Simulation with observer based state feedback of the reaction-convection-diffusion equation¶

Implementation of the approximation scheme presented in [RW2018b]. The system

$\begin{align*} \dot x(z,t) &= a_2 x''(z,t) + a_1 x'(z,t) + a_0 x(z,t) \\ x'(0,t) &= \alpha x(0,t) \\ x'(1,t) &= -\beta x(1,t) + u(t) \\ \end{align*}$

and the observer

$\begin{align*} \dot{\hat{x}}(z,t) &= a_2 \hat x''(z,t) + a_1 \hat x'(z,t) + a_0 \hat x(z,t) + l(z) \tilde y(t)\\ \hat x'(0,t) &= \alpha \hat x(0,t) + l_0 \tilde y(t) \\ \hat x'(1,t) &= -\beta \hat x(1,t) + u(t) \\ \end{align*}$

are approximated with LagrangeFirstOrder (FEM) shapefunctions and the backstepping controller and observer are approximated with the eigenfunctions respectively the adjoint eigenfunction of the system operator, see [RW2018b].

Note

For now, only a0 = 0 and a0_t_o = 0 are supported, because of some limitations of the automatic observer gain transformation, see evaluate_transformations() docstring.

References

RW2018b(1,2): Marcus Riesmeier and Frank Woittennek; On approximation of backstepping observers for parabolic systems with robin boundary conditions; In: Proceedings of the 57th IEEE, International Conference on Decision and Control (CDC), Miami, Florida, USA, December 17-19, 2018.

class ReversedRobinEigenfunction(om, param, l, scale=1, max_der_order=2)¶

Bases: pyinduct.SecondOrderRobinEigenfunction

This class provides an eigenfunction $\varphi(z)$ to the eigenvalue problem given by

$a_2\varphi''(z) + a_1&\varphi'(z) + a_0\varphi(z) = \lambda\varphi(z) \varphi'(0) &= \alpha \varphi(0) \varphi'(l) &= -\beta \varphi(l) .$

The eigenfrequency $\omega = \sqrt{-\frac{a_1^2}{4a_2^2}+\frac{a_0-\lambda}{a_2}}$ must be provided (for example with the eigfreq_eigval_hint() of this class).

Parameters

om (numbers.Number) – eigenfrequency $\omega$
param (array_like) – $\Big( a_2, a_1, a_0, \alpha, \beta \Big)^T$
l (numbers.Number) – End of the domain $z\in [0,l]$ .
scale (numbers.Number) – Factor to scale the eigenfunctions (corresponds to $\varphi(0)=\text{phi\_0}$ ).
max_der_order (int) – Number of derivative handles that are needed.

static eigfreq_eigval_hint(param, l, n_roots, show_plot=False)¶

Return the first n_roots eigenfrequencies $\omega$ and eigenvalues $\lambda$ .

$\omega_i = \sqrt{ - \frac{a_1^2}{4a_2^2} + \frac{a_0 - \lambda_i}{a_2}} \quad i = 1, \dotsc, \text{n\_roots}$

to the considered eigenvalue problem.

Parameters

param (array_like) – Parameters $\big( a_2, a_1, a_0, \alpha, \beta \big)^T$
l (numbers.Number) – Right boundary value of the domain $[0,l]\ni z$ .
n_roots (int) – Amount of eigenfrequencies to compute.
show_plot (bool) – Show a plot window of the characteristic equation.

Returns

$\Big(\big[\omega_1, \dotsc, \omega_{\text{n\_roots}}\Big], \Big[\lambda_1, \dotsc, \lambda_{\text{n\_roots}}\big]\Big)$

Return type

tuple –> booth tuple elements are numpy.ndarrays of length nroots

function_handle_factory(self, old_handle, l, der_order=0)¶

approximate_observer(obs_params, sys_params, sys_domain, sys_lbl, obs_sys_lbl, test_lbl, tar_test_lbl, system_input)¶

run(show_plots)¶