Sensitivity-Driven Experimental Design to Facilitate Control of Dynamical Systems

Abstract

Control of nonlinear dynamical systems is a complex and multifaceted process. Essential elements of many engineering systems include high-fidelity physics-based modeling, offline trajectory planning, feedback control design, and data acquisition strategies to reduce uncertainties. This article proposes an optimization-centric perspective which couples these elements in a cohesive framework. We introduce a novel use of hyper-differential sensitivity analysis to understand the sensitivity of feedback controllers to parametric uncertainty in physics-based models used for trajectory planning. These sensitivities provide a foundation to define an optimal experimental design which seeks to acquire data most relevant in reducing demand on the feedback controller. Our proposed framework is illustrated on the Zermelo navigation problem and a hypersonic trajectory control problem using data from NASA’s X-43 hypersonic flight tests.

Appendix A

This appendix provides details on the discretization of optimal control problems using adaptive pseudospectral methods. The fundamental ideas are common in fluid dynamics [5] and became popular in trajectory planning thanks to the pioneering work of Fahroo and Ross [8, 22]. Adaptive pseudospectral methods have gained significant attention in recent years thanks to the work of Rao and collaborators [7, 19].

For simplicity of the exposition, we consider problems of the form

$$\begin{aligned}&\min _{{\varvec{x}},{\varvec{u}}} \int _0^{T} C_{run}({\varvec{x}}(t),{\varvec{u}}(t),t)dt + C_{final}({\varvec{x}}(T),{\varvec{u}}(T))\qquad \qquad \qquad \qquad \qquad \qquad (\hbox {OC})\nonumber \\&s.t. \nonumber \\&{\left\{ \begin{array}{ll} \dot{{\varvec{x}}}(t) = {\varvec{f}}(t,{\varvec{x}},{\varvec{u}}) \qquad t \in (0,T) \\ {\varvec{x}}(0) = {\varvec{x}}_0, \end{array}\right. } \end{aligned}$$

(13)

and note that the subsequent developments may be easily extended for the more general case with inequality constraints, final time constraints, and a free final time. We have suppressed dependence on ${\varvec{g}}$ for notational simplicity.

Pseudospectral methods discretize the dynamics (13) by representing the state and control in a finite-dimensional basis and collocating the ODE system at a finite number of nodes in time. In particular, we consider a partition of the time interval [0, T] into p subintervals

$$\begin{aligned} (0,T] = \bigcup _{i=1}^p (t_{i-1},t_i], \end{aligned}$$

(14)

where $0=t_0< t_1< t_2< \cdots < t_p = T$. We approximate the state variables using N local (in the subintervals) Lagrange polynomials at Gauss–Lobatto nodes. Let $\{\mathcal Y_i\}_{i=1}^N$ denote this set of basis functions and ${\varvec{y}}=(y_1^1,y_2^1,\dots ,y_N^1,y_1^2,\dots ,y_N^2,\dots ,y_N^n)^T \in {\mathbb R}^{K_n}$, $K_n=nN$, denote the coordinate representation of the approximation, i.e.,

$$\begin{aligned} x_k(t) \approx \sum \limits _{i=1}^N y_i^k \mathcal Y_i(t). \end{aligned}$$

(15)

The controller is also discretized via an expansion in basis functions $\{\mathcal Z_j\}_{j=1}^M$; however, they may and in many cases are different than $\{\mathcal Y_i\}_{i=1}^N$. In particular, the state basis functions are adapted to resolve the dynamics, whereas the control basis is designed to enforce the users desired smoothness in the control solution, although it may also be adapted to focus nodes in regions with fast time scales if the user desires. Let ${\varvec{z}}=(z_1^1,z_2^1,\dots ,z_M^1,z_1^2,\dots ,z_M^2,\dots ,z_M^m)^T \in {\mathbb R}^{K_m}$, $K_m=mM$, be the coordinates for the control, i.e.,

$$\begin{aligned} u_k(t) \approx \sum \limits _{j=1}^M z_j^k \mathcal Z_j(t). \end{aligned}$$

