# Difference between revisions of "CDS 140a Winter 2014 Homework 9"

 R. Murray Issued: 25 Feb 2014 (Tue) ACM 101b/AM 125b/CDS 140a, Winter 2014 Due: 5 Mar 2014 (Wed) @ noon

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1. Perko, Section 4.4, Problem 1a
2. 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.

3. 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.

(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) Suppose that we restrict the dynamics of the system to the invariant set given by $J = 0$. Show that the system undergoes a subcritical Hopf bifurcation (this phenomena is called "surge").