[Paper Review] Delayed Recursive State and Input Reconstruction

Roshan A. Chavan, Harish J. Palanthandalam‐Madapusi|arXiv (Cornell University)|Sep 21, 2015

Fault Detection and Control Systems26 references3 citations

TL;DR

This paper proposes a delayed recursive filter for joint state and unknown input reconstruction in linear dynamical systems, using current measurements to estimate past states and inputs with a time delay. The key contribution is establishing necessary and sufficient conditions for filter convergence linked to system multivariable zeros, generalizing existing unbiased minimum-variance filters and relaxing the restrictive full-column-rank assumption on $CH$. The method applies to both square and non-square systems and ensures asymptotic convergence when system zeros are stable and non-minimum phase.

ABSTRACT

The unknown inputs in a dynamical system may represent unknown external drivers, input uncertainty, state uncertainty, or instrument faults and thus unknown-input reconstruction has several wide-spread applications. In this paper we consider delayed recursive reconstruction of states and unknown inputs for both square and non-square systems. That is, we develop filters that use current measurements to estimate past states and reconstruct past inputs. We further derive necessary and sufficient conditions for convergence of filter estimates and show that these convergence properties are related to multivariable zeros of the system. With the help of illustrative examples we highlight the key contributions of this paper in relation with the existing literature. Finally, we also show that existing unbiased minimum-variance filters are special cases of the proposed filters and as a consequence the convergence results in this paper also apply to existing unbiased minimum-variance filters.

Motivation & Objective

• Address the limitation of existing filters that require $CH$ to have full column rank for unknown input and state estimation.
• Enable recursive estimation of past states and inputs using current output measurements, introducing a reconstruction delay $r$.
• Establish necessary and sufficient conditions for filter convergence, linking convergence to the location of multivariable system zeros.
• Demonstrate that existing unbiased minimum-variance filters are special cases of the proposed framework when $r=0$.
• Provide a theoretical foundation for delayed input and state reconstruction applicable to non-square systems where $CH$ may not be full rank.

Proposed method

• Formulate a recursive filter that estimates states $\hat{x}_{k-r|k}$ and inputs $\hat{e}_{k-r-1|k}$ at a delayed time $k-r$ using measurements up to time $k$.
• Derive filter gain $L_k$ using a minimum-variance criterion, ensuring unbiasedness through the condition $L_k C H = H$.
• Use a prediction step to propagate the state estimate forward from $k-r-1$ to $k$ using the system dynamics $x_{k+1} = A x_k + H e_k$.
• Construct the input reconstruction as $\hat{e}_{k-r-1} = (CH)^{-1}(y_k - C \hat{x}_{k|k-1})$, valid when $CH$ is invertible.
• Introduce a delay $r$ to allow estimation of past states and inputs even when $CH$ is not full column rank, enabling broader system applicability.
• Apply the filter to both square and non-square systems, with convergence conditions derived based on the invariant zeros of the system.

Experimental results

Research questions

• RQ1 Under what conditions can past states and unknown inputs be recursively reconstructed from current output measurements?
• RQ2 How does the reconstruction delay $r$ affect the convergence and stability of the filter?
• RQ3 What is the relationship between filter convergence and the multivariable zeros of the system?
• RQ4 Can existing unbiased minimum-variance filters be recovered as special cases of the proposed delayed filter framework?
• RQ5 What are the necessary and sufficient conditions for unbiasedness and convergence in non-square systems with $CH$ not full column rank?

Key findings

• The filter provides unbiased estimates of past states and inputs when the system's multivariable zeros are stable and non-minimum phase, ensuring convergence.
• For square systems, the sufficient conditions for convergence are equivalent to the absence of non-minimum phase zeros, linking filter performance directly to system structural properties.
• The filter converges asymptotically when the system has stable invariant zeros; convergence fails if there are unstable or non-minimum phase zeros.
• The proposed filter generalizes existing unbiased minimum-variance filters, which are recovered as a special case when $r=0$ and $CH$ is invertible.
• The reconstruction delay $r$ is not user-selected but determined by the system's structural properties, with an upper bound derived based on system zeros.
• Numerical examples demonstrate successful input reconstruction in non-square systems where traditional methods fail due to $CH$ not being full rank.