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Computation of Maximal Admissible Robust Positive Invariant Sets for Linear Systems with Parametric and Additive Uncertainties

Anchita Dey, Shubhendu Bhasin

TL;DR

This work tackles computing the maximal admissible robust positive invariant set ($\mathbf{S}$) for discrete-time linear time-varying systems with polytopic parametric uncertainties and additive disturbances under hard state and input constraints. It recasts the problem into an autonomous framework using a quadratically stabilizing feedback gain $K$, representing the uncertain dynamics as $x_{t+1}=\phi_t x_t+d_t$ with $\phi_t\in\Phi$ and $d_t\in\mathbf{D}$, and builds backward reachable sets from polyhedral descriptions. A vertex-based, finite-intersection algorithm is developed, with verifiable existence conditions via the convex hull of the minimal RPI set and a stopping criterion that guarantees finite termination under mild conditions; an LP-based precursory procedure further enhances tractability. The approach yields a convex MARPI, suitable as a robust MPC terminal set or for safe controller design, and is demonstrated on a two-dimensional example showing rapid convergence and a compact halfspace representation. Potential extensions include nonconvex constraints and enlarged precursor sets to broaden applicability in robust control contexts.

Abstract

In this paper, we address the problem of computing the maximal admissible robust positive invariant (MARPI) set for discrete-time linear time-varying systems with parametric uncertainties and additive disturbances. The system state and input are subjected to hard constraints, and the system parameters and the exogenous disturbance are assumed to belong to known convex polytopes. We provide necessary and sufficient conditions for the existence of the non-empty MARPI set, and explore relevant features of the set that lead to an efficient finite-time converging algorithm with a suitable stopping criterion. The analysis hinges on backward reachable sets defined using recursively computed halfspaces and the minimal RPI set. A numerical example is used to validate the theoretical development.

Computation of Maximal Admissible Robust Positive Invariant Sets for Linear Systems with Parametric and Additive Uncertainties

TL;DR

This work tackles computing the maximal admissible robust positive invariant set () for discrete-time linear time-varying systems with polytopic parametric uncertainties and additive disturbances under hard state and input constraints. It recasts the problem into an autonomous framework using a quadratically stabilizing feedback gain , representing the uncertain dynamics as with and , and builds backward reachable sets from polyhedral descriptions. A vertex-based, finite-intersection algorithm is developed, with verifiable existence conditions via the convex hull of the minimal RPI set and a stopping criterion that guarantees finite termination under mild conditions; an LP-based precursory procedure further enhances tractability. The approach yields a convex MARPI, suitable as a robust MPC terminal set or for safe controller design, and is demonstrated on a two-dimensional example showing rapid convergence and a compact halfspace representation. Potential extensions include nonconvex constraints and enlarged precursor sets to broaden applicability in robust control contexts.

Abstract

In this paper, we address the problem of computing the maximal admissible robust positive invariant (MARPI) set for discrete-time linear time-varying systems with parametric uncertainties and additive disturbances. The system state and input are subjected to hard constraints, and the system parameters and the exogenous disturbance are assumed to belong to known convex polytopes. We provide necessary and sufficient conditions for the existence of the non-empty MARPI set, and explore relevant features of the set that lead to an efficient finite-time converging algorithm with a suitable stopping criterion. The analysis hinges on backward reachable sets defined using recursively computed halfspaces and the minimal RPI set. A numerical example is used to validate the theoretical development.
Paper Structure (9 sections, 7 theorems, 14 equations, 1 figure, 1 algorithm)

This paper contains 9 sections, 7 theorems, 14 equations, 1 figure, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Reformulation of the objective
  4. Construction of the MARPI Set
  5. Tractable method to compute the MARPI Set
  6. Simulation Result
  7. Conclusion
  8. Proof of Lemma \ref{['lemma1']}
  9. Proof of Lemma \ref{['lemma2']}

Key Result

Theorem 1

The MARPI set $\mathbf{S}$ for invdyn is given by where each $\mathbf{S}_k$ is defined in noco_Sk. Also, $\mathbf{S}$ is convex.

Figures (1)

  • Figure 1: Here $x=[x_{(1)}\;\;x_{(2)} ]^\intercal$. (a) The sets $\hat{\mathbf{S}}_{(i)}$ converge to the MARPI set $\mathbf{S}$ in $2$ iterations. No new halfspaces are added in the $3^\text{rd}$ iteration. (b) The invariance property of $\mathbf{S}$ is verified since $\phi^{[j]}\mathbf{S}\oplus \mathbf{D}\subset \mathbf{S}$$\forall j\in\mathbb{I}_1^3$.

Theorems & Definitions (18)

  • Theorem 1
  • proof
  • Lemma 1
  • proof
  • Theorem 2
  • proof
  • Lemma 2
  • proof
  • Theorem 3
  • proof
  • ...and 8 more