Feedforward¶

class InterpolationTrajectory(t, u, **kwargs)¶

Bases: pyinduct.simulation.SimulationInput

Provides a system input through one-dimensional linear interpolation in the given vector $u$ .

Parameters:

t (array_like) – Vector $t$ with time steps.
u (array_like) – Vector $u$ with function values, evaluated at $t$ .
**kwargs – see below

Keyword Arguments:

show_plot (bool) – to open a plot window, showing u(t).
scale (float) – factor to scale the output.

get_plot()¶

Create a plot of the interpolated trajectory.

Todo

the function name does not really tell that a QtEvent loop will be executed in here

Returns:: the PlotWindow widget.
Return type:: (pg.PlotWindow)

scale(scale)¶

class RadFeedForward(l, T, param_original, bound_cond_type, actuation_type, n=80, sigma=None, k=None, length_t=None, y_start=0, y_end=1, **kwargs)¶

Bases: pyinduct.trajectory.InterpolationTrajectory

Class that implements a flatness based control approach for the reaction-advection-diffusion equation

$\dot x(z,t) = a_2 x''(z,t) + a_1 x'(z,t) + a_0 x(z,t)$

with the boundary condition

bound_cond_type == "dirichlet": $x(0,t)=0$

A transition from $x'(0,0)=y0$ to $x'(0,T)=y1$ is considered.

With $x'(0,t) = y(t)$ where $y(t)$ is the flat output.

bound_cond_type == "robin": $x'(0,t) = \alpha \, x(0,t)$

A transition from $x(0,0)=y0$ to $x(0,T)=y1$ is considered.

With $x(0,t) = y(t)$ where $y(t)$ is the flat output.

and the actuation

actuation_type == "dirichlet": $x(l,t)=u(t)$

actuation_type == "robin": $x'(l,t) = -\beta x(l,t) + u(t)$ .

The flat output trajectory $y(t)$ will be calculated with gevrey_tanh().

Parameters:

l (float) – Domain length.
t_end (float) – Transition time.
param_original (tuple) – Tuple holding the coefficients of the pde and boundary conditions.
bound_cond_type (string) – Boundary condition type. Can be dirichlet or robin, see above.
actuation_type (string) – Actuation condition type. Can be dirichlet or robin, see above.
n (int) – Derivative order to provide (defaults to 80).
sigma (number.Number) – sigma value for gevrey_tanh().
k (number.Number) – K value for gevrey_tanh().
length_t (int) – length_t value for gevrey_tanh().
y0 (float) – Initial value for the flat output.
y1 (float) – Desired value for the flat output after transition time.
**kwargs – see below. All arguments that are not specified below are passed to InterpolationTrajectory .

class SecondOrderOperator(a2=0, a1=0, a0=0, alpha1=0, alpha0=0, beta1=0, beta0=0, domain=(-np.inf, np.inf))¶

Interface class to collect all important parameters that describe a second order ordinary differential equation.

Parameters:

a2 (Number or callable) – coefficient $a_2$ .
a1 (Number or callable) – coefficient $a_1$ .
a0 (Number or callable) – coefficient $a_0$ .
alpha1 (Number) – coefficient $\alpha_1$ .
alpha0 (Number) – coefficient $\alpha_0$ .
beta1 (Number) – coefficient $\beta_1$ .
beta0 (Number) – coefficient $\beta_0$ .

static from_dict(param_dict, domain=None)¶

static from_list(param_list, domain=None)¶

get_adjoint_problem()¶

Return the parameters of the operator $A^*$ describing the the problem

$(\textup{A}^*\psi)(z) = \bar a_2 \partial_z^2 \psi(z) + \bar a_1 \partial_z \psi(z) + \bar a_0 \psi(z) \:,$

where the $\bar a_i$ are constant and whose boundary conditions are given by

$\bar\alpha_1 \partial_z \psi(z_1) + \bar\alpha_0 \psi(z_1) &= 0 \\ \bar\beta_1 \partial_z \psi(z_2) + \bar\beta_0 \psi(z_2) &= 0 .$

The following mapping is used:

$\bar a_2 = a_2, \quad\bar a_1 = -a_1, \quad\bar a_0 = a_0, \\ \bar\alpha_1 = -1, \quad \bar\alpha_0 = \frac{a_1}{a_2} - \frac{\alpha_0}{\alpha_1} , \\ \bar\beta_1 = -1, \quad \bar\beta_0 = \frac{a_1}{a_2} - \frac{\beta_0}{\beta_1} \:.$

Returns:: Parameter set describing $A^*$ .
Return type:: SecondOrderOperator

eliminate_advection_term(param, domain_end)¶

This method performs a transformation

$\tilde x(z,t)=x(z,t) e^{\int_0^z \frac{a_1(\bar z)}{2 a_2}\,d\bar z} ,$

on the system, which eliminates the advection term $a_1 x(z,t)$ from a reaction-advection-diffusion equation of the type:

$\dot x(z,t) = a_2 x''(z,t) + a_1(z) x'(z,t) + a_0(z) x(z,t) .$

The boundary can be given by robin

$x'(0,t) = \alpha x(0,t), \quad x'(l,t) = -\beta x(l,t) ,$

dirichlet

$x(0,t) = 0, \quad x(l,t) = 0$

or mixed boundary conditions.

