Optimal Set-Membership Smoothing

TL;DR

This work addresses optimal Set-Membership Smoothing (SMSing) for non-stochastic Hidden Markov Models by adopting the uncertain-variable framework. It establishes an optimal SMSing framework including a backward smoothing equation and demonstrates its relation to optimal SMFing, yielding a closed-form CZ-based solution for linear systems and a nonlinear SMS for a specific class of systems. The approach produces the smallest guaranteed smoothed ranges $\llbracket \mathbf{x}_k|y_{0:T} \rrbracket$ and consistently tightens estimates compared with stochastic smoothing when noise statistics are unknown. Numerically, SMSing improves posterior diameters and point-estimation accuracy relative to SMFing and RTS smoothers, underscoring its practical value for non-stochastic settings with uncertain data.

Abstract

This article studies the Set-Membership Smoothing (SMSing) problem for non-stochastic Hidden Markov Models. By adopting the mathematical concept of uncertain variables, an optimal SMSing framework is established for the first time. This optimal framework reveals the principles of SMSing and the relationship between set-membership filtering and smoothing. Based on the design principles, we put forward two SMSing algorithms: one for linear systems with zonotopic constrained uncertainties, where the solution is given in a closed form, and the other for a class of nonlinear systems. Numerical simulations corroborate the effectiveness of our theoretical results.

Figures (6)

• Figure 1: Timelines of stochastic and set-membership smoothing methods. Various stochastic smoothing methods such as the RTS smoother, unscented RTS smoother, two-filter smoother, and particle smoother have been extensively studied over the past few decades. In contrast, very few studies were conducted on SMSing, and importantly, the knowledge on optimal SMSing is still lacking.
• Figure 2: Illustration of deriving the smoothed range $\llbracket\mathbf{x}_k|y_{0:T}\rrbracket$ at $k$ (the red hexagon), based on the smoothed range $\llbracket\mathbf{x}_{k+1}|y_{0:T}\rrbracket$ derived in the previous smoothing step $k+1$ (the darker red triangle on the RHS) and the posterior range $\llbracket {{{\mathbf{x}}_k}|{y_{0:k}}}\rrbracket$ at $k$ (the grey triangle on the LHS). For each $x_{k+1}$ in $\llbracket\mathbf{x}_{k+1}|y_{0:T}\rrbracket$, we obtain the preimage of $\{x_{k+1}\}$ under the map $f_{k,w_k}$ and take the union for all $w_k \in \llbracket{{\mathbf{w}}_k}\rrbracket$ to derive $\bigcup\nolimits_{{w_k} \in {\llbracket{\mathbf{w}}_k}\rrbracket} {f_{k,{w_k}}^{ - 1}(\{ {x_{k + 1}}\} )}$, i.e., the quadrangle with dashed edges; all such "quadrangles" form the lighter red triangle, i.e., $\bigcup\nolimits_{{w_k} \in {\llbracket{\mathbf{w}}_k}\rrbracket} f_{k,{w_k}}^{ - 1}(\llbracket {{{\mathbf{x}}_{k + 1}}|{y_{0:T}}}\rrbracket)$, which intersecting the posterior range $\llbracket {{{\mathbf{x}}_k}|{y_{0:k}}}\rrbracket$ gives the smoothed range $\llbracket\mathbf{x}_k|y_{0:T}\rrbracket$.
• Figure 3: Comparison between optimal SMFing and optimal SMSing: (a) interval hulls of the smoothed range $\llbracket{{{\mathbf{x}}_k}|{y_{0:T}}}\rrbracket$ (red rectangles) and posterior range $\llbracket{{{\mathbf{x}}_k}|{y_{0:k}}}\rrbracket$ (green rectangles) derived by optimal SMSing and optimal SMFing over $k\in [0,10]$, respectively, and the real state trajectory marked by dashed lines; (b) diameters (in the sense of $\infty$-norm) of $\llbracket{{{\mathbf{x}}_k}|{y_{0:k}}}\rrbracket$ and $\llbracket{{{\mathbf{x}}_k}|{y_{0:T}}}\rrbracket$ from the optimal SMS Algorithm \ref{['alg:linearsms']} and the optimal SMF cong2021rethinking over $k\in [0,50]$, averaged under 5000 simulation runs, which is shown by red curve (with star markers) and green curve (with circle markers), respectively.
• Figure 4: Average MSE (mean square error) of RTS smoothers with different parameters $q$ and $r$ in \ref{['eq_RTSpara']}. The MSE is computed for $k\in[0,50]$ and is averaged over 100 simulation runs for each parameter pair $(q, r)$.
• Figure 5: Comparison of point-estimation accuracy (in the sense of MSE) between the optimal SMS and the RTS smoother. The MSE is computed for $k\in [0,50]$ over 5000 times simulation runs, where the MSEs of the optimal SMS and the RTS smoother are shown by the red curve (with square markers) and the blue curve (with star markers), respectively.

Theorems & Definitions (11)

• Lemma 1: Law of Total range cong2021rethinking
• Lemma 2: Bayes' Rule for uncertain variables cong2021rethinking
• Definition 1: Optimal SMSing
• Lemma 3: Optimal SMFing cong2021rethinking
• Theorem 1: Optimal smoothing equation
• Remark 1
• Corollary 1: Optimal smoothing equation for linear systems
• Definition 2: cong2022stability
• Proposition 1
• Remark 2