Data-Driven Observability Analysis for Nonlinear Stochastic Systems

TL;DR

This paper introduces a data-driven method to analyze the observability of nonlinear stochastic systems by quantifying distributional distinguishability from measurement data, addressing challenges posed by noise and lack of analytical models.

Contribution

It presents a novel approach to assess and quantify distributional distinguishability directly from data, including a statistical test for high-confidence state differentiation, applicable to nonlinear stochastic systems.

Findings

Distributional distinguishability is equivalent to classical for linear systems.

The method quantifies how much data is needed to differentiate initial states.

The approach successfully maps distinguishability in simulation and hardware experiments.

Abstract

Distinguishability and, by extension, observability are key properties of dynamical systems. Establishing these properties is challenging, especially when no analytical model is available and they are to be inferred directly from measurement data. The presence of noise further complicates this analysis, as standard notions of distinguishability are tailored to deterministic systems. We build on distributional distinguishability, which extends the deterministic notion by comparing distributions of outputs of stochastic systems. We first show that both concepts are equivalent for a class of systems that includes linear systems. We then present a method to assess and quantify distributional distinguishability from output data. Specifically, our quantification measures how much data is required to tell apart two initial states, inducing a continuous spectrum of distinguishability. We propose a statistical test to determine a threshold above which two states can be considered distinguishable with high confidence. We illustrate these tools by computing distinguishability maps over the state space in simulation, then leverage the test to compare sensor configurations on hardware.

Tables

Table 1. TABLE I: Results for the Furuta pendulum, averaged over ten tests. Only the first two configurations are identified as distinguishable.

Measurement	$y=({\theta }_1,{\theta }_2)$	$y={\theta }_1$	$y={\theta }_2$
$\sigma$ ($\mathrm{rad}$)	1300	1300	2500
$\frac{{\mathrm{MMD}}_b}{\kappa }$	1.40	2.50	0.34
Trigger (% of trials)	$100$	$100$	$0$

Equations

$$\begin{array}{rl}{x}_{t+1}& =f(x_t),\\ y_t& =h(x_t),\end{array}$$

$$\gamma(T,x_{0})=\big{(}h(\phi(0,x_{0})),\,\dots,\,h(\phi(T,x_{0}))\big{)}\in\mathbb{Y}^{T+1}.$$

$$\begin{array}{rl}{X}_{t+1}& =F(X_t,{\eta }_t),\\ Y_t& =H(X_t,{ϵ}_t).\end{array}$$

$$\Gamma(T,\mu)=\big{(}H(\Phi(0,\mu),\epsilon_{0}),\,\dots,\,H(\Phi(T,\mu),\epsilon_{T})\big{)}\in\mathbb{Y}^{T+1},$$

$$\begin{array}{r}{\int }_{{\mathbb{Y}}^{T+1}}\mathbb{P}[K(\hat{\gamma },ϵ)\in A]\mathrm{d}{\nu }_{\mathrm{a}}(\hat{\gamma })\ne {\int }_{{\mathbb{Y}}^{T+1}}\mathbb{P}[K(\hat{\gamma },ϵ)\in A]\mathrm{d}{\nu }_{\mathrm{b}}(\hat{\gamma }).\end{array}$$

$${\int }_{{\mathbb{Y}}^{T+1}}\mathbb{P}[K(\hat{\gamma },ϵ)\in A]\mathrm{d}{\nu }_i(\hat{\gamma })=\mathbb{P}[K({\hat{\Gamma }}_i,ϵ)\in A],$$

$${\mathbb{P}}_{{\mu }_i}^T[A]={\int }_{{\mathbb{Y}}^{T+1}}\pi (\hat{\gamma },A)\mathrm{d}{\nu }_i(\hat{\gamma }),$$

$$\Gamma (T,\mu )=\Gamma (T,0)+\gamma (T,\mu ),$$

$$\mathrm{MMD}[\mathbb{P},\mathbb{Q}]={\Vert {\mu }_{\mathbb{P}}-{\mu }_{\mathbb{Q}}\Vert }_{\mathcal{H}},$$

$$k({\gamma }_{\mathrm{a}},{\gamma }_{\mathrm{b}})=\exp \left[-\frac{1}{2{\sigma }^2}\sum _{t=0}^T\sum _{i=1}^{n_y}({\gamma }_{\mathrm{a},i,t}-{\gamma }_{\mathrm{b},i,t}{)}^2\right],$$

$${\mathrm{MMD}}_b^2[m,n]$$

$$-\frac{2}{mn}\sum _{i,j=1}^{m,n}k({\Gamma }_{\mathrm{a},i},{\Gamma }_{\mathrm{b},j}),$$

$$\mathbb{P}$$

$$\le 2\exp \left[-\frac{{ϵ}^2}{2K}\cdot \frac{mn}{m+n}\right].$$

$$H_0:{\mathbb{P}}_{{\mu }_{\mathrm{a}}}^T={\mathbb{P}}_{{\mu }_{\mathrm{b}}}^T\text{vs.}{H}_1:{\mathbb{P}}_{{\mu }_{\mathrm{a}}}^T\ne {\mathbb{P}}_{{\mu }_{\mathrm{b}}}^T.$$

$${\mathrm{MMD}}_b[m,m]\ge \sqrt{\frac{2K}{m}}\left(1+\sqrt{2\ln {\alpha }^{-1}}\right)\dot{=}\kappa ,$$

$$m>\frac{K}{z^2}{\left(4+\sqrt{2}+2\left(\sqrt{\ln \frac{1}{\beta }}+\sqrt{\ln \frac{2}{\beta }}\right)\right)}^2.$$

$$ϵ=2\sqrt{\frac{K}{m}\ln \frac{2}{\beta }},\text{and}\delta =2\sqrt{\frac{K}{m}}\left(2+\sqrt{\ln \frac{2}{\beta }}\right).$$

$$\begin{array}{rl}\mathrm{d}{X}_t& =A{X}_t\mathrm{d}t+A_0\sin (\omega t)\mathrm{d}t+\Sigma \mathrm{d}{W}_t,\\ Y_t& =C{X}_t+{ϵ}_t,\\ A& =\left(\begin{array}{cc}-2& -1\\ -1& -2\end{array}\right),C=\left(\begin{array}{cc}-1& 1\end{array}\right),\end{array}$$

$$\mathrm{d}{X}_1$$

$$\mathrm{d}{X}_2$$

$$h(x)$$

Full text

††publicationid: pubid:

Under review. May be subject to changes.

I Introduction

Distinguishability is the property that allows telling apart different initial states from output measurements; observability then refers to all states being distinguishable. It is a core assumption in observer design, and there exist multiple sufficient criteria to establish observability from analytical models [1]. However, in many cases, such an analytical model is either unavailable or impractical to work with. Examples involve engineering systems with intractable models, black-box simulators, or systems that are only partially or imperfectly modeled such that the derived properties may not transfer to the real world [2]. In contrast, we often have access to measurement data generated by these systems. In this paper, we propose to infer observability of the underlying system from such data, without relying on an explicit model of the dynamics.

In many cases, measurements are corrupted by noise. Yet, the standard definition of observability is tailored to deterministic systems [3], and noise is often treated as a perturbation against which an observer should be robust [4]. This perspective circumvents defining stochastic observability by focusing on the robustness of the specific observer. In contrast, we start by defining an inherent notion of observability for stochastic systems. We argue that there is a broad class of reasonable definitions, each corresponding to a way of comparing the underlying stochastic processes [5, 6, 7, 8, 9, 10]. We generalize distributional distinguishability, first defined for probabilistic Boolean networks [5], to arbitrary nonlinear systems. It consists in comparing the distributions of the output. We show that distributional distinguishability is a reasonable generalization of the deterministic notion by proving that the stochastic and nominal systems share the same classes of indistinguishability under some assumptions, which include linear systems.

