Control¶

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

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

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

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

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

where $\bar a_1,\, \bar a_0,\, \bar\alpha,\, \bar\beta$ are controller parameter.

For this purpose the backstepping method is used for controller design, where the backstepping transformation

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

is used to transform the origninal system into the target system.

Note

For more details see:

Example pyinduct.examples.rad_eq_const_coeff

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

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_zz (`numbers.Number`) – Integral kernel $k(z,z) = \bar\alpha - \alpha + \frac{a_0-\bar a_0}{a_2} z$ original_beta (`numbers.Number`) – $\beta$ target_beta (`numbers.Number`) – $\bar\beta$ 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`

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 $a+b=l$ .

$k1+k2=n$ is calculated such that $n$ is odd and $a=k1*l_0$ is close to a_desired. Three modes are available:

‘force_k2_as_prime_number’: k2 is an prime number (k1, k2 are coprime)

‘coprime’: k1, k2 are coprime (default)

‘one_even_one_odd’: one is even one is odd.