Scaled-Zonotope Inclusion Conditions

• Scaled-zonotope inclusion conditions are algebraic certificates that use independently scaled generators to verify set containment in zonotopes and constrained zonotopes.
• They transform nonlinear or bilinear checks into linear constraints, greatly enhancing tractability in applications like elastic tube MPC and data-driven safe control.
• These conditions underpin robust invariant set approximations, viability kernel computations, and convex geometry approximations by leveraging scaling variables and auxiliary formulations.

Scaled-zonotope inclusion conditions are algebraic set-containment certificates in which the relevant set is a zonotope or constrained zonotope whose size is modulated by explicit scaling variables. In recent control-oriented formulations, the notion appears in two closely related forms: generator-wise scaling of a zonotope, written as $\langle c,G\Delta\rangle$ with $\Delta=\operatorname{diag}(\delta)$, and $\lambda$-scaled level sets of constrained zonotopes, written as $\langle \lambda G_x,c_x,A_x,\lambda b_x\rangle$. These conditions are used to replace nonlinear or bilinear containment checks by linear or optimization-friendly certificates, especially in elastic tube MPC and in data-driven safe control of uncertain linear systems (Diaconescu et al., 24 Sep 2025, Modares et al., 6 Feb 2025). More broadly, zonotope scaling also appears in viability and discriminating-kernel under-approximation, in oracle-model containment algorithms for zonotopes, and in nonlinear generalizations based on constrained polynomial zonotopes (Mitchell et al., 2019, Eisenbrand et al., 5 May 2026, Gheorghe et al., 27 Mar 2026).

1. Foundational set representations and the meaning of scaling

A polytope is represented as

$\mathcal P(H,h)=\{x\in\mathbb R^n:Hx\le h\}.$

A zonotope with center $c$ and generator matrix $G$ is

$$\mathcal Z=\langle G,c\rangle=\left\{x\in\mathbb R^n:x=G\zeta+c,\ \|\zeta\|_\infty\le 1\right\}.$$

A constrained zonotope augments this with linear equalities,

$$\mathcal C=\langle G,c,A_c,b_c\rangle =\left\{x\in\mathbb R^n:x=G\zeta+c,\ \|\zeta\|_\infty\le 1,\ A_c\zeta=b_c\right\}.$$

For data-driven system identification and robust control, the same structure is lifted to matrices. A matrix zonotope is

$$\mathcal M=\langle G,C\rangle =\left\{X\in\mathbb R^{n\times p}: X=\sum_{i=1}^s G_i\zeta_i + C,\ \|\zeta\|_\infty\le 1\right\},$$

and a constrained matrix zonotope is

$\Delta=\operatorname{diag}(\delta)$0

These are the basic set classes used to formulate scaled inclusion in safe control (Modares et al., 6 Feb 2025).

In elastic tube MPC, the scaled object is a zonotope whose generators are independently rescaled. If

$\Delta=\operatorname{diag}(\delta)$1

then a scaled zonotope is

$\Delta=\operatorname{diag}(\delta)$2

The tube center $\Delta=\operatorname{diag}(\delta)$3 and scaling vector $\Delta=\operatorname{diag}(\delta)$4 are treated as optimization variables, so each generator direction can expand or contract independently (Diaconescu et al., 24 Sep 2025).

A distinct but related scaling appears in contractive safe control. For a constrained-zonotope safe set

$\Delta=\operatorname{diag}(\delta)$5

the $\Delta=\operatorname{diag}(\delta)$6-scaled level set is defined as

$\Delta=\operatorname{diag}(\delta)$7

Here $\Delta=\operatorname{diag}(\delta)$8 enforces contraction toward the interior of the safe set; the paper explicitly states that smaller $\Delta=\operatorname{diag}(\delta)$9 means stronger contraction, $\lambda$0 guarantees contraction and thus invariant safety, and smaller $\lambda$1 is safer/faster but harder to satisfy (Modares et al., 6 Feb 2025).

Outside control, scaled inclusion can be posed as the optimization problem

$\lambda$2

where $\lambda$3 is a convex body and $\lambda$4 is a zonotope. In that setting, the scaling factor is global rather than generator-wise, and the emphasis shifts from exact certificates to approximation guarantees in the oracle model (Eisenbrand et al., 5 May 2026).

