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Backstepping Design of Dynamic State Feedback Controllers for Parabolic Systems
Nicole Gehring, Benedikt Schwämmle, Abdurrahman Irscheid, Joachim Deutscher
TL;DR
Static-feedback designs for parabolic systems with differing diffusion coefficients face target-system restrictions, and infinite-dimensional dynamic feedback for general diffusion-reaction systems had not been available. This paper introduces a dynamic extension that homogenizes diffusion coefficients and applies backstepping to the extended system, yielding dynamic feedback; simulations illustrate stabilization and decoupled input-output behavior.
Problem
Static backstepping with differing diffusion coefficients requires target-system terms for kernel well-posedness, while infinite-dimensional dynamic state feedback for general diffusion-reaction systems was not available.
Source: §I · p. 1, B. Static State Feedback · p. 2Method
The design introduces a dynamic extension to homogenize diffusion coefficients, then applies backstepping-based static feedback to the extended system.
Source: C. Dynamic State Feedback · p. 2, §V · p. 5Results
Simulations show the state norm decaying to zero in a stabilization example and decoupled input-output behavior with dynamic feedback.
Source: A. Stabilization · p. 5, B. Decoupling · p. 5Takeaways & Limitations
The design supports boundary-coupled parabolic systems and broader closed-loop dynamics, including decoupled transfer behavior.
Source: D. Backstepping Controller · p. 5, §V · p. 6
Abstract
from arXiv · showhide
Recently, dynamic state feedback controllers that are based on dynamic extensions have been presented for heterodirectional hyperbolic systems. In this paper, a similar concept for the control of coupled diffusion-reaction systems is suggested. The introduction of a specific controller dynamics leads to homogenized diffusion coefficients for the extended system. Then, a backstepping-based static state feedback for the dynamically extended system is designed, which, overall, results in a dynamic state feedback. Such a design allows stabilizing a more general class of parabolic systems as well as assigning arbitrary closed-loop dynamics. This can be used, e.g., to achieve a decoupled input-output behavior, which is, in general, not possible with a static state feedback. A simulation example illustrates the results.
I. INTRODUCTION
The introduction situates backstepping for coupled parabolic systems and identifies the limits of static feedback. It motivates dynamic state feedback and introduces the diffusion-reaction setting considered.
I. INTRODUCTION
- Backstepping was extended from scalar parabolic systems to coupled systems with identical or differing diffusion coefficients and to spatially varying coefficients under Dirichlet or Robin conditions.The cited developments are –, following general overviews in and.Source: §I · p. 1
- Static feedback uses the current state, but differing diffusion coefficients impose target-system terms for kernel well-posedness; dynamic feedback adds controller dynamics.Dynamic controllers can support input-output decoupling, which is generally infeasible with static feedback; observer-based compensators instead use system output as their dynamic input.Source: §I · p. 1
- Infinite-dimensional dynamic state feedback was scarce, while delay-based backstepping and dynamic extensions had been developed for hyperbolic systems –.Dynamic extensions in homogenize transport delays and relax target-system restrictions.Source: §I · p. 1
- The paper extends dynamic-extension backstepping to general linear diffusion-reaction systems, allowing boundary coupling at the unactuated boundary and greater freedom in closed-loop dynamics.Unlike static designs, the target system needs no additional terms for kernel-equation well-posedness.Source: §I · p. 1
A. System Definition
- The study introduces the system and static-feedback limitations before considering a coupled diffusion-reaction model with Robin boundary conditions and boundary coupling matrices.The system has n state components on the unit interval, with positive, ordered diffusion coefficients and in-domain reaction coupling.Source: §II · p. 1, A. System Definition · p. 2
B. Static State Feedback
The static state-feedback design transforms the system with a Volterra integral map and chooses a decay parameter. Its kernel equations require structural and boundary conditions.
B. Static State Feedback
- For static state-feedback design, the approach follows, restricts Q0 in (1b) to a diagonal matrix, and transforms system (1) into a target system using a Volterra integral transformation.The kernel K(z, ζ) is defined on 0 ≤ ζ ≤ z ≤ 1.Source: B. Static State Feedback · p. 2
- The parameter μ > 0 prescribes the target system's decay rate, and the input v̄(t) is set to zero for exponential stabilization.Source: B. Static State Feedback · p. 2
- The kernel equations are well-posed and admit a unique piecewise C2 solution when A0(z) is strictly lower triangular, Q0 is diagonal, and additional boundary conditions are imposed; with equal diffusivities, A0(z) is unnecessary.Source: B. Static State Feedback · p. 2
C. Dynamic State Feedback
The dynamic-feedback design homogenizes diffusion coefficients through a dynamic extension, then applies backstepping to the extended system. It removes a restriction on boundary coupling and broadens the available closed-loop dynamics.