Our main contribution is then a method to assess and quantify this distributional distinguishability from measurement data. To achieve this, we propose to use a metric between the distributions of output measurements; its value reflects how different these distributions are. We choose the maximum mean discrepancy (MMD) [11] as this metric, for three reasons. First, it takes values in a continuous set, which induces a continuous spectrum of relative distinguishability on which some pairs are more distinguishable than others. Second, it can be estimated from data, hereby establishing distinguishability of initial distributions with high confidence given the data (absolute distinguishability) through statistical testing. Third, its continuous value is interpreted as how easily two initial distributions can be told apart, in the sense of how much data the test requires.

We validate the proposed tools in simulation and on hardware. We first demonstrate using the finite-sample MMD to evaluate and quantify distinguishability. We experimentally recover the nominal system’s analytical classes of indistinguishability in the cases covered by our theoretical results, and empirically demonstrate how the metric increases as we get farther from a state’s class of indistinguishability. This also enables evaluating the influence of the noise on said classes. Further, we illustrate the difference between our test and empirical Gramians [12], a tool for data-driven weak observability analysis. We then use the proposed test on a Furuta pendulum for experiment design by checking whether removing specific sensors harms observability. Finally, we begin bridging the gap between the proposed tools and observer design by showcasing how an observer outputs the same distribution over estimated trajectories when the system is initialized in indistinguishable states. This is a first step towards using our tools for a priori error analysis in observer design; the MMD indicates which states observers will fail to distinguish.

I-A Related Work

We propose a method to assess and quantify observability from output data without explicitly relying on a dynamics model. To the best of our knowledge, such a model-free, data-driven analysis has never been proposed for nonlinear stochastic systems. We start by reviewing existing notions of stochastic observability and distinguishability and corresponding metrics. Finally, we present an essential concept for the metric we propose: kernel mean embeddings.

I-A1 Stochastic Observability

Noise is often treated as a perturbation against which observers should be robust [4]; stochastic observability then reduces to (approximate) convergence of usual observers, possibly up to some probability [13]. Despite its practical appeal, this approach ties observability (a system property) to the observer (a design choice), and neglects the question of how noise affects observability independently of the chosen observer. Answering it requires defining observability for a stochastic system; yet, there is no consensus over such a definition [12].

Many approaches deal with linear systems and rely on the superposition principle to define observability of the whole system directly [14]. Extensions to nonlinear systems typically result in local notions of observability. A relevant example is stochastic Gramians [12], a generalization of empirical Gramians of nonlinear systems [15, 16]. Similarly to our method, they enable a data-driven analysis of observability with the important difference that they build on local weak observability instead of distinguishability, which only provides local information, as we illustrate in Section IV.

Competing approaches aim at generalizing distinguishability [3] of (distributions over) initial states, rather than observability. Core to these approaches is a way to compare the underlying stochastic processes induced by the stochastic initial state, dynamics, and measurement. Since stochastic processes can be compared in many more ways than deterministic trajectories, there exists different notions of distinguishability for stochastic systems. Examples involve PBNs [5, 6, 7] and other linear systems [8, 9, 10]; their corresponding definitions of distinguishability are not always explicit, but can be straightforwardly adapted from those of observability. Interestingly, the relationship between such definitions and the existence of a probabilistic observer has not been studied, unlike for deterministic systems [1]. Our approach builds on distributional distinguishability [5], and we show that it is equivalent to observability of the nominal system in certain cases, such as linear dynamics or additive measurement noise. One advantage of this notion is also its amenability to statistical tests provided independent realizations of the system.

I-A2 Metricizing Observability

A practical way to check observability is to summarize it in a scalar value or function. For linear systems, such a metric is given by the rank of the observability matrix or, equivalently, of the observability Gramian [17]. An alternative is the largest singular value of the Gramian, introducing a continuous scale and enabling a quantification of observability. These metrics translate locally to nonlinear systems with empirical Gramians [15]; they inform on the weak observability of each state. Gramians and their corresponding metrics have also been extended for stochastic systems [12]. They are formally defined for linear systems with additive or multiplicative noise, and extensions towards arbitrary nonlinear systems are only hinted at [12, Section 5]. While our theoretical analysis shares similar restrictions, our approach differs in that

1. our definition and quantification are readily applicable to both linear and nonlinear systems; and
2. we quantify distinguishability of two initial states, instead of weak observability.

Finally, there is a body of research that defines observability [18] and observability metrics [19] based on information-theoretic properties of the output. This definition strongly differs from our approach, which is thus of independent interest.

I-A3 Kernel Mean Embeddings

Our quantification of observability relies on KMEs of probability distributions [11]. In short, these embeddings represent probability distributions in a high-dimensional reproducing kernel Hilbert space (RKHS) where scalar products can be performed efficiently and often estimated from data. Therefrom developed many methods to model and compare random variables, such as modeling of conditional relations [20] or Bayesian inference [21]. Essential for this work is the MMD, a distance between probability distributions that can be estimated from data [22]. The MMD was leveraged in a distribution-free two-sample test able to compare probability distributions from independent samples [23]. We heavily rely on both the MMD and this test, which requires independent samples from the same distributions. This can be problematic for dynamical systems, where the future greatly depends on the past. Some approaches alleviate this problem with mixing [24]; in contrast, we assume independent reinitializations of the system.

I-B Notations

The Borel $\sigma$-algebra of an open set $\mathbb{S}\subset\mathbb{R}^{n}$ is $\mathcal{B}(\mathbb{S})$, and $\mathcal{M}^{+}_{1}(\mathbb{S})$ is the set of probability measures on $(\mathbb{S},\mathcal{B}(\mathbb{S}))$. We assume open sets $\mathbb{X}\subset\mathbb{R}^{n_{x}}$ and $\mathbb{Y}\subset\mathbb{R}^{n_{y}}$, respectively called state and output spaces, and equip them with their Borel $\sigma$-algebras. All random variables are defined on a probability space $(\Omega,\mathcal{A},\mathbb{P})$. For a random variable $Z$, the notation $Z\in\mathbb{S}$ means that $Z$ takes values in $(\mathbb{S},\mathcal{B}(\mathbb{S}))$, and {\mathchoice{\raisebox{0.0pt}{\displaystyle\chi}}{\raisebox{0.0pt}{\textstyle\chi}}{\raisebox{0.0pt}{\scriptstyle\chi}}{\raisebox{0.0pt}{\scriptscriptstyle\chi}}}_{Z} is the characteristic function of $Z$.

II Distinguishability

We first recall a definition of observability for deterministic systems (Section II-A), then extend this notion to nonlinear stochastic systems by generalizing distributional distinguishability of PBNs (Section II-B). This sets the stage for our first contribution: showing that, for linear dynamics or certain nonlinear systems, the two notions are equivalent (Section II-C).

For simplicity, we focus on autonomous systems. All concepts and results translate to control systems by considering distinguishability for a specific controller, as is commonly done [3, 15].