2. Algebraic containment certificates for zonotopes and constrained zonotopes

A central sufficient condition for inclusion of a Minkowski sum of zonotopes into another zonotope is the existence of matrices $\lambda$5 and a vector $\lambda$6 such that, for $\lambda$7,

$\lambda$8

holds if

$\lambda$9

and

$\langle \lambda G_x,c_x,A_x,\lambda b_x\rangle$0

The proof is constructive: every point in the sum is rewritten in $\langle \lambda G_x,c_x,A_x,\lambda b_x\rangle$1-coordinates, and the absolute-value bound ensures that the new coefficient vector remains in the unit cube (Diaconescu et al., 24 Sep 2025).

The same condition admits an equivalent linear encoding with auxiliary variables $\langle \lambda G_x,c_x,A_x,\lambda b_x\rangle$2. Let

$\langle \lambda G_x,c_x,A_x,\lambda b_x\rangle$3

Then the inclusion condition is equivalently written as

$\langle \lambda G_x,c_x,A_x,\lambda b_x\rangle$4

with

$\langle \lambda G_x,c_x,A_x,\lambda b_x\rangle$5

The paper interprets this as expressing each generator-row coefficient vector as a convex combination of vertices of the unit cross-polytope, the polar of the hypercube. This formulation is emphasized as more convenient for embedding in optimization, even though it introduces many auxiliary variables (Diaconescu et al., 24 Sep 2025).

For scaled zonotopes, each $\langle \lambda G_x,c_x,A_x,\lambda b_x\rangle$6 is replaced by $\langle \lambda G_x,c_x,A_x,\lambda b_x\rangle$7, and the certificate becomes

$\langle \lambda G_x,c_x,A_x,\lambda b_x\rangle$8

together with

$\langle \lambda G_x,c_x,A_x,\lambda b_x\rangle$9

This is the main scaled-zonotope inclusion condition of elastic tube MPC, and its central advantage is that it is linear in both the scaling variables $\mathcal P(H,h)=\{x\in\mathbb R^n:Hx\le h\}.$0 and the certificate $\mathcal P(H,h)=\{x\in\mathbb R^n:Hx\le h\}.$1 (Diaconescu et al., 24 Sep 2025).

For constrained zonotopes, inclusion is encoded by a different certificate. If

$\mathcal P(H,h)=\{x\in\mathbb R^n:Hx\le h\}.$2

then $\mathcal P(H,h)=\{x\in\mathbb R^n:Hx\le h\}.$3 holds if there exist $\mathcal P(H,h)=\{x\in\mathbb R^n:Hx\le h\}.$4, $\mathcal P(H,h)=\{x\in\mathbb R^n:Hx\le h\}.$5, and $\mathcal P(H,h)=\{x\in\mathbb R^n:Hx\le h\}.$6 such that

$\mathcal P(H,h)=\{x\in\mathbb R^n:Hx\le h\}.$7

$\mathcal P(H,h)=\{x\in\mathbb R^n:Hx\le h\}.$8

$\mathcal P(H,h)=\{x\in\mathbb R^n:Hx\le h\}.$9

This certificate aligns the center, generators, and equality constraints while simultaneously preserving the unit-box admissibility of latent coefficients (Modares et al., 6 Feb 2025).

These formulations clarify a common distinction. In the zonotopic elastic-tube setting, the inclusion rules are presented as sufficient conditions. In the constrained-zonotope safe-control setting, the paper states an exact inclusion condition for the specific $c$0 construction. In the CPZ setting discussed later, the certificate is again explicitly sufficient rather than necessary.

3. Elastic tube MPC and generator-wise scaled-zonotope inclusion

In tube-based MPC for constrained linear systems under additive disturbance, the closed-loop model is

$c$1

with disturbance

$c$2

and feedback law

$c$3

The tube cross-section at time $c$4 is parameterized as

$c$5

The one-step reachability condition is imposed by requiring

$c$6

where $c$7. The online certificate is then written as the linear constraints

$c$8

with

$c$9

This is the zonotopic-elastic realization of scaled-zonotope containment in the MPC loop (Diaconescu et al., 24 Sep 2025).

