Elastically-Scaled Zonotopic Sets

• Elastically-scaled zonotopic sets are convex symmetric structures with adjustable generator magnitudes that encode directional geometry and uncertainty.
• They leverage Feret diameter measurements to reconstruct face lengths, providing accurate zonotopic approximations of symmetric convex random sets with explicit convergence bounds.
• In robust MPC, elastic scaling creates flexible tube representations that enhance feasibility and scalability by avoiding complex bilinear constraints.

Elastically-scaled zonotopic sets are zonotopes whose defining segment lengths or generator magnitudes are adjusted in a structured, non-uniform manner to encode geometry or uncertainty. In the geometric setting of symmetric convex random closed sets in $\mathbb R^2$, the relevant scaling is determined from the Feret diameter process, yielding a zonotopic representation whose face lengths are directionally calibrated by sampled widths (Rahmani et al., 2017). In the control setting of tube-based Model Predictive Control (MPC), the term denotes a sequence of zonotopes whose generators are scaled componentwise by time-varying nonnegative vectors along the prediction horizon, producing an “elastic tube” representation for robust constrained linear systems (Diaconescu et al., 24 Sep 2025).

1. Geometric and algebraic foundations

A random closed set (RACS) is a random variable taking values in the family of closed subsets of $\mathbb R^2$. The geometric framework in (Rahmani et al., 2017) restricts attention to random convex sets, meaning that almost every realization $X(\omega)$ is compact and convex, and then further to symmetric sets satisfying

$X=\breve X:=-X,$

or equivalently

$X=\frac12(X\oplus \breve X).$

This symmetry is structurally decisive because the Feret diameter of a convex set determines only its symmetrized version in general; for symmetric sets it determines the set itself (Rahmani et al., 2017).

A zonotope is a Minkowski sum of line segments,

$$X=\bigoplus_{i=1}^N \alpha_i S_{\theta_i}, \qquad \alpha_i\ge 0,\ \theta_i\in[0,\pi),$$

where $S_0=[-\tfrac12,\tfrac12]$ and $S_\theta$ is its rotation by angle $\theta$. In this representation, a zonotope is a compact convex symmetric polygon with at most $2N$ faces. For regular angular grids

$\mathbb R^2$0

the paper defines the class $\mathbb R^2$1 of 0-regular zonotopes, its rotated analogue $\mathbb R^2$2, and the union

$\mathbb R^2$3

the class of regular zonotopes with $\mathbb R^2$4 faces (Rahmani et al., 2017).

In the MPC formulation of (Diaconescu et al., 24 Sep 2025), the same object is expressed in generator form,

$\mathbb R^2$5

with center $\mathbb R^2$6, generator matrix $\mathbb R^2$7, and coefficient vector $\mathbb R^2$8. A scaled zonotope is obtained by componentwise scaling of the generators,

$\mathbb R^2$9

so that

$X(\omega)$0

This algebraic form makes explicit that zonotopic deformation can be encoded entirely by a nonnegative scaling vector.

2. Feret-diameter description and directional reconstruction

The Feret diameter, or caliper diameter, of a convex set $X(\omega)$1 in direction $X(\omega)$2 is the width of $X(\omega)$3 in that direction. In (Rahmani et al., 2017) it is written through the support function $X(\omega)$4 as

$X(\omega)$5

and for symmetric $X(\omega)$6, $X(\omega)$7 fully characterizes $X(\omega)$8. For a random convex set $X(\omega)$9, the family $X=\breve X:=-X,$0 is the Feret diameter random process; it is $X=\breve X:=-X,$1-periodic and continuous along each realization.

For a zonotope

$X=\breve X:=-X,$2

its Feret diameter satisfies the deterministic identity

$X=\breve X:=-X,$3

This identity is the central bridge between geometry and algebra: sampling $X=\breve X:=-X,$4 at the angles $X=\breve X:=-X,$5 produces a linear system in the face lengths $X=\breve X:=-X,$6 (Rahmani et al., 2017).

The paper’s “elastic scaling” interpretation follows from this reconstruction formula. In the approximation

$X=\breve X:=-X,$7

the face lengths are not chosen freely; they are the coefficients required to match the sampled caliper widths exactly. Under a global homothety $X=\breve X:=-X,$8, the Feret diameter scales linearly,

$X=\breve X:=-X,$9

and therefore

$X=\frac12(X\oplus \breve X).$0

This suggests that the elastic aspect, in the geometric sense, is a directional calibration mechanism rather than a uniform rescaling: each face length responds to the directional content of the observed width process (Rahmani et al., 2017).

