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Explicit feedback synthesis for nonlinear robust model predictive control driven by quasi-interpolation

Siddhartha Ganguly, Debasish Chatterjee

TL;DR

This work presents QuIFS, a grid-based, one-shot explicit synthesis method for nonlinear robust minmax MPC that guarantees uniform approximation of the optimal feedback within any pre-specified tolerance $\\varepsilon$. It departs from conventional explicit MPC approaches by using a quasi-interpolation backbone and a Lipschitz extension to provide a computable policy over the whole state space with stability and recursive feasibility guarantees. The main contributions are the four-part Lipschitz-extension/interpolation algorithm, the uniform-error guarantees, and ISS-like stability results for the closed-loop under the approximate policy, applicable to both nonlinear and linear MPC. Numerical experiments on linear and nonlinear MPC demonstrate tight approximation to online receding-horizon trajectories and favorable performance relative to prior explicit approaches, with a clear pathway to scalable one-shot synthesis for constrained robust control in practice.

Abstract

We present QuIFS (Quasi-Interpolation driven Feedback Synthesis): an offline feedback synthesis algorithm for explicit nonlinear robust minmax model predictive control (MPC) problems with guaranteed quality of approximation. The underlying technique is driven by a particular type of grid-based quasi-interpolation scheme. The QuIFS algorithm departs drastically from conventional approximation algorithms that are employed in the MPC industry (in particular, it is neither based on multi-parametric programming tools and nor does it involve kernel methods), and the essence of its point of departure is encoded in the following challenge-answer approach: Given an error margin $\varepsilon>0$, compute in a single stroke a feasible feedback policy that is uniformly $\varepsilon$-close to the optimal MPC feedback policy for a given nonlinear system subjected to constraints and bounded uncertainties. Closed-loop stability and recursive feasibility under the approximate feedback policy are also established. We provide a library of numerical examples to illustrate our results.

Explicit feedback synthesis for nonlinear robust model predictive control driven by quasi-interpolation

TL;DR

This work presents QuIFS, a grid-based, one-shot explicit synthesis method for nonlinear robust minmax MPC that guarantees uniform approximation of the optimal feedback within any pre-specified tolerance . It departs from conventional explicit MPC approaches by using a quasi-interpolation backbone and a Lipschitz extension to provide a computable policy over the whole state space with stability and recursive feasibility guarantees. The main contributions are the four-part Lipschitz-extension/interpolation algorithm, the uniform-error guarantees, and ISS-like stability results for the closed-loop under the approximate policy, applicable to both nonlinear and linear MPC. Numerical experiments on linear and nonlinear MPC demonstrate tight approximation to online receding-horizon trajectories and favorable performance relative to prior explicit approaches, with a clear pathway to scalable one-shot synthesis for constrained robust control in practice.

Abstract

We present QuIFS (Quasi-Interpolation driven Feedback Synthesis): an offline feedback synthesis algorithm for explicit nonlinear robust minmax model predictive control (MPC) problems with guaranteed quality of approximation. The underlying technique is driven by a particular type of grid-based quasi-interpolation scheme. The QuIFS algorithm departs drastically from conventional approximation algorithms that are employed in the MPC industry (in particular, it is neither based on multi-parametric programming tools and nor does it involve kernel methods), and the essence of its point of departure is encoded in the following challenge-answer approach: Given an error margin , compute in a single stroke a feasible feedback policy that is uniformly -close to the optimal MPC feedback policy for a given nonlinear system subjected to constraints and bounded uncertainties. Closed-loop stability and recursive feasibility under the approximate feedback policy are also established. We provide a library of numerical examples to illustrate our results.
Paper Structure (18 sections, 5 theorems, 58 equations, 10 figures, 7 tables)

This paper contains 18 sections, 5 theorems, 58 equations, 10 figures, 7 tables.

Table of Contents

  1. Introduction
  2. Problem formulation
  3. From the baseline MPC to the approximation-ready MPC: motivation and formulation
  4. Review of quasi-interpolation
  5. Main result: Algorithm and theory
  6. Lipschitz extension algorithm
  7. Calculation of the approximation-ready policy at points
  8. Extension to $\mathbb{R}^d$
  9. Approximation
  10. Restriction
  11. Stability guarantees under the Lipschitz extension algorithm
  12. Discussion
  13. Linear MPC
  14. Numerical experiments
  15. Linear MPC
  16. ...and 3 more sections

Key Result

Theorem 3.1

ref:mazyabook Consider a Lipschitz continuous function $u:\mathbb{R}^d \longrightarrow \mathbb{R}$ of Lipschitz rank $L_0$, i.e., $u(\cdot)$ satisfies the inequality $\|u(x+y)-u(x)\| \leqslant L_0 \|y\|$ for all $x,y \in \mathbb{R}^d$. Let $h>0$ and suppose that $\aset[]{mh \suchthat m \in \mathbb{Z where $\Delta_0(\psi,\mathcal{D}) \coloneqq \mathcal{E}_0(\psi,\mathcal{D})\left\lVert u(\cdot) \ri

Figures (10)

  • Figure 1: A flowchart explaining the QuIFS algorithm.
  • Figure 2: The absolute error between ${\mu}_{0}^*(\cdot)$ and $\mu^{\dagger}_0(\cdot)$ in Example \ref{['exmp:lmpc_1']} without the employment of the Lipschitz extension algorithm on the left and with it on the right. Notice that the vertical axis is scaled by the factor of $10^{-3}$. While in the first case the preassigned error margin $\varepsilon$ was respected, this may not be typical, and the extension procedure should be carried out in order to conform to the theoretical guarantees.
  • Figure 3: Closed-loop state trajectories starting from $x(0) \coloneqq (1,4,1,-1)^{\top}$ for Example \ref{['exmp:lmpc_0']}. QuIFS performs better in terms of closeness to the online RHC trajectories in comparison to the trajectories reported in ref:summers-multires; see the state trajectories, and especially the errors in there.
  • Figure 4: The online receding horizon control and the solution obtained from QuIFS for Example \ref{['exmp:lmpc_0']}. See ref:summers-multires for a comparison.
  • Figure 5: The feedback ${\mu}_{0}^*(\cdot)$, the explicit feedback $\mu^{\dagger}_0(\cdot)$, and the absolute error between ${\mu}_{0}^*(\cdot)$ and $\mu^{\dagger}_0(\cdot)$ for Example \ref{['exmp:nmpc_3']} with $\varepsilon=0.05$, and $\bigl(h,\mathcal{D} \bigr)=(0.01,2)$.
  • ...and 5 more figures

Theorems & Definitions (22)

  • Remark 2.2: Motivation, robust formulation and uniform error
  • Theorem 3.1
  • Remark 3.2
  • Remark 3.3
  • Proposition 4.1
  • proof
  • Theorem 4.3
  • proof
  • Remark 4.4
  • Remark 4.5
  • ...and 12 more