Difference between revisions of "CDS 140a Winter 2014 Homework 8"

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  | pdf = cds140-wi14_hw8.pdf
 
  | pdf = cds140-wi14_hw8.pdf
 
}} __MATHJAX__
 
}} __MATHJAX__
 
{{warning|This homework set is still in preparation.  This banner will be removed with the homework set is finalized.}}
 
  
 
'''Note:''' In the upper left hand corner of the ''second'' page of your homework set, please put the number of hours that you spent on
 
'''Note:''' In the upper left hand corner of the ''second'' page of your homework set, please put the number of hours that you spent on
Line 15: Line 13:
  
 
<ol>
 
<ol>
 +
<!-- 2014 TA comments: On problem 1, there were a few questions on how to use the hint for part b. I suggested to most people to instead use the result to the problem in HW6 showing that the eigenvalues of the linearization are off by a positive constant. -->
 +
 
<li> '''Perko, Section 4.1, Problem 1''':<br>
 
<li> '''Perko, Section 4.1, Problem 1''':<br>
 
(a) Consider the two vector fields
 
(a) Consider the two vector fields
Line 33: Line 33:
 
</li>
 
</li>
  
 +
<!-- 2014 TA comments: some people were confused about setting up the ODE's for the sensitivity. A lot more people were confused about what to plot in part c. -->
 
<li> Consider the dynamical system
 
<li> Consider the dynamical system
 
<center><amsmath>
 
<center><amsmath>
Line 48: Line 49:
 
</li>
 
</li>
  
<li> '''Perko, Section 4.2, Problem 7''': Consider the two dimensional system
+
<!-- 2014 TA comments: people were confused about how to write out D^2 f. People were also wary about how to compute the dimensions of the manifolds. A lot of people were surprised that they only needed to use the linearization to assess stability and dimension of the stable/unstable manifolds. It might be good to have the students explicitly write out the linearization.-->
 +
<li> '''Perko, Section 4.2, Problem 4''': Consider the planar system
 
<center><amsmath>
 
<center><amsmath>
 
   \aligned
 
   \aligned
     \dot x &= -x^4 + 5 \mu x^2 - 4 \mu^2 \\
+
     \dot x &= \mu x - x^2 \\
 
     \dot y &= -y.
 
     \dot y &= -y.
 
   \endaligned
 
   \endaligned
 
</amsmath></center>
 
</amsmath></center>
Determine the critical points and the bifurcation diagram for this system. Draw phase portraits for the various values of $\mu$ and draw the bifurcation diagram.
+
Verify that the system satisfies the conditions for a transcritical bifurcation (equation (3) in Section 4.2) and determine the dimensions of the various stable, unstable and center manifolds that occur.
</li>
 
  
<li> '''Perko, Section 4.4, Problem 1a'''</li> Show that for $a + b \neq 0$ the system
+
<!-- 2014 TA comments: it might be good to have the students explicitly write out the linearization  the reason it might be helpful for future homework assignments to ask the students to compute the linearization for 3 and 4 (especially 4) in order to do the stability analysis, is because it seems that by the time the students get to this part of the class, they are so used to using the many advanced theorems and methods to analyze stability that they forget how useful and important trying the linearization is first. -->
 +
<li> '''Perko, Section 4.2, Problem 7''': Consider the two dimensional system
 
<center><amsmath>
 
<center><amsmath>
 
   \aligned
 
   \aligned
     \dot x &= \mu x - y + a (x^2 + y^2) x - b (x^2 + y^2) y + O(|x|^4) \\
+
     \dot x &= -x^4 + 5 \mu x^2 - 4 \mu^2 \\
     \dot y &= x + \mu y + a (x^2 + y^2) x + b (x^2 + y^2) y + O(|x|^4)
+
     \dot y &= -y.
 
   \endaligned
 
   \endaligned
 
</amsmath></center>
 
</amsmath></center>
has a Hopf bifurcaton at the origin at the bifurcation value $\mu = 0$.  Determine whether it is supercritical or subcritical.
+
Determine the critical points and the bifurcation diagram for this system.  Draw phase portraits for the various values of $\mu$ and draw the bifurcation diagram.
 
