Minimal Robust Positive Invariant Set

• The mRPI set is the minimal robust positively invariant set capturing the ultimate effect of bounded disturbances on discrete-time systems.
• It is computed via infinite Minkowski sums and approximated by finite truncations with explicit Hausdorff error bounds based on contraction metrics.
• mRPI sets are pivotal in robust control and MPC, enabling constraint satisfaction with scalable polyhedral representations and convex optimization methods.

A minimal robust positive invariant (mRPI) set is a fundamental construct in the analysis and synthesis of constrained control systems subject to bounded disturbances. For a given discrete-time system with Schur-stable dynamics and compact, convex disturbance set, the mRPI set is the smallest set that remains invariant under all admissible disturbances and captures the ultimate effect of these disturbances on the system state. Its computation and approximation are central to constraint satisfaction in model predictive control (MPC), robust control law design, and uncertainty propagation in autonomous systems. The mRPI set is characterized by properties of invariance, minimality, and dependence on system contraction and disturbance geometry.

1. Formal Definition and Properties

Consider the discrete-time linear system

$x_{k+1} = A x_k + w_k, \quad w_k \in \mathcal{W}$

where $A \in \mathbb{R}^{n \times n}$ is Schur-stable ($\rho(A) < 1$), and $\mathcal{W} \subset \mathbb{R}^n$ is a compact, convex disturbance set. The minimal robust positively invariant set is given by the infinite Minkowski sum:

$$\mathcal{E}_\infty = \bigoplus_{i=0}^\infty A^i \mathcal{W}$$

$\mathcal{E}_\infty$ is the unique smallest set such that $$A \mathcal{E}_\infty \oplus \mathcal{W} \subseteq \mathcal{E}_\infty$$. Any proper RPI set contains $\mathcal{E}_\infty$, but none can be strictly smaller. For systems with parametric uncertainty or time-varying dynamics $x_{k+1} = \phi_k x_k \oplus d_k$, an analogous recursive construction yields the mRPI set as the limit of iterated disturbance-reach sets (Dey et al., 2024).

2. Truncated Approximations and Explicit Error Bounds

Direct computation of $\mathcal{E}_\infty$ is infeasible for most high-dimensional or non-polyhedral systems. Instead, finite-horizon approximations via truncated Minkowski series,

$A \in \mathbb{R}^{n \times n}$0

are used in practice. The recently established explicit bound (Sun, 23 Nov 2025) quantifies the error between $A \in \mathbb{R}^{n \times n}$1 and $A \in \mathbb{R}^{n \times n}$2 in the Hausdorff metric:

$A \in \mathbb{R}^{n \times n}$3

where $A \in \mathbb{R}^{n \times n}$4 is the induced norm contraction factor and $A \in \mathbb{R}^{n \times n}$5. This bound is sharp, analytic, and requires no iterative set computations. The rate $A \in \mathbb{R}^{n \times n}$6 can be minimized via norm design (Euclidean, scaled, or Lyapunov norms), providing direct control over the approximation tightness and computational horizon for robust controller synthesis.

3. Polyhedral and Convex Representations

When $A \in \mathbb{R}^{n \times n}$7 is a polytope, $A \in \mathbb{R}^{n \times n}$8 and $A \in \mathbb{R}^{n \times n}$9 are (possibly nonconvex) polytopes or unions thereof. To obtain tractable representations suitable for control synthesis, one often restricts to a family of polytopic sets defined by fixed normal directions:

$\rho(A) < 1$0

where $\rho(A) < 1$1 is fixed and $\rho(A) < 1$2 is sought to minimize set size. The minimal RPI set within this family can be computed via a single LP whose solution $\rho(A) < 1$3 gives the minimal $\rho(A) < 1$4-template RPI set. This approach achieves strong scalability, and the uniqueness and minimality of $\rho(A) < 1$5 are established under standard conditions (Trodden, 2015). Choosing a richer set of normals improves approximation at the expense of computational cost.

