Source-linked AI summary
A Unified Filter for Simultaneous Input and State Estimation of Linear Discrete-time Stochastic Systems
Sze Zheng Yong, Minghui Zhu, Emilio Frazzoli
TL;DR
Unknown inputs may invalidate standard Kalman assumptions, motivating simultaneous state and input estimation for stochastic linear systems. The paper proposes a unified unbiased minimum-variance filter without direct-feedthrough restrictions, derives detectability-linked stability conditions, and proves global optimality for one variant.
Problem
Existing filters do not provide a unified unbiased minimum-variance solution for simultaneous state and input estimation across rank-deficient direct-feedthrough cases.
Method
The paper develops two unified filter variants for linear discrete-time stochastic systems and analyzes their optimality, stability, and convergence through strong detectability conditions.
Results
ULISE is globally optimal over all linear state and input estimators, and simulations found it was the best estimator in all test trials.
Takeaways & Limitations
The unified framework covers arbitrary direct feedthrough matrices and connects stable unbiased estimation with strong detectability.
Takeaways & Limitations
Stability analysis relies on conditions involving matrices that depend on the state-estimation covariance, which may not be computable before applying ULISE.
Abstract
from arXiv · showhide
In this paper, we present a unified optimal and exponentially stable filter for linear discrete-time stochastic systems that simultaneously estimates the states and unknown inputs in an unbiased minimum-variance sense, without making any assumptions on the direct feedthrough matrix. We also derive input and state observability/detectability conditions, and analyze their connection to the convergence and stability of the estimator. We discuss two variations of the filter and their optimality and stability properties, and show that filters in the literature, including the Kalman filter, are special cases of the filter derived in this paper. Finally, illustrative examples are given to demonstrate the performance of the unified unbiased minimum-variance filter.
1 Introduction
The paper addresses simultaneous state and unknown-input estimation when unknown inputs are not well modeled as Gaussian noise. It develops a unified unbiased minimum-variance filter covering arbitrary direct feedthrough matrices and connects filter stability with detectability.
- Motivation: Unknown inputs can make standard Kalman filtering unreliable when they are not modeled as zero-mean Gaussian white noise.The motivation includes inaccessible vehicle inputs and applications such as collision avoidance, precipitation estimation, and fault diagnosis.
- Research gap: Existing work provides optimal simultaneous state-and-input filters only for restricted direct-feedthrough cases, leaving the rank-deficient unbiased minimum-variance problem open.The literature also lacks a unified minimum-variance unbiased filter that works across all direct-feedthrough cases.
- Stability and detectability: The paper derives stability conditions for linear time-varying systems and convergence conditions for linear time-invariant systems, relating exponential filter stability to strong detectability.The paper identifies stability and observability/detectability as connected aspects of unknown-input filtering.
- Contribution: The paper introduces a unified filter for unbiased minimum-variance estimation of states and unknown inputs without restrictions on the direct feedthrough matrix.The framework combines and extends ideas from prior filters and the Kalman filter.
2 Problem Statement
The paper considers linear discrete-time stochastic systems with known and unknown inputs, noisy measurements, and unrestricted direct feedthrough. It formulates the design of a globally optimal, stable, unbiased minimum-variance filter for simultaneous state and unknown-input estimation.
- System model: The system contains state, known input, unknown input, and measurement vectors, together with process and measurement noise.The state and unknown input are estimated from the measured output while the known system matrices are assumed available.
- Assumptions: Process and measurement noises are assumed mutually uncorrelated, zero-mean, white random signals with known covariance matrices.The initial state is also assumed independent of the process and measurement noises.
- Direct feedthrough: No restriction is imposed on the direct feedthrough matrix, which may be zero or rank deficient.The paper assumes the relevant input directions are linearly independent without requiring full column rank of the direct feedthrough matrix.
- Design objective: The design objective is a globally optimal and stable filter that simultaneously estimates states and unknown inputs in an unbiased minimum-variance manner.This objective defines the central problem addressed by the paper.