II-A Deterministic Distinguishability

Consider a deterministic system

[TABLE]

with arbitrary initial condition $x_{0}\in\mathbb{X}$. The function $f:\mathbb{X}\to\mathbb{X}$ is the dynamics, $h:\mathbb{X}\to\mathbb{Y}$ is the output map. We denote by $\phi(t,x_{0})\in\mathbb{X}$ the state of (1) initialized in $x_{0}$, at time $t$. Additionally, define the output trajectory up to time $T\in\mathbb{N}$ of (1) initialized in $x_{0}$ as

[TABLE]

We can then define distinguishability and observability as:

Definition 1[Chapter 6]Son1998. (Sontag [3, Chapter 6])

Let $T\in\mathbb{N}$, and let $x_{\mathrm{a}},x_{\mathrm{b}}\in\mathbb{X}$ be two initial states of (1). We say that $x_{\mathrm{a}}$ and $x_{\mathrm{b}}$ are distinguishable (in time $T$) if $\gamma(T,x_{\mathrm{a}})\neq\gamma(T,x_{\mathrm{b}})$. Otherwise, we say that they are indistinguishable (in time $T$). Finally, the system is observable (in time $T$) if all different initial states are distinguishable.

Intuitively, indistinguishable states produce the same output until time $T$. It is thus impossible to tell them apart by looking only at the output sequence. In contrast, distinguishability means that the outputs differ after some time.

Definition 1 is binary, meaning that an infinitesimal difference between trajectories can make two states distinguishable. This becomes problematic in practice, in particular for stochastic systems, where two sample paths from the same initial point will differ almost surely (a.s.) for nontrivial noise, dynamics, and measurement.

II-B Distributional Distinguishability

We therefore need a generalized notion of distinguishability to handle stochastic systems. They come with the additional hurdle that output measurements are now random variables. Hence, the question of distinguishability pertains to the more general problem of how to compare stochastic processes. There are many suitable ways to define the “equivalence” of stochastic processes, each leading to a specific notion of stochastic observability [7, 5]. We focus on distributional distinguishability, since the similarity with the deterministic case makes it a natural extension and because, as we will show, it lends itself to data-driven evaluation.

Consider a stochastic dynamical system of the form

[TABLE]

The state $X_{t}$ and output $Y_{t}$ are now RVs taking values in $\mathbb{X}$ and $\mathbb{Y}$, respectively. The RV $\eta_{t}\in\mathbb{R}^{n_{\eta}}$ represents the stochastic part of the dynamics, and $\epsilon_{t}\in\mathbb{R}^{n_{\epsilon}}$ the one of the measurement. We assume that (3) is a noisy version of (1), that is, $F(x,0)=f(x)$ and $H(x,0)=h(x)$.

We allow stochastic initializations, meaning that there exists an initial distribution $\mu$ defined on $\mathbb{X}$ such that $X_{0}\sim\mu$. We assume that $\eta_{t}$ and $\epsilon_{t}$ are independent of the past states $(X_{s})_{s\leq t}$. We do not make further assumptions on the noise; e.g., we allow non-centered or autocorrelated noise. Finally, we denote by $\Phi(t,\mu)$ the random state of (3) initialized as per $\mu$, at time $t\in\mathbb{N}$, and define

[TABLE]

the random output trajectory up to time $T\in\mathbb{N}$. We denote its law by $\mathbb{P}_{\mu}^{T}$. We are now ready to generalize distributional distinguishability, first introduced for PBNs [5], as follows:

Definition 2** (Distributional Distinguishability)**

Let $T\in\mathbb{N}$, and let $\mu_{\mathrm{a}},\mu_{\mathrm{b}}$ be two initial distributions of (3). We say that $\mu_{\mathrm{a}}$ and $\mu_{\mathrm{b}}$ are distributionally distinguishable (in time $T$) if $\mathbb{P}_{\mu_{\mathrm{a}}}^{T}\neq\mathbb{P}_{\mu_{\mathrm{b}}}^{T}$. Otherwise, we say that they are distributionally indistinguishable (in time $T$). Finally, the system is distributionally observable (in time $T$) if all different initial distributions are distributionally distinguishable.

Distributional observability thus compares the laws of the random trajectories instead of comparing the trajectories themselves, as Definition 1 does. Intuitively, it focuses on whether one can tell apart different initial distributions by repeating the experiment.

Since distributional distinguishability is defined for arbitrary initial distributions, it is also meaningful to talk about distributional distinguishability of initial states; the initial distributions involved are then accordingly-centered Diracs. In what follows, we abuse notations and use $x$ directly instead of $\delta_{x}$ in $\Phi(t,\cdot),\Gamma(T,\cdot)$, and $\mathbb{P}_{\cdot}^{T}$.

II-C Noise and Distinguishability

Distributional observability reduces to the classical, deterministic notion when the system is deterministic with deterministic initializations; the distributions of $\Phi(T,x_{\mathrm{a}})$ and $\Phi(T,x_{\mathrm{b}})$ are then Diracs and are thus equal if, and only if, (iff) the variables are equal a.s. Therefore, we study whether noise influences distinguishability. In particular, can introducing noise make two states distinguishable? Conversely, can noise hide the difference between nominally-distinguishable initial states? In general, yes; the classes of indistinguishability of the nominal and noisy systems differ, as we illustrate in Section IV. In this section, we study conditions under which the noise does not influence distinguishability. We start with measurement noise and show that, under a technical assumption, it does not affect distinguishability. This implies that the deterministic and stochastic notions coincide in the absence of process noise. We then treat linear systems, for which this equivalence always holds.

II-C1 Measurement Noise

We start with the following assumption to restrict the influence of measurement noise to the output of $h$:

Assumption 1

There exists a measurable function $K:\mathbb{Y}\times\mathbb{R}^{n_{\epsilon}}\to\mathbb{Y}$ such that $H(x,e)=K(h(x),e)$.

This enables introducing the output before its corruption by measurement noise, @iaciRV RV denoted by $\hat{Y}_{t}=h(X_{t})$ independent of $\epsilon_{t}$.

Assumption 2

For all $\nu_{\mathrm{a}},\nu_{\mathrm{b}}\in\mathcal{M}^{+}_{1}(\mathbb{Y}^{T+1})$ with $\nu_{\mathrm{a}}\neq\nu_{\mathrm{b}}$, there exists $A\in\mathcal{B}(\mathbb{Y}^{T+1})$ such that, by applying $K$ element-wise,

[TABLE]

This assumption is a generalized injectivity condition: it requires the mapping from a distribution $\nu$ of $\hat{\Gamma}$ to the output distribution after corruption by measurement noise to be injective, where $\hat{\Gamma}=(\hat{Y}_{0},\dots,\hat{Y}_{T})$ is the output trajectory before the measurement noise. It ensures that if two distributions of $\hat{\Gamma}$ are distinct, they remain different after applying measurement noise.

