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 |
__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).
- Perko, Section 4.4, Problem 1a Show that for $a + b \neq 0$ the system
- 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").
\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.
