Homework 2
UC Berkeley, Advanced Control Systems II (ME233)
Due: Feb. 18 2014 before class
1. Consider a (rst-order) discrete-time system described by
x(k + 1) = ax(k) + w(k) + c
where x(0) and c are random variables, and w(k) is a white random process (s
ME 233, UC Berkeley, Spring 2014
Xu Chen
Lecture 2: Discrete-time Linear Quadratic Optimal
Control
Big picture
Example
Convergence of finite-time LQ solutions
Big picture
I
previously: dynamic programming and finite-horizon discrete-time
LQ
I
this lecture
ME 233, UC Berkeley, Spring 2014
Xu Chen
Lecture 14: Disturbance Observer
Big picture
Disturbance and uncertainties in mechanical systems:
I system models are important in design: e.g., in ZPET, observer,
and preview controls
I inevitable to have uncertai
University of California at Berkeley
Department of Mechanical Engineering
ME 233: Advanced Control Systems II
Spring 2014
ME233 discusses advanced control methodologies and their applications to engineering systems.
Methodologies include but are not limit
Homework 7
UC Berkeley, Advanced Control Systems II (ME233), Spring 2014 Due: May 5th 2014 11pm
1. In this problem, you will study various aspects of the parameter adaptation algorithms (PAAs) discussed in the
class. For this purpose, two MATLAB les (sp_p
Homework 5
UC Berkeley, Advanced Control Systems II (ME233), Spring 2014
Due: Apr. 1, 2014 8am
Instructions:
This set of homework accounts for 35% of the total homework scores in the course.
Start the homework early.
For Problem 3, you can work within
Homework 6
UC Berkeley, Advanced Control Systems II (ME233), Spring 2014
Due: Apr. 17 2014 8am
1. Verify the matrix inversion lemma: if A is nonsingular, B and C have compatible dimensions, then
(A + BC)
1
= A1 A1 B CA1 B + I
1
CA1
Let A = I and BC = T .
Homework 4
UC Berkeley, Advanced Control Systems II (ME233)
Due: Mar. 18, 2014 8am
1. In this problem, we design a compensator for the disk drive system in problem 4 of homework 3, by applying the
LQR with frequency shaped cost functional.
(a) First let u
Homework 3
UC Berkeley, Advanced Control Systems II (ME233)
Due: Mar. 4, 2014, 8am
1. A discrete time system is described by
x(k + 1) = Ax(k) + Bu(k) + Bw w(k)
where w(k) is a colored noise given by
w(k) = Cw xw (k)
xw (k + 1) = Aw xw (k) + Bn n(k)
x(0),
Homework 1
UC Berkeley, Advanced Control Systems II (ME233)
Due: Feb. 4 2014 before class
1. How do you split one number, say xf , to N pieces so that the product of the N pieces is maximized? Think in the
following way. We have an integrator described by
ME 233, UC Berkeley, Spring 2014
Xu Chen
Lecture 19: Adaptive Control based on Pole
Assignment
Big picture
reasons for adaptive control:
unknown or time-varying plants
unknown or time-varying disturbance (with known structure but
unknown coecients)
two ma
ME 233, UC Berkeley, Spring 2014
Xu Chen
Lecture 18: Parameter Convergence in PAAs
Big picture
why are we learning this:
Consider a series-parallel PAA
/ B (z 1 ,k+1)
O
+
u(k)
1
/ B (z )
y (k+1)
A(z 1 )
(k+1)
/
/ A(z 1 ,k+1)
where the plant is stable.
(Hy
ME 233, UC Berkeley, Spring 2014
Xu Chen
Lecture 17: PAA with Parallel Predictors
Big picture: we know now.
u (k)
B z 1
/
A (z 1 )
/
y (k + 1)
simply means:
y (k + 1) = B z 1 u (k) A z 1 1 y (k + 1)
= T (k)
In RLS:
y o (k + 1) = T (k) (k) = B z 1 , k u (k
ME 233, UC Berkeley, Spring 2014
Xu Chen
Lecture 1: Dynamic Programming
General problem
Multivariable derivative
Discrete-time LQ
Dynamic programming (DP)
introduction:
I
history: developed in the 1950s by Richard Bellman
programming: ~planning (has nothi
ME 233, UC Berkeley, Spring 2014
Xu Chen
Lecture 12: Preview Control
Big picture
Problem formulation
Relationship to LQ
Solution
Review: optimal tracking
We consider controlling the system
x (k + 1) = Ax (k) + Bu (k)
y (k) = Cx (k)
(1)
where
x Rn , u Rm ,
UC Berkeley
Lecture Notes for ME233
Advanced Control Systems II
Xu Chen and Masayoshi Tomizuka
Spring 2014
Copyright: Xu Chen and Masayoshi Tomizuka 2013~. Limited copying or use for educational
purposes allowed, but please make proper acknowledgement, e.