Example 1** (Additive Noise)**

Albeit technical, Assumption 2 is satisfied for additive noise. Indeed, assume $K(\hat{\gamma},e)=\hat{\gamma}+e$, and take $\nu_{\mathrm{a}},\nu_{\mathrm{b}}\in\mathcal{M}^{+}_{1}(\mathbb{Y}^{T+1})$, distinct. Let $\hat{\Gamma}_{\mathrm{a}}\sim\nu_{\mathrm{a}}$, $\hat{\Gamma}_{\mathrm{b}}\sim\nu_{\mathrm{b}}$, and assume that $\hat{\Gamma}_{\mathrm{a}},\hat{\Gamma}_{\mathrm{b}}$, and $\epsilon$ are independent. We have, for all $A\in\mathcal{B}(\mathbb{Y}^{T+1})$:

[TABLE]

where $i\in\{a,b\}$. Therefore, Assumption 2 is satisfied iff $\hat{\Gamma}_{\mathrm{a}}+\epsilon$ and $\hat{\Gamma}_{\mathrm{b}}+\epsilon$ have different laws, iff {\mathchoice{\raisebox{0.0pt}{\displaystyle\chi}}{\raisebox{0.0pt}{\textstyle\chi}}{\raisebox{0.0pt}{\scriptstyle\chi}}{\raisebox{0.0pt}{\scriptscriptstyle\chi}}}_{\hat{\Gamma}_{\mathrm{a}}+\epsilon}\neq{\mathchoice{\raisebox{0.0pt}{\displaystyle\chi}}{\raisebox{0.0pt}{\textstyle\chi}}{\raisebox{0.0pt}{\scriptstyle\chi}}{\raisebox{0.0pt}{\scriptscriptstyle\chi}}}_{\hat{\Gamma}_{\mathrm{b}}+\epsilon}. By independence, {\mathchoice{\raisebox{0.0pt}{\displaystyle\chi}}{\raisebox{0.0pt}{\textstyle\chi}}{\raisebox{0.0pt}{\scriptstyle\chi}}{\raisebox{0.0pt}{\scriptscriptstyle\chi}}}_{\hat{\Gamma}_{i}+\epsilon}={\mathchoice{\raisebox{0.0pt}{\displaystyle\chi}}{\raisebox{0.0pt}{\textstyle\chi}}{\raisebox{0.0pt}{\scriptstyle\chi}}{\raisebox{0.0pt}{\scriptscriptstyle\chi}}}_{\hat{\Gamma}_{i}}\cdot{\mathchoice{\raisebox{0.0pt}{\displaystyle\chi}}{\raisebox{0.0pt}{\textstyle\chi}}{\raisebox{0.0pt}{\scriptstyle\chi}}{\raisebox{0.0pt}{\scriptscriptstyle\chi}}}_{\epsilon}. Therefore, Assumption 2 holds iff {\mathchoice{\raisebox{0.0pt}{\displaystyle\chi}}{\raisebox{0.0pt}{\textstyle\chi}}{\raisebox{0.0pt}{\scriptstyle\chi}}{\raisebox{0.0pt}{\scriptscriptstyle\chi}}}_{\hat{\Gamma}_{\mathrm{a}}}\neq{\mathchoice{\raisebox{0.0pt}{\displaystyle\chi}}{\raisebox{0.0pt}{\textstyle\chi}}{\raisebox{0.0pt}{\scriptstyle\chi}}{\raisebox{0.0pt}{\scriptscriptstyle\chi}}}_{\hat{\Gamma}_{\mathrm{b}}}, which is true since $\nu_{\mathrm{a}}\neq\nu_{\mathrm{b}}$.

Importantly, Assumption 2 enables decoupling the effect of measurement and process noise on distributional distinguishability; measurement noise then does not affect distinguishability.

Theorem 1

Let $\mu_{\mathrm{a}},\mu_{\mathrm{b}}\in\mathcal{M}^{+}_{1}(\mathbb{X})$ be two initial distributions, and let Assumption 1 hold. If $\mu_{\mathrm{a}}$ and $\mu_{\mathrm{b}}$ are distributionally distinguishable, the distributions of $\hat{\Gamma}_{\mathrm{a}}\,\dot{=}\,(h(\Phi(t,\mu_{\mathrm{a}}))_{t=0}^{T}$ and $\hat{\Gamma}_{\mathrm{b}}\,\dot{=}\,(h(\Phi(t,\mu_{\mathrm{b}}))_{t=0}^{T}$ differ. Under Assumption 2, the converse is also true.

Proof:

Let $\mu_{\mathrm{a}},\mu_{\mathrm{b}}\in\mathcal{M}^{+}_{1}(\mathbb{X})$ be distinguishable, and take $A\in\mathcal{B}(\mathbb{Y}^{T+1})$ such that $\mathbb{P}_{\mu_{\mathrm{a}}}^{T}[A]\neq\mathbb{P}_{\mu_{\mathrm{b}}}^{T}[A]$. Let $i\in\{a,b\}$, and $\nu_{i}$ be the distribution of $\hat{\Gamma}_{i}$. Marginalizing $\Gamma(T,\mu_{i})$ on $\hat{\Gamma}_{i}$ yields

[TABLE]

where $\pi(\hat{\gamma},A)=\mathbb{P}\left[K(\hat{\gamma},\epsilon)\in A\right]$ is the probability of the observation lying in $A$ given that the nominal observation is $\hat{\gamma}$; it is thus independent of $\mu_{\mathrm{a}}$ and $\mu_{\mathrm{b}}$. Since $\mathbb{P}_{\mu_{\mathrm{a}}}^{T}[A]\neq\mathbb{P}_{\mu_{\mathrm{b}}}^{T}[A]$, (7) implies that the integral of $\pi(\cdot,A)$ differs when considered w.r.t. $\nu_{\mathrm{a}}$ and $\nu_{\mathrm{b}}$. Therefore, $\nu_{\mathrm{a}}\neq\nu_{\mathrm{b}}$. Conversely, assume that $\nu_{\mathrm{a}}$ and $\nu_{\mathrm{b}}$ differ. Assumption 2 ensures the existence of $A\in\mathcal{B}(\mathbb{Y}^{T+1})$ such that the right-hand side of (7) differ for $i=a$ and $i=b$; this shows $\mathbb{P}_{\mu_{\mathrm{a}}}^{{}^{T}}\neq\mathbb{P}_{\mu_{\mathrm{b}}}^{T}$ and concludes the proof. ∎

Corollary 1

Let Assumptions 1 and 2 hold. Assume that there is no process noise, i.e., $\eta_{t}=0$ a.s. for all $t$. Let $x_{\mathrm{a}},x_{\mathrm{b}}\in\mathbb{X}$. Then, $x_{\mathrm{a}}$ and $x_{\mathrm{b}}$ are distinguishable for (1) iff they are distributionally distinguishable for (3).

Proof:

By Theorem 1, $x_{\mathrm{a}}$ and $x_{\mathrm{b}}$ are distributionally distinguishable iff $\hat{\Gamma}_{\mathrm{a}}$ and $\hat{\Gamma}_{\mathrm{b}}$ have different distributions. From $\eta=0$ a.s., we have $\forall x\in\mathbb{X},(h(\Phi(t,x))_{t=0}^{T}=\gamma(T,x)$ a.s. The result follows from Definition 1. ∎

Unfortunately, this corollary does not generalize to non-zero process noise, as we illustrate on an example in Section IV. We expect that such a generalization requires strong joint assumptions on the noise and dynamics, in the absence of which the trajectories of the nominal and noisy systems need not correlate.

II-C2 Linear Systems

For linear systems, however, the superposition principle enables such a generalization.

Theorem 2

Assume that (3) is linear, i.e., $F(x,n)=Ax+Qn$ and $H(x,e)=Cx+Rn$. Then, two initial states $x_{\mathrm{a}},x_{\mathrm{b}}\in\mathbb{X}$ are distributionally distinguishable iff they are distinguishable for (1). Additionally, (3) is distributionally observable iff (1) is observable.

