SIFt-RLS: Subspace of Information Forgetting Recursive Least Squares

TL;DR

SIFt-RLS tackles covariance windup in Recursive Least Squares under nonuniform excitation by introducing a directional forgetting mechanism that acts only in the information subspace spanned by the regressor row space. It achieves this via a subspace decomposition of positive definite matrices, coupled with an SVD-based information filtering step to form a filtered regressor, and a subspace-aware update that forgets parallel to the information subspace while preserving orthogonal components. The paper provides explicit eigenvalue bounds for the information matrix without requiring persistent excitation, proves uniform Lyapunov stability of the estimation error (and global uniform exponential stability under PE), and demonstrates favorable tracking in a vector-measurement (MIMO) setting through a detailed numerical example. Overall, SIFt-RLS offers robust, computationally tractable parameter estimation when excitation is nonuniform, with practical implications for adaptive control and online system identification. $R_k$, $P_k$, and related matrices$ remain well-conditioned despite nonuniform excitation, thanks to information filtering and subspace forgetting.$

Abstract

This paper presents subspace of information forgetting recursive least squares (SIFt-RLS), a directional forgetting algorithm which, at each step, forgets only in row space of the regressor matrix, or the \textit{information subspace}. As a result, SIFt-RLS tracks parameters that are in excited directions while not changing parameter estimation in unexcited directions. It is shown that SIFt-RLS guarantees an upper and lower bound of the covariance matrix, without assumptions of persistent excitation, and explicit bounds are given. Furthermore, sufficient conditions are given for the uniform Lyapunov stability and global uniform exponential stability of parameter estimation error in SIFt-RLS when estimating fixed parameters without noise. SIFt-RLS is compared to other RLS algorithms from the literature in a numerical example without persistently exciting data.

Figures (5)

• Figure 1: Comparison of computation time to compute $P_{k+1}$ in SIFt-RLS, tested in MATLAB on an i7-6600U processor with 16 GB of RAM. Blue x indicates that, for the given values of $q_k$ and $n$, it is faster, on average, to compute $P_{k+1} \in {\mathbb R}^{n \times n}$ using the matrix inversion lemma via \ref{['eqn: SIFt Pkbar update']} and \ref{['eqn: SIFt Pk update']} than by direct inversion of $R_{k+1}$. However, further testing has shown that these results may differ greatly between different machines.
• Figure 2: Estimated parameters $\theta^1_k$, $\theta^2_k$, $\theta^3_k$, $\theta^4_k$, $\theta^\parallel_k$, and $\theta^\perp_k$ using SIFt-RLS (blue) and true parameters $\theta^1_{{\rm true},k}$, $\theta^2_{{\rm true},k}$, $\theta^3_{{\rm true},k}$, $\theta^4_{{\rm true},k}$, $\theta^\parallel_{{\rm true},k}$, and $\theta^\perp_{{\rm true},k}$ (black).
• Figure 3: Over $0 \le k \le 800$, $e_k^{12}$ and $e_k^{34}$ show the parameter estimation error in the subspaces $S^{34} \triangleq \{[x_1 \ x_2 \ 0 \ 0]^{\rm T} \colon x_1,x_2 \in {\mathbb R}\}$ and $S_{34} \triangleq \{[0 \ 0 \ x_3 \ x_4]^{\rm T} \colon x_3,x_4 \in {\mathbb R}\}$, respectively. Subspace $S^{12}$ is excited over $0 \le k < 400$ while $S^{34}$ is not, and vice versa over $400 \le k < 800$. Next, over $800 \le k \le 1200$, $e_k^{\parallel}$ and $e_k^{\perp}$ show the parameter estimation error in the subspaces $S^\parallel \triangleq \{[0 \ 2x \ x \ 0] \colon x \in {\mathbb R}\}$ and $S^\perp \triangleq \{[x_1 \ x \ -2 x \ x_4] \colon x_1,x,x_4 \in {\mathbb R}\}$, respectively. Subspace $S^\parallel$ is excited over $800 \le k \le 1200$ while $S^\perp$ is not.
• Figure 4: $\Delta_k^{12}$, $\Delta_k^{34}$, and $\Delta_k^\perp$ show the magnitude of the change in parameter estimates in the unexcited subspace over $0 \le k < 400$, $400 \le k < 800$, and $800 \le k \le 1200$, respectively.
• Figure 5: Spectral radius of the covariance matrix at step $k$.

Theorems & Definitions (11)

• Theorem 1: Properties of $A^{\parallel S}$ and $A^{\perp S}$
• Theorem 2: Uniqueness of $A^{\parallel S}$
• Theorem 3
• Theorem 4
• Lemma 1: Matrix Inversion Lemma
• Lemma 2: min-max theorem
• Lemma 3
• Lemma 4
• Lemma 5
• Lemma 6