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Data-driven design of deadbeat observers for higher-order LTI systems
Aishwarya Bharti, Debasattam Pal
TL;DR
Higher-order difference-equation systems extend the scope of data-driven observer design beyond first-order formulations. The paper develops finite-data conditions and procedures for DBOs, and illustrates the approach with state estimation in the presence of unknown inputs.
Problem
Prior data-driven estimation work was limited to first-order representations and specific estimation problems, restricting the class of observers considered.
Source: §I · p. 1Method
The paper uses local reconstructibility and finite historical data to test for DBO existence, construct a consistent DBO, and parametrize DBOs.
Source: §I · p. 1, §III · p. 3Results
The paper establishes a necessary and sufficient finite-data test for global reconstructibility under a suitable informativity assumption and illustrates the approach with unknown-input state estimation.
Source: §I · p. 1, §VII · p. 6Takeaways & Limitations
The approach provides a finite-data route to DBO construction and parametrization for higher-order systems, illustrated for state estimation with unknown inputs.
Source: §VII · p. 6
Abstract
from arXiv · showhide
In this letter, we derive a necessary and sufficient condition for the existence of a deadbeat observer (DBO) for systems described by higher-order difference equations based on finite historical data. Furthermore, we develop an algorithm to construct a consistent DBO and to parametrise all DBOs directly from the historical data. Finally, we apply the proposed approach to the problem of state estimation in the presence of unknown inputs.
I. INTRODUCTION
The paper extends data-driven observer design from first-order formulations to higher-order difference-equation systems using finite historical data. It uses reconstructibility to ground finite-data tests for DBO existence and procedures for constructing and parametrizing observers.
I. INTRODUCTION
- Earlier data-driven estimation work was limited to first-order representations and specific estimation problems, restricting analysis to a class of admissible observers.The paper instead considers higher-order difference equations and estimation from measured variables to relevant variables.Source: §I · p. 1
- Selecting relevant and measured variables provides a unified platform for classical estimation problems, including state estimation, disturbance estimation, fault detection, and input reconstruction.The paper presents higher-order difference equations as the basis for this approach.Source: §I · p. 1
- Local reconstructibility is introduced as the bridge that lets finite data test DBO existence and supports construction from historical data.Reconstructibility is identified as equivalent to DBO existence [9, Proposition 4.2].Source: §I · p. 1
- A necessary and sufficient condition tests global reconstructibility from a finite historical dataset under a suitable informativity assumption.Source: §I · p. 1
- The approach constructs a consistent DBO and parametrizes the complete family of DBOs from finite historical data.Source: §I · p. 1
II. PRELIMINARIES
The preliminaries formulate the system as trajectories satisfying higher-order difference equations and define estimation of relevant variables from measured variables. They relate DBO existence to reconstructibility and establish an order-based bound on the convergence time.
II. PRELIMINARIES
- The system behaviour comprises trajectories partitioned into measured and relevant variables that satisfy the kernel equation R1(σ)w1 + R2(σ)w2 = 0.Source: §II · pp. 1–2
- A DBO estimates relevant variables from measured variables after a finite delay, while consistency requires true trajectories to satisfy the observer equation as well.Source: §II · p. 2
- Global reconstructibility of relevant variables from measured variables is equivalent to right monomicity and to the existence of a polynomial-form consistent DBO.Source: §II · p. 2
- When global reconstructibility holds, a DBO exists with convergence within n steps, where n is the system order.Source: §II · p. 2
III. LOCAL RECONSTRUCTIBILITY
The section connects finite-window local reconstructibility to global reconstructibility, providing the basis for testing the latter from finite data. For a system of order n, a 2n-length window gives an exact characterization.
III. LOCAL RECONSTRUCTIBILITY
- A restricted behaviour consists of trajectories truncated to a finite interval, providing the finite-dimensional setting for relating local and global reconstructibility.The section uses this relation as the basis for its finite-data approach.Source: §III · p. 3
- Global reconstructibility need not imply k-local N-reconstructibility for every k; the example fails the local condition at k = 0 and 1.The example illustrates why an arbitrary short window cannot stand in for the section’s bounded-threshold result.Source: §III · p. 3
- k-local N-reconstructibility is equivalent to a linear map from measured samples w1(0) through w1(k) that recovers w2(N).This local observer relation is the step that connects finite-window reconstructibility to observer construction.Source: §III · p. 3
- Eventual local N-reconstructibility is equivalent to global reconstructibility, while failure at one k alone does not establish global failure.The section supplies a threshold on k for which failure of the local test does imply failure of global N-reconstructibility.Source: §III · p. 3
- For a system of order n, global reconstructibility holds exactly when w2 is 2n-locally N-reconstructible for some N in {0, 1, ..., n}.The threshold 2n is a general bound; the section notes it is often weaker than necessary.Source: §III · p. 4
IV. DATA-DRIVEN TEST FOR RECONSTRUCTIBILITY
The section gives a finite-data test for global reconstructibility using historical trajectories and a suitable informativity condition. The test uses Hankel row spaces, with the known system order setting the finite window.
