Category:GIOC

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

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

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

On the outer level, a norm $\| \cdot \|$ and the regularization term $R$ 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 $g_i$ with $w_i$ 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 $\mathcal{X}$ and $\mathcal{U}$ , 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 $t$ .
The differential equations might depend on state-dependent switches.
The variables may include boolean variables.
The underlying process might be a multistage process.
The dynamics might be unstable.
There might be an underlying network topology.
The integer control functions might have been (re)formulated by means of an outer convexification.

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

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Category:GIOC

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