4. Recursive and Fixed-Point Algorithms

The mRPI set admits a recursive construction via forward disturbance reachability:

• Initialize $\rho(A) < 1$6
• For $\rho(A) < 1$7: $\rho(A) < 1$8
• $\rho(A) < 1$9

The minimal set is approached monotonically as $\mathcal{W} \subset \mathbb{R}^n$0, and convex hulls can be taken at each stage to preserve polyhedral properties (Dey et al., 2024). For systems with polytopic uncertainty $\mathcal{W} \subset \mathbb{R}^n$1, the recursion generalizes with set-valued images under all vertices.

In practical computation, finite-time convergence is reached once $\mathcal{W} \subset \mathbb{R}^n$2, monitored via support function convergence. This enables automated stopping criteria and guarantees on coverage for constrained control pipelines.

5. Parameterization and Approximation Schemes

Approximating the mRPI set for use in optimization and synthesis tasks motivates explicit parameterizations and a priori error control. The rigid $\mathcal{W} \subset \mathbb{R}^n$3-RPI approximation—an outer set $\mathcal{W} \subset \mathbb{R}^n$4 with

$\mathcal{W} \subset \mathbb{R}^n$5

—can be constructed via parametrized reach-set sums and set-inclusion LMIs, as in data-driven and implicit set approaches. Techniques such as axis-aligned box convex hulls and polyhedral combinations admit linear or SOC constraints in performance-orientated optimization routines (Mulagaleti et al., 2023, Mulagaleti et al., 2021).

An example algorithm computes index, scaling, and remainder parameters $\mathcal{W} \subset \mathbb{R}^n$6 via SOCP, then optimizes over box-vertex parameters in a smooth NLP, yielding efficient and non-conservative disturbance set computation.

6. Impact of Norm Selection and Computational Considerations

The contraction rate $\mathcal{W} \subset \mathbb{R}^n$7 in the explicit truncation bound is norm-dependent. Systematic norm design (diagonal scaling, Lyapunov metrics) enables $\mathcal{W} \subset \mathbb{R}^n$8 arbitrarily close to $\mathcal{W} \subset \mathbb{R}^n$9, drastically reducing the truncation horizon $$\mathcal{E}_\infty = \bigoplus_{i=0}^\infty A^i \mathcal{W}$$0 for a given error. This design lever is both algorithmic and theoretical: it optimally shapes convergence rates, tightens tube-MPC tubes, and reduces conservatism relative to worst-case geometric bounds (Sun, 23 Nov 2025).

Efficient polyhedral set computation is greatly improved by one-step LP or SDP approaches over classical iterative Minkowski summation, which can become intractable due to facet explosion. Modern methods decouple the disturbance and state template selection, solve the fixed-point inclusion via convex optimization, and easily accommodate data-driven model uncertainty (Trodden, 2015, Mulagaleti et al., 2021).

7. Applications and Numerical Insights

Applications of mRPI set computation include robust tube-based MPC, output constraint satisfaction, decentralized control, and verification pipelines. Explicit remainder bounds allow direct synthesis of low-conservatism feasible regions and constraint tightening, outperforming legacy methods in speed and solution quality. Extensive numerical results confirm sharpness and scalability for moderate- to high-dimensional systems and in hybrid optimization contexts (Sun, 23 Nov 2025, Trodden, 2015, Mulagaleti et al., 2021).

Numerical experiments illustrate that norm shaping, polyhedral template enrichment, or advanced data-driven parameterization can improve approximation and computational tractability by orders of magnitude.

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References:

• "Explicit Bounds on the Hausdorff Distance for Truncated mRPI Sets via Norm-Dependent Contraction Rates" (Sun, 23 Nov 2025)
• "A One-step Approach to Computing a Polytopic Robust Positively Invariant Set" (Trodden, 2015)
• "Computation of Maximal Admissible Robust Positive Invariant Sets for Linear Systems with Parametric and Additive Uncertainties" (Dey et al., 2024)
• "Computation of safe disturbance sets using implicit RPI sets" (Mulagaleti et al., 2023)
• "Data-driven synthesis of Robust Invariant Sets and Controllers" (Mulagaleti et al., 2021)