[Paper Review] Dynamics Compensation in Observation of Abstract Linear Systems

Hongyinping Feng, Xiaohui Wu+1 more|arXiv (Cornell University)|Sep 3, 2020

Stability and Controllability of Differential Equations23 references4 citations

TL;DR

This paper proposes a unified observer design framework for abstract cascade systems to solve sensor dynamics compensation and output regulation problems in linear systems. By formulating both issues within a common abstract linear system structure and using a Luenberger-like observer with tuning gain operators, the method ensures exponential convergence of the observation error without requiring target system design or Lyapunov function construction, as validated through ODEs with time-delay and an unstable heat equation with ODE sensor dynamics.

ABSTRACT

This is the second part of four series papers, aiming at the problem of sensor dynamics compensation for abstract linear systems. Two major issues are addressed. The first one is about the sensor dynamics compensation in system observation and the second one is on the disturbance dynamics compensation in output regulation for linear system. Both of them can be described by the problem of state observation for an abstract cascade system. We consider these two apparently different problems from the same abstract linear system point of view. A new scheme of the observer design for the abstract cascade system is developed and the exponential convergence of the observation error is established. It is shown that the error based observer design in the problem of output regulation can be converted into a sensor dynamics compensation problem by the well known regulator equations. As a result, a tracking error based observer for output regulation problem is designed by exploiting the developed method. As applications, the ordinary differential equations (ODEs) with output time-delay and an unstable heat equation with ODE sensor dynamics are fully investigated to validate the theoretical results. The numerical simulations for the unstable heat system are carried out to validate the proposed method visually.

Motivation & Objective

• To address sensor dynamics compensation in indirect system observation for abstract linear systems.
• To solve the disturbance dynamics compensation problem in output regulation using a unified observer framework.
• To develop a systematic, general-purpose observer design method applicable to both ODE and PDE systems.
• To avoid reliance on target system selection and Lyapunov function construction, common challenges in PDE backstepping.
• To validate the method on finite-dimensional systems with time-delay and infinite-dimensional unstable heat equations.

Proposed method

• Formulate the sensor dynamics and output regulation problems as a cascade system governed by abstract linear evolution equations in Hilbert spaces.
• Introduce a Luenberger-like observer with tuning gain operators F₁ and F₂ to estimate the full state from delayed or indirect measurements.
• Establish well-posedness and exponential convergence of the observation error using spectral analysis and invertible transformation via regulator equations.
• Use the regulator equations to convert the tracking error-based observer design into a sensor dynamics compensation problem, enabling unified treatment.
• Apply the method to ODEs with output time-delay and an unstable heat equation with ODE sensor dynamics, using finite-difference spatial discretization for numerical validation.
• Characterize observer gain existence via the observability of finite-dimensional projections and detectability of infinite-dimensional subsystems.

Experimental results

Research questions

• RQ1 Can sensor dynamics compensation and output regulation be treated as instances of a single abstract cascade system observer problem?
• RQ2 How can a Luenberger-like observer be designed for abstract cascade systems without requiring a priori target system selection?
• RQ3 What conditions ensure exponential convergence of the observation error in such systems?
• RQ4 Can the method be applied to systems with ODE sensor dynamics and infinite-dimensional plants, such as unstable heat equations?
• RQ5 How can the regulator equations be used to unify observer design for output regulation and sensor dynamics compensation?

Key findings

• The proposed observer design ensures exponential convergence of the observation error for the abstract cascade system, as proven via spectral analysis and invertible transformation.
• The method avoids the need for target system construction and Lyapunov function derivation, offering an alternative to the PDE backstepping method.
• For the unstable heat equation with ODE sensor dynamics, the observer gains were successfully computed and numerical simulations confirmed smooth, effective convergence of both state and output estimation errors.
• The finite-dimensional approximation of the system (N=1) was observable, and a Hurwitz gain matrix was obtained, ensuring stability of the error dynamics.
• The transformation via regulator equations successfully linked the output regulation problem to sensor dynamics compensation, enabling a unified observer design.
• Numerical results with time step 4×10⁻⁵ and space step 10⁻² showed effective and smooth convergence of both state and output estimation errors.