Control¶

get_parabolic_robin_backstepping_controller(state, approx_state, d_approx_state, approx_target_state, d_approx_target_state, integral_kernel_ll, original_beta, target_beta, scale=None)[source]¶

Build a modal approximated backstepping controller $u(t)=(Kx)(t)$ , for the (open loop-) diffusion system with reaction term, robin boundary condition and robin actuation

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

such that the closed loop system has the desired dynamic of the target system

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

where $\bar a_0,\, \bar\alpha,\, \bar\beta$ are controller parameters.

The control design is performed using the backstepping method, whose integral transform

$\bar x(z) = x(z) + \int_0^z k(z, \bar z) x(\bar z) \, d\bar z$

maps from the original system to the target system.

Note

For more details see the example script pyinduct.examples.rad_eq_const_coeff that implements the example from [WoiEtAl17] .

Parameters:	state (list of `ScalarTerm`’s) – Measurement / value from simulation of $x(l)$ . approx_state (list of `ScalarTerm`’s) – Modal approximated $x(l)$ . d_approx_state (list of `ScalarTerm`’s) – Modal approximated $x'(l)$ . approx_target_state (list of `ScalarTerm`’s) – Modal approximated $\bar x(l)$ . d_approx_target_state (list of `ScalarTerm`’s) – Modal approximated $\bar x'(l)$ . integral_kernel_ll (`numbers.Number`) – Integral kernel evaluated at $\bar z = z = l$ : $k(l, l) = \bar\alpha - \alpha + \frac{a_0-\bar a_0}{a_2} l \:.$ original_beta (`numbers.Number`) – Coefficient $\beta$ of the original system. target_beta (`numbers.Number`) – Coefficient $\bar\beta$ of the target system. scale (`numbers.Number`) – A constant $c \in \mathbb R$ to scale the control law: $u(t) = c \, (Kx)(t)$ .
Returns:	$(Kx)(t)$
Return type:	`Controller`

[WoiEtAl17]

Frank Woittennek, Marcus Riesmeier and Stefan Ecklebe; On approximation and implementation of transformation based feedback laws for distributed parameter systems; IFAC World Congress, 2017, Toulouse

split_domain(n, a_desired, l, mode='coprime')[source]¶

Consider a domain $[0,l]$ which is divided into the two sub domains $[0,a]$ and $[a,l]$ with the discretization $l_0 = l/n$ and a partition $a + b = l$ .

Calculate two numbers $k_1$ and $k_2$ with $k_1 + k_2 = n$ such that $n$ is odd and $a = k_1l_0$ is close to a_desired.

Parameters:

n (int) – Number of sub-intervals to create (must be odd).
a_desired (float) – Desired partition size $a$ .
l (float) – Length $l$ of the interval.
mode (str) –
Operation mode to use:
- ’coprime’: $k_1$ and $k_2$ are coprime (default) .
- ’force_k2_as_prime_number’: $k_2$ is a prime number ( $k_1$ and $k_2$ are coprime)
- ’one_even_one_odd’: One is even and one is odd.