</li>
 
</li>
 
+
</ol>
<li> The Moore-Greitzer equations model rotating stall and surge in gas turbine engines describe the dynamics of a compression system, such as those in gas turbine engines.  The three-state "MG3" equations have the form:
 
<center><amsmath>
 
  \aligned
 
    \frac{d\psi}{dt} &= \frac{1}{4 B^2 l_c}\left(\phi - \Phi_T(\psi) \right), \\
 
    \frac{d\phi}{dt} &= \frac{1}{l_c}\left( \Psi_c(\phi) - \psi + \frac{J}{8}
 
      \frac{\partial^2 \Psi_c}{\partial \phi^2} \right), \\
 
    \frac{dJ}{dt} &= \frac{2}{\mu + m} \left(
 
      \frac{\partial \Psi_c}{\partial \phi} + \frac{J}{8}
 
          \frac{\partial^3 \Psi_c}{\partial \phi^3} \right) J,
 
  \endaligned
 
</amsmath></center>
 
where <amsmath>\psi</amsmath> represents the pressure rise across the compressor, <amsmath>\phi</amsmath> represents the mass flow through the compressor and <amsmath>J</amsmath> represents the amplitude squared of the first modal flow perturbation (corresponding to a rotating stall disturbance).  For the Caltech compressor rig, the parameters and
 
characteristic curves are given by:
 
<center><amsmath>
 
  \aligned
 
      B &= 0.2, & \Phi_T(\psi) &= \gamma \sqrt{\psi},\\
 
      l_c &= 6, & \Psi_c(\phi) &= 1 + 1.5 (\phi-1) - 0.5 (\phi-1)^3, \\
 
      \mu &= 1.25, &\qquad\qquad m &= 2.
 
  \endaligned
 
</amsmath></center>
 
The parameter <amsmath>\gamma</amsmath> represents the throttle setting and typically
 
varies between 1 and 2.
 
 
 
[[Image:MG3-bifdiag.png|right|200px|Test]]
 
(a) Compute the bifurcation diagram for the system showing the
 
equilibrium value(s) for <amsmath>J</amsmath> as a function of <amsmath>\gamma</amsmath>.  Your answer should match what was shown in class (i.e., make sure to get capture the hysteresis loop).
 
 
 
(b) Suppose that we can modulate the throttle, so that <amsmath>\gamma = \gamma_0 + u</amsmath>.  Analyze the performance of the system using the Liaw-Abed control law <amsmath>u = k J</amsmath>.  Show that if we choose <amsmath>k</amsmath> sufficiently large to cause the bifurcation to stall to be supercritical.
 
 
 
(c) (Optional*) Suppose that we impose magnitude and rate limits on <amsmath>u</amsmath>:
 
<center><amsmath>
 
  |u| \leq 1, \qquad |\dot u| \leq 1.
 
</amsmath></center>
 
Assume that we implement the control law
 
<center><amsmath>
 
  \dot u = \alpha(J) = \begin{cases}
 
    \text{sat}(\frac{1}{\epsilon} (k J - \text{sat}(u))) & |u| < 1, \\
 
    0 & |u| = 1,
 
  \end{cases}
 
</amsmath></center>
 
where <amsmath>\text{sat}(\cdot)</amsmath> is a saturation function of magnitude 1 and
 
<amsmath>\epsilon</amsmath> is a small constant.  This
 
control law limits both the magnitude and rate of the input.
 
Using the center manifold theorem, compute an approximate model of the
 
system at the bifurcation point in terms of <amsmath>u</amsmath> and <amsmath>J</amsmath> and use a
 
phase portrait (computed with MATLAB or a similar tool) to describe
 
the set of initial conditions for <amsmath>J</amsmath> (assuming <amsmath>u(0) = 0</amsmath>) for which
 
the system avoids hysteresis.
 
 
 
<nowiki>*</nowiki> Part c is ''optional'', not extra credit.  If you attempt this problem, one of the instructors will look at your solution and give you feedback.  This problem is quite difficult, so you should only do this if you complete the rest of the set quickly and don't have any other homework or studying that is a higher priority before the exam is due.  This problem is related to a patent that was filed by Caltech on how to mitigate magnitude and rate limits in active control of rotating stall.
 
