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Sampled-Data Boundary Feedback Control of 1-D Parabolic PDEs
Iasson Karafyllis, Miroslav Krstic
TL;DR
Linear parabolic PDEs lack a general robustness result for sampled-data boundary feedback under sufficiently small, nonperiodic sampling schedules. This paper applies two continuous-time designs in sample-and-hold form and establishes exponential stability estimates robust to schedule perturbations.
Problem
A general robustness result for continuous-time linear boundary feedback under ZOH and sufficiently small, not necessarily periodic sampling schedules is missing for linear parabolic PDEs.
Source: §1 · p. 1Method
The paper emulates the Reduced Model Design and Backstepping Design by applying their boundary feedback laws with zero-order hold.
Source: §1 · p. 2, §6 · p. 17Results
Both designs yield closed-loop exponential stability estimates in weighted 2-norms when the sampling period is sufficiently small, with robustness to sampling-schedule perturbations.
Source: §3 · p. 5, §6 · p. 17Takeaways & Limitations
For the studied linear parabolic systems, both boundary feedback designs retain exponential stability under sufficiently fast sample-and-hold implementation.
Source: §1 · p. 2, §6 · p. 17For Backstepping Design, the results cover Dirichlet actuation and L2 estimates; exponential estimates in other norms and the pointwise estimates needed for Neumann or Robin actuation are not known.
Source: §3 · p. 7
Abstract
from arXiv · showhide
The paper provides results for the application of boundary feedback control with Zero-Order-Hold (ZOH) to 1-D linear parabolic systems on bounded domains. It is shown that the continuous-time boundary feedback applied in a sample-and-hold fashion guarantees closed-loop exponential stability, provided that the sampling period is sufficiently small. Two different continuous-time feedback designs are considered: the reduced model design and the backstepping design. The obtained results provide stability estimates for weighted 2-norms of the state and robustness with respect to perturbations of the sampling schedule is guaranteed.
1. Introduction
The introduction identifies a gap in robust sampled-data boundary feedback results for linear parabolic PDEs. It presents two feedback designs and establishes exponential stability under sample-and-hold implementation when the sampling period is sufficiently small.
1. Introduction
- A general robustness result for continuous-time linear boundary feedback under ZOH and sufficiently small, not necessarily periodic sampling is missing for linear parabolic PDEs.The paper targets this gap in sampled-data control for infinite-dimensional systems.Source: §1 · p. 1
- Both the Reduced Model Design and Backstepping Design preserve closed-loop exponential stability under sample-and-hold control for a sufficiently small sampling period.The results give weighted 2-norm stability estimates and robustness to sampling-schedule perturbations.Source: §1 · p. 2
- The paper uses two boundary feedback designs: Reduced Model Design, proposed in and studied in, and Backstepping Design, developed in.Source: §1 · p. 2
- A general Lyapunov-like proof may not be available for every linear parabolic PDE because boundary inputs act through unbounded operators.The difference between continuous feedback and the held control action also involves unbounded operators.Source: §1 · p. 2
2. The notion of the Solution for a Parabolic System Under Boundary Sampled-Data Control
The paper formalizes solutions for parabolic PDEs driven by boundary inputs held constant between sampling times. The solution is unique and continuous in a weighted L2 state space, while continuity in L∞ is not generally guaranteed for Dirichlet actuation with changing inputs.
2. The notion of the Solution for a Parabolic System Under Boundary Sampled-Data Control
- The boundary input is held constant on each interval between consecutive sampling times, and the state remains continuous at sampling instants.Source: §2 · pp. 3–4
- For each admissible initial state and sampled input sequence, the system has a unique solution whose state profile is continuous in weighted L2.The solution is differentiable on open intervals between consecutive sampling times.Source: §2 · p. 4
- For Dirichlet actuation with a nonconstant applied-input sequence, the solution mapping is not generally continuous in L∞, making the state-space choice crucial.The stated continuity is in weighted L2.Source: §2 · p. 4
3. Stability Under Sampled-Data Implementation
The paper studies sample-and-hold implementations of reduced-model and backstepping boundary feedback designs for linear parabolic systems. For sufficiently small sampling periods, both designs yield exponential stability estimates, robust to perturbations in the sampling schedule.