3 Preliminary Material
The paper transforms systems with arbitrary direct feedthrough into components that support unified input and state estimation. It then characterizes observability and detectability conditions connected to estimator convergence and stability.
- System Transformation: Singular value decomposition separates the direct-feedthrough matrix into full-rank and no-direct-feedthrough components.The transformed output equation contains separate unknown-input components, enabling filter designs for both cases.
- System Transformation: The transformation also decouples measurement-noise components and characterizes their covariance relationships.The decoupled noise terms are constructed to be uncorrelated, with covariances derived from the transformed measurement-noise covariance.
- Input and State Detectability: The paper connects zero-output state convergence to strong detectability and uses input and state observability or detectability to analyze unified-filter convergence and stability.For zero inputs and noise, the relevant implication is that y_k = 0 for all k ≥ 0 leads to x_k → 0 as k → ∞.
- Input and State Observability: Strong observability means that the initial state and a finite sequence of unknown inputs can be uniquely determined from measured outputs.For time-varying systems, the paper gives rank conditions involving observability and invertibility matrices; the time-invariant case has an equivalent characterization.
- Input and State Observability: The time-invariant observability theorem identifies rank conditions equivalent to input and state observability, with observability of (A, C) necessary.The proof relates full-rank conditions to the transformed observable subspace and associated matrix structures.
- Input and State Detectability: Strong detectability is a weaker condition than strong observability and is characterized for time-invariant systems by equivalent algebraic conditions.These conditions are equivalent to the system being minimum-phase, meaning its invariant zeros are stable.
4 Algorithms for Minimum-variance Unbiased Filter for Simultaneous Input and State Estimation
The paper develops a recursive three-step unified filter that estimates unknown inputs and system states simultaneously with unbiased minimum-variance properties, for arbitrary direct feedthrough matrices. Its ULISE variant is globally optimal, while both variants have stability guarantees under stated observability and detectability conditions.
- Filter variants: The recursive design consists of unknown-input estimation, time update, and measurement update steps.ULISE estimates d2,k−1 before the time update, whereas PLISE uses a different ordering and propagated state estimate.
- Unified filter: The filter decomposes unknown inputs and outputs into components, enabling simultaneous estimation for systems with arbitrary direct feedthrough matrices.The d1,k component is estimated using one projected output, while d2,k is estimated using the component with full-rank feedthrough.
- Optimality: Both ULISE and PLISE provide unbiased best linear estimates of unknown inputs and minimum-variance unbiased state estimates under the stated rank and initialization conditions.The construction chooses gain matrices to minimize state and input error covariances.
- ULISE guarantees: ULISE is globally optimal over all linear state and input estimators, and its initial state and input biases decay exponentially when the required uniform stabilizability and detectability conditions hold.The corresponding theorem also establishes bounded error covariance under uniform detectability.
- Time-invariant properties: For time-invariant systems, ULISE gain convergence to a unique stationary solution is linked to strong observability or detectability conditions.The paper states strong detectability as a condition for steady-state convergence and relates filter stability to strong detectability.
- Filter variants: PLISE is suboptimal relative to ULISE but retains bounded estimate errors and error covariances under time-invariant stability conditions.The paper explicitly contrasts PLISE’s suboptimal structure with its stability guarantees.
5 Filter Description and Analysis
The filter constructs unbiased state and unknown-input estimates, then chooses gains and least-squares estimators to minimize estimation-error variance. Its convergence and stability depend on observability, detectability, and conditions that differ across ULISE and PLISE.
- Unbiasedness: The state and unknown-input estimates are unbiased for all time when the gain and estimator matrices satisfy the stated rank and consistency conditions.These include M1,kΣk = I, M2,kC2,kG2,k−1 = I, and LkU1,k = 0.
- Unknown-input estimation: The unknown-input estimates use generalized least squares to obtain best linear unbiased estimates with minimum variance.The construction estimates both input components separately and combines their optimal estimates.
- Optimality: The unified unknown-input estimate has minimum variance when its component estimates each have minimum variance.Under Gaussian white-noise conditions, the resulting input estimate is also minimum-variance unbiased.
