ASEN 5044 Statistical Estimation for Dynamical Systems Homework #2 Solution

Show all your work and explain your reasoning.

 

 

1.   Consider the  equations  of motion  for a unit  mass subjected  to an inverse square  law force field, e.g. a satellite  orbiting a planet,

 



−
 
r¨ = rθ˙2         k r2


 

+ u1 (t)                                                (1)



˙
 
θ¨ = − 2θr˙ +  1 u (t)                                                  (2)

r       r  2

where r represents  the  radius  from the  center  of the  force field, θ gives the  angle with respect to a reference direction in the orbital plane, k is a constant, and u1  and u2  represent radial and tangential thrusts, respectively.  It is easily shown that  for the initial conditions r(0) = r0 ,  θ(0) = 0, r˙(0) = 0, and θ˙(0) = ω0 with nominal thrusts u1(t) = 0 and u2(t) = 0 for all t ≥ 0, the equations  of motion have as a solution the circular orbit given by

 

r(t) = r0 = constant                                                              (3)

s k

θ˙(t) = ω0 = constant =


3 ,                                              (4)



r
 
0

θ(t) = ω0 t + constant                                                            (5) (a)  Pick a state  vector for this system, and express the original nonlinear ODEs in ‘stan-

dard’ nonlinear state  space form.

 

(b)  Linearize this system’s nominal equations of motion about the nominal solution r(t) =

r0,


r˙(0)  = 0, θ(t) =  ω0 t + constant  and


θ˙(t)  = ω0  with  u1(t) = 0 and  u2(t) = 0.

Find  (A, B, C, D) matrices  for output y(t) = [r(t), θ(t)]T  for the linearized system of

equations  about  the nominal solution.

 

(c)  Convert the continuous time (A, B, C, D) matrices you found from part (b) into discrete time (F, G, H, M ) matrices,  using a discretization  step  size of ∆t = 10s and  setting k = 398600 km3/s2 and r0 = 6678 km.

 

(d)  Interpret the results for the STM in part  (c), i.e. what is the physical meaning of each column vector that  makes up F ?

 

 



2
 
2.   Simon, Problem  1.13 (for part (b):  use only the first and last expressions in eq.  1.71, i.e. you can skip the Laplace transform  technique).  Note: there is an error in the problem statement: the equation  for p should be p = p0 + p˙0t +  1 p¨0 t2. The problem statement for part  (a) should also be clarified as asking for the state  space equation  that  describes the system whose solution is given by this formula for p.

3.   Simon, Problem 1.14.

 

 

4.   Simon, Problem  1.19.  For part  (d),  use Matlab’s  ode45 command  (or a similar com- mand in whatever language you are using) to simulate the full nonlinear dynamics and the linearized model – see the doc ode45 file for more details and examples of how to use this function  correctly  if you are using Matlab  (note  that  you will need to express both  the nonlinear and linearized dynamics in ‘standard’ state  space form).  Also note:  for part  d of this problem, the book errata  says there should be an absolute value in the input  u(t), so that  it should be u(t) = u0(t) + ∆u · abs(cos(t)) – you need this to obtain  physical results. Also assume that  the initial conditions to be used in the simulation  should be the nominal initial conditions,  with x3,0  = 1000 kg.

 

 

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.  Simon, Problem 1.6.

 

 

AQ2.  Simon,  Problem  1.11.   Note:  there  is a mistake  in the  problem  statement:  the expression to be proven  should read  | exp(At)|  = exp(tr(A)t) (also stated  in the  posted errata  for the Simon book).