3. Approximation of symmetric convex random closed sets by zonotopes

The main deterministic approximation result in (Rahmani et al., 2017) states that any symmetric convex set $X=\frac12(X\oplus \breve X).$1 can be approximated as closely as desired by a 0-regular zonotope constructed from Feret diameter samples. Define

$X=\frac12(X\oplus \breve X).$2

and

$X=\frac12(X\oplus \breve X).$3

Then the $X=\frac12(X\oplus \breve X).$4-approximation is

$X=\frac12(X\oplus \breve X).$5

This approximation has three stated properties. First, $X=\frac12(X\oplus \breve X).$6. Second, it is the unique element of $X=\frac12(X\oplus \breve X).$7 satisfying

$X=\frac12(X\oplus \breve X).$8

Third, it converges to $X=\frac12(X\oplus \breve X).$9 in Hausdorff distance with the explicit bound

$$X=\bigoplus_{i=1}^N \alpha_i S_{\theta_i}, \qquad \alpha_i\ge 0,\ \theta_i\in[0,\pi),$$0

hence

$$X=\bigoplus_{i=1}^N \alpha_i S_{\theta_i}, \qquad \alpha_i\ge 0,\ \theta_i\in[0,\pi),$$1

In the random setting, the same construction is applied realization-wise, producing a random zonotope approximation with consistency as $$X=\bigoplus_{i=1}^N \alpha_i S_{\theta_i}, \qquad \alpha_i\ge 0,\ \theta_i\in[0,\pi),$$2. The paper also defines a rotationally optimized approximation $$X=\bigoplus_{i=1}^N \alpha_i S_{\theta_i}, \qquad \alpha_i\ge 0,\ \theta_i\in[0,\pi),$$3, obtained by selecting the best orientation among all rotations. This removes the orientation sensitivity of the class $$X=\bigoplus_{i=1}^N \alpha_i S_{\theta_i}, \qquad \alpha_i\ge 0,\ \theta_i\in[0,\pi),$$4 (Rahmani et al., 2017).

A common misconception is that zonotopic approximation in this framework is merely polygonal curve fitting. The deterministic statement is stronger: the approximation is explicitly tied to the Feret diameter samples, is unique within the chosen regular class, and comes with a Hausdorff-distance estimate. The role of symmetry is equally non-optional; without symmetry, the Feret diameter determines the symmetrized body rather than the original set.

4. Tube parameterization and elastic scaling in robust MPC

In (Diaconescu et al., 24 Sep 2025), elastic scaling is formalized for robust tube MPC. The scaling vector is allowed to vary along the prediction horizon,

$$X=\bigoplus_{i=1}^N \alpha_i S_{\theta_i}, \qquad \alpha_i\ge 0,\ \theta_i\in[0,\pi),$$5

so the tube cross-section at time $$X=\bigoplus_{i=1}^N \alpha_i S_{\theta_i}, \qquad \alpha_i\ge 0,\ \theta_i\in[0,\pi),$$6 is a zonotope whose generators are stretched or shrunk over time. This differs from rigid tubes, which use fixed-shape translation only, from homothetic tubes, which use one scalar scaling factor for all generators, and from elastic polyhedral tubes, where half-space offsets vary but inclusion conditions are generally more complex and can become bilinear (Diaconescu et al., 24 Sep 2025).

The tube is parameterized as

$$X=\bigoplus_{i=1}^N \alpha_i S_{\theta_i}, \qquad \alpha_i\ge 0,\ \theta_i\in[0,\pi),$$7

where $$X=\bigoplus_{i=1}^N \alpha_i S_{\theta_i}, \qquad \alpha_i\ge 0,\ \theta_i\in[0,\pi),$$8 is the nominal trajectory, $$X=\bigoplus_{i=1}^N \alpha_i S_{\theta_i}, \qquad \alpha_i\ge 0,\ \theta_i\in[0,\pi),$$9 is the tube center offset, $S_0=[-\tfrac12,\tfrac12]$0 is a fixed seed generator matrix, and $S_0=[-\tfrac12,\tfrac12]$1 is the scaling vector. The tuple $S_0=[-\tfrac12,\tfrac12]$2 is admissible if there exists $S_0=[-\tfrac12,\tfrac12]$3 such that one-step reachability and state admissibility hold, namely

$S_0=[-\tfrac12,\tfrac12]$4

and

$S_0=[-\tfrac12,\tfrac12]$5

Input admissibility is reduced to

$S_0=[-\tfrac12,\tfrac12]$6

The applied control law is

$S_0=[-\tfrac12,\tfrac12]$7

with stabilizing feedback gain $S_0=[-\tfrac12,\tfrac12]$8 and $S_0=[-\tfrac12,\tfrac12]$9. Under this feedback, one-step reachability is guaranteed if

$S_\theta$0

State and input admissibility are ensured by

$S_\theta$1

and

$S_\theta$2

The essential point is that the tube shape is not fixed a priori; it is parameterized by the horizon-indexed vectors $S_\theta$3, which act as elastic degrees of freedom (Diaconescu et al., 24 Sep 2025).