</li></ol>
 

Latest revision as of 20:58, 14 February 2015

R. Murray Issued: 25 Feb 2014 (Tue)
ACM 101b/AM 125b/CDS 140a, Winter 2014

(PDF)

Due: 5 Mar 2014 (Wed) @ noon

__MATHJAX__

Note: In the upper left hand corner of the second page of your homework set, please put the number of hours that you spent on this homework set (including reading).

  1. Perko, Section 4.1, Problem 1:
    (a) Consider the two vector fields
    <amsmath>
     f(x) = \begin{bmatrix} -x_2 \\ x_1 \end{bmatrix}, \qquad
     g(x) = \begin{bmatrix} -x_2 + \mu x_1 \\ x_1 + \mu x_2 \end{bmatrix}.
    
    </amsmath>

    Show that $\|f-g\|_1 = |\mu| (\max_{x \in K} \|x\| + 1)$, where $K \subset {\mathbb R}^2$ is a compact set containing the origin in its interior.

    (b) Show that for $\mu \neq 0$ the systems

    <amsmath>
     \aligned \dot x_1 &= -x_2 \\ \dot x_2 &= x_1 \endaligned \quad\text{and}\quad
     \aligned \dot x_1 &= -x_2 + \mu x_1 \\ \dot x_2 &= x_1 + \mu x_2 \endaligned
    
    </amsmath>

    are not topologically equivalent.

    Hint: Let $\phi_t$ and $\psi_t$ be the flows defined by these two systems and assume that there is a homeomorphism $H:{\mathbb R}^2 \to {\mathbb R}^2$ and a strictly increasing, continuous function $t(\tau)$ mapping $\mathbb R$ onto $\mathbb R$ such that $\phi_{t(\tau)} = H^{-1} \circ \psi_\tau \circ H$. Use the fact that $\lim_{t\to\infty} \phi_t(1,0) \neq 0$ and that for $\mu < 0$, $\lim_{t \to \infty} \psi_t(x) = 0$ for all $x \in {\mathbb R}^2$ to arrive at a contradiction.

  2. Consider the dynamical system
    <amsmath>
     m \ddot q + b \dot q + k q = u(t), \qquad
     u(t) = \begin{cases} 0 & t = 0, \\ 1 & t > 0, \end{cases} \qquad
     q(0) = \dot q(0) = 0,
    
    </amsmath>

    which describes the "step response" of a mass-spring-damper system.

    (a) Derive the differential equations for the sensitivities of <amsmath>q(t) \in {\mathbb R}</amsmath> to the parameters <amsmath>b</amsmath> and <amsmath>k</amsmath>. Write out explicit systems of ODEs for computing these, including any initial conditions. (You don't have to actually solve the differential equations explicitly, though it is not so hard to do so.)

    (b) Compute the sensitivities and the relative (normlized) sensitivies of the equilibrium value of <amsmath>q_e</amsmath> to the parameters <amsmath>b</amsmath> and <amsmath>k</amsmath>. You should give explicit formulas in terms of the relevant parameters and initial conditions.

    (c) Sketch the plots of the relative sensitivities <amsmath>S_{q,b}</amsmath> and <amsmath>S_{q,k}</amsmath> as a function of time for the nominal parameter values <amsmath>m = 1</amsmath>, <amsmath>b = 2</amsmath>, <amsmath>k = 1</amsmath>.

  3. Perko, Section 4.2, Problem 4: Consider the planar system
    <amsmath>
     \aligned
       \dot x &= \mu x - x^2 \\
       \dot y &= -y.
     \endaligned
    
    </amsmath>

    Verify that the system satisfies the conditions for a transcritical bifurcation (equation (3) in Section 4.2) and determine the dimensions of the various stable, unstable and center manifolds that occur.

  4. Perko, Section 4.2, Problem 7: Consider the two dimensional system
    <amsmath>
     \aligned
       \dot x &= -x^4 + 5 \mu x^2 - 4 \mu^2 \\
       \dot y &= -y.
     \endaligned
    
    </amsmath>

    Determine the critical points and the bifurcation diagram for this system. Draw phase portraits for the various values of $\mu$ and draw the bifurcation diagram.