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1. Compute  determinants for the  following matrices  by  hand (i.e.  do not  use Matlab, Mathematica, etc.), and state  whether or not each is invertible.

1 2 3

A =      6 5 4

8 9 7

A =      11 26 0 0

63 57 0 10

83 91 1 71

54 23 0 71

A =      -34 16 17

22 10 -11 e

-64 31 32 /2

76 37 38 42

A =      1 1   2   3   5

0 8 13 21 34

0 0 58 89 144

0 0 0  233 377

0 0 0    0   610

1. Prove each of the following statements (stick to solid mathematical facts and reasoning;

eschew numerical or hand-wavy  arguments):

(a)  If a and b are non-zero n × 1 vectors, then  matrix  abT  has rank = 1.

(b)  tr(AB) = tr(BA) if A is an m × n matrix  and B is n × m (hint:  consider expressing

A and B in terms of stacked row or column vectors)

 |    |

(c)  If A is invertible,  then   A−1    =   1

|A|

(hint:  note that  |AB| = |BA| = |B| |A| if A and B

are any two compatible  square matrices).

1. Consider the equations  of motion for the coupled 2 mass/3  spring system, like the one discussed in lecture,

m1 q¨1  = −k1q1  − k2(q1 − q2 ) − u1

m2 q¨2  = −k2(q2  − q1 ) − k3q2 + u1 + u2,

where q1  is the  displacement of mass m1, q2  is the  displacement of mass m2, the k’s are spring constants,  and the u’s are external  forces. Find a set of A,B,C ,D matrices  for the state  vector definition,

x = [q1 − q2, q˙1 − q˙2, q1 + q2 , q˙1 + q˙2 ]T ,

and  for observations  y = [q1 , q2]T   and  inputs  u = [u1 , u2]T .  For  this  problem,  you may use Matlab  or other  software to assist with any required  intermediate numerical  matrix calculations  (i.e. no symbolic tools allowed).

1. The  linearized  equations  of motion  for an  orbiting  satellite  spinning  with  nominal angular  rate  po  about  the x axis are

Image missing

Figure 1: Spinning satellite  variables.

∆p˙ =

∆q˙ =

∆r˙ =

Mx

Ix

po (Ix  Iz )∆r + My

Iy

po (Iy Ix)∆q + Mz

Iz

where Ix, Iy , and Iz are the moments of inertia about the roll, pitch, and yaw axes; Mx, My

and Mz   are the corresponding  input  (control  and disturbance) torques;  and ∆p, ∆q  and

∆r are perturbations in rolling, pitching,  and yawing rates from the linearization  point.

(a)  Using the state  vector x = [∆p, ∆q, ∆r]T , input vector u = [Mx, My , Mz ]T , and output vector y = x, put this system into state  space form (i.e. find the A, B, C, D parameter matrices).

(b)  Use Matlab’s  expm function  to compute  the  state  transition matrix  for this  system, assuming that  Iy = 750 kg m3 , Iz = 1000 kg m3, Ix = 500 kg m3, po  = 20 rad/s, and

∆t = 0.1 s.

(c)  Use the  state  transition matrix  to compute  and  plot  the  state  time  history  for 5 s, assuming  zero inputs  and  assuming  initial  states  ∆q(0)  = 0.1 rad/s, and  ∆p(0) =

∆r(0) = 0. Be sure to label each of the state  plots carefully.  What  can you say about the  behavior  of this  system  in terms  of stability,  i.e.  does it  want to stay  near  the conditions used to linearize the equations  of motion?

1. Do the following end of chapter  problems from the Simon textbook: (a)  Problem 1.5

(b)  Problem  1.8

Advanced Questions  PhD students in the  class MUST answer ALL  questions below  in addition to  regular  homework questions above – non-PhD  students  are welcome to  try  any  of these  for  extra  credit  (only  given if all regular  problems  turned in on time  as well).  In either  case,  Submit your responses  for these questions with rest of your  homework,  but make sure  these  are  clearly  labeled and  start  on  separate  pages

– indicate on  the  top  of  the  front page of  your assignment if you  answered these questions (as a PhD student, or for extra credit) so  they can be spotted, graded and recorded more easily.

AQ1.  Explain why matrix  element (4,4) in problem 1c belongs in the last column (hint: Google it, if you must…).

AQ2.  Using the  infinite  series definition  of the  matrix  exponential,  find the  analytical expression for the state transition matrix (STM) of the spinning satellite system in problem

4, assuming a generic value for ∆t and using the values for Ix, Iy , Iz , and po  given in that problem.  Your answer should be a function of ∆t. Note: do not use a symbolic calculation tool (e.g.  Matlab’s  symbolic toolbox,  Maple, Mathematica, etc.)   – you must  derive the STM entirely by hand using the series definition of the matrix  exponential,  and show your work.  Hint:  it may help to simplify certain  terms  by defining intermediate variables and by recalling some basic Taylor  series expansions  for certain  trigonometric  functions.   It’s also wise to verify that  your answer agrees with your results from #4.