A second formulation precomputes a fixed certificate $G$0. If $G$1 satisfies the unscaled inclusion condition, then a sufficient condition for the scaled case is

$G$2

The derivation uses

$G$3

rewrites the certificate with

$G$4

and then enforces $G$5. The practical interpretation given in the paper is that $G$6 can be computed offline, after which the online MPC problem uses only a compact linear inequality in the scaling factors. The stated trade-off is conservatism: fixing $G$7 can make some otherwise feasible scaling choices infeasible (Diaconescu et al., 24 Sep 2025).

The same inclusion machinery is used to compute a zonotopic approximation of the robust positively invariant set for the error dynamics

$G$8

With

$G$9

the proposed LP is

$$\mathcal Z=\langle G,c\rangle=\left\{x\in\mathbb R^n:x=G\zeta+c,\ \|\zeta\|_\infty\le 1\right\}.$$0

subject to

$$\mathcal Z=\langle G,c\rangle=\left\{x\in\mathbb R^n:x=G\zeta+c,\ \|\zeta\|_\infty\le 1\right\}.$$1

The result is a zonotopic RPI set $$\mathcal Z=\langle G,c\rangle=\left\{x\in\mathbb R^n:x=G\zeta+c,\ \|\zeta\|_\infty\le 1\right\}.$$2 (Diaconescu et al., 24 Sep 2025).

The complexity motivation is explicit. Compared with polyhedral elastic tubes, the zonotopic parameterization uses only the number of generators $$\mathcal Z=\langle G,c\rangle=\left\{x\in\mathbb R^n:x=G\zeta+c,\ \|\zeta\|_\infty\le 1\right\}.$$3 as scaling factors, whereas a polyhedral elastic representation may require $$\mathcal Z=\langle G,c\rangle=\left\{x\in\mathbb R^n:x=G\zeta+c,\ \|\zeta\|_\infty\le 1\right\}.$$4 scaling parameters after converting a zonotope to half-space form via Buck’s formula. Among the zonotopic variants, the $$\mathcal Z=\langle G,c\rangle=\left\{x\in\mathbb R^n:x=G\zeta+c,\ \|\zeta\|_\infty\le 1\right\}.$$5-form is the most flexible but introduces many auxiliary variables scaling roughly with $$\mathcal Z=\langle G,c\rangle=\left\{x\in\mathbb R^n:x=G\zeta+c,\ \|\zeta\|_\infty\le 1\right\}.$$6; the $$\mathcal Z=\langle G,c\rangle=\left\{x\in\mathbb R^n:x=G\zeta+c,\ \|\zeta\|_\infty\le 1\right\}.$$7 and $$\mathcal Z=\langle G,c\rangle=\left\{x\in\mathbb R^n:x=G\zeta+c,\ \|\zeta\|_\infty\le 1\right\}.$$8 forms are more compact because the certificates are precomputed offline, but are more restrictive and can lose feasible scaling combinations. The numerical experiments reported in the paper state that the fully online $$\mathcal Z=\langle G,c\rangle=\left\{x\in\mathbb R^n:x=G\zeta+c,\ \|\zeta\|_\infty\le 1\right\}.$$9 variant yields the largest domain of attraction but the highest runtime, while the $$\mathcal C=\langle G,c,A_c,b_c\rangle =\left\{x\in\mathbb R^n:x=G\zeta+c,\ \|\zeta\|_\infty\le 1,\ A_c\zeta=b_c\right\}.$$0 variant is slightly more conservative but considerably cheaper (Diaconescu et al., 24 Sep 2025).

4. $$\mathcal C=\langle G,c,A_c,b_c\rangle =\left\{x\in\mathbb R^n:x=G\zeta+c,\ \|\zeta\|_\infty\le 1,\ A_c\zeta=b_c\right\}.$$1-scaled safe sets in data-driven safe control

A different use of scaled inclusion arises in direct learning of safe controllers for linear uncertain systems under disturbances. The unknown true system is

$$\mathcal C=\langle G,c,A_c,b_c\rangle =\left\{x\in\mathbb R^n:x=G\zeta+c,\ \|\zeta\|_\infty\le 1,\ A_c\zeta=b_c\right\}.$$2

with additive disturbance $$\mathcal C=\langle G,c,A_c,b_c\rangle =\left\{x\in\mathbb R^n:x=G\zeta+c,\ \|\zeta\|_\infty\le 1,\ A_c\zeta=b_c\right\}.$$3, and the controller is linear state feedback,

$$\mathcal C=\langle G,c,A_c,b_c\rangle =\left\{x\in\mathbb R^n:x=G\zeta+c,\ \|\zeta\|_\infty\le 1,\ A_c\zeta=b_c\right\}.$$4

