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State-Dependent Uncertainty Modeling in Robust Optimal Control Problems through Generalized Semi-Infinite Programming

J. Wehbeh, E. C. Kerrigan

TL;DR

This work addresses robust optimal control under state- and input-dependent uncertainty by formulating the problem as a generalized semi-infinite program ($GSIP$). It introduces a smoothing-based transformation to convert a $GSIP$ into an existence-constrained SIP ($ESIP$) and applies a local reduction algorithm to solve it efficiently, with guarantees under standard continuity and compactness assumptions. The method is validated on a nonlinear planar quadrotor, where the $ESIP$ matches the exact $GSIP$ solution and outperforms conservative SIPs that assume a uniform uncertainty bound. The results demonstrate a practical, provably exact approach for nonconservative robustness in nonlinear control, with potential for broader application beyond the quadrotor example.

Abstract

Generalized semi-infinite programs (generalized SIPs) are problems featuring a finite number of decision variables but an infinite number of constraints. They differ from standard SIPs in that their constraint set itself depends on the choice of the decision variable. Generalized SIPs can be used to model robust optimal control problems where the uncertainty itself is a function of the state or control input, allowing for a less conservative alternative to assuming a uniform uncertainty set over the entire decision space. In this work, we demonstrate how any generalized SIP can be converted to an existence-constrained SIP through a reformulation of the constraints and solved using a local reduction approach, which approximates the infinite constraint set by a finite number of scenarios. This transformation is then exploited to solve nonlinear robust optimal control problems with state-dependent uncertainties. We showcase our proposed approach on a planar quadrotor simulation where it recovers the true generalized SIP solution and outperforms a SIP-based approach with uniform uncertainty bounds.

State-Dependent Uncertainty Modeling in Robust Optimal Control Problems through Generalized Semi-Infinite Programming

TL;DR

This work addresses robust optimal control under state- and input-dependent uncertainty by formulating the problem as a generalized semi-infinite program (). It introduces a smoothing-based transformation to convert a into an existence-constrained SIP () and applies a local reduction algorithm to solve it efficiently, with guarantees under standard continuity and compactness assumptions. The method is validated on a nonlinear planar quadrotor, where the matches the exact solution and outperforms conservative SIPs that assume a uniform uncertainty bound. The results demonstrate a practical, provably exact approach for nonconservative robustness in nonlinear control, with potential for broader application beyond the quadrotor example.

Abstract

Generalized semi-infinite programs (generalized SIPs) are problems featuring a finite number of decision variables but an infinite number of constraints. They differ from standard SIPs in that their constraint set itself depends on the choice of the decision variable. Generalized SIPs can be used to model robust optimal control problems where the uncertainty itself is a function of the state or control input, allowing for a less conservative alternative to assuming a uniform uncertainty set over the entire decision space. In this work, we demonstrate how any generalized SIP can be converted to an existence-constrained SIP through a reformulation of the constraints and solved using a local reduction approach, which approximates the infinite constraint set by a finite number of scenarios. This transformation is then exploited to solve nonlinear robust optimal control problems with state-dependent uncertainties. We showcase our proposed approach on a planar quadrotor simulation where it recovers the true generalized SIP solution and outperforms a SIP-based approach with uniform uncertainty bounds.

Paper Structure

This paper contains 11 sections, 2 theorems, 41 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Background and Motivation
  3. State of the Art
  4. Contributions
  5. System Description
  6. Generalized SIP Equivalence
  7. Local Reduction on Generalized SIPs
  8. Proof of Convergence
  9. Local Reduction Implementation
  10. Simulation Results
  11. Conclusions

Key Result

Theorem 1

Any generalized SIP of the form of eq:gsip_base can be rewritten as the existence-constrained SIP for some $\epsilon > 0$ and where $\lambda \in \Lambda \subset \mathbb{R}^2$ is a member of the 2-dimensional simplex set with elements $\ell_1$ and $\ell_2$, such that The problem of eq:existence_sip will have the same solution $\bar{u}^*$ and associated objective value $\bar{J}(\bar{u}^*)$ as that

Figures (3)

  • Figure 1: Illustration of the quadrotor's horizontal position ($r$), altitude ($s$), and tilt angle ($\psi$) taken from wehbeh2024robust.
  • Figure 2: Quadrotor trajectories (in gray) for the solution obtained from the ESIP or RSIP. Final positions are shown in red and the target in green.
  • Figure 3: Quadrotor trajectories for the solution obtained from SIP 2. Colors are as in Figure \ref{['fig:sip2_sol']}.

Theorems & Definitions (4)

  • Theorem 1
  • proof
  • Theorem 2
  • proof