IV. DATA-DRIVEN TEST FOR RECONSTRUCTIBILITY
- The test assumes a known system order n and previously collected finite data for both measured and relevant variables.The paper notes that knowing n is restrictive but enables its data-driven solution.Source: §IV · p. 4
- The approach does not assume controllability; instead, it requires multiple trajectories to satisfy the suitable informativity condition.The section gives this as a distinction from most existing works.Source: §IV · p. 4
- Under Assumption 1, global reconstructibility holds exactly when, for some N in {0, ..., n}, the w2 Hankel block-row space is contained in the w1 Hankel row space.The test uses Hankel matrices of depth 2n.Source: §IV · p. 5
- Dualizing the row-space condition yields a local DBO, from which a global DBO can be obtained.The section points to Algorithm 1 for the global observer construction.Source: §IV · p. 5
- If the estimated order is below n, a satisfied test implies global reconstructibility but a failed test is inconclusive; if it is at least n, either outcome is decisive.The latter case allows a failed test to conclude failure of global reconstructibility.Source: §IV · p. 5
V. DEADBEAT OBSERVER DESIGN FROM FINITE DATA
Using a dataset satisfying (6), the section gives an algorithm that constructs a consistent DBO and parametrises all DBOs. Its construction combines a finite-data observer with a parametrisation of the remaining observers.
V. DEADBEAT OBSERVER DESIGN FROM FINITE DATA
- The model-based characterisation requires a DBO relation whose observer matrix Q(σ) is monomic and whose polynomial row is factored through the system kernel representation.The section uses this characterisation as the basis for constructing the data-based family of observers.Source: §V · p. 5
- The algorithm parametrises the full family by combining a constructed DBO with Laurent-polynomial matrices that yield a square monomic observer matrix.It uses a polynomial generator for the left kernel of the measured-data Hankel matrix, whose row space is specified in the algorithm.Source: §V · p. 5
- For a historic dataset satisfying (6), Algorithm 1 computes a consistent DBO and a parametrisation of all DBOs for the system.The construction step finds N* and polynomial coefficients satisfying a Hankel-matrix relation, then defines the observer polynomial.Source: §V · p. 5
VI. EXAMPLE
The example applies the data-driven algorithm to estimate a two-dimensional state from an output in a system with an unknown disturbance. It constructs a DBO and its parametrisation, while noting that measurement noise requires a different treatment.
VI. EXAMPLE
- The example illustrates state estimation of a two-dimensional state x from output y in a state-space system with an unknown input disturbance, framed as a DBO of x from y.The disturbance d(t) is chosen as a zero-mean, unit-variance Gaussian sequence.Source: §VI · pp. 5–6
- For N=1, the example constructs R_DBO(σ)=[σ −1−σ] as a DBO and obtains the same DBO across several datasets from different initial conditions.The reported Hankel ranks are rank(H4(W2))=2 and rank(H4(W1))=5; the dataset satisfies condition (6).Source: §VI · p. 6
- Every DBO in this example is obtained from the constructed observer using X(σ±1) with σX(σ±1) monomic.Because Hn(W1) is full row rank, L(σ)=0; the example also gives a second DBO, Σ2.Source: §VI · p. 6
- Measurement noise inflates Hankel-matrix ranks, so Theorem 15’s condition does not work directly; analysis of noisy historical data is left to future work.The authors suggest reformulating the condition as an error-in-variables problem and using structured low-rank approximation while preserving Hankel structure.Source: §VI · p. 6
VII. CONCLUSION
The paper studies deadbeat observers for higher-order LTI systems using historical data, establishing a finite-data existence condition and an algorithm for construction and parametrisation.
VII. CONCLUSION
- For higher-order LTI systems, the paper studies constructing DBOs for relevant variables from measured variables using historical data.Source: §VII · p. 6
- Local reconstructibility yields a necessary and sufficient finite-data condition for DBO existence, and a suitable data assumption supports an algorithm for obtaining a DBO and parametrising all DBOs.Source: §VII · p. 6
- The results are illustrated through state estimation with unknown inputs; future work includes lower-complexity non-consistent DBOs and causal DBOs for real-time implementation.Source: §VII · p. 6