This guarantees that, for linear systems, noise preserves the classes of indistinguishability under the single assumption that the noise is independent of the past trajectory. Additionally, Kalman’s criterion on the observability matrix enables checking distributional observability. We emphasize that the second claim is not a consequence of the first one; distributional observability requires all initial distributions to be distinguishable rather than initial states. This statement thus means that, for a linear system, distinguishability of all initial states ensures observability; there is no gain in considering initial distributions.

Proof:

For $\mu\in\mathcal{M}^{+}_{1}(\mathbb{X})$, the superposition principle holds:

[TABLE]

where we abusively denote $\gamma(T,X_{0})$ with $X_{0}\sim\mu$ by $\gamma(T,\mu)$. Crucially, $\Gamma(T,0)$ and $\gamma(T,\mu)$ are independent, since the former is driven by noise and the latter by its random initialization. Hence, we have {\mathchoice{\raisebox{0.0pt}{\displaystyle\chi}}{\raisebox{0.0pt}{\textstyle\chi}}{\raisebox{0.0pt}{\scriptstyle\chi}}{\raisebox{0.0pt}{\scriptscriptstyle\chi}}}_{\Gamma(T,\mu)}={\mathchoice{\raisebox{0.0pt}{\displaystyle\chi}}{\raisebox{0.0pt}{\textstyle\chi}}{\raisebox{0.0pt}{\scriptstyle\chi}}{\raisebox{0.0pt}{\scriptscriptstyle\chi}}}_{\Gamma(T,0)}\cdot{\mathchoice{\raisebox{0.0pt}{\displaystyle\chi}}{\raisebox{0.0pt}{\textstyle\chi}}{\raisebox{0.0pt}{\scriptstyle\chi}}{\raisebox{0.0pt}{\scriptscriptstyle\chi}}}_{\gamma(T,\mu)}. We thus have the following equivalences, for $x_{\mathrm{a}},x_{\mathrm{b}}\in\mathbb{X}$: they are distributionally indistinguishable iff $\mathbb{P}_{x_{\mathrm{a}}}^{T}=\mathbb{P}_{x_{\mathrm{b}}}^{T}$, iff {\mathchoice{\raisebox{0.0pt}{\displaystyle\chi}}{\raisebox{0.0pt}{\textstyle\chi}}{\raisebox{0.0pt}{\scriptstyle\chi}}{\raisebox{0.0pt}{\scriptscriptstyle\chi}}}_{\Gamma(T,x_{\mathrm{a}})}={\mathchoice{\raisebox{0.0pt}{\displaystyle\chi}}{\raisebox{0.0pt}{\textstyle\chi}}{\raisebox{0.0pt}{\scriptstyle\chi}}{\raisebox{0.0pt}{\scriptscriptstyle\chi}}}_{\Gamma(T,x_{\mathrm{b}})}, iff {\mathchoice{\raisebox{0.0pt}{\displaystyle\chi}}{\raisebox{0.0pt}{\textstyle\chi}}{\raisebox{0.0pt}{\scriptstyle\chi}}{\raisebox{0.0pt}{\scriptscriptstyle\chi}}}_{\gamma(T,\delta_{x_{\mathrm{a}}})}={\mathchoice{\raisebox{0.0pt}{\displaystyle\chi}}{\raisebox{0.0pt}{\textstyle\chi}}{\raisebox{0.0pt}{\scriptstyle\chi}}{\raisebox{0.0pt}{\scriptscriptstyle\chi}}}_{\gamma(T,\delta_{x_{\mathrm{b}}})}, iff $\gamma(T,x_{\mathrm{a}})=\gamma(T,x_{\mathrm{b}})$. This shows that $x_{\mathrm{a}}$ and $x_{\mathrm{b}}$ are distributionally indistinguishable iff they are deterministically indistinguishable.

We now show that deterministic observability implies distributional observability, since the converse follows immediately from what precedes. Assume deterministic observability, and let $\mu_{\mathrm{a}},\mu_{\mathrm{b}}\in\mathcal{M}^{+}_{1}(\mathbb{X})$ be two initial distributions with $\mathbb{P}_{\mu_{\mathrm{a}}}^{T}=\mathbb{P}_{\mu_{\mathrm{b}}}^{T}$. We show $\mu_{\mathrm{a}}=\mu_{\mathrm{b}}$. A similar reasoning as previously shows {\mathchoice{\raisebox{0.0pt}{\displaystyle\chi}}{\raisebox{0.0pt}{\textstyle\chi}}{\raisebox{0.0pt}{\scriptstyle\chi}}{\raisebox{0.0pt}{\scriptscriptstyle\chi}}}_{\gamma(T,\mu_{\mathrm{a}})}={\mathchoice{\raisebox{0.0pt}{\displaystyle\chi}}{\raisebox{0.0pt}{\textstyle\chi}}{\raisebox{0.0pt}{\scriptstyle\chi}}{\raisebox{0.0pt}{\scriptscriptstyle\chi}}}_{\gamma(T,\mu_{\mathrm{b}})}, i.e., $\gamma(T,\mu_{\mathrm{a}})$ and $\gamma(T,\mu_{\mathrm{b}})$ have the same law and, thus, $\mu_{\mathrm{a}}$ and $\mu_{\mathrm{b}}$ coincide on all sets $A$ in the $\sigma$-algebra $\mathcal{S}=\gamma^{-1}(T,\mathcal{B}(\mathbb{Y}^{T+1}))\subset\mathcal{B}(\mathbb{X})$. We show $\mathcal{S}=\mathcal{B}(\mathbb{X})$. From observability of (1), $\gamma(T,\cdot)$ is injective. Since it is also linear, it admits a continuous (and thus, measurable) left inverse, say, $\lambda:\mathbb{Y}^{T+1}\to\mathbb{X}$. We are now ready to conclude. Take $A\in\mathcal{B}(\mathbb{X})$, and define $B=\lambda^{-1}(A)\in\mathcal{B}(\mathbb{Y}^{T+1})$. We have, by definition of $\lambda$ as a left inverse of $\gamma$, $\gamma^{-1}(T,B)=\gamma^{-1}(T,\lambda^{-1}(A))=A$. This shows that $A\in\mathcal{S}$, i.e., that $\mathcal{B}(\mathbb{X})=\mathcal{S}$. Therefore, $\mu_{\mathrm{a}}(A)=\mu_{\mathrm{b}}(A)$ for all $A\in\mathcal{B}(\mathbb{X})$, which shows distributional observability of (3). ∎

Albeit distributional and deterministic distinguishability differ in general, it is reasonable to expect that a small amount of noise only mildly affects the distinguishability of two states. In the next section, we propose a quantification of distinguishability to formalize the above intuition, and a method to estimate it from output data.

III Measuring Distributional Observability

Distributional observability is defined as the equality of certain probability distributions. We propose to extend this binary notion to a continuous spectrum of relative distinguishability by quantifying it with the MMD, a distance between said distributions measuring how easily they can be told apart. Additionally, the MMD can be estimated from data. This leads to our main methodological contribution: use output measurements to

1. approximate relative distinguishability; and

2. leverage a two-sample test [23] to assess distributional observability with high confidence from output measurements.

III-A The MMD to Measure Relative Distinguishability

We start with background necessary to define the MMD and state the results that justify the method we propose.

III-A1 Background

Before introducing the MMD, we need to define RKHSs and KMEs. We focus on a minimal treatment and refer the interested reader to a dedicated survey [11] for an overview.

