Homework 5 - MAE143B Spring 2010: due Thursday May 13 Question 1: State reachability
Part i: Consider the linear system with xt I n , R xt = Axt + But . For T > 0, define the controllability Gramian as the n n matrix
T
Wc (0, T ) =
0
e-A BB T e-A
T
d .
Su
Question 1
-1 Part i: Show that ut = -B e-A t Wc (0, T )i transfers the state x0 = xi to xT = 0. Note that the solution x to the ODE, xt = Axt + But ,
is, as was shown repeatedly and in various contexts, x(t) = eAt x0 + eAt 0 e-A Bu( )d . To prove the re
Homework 6 - MAE143B Spring 2010: due Thursday May 20
This week's homework will consist of three parts all dealing with the same problem; the control of the nonlinear predator-prey model described in Astr m & Murray pp. 90-91, 181-183. This model o descri
Homework 7 - MAE143B Spring 2010: due Thursday May 27 Question 1: Frequency response
Consider the linear system dn yt dn-1 yt dn-2 yt dyt dn-2 ut dut dn-1 ut + a1 n-1 + a2 n-2 + + an-1 1 + an yt = b1 n-1 + b2 n-2 + + bn-1 1 + bn ut . n dt dt dt dt dt dt d
Cruise control example calculations
Open-loop system dynamics mv(t) = -bv(t) + u(t) + Fhill (t). Denote by vdes the desired highway speed of the car.
Open-loop behavior
Consider the following situation. The car has been traveling across flat terrain for a
Bob's Code from class April 6 and 8, 2010
Here are the matlab examples that I provided in class. Mass-Spring-Damper system from car, function: function dydt = spring_dydt(t,y, k1, k2, k3 ,m1, m2, b, omega) u = 0.00315*cos(omega*t); dydt = [y(3); y(4); -(k
Pendulum Example
Pendulums are very common to study in controls classes because they exhibit many interesting dynamic phenomena common to more complex systems, yet are very easy to model. Wikipedia has a thorough overview of the pendulum and history in th
Linearization
Bob Bitmead, April 19, 2010.
Linearized ODEs
Consider the linearization of the first-order vector ordinary differential equation x = f (x, u). We use Taylor's theorem to linearize the solution about some (t , ut ) functions. Often the x choi
Midterm Exam - MAE143B Spring 2010; Tuesday May 4
Instructions
1. You have 75 minutes; 9:30 to 10:45. 2. The exam is open-book and open-notes. 3. Answer all the questions you are able to. 4. The questions are of equal value. The question parts are not. 5.
Introduction to the Sugar Cane Crushing Mill
A video extravaganza
APC04 Vancouver
Joint Modeling & Control Design
Sugar cane crushing mill
Two-input two-output control problem Part of a sequence of five mills crushing the same material Separation of fiber
Data preparation for System Identification
Carpentry
Preparation and prejudice
The sugar mill is available for experiment We need to get some prejudicial decision made before we can start delving into System Identification via matlab toolbox Almost all ou
Iterative modeling & control design
Tying together the threads
Modern model-based control needs two things A local performance objective using the nominal model A robustness measure involving the rough model error Modeling with closed-loop data tells us a