Rather than proving invariance directly, the safe set is made $$\mathcal C=\langle G,c,A_c,b_c\rangle =\left\{x\in\mathbb R^n:x=G\zeta+c,\ \|\zeta\|_\infty\le 1,\ A_c\zeta=b_c\right\}.$$5-contractive for some $$\mathcal C=\langle G,c,A_c,b_c\rangle =\left\{x\in\mathbb R^n:x=G\zeta+c,\ \|\zeta\|_\infty\le 1,\ A_c\zeta=b_c\right\}.$$6, meaning

$$\mathcal C=\langle G,c,A_c,b_c\rangle =\left\{x\in\mathbb R^n:x=G\zeta+c,\ \|\zeta\|_\infty\le 1,\ A_c\zeta=b_c\right\}.$$7

The paper states two benefits: safety, because if the set is contractive then it is invariant, and convergence, because the state is driven toward the origin at a rate tied to $$\mathcal C=\langle G,c,A_c,b_c\rangle =\left\{x\in\mathbb R^n:x=G\zeta+c,\ \|\zeta\|_\infty\le 1,\ A_c\zeta=b_c\right\}.$$8 (Modares et al., 6 Feb 2025).

Prior knowledge enters through a matrix zonotope restriction on the unknown parameter matrix

$$\mathcal C=\langle G,c,A_c,b_c\rangle =\left\{x\in\mathbb R^n:x=G\zeta+c,\ \|\zeta\|_\infty\le 1,\ A_c\zeta=b_c\right\}.$$9

From data $$\mathcal M=\langle G,C\rangle =\left\{X\in\mathbb R^{n\times p}: X=\sum_{i=1}^s G_i\zeta_i + C,\ \|\zeta\|_\infty\le 1\right\},$$0, the construction uses

$$\mathcal M=\langle G,C\rangle =\left\{X\in\mathbb R^{n\times p}: X=\sum_{i=1}^s G_i\zeta_i + C,\ \|\zeta\|_\infty\le 1\right\},$$1

and the right-inverse parameterization

$$\mathcal M=\langle G,C\rangle =\left\{X\in\mathbb R^{n\times p}: X=\sum_{i=1}^s G_i\zeta_i + C,\ \|\zeta\|_\infty\le 1\right\},$$2

The set of all closed-loop systems consistent with data and prior knowledge is

$$\mathcal M=\langle G,C\rangle =\left\{X\in\mathbb R^{n\times p}: X=\sum_{i=1}^s G_i\zeta_i + C,\ \|\zeta\|_\infty\le 1\right\},$$3

and the key theorem gives its exact constrained-matrix-zonotope representation,

$$\mathcal M=\langle G,C\rangle =\left\{X\in\mathbb R^{n\times p}: X=\sum_{i=1}^s G_i\zeta_i + C,\ \|\zeta\|_\infty\le 1\right\},$$4

with

$$\mathcal M=\langle G,C\rangle =\left\{X\in\mathbb R^{n\times p}: X=\sum_{i=1}^s G_i\zeta_i + C,\ \|\zeta\|_\infty\le 1\right\},$$5

The equality conformity constraint involving $$\mathcal M=\langle G,C\rangle =\left\{X\in\mathbb R^{n\times p}: X=\sum_{i=1}^s G_i\zeta_i + C,\ \|\zeta\|_\infty\le 1\right\},$$6 is the mechanism that filters out models that cannot simultaneously explain the data and prior knowledge (Modares et al., 6 Feb 2025).

