Distance Between Stochastic Linear Systems

TL;DR

The paper introduces two new distance measures between stochastic LTI systems: a frequency-domain distance based on stereographic projection of uncertain frequency responses onto the Riemann sphere and a type-$q$ Wasserstein distance, and a time-domain distance based on the gap metric induced Wasserstein distance between perturbation distributions. It establishes upper and lower bounds for both domains, and proves that the frequency-domain distance never exceeds the time-domain distance, mirroring known results for deterministic $ u$-gap and gap metrics. The authors provide empirical demonstrations and discuss extensions to MIMO systems, along with practical computation via linear programming and push-forward distributions. The work lays groundwork for probabilistic robustness guarantees and controller-performance analysis under stochastic plant uncertainties, with open-source code accompanying the results.

Abstract

While the existing stochastic control theory is well equipped to handle dynamical systems with stochastic uncertainties, a paradigm shift using distance measure based decision making is required for the effective further exploration of the field. As a first step, a distance measure between two stochastic linear time invariant systems is proposed here, extending the existing distance metrics between deterministic linear dynamical systems. In the frequency domain, the proposed distance measure corresponds to the worst-case point-wise in frequency Wasserstein distance between distributions characterising the uncertainties using inverse stereographic projection on the Riemann sphere. For the time domain setting, the proposed distance corresponds to the gap metric induced type-q Wasserstein distance between the distributions characterising the uncertainty of plant models. Apart from providing lower and upper bounds for the proposed distance measures in both frequency and time domain settings, it is proved that the former never exceeds the latter. The proposed distance measures will facilitate the provision of probabilistic guarantees on system robustness and controller performances.

Figures (2)

• Figure 1: The Riemann sphere tangent to $\mathbb{C}$ is shown in shaded dark brown. An instance of the two stochastic systems $P_1$ and $P_2$ are depicted using their Nyquist plot in blue & red curves respectively. The distributions $\mathbb{P}_{P_{1}}$ and $\mathbb{P}_{P_{2}}$ characterizing the uncertainties of $P_1$ and $P_2$ at a frequency are shown as shaded blue & red colours with compact support sets $\mathcal{S}_{P_{1}}$ and $\mathcal{S}_{P_{2}}$ in $\mathbb{C}$ respectively. The corresponding inverse stereographic projections of the support sets onto the Riemann sphere are shown as sets $\mathcal{R}_{P_{1}}$ and $\mathcal{R}_{P_{2}}$ respectively. The known nominal models $\bar{P}_{1}, \bar{P}_{2}$ along with their projected counterparts on the Riemann sphere $\phi^{-1}(\bar{P}_{1}), \phi^{-1}(\bar{P}_{2})$ are also shown here.
• Figure 2: The chordal metric based type-1 Wasserstein distance $\hat{d}_1(P_1, P_2)$ between two systems $P_{1}(s) = \frac{1}{1+0.5s}$ and $P_{2}(s) = \frac{1}{(1+0.2s)(1+0.7s)}$ is shown here in blue colour. The upper bound using the support distance is given in red colour and its lower bound from Theorem \ref{['theorem_triangle_lower_bound_time_domain_empirical']} is shown in magenta colour. Quantities that are functions of frequency is given by solid lines and their respective maximum values are shown in dashed horizontal lines in the same colour.

Theorems & Definitions (39)

• Definition II.1
• Definition II.2
• Definition II.3
• Definition II.4
• Proposition II.1
• Remark III.1
• Definition III.1
• Proposition III.1
• proof
• Definition III.2