Statistical Guarantees in Data-Driven Nonlinear Control: Conformal Robustness for Stability and Safety

We present a true-dynamics-agnostic, statistically rigorous framework for establishing exponential stability and safety guarantees of closed-loop, data-driven nonlinear control. In particular, we consider a nonlinear dynamical system given by \begin{align} \label{eq:true_dyn} \dot{x}=f(x, u) \end{align} where $x$ is the state, $u$ is the control input, and $f$ is an unknown locally Lipschitz function.

Central to our approach is the novel concept of conformal robustness, which robustifies the Lyapunov and zeroing barrier certificates of data-driven dynamical systems against model prediction uncertainties using conformal prediction. It quantifies these uncertainties by leveraging rank statistics of prediction scores over system trajectories, without assuming any specific underlying structure of the prediction model or distribution of the uncertainties.

With the quantified uncertainty information, we further construct the conformally robust control Lyapunov function (CR-CLF) and control barrier function (CR-CBF), data-driven counterparts of the CLF and CBF, for fully data-driven control with statistical guarantees of finite-horizon exponential stability and safety. The performance of the proposed concept is validated in numerical simulations with four benchmark nonlinear control problems.

Suppose we have a random dataset $S:= \left\{(X_i, Y_i) \right\}_{i\in\mathcal{I}}$, where $X_i$ and $Y_i$ are the features and labels with $N:= |\mathcal{I}|$. Given any regression model $\hat{Y}_i = \hat{\mathcal{F}}(X_i)$, let us define the nonconformity scores as follows: \begin{align} s_i := \left\|\hat{\mathcal{F}}(X_i)-Y_i\right\|,~\forall i\in \mathcal{I}. \end{align} Intuitively, smaller scores imply a more accurate regression model. Given a failure probability $\delta\in(0,1)$, we define the conformal quantile as the $(1-\delta)$-th quantile of the empirical distribution of $\{s_i\}_{i\in\mathcal{I}}\cup\infty$, denoted by \begin{align} \label{eq:quantile} q_{1-\delta} := \mathrm{Quantile}\left(1-\delta; \sum_{i=1}^{N} \frac{1}{1+N} \delta_{s_i} + \frac{1}{1+N} \delta_{\infty}\right), \end{align} in which $\delta_a$ denotes the point mass probability at $a\in\mathbb{R}\cup\{-\infty, \infty\}$. This quantile can be equivalently interpreted as the $\lceil (1-\delta)(N+1)\rceil$-th smallest nonconformity score in $\{s_i\}_{i\in\mathcal{I}}$, where $\lceil\cdot\rceil$ denotes the ceiling function. If, for any given new sample $(X_{i_{\mathrm{new}}}, Y_{i_{\mathrm{new}}})$, all the nonconformity scores in $\{s_i\}_{i\in\mathcal{I}\cup i_{\mathrm{new}}}$ are exchangeable, then we have the following statistical bound: \begin{align} \mathrm{Pr}\left\{\left\|\hat{\mathcal{F}}(X_{i_{\mathrm{new}}})-Y_{i_{\mathrm{new}}}\right\| \leq q_{1-\delta} \right\} \geq 1-\delta. \end{align}

Suppose we are given a data-driven model $\hat{f}(x,u)$ of an unknown dynamical system and a conformal quantile $q_{1-\delta}$ defined in \eqref{eq:quantile}. A $C^1$ function $V:\mathcal{X}\rightarrow\mathbb{R}$ is a conformally robust control Lyapunov function (CR-CLF) if it satisfies $c_1\|x\|^2 \leq V(x) \leq c_2\|x\|^2$, and $\forall x\in\mathcal{X}$, $\exists u\in\mathcal{U}$ s.t. \begin{align} \label{eq:crclf_cond} \frac{\partial V}{\partial x}\hat{f}(x,u) + c_3 V(x) + \left\|\frac{\partial V}{\partial x} \right\| \cdot q_{1-\delta} \leq 0. \end{align} Consider a set of control inputs given by \begin{align} \mathbf{K_{CL}}(x) = \left\{u\in\mathcal{U} ~\middle\vert \frac{\partial V}{\partial x}\hat{f}(x,u) + c_3 V(x) +\left\|\frac{\partial V}{\partial x}\right\| q_{1-\delta} \leq 0 \right\}. \end{align} Any locally Lipschitz continuous control policy $u(x)\in \mathbf{K_{CL}}(x)$ renders \begin{align} \mathrm{Pr}\left\{ V(x_t) \leq V(x_0) e^{-c_3 t}, \forall t\in[0,T]\right\} \geq 1-\delta \end{align} and then \begin{align} \mathrm{Pr}\left\{\|x_t\| \leq \sqrt{c_2/c_1}\|x_0\| e^{-c_3 t/2}, \forall t\in[0,T]\right\} \geq 1-\delta,~\forall x_0 \in \mathcal{D}. \end{align}

Suppose we are given a data-driven model $\hat{f}$ and a conformal quantile $q_{1-\delta}$ defined in \eqref{eq:quantile}. Let a safe set $\mathcal{C}\subset\mathcal{X}$ be the 0-superlevel set of a $C^1$ function $h: \mathcal{X}\rightarrow \mathbb{R}$, i.e., $\mathcal{C}=\{ x\in\mathcal{X} | h(x)\geq 0\}$. A $C^1$ function $h:\mathcal{X}\rightarrow\mathbb{R}$ is a conformally robust control barrier function (CR-CBF) if the safe set $\mathcal{C}$ is its 0-superlevel set, and $\forall x\in\mathcal{X},~\exists u\in\mathcal{U}$ s.t. \begin{align} \label{eq:crcbf_cond} \frac{\partial h}{\partial x} \hat{f}(x, u) + \gamma h(x) - \left\|\frac{\partial h}{\partial x} \right\| \cdot q_{1-\delta} \geq 0. \end{align} Consider a set of control inputs given by \begin{align} \mathbf{K_{CB}}(x) = \left\{u \in \mathcal{U} ~\middle\vert \frac{\partial h}{\partial x} \hat{f}(x, u) + \gamma h(x) - \left\|\frac{\partial h}{\partial x} \right\| q_{1-\delta} \geq 0 \right\}. \end{align} Any locally Lipschitz continuous control policy $u(x)\in \mathbf{K_{CB}}(x)$ renders \begin{align} \mathrm{Pr}\left\{ x_t \in \mathcal{C},~\forall t\in[0,T] \mid x_0\in\mathcal{C} \right\} \geq 1-\delta. \end{align}

Note that (1) our CR-CLFs and CR-CBFs also accommodate constraint violations, meaning the Lyapunov and barrier functions themselves need not be exact.

In the following numerical simulations with four benchmark nonlinear control problems, we demonstrate the CR-CLFs and CR-CBFs in fully data-driven settings. We also show that they can be synthesized by quadratic programs and neural networks.