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Successive design of backstepping observers for parabolic PDE-ODE systems and its duality to state feedback stabilization
Nicole Gehring
TL;DR
The paper addresses observer design for strictly feedforward parabolic PDE-ODE systems, where coupling structure determines the stabilization order. It proposes successive transformations that stabilize the distal ODE and then PDE error subsystems, and shows duality with multi-step state feedback design. The resulting cascade consists of exponentially stable ODE and PDE subsystems, while the duality explains the observer construction.
Problem
Observer design for strictly feedforward parabolic PDE-ODE systems must account for coupling structure when stabilizing PDE and ODE estimation errors from boundary measurements.
Method
The paper uses successive backstepping transformations: a virtual-measurement-based step for the distal ODE error and a Volterra step for the PDE error.
Results
The transformations map the error dynamics into a cascade of exponentially stable ODE and PDE subsystems and are dual to the corresponding multi-step state feedback design.
Takeaways & Limitations
The duality shows that the observer transformations are dual counterparts of controller transformations rather than ad hoc constructions.
Abstract
from arXiv · showhide
The paper introduces a successive backstepping observer design for strictly feedforward parabolic PDE-ODE systems, in which the coupling structure determines the order of error stabilization and the corresponding transformations. First, a transformation based on a virtual measurement stabilizes the ODE observer error subsystem, which is most distal from the measurement, while decoupling it from the PDE error state. Second, a Volterra integral transformation is employed to stabilize the PDE error subsystem and to map the overall error dynamics into a cascade of exponentially stable ODE and PDE subsystems. The design is shown to be dual to a recently proposed multi-step state feedback design for parabolic PDE-ODE systems in strict feedback form, thus explaining the structure of the presented observer design.
I. INTRODUCTION
The paper motivates successive observer design by the strict feedforward structure of parabolic PDE-ODE systems and its duality with multi-step state feedback design.
- Backstepping observers commonly exploit strict feedforward forms relative to boundary measurements, whereas bidirectionally coupled PDE-ODE systems are more challenging.
- The coupling structure of many bidirectionally coupled systems suggests successive rather than monolithic observer transformations.
- The paper proposes a two-step design that first stabilizes the distal ODE error through a virtual measurement and then stabilizes the PDE error using a Volterra transformation.
- The observer transformations and injection functions are dual to those in a multi-step state feedback design, explaining the observer structure.
II. PROBLEM STATEMENT
The problem concerns estimating PDE and ODE states from a boundary measurement in a bidirectionally interconnected parabolic PDE-ODE system under structural and detectability assumptions.
- The system couples an ODE subsystem with a parabolic PDE subsystem, with positive ordered diffusion coefficients and specified boundary matrices.
- The objective is to make the ODE and PDE estimation errors converge to zero using the boundary measurement.
- Detectability of (F, C) ensures that the unstable ODE component affects the PDE subsystem and can be detected through the measurement.
- The observer is a copy of the plant with injection functions depending only on known measurements and observer states.
- The strict feedforward error structure permits a multi-step design in which the ODE dynamics may depend on both states, while downstream PDE dynamics have restricted dependence.
III. SUCCESSIVE BACKSTEPPING OBSERVER DESIGN
The observer exploits strict feedforward error dynamics through successive transformations, stabilizing the distal ODE subsystem before the PDE subsystem.
- The first step stabilizes the ODE error subsystem most distal from the measurement by using a virtual measurement.
- A second standard Volterra transformation moves destabilizing effects to locations where injection functions can compensate them.
- The resulting observer error dynamics form a cascade of exponentially stable ODE and PDE subsystems.
A. Step 1: Stabilization of the ODE subsystem via a virtual measurement
The first design step replaces the unavailable virtual ODE measurement with a boundary-measurement-based transformation that stabilizes and decouples the ODE error subsystem.
- The first step targets the ODE subsystem, which is most distal from the boundary measurement.
- The virtual measurement represents the ODE's impact on the PDE subsystem and motivates the ODE stabilization procedure.
- The construction replaces the unavailable virtual measurement with the available boundary measurement without destroying the desired ODE stabilization.
- The transformation introduces a matrix function N(z) to remove PDE-error influence from the ODE dynamics and stabilize the ODE subsystem.
- Choosing the transformation conditions appropriately ensures exponential convergence of the transformed ODE error.
- After ODE stabilization and decoupling, stabilizing the PDE subsystem is sufficient for overall stability.
B. Step 2: Stabilization of the PDE subsystem
The second step uses a Volterra transformation to stabilize the PDE error after the first step has isolated an exponentially stable ODE subsystem. The transformed dynamics concentrate destabilizing terms in the measurement error, enabling compensating injections and exponential stability.
- PDE stabilization: The first step leaves an exponentially stable ODE error subsystem driven only by the PDE boundary error, reducing the remaining task to PDE stabilization.This decomposition permits a standard Volterra transformation in the second step.
- PDE stabilization: A Volterra integral transformation with kernel K(z, ζ) is defined on the triangular domain D to map the observer error dynamics into a target form.The kernel is associated with equations involving a strictly upper triangular matrix A0(z) and design parameter µ.
- PDE stabilization: The kernel equations admit a unique C2(D) solution, which also determines the non-zero elements of A0(z).The equations are traced back to earlier kernel constructions.
- PDE stabilization: After transformation, all destabilizing elements are contained in terms proportional to the measurement error ˜e(1, t), so simple compensation suffices.The ODE subsystem is already exponentially stable from the first step.
