SebastianSager: integer controls

2023-10-19T12:53:11Z

integer controls

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	== Extensions ==		== Extensions ==
			* The functions <math>u \in \mathcal{U}</math> may also include integer controls.
	* For some problems the functions may as well depend explicitely on the time <math>t</math>.		* For some problems the functions may as well depend explicitely on the time <math>t</math>.
	* The differential equations might depend on [[:Category:State dependent switches \| state-dependent switches]].		* The differential equations might depend on [[:Category:State dependent switches \| state-dependent switches]].

SebastianSager: Initial setup of gIOC class

2023-10-19T12:24:08Z

Initial setup of gIOC class

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This category includes all generalized inverse optimal control problems.
To formalize this problem class, we define the following bilevel problem with differential states <math>x \in \mathcal{X}</math>, controls <math>u \in \mathcal{U}</math>, model parameters <math>p \in \mathbb{R}^{n_p}</math>, and convex multipliers <math>w \in \mathcal{W}</math>

<math>
\displaystyle \min_{(p, w, x^*, u^*) \in \Omega_1} \; \| h(x^*, u^*) - \eta \| + R(p,w)
</math>
subject to
<math>
(x^*,u^*) \in \displaystyle \arg \min_{x, u} \sum_{i \in M_1} w_i \; \phi_i[x,u,p]
</math>
subject to
<math>
\begin{array}{rcl}
\dot{x}(t) &=& \displaystyle \sum_{i \in M_2} w_i f_i(x(t), u(t), p) \\
0 &\le& \displaystyle w_i \; g_i(x(t), u(t), p) \quad \forall \; i \in M_3\\
(x,u,p,w) &\in& \Omega_2
\end{array}
</math>

as a generalized inverse optimal control problem. Here <math>\mathcal{X}</math> and <math>\mathcal{U}</math> are properly defined function spaces. The variables <math>w</math> indicate which objective functions, right hand side functions, and constraints are relevant in the inner problem. The variables are normalized for given index sets <math>M_1, M_2, M_3</math> that partition the indices from <math>1</math> to <math>n_w</math>. For normalization, we define the feasible set <math>\mathcal{W} := \{ w \in [0,1]^{n_w}: \; \textstyle \sum^{i \in M_j} w_i = 1 \text{ for } j \in \{1,2,3\}\}</math>. On the outer level, the feasible set is <math>\Omega_1 := \mathbb{R}^{n_p} \times \mathcal{W} \times \mathcal{X} \times \mathcal{U}</math>, while on the inner level <math>\Omega_2</math> contains bounds, boundary conditions, mixed path and control constraints, and more involved constraints such as dwell time constraints. We have observational data <math>\eta \in \mathbb{R}^{n_\eta}</math>, a measurement function <math>h: \mathcal{X} \times \mathcal{U} \mapsto \mathbb{R}^{n_\eta}</math>, a regularization function with a priori knowledge on parameters and weights <math>R: \mathbb{R}^{n_p} \times \mathcal{W} \mapsto \mathbb{R}</math> and candidate functionals <math>\phi_i: \mathcal{X} \times \mathcal {U} \times \mathbb{R}^{n_p} \mapsto \mathbb{R}</math> and functions <math>f_i: \mathcal{X} \times \mathcal {U} \times \mathbb{R}^{n_p} \mapsto \mathbb{R}^{n_x}</math> and <math>g_i: \mathcal{X} \times \mathcal {U} \times \mathbb{R}^{n_p} \mapsto \mathbb{R}^{n_g}</math> for the unknown objective function, dynamics, and constraints, respectively. Two cases are of practical interest: first, the manual, often cumbersome and trial-and-error based a priori definition of all candidates <math>\phi_i, f_i, g_i</math> by experts and second, a systematic, but often challenging automatic symbolic regression of these unknown functions.

On the outer level, a norm <math>\| \cdot \|</math> and the regularization term <math>R</math> define a data fit (regression) problem and relate to prior knowledge and statistical assumptions. On the inner level, the above bilevel problem is constrained by a possibly nonconvex optimal control problem. The unknown parts of this inner level optimal control problem are modeled as convex combinations of a finite set of candidates (and a multiplication of constraints <math>g_i</math> with <math>w_i</math> that can be either zero or strictly positive). On the one hand the problem formulation is restrictive in the interest of a clearer presentation and might be further generalized, e.g., to multi-stage formulations involving differential-algebraic or partial differential equations. On the other hand, the problem class is quite generic and allows, e.g., the consideration of switched systems, periodic processes, different underlying function spaces <math>\mathcal{X}</math> and <math>\mathcal{U}</math>, and the usage of universal approximators such as neural networks as candidate functions.

== Extensions ==
* For some problems the functions may as well depend explicitely on the time <math>t</math>.
* The differential equations might depend on [[:Category:State dependent switches | state-dependent switches]].
* The variables may include [[:Category:Boolean variables | boolean variables]].
* The underlying process might be a [[:Category:Multistage process | multistage process]].
* The dynamics might be [[:Category:Unstable | unstable]].
* There might be an underlying [[:Category:Network topology | network topology]].
* The integer control functions might have been (re)formulated by means of an [[:Category:Outer convexification|outer convexification]].

Note that a Lagrange term <math>\int_{t_0}^{t_f} L( x(t), u(t), v(t), q, \rho) \; \mathrm{d} t</math> can be transformed into a Mayer-type objective functional.

[[Category:Model characterization]]

Category:GIOC - Revision history

SebastianSager: integer controls

SebastianSager: Initial setup of gIOC class