Stable State Space SubSpace (S$^5$) Identification

TL;DR

This work introduces S$^5$, a closed-form, tuning-parameter-free stable state-space subspace identification method for IO-SS and N-SS with VAR(1) inputs. By extracting a Markovian state via a Sylvester equation and applying correlation-stable estimators, S$^5$ yields stable estimates of $A$ and $A_u$ with proven consistency under standard assumptions, while remaining computationally efficient even at high state dimensions. Theoretical results establish strong consistency for both the stabilized input and system matrices, and extensive simulations show superior accuracy and speed compared with existing stability-guaranteed methods, especially as model order grows. The approach offers a practically impactful solution for reliable, large-scale system identification where guaranteeing stability is essential.

Abstract

State space subspace algorithms for input-output systems have been widely applied but also have a reasonably well-developedasymptotic theory dealing with consistency. However, guaranteeing the stability of the estimated system matrix is a major issue. Existing stability-guaranteed algorithms are computationally expensive, require several tuning parameters, and scale badly to high state dimensions. Here, we develop a new algorithm that is closed-form and requires no tuning parameters. It is thus computationally cheap and scales easily to high state dimensions. We also prove its consistency under reasonable conditions.

Figures (6)

• Figure 1: Locations of the estimated complex poles for $n=5$: Poles for S$^5$, rand-FGM and SDP cluster closely around the true ones. Poles for rand-FGM and SDP almost overlap. However, a large number of poles for Orth lie on the real axis yielding a large discrepancy.
• Figure 2: Histograms of the estimated real dominating poles for $n=5$: We select LS estimates that are unstable. The '$*$' marks are the true dominating pole. All four algorithms guarantee system stability. Orth and rand-FGM tend to put the eigenvalues on the unit circle, yielding inaccurate system dynamics. SDP also has such a trend as $\bar{T}$ grows. However, S$^5$ does not.
• Figure 3: Histograms of the soft $H_\infty$ error $e_3=\sup_{\omega\in[0,3]} \bar{\sigma}(\hat{F}(\omega)-F(\omega))$ for $n=5$: The medians are indicated by "$*$" marks and the number at the upper right corner of each histogram. The upper and lower quartiles are indicated by "$|$" marks.
• Figure 4: $100$ estimated bode plots in frequency range $\omega\in[0,3]$ for $\bar{T}=1280$ for $n=5$: True bode plots are black. The bode plots for rand-FGM and SDP resemble closely to those of S$^5$ and are thus omitted. In the undisplayed range $\omega\in(3,\pi]$, the peak magnitudes are around $45$dB, $300$dB, $200$dB and $110$dB for S$^5$, Orth, rand-FGM and SDP, respectively.
• Figure 5: Histograms of the pole magnitudes in high dimensional simulations: The "$*$" marks are the true pole magnitude. The histograms for rand-FGM for $n=256$ and $n=1024$ use the $26$ and the $37$ converged estimates, respectively.