A new $ν$-metric computational example for the diffusion equation with boundary control and point observation

Abstract: The $ν$-metric used in robust control is computed for control systems with parametric uncertainty, governed by a diffusion equation in a bounded one-dimensional spatial region with boundary control and point observation.

• The paper presents an explicit coprime factorization for a diffusion PDE, enabling a rigorous ν-metric analysis for robust stabilizability.
• It utilizes Hardy space techniques and the corona theorem to extend traditional ν-metric methods to infinite-dimensional systems.
• Numerical results, particularly a ν-metric value of approximately 0.12, illustrate the method’s practical effectiveness in handling parametric uncertainty.

Authoritative Summary of “A new $ν$-metric computational example for the diffusion equation with boundary control and point observation” (2512.11428)

Background and Motivation

Robust control theory is fundamentally concerned with ensuring stability of controllers for uncertain plants—the realistic scenario where system parameters may vary due to modeling errors or disturbances. The analysis leans heavily on algebraic factorization of transfer functions within an appropriate complex normed algebra $R$ (e.g., $H^\infty$, $A_+$) and utilizes metrics to quantify “closeness” of transfer functions; the $\nu$-metric, first introduced by Vinnicombe, ensures that robust stabilizability can be characterized via neighborhoods of the nominal plant in this metric space.

Initial developments of the $\nu$-metric relied upon normalised coprime factorizations, which for infinite-dimensional systems are not always constructible or computationally tractable. Sasane and collaborators refined the extension of the $\nu$-metric to circumvent the normalization requirement and broaden applicability to more general classes of plants (e.g., those described by PDEs, or with transfer functions in $H^\infty(D)$).

Mathematical Foundations and the $ν$-metric on $H^\infty(D)$

The transfer functions of interest reside in $H^\infty(D)$—the Banach algebra of bounded holomorphic functions on the unit disk. The field of fractions $F(H^\infty(D))$ contains all rational transfer functions, and $S(H^\infty(D))$ is the subset admitting coprime factorization, which is preserved under holomorphic functional operations.

The metric is operationalized via the following:

• Chordal Distance $\kappa(p_1, p_2)$: This is computed via the supremum over the maximal ideal space of a direct/inductive limit of $C^*$-algebras corresponding to concentric annuli in $D$, translating to the essential supremum norm on the boundary in $L^\infty(\mathbb{T})$.
• $ν$-Metric Definition: For plants $p_1, p_2 \in S(H^\infty(D))$ with coprime factorizations $p_1 = n_1/d_1$, $p_2 = n_2/d_2$, the metric $d(p_1, p_2)$ equals $\kappa(p_1, p_2)$ if an invertibility and winding number condition is satisfied for $n_1^* n_2 + d_1^* d_2$ in the relevant $C^*$-algebra; otherwise, $d(p_1, p_2) = 1$.

PDE System and Control Configuration

The work considers a one-dimensional diffusion equation with a Neumann boundary control input and point observation. The plant is specified by

$$\frac{\partial w}{\partial t}(x, t) = \frac{\partial^2 w}{\partial x^2}(x, t), \quad 0 < x < 1, t \ge 0$$

with boundary conditions:

$$\frac{\partial w}{\partial x}(0,t) = 0, \quad \frac{\partial w}{\partial x}(1,t) = u(t), \quad w(x, 0) = 0$$

and output:

$y(t) = w(a, t)$

with $a \in (0, 1)$. The transfer function from Laplace-domain input to output is:

$$p_a(s) = \frac{\cosh(a \sqrt{s})}{\sqrt{s} \sinh \sqrt{s}}$$

Computational Results: $ν$-Metric for Parametric Uncertainty

Coprime Factorization Construction

The transfer function $p_a$ is shown to admit a coprime factorization as:

$$n_a(s) = \frac{1}{\sqrt{s} + 1} \cdot \frac{\sinh(a \sqrt{s})}{\sinh \sqrt{s}}, \quad d_a(s) = \frac{\sqrt{s}}{\sqrt{s} + 1} \cdot \frac{\sinh(a \sqrt{s})}{\cosh(a \sqrt{s})}$$

The paper demonstrates, with explicit bounds via function properties in the angular domain $\Delta$ and analytic arguments, that $n_a, d_a \in H^\infty$ and are coprime, leveraging the corona theorem for uniform algebras. The continuity of $a \mapsto n_a, d_a$ is established, ensuring well-behaved metric properties under small parametric variation.

$ν$-Metric Computation

For nominal plant $p_a$ and perturbed plant $p_{\widetilde{a}}$ (where $\widetilde{a}$ is close to $a$):

• The invertibility and index condition on $n_a^* n_{\widetilde{a}} + d_a^* d_{\widetilde{a}}$ is verified for sufficiently small $|\widetilde{a} - a|$.
• The $\nu$-metric reduces to the chordal distance, computed explicitly (in the absence of analytic formulas) via numerical evaluation over the imaginary axis.

A concrete result is given for $a=1/2$, $\widetilde{a}=3/4$, where the metric $d_\nu(p_a, p_{\widetilde{a}}) \approx 0.12$ is reported, providing a quantitative measure of robust neighborhood size.

Key Numerical Results and Claims

Strong Technical Claims:

• The paper asserts the explicit construction of coprime factorizations for the class of transfer functions arising from the boundary-controlled diffusion equation.
• Continuity of the mapping $a \mapsto n_a, d_a$ ensures stability of the robust control properties under small parameter perturbations.

Numerical Results:

• For the given parametric variation, the $\nu$-metric’s computed value is approximately $0.12$, which quantifies how much uncertainty in the control location $a$ can be tolerated without loss of robust stabilizability.

Theoretical and Practical Implications

The results tighten the connection between abstract robust control theory—formerly limited largely to ODE or delay-differential models—and infinite-dimensional systems governed by PDEs. The established coprime factorizations and metric continuity imply that for physically relevant PDE plants, robust stabilizability can be quantitatively assessed in terms of allowable model perturbations in the spatial parameter $a$.

In practice, this opens the possibility for direct application of algebraic robust control techniques to distributed parameter systems where control and observation are spatially distributed—critical for applications in process engineering, thermal systems, and spatially diffusive models in biological or environmental systems.

From a theoretical perspective, the extension underlines the critical role of functional analytic machinery (Hardy algebras, $C^*$-algebraic direct/inductive limits) in bridging finite- and infinite-dimensional system theory.

Speculation on Future Developments

Potential future work includes:

• Generalization to multi-input/multi-output and higher-dimensional PDE control systems, exploring the computational feasibility of these algebraic techniques.
• Development of efficient numerical schemes for evaluating the $\nu$-metric in large classes of infinite-dimensional systems.
• Investigation into the impact of model uncertainty beyond parametric variations, such as structural or functional perturbations in the PDE itself.
• Integration of these results into robust controller synthesis algorithms for distributed systems.

Conclusion

This work provides a rigorous technical computation of the $\nu$-metric for a class of PDE-driven plants, establishing the feasibility of robust control analysis for boundary-controlled diffusion equations under point observation. Explicit coprime factorizations, continuity proofs, and numerical evaluation of the metric demonstrate both the practical computability and theoretical soundness of the approach, marking a significant extension of the robust control framework to infinite-dimensional systems.