CDS 140a Winter 2013 Homework 8

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R. Murray, D. MacMartin Issued: 28 Feb 2013 (Thu)
ACM 101/AM 125b/CDS 140a, Winter 2013 Due: 12 Mar 2013 (Tue)
__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 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.

  4. Perko, Section 4.4, Problem 1a
  5. Show that for $a + b \neq 0$ the system
    <amsmath>
     \aligned
       \dot x &= \mu x - y + a (x^2 + y^2) x - b (x^2 + y^2) y + O(|x|^4) \\
       \dot y &= x + \mu y + a (x^2 + y^2) x + b (x^2 + y^2) y + O(|x|^4)
     \endaligned
    
    </amsmath>

    has a Hopf bifurcaton at the origin at the bifurcation value $\mu = 0$. Determine whether it is supercritical or subcritical.

  6. 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:
    <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>

    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:

    <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>

    The parameter <amsmath>\gamma</amsmath> represents the throttle setting and typically varies between 1 and 2.

    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>:

    <amsmath>
     |u| \leq 1, \qquad |\dot u| \leq 1.
    
    </amsmath>

    Assume that we implement the control law

    <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>

    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.

    * 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.