⚡ electrical engineering

Observer Design for a Class of Systems Described by Differential-Algebraic Equations and Parameter Identification of an Unmeasured Disturbance

This paper proposes a novel observer design and parameter identification method for linear descriptor systems subject to unmatched disturbances, utilizing structural assumptions and a new parameterization technique to reconstruct state vectors and estimate unknown disturbance parameters, as validated by numerical simulations.

Original authors: Olga Oskina, Alexey Bobtsov

Published 2026-07-13

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Olga Oskina, Alexey Bobtsov

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to solve a giant, complex puzzle where some pieces are moving, some are stuck, and a mischievous ghost is constantly shaking the table. This paper is about building a super-smart detective (called an "observer") that can figure out exactly where every puzzle piece is, even when the ghost is hiding them, and then identify exactly what kind of ghost is causing the trouble.

The Puzzle: A Mix of Moving and Stuck Pieces
Most systems you learn about in school are like a car driving down a road: everything changes smoothly over time. But this paper deals with a special class of systems called "descriptor systems" (or differential-algebraic equations). Think of these as a system with two types of variables:

Dynamic pieces: These are like the car's speed; they change and move over time.
Algebraic pieces: These are like a rule that says, "If the car is at mile marker 5, the fuel gauge must read 20%." They don't move on their own; they are locked in a relationship with the moving pieces.

The authors are looking at a system where the "ghost" (an unknown disturbance) is messing with the puzzle. The goal is to build a detective that can guess the position of the moving pieces, figure out what the ghost is doing, and even identify the ghost's secret ID card (its unknown parameters).

The Detective's Strategy: Cleaning the Clutter
The paper argues that you can't just use standard detective tools for these mixed systems; they get confused by the "stuck" algebraic rules. The authors propose a specific set of rules to make the job possible:

The "Ghost-Free" Output: They assume the ghost doesn't directly touch the final report (the output signal) in a way that can't be traced back. If the ghost could magically change the report without leaving a trace in the system's rules, the detective would be stuck. The paper explicitly requires that the ghost's influence can be mathematically separated out.
Solving the Lock: They show that if the "stuck" rules are solvable (a specific mathematical condition called "rank"), you can rewrite the whole puzzle. You can express the stuck pieces in terms of the moving ones and the ghost. This turns the messy mixed puzzle into a cleaner, standard moving puzzle.

The Magic Trick: Turning a Ghost into a Line
Here is the paper's most creative move. Usually, if a ghost's behavior depends on unknown numbers in a complicated, curved way (nonlinear), it's a nightmare to find those numbers. The authors suggest a clever trick using "time delays."

Imagine the ghost is shouting a message. If you record the message, wait a second, record it again, wait another second, and record it a third time, you have three different versions of the same shout. The authors use these delayed versions to create a "mixing" procedure.

They take the reconstructed ghost signal and mix it with its delayed copies.
Through some algebraic magic (using something called the DREM method), they transform the complicated, curved relationship into a straight line (a linear regression).
Suddenly, the unknown numbers the ghost is hiding become easy to spot, just like finding a straight line on a graph.

The Simulation: Did It Work?
The authors didn't just dream this up; they built a computer model to test it. They created a fake system with 3 variables (two moving, one stuck) and a ghost that was shouting a mix of two exponential signals (like e raised to a power).

The Setup: They set the ghost's secret numbers to -0.1 and -0.3.
The Test: They ran a simulation where their detective tried to guess the moving parts and the ghost's identity.
The Result: The simulation showed that the detective successfully guessed the moving parts, and the error (the difference between the guess and the truth) shrank down to almost zero. The ghost's secret numbers were also identified correctly, but only after a short "waiting period."

Why the Waiting Period?
The paper is careful to note that you can't start guessing the ghost's ID immediately. The detective needs time to settle down, and the filters used to guess the "speed" of the signals (since you can't measure speed directly) need time to stop wobbling. The authors calculated a specific time, $T_{id}$ , which is the sum of the time for the observer to settle, the time for the filters to settle, and the time for the longest delay used in the mixing trick. Only after this time passes should you trust the ID guess.

The Bottom Line
This paper suggests a way to build a detective for a very specific, tricky type of system. It proves (through computer simulations) that if you follow their structural rules—specifically ensuring the ghost doesn't bypass the system's algebraic locks—you can:

Reconstruct the hidden moving parts of the system.
Reconstruct the ghost's signal.
Turn the ghost's complicated, unknown personality into a simple list of numbers you can calculate.

The authors conclude that once you know the ghost's ID, you could theoretically build a controller to cancel it out, but for now, they have only shown that the identification part works in their computer simulations. The method is a promising new tool for handling systems where the rules are as important as the motion.