- State estimation: The state gain is obtained by minimizing the trace of the state-error covariance subject to the unbiasedness constraint LkU1,k = 0.The trace represents the sum of state-estimation error variances.
- Global optimality: Relaxing recursivity shows that ULISE is globally optimal over the class of linear unbiased state and input estimators.Its state update has the optimal form established in related work, and the input estimate is also proved globally optimal.
- Stability: ULISE stability follows under uniform detectability and bounded gains, whereas PLISE stability is established only for linear time-invariant systems.For ULISE, covariance convergence and exponential stability are tied to detectability; the time-varying PLISE proof does not use the same approach.
6 Connection to existing literature
The unified framework contains several existing filters as special cases and extends filters designed for systems without direct feedthrough. It also recovers the Kalman filter when unknown-input channels vanish.
- Special cases: ULISE and PLISE reduce to estimators closely related to filters in the existing literature under appropriate structural special cases.The relationships depend on the direct-feedthrough structure and the choice of input-estimation matrices.
- Full-row-rank feedthrough: When the direct-feedthrough matrix has full row rank, the ULISE and PLISE expressions become equivalent to corresponding existing filters.The equivalence conditions are stated through the rank of Hk and the resulting matrix identities.
- No direct feedthrough: For systems without direct feedthrough, ULISE and PLISE reduce to the filter derived in and generalize it to systems with direct feedthrough.The same relation extends, by implication, to filters in.
- Kalman-filter case: When Gk = 0 and Hk = 0, the unified filter reduces to the Kalman filter equations.The gain also reduces to the Kalman filter gain in this case.
7 Illustrative Examples
The examples evaluate simultaneous state and unknown-input estimation for fault identification and multi-vehicle tracking. The proposed filters estimate both quantities successfully, with ULISE generally outperforming alternatives and PLISE converging more slowly in the tracking example.
- Fault identification: The fault-identification example compares CYWZ, ULISE, PLISE, GDM, FSY, and YZF using steady-state traces of state and input error covariances.The comparison includes filters for both general and full-rank direct-feedthrough systems.
- Fault identification: The first three MVU estimators successfully estimate the states and unknown inputs in the first fault-identification system.Figure 1 displays actual states and inputs together with their estimates.
- Fault identification: ULISE is consistently the best-performing filter across the six fault-identification systems, while CYWZ performs comparably in this example.The reported comparison attributes the result to ULISE’s global optimality and finds little impact from replacing GLS with OLS for d2,k here.
- Fault identification: For full-rank direct-feedthrough systems H2 and H6, GDM and YZF perform as well as CYWZ and ULISE.FSY estimates unknown inputs better than PLISE but estimates system states worse than PLISE in both examples.
- Multi-vehicle tracking: Both ULISE and PLISE successfully estimate the states and unknown inputs in the multi-vehicle tracking example.The unknown inputs are the uncontrolled vehicle’s input and a time-varying measurement bias.
- Multi-vehicle tracking: PLISE has a slightly slower convergence rate for the trace of the unknown-input estimation-error covariance than ULISE.This difference is reported in the comparison associated with Figure 4.
8 Conclusion
The paper presents a unified filter for unbiased minimum-variance simultaneous state and unknown-input estimation without restricting the direct feedthrough matrix. It proposes two variants, establishes stability conditions, and identifies extensions to continuous-time, switched, and nonlinear systems as future work.
- The unified filter simultaneously estimates states and unknown inputs with unbiased minimum-variance performance without restricting the direct feedthrough matrix.
- Figure 4 compares the traces of state and unknown-input estimate error covariances over the first 0.25s.
- Two variants are proposed: PLISE uses the propagated state estimate, while ULISE uses the updated state estimate for unknown-input estimation.
- ULISE is globally optimal among linear unbiased state and input estimates, while PLISE is not globally optimal but performs reasonably well in simulations.
- Extending the unified filter to continuous-time, switched, and nonlinear systems is identified as future work.