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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 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
                 \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

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.


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.

Pages in category "GIOC"

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