Anti-windup design for internal model online constrained optimization

TL;DR

The paper tackles online constrained optimization with time-varying quadratic costs under nonnegativity constraints, formulating min_{x ≥ 0} f_t(x) with f_t(x) = 1/2 x^T A x + b(t)^T x + c(t) and A ≻ 0. It develops the Projected-Internal Model Anti-Windup (P-IMAW) gradient descent, which combines an internal-model-based controller to achieve unconstrained convergence with an anti-windup layer to manage projection nonlinearity. The synthesis proceeds in three steps: decoupling via eigendecomposition, IMP-based unconstrained design with robust LMIs, and anti-windup synthesis using Φ_I nonlinearity and LMI-based gain computation. Simulations show that P-IMAW yields superior tracking of the time-varying optimizer x*(t) and reduced windup compared to standard projected gradient methods and non-optimized variants, offering stability guarantees under the projected dynamics. This framework advances constrained online optimization by providing structured, certifiable performance in dynamic environments with partial knowledge of problem evolution.

Abstract

This paper proposes a novel algorithmic design procedure for online constrained optimization grounded in control-theoretic principles. By integrating the Internal Model Principle (IMP) with an anti-windup compensation mechanism, the proposed Projected-Internal Model Anti-Windup (P-IMAW) gradient descent exploits a partial knowledge of the temporal evolution of the cost function to enhance tracking performance. The algorithm is developed through a structured synthesis procedure: first, a robust controller leveraging the IMP ensures asymptotic convergence in the unconstrained setting. Second, an anti-windup augmentation guarantees stability and performance in the presence of the projection operator needed to satisfy the constraints. The effectiveness of the proposed approach is demonstrated through numerical simulations comparing it against other classical techniques.

Figures (8)

• Figure 1: General online projected gradient descent scheme.
• Figure 2: Projected Internal Model Anti-Windup (P-IMAW) gradient descent scheme proposed in this work. Here, ${\mathbold{C}}{\left(s\right)}$ denotes a robust internal model controller, $\mathop{\mathrm{proj}}\nolimits_{\geq 0}$ the projection onto the nonnegative orthant, and ${\mathbold{T}}$ the anti-windup compensation mechanism. The algorithm assumes access to an oracle providing the gradient ${\nabla f_t{\left({\mathbold{x}}\right)}}$.
• Figure 3: Equivalent block diagram of Fig. \ref{['fig:algGen']} after applying the substitutions \ref{['eq:gradf']}, ${\mathbold{C}}{\left(s\right)} = c{\left(s\right)} {\mathbold{I}}_n$ and ${\mathbold{T}} = \rho {\mathbold{I}}_n$.
• Figure 4: Intermediate equivalent block diagram of Fig. \ref{['fig:alg0']}, obtained by exploiting the eigendecomposition ${\mathbold{A}} = {\mathbold{V}} {\mathbold{\Lambda}} {\mathbold{V}}^\top$ and denoting by $\overline{{\mathbold{x}}}{\left(t\right)}$, $\overline{{\mathbold{b}}}{\left(t\right)}$, and $\overline{{\mathbold{w}}}{\left(t\right)}$ as in \ref{['eq:transfSignals']}.
• Figure 5: Final decoupled block diagram equivalent to Fig. \ref{['fig:alg0']} and \ref{['fig:alg1']} with the transformed signals $\overline{{\mathbold{x}}}{\left(t\right)}$, $\overline{{\mathbold{b}}}{\left(t\right)}$, and $\overline{{\mathbold{w}}}{\left(t\right)}$.

Theorems & Definitions (12)

• Remark 1
• Remark 2
• Remark 3
• Remark 4
• Remark 5
• Definition III.1
• Proposition 1
• proof
• Definition III.2
• Remark 6