Parameters:

param (array_like) – $\Big( a_2, a_1, a_0, \alpha, \beta \Big)^T$
domain_end (float) – upper bound of the spatial domain

Raises:

TypeError – If $a_1(z)$ is callable but no derivative handle is
defined for it. –

Returns:

Parameters

$\big(a_2, \tilde a_1=0, \tilde a_0(z), \tilde \alpha, \tilde \beta \big) for$

the transformed system

$\dot{\tilde{x}}(z,t) = a_2 \tilde x''(z,t) + \tilde a_0(z) \tilde x(z,t)$

and the corresponding boundary conditions ( $\alpha$ and/or $\beta$ set to None by dirichlet boundary condition).

Return type:

SecondOrderOperator or tuple

gevrey_tanh(T, n, sigma=1.1, K=2, length_t=None)¶

Provide Gevrey function

$\eta(t) = \left\{ \begin{array}{lcl} 0 & \forall & t<0 \\ \frac{1}{2} + \frac{1}{2}\tanh \left(K \frac{2(2t-1)} {(4(t^2-t))^\sigma} \right) & \forall & 0\le t \le T \\ 1 & \forall & t>T \end{array} \right.$

with the Gevrey-order $\rho=1+\frac{1}{\sigma}$ and the derivatives up to order n.

Note

For details of the recursive calculation of the derivatives see:

Rudolph, J., J. Winkler und F. Woittennek: Flatness Based Control of Distributed Parameter Systems: Examples and Computer Exercises from Various Technological Domains (Berichte aus der Steuerungs- und Regelungstechnik). Shaker Verlag GmbH, Germany, 2003.

Parameters:

T (numbers.Number) – End of the time domain=[0, T].
n (int) – The derivatives will calculated up to order n.
sigma (numbers.Number) – Constant $\sigma$ to adjust the Gevrey order $\rho=1+\frac{1}{\sigma}$ of $\varphi(t)$ .
K (numbers.Number) – Constant to adjust the slope of $\varphi(t)$ .
length_t (int) – Ammount of sample points to use. Default: int(50 * T)

Returns:

numpy.array([[ $\varphi(t)$ ], … , [ $\varphi^{(n)}(t)$ ]])
t: numpy.array([0,…,T])

Return type:

tuple

power_series_flat_out(z, t, n, param, y, bound_cond_type)¶

Provides the solution $x(z,t)$ (and the spatial derivative $x'(z,t)$ ) of the pde

$\dot x(z,t) = a_2 x''(z,t) + \underbrace{a_1 x'(z,t)}_{=0} + a_0 x(z,t), \qquad a_1 = 0, \qquad z\in(0,l), \qquad t\in(0,T)$

as power series approximation:

for the boundary condition (bound_cond_type == "dirichlet") $x(0,t)=0$ and the flat output $y(t) = x'(0,t)$ with

$x(z,t) = \sum_{n=0}^\infty \frac{z^{2n+1}}{a_2^n(2n+1)!} \sum_{k=0}^n \binom{n}{k}(-a_0)^{n-k}y^{(k)}(t)$

for the boundary condition (bound_cond_type == "robin") $x'(0,t) = \alpha \, x(0,t)$ and the flat output $y(t) = x(0,t)$ with

$x(z,t) = \sum_{n=0}^\infty \left( 1 + \alpha\frac{z}{2n+1}\right) \frac{z^{2n}}{a_2^{n}(2n)!} \sum_{k=0}^{n} \binom{n}{k}(-a_0)^{n-k}y^{(k)}(t).$

Parameters:

z (array_like) – $[0, ..., l]$
t (array_like) – $[0, ... , T]$
n (int) – Series termination index.
param (array_like) –
Parameters

$[a_2, a_1, a_0, \alpha, \beta]$
- $\alpha=\text{None}$ for bound_cond_type == dirichlet
- $beta$ is not used from this function but has to be provided (for now)
y (array_like) –
Flat output $y(t)$ and derivatives:

$[[y(0), ..., y(T)],...,[y^{(n/2)}(0), ..., y^{(n/2)}(T)]].$
bound_cond_type (str) – dirichlet or robin

Returns:

Solution $x(z,t)$ of the pde and the spatial derivative $x'(z,t)$ .

Return type:

tuple

Feedforward¶

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