3. Stability Under Sampled-Data Implementation
- The Reduced Model Design includes all unstable eigenvalues in a finite-dimensional system, whereas Backstepping Design affects the full Sturm–Liouville spectrum without assigning eigenvalues individually.Source: §3 · pp. 5, 7
- For sufficiently small sampling periods, the emulated Reduced Model Design yields exponential decay in a weighted state norm for every covered sampling sequence.The result permits schedule perturbations subject to the stated upper bound on sampling intervals.Source: §3 · p. 5
- For sufficiently small sampling periods, the emulated Backstepping Design yields exponential decay in the L2 norm for the stated Dirichlet-actuation system.The continuous-time backstepping feedback is first shown exponentially stable in L2.Source: §3 · pp. 5–6
- For Backstepping Design, exponential estimates in norms other than L2 are not known; extending the analysis to Neumann or Robin actuation would require pointwise solution estimates.The results considered here use Dirichlet actuation.Source: §3 · pp. 5, 7
- When the reduced-model dimension is large, designing its feedback gains can become sensitive to errors in eigenvalues and coefficients, especially for strongly unstable systems.Source: §3 · p. 7
4. Proofs of Main Results
The section develops proofs of stability results for sampled-data systems and the PDE setting, using mode decompositions and estimates on transformed states, control mismatch, and integral terms. For the results that require it, the estimates are closed by taking the sampling bound sufficiently small.
Proof of Lemma 4.1
- The finite-dimensional system is controllable, and any feedback gain making its closed-loop matrix Hurwitz yields exponential stability when the sampling period is sufficiently small.The lemma bounds the state and input by a decaying exponential for sampling schedules with intervals bounded by T; the proof selects T small enough to obtain the estimate.Source: §4 · pp. 7–9
- A change of coordinates converts the finite-dimensional system to a diagonal modal system with a single input, whose controllability follows from the distinct eigenvalues.The transformed system's controllability matrix is Vandermonde, so its determinant is nonzero when the eigenvalues are distinct.Source: §4 · p. 8
Proof of Theorem 3.1
- The proof of Theorem 3.1 expands the PDE state in the Sturm–Liouville eigenfunction basis and uses Parseval’s identity to relate the state norm to its modal coefficients.The proof also starts deriving the modal dynamics from the PDE and boundary conditions.Source: §4 · pp. 9–11
Proof of Theorem 3.2
- Combining the transformed-state and mismatch estimates yields the final bound after the sampling bound is chosen sufficiently small.The proof uses continuity of the relevant expression in the sampling bound to select a sufficiently small value.Source: §4 · p. 14
Discussion of the proof of Theorem 3.2
- The stability analysis is performed on the transformed system, where the difference between applied and nominal feedback acts as a boundary perturbation.This mismatch is defined as the signal v(t).Source: §4 · p. 15
- The derived equation for the mismatch signal does not yield a bound on its time derivative because it contains terms that cannot be bounded.The discussion identifies this as a limitation of that equation for estimating v̇(t).Source: §4 · p. 15
4. Proofs of Main Results
- Decomposition (4.37) is used to estimate signal w by bounding its time derivative and signal v.The passage also refers to estimating a term as part of estimating the signal magnitude.Source: §4 · p. 15
- An integral term can become arbitrarily small by controlling the magnitude of the associated quantity.Source: §4 · p. 15
5. Illustrative Example
The example applies sampled boundary feedback to a 1-D parabolic system and gives exponential stability estimates under different parameter and feedback-design conditions.
5. Illustrative Example
- For the reduced-model design, the sampled system satisfies the exponential estimate (5.10) when the sampling period is sufficiently small and the example’s parameter conditions hold.The sampling-period bound is conservative, and the convergence rate has conservative bounds stated in (5.6) and the accompanying inequalities.Source: §5 · p. 16
- The alternative feedback law (5.11), derived using Theorem 3.2, guarantees the estimate (5.10) for all positive p and q.The example specifies this law for every c ≥ 0.Source: §5 · p. 16
- The reduced-model feedback law (4.8) applies only to positive p and q satisfying inequality (5.6).For this example, p and q determine the system’s coefficients and the eigenvalues are given by λ_n = n^2π^2 − p − q.Source: §5 · pp. 15–16
6. Concluding Remarks
The paper establishes sampled-data exponential stability results for two continuous-time boundary-feedback designs and states a future research direction involving simultaneous space-and-time sampling.
6. Concluding Remarks
- For both the reduced-model and backstepping designs, sufficiently small sampling periods preserve closed-loop exponential stability, with weighted 2-norm estimates and robustness to sampling-schedule perturbations.The paper identifies development of boundary feedback designs for simultaneous space-and-time sampling, and sampled-data observers, as future work.Source: §6 · p. 17
Appendix
The appendix constructs solutions for the transformed system and establishes continuity and uniqueness using a series representation.
Appendix
- A transformation defines x from y and the time signal v, with the transformation’s spatial coefficient functions specified in the proof.The proof introduces y through a transformed PDE and boundary conditions before recovering x.Source: Appendix · p. 19
- The series used to define y is uniformly and absolutely convergent on each stated finite time interval away from zero.The proof then defines y by that formula and sets its initial condition.Source: Appendix · p. 21
- The constructed solution mapping is continuous in the weighted 2-norm, and uniqueness follows from the cited strong-solution result.The proof establishes continuity and then invokes Corollary 2.2 on page 106 of for uniqueness.Source: Appendix · p. 21