Given the closed-loop uncertainty $$\mathcal M=\langle G,C\rangle =\left\{X\in\mathbb R^{n\times p}: X=\sum_{i=1}^s G_i\zeta_i + C,\ \|\zeta\|_\infty\le 1\right\},$$7, a current state $$\mathcal M=\langle G,C\rangle =\left\{X\in\mathbb R^{n\times p}: X=\sum_{i=1}^s G_i\zeta_i + C,\ \|\zeta\|_\infty\le 1\right\},$$8, and disturbance $$\mathcal M=\langle G,C\rangle =\left\{X\in\mathbb R^{n\times p}: X=\sum_{i=1}^s G_i\zeta_i + C,\ \|\zeta\|_\infty\le 1\right\},$$9, the next state lies in a constrained zonotope

$\Delta=\operatorname{diag}(\delta)$00

with

$\Delta=\operatorname{diag}(\delta)$01

$\Delta=\operatorname{diag}(\delta)$02

$\Delta=\operatorname{diag}(\delta)$03

Safety is then reduced to the inclusion problem

$\Delta=\operatorname{diag}(\delta)$04

Using the constrained-zonotope inclusion lemma, the paper states that the safe control problem is solved if there exist $\Delta=\operatorname{diag}(\delta)$05 satisfying

$\Delta=\operatorname{diag}(\delta)$06

$\Delta=\operatorname{diag}(\delta)$07

$\Delta=\operatorname{diag}(\delta)$08

$\Delta=\operatorname{diag}(\delta)$09

$\Delta=\operatorname{diag}(\delta)$10

This is the exact inclusion condition for the constrained-zonotope case (Modares et al., 6 Feb 2025).

When the safe set is instead represented as a polytope

$\Delta=\operatorname{diag}(\delta)$11

the same contractive requirement becomes

$\Delta=\operatorname{diag}(\delta)$12

and the paper derives an LP: $\Delta=\operatorname{diag}(\delta)$13 subject to

$\Delta=\operatorname{diag}(\delta)$14

$\Delta=\operatorname{diag}(\delta)$15

$\Delta=\operatorname{diag}(\delta)$16

where

$\Delta=\operatorname{diag}(\delta)$17

and

$\Delta=\operatorname{diag}(\delta)$18

subject to auxiliary constrained-zonotope feasibility conditions on $\Delta=\operatorname{diag}(\delta)$19. This polytope case is described as more computationally oriented because inclusion is checked by solving a linear program rather than by directly solving the constrained-zonotope inclusion equations (Modares et al., 6 Feb 2025).

5. Approximate scaled containment of zonotopes in convex geometry

In convex geometry and oracle-model algorithms, the scaled-zonotope question is formulated as

$\Delta=\operatorname{diag}(\delta)$20

where the outer body $\Delta=\operatorname{diag}(\delta)$21 is described by a membership oracle and the inner body $\Delta=\operatorname{diag}(\delta)$22 is a zonotope

$\Delta=\operatorname{diag}(\delta)$23

The central issue is no longer an exact linear certificate but the best efficiently computable approximation factor $\Delta=\operatorname{diag}(\delta)$24 and the ability to find a point in $\Delta=\operatorname{diag}(\delta)$25 when containment fails (Eisenbrand et al., 5 May 2026).

The main algorithmic guarantee is a randomized polynomial-time algorithm with approximation factor

$\Delta=\operatorname{diag}(\delta)$26

More precisely, if $\Delta=\operatorname{diag}(\delta)$27, the algorithm finds a point in

$\Delta=\operatorname{diag}(\delta)$28

with high probability. The method is sampling-based and combines two ingredients: sparsification of the zonotope using Talagrand’s theorem and sampling a random hypercube vertex. After reducing to a zonotope with $\Delta=\operatorname{diag}(\delta)$29 generators, the core probabilistic step states that if $\Delta=\operatorname{diag}(\delta)$30, then for a random $\Delta=\operatorname{diag}(\delta)$31,

$\Delta=\operatorname{diag}(\delta)$32

with probability at least $\Delta=\operatorname{diag}(\delta)$33, so repeated sampling finds a violating point efficiently (Eisenbrand et al., 5 May 2026).

The lower-bound theory is nearly tight. The paper proves a universal oracle-model lower bound

$\Delta=\operatorname{diag}(\delta)$34

for all zonotopes, strengthening earlier tightness results that were known for the hypercube case $\Delta=\operatorname{diag}(\delta)$35. Under Talagrand’s conjecture, which predicts sparsification with only $\Delta=\operatorname{diag}(\delta)$36 generators, the approximation improves to the near-optimal factor

$\Delta=\operatorname{diag}(\delta)$37

For constant-$\Delta=\operatorname{diag}(\delta)$38 modular zonotopes, the paper proves this sparsification theorem and obtains the containment approximation factor

$\Delta=\operatorname{diag}(\delta)$39

which is essentially optimal up to constants when $\Delta=\operatorname{diag}(\delta)$40 is constant (Eisenbrand et al., 5 May 2026).

