Moore-Greitzer:
We consider the Moore-Greitzer jet engine model, the differential equations of the system are given below (See [1]):
\begin{cases}
\overset{.}{\phi} = -\psi - (3/2)\phi^2 - (1/2)\phi^3 + \Delta\\
\overset{.}{\psi} = 3\phi - \psi \\
\end{cases}
where \((\phi, \psi) \in \mathbb{R}^2\) are the state variables.
We consider the parameter value \(\Delta = -1.1\) and initial condition
\(x(0) = (0, -0.75) \)
Results:
We consider a system with uncertainty \(w=0.1\), set of initial conditions \(B(x_0,\varepsilon)\) with \(\varepsilon=0.2\), the time-step used in Euler's method is \(\tau = 10^{-4}\), and we take \(T=k\tau\) with \(k=45307\) as an approximate period.
Using the figures shown below, we check that:
- \(B((i_0+1)T)\subset B(i_0T)\) for \(i_0=0\)
- and \(\Sigma_{i=1}^k\lambda_i\approx\) \(-7871< 0\).
Then, we can conclude that the system
converges towards an attractive LC
contained in
\([B(0),B(T)]\).
The figures below show respectively the simulation of \(\phi(t), \psi(t)\) and \(\delta_{{\cal W}}(t)\) with perturbation (w=0.1) over 5 periods (5T=22.6536) for dt=0.0001.
In the figrues \(\phi(t)\) and \(\psi(t)\), the red curves represent the Euler approximation and the green curves correspond to the borders of tube \(B\)
References:
[1] AYLWARD, Erin M., PARRILO, Pablo A., and SLOTINE, Jean-Jacques E. Stability and robustness analysis of nonlinear systems via contraction metrics and SOS programming. Automatica, 2008, vol. 44, no 8, p. 2163-2170.