We collocate the dynamics (13) by differentiating (15) and evaluating the derivative at the time nodes. Enforcing that $\dot{{\varvec{x}}}(t) = {\varvec{f}}(t,{\varvec{x}},{\varvec{u}})$ at the time nodes yields the system of equations

$$\begin{aligned} \textbf{r}_{coll}({\varvec{y}},{\varvec{z}}) = Y_t {\varvec{y}}- \varvec{\xi }({\varvec{y}},{\varvec{z}}) = \varvec{0}, \end{aligned}$$

where $\varvec{\xi }({\varvec{y}},{\varvec{z}})$ is a vector corresponding to stacking together evaluations of ${\varvec{f}}(t,{\varvec{x}},{\varvec{u}})$ at the time nodes and $Y_t$ is a matrix populated with time derivatives of the basis functions $\{\mathcal Y_i\}$. The vector $\textbf{r}_{coll}({\varvec{y}},{\varvec{z}})$ contains $(N-p-1)n$ nonlinear equations as we do not collocate at $\{t_i\}_{i=0}^p$, the interface, initial, and terminal time nodes. Rather, the initial conditions are enforced directly yielding n equations which we denote as $\varvec{r}_{init}({\varvec{y}},{\varvec{z}}) \in {\mathbb R}^n$, and the terminal nodes on each subinterval $\{t_i\}_{i=1}^p$ are enforced by requiring that ${\varvec{x}}(t_i)-{\varvec{x}}(t_{i-1})=\int _{t_{i-1}}^{t_i} {\varvec{f}}(t,{\varvec{x}},{\varvec{u}})$. This yields pn additional equations which we denote as $\varvec{r}_{inter}({\varvec{y}},{\varvec{z}}) \in {\mathbb R}^{pn}$. Combining $\textbf{r}_{coll}$, $\varvec{r}_{init}$, and $\varvec{r}_{inter}$ yields a system of nN nonlinear equations

$$\begin{aligned} \textbf{r}({\varvec{y}},{\varvec{z}}) = \left( \begin{array}{c} \textbf{r}_{coll}({\varvec{y}},{\varvec{z}}) \\ \varvec{r}_{init}({\varvec{y}},{\varvec{z}}) \\ \varvec{r}_{inter}({\varvec{y}},{\varvec{z}}) \\ \end{array} \right) = \varvec{0} \end{aligned}$$

whose solution is the coordinates for an approximate solution of the dynamics (13). To ensure a quality approximation, the time interval partition (14) is adapted to allocate time nodes in regions of faster dynamics.

The objective function is discretized by approximating the integral term

$$\begin{aligned} \int _0^{T} C_{run}({\varvec{x}}(t),{\varvec{u}}(t),t)dt \approx \sum \limits _{i=1}^N w_i c_{run}^i({\varvec{y}},{\varvec{z}}), \end{aligned}$$

where $\{w_i\}_{i=1}^N$ are the integration weights corresponding to applying Gauss–Lobatto integration on the subintervals and $c_{run}^i({\varvec{y}},{\varvec{z}})$ is the evaluation of $C_{run}$ at the $i^{th}$ time node using the discretized state and control. We represent the final time objective $C_{final}({\varvec{x}}(T),{\varvec{u}}(T))$ with $c_{final}({\varvec{y}},{\varvec{z}})$ by evaluating $C_{final}$ using the coordinates for the final time node.

The discretized optimal control problem is

$$\begin{aligned}&\min _{{\varvec{y}},{\varvec{z}}} \sum \limits _{i=1}^N w_i c_{run}^i({\varvec{y}},{\varvec{z}}) + c_{final}({\varvec{y}},{\varvec{z}})\nonumber \\&s.t. \nonumber \\&\varvec{r}({\varvec{y}},{\varvec{z}}) = 0 \end{aligned}$$

and may be solved using standard nonlinear programming methods.