Technical Summary: Observer Design and Parameter Identification for Descriptor Systems with Unmeasured Disturbances

Problem Statement
This paper addresses the challenge of state estimation and parameter identification for a class of linear time-invariant descriptor systems (also known as differential-algebraic equations) subject to unknown, unmatched disturbances. In many practical applications—ranging from electrical circuits to chemical processes—systems are governed by both dynamic differential equations and static algebraic constraints. A significant difficulty arises when only a subset of state variables is measurable, and the system is affected by external disturbances that are not directly measurable and do not enter the system through the same channel as the control input (unmatched). The primary objective is to construct an observer that estimates the dynamic components of the state vector and subsequently reconstructs the disturbance signal to identify its unknown constant parameters.

Methodology
The proposed solution follows a structured, multi-stage approach based on specific structural assumptions:

System Reduction: The authors assume the singular matrix $E$ has a specific block structure, allowing the system to be decomposed into dynamic ( $x_a$ ) and algebraic ( $x_b$ ) components. Under the assumption that the algebraic subsystem is solvable with respect to $x_b$ , the authors derive an explicit expression for $x_b$ in terms of $x_a$ , the known input $u$ , and the disturbance $\delta$ . Substituting this back into the dynamic equations yields a reduced-order ordinary differential equation (ODE) for $x_a$ .
Disturbance Elimination and Observer Design: A critical structural assumption is introduced: the reduced output equation must not have a direct feedthrough channel for the disturbance (i.e., $C_2 A_{22}^{-1} B_{22} = 0$ ). This allows the output equation to depend only on $x_a$ and $u$ . By differentiating the output and utilizing the condition that the disturbance channel in the state equation is non-zero ( $C_3 B_{2a} \neq 0$ ), the disturbance $\delta(t)$ is explicitly expressed in terms of the state, input, and output derivatives.
Substituting this expression back into the state equation eliminates the disturbance term, resulting in a disturbance-free ODE for $x_a$ . An unknown-input observer is then constructed for this reduced system. To avoid the practical difficulty of differentiating noisy signals, the observer is reformulated using a coordinate transformation that eliminates the need for direct output derivatives, and stable first-order filters are employed to estimate the necessary derivatives of the output and input signals.
Disturbance Reconstruction and Parameter Identification: Once the state estimate $\hat{x}_a$ is obtained, the disturbance signal $\hat{\delta}$ is reconstructed. The paper focuses on a specific class of disturbances that depend nonlinearly on unknown constant parameters but can be parameterized using known time-varying functions and time delays.
To identify these parameters, the authors propose a novel parameterization method. By introducing multiple time delays to the reconstructed disturbance signal, they form an extended regression system. Through a "mixing" procedure (utilizing the adjugate matrix and determinant of the regression matrix), they transform the original nonlinear dependence on parameters into a linear regression equation with respect to new, constant parameter combinations. These combinations are then estimated using standard adaptive algorithms, such as the gradient method or recursive least squares. Finally, the original unknown parameters are recovered algebraically from these estimates.

Key Contributions

Observer Design for Descriptor Systems: The paper provides a systematic method to design observers for linear descriptor systems with unmatched disturbances, avoiding the loss of information often associated with reducing descriptor systems to standard ODEs.
Structural Conditions for Disturbance Decoupling: The authors identify specific structural conditions (Assumptions 4 and 5) under which the disturbance can be algebraically eliminated from the state estimation dynamics, enabling the construction of a standard observer for the dynamic part of the system.
Nonlinear-to-Linear Parameterization: A significant contribution is the proposed method for handling disturbance signals that are nonlinear in their unknown parameters. By leveraging time delays and the Dynamic Regressor Extension and Mixing (DREM) concept, the authors successfully convert a nonlinear identification problem into a linear regression problem, facilitating the use of robust, standard estimation techniques.
Implementation Strategy: The work details practical implementation issues, including the use of filters to approximate derivatives and the calculation of a specific "start time" ( $T_{id}$ ) for the identification algorithm to ensure that observer and filter transients have decayed before parameter estimation begins.

Results
The effectiveness of the proposed procedures is demonstrated through a numerical simulation of an academic example involving a third-order descriptor system. The simulation results show:

State Estimation: The estimation error for the dynamic state vector $x_a$ converges to zero.
Disturbance Reconstruction: The reconstructed disturbance signal $\hat{\delta}$ accurately tracks the actual disturbance.
Parameter Identification: The unknown parameters of the disturbance ( $\theta_1, \theta_2$ ) converge to their true values after the initial transient period, validating the proposed linearization and identification scheme.

Significance and Claims
The paper claims that the proposed methodology offers a theoretically sound and practically applicable framework for observing and identifying disturbances in descriptor systems. The authors emphasize that their approach allows for the reconstruction of the disturbance signal and the identification of its parameters, which is a prerequisite for subsequent tasks such as disturbance compensation and the design of stabilizing controllers based on the internal model principle. The work highlights that by carefully analyzing the algebraic structure of the system, one can overcome the limitations of standard observer designs that rely on reduction to ordinary differential equations, thereby preserving full system information and improving estimation accuracy. The paper concludes modestly, stating that numerical simulations confirm the convergence of estimation errors to zero (for states) or a neighborhood of zero (for disturbance and parameters), under the introduced assumptions.

More like this