First, one picks a positive definite kernel function $k:\mathbb{Y}^{T+1}\times\mathbb{Y}^{T+1}\to\mathbb{R}$. Such a kernel function uniquely determines an underlying Hilbert space of functions $\mathcal{H}$ defined on $\mathbb{Y}^{T+1}$, called @iaciRKHS RKHS. Then, one embeds a given probability distribution $\mathbb{P}\in\mathcal{M}^{+}_{1}(\mathbb{Y}^{T+1})$ into this RKHS through its KME $\mu_{\mathbb{P}}=\mathbb{E}_{X\sim\mathbb{P}}[k(\cdot,X)]$. The MMD between two distributions $\mathbb{P}$ and $\mathbb{Q}$ is then the distance in the RKHS between their embeddings:

[TABLE]

where ${\lVert\cdot\rVert}_{\mathcal{H}}$ is the norm in the RKHS $\mathcal{H}$. In general, the MMD is only a semi-metric, since the embedding $\mathbb{P}\mapsto\mu_{\mathbb{P}}$ may not be injective. Kernels for which it is injective are called characteristic [25]; they include the Gaussian kernel. From now on, we assume a characteristic kernel.

Remark 1** (A Kernel on Trajectories)**

We emphasize that the considered kernel $k$ is defined on the output trajectory space $\mathbb{Y}^{T+1}$, and not on the state space $\mathbb{X}$ nor on the output space $\mathbb{Y}$. For instance, the expression of a Gaussian kernel on $\mathbb{Y}^{T+1}$ with scalar width $\sigma>0$, which is known to be characteristic [11, Table 3.1], is

[TABLE]

where $\gamma_{\mathrm{a}},\gamma_{\mathrm{b}}\in\mathbb{R}^{(T+1)\times{n_{y}}}$ are output trajectories.

III-A2 A Metric of Distinguishability

The value of the MMD thus indicates whether initial distributions are distinguishable.

Proposition 1

Let $\mu_{\mathrm{a}},\mu_{\mathrm{b}}$ be two initial distributions. Assume that the kernel $k$ is characteristic. Then, $\mathrm{MMD}[\mathbb{P}_{\mu_{\mathrm{a}}}^{T},\mathbb{P}_{\mu_{\mathrm{b}}}^{T}]=0$ iff $\mu_{\mathrm{a}}$ and $\mu_{\mathrm{b}}$ are distributionally indistinguishable.

Proof:

The MMD is a metric when the kernel is characteristic [25], which implies that $\mathrm{MMD}^{2}[\mathbb{P}_{\mu_{\mathrm{a}}}^{T},\mathbb{P}_{\mu_{\mathrm{b}}}^{T}]=0$ iff $\mathbb{P}_{\mu_{\mathrm{a}}}^{T}=\mathbb{P}_{\mu_{\mathrm{b}}}^{T}$. The result follows from Definition 2. ∎

The information the MMD carries is richer than a simple yes-or-no indication. Indeed, it takes continuous values; low values reveal similar distributions. Therefore, the value of the MMD extends the binary notion of absolute distinguishability to a continuous spectrum of relative distinguishability on which some pairs of initial distributions are more (or less) distinguishable than others. We postpone further interpreting this value to Section III-B3

III-B Finite-Sample Approximation

To access both the relative and absolute notions, one needs to evaluate the MMD. We rely on a finite-sample approximation [23] for this and use a two-sample test to conclude on (absolute) distinguishability, since the approximation is never exactly [math]. Finally, this test helps refine the interpretation of the MMD value.

III-B1 Background – Finite-Sample MMD

While a closed-form formula is often unattainable or requires perfect system knowledge, the MMD can be approximated from independent samples of output trajectories by the following empirical estimator.

Theorem 3** (Gretton et al. [23])**

Let $\mu_{\mathrm{a}},\mu_{\mathrm{b}}$ be two initial distributions. Let $\Gamma_{\mathrm{a},1},\dots,\Gamma_{\mathrm{a},m}$ be independent copies of $\Gamma(\mu_{\mathrm{a}},T)$, and $\Gamma_{\mathrm{b},1},\dots,\Gamma_{\mathrm{b},n}$ independent copies of $\Gamma(\mu_{\mathrm{b}},T)$. Assume that $k$ is bounded, i.e., $0\leq k\leq K$ for some $K\in\mathbb{R}$. Then,

[TABLE]

converges in probability to $\mathrm{MMD}^{2}[\mathbb{P}_{\mu_{\mathrm{a}}}^{T},\mathbb{P}_{\mu_{\mathrm{b}}}^{T}]$ with the following concentration bound for $\epsilon>0$:

[TABLE]

The outcome of (11) is a noisy, biased, and finite-sample approximation of the population value $\mathrm{MMD}[\mathbb{P}_{\mu_{\mathrm{a}}}^{T},\mathbb{P}_{\mu_{\mathrm{b}}}^{T}]$. We can rely on the value of the estimator to approximate relative distinguishability, however, we cannot use it directly to check absolute distinguishability. Instead, we use a statistical test.

III-B2 A Two-Sample Test of Distinguishability

To determine whether two initial distributions $\mu_{\mathrm{a}}$ and $\mu_{\mathrm{b}}$ are distinguishable from the value of (11), we perform a statistical test of the null hypothesis

[TABLE]

Such a test for equality of distributions is called a two-sample test. We propose to use one based on the concentration bound (12) [23]. One specifies an acceptable Type I risk $\alpha\in(0,1)$, and the test provides an acceptance region threshold; if the outcome of (11) falls outside of that region, one rejects the null hypothesis $H_{0}$ with confidence level at least $1-\alpha$.

Proposition 2

Let $\alpha\in(0,1)$. Under the setting of Theorem 3 with $n=m$, consider a realization of $\mathrm{MMD}_{b}[m,m]$. If this empirical estimate is such that

[TABLE]

then $\mu_{\mathrm{a}}$ and $\mu_{\mathrm{b}}$ are distributionally distinguishable in time $T$ with confidence level $1-\alpha$.

Proof:

This immediately follows from [23, Corollary 9]. ∎

The threshold $\kappa$ is conservative for many practical applications. It is common to use bootstrapping instead, as suggested in [23] and implemented in their original code. We use this bootstrapped value for $\kappa$ in Section IV.

III-B3 From Qualitative Properties to Sample Bounds

In Section III-A2, we interpreted the MMD as a measure of distinguishability based on the argument that it takes values in a continuous set. The two-sample test of Proposition 2 allows us to refine this interpretation: the value of the MMD indicates how much data is required for the test to successfully reject the null hypothesis $H_{0}$ with high confidence. In other words, the MMD provides a sample bound for the test to achieve a specific Type II error $\beta\in(0,1)$.

Corollary 2

Let $\beta\in(0,1)$. Under the setting of Theorem 3 with $n=m$, assume that the null hypothesis is incorrect and let $z=\mathrm{MMD}[\mathbb{P}_{\mu}^{T},\mathbb{P}_{\nu}^{T}]>0$. Assume that

[TABLE]

Then, the null hypothesis is rejected with probability at least $1-\beta$.