C. Dynamic State Feedback
- Because mutually different diffusion coefficients constrain static-feedback target dynamics, the design introduces dynamic feedback to homogenize them at the smallest coefficient λn.The controller dynamics arise from a dynamic extension of the original system.Source: C. Dynamic State Feedback · p. 2
- The four steps eliminate in-domain coupling, scale each component to diffusion coefficient λn, extend the scaled domains to [0, 1], and backstep on the extended system.The resulting feedback is dynamic for the original system.Source: C. Dynamic State Feedback · p. 2
- The design places no restriction on Q0, covers boundary-coupled systems, and permits general closed-loop dynamics that can yield decoupled input-output behavior.This contrasts with the diagonal-Q0 restriction in.Source: C. Dynamic State Feedback · p. 2
A. Preliminary Transformation
The preliminary transformation removes in-domain reaction coupling to simplify the dynamic extension. The transformed dynamics and boundary input prepare the system for the subsequent design.
A. Preliminary Transformation
- A Volterra integral transformation and a new input are introduced to eliminate the in-domain reaction coupling before adding the dynamic extension.The transformed system is presented after defining the kernel and new input.Source: A. Preliminary Transformation · p. 2
- The preliminary transformed dynamics do not include the stabilizing term −μx̃, because stabilization is not yet the objective.Source: A. Preliminary Transformation · p. 2
- The first component is defined on a scaled domain and complemented by controller state, while the second component remains unchanged; a new input is introduced at z = 1.Source: A. Preliminary Transformation · p. 3
- Kernel well-posedness still requires strictly lower-triangular A0(z) and additional boundary conditions, but the design imposes no restriction on Q0.Existence of a unique piecewise C2 solution is attributed to.Source: A. Preliminary Transformation · p. 3
B. Homogenization
The spatial domain for each state component is rescaled so every diffusion process has the smallest diffusion coefficient. The resulting components are defined on intervals of different lengths, with the last component’s domain unchanged.
B. Homogenization
- The rescaled state is defined by representing each original component on its new spatial coordinate, with the last component’s coordinate unchanged.The last component already has diffusion coefficient λ_n, so its domain remains unscaled.Source: B. Homogenization · p. 3
- The coordinate transformations are strictly positive and monotonically increasing, with inverse maps; only the last component’s transformed interval reaches 1.For each component, σ_i(0)=0 and 0<σ_i(1)≤1, with σ_n(1)=1.Source: B. Homogenization · p. 3
- Every rescaled state component has the same diffusion coefficient, λ_n, on its new spatial interval.The intervals need not have the same length.Source: B. Homogenization · p. 3
C. Dynamic Extension
A dynamic extension fills the portions of the rescaled domains that do not span [0,1]. The controller dynamics and boundary connections produce a composite state on the full domain with homogenized diffusion coefficients.
C. Dynamic Extension
- For the first n−1 components, extension states are introduced on the intervals beyond the rescaled state domains.The composite state combines each rescaled state with its extension, while the last component needs no extension.Source: C. Dynamic Extension · p. 3
- The composite state spans [0,1], and the chosen controller dynamics gives it the common diffusion coefficient λ_n.The special controller dynamics is selected to preserve homogenized diffusion across the composite domain.Source: C. Dynamic Extension · p. 3
- Boundary conditions connect the homogenized system and controller PDEs to ensure well-posedness of the resulting system.New inputs are introduced at the boundary z̄_i=1.Source: C. Dynamic Extension · p. 4
- The controller dynamics consists of PDEs with boundary conditions, and together with the homogenized system defines the dynamically extended system.The composite state χ is formed from the controller dynamics and homogenized system.Source: C. Dynamic Extension · p. 4
D. Backstepping Controller
Backstepping maps the dynamically extended system to a target system using a Volterra transformation and a designed control law. Under the theorem’s well-posedness, stability, and initial-condition conditions, the original state converges exponentially to zero.