ME 233, UC Berkeley, Spring 2014
Xu Chen
Lecture 5: Stochastic State Estimation
(Kalman Filter)
Big picture
Problem statement
Discrete-time Kalman Filter
Properties
Continuous-time Kalman Filter
Properties
Example
Big picture
why are we learning this?
I
s
ME 233, UC Berkeley, Spring 2014
Xu Chen
Lecture 3: Review of Probability Theory
Connection with control systems
Random variable, distribution
Multiple random variables
Random process, filtering a random process
Big picture
why are we learning this:
We ha
ME 233, UC Berkeley, Spring 2014
Xu Chen
Lecture 4: Least Squares (LS) Estimation
Background and general solution
Solution in the Gaussian case
Properties
Example
Big picture
general least squares estimation:
I
I
I
given: jointly distributed x (n-dimensio
ME 233, UC Berkeley, Spring 2014
Xu Chen
Lecture 13: Internal Model Principle and
Repetitive Control
Big picture
review of integral control in PID design
example:
D(s)
0
E (s)
/
/O
C (s)
/ +
+
/ P (s)
/ Y (s)
where
1
1
, C (s) = kp + ki + kd s, kp , ki ,
ME 233, UC Berkeley, Spring 2014
Xu Chen
Lecture 17: PAA with Parallel Predictors
Big picture: we know now.
u (k)
/
z 1
B
A (z 1 )
/
y (k + 1)
simply means:
y (k + 1) = B z
1
u (k) A z
1
1 y (k + 1)
= T (k)
In RLS:
z
y (k + 1) = (k) (k) = B
o
T
1
1
z ,
ME 233, UC Berkeley, Spring 2014
Xu Chen
Lecture 16: Stability of Parameter Adaptation
Algorithms
Big picture
I
For
(k + 1) = (k) + [correction term]
we havent talked about whether (k) will converge to the true
value if k . We havent even talked about wh
ME 233, UC Berkeley, Spring 2014
Xu Chen
Lecture 15: System Identification and Recursive
Least Squares
Big picture
We have been assuming knwoledge of the plant in controller design.
In practice, plant models come from:
I modeling by physics: Newtons law,
ME 233, UC Berkeley, Spring 2014
Xu Chen
Lecture 11: Feedforward Control
Zero Phase Error Tracking
Big picture
Stable pole-zero cancellation
Phase error
Zero phase error tracking
Big picture
why are we learning this:
r (k)
ks
+/
O
/
y (k)
/P
Feedback C
+
ME 233, UC Berkeley, Spring 2014
Xu Chen
Lecture 7: Principles of Feedback Design
MIMO closed-loop analysis
Robust stability
MIMO feedback design
Big picture
I
I
I
we are pretty familiar with SISO feedback system design and
analysis
state-space designs (L
ME 233, UC Berkeley, Spring 2014
Xu Chen
Lecture 6: Linear Quadratic Gaussian (LQG)
Control
Big picture
LQ when there is Gaussian noise
LQG
Steady-state LQG
Big picture
in deterministic control design:
I state feedback: arbitrary pole placement for contro
ME 233, UC Berkeley, Spring 2014
Xu Chen
Lecture 8: Discretization and Implementation
of Continuous-time Design
Big picture
Discrete-time frequency response
Discretization of continuous-time design
Aliasing and anti-aliasing
Big picture
why are we learnin
ME 233, UC Berkeley, Spring 2014
Xu Chen
Lecture 16: Stability of Parameter Adaptation
Algorithms
Big picture
For
(k + 1) = (k) + [correction term]
we havent talked about whether (k) will converge to the true
value if k . We havent even talked about whet
ME 233, UC Berkeley, Spring 2014
Xu Chen
Lecture 15: System Identication and Recursive
Least Squares
Big picture
We have been assuming knwoledge of the plant in controller design.
In practice, plant models come from:
modeling by physics: Newtons law, cons
ME 233, UC Berkeley, Spring 2014
Lecture 14: Disturbance Observer
Xu Chen
Big picture
Disturbance and uncertainties in mechanical systems:
system models are important in design: e.g., in ZPET, observer,
and preview controls
inevitable to have uncertainty