Proof:

Introduce $A_{m}=\{\mathrm{MMD}_{b}[m,m]<\kappa\}$ the test’s acceptance region of level $\beta$ and $B_{m}=\{\lvert\mathrm{MMD}_{b}[m,m]-z\rvert\leq\delta\}$ the high-probability region of the concentration bound (12), with

[TABLE]

If $A_{m}\cap B_{m}=\emptyset$, then $H_{0}$ is rejected with probability at least $1-\beta$; indeed, we then have $\mathbb{P}[A_{m}]\leq\mathbb{P}[B_{m}^{\complement}]\leq\beta$. It is sufficient for $A_{m}\cap B_{m}=\emptyset$ that $\kappa<z-\delta$. Rearranging yields (15). ∎

Albeit the threshold (15) depends on $z$ and thus cannot be used in practice to guide data collection, it sheds light on the interpretation of the MMD by translating a qualitative property (it measures how different outputs from two initial distributions are) into practical considerations (how much data is required to tell them apart). Intuitively, initial distributions leading to a high MMD can be told apart with confidence given only a small number of output trajectories.

IV Experimental Results

The proposed tools enable a data-driven observability analysis of nonlinear stochastic systems. We first verify that adding noise preserves distributional observability for linear systems, as stated in Theorem 2. We then illustrate the real-valued quantification by computing it on the whole state space for a nonlinear Duffing oscillator, revealing a continuous increase as we get away from the class of indistinguishability. This also enables measuring the effects of both measurement and process noise on said classes, illustrating the results of Section II-C. Finally, we demonstrate on a Furuta pendulum how to test whether a sensor configuration distinguishes certain states, with implications on experiment design111Code to reproduce the results is available at https://github.com/Data-Science-in-Mechanical-Engineering/data-obs.git..

Remark 2

We have focused on discrete-time systems; yet, dynamical systems are often described in continuous-time and only the sensor measurements are discrete-time. This discrepancy is not problematic as the results of Section III still hold if the state is a continuous-time process.

In the following, we use $\alpha=0.05$ and a standard Gaussian kernel (10). A common heuristic to select the width parameter $\sigma$ is to pick the median pairwise distance between the data sets at hand [23]. It is not directly applicable here, however: our interpretation of the MMD as a measure of relative distinguishability (Section III-A) requires that $\sigma$ be the same for all points to be compared. Indeed, comparing different values of MMD is only meaningful if the underlying embeddings are all in the same RKHS, that is, use the same kernel function and parameter values. Therefore, we suggest the following meta-heuristic. We start by computing the above heuristic for each pair of data sets of interest. Then, we set $\sigma$ as the 10-quantile of all previous values. These low values correspond to distributions that are indistinguishable, or hardly distinguishable, so $\sigma$ has the correct order of magnitude for the noise of the system. We run all experiments with the fixed value of $\sigma$ given by this meta-heuristic, which yields $\sigma=5$, 1500, and the values given in Table I, respectively for the following sections. This leads to satisfying results in all our use cases, however, it is but an arbitrary choice; one should ensure $\sigma$ is set to a suitable order of magnitude or the problem might be ill-conditioned and all MMD values close to [math].

IV-A Influence of Noise on Observability

Theorem 2 states that two states are distributionally distinguishable for a linear stochastic system iff they are distinguishable for the corresponding deterministic system. This can be interpreted as follows: for linear systems, the presence of noise does not alter distinguishability between two states and, by extension, observability. We verify this numerically on the following linear system:

[TABLE]

where $\mathrm{d}W_{t}$ is a 2-dimensional independent Wiener process, $A_{0}=(3,\,3)$, $\omega=2$, $\Sigma=0.1I_{2}$, and $\epsilon_{t}\sim\mathcal{N}\left(0,0.01\right)$ is Gaussian measurement noise. The term $A_{0}\sin(\omega t)\mathrm{d}t+\Sigma\mathrm{d}W_{t}$ is a non-centered process perturbation. The nominal linear system $(A,C)$ is not observable. More specifically, the direction $C^{\perp}=\operatorname{span}\{(1,1)\}$ cannot be observed, meaning that the set of states indistinguishable from $x\in\mathbb{R}^{2}$ is $x+C^{\perp}$. This set is called the class of indistinguishability of $x$. We show experimentally that the non-centered noise in (17) preserves this class of indistinguishability. We emphasize that the process noise has nonzero mean over the simulation time.

We select a reference point $x_{\mathrm{a}}=(1.5,0.5)$ and simulate $n=30$ output trajectories for $2\text{\,}\mathrm{s}$ starting from $x_{\mathrm{a}}$, with time steps of length $\Delta t=$0.01\text{,}\mathrm{s}$$, yielding discrete trajectories of length $T=200$. These trajectories constitute a data set of samples from $\mathbb{P}_{x_{\mathrm{a}}}^{T}$.

We then build a grid of $50\times 50$ points and simulate $m=30$ output trajectories starting from each point $x_{\mathrm{b}}$ in the grid. We compute the MMD between the data set from $x_{\mathrm{a}}$ and the one for each $x_{\mathrm{b}}$. This yields a heatmap of relative distinguishability depicted in Figure 1. We observe that the MMD is minimal along the diagonal $x+C^{\perp}$, and increases as the query point gets farther. This shows that the farther from the diagonal a state is, the more distinguishable it is from $x_{\mathrm{a}}$. In fact, the test of Proposition 2 does not trigger along this diagonal (red dots), and triggers everywhere else, empirically identifying this set as the class of distributional indistinguishability of $x_{\mathrm{a}}$ given the output data. This class indeed corresponds to the class of deterministic indistinguishability of $x_{\mathrm{a}}$ for the unperturbed version of (17) (computed analytically), as predicted by Theorem 2.

IV-B Analyzing Observability in the State Space

The proposed approach can be used to determine empirically the class of (distributional) indistinguishability of any state for an arbitrary nonlinear system, and to observe the continuous increase in relative distinguishability as we get farther from that class. We illustrate this with an undamped, unforced Duffing oscillator, often used in nonlinear observer design [26, 27]:

[TABLE]

where $\mathrm{d}W_{1,t}$, $\mathrm{d}W_{2,t}$ are independent Wiener processes. The nominal system is Hamiltonian and conserves the quantity

[TABLE]

along trajectories. We consider the output map $Y_{t}=h(X_{t})+\epsilon_{t}$ with $\epsilon$ centered Gaussian measurement noise of variance $0.5$, and two noise configurations: $b_{1}=b_{2}=0.05$ (low process noise), and $b_{1}=b_{2}=0.5$ (high process noise). Given an arbitrary point $x_{\mathrm{a}}$, we aim to determine its class of indistinguishability from samples of the stochastic system. For this, we generate $n=50$ output trajectories of $1\text{\,}\mathrm{s}$ initialized in $x_{\mathrm{a}}$, and $m=50$ trajectories in each other point $x_{\mathrm{b}}$ on a $100\times 100$ grid, with $\Delta t=$0.001\text{,}\mathrm{s}$$ ($T=1000$). We then compute the empirical MMD (11) between the resulting data sets.

Fig. 2 shows the resulting MMD maps for two values of $x_{\mathrm{a}}$, and the two levels of process noise. For low process noise (a and b), the MMD increases as the discrepancy in $h$ in the initial states does; the states get more easily distinguishable. Additionally, the empirical class of indistinguishability (red squares) recovers the nominal system’s one, which indeed contains the deterministic trajectory starting from $x_{\mathrm{a}}$ (orange curve), where $h$ is constant. For higher levels of process noise, the empirical class is deformed, illustrating that Corollary 1 does not generalize without further assumptions. This continuous effect of the level of process noise on the classes highlights the interest of a continuous-valued metric of relative distinguishability; states where test triggers may still be difficult to distinguish.