D. Backstepping Controller
- A boundedly invertible Hopf-Cole-type transformation removes the advection term introduced by scaling the spatial domains.The advection term appears in the scaled system before this transformation is applied.Source: D. Backstepping Controller · p. 4
- A Volterra integral transformation with kernel L maps the dynamically extended system into the desired target system, with the control law implementing the mapping.The new input v̄(t) and target-system parameters are design parameters.Source: D. Backstepping Controller · p. 4
- Homogenized diffusion coefficients yield kernel equations with a unique piecewise C2 solution and no additional terms required for well-posedness.With diagonal B(z) and B0, the kernel equations separate into n independent scalar equations, allowing more freedom in choosing the target system.Source: D. Backstepping Controller · p. 4
- The original state converges exponentially to zero when the target system is well-posed and exponentially stable and the closed-loop initial conditions are compatible.The controller and system states must have compatible L2 initial conditions.Source: D. Backstepping Controller · pp. 4–5
- Diagonal B(z) and B0 produce decoupled scalar diffusion-reaction equations and decoupled input-output behavior, generally unavailable with static feedback.The target system’s input is v̄(t) and its output is y(t)=χ̄(0,t)=x(0,t).Source: D. Backstepping Controller · p. 5
IV. NUMERICAL RESULTS
The numerical section describes simulations for stabilization and input-output decoupling potential in a three-state diffusion-reaction system. It also specifies the plotted state norm and simulation approach, but the supplied passages do not report the plotted outcome.
IV. NUMERICAL RESULTS
- The numerical example uses a three-state system with Λ(z)=diag(3,2,1), A(z)=e^z·1_{3×3}, and Q1=0.1I; Q0 is specified later.Source: §IV · p. 5
- The simulations use MATLAB, finite differences for the system dynamics, and successive approximation for the dynamic controller’s kernel equations.The finite-difference scheme uses forward Euler time differences and second-order central spatial differences.Source: §IV · p. 5
- The simulations examine stabilization and input-output decoupling potential; the uncontrolled system is inherently unstable under the stated signs of A(z), Q0, and Q1.Source: §IV · p. 5
- Figure 2 plots the weighted L2 norm of the closed-loop state for v̄(z)=0, B0=0, and B(z)=−I.Source: Figure 2 · p. 5
A. Stabilization
With Q0 = −0.1 · 13×3, the chosen target-system parameters yield a decaying weighted L2 norm that converges to zero, confirming stabilization of system (1).
A. Stabilization
- With Q0 = −0.1 · 13×3, the target parameters are v̄(t) = 0, B0 = 0, and B(z) = −I.Source: A. Stabilization · p. 5
- The initial conditions are chosen to satisfy boundary conditions (1b) and (1c), ensuring a classical solution.Source: A. Stabilization · p. 5
- The weighted L2 norm of x(z, t) decays and converges to zero, confirming stabilization of the unstable system (1).Source: A. Stabilization · p. 5
B. Decoupling
For a unit step in the first input, dynamic feedback decouples the output components, unlike the compared static feedback, whose state coupling produces responses in the other components.
B. Decoupling
- Only x1(0, t) responds to a unit step in v̄1(t) under dynamic feedback; the other output components remain zero.Source: B. Decoupling · p. 5
- The static feedback does not decouple the outputs because the strictly lower triangular A0(z) couples the state components.Source: B. Decoupling · p. 5
- The comparison uses Q0 = −0.1I, zero initial state, and a unit step in v̄1(t), with v̄2(t) and v̄3(t) held at zero.Source: B. Decoupling · p. 5
V. CONCLUDING REMARKS
The paper designs dynamic state feedback for coupled diffusion-reaction systems using a dynamic extension and backstepping-based static feedback on the extended system. The design supports a broader class of boundary-coupled parabolic systems and more general closed-loop dynamics, including decoupled input-output behavior.
V. CONCLUDING REMARKS
- A dynamic extension and backstepping-based static state feedback for the extended system together produce a dynamic controller for coupled diffusion-reaction systems.Source: §V · pp. 5–6
- Figure 3 compares a unit step in v̄1(t), with v̄2(t) = v̄3(t) = 0, against the step response of x(0, t).Source: Figure 3 · p. 6
- The design has no restrictions on the form of Q0, allowing stabilization of a larger class of boundary-coupled parabolic systems.Source: §V · p. 6
- Greater freedom in target-system choice permits more general closed-loop dynamics, including decoupling the transfer behavior from v̄(t) to x(0, t).Source: §V · p. 6