- PDE stabilization: With an appropriate choice of µ, the injection design guarantees convergence of the PDE state ˜e(z, t) to zero.The resulting observer error dynamics is exponentially stable in an appropriate norm.
C. State observer
The observer is exponentially stable in an appropriate norm when the finite-dimensional injection is chosen through a Hurwitz matrix condition and µ is positive. Invertibility of the transformations transfers this stability to the original error dynamics.
- State observer: Choosing L so that F + LC is Hurwitz and selecting µ > 0 makes the transformed observer error dynamics exponentially stable.The condition is stated for all initial ODE and PDE errors.
- State observer: Invertibility of the transformations implies exponential stability for the original observer error dynamics.The stability result therefore applies to the original errors ε(0) and e(z, 0).
- State observer: The observer injection functions depend only on the measurement error and therefore on known quantities.They are obtained from the two design steps and the transformation relations.
IV. DUALITY TO CONTROLLER DESIGN
The observer design is dual to a multi-step state feedback design. This duality explains why the observer is constructed successively according to the system’s coupling structure.
- Duality to controller design: The paper establishes the proposed observer design as the dual counterpart of a multi-step state feedback design.The duality also explains the successive observer construction.
- Duality to controller design: The strict feedforward structure of the primal system becomes a strict feedback structure in the dual system.This structural correspondence connects the observer and controller designs.
- Duality to controller design: The observer’s first step corresponds to the virtual control step of the state feedback design.The correspondence extends to transformations, kernel equations, injection functions, and state feedback.
A. Duality and dual system
The paper defines the dual system through adjoint operators and applies this construction to the parabolic PDE-ODE system. The resulting duality converts feedforward observability into feedback stabilizability and pairs observer injection with state feedback.
- Dual system: Dual systems are defined using the state, input, output, and adjoint-operator representation of a primal system.The definition is formulated through the corresponding function spaces and inner products.
- Dual system: For the PDE-ODE system, the state space is X = Rn0 × (L2([0, 1]))n with a weighted inner product for the distributed component.The weight Λ−1(z) is introduced for simplification.
- Dual system: The dual PDE coefficient is A∗(z) = Λ(z)AT(z)Λ−1(z), with the corresponding boundary relation involving Q1 and its transpose.The coefficient relation follows from applying the adjoint construction.
- Dual system: The primal system’s strict feedforward form with respect to its output produces a strict feedback form in the dual system with respect to its input.Because the primal system has no input, the dual system has no output.
- Dual system: Detectability of (F, C) implies stabilizability of (F∗, B∗), linking observer design conditions to dual state-feedback conditions.Stabilizing the dual system by state feedback is dual to stabilizing the observer error dynamics through injection functions.
- Dual system: The observer injections and dual state feedback are related by treating known quantities as measurements in the duality construction.The correspondence is developed using the system and error dynamics representations.
B. 1st design step
The first design step transforms the observer so the ODE subsystem is stabilized through a virtual measurement while the ODE state remains unchanged in the dual feedback design. The transformed observer and controller structures are dual.
- B. 1st design step: The dual feedback design treats x∗(0, t) as a virtual control input for the ODE subsystem, but replaces the unavailable state feedback with a state transformation.The transformation leaves the ODE state unchanged and is defined through an initial value problem for N∗(z).
- B. 1st design step: The transformation maps the system into a form where the ODE is exponentially stable when the boundary input x̄∗(0, t) is zero.
- B. 1st design step: This feedback-design step is dual to the observer step that uses a virtual measurement.
- B. 1st design step: The observer-controller duality follows from the transformation relation and the corresponding substitutions for the input, output, and system matrices.
- B. 1st design step: The adjoint-operator interpretation uses a simplified transformation relation because the observer maps are inverse transformations whereas the controller maps are not.
C. 2nd design step
The second design step uses a Volterra transformation to map the PDE dynamics into a stable target form, with its kernel equations matching those of the dual controller design. This establishes duality at the kernel and transformed-dynamics levels.
- C. 2nd design step: The Volterra integral transformation maps the PDE-ODE system into a form with a strictly lower triangular matrix A∗.
- C. 2nd design step: The inverse transformation is characterized by a kernel K_I(z, ζ), while stabilization uses a sufficiently large design parameter µ and a boundary condition on the transformed PDE state.
- C. 2nd design step: The transformations establish dual state representations for the observer and state-feedback systems in the shared state space.
- C. 2nd design step: The controller kernel equations become identical to the observer kernel equations when A∗_0(z) = −Λ(z)A^T_0(z) is imposed.
- C. 2nd design step: The observer error dynamics and closed-loop dynamics are dual, and the injection functions are dual to the state-feedback law.
V. CONCLUDING REMARKS
The paper presents successive observer design as a general principle for strictly feedforward systems, supported by its duality with multi-step state feedback. The strategy is applicable beyond the specific parabolic PDE-ODE setting, subject to the required feedforward structure.
- V. CONCLUDING REMARKS: The main result is a design principle for observers of strictly feedforward systems, developed here for parabolic PDE-ODE systems.
- V. CONCLUDING REMARKS: Observer-controller duality shows that the observer transformations are dual counterparts of the transformations used in multi-step state-feedback design.
- V. CONCLUDING REMARKS: The strategy can extend to other boundary-condition configurations, parabolic ODE-PDE-ODE systems, and hyperbolic distributed-parameter systems with finite-dimensional boundary dynamics.
- V. CONCLUDING REMARKS: The extension requires a strict feedforward form with respect to the boundary measurement; the stated system is not in that form for x(0, t) or an ODE-state measurement.