Remark 3

For comparison, we also compute the nominal system’s empirical observability Gramian at $x_{\mathrm{a}}$ [15, Eq. 4]. If the null space $N$ of this empirical Gramian is $\{0\}$ in the limit $\epsilon\to 0$, then the system is weakly observable in $x_{\mathrm{a}}$ [15, Thm. 2]. Based on [15, Eq. (14)], we thus interpret $N$ as the “direction of weak unobservability” in $x_{\mathrm{a}}$. We approximate the assumption $\epsilon\to 0$ by taking $\epsilon=0.1$ and considering the eigenspace of the smaller eigenvalue. Fig. 2 shows the resulting approximation of $N$ by plotting a generating vector. We find that null space is the tangent space to the class of indisinguishability our test finds. In other words, empirical Gramians provide the local directions of indistinguishability, while our test recovers the whole class. Recovering the information on local observability of $x_{\mathrm{a}}$ could also be achieved with our test; e.g., by checking the test outcome in an $x_{\mathrm{b}}$-spherical shell centered in $x_{\mathrm{a}}$. Indeed, states very close to $x_{\mathrm{a}}$ will never trigger it due to finite-sample approximations, and should thus be excluded. We leave this connection to local observability for future work.

IV-C Analyzing Sensor Configurations on Hardware

When studying an experimental system, one can sometimes choose between possible sensor configurations whose benefits can be ambiguous. The system may be observable and allow building an observer to achieve further downstream tasks in some of these configurations but not in others. We now demonstrate how the proposed tools can help compare these configurations from measurements. For this, we collect hardware data on a Furuta pendulum: the Qube Servo 2 by Quanser (Fig. 3(a)). The pendulum consists of two arms, the first of which is actuated, with respective angular positions $(\theta_{1},\theta_{2})\in\mathbb{R}^{2}$ (in radians), where $\theta_{1}=0$ is arbitrary and $\theta_{2}=0$ corresponds to the upright position. The state is then $(\theta_{1},\theta_{2},\dot{\theta}_{1},\dot{\theta}_{2})$. We collect a set of output trajectories by initializing the system by hand close to $x_{\mathrm{a}}=(0,0,0,0)$ and $x_{\mathrm{b}}=(\pi/4,0,0,0)$, respectively. We measure both angles with encoders and corrupt these measurements with centered Gaussian noise of variance $0.001\text{\,}{\mathrm{rad}}^{2}$ independently on each dimension, in order to further complicate the experiment. The trajectories of $9.75\text{\,}\mathrm{s}$ are sampled with $\Delta t=$0.01\text{,}\mathrm{s}$$.

IV-C1 Choosing a Sensor Configuration

We consider three possible settings: $y=(\theta_{1},\theta_{2})$, $y=\theta_{1}$ and $y=\theta_{2}$. Our goal is to determine from the collected samples which of these configurations enable distinguishing the initial states, a necessary condition to provably reconstruct the whole state of the system.

For this, we run the kernel two-sample test of Theorem 2 ten times in all three settings, and present the results in Table I. The test triggers every time when $y=(\theta_{1},\theta_{2})$ or $y=\theta_{1}$, indicating that the initial distributions are distinguishable with these outputs with high confidence. In contrast, it never triggers when $y=\theta_{2}$; the data does not allow to distinguish between $x_{\mathrm{a}}$ and $x_{\mathrm{b}}$ when only measuring $\theta_{2}$. This generalizes to the nonlinear system the conclusion obtained by linearizing the dynamics; Kalman’s observability criterion states that the model is not locally observable around the origin for $y=\theta_{2}$, while it is in the other configurations.

Remark 4

The test evaluates the distinguishability of the initial distributions $\mu_{\mathrm{a}}$ and $\mu_{\mathrm{b}}$ whereas, in practice, one is usually concerned with the distinguishability of the initial states $x_{\mathrm{a}}$ and $x_{\mathrm{b}}$; yet, $\mu_{\mathrm{a}}$ and $\mu_{\mathrm{b}}$ are not Diracs. This could lead to a false interpretation of the test result. For example, if $x_{\mathrm{a}}=x_{\mathrm{b}}$ but $\mu_{\mathrm{a}}\neq\mu_{\mathrm{b}}$ (e.g., one is much more spread than the other), the output distributions may differ: the test would then trigger, although $x_{\mathrm{a}}$ and $x_{\mathrm{b}}$ are indistinguishable. To interpret the test result as distinguishability induced by the dynamics and not as an artifact introduced by non-ideal initial distributions, the support of the latter should be as narrow as possible around $x_{\mathrm{a}}$ and $x_{\mathrm{b}}$. This is theoretically sound due to the continuity of the MMD: if the support of $\mu_{\mathrm{a}}$ and $\mu_{\mathrm{b}}$ is sufficiently small then $\mathrm{MMD}[\mathbb{P}_{\mu_{\mathrm{a}}}^{T},\mathbb{P}_{\mu_{\mathrm{b}}}^{T}]$ is close to $\mathrm{MMD}[\mathbb{P}_{x_{\mathrm{a}}}^{T},\mathbb{P}_{x_{\mathrm{b}}}^{T}]$.

In practice, several methods can ensure a small spread, such as careful reinitializations of the system or discarding trajectories that are too far from the desired initial states. We initialize the system by hand, such that some randomness is always present, but stay in the same experimental conditions. To demonstrate robustness to this issue, we run our experiments multiple times by selecting only $n=m=40$ trajectories randomly from each initial distribution.

IV-C2 Consequences for Observer Behavior

Beyond choosing sensor configurations, the information provided by our test can also help in understanding the behavior of observers that fail to converge. An observer initialized in two distributionally indistinguishable initial states takes as input the same output distributions. Therefore, the output of the observer itself has the same distribution; its estimates are the same on the whole class of indistinguishability. In other words, the observer fails to distinguish the states.

We demonstrate this by implementing @iaciEKF EKF based on a model of the pendulum and taking $y$ as input. We run the EKF on our data set in the two settings $y=\theta_{1}$ (distinguishable) and $y=\theta_{2}$ (indistinguishable). The observer converges in the first case, but not in the second one (Fig. 3, top row). More interestingly, we estimate the evolution of the distribution of $\sin(\theta_{1})$ as estimated by the EKF (Fig. 3, bottom row). In the distinguishable case (Fig. 3(b)), the two output distributions of the EKF differ, whereas they appear to be the same in the indistinguishable case (Fig. 3(c)). This gives a precise meaning to the idea that an observer can estimate the class of indistinguishability of the initial state, which our test can determine.

V Conclusion and Outlook

In this paper, we present a method to assess the observability of stochastic systems from data without explicitly relying on a dynamics model; this is the first such method to the best of our knowledge.

We start by extending distributional distinguishability to arbitrary nonlinear stochastic systems and investigate its relationship to deterministic observability, showing that both notions are equivalent for certain classes of systems which includes linear systems. We then introduce the MMD and a finite-sample estimate thereof to quantify distinguishability of initial states from noisy output trajectories. We leverage the MMD for a statistical test of distinguishability, hereby interpreting its value as how much data is required to tell initial states apart with confidence. We illustrate the proposed tools and notions in simulation and on hardware. We reveal experimentally the relationship between noise and distinguishability, emphasizing the interest of a continuous metric, and showcase how such an approach could benefit observer design by allowing for a priori error analysis.

The theoretical analysis of distributional observability that we propose can be further pushed, e.g., by characterizing other classes of measurement noise that satisfy Assumption 2, or by finding sufficient conditions on the dynamics such that classes of process noise preserve distinguishability.

Acknowledgment

We thank Sebastian Giedyk for his help with the experiments.