTI systems"
First order systems"
Higher order systems"
ME 232 Advanced Control Systems I
Lecture 5
Solution of LTI State Equations"
The transition matrix"
Systems with a diagonal A matrix"
Systems with a Jor
1
2
Summary"
ME 232 Advanced Control Systems I
Lecture 7
Solution Matrix via Inverse
Laplace and Z- Transforms
"
We previously derived the solution of LTI state
equations using the time domain approach both for"
continuous time"
discrete time."
In th
Static System
ME 232 Advanced Control I
Lecture 3
Its present output depends only on its present
input
State Space Models of Dynamic
Systems
(ME232 Class Notes pp. 9-11)
and
(Additional Notes pp. 20-22)
memory-less systems
described by algebraic equations
1
2
Outline"
ME 232 Advanced Control Systems I
Lecture 2
The Z Transformation"
Introduction"
Discrete time sequence"
Z transform definition"
conditions and region of existence"
Examples"
Properties of the Z transform "
Applications of Z transformati
1
2
Summary
ME 232 Advanced Control Systems I
Lecture 16
Singular value decomposition
meaning
algorithm
Balanced realizations
meaning
algorithm
proof
Singular Value Decomposition &
Balanced Realizations
(ME232 Class Notes pp. 92-96)
3
Matrix induce
1
2
Summary
ME 232 Advanced Control Systems I
Lecture 11
Continuous time LTI systems
Lyapunovs direct method for LTI systems
The Lyapunov equation
Stability
Part III The Lyapunov Equation
Existence ad uniqueness of a solution
The exponential stabilit
23
Stability of LTI Systems
24
Stability of Continuous Time LTI Systems
The stability of the equilibrium state 0 for a linear systems
Unstable
Recfw_ i > 0 for at least one i, or
Recfw_ i 0 for all i's, but for a repeated j
on the imaginary axis with mu
1
2
Outline
ME 232 Advanced Control Systems I
Introduction
Continuous time function
Laplace transform definition
conditions and region of existence
Examples
Laplace transform properties
Applications of Laplace transform
Lecture 2
The Laplace Transf
1
ME 232 Advanced Control Systems I
Lecture 20
Properties of
Optimal Linear Quadratic Regulators (LQR)
Based on Return Difference Equality
Continuous Time
2
Outline
Brief review of SISO control systems
gain and phase margins
root locus
LQR problem
R
1
2
Outline
ME 232 Advanced Control Systems I
Lecture 9
Review of finite dimensional vector norms
Equilibrium state of CT unforced systems
Stability
Lyapunovs definitions of stability
Part I Definitions & Routh Hurwitz Test
Stability of CT LTI Systems
1
2
Summary
ME 232 Advanced Control Systems I
Lecture 19
Definition of the infinite horizon (IH) linear
quadratic regulator (LQR)
Conditions for the existence of an IH LQR
solution
Solution of the optimal LQR
Solution via the algebraic Riccati equatio
1
2
Matlab Commands Used
ME 232 Advanced Control Systems I
These examples make use of basic matlab
commands including:
Complement to Lectures 17 & 18
(new version)
ss, place, connect, sumblk
Commands such as
State Variable and State Observer
Feedback Con
1
2
Summary
ME 232 Advanced Control Systems I
Lecture 15
Controllable and Unobservable Subspaces
Kalman Canonical Staircase Decompositions
(ME232 Class Notes pp. 97-102)
and
(Additional Notes section 5)
Subspaces, range and null spaces (review)
The contro
1
2
Summary
ME 232 Advance Control I
Lecture 3A
Mathematical notation
Functions
Fields
Vector spaces
Linear independence
Dimension, span and basis
Normed vector spaces
Inner product vector spaces
Linear operators
Coordinate representation
Range and
1
2
Outline
1. Preliminary review material
1. Matrix inversion
2. Block diagram algebra
2. Continuous time Linear Time Invariant (LTI)
dynamic system description
1. State space
2. Input /output in time and Laplace domains
3. Conversion from state space to
ME C232 / EE C220A
Advanced Control Systems I
Instruction
Introduction
Lecture 1"
Instructor: Roberto Horowitz
Room 5138 EH, horowitz@berkeley.edu
Tentative Office Hours: M, W 9:00-10:30
GSI: Behrooz Shahsavari, behrooz@berkeley.edu
Office Hours: TBD
1
2
Outline"
ME 232 Advanced Control Systems I
Lecture 10
Stability
Part II Lyapunovs Direct Method"
Positive Definite and Quadratic Functions"
The Direct Method of Lyapunov "
Lyapunov Stability Theorems"
Stability in the sense of Lyapunov"
Asymptoti
UNIVERSITY OF CALIFORNIA AT BERKELEY
Department of Mechanical Engineering
ME C232 EE C220A Advanced Control Systems I
Fall 2014
Homework #9
Assigned:
Due:
Midterm Exam II Date: Tuesday November 24
Tuesday, November 17
Thursday, December 3
(Before Thanksg
UNIVERSITY OF CALIFORNIA, BERKELEY
Department of Mechanical Engineering
ME C232 / EE C220A Advanced Control Systems I
Fall 2015
Homework #2
Assigned:
Due:
Tuesday, September 8
Thursday, September 17
1) A second order system is described by
dx1 (t)
= x2 (t
UNIVERSITY OF CALIFORNIA AT BERKELEY
Department of Mechanical Engineering
ME C232 EE C220A Advanced Control Systems I
Fall 2015
Homework #9
Assigned:
Due:
1) Consider the following third order LTI system
1
1 0 2
x1
x1
d
x2 = 4 1 6 x2 + 1 u
dt
1
3 0 4
UNIVERSITY OF CALIFORNIA AT BERKELEY
Department of Mechanical Engineering
ME C232 / EE C220A Advanced Control Systems I
Fall 2015
Homework #4
Assigned:
Due:
Thursday, September 24
Tuesday, October 6
1. Prove parts (d) and (e) of Theorem 2.3.6. of the ME23
UNIVERSITY OF CALIFORNIA AT BERKELEY
Department of Mechanical Engineering
ME C232 / EE C220A Advanced Control Systems I
Fall 2015
Homework #7
Assigned:
Due:
Tuesday, October 20
Tuesday, November 3
Midterm Examination on Thursday October 29:
The exam will
UNIVERSITY OF CALIFORNIA AT BERKELEY
Department of Mechanical Engineering
ME C232 EE C220A Advanced Control Systems I
Fall 2015
Homework #11
Assigned:
Due:
Tuesday, December 3
Thursday, December 10 (before 4:00 PM)
Please turn your assignment to Emily Hig
UNIVERSITY OF CALIFORNIA AT BERKELEY
Department of Mechanical Engineering
ME C232 EE C220A Advanced Control Systems I
Fall 2015
Homework #8
Assigned:
Due:
Tuesday, November 3
Tuesday, November 10
1) Given the LTI system 1
x1 (k + 1)
1.1 0.34 0.32
x1 (