5. Linear inclusion conditions for scaled zonotopes

A major contribution of (Diaconescu et al., 24 Sep 2025) is a linear inclusion test for zonotopes. For zonotopes $S_\theta$4, a sufficient condition for

$S_\theta$5

is the existence of matrices $S_\theta$6 and a vector $S_\theta$7 such that

$S_\theta$8

and

$S_\theta$9

The paper then rewrites this containment criterion as a linear condition using an auxiliary matrix $\theta$0: $\theta$1

$\theta$2

with

$\theta$3

For scaled zonotopes

$\theta$4

the inclusion condition becomes

$\theta$5

$\theta$6

The paper emphasizes that this encoding is linear in $\theta$7 and linear in the scaling factors $\theta$8, so it can be embedded directly into optimization problems without bilinearities.

To reduce online complexity, the paper proposes a precomputed matrix $\theta$9 satisfying the unscaled inclusion condition, and then imposes the sufficient condition

$2N$0

The corresponding precomputation problem is

$2N$1

This yields the stated trade-off: $2N$2 is more flexible and more expensive, whereas $2N$3 is cheaper but more restrictive (Diaconescu et al., 24 Sep 2025).

6. Moment formulas, inversion, and optimization structure

In the geometric setting, zonotopic representation is constructive. Once the Feret diameters $2N$4 are known at regular angles, the face-length vector is recovered from

$2N$5

For the 0-regular approximation, the paper gives the direct inversion formulas

$2N$6

$2N$7

$2N$8

For a zonotope

$2N$9

its perimeter and area are

$\mathbb R^2$00

and

$\mathbb R^2$01

For isotropic random regular zonotopes, (Rahmani et al., 2017) introduces the kernel

$\mathbb R^2$02

with explicit form on $\mathbb R^2$03,

$\mathbb R^2$04

and derives the second-order Feret relation

$\mathbb R^2$05

For a central face-length vector $\mathbb R^2$06 of an isotropic regular random zonotope,

$\mathbb R^2$07

and

$\mathbb R^2$08

If noisy data are available, the paper suggests the least-squares estimation

$\mathbb R^2$09

In the MPC setting, the full elastic-tube optimization problem is

$\mathbb R^2$10

subject to constraints including

$\mathbb R^2$11

$\mathbb R^2$12

$\mathbb R^2$13

$\mathbb R^2$14

$\mathbb R^2$15

$\mathbb R^2$16

and terminal admissibility

$\mathbb R^2$17

The stage and terminal costs are

$\mathbb R^2$18

and

$\mathbb R^2$19

These formulas show that elastic scaling enters not only the feasible-set description but also the optimization objective (Diaconescu et al., 24 Sep 2025).

7. Computational trade-offs, invariance, and applications

The control-theoretic treatment in (Diaconescu et al., 24 Sep 2025) makes the computational trade-offs explicit. The paper compares polyhedral and zonotopic tubes, together with rigid, homothetic, and elastic parameterizations. Its stated conclusions are that polyhedral elastic tubes are expensive because the polyhedral robust positively invariant representation can have

$\mathbb R^2$20

constraints; zonotopic formulations use only $\mathbb R^2$21 generators, so the number of scaling variables is much smaller; $\mathbb R^2$22 becomes impractical for $\mathbb R^2$23; $\mathbb R^2$24 has the most flexibility but highest online complexity; $\mathbb R^2$25 and $\mathbb R^2$26 are more compact; and $\mathbb R^2$27 removes the need for online auxiliary variables. The resulting three-way trade-off is among size of the domain of attraction, computational burden, and flexibility of the tube shape.

The same paper gives a zonotopic approximation of the minimal robust positively invariant (RPI) set for the error dynamics

$\mathbb R^2$28

With

$\mathbb R^2$29

the associated program is

$\mathbb R^2$30

subject to

$\mathbb R^2$31

$\mathbb R^2$32

This returns a zonotope $\mathbb R^2$33 that is robust positively invariant. The paper states that this LP uses

$\mathbb R^2$34

decision variables, fewer than the earlier CDC’24 formulation with

$\mathbb R^2$35

variables.

For terminal conditions, the nominal terminal control law is

$\mathbb R^2$36

with terminal closed-loop dynamics

$\mathbb R^2$37

and

$\mathbb R^2$38

Terminal admissibility is encoded by

$\mathbb R^2$39

and

$\mathbb R^2$40

while the Lyapunov-like conditions are guaranteed by the LMIs

$\mathbb R^2$41

and

$\mathbb R^2$42

The practical implications differ across the two literatures but are conceptually aligned. In stochastic geometry and image-based shape analysis, the geometric construction replaces an arbitrary symmetric convex random set by a finite-dimensional random vector of face lengths linked to observable Feret-diameter data (Rahmani et al., 2017). In robust MPC, elastic generator scaling yields a tube description that is more scalable than polyhedral elastic tubes and, in the numerical examples reported, maintains feasibility in higher-dimensional settings where polyhedral methods fail (Diaconescu et al., 24 Sep 2025). A plausible implication is that elastically-scaled zonotopic sets are best understood not as a single narrowly defined object, but as a common zonotopic principle: fixed combinatorial structure combined with adaptable directional magnitudes, calibrated either by geometric width data or by horizon-dependent robustness requirements.