Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Feedback control solves pseudoconvex optimal tracking problems in nonlinear dynamical systems

Tingli Hu, Sami Haddadin

TL;DR

This work introduces a closed-form Optimal Tracking Controller (OTC) that solves pseudoconvex tracking problems for time-variant nonlinear dynamics by integrating barrier-based interior-point optimization with a feedback control law. It proves exponential convergence of the closed-loop to time-varying local minimizers under mild conditions and provides a detailed complexity analysis showing OTC is well-suited for real-time high-dimensional settings, outperforming IPA and SQP in both speed and accuracy. The paper also extends the framework to state-constrained systems via a transformation that preserves optimality properties (SCLQR) and outlines an outlook for optimal trajectory planning and multi-agent cartography. Together, these results offer a causally deterministic optimization-control synthesis with practical real-time applicability and broad potential impact across robotics and engineered systems.

Abstract

Achieving optimality in controlling physical systems is a profound challenge across diverse scientific and engineering fields, spanning neuromechanics, biochemistry, autonomous systems, economics, and beyond. Traditional solutions, relying on time-consuming offline iterative algorithms, often yield limited insights into fundamental natural processes. In this work, we introduce a novel, causally deterministic approach, presenting the closed-form optimal tracking controller (OTC) that inherently solves pseudoconvex optimization problems in various fields. Through rigorous analysis and comprehensive numerical examples, we demonstrate OTC's capability of achieving both high accuracy and rapid response, even when facing high-dimensional and high-dynamical real-world problems. Notably, our OTC outperforms state-of-the-art methods by, e.g., solving a 1304-dimensional neuromechanics problem 1311 times faster or with 113 times higher accuracy. Most importantly, OTC embodies a causally deterministic system interpretation of optimality principles, providing a new and fundamental perspective of optimization in natural and artificial processes. We anticipate our work to be an important step towards establishing a general causally deterministic optimization theory for a broader spectrum of system and problem classes, promising advances in understanding optimality principles in complex systems.

Feedback control solves pseudoconvex optimal tracking problems in nonlinear dynamical systems

TL;DR

This work introduces a closed-form Optimal Tracking Controller (OTC) that solves pseudoconvex tracking problems for time-variant nonlinear dynamics by integrating barrier-based interior-point optimization with a feedback control law. It proves exponential convergence of the closed-loop to time-varying local minimizers under mild conditions and provides a detailed complexity analysis showing OTC is well-suited for real-time high-dimensional settings, outperforming IPA and SQP in both speed and accuracy. The paper also extends the framework to state-constrained systems via a transformation that preserves optimality properties (SCLQR) and outlines an outlook for optimal trajectory planning and multi-agent cartography. Together, these results offer a causally deterministic optimization-control synthesis with practical real-time applicability and broad potential impact across robotics and engineered systems.

Abstract

Achieving optimality in controlling physical systems is a profound challenge across diverse scientific and engineering fields, spanning neuromechanics, biochemistry, autonomous systems, economics, and beyond. Traditional solutions, relying on time-consuming offline iterative algorithms, often yield limited insights into fundamental natural processes. In this work, we introduce a novel, causally deterministic approach, presenting the closed-form optimal tracking controller (OTC) that inherently solves pseudoconvex optimization problems in various fields. Through rigorous analysis and comprehensive numerical examples, we demonstrate OTC's capability of achieving both high accuracy and rapid response, even when facing high-dimensional and high-dynamical real-world problems. Notably, our OTC outperforms state-of-the-art methods by, e.g., solving a 1304-dimensional neuromechanics problem 1311 times faster or with 113 times higher accuracy. Most importantly, OTC embodies a causally deterministic system interpretation of optimality principles, providing a new and fundamental perspective of optimization in natural and artificial processes. We anticipate our work to be an important step towards establishing a general causally deterministic optimization theory for a broader spectrum of system and problem classes, promising advances in understanding optimality principles in complex systems.

Paper Structure

This paper contains 27 sections, 19 theorems, 100 equations, 4 figures, 1 algorithm.

Table of Contents

  1. System and problem class
  2. Feedback controller for optimality
  3. Interior-point iteration
  4. Output tracking controller with full-state feedback
  5. Exponential convergence
  6. Optimal tracking controller
  7. Computational complexity
  8. Optimal tracking controller
  9. Time complexity
  10. Space complexity
  11. Interior-point algorithm
  12. Time complexity
  13. Space complexity
  14. Sequential quadratic programming
  15. Time complexity
  16. ...and 12 more sections

Key Result

Lemma 1

Let $\mathcal{E}^{ \mathbin{ \ooalign{$⊕$\cr$⊗$\cr}}}(t) \subset \mathbb{R} ^{ d_{\mathrm{x}} }$ be the set of all local minimizers of (eq:constrained optimization) at instant $t$, and $\mathcal{E}^{ \mathbin{ \ooalign{$⊕$\cr$⊗$\cr}}}_{\textup{B}}(t)$ the set of all local minimizers (eq:constrained

Figures (4)

  • Figure S1: The workflow of deriving the OTC by addressing three interrelated aspects: pseudoconvex optimization (cyan), feedback control (yellow), and optimal tracking control (green). Each aspect contributes through a series of deductions, lemmata, and theorems, ultimately revealing $\bm{\mu}$ as an optimal tracking controller.
  • Figure S2: Barrier function $\beta$ and its derivative-related functions $\xi$ and $\varXi$ with various values for parameter $p_{1}$.
  • Figure S3: Values of absolute accuracy metrics $E_{\text{OTC}}$ and $E_{\text{SQP}}$, cf. (\ref{['eq:error metrics for real-time accuracy']}), for all Examples R1--R3 of real-world problems. Each plot represents a numerical scenario characterized by the system's dimension $d_{\mathrm{x}}$ and the cost function's Hessian matrix ${\bm Q}$ (identity matrices ${\bm I}$, diagonal matrices, and three different types of symmetric and positive definite matrices).
  • Figure S4: Exemplary three-dimensional state-constrained linear-quadratic problem (\ref{['eq:state-constrained LQ problem']}) solved numerically and by SCLQR (\ref{['eq:state-constrained LQR']}). The difference between the resulting state trajectories, colored with blue and red, respectively, is negligible, being tested with two initial conditions ${\bm x} (0)$. (a) State trajectories in the 3-D coordinate system (left) and in the 2-D parametric coordinate system (right). (b) Time evolution of relevant quantities. Refer to texts for detailed description.

Theorems & Definitions (29)

  • Definition 1: Optimal tracking controller
  • Definition 2: Barrier function, augmented cost function
  • Lemma 1: Bound constraint as penalty
  • Definition 3: Quotient-linear convergence Kelley1995NocedalWri2006Marinov2009
  • Lemma 2: Chord method Kelley1995
  • Lemma 3
  • Lemma 4
  • Theorem 1: Interior-point iteration
  • Definition 4: Equilibrium points at instant $t$
  • Lemma 5: Local minimizers
  • ...and 19 more