Finite Sample Identification of Partially Observed Bilinear Dynamical Systems

TL;DR

The paper tackles the problem of identifying partially observed BLDS from a single trajectory and derives finite-sample guarantees for learning the BLDS's Markov-like parameters with a $\tilde{\mathcal{O}}(1/\sqrt{T})$ rate. It introduces a novel $(\mathcal{U},\kappa,\rho)$-uniform stability framework and constructs a nonlinear feature map $\tilde{\boldsymbol{u}}_t$ to form the Markov-like parameter tensor $\boldsymbol{G}$, estimated via least squares and recoverable to the state-space via Ho-Kalman. The analysis reveals an exponential-in-history dependence on the number of parameters, governed by history length $L$ and input dimension $p$, and provides a persistence-of-excitation result under $(4,2,\gamma)$-hypercontractive, isotropic inputs. Numerical experiments with synthetic data show that isotropic-sphere inputs can outperform Gaussian inputs for Markov-like parameter estimation and illustrate the impact of $L$ and system stability on learning accuracy.

Abstract

We consider the problem of learning a realization of a partially observed bilinear dynamical system (BLDS) from noisy input-output data. Given a single trajectory of input-output samples, we provide a finite time analysis for learning the system's Markov-like parameters, from which a balanced realization of the bilinear system can be obtained. Our bilinear system identification algorithm learns the system's Markov-like parameters by regressing the outputs to highly correlated, nonlinear, and heavy-tailed covariates. Moreover, the stability of BLDS depends on the sequence of inputs used to excite the system. These properties, unique to partially observed bilinear dynamical systems, pose significant challenges to the analysis of our algorithm for learning the unknown dynamics. We address these challenges and provide high probability error bounds on our identification algorithm under a uniform stability assumption. Our analysis provides insights into system theoretic quantities that affect learning accuracy and sample complexity. Lastly, we perform numerical experiments with synthetic data to reinforce these insights.

Figures (1)

• Figure 1: We plot the estimation error $\|{{\boldsymbol{G}} - \hat{{\boldsymbol{G}}}} \|_{\textup{op}}^2$ over different values of $T$, $L$, $\rho({\boldsymbol{A}}_0)$, $\rho({\boldsymbol{A}}_1)$, $\rho({\boldsymbol{A}}_2)$ while fixing $n{=}5$, $p{=}2$ and $m{=}2$. Figure \ref{['fig1a']} and Figure \ref{['fig1c']} correspond to estimation with Gaussian inputs $\{{\boldsymbol{u}}_t\}_{t\geq 0} \overset{\text{i.i.d.}}{\sim} \mathcal{N}(0, {\bm{I}}_p)$, whereas, Figure \ref{['fig1b']} and Figure \ref{['fig1d']} correspond to estimation with uniformly distributed inputs $\{{\boldsymbol{u}}_t\}_{t\geq 0} \overset{\text{i.i.d.}}{\sim} \mathrm{Unif}(\sqrt{p}\cdot \mathcal{S}^{d-1})$. Our plots show that the later input choice gives better estimation of ${\boldsymbol{G}}$ as compared to the first one. Moreover, the estimation error increases as $L$ and $\rho$ increases, whereas, it decreases as $T$ increases.

Theorems & Definitions (21)

• Definition 1: $( \mathcal{U}, \kappa, \rho)$-uniform-stability
• Theorem 1: Learning Markov-like parameters
• Remark 1: The BLDS parameter recovery.
• Definition 2: Hypercontractivity
• Theorem 2: Persistence of Excitation
• Proposition 1
• Proposition 2
• Lemma 1
• proof
• Lemma 2