---
title: Scaled-Zonotope Inclusion Conditions
url: https://www.emergentmind.com/topics/scaled-zonotope-inclusion-conditions
type: topic
---

# Scaled-Zonotope Inclusion Conditions

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 [2509.19824, 2502.04195]. 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 [1901.01006, 2605.04183, 2603.26621].

## 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
\[
\mathcal K=\langle G,C,A_C,B_C\rangle
=\left\{X\in\mathbb R^{n\times p}: X=\sum_{i=1}^s G_i\zeta_i+C,\ \sum_{i=1}^s (A_C)_i\zeta_i=B_C,\ \|\zeta\|_\infty\le 1 \right\}.