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Learning Quasi-LPV Models and Robust Control Invariant Sets with Reduced Conservativeness

Sampath Kumar Mulagaleti, Alberto Bemporad

TL;DR

The work addresses reliable data-driven control of nonlinear plants by jointly identifying a quasi-LPV model and synthesizing a robust controller that guarantees constraint satisfaction. It introduces a reduced-conservativeness control-oriented regularization derived from an uncertain LTI bound via interval bound propagation, and solves a nonlinear robust optimization with a differentiable algorithm integrated into a concurrent identification-synthesis framework. Key contributions include a tightened multiplicative-uncertainty formulation, an IBP-enabled bound computation, and a differentiable solver enabling end-to-end gradient-based optimization, with numerical validation showing smaller RCIs without sacrificing predictive performance. The approach advances data-driven constrained control by enabling less conservative yet reliable RCIs, potentially improving feasibility and performance in real-world nonlinear systems.

Abstract

We present an approach to identify a quasi Linear Parameter Varying (qLPV) model of a plant, with the qLPV model guaranteed to admit a robust control invariant (RCI) set. It builds upon the concurrent synthesis framework presented in [1], in which the requirement of existence of an RCI set is modeled as a control-oriented regularization. Here, we reduce the conservativeness of the approach by bounding the qLPV system with an uncertain LTI system, which we derive using bound propagation approaches. The resulting regularization function is the optimal value of a nonlinear robust optimization problem that we solve via a differentiable algorithm. We numerically demonstrate the benefits of the proposed approach over two benchmark approaches.

Learning Quasi-LPV Models and Robust Control Invariant Sets with Reduced Conservativeness

TL;DR

The work addresses reliable data-driven control of nonlinear plants by jointly identifying a quasi-LPV model and synthesizing a robust controller that guarantees constraint satisfaction. It introduces a reduced-conservativeness control-oriented regularization derived from an uncertain LTI bound via interval bound propagation, and solves a nonlinear robust optimization with a differentiable algorithm integrated into a concurrent identification-synthesis framework. Key contributions include a tightened multiplicative-uncertainty formulation, an IBP-enabled bound computation, and a differentiable solver enabling end-to-end gradient-based optimization, with numerical validation showing smaller RCIs without sacrificing predictive performance. The approach advances data-driven constrained control by enabling less conservative yet reliable RCIs, potentially improving feasibility and performance in real-world nonlinear systems.

Abstract

We present an approach to identify a quasi Linear Parameter Varying (qLPV) model of a plant, with the qLPV model guaranteed to admit a robust control invariant (RCI) set. It builds upon the concurrent synthesis framework presented in [1], in which the requirement of existence of an RCI set is modeled as a control-oriented regularization. Here, we reduce the conservativeness of the approach by bounding the qLPV system with an uncertain LTI system, which we derive using bound propagation approaches. The resulting regularization function is the optimal value of a nonlinear robust optimization problem that we solve via a differentiable algorithm. We numerically demonstrate the benefits of the proposed approach over two benchmark approaches.
Paper Structure (16 sections, 5 theorems, 36 equations, 3 figures, 2 algorithms)

This paper contains 16 sections, 5 theorems, 36 equations, 3 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Problem setup
  3. Concurrent synthesis framework
  4. Control-oriented regularization
  5. Characterization of $\mathbb{W}$
  6. Characterization of $X$
  7. Size of the RCI set
  8. Control-oriented regularization with reduced conservativeness
  9. RCI sets with reduced conservativeness
  10. Solving Problem \ref{['eq:new_r']}
  11. Interval bound propagation
  12. Concurrent synthesis algorithm
  13. Numerical example
  14. Concurrent identification
  15. Controller synthesis
  16. ...and 1 more sections

Key Result

Proposition 1

($i$) Suppose that the behavior of system eq:underlying_nonlinear is described by the model for some $\mathbb{V} \subset \mathbb{R}^{n_y}$, and there exists some set $\mathcal{E} \subseteq \mathbb{R}^{n_x}$ that satisfies $\hat{x}_0 - z_0 \in \mathcal{E} \ \Rightarrow \ \hat{x}_t - z_t \in \mathcal{E}$ for all $t> 0$ when eq:state_observer and eq:underlying_fake are excited by the same inp and de

Figures (3)

  • Figure 1: Output of Algorithm \ref{['alg:RCI']} for different $\zeta$, simulated using models $\Theta_{\zeta}$ identified by Algorithm \ref{['alg:concurrent']}. Observe that for chosen $\zeta$, we obtain $\mathrm{d}_{\zeta}$ significantly lesser than $\mathrm{d}_{\mathrm{seq}}=22.6924$ and $\mathrm{d}_{\mathrm{base}}=5.0936$.
  • Figure 2: Iterations of Algorithm \ref{['alg:RCI']} for $\Theta=\Theta_{0.07}$ and $\zeta=0.07.$
  • Figure 3: Closed-loop trajectories using tracking controller \ref{['eq:tracking_controller']}. The black region in top figure denotes boundaries of $Y$.

Theorems & Definitions (10)

  • Remark 1
  • Proposition 1
  • Proposition 2
  • Remark 2
  • Proposition 3
  • Proof 1
  • Corollary 1
  • Proof 2
  • Proposition 4
  • Proof 3