The geometric mechanism differs sharply from control-oriented inclusion conditions. Inclusion is checked via support functions on facet normals, and the analysis is driven by zonoid sparsification, spectral sparsification, and oracle complexity rather than by explicit state-space constraints or recursive-feasibility requirements. This suggests a useful conceptual split: in control, scaled-zonotope inclusion is typically a tractable certificate embedded in an optimization problem; in oracle-model geometry, it is an approximation problem with provable dimensional barriers.

6. Extensions beyond convex zonotopes and antecedents in safety verification

Constrained polynomial zonotopes extend zonotopes and constrained zonotopes to non-convex sets with polynomial latent-variable structure. A CPZ is

$\Delta=\operatorname{diag}(\delta)$41

The inclusion problem $\Delta=\operatorname{diag}(\delta)$42 is certified by the existence of

$\Delta=\operatorname{diag}(\delta)$43

satisfying

$\Delta=\operatorname{diag}(\delta)$44

$\Delta=\operatorname{diag}(\delta)$45

together with the nonlinear boundedness conditions

$\Delta=\operatorname{diag}(\delta)$46

$\Delta=\operatorname{diag}(\delta)$47

The paper presents this as a sufficient, optimization-friendly certificate and interprets it as a nonlinear generalization of zonotope inclusion based on mapping the latent domain of $\Delta=\operatorname{diag}(\delta)$48 into that of $\Delta=\operatorname{diag}(\delta)$49 (Gheorghe et al., 27 Mar 2026).

When the polynomial structure is removed, the CPZ conditions reduce to the familiar constrained-zonotope form: $\Delta=\operatorname{diag}(\delta)$50

$\Delta=\operatorname{diag}(\delta)$51

$\Delta=\operatorname{diag}(\delta)$52

This reduction makes explicit that scaled-zonotope and constrained-zonotope inclusion tests are the linear core from which the CPZ construction generalizes (Gheorghe et al., 27 Mar 2026).

An earlier control-theoretic precursor uses zonotope scaling to under-approximate invariant, viable, and discriminating kernels for discrete-time affine systems with adversarial inputs. There, a state set is parameterized as

$\Delta=\operatorname{diag}(\delta)$53

with fixed generator directions and optimized scaling factors. The key containment device is box inclusion: $\Delta=\operatorname{diag}(\delta)$54 and for a box $\Delta=\operatorname{diag}(\delta)$55,

$\Delta=\operatorname{diag}(\delta)$56

is equivalent to

$\Delta=\operatorname{diag}(\delta)$57

This converts reachability and safety constraints into linear or convex programs for invariant, viable, and discriminating under-approximations (Mitchell et al., 2019).

For viability and discriminating-kernel computation, the same paper introduces scaled control-authority zonotopes,

$\Delta=\operatorname{diag}(\delta)$58

and optimizes over $\Delta=\operatorname{diag}(\delta)$59, $\Delta=\operatorname{diag}(\delta)$60, and auxiliary control parameters. The best-case control and worst-case disturbance distinction is explicit: viable sets use best-case control, discriminating sets use best-case control together with worst-case adversarial disturbance. This earlier framework does not formulate the modern $\Delta=\operatorname{diag}(\delta)$61- or $\Delta=\operatorname{diag}(\delta)$62-scaled inclusion certificates, but it establishes the same basic principle that zonotope scaling can encode safety-relevant set under-approximations through convex constraints (Mitchell et al., 2019).

Taken together, these developments identify scaled-zonotope inclusion conditions as a family of containment mechanisms rather than a single formula. In ordinary zonotopes they appear as linear certificates for sums and generator scalings; in constrained zonotopes they encode equality-compatible contractive inclusion; in CPZs they become nonlinear boundedness constraints on monomial parameter maps; and in oracle-model convex geometry they define an approximation problem whose optimal scale depends on dimension. The shared structure is the same throughout: inclusion is certified, or approximately decided, by controlling how latent coefficients or support-function values transform under scaling.