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 ${\displaystyle x\in {𝒳}}$, controls ${\displaystyle u\in {𝒰}}$, model parameters ${\displaystyle p\in {\mathbb{ℝ}}^{n_p}}$, and convex multipliers ${\displaystyle w\in {𝒲}}$

  ${\displaystyle {\displaystyle \underset{(p,w,x^{*},u^{*})\in {\mathrm{\Omega }}_1}{\min }\Vert h(x^{*},u^{*})-\eta \Vert +R(p,w)}}$
     subject to
     ${\displaystyle (x^{*},u^{*})\in {\displaystyle \arg \underset{x,u}{\min }\sum _{i\in M_1}{w}_i{\varphi }_i[x,u,p]}}$
                subject to
                ${\displaystyle \begin{array}{rcl}\dot{x}(t)& =& {\displaystyle \sum _{i\in M_2}{w}_i{f}_i(x(t),u(t),p)}\\ 0& \le & {\displaystyle w_i{g}_i(x(t),u(t),p)\mathrm{\forall }i\in M_3}\\ (x,u,p,w)& \in & {\mathrm{\Omega }}_2\end{array}}$

as a generalized inverse optimal control problem. Here ${\displaystyle {𝒳}}$ and ${\displaystyle {𝒰}}$ are properly defined function spaces. The variables ${\displaystyle w}$ indicate which objective functions, right hand side functions, and constraints are relevant in the inner problem. The variables are normalized for given index sets ${\displaystyle M_1,M_2,M_3}$ that partition the indices from ${\displaystyle 1}$ to ${\displaystyle n_w}$. For normalization, we define the feasible set ${\displaystyle {𝒲}:=\{w\in [0,1{]}^{n_w}:{\sum }^{i\in M_j}{w}_i=1\text{ for }j\in \{1,2,3\}\}}$. On the outer level, the feasible set is ${\displaystyle {\mathrm{\Omega }}_1:={\mathbb{ℝ}}^{n_p}\times {𝒲}\times {𝒳}\times {𝒰}}$, while on the inner level ${\displaystyle {\mathrm{\Omega }}_2}$ contains bounds, boundary conditions, mixed path and control constraints, and more involved constraints such as dwell time constraints. We have observational data ${\displaystyle \eta \in {\mathbb{ℝ}}^{n_{\eta }}}$, a measurement function ${\displaystyle h:{𝒳}\times {𝒰}\mapsto {\mathbb{ℝ}}^{n_{\eta }}}$, a regularization function with a priori knowledge on parameters and weights ${\displaystyle R:{\mathbb{ℝ}}^{n_p}\times {𝒲}\mapsto \mathbb{ℝ}}$ and candidate functionals ${\displaystyle {\varphi }_i:{𝒳}\times {𝒰}\times {\mathbb{ℝ}}^{n_p}\mapsto \mathbb{ℝ}}$ and functions ${\displaystyle f_i:{𝒳}\times {𝒰}\times {\mathbb{ℝ}}^{n_p}\mapsto {\mathbb{ℝ}}^{n_x}}$ and ${\displaystyle g_i:{𝒳}\times {𝒰}\times {\mathbb{ℝ}}^{n_p}\mapsto {\mathbb{ℝ}}^{n_g}}$ 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 ${\displaystyle {\varphi }_i,f_i,g_i}$ by experts and second, a systematic, but often challenging automatic symbolic regression of these unknown functions.

On the outer level, a norm ${\displaystyle \Vert \cdot \Vert }$ and the regularization term ${\displaystyle R}$ 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 ${\displaystyle g_i}$ with ${\displaystyle w_i}$ 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 ${\displaystyle {𝒳}}$ and ${\displaystyle {𝒰}}$, and the usage of universal approximators such as neural networks as candidate functions.

Extensions

• The functions ${\displaystyle u\in {𝒰}}$ may also include integer controls.
• For some problems the functions may as well depend explicitely on the time ${\displaystyle t}$.
• 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 ${\displaystyle {\displaystyle \underset{t_0}{\overset{t_f}{\int }}}L(x(t),u(t),v(t),q,\rho )\mathrm{d}t}$ can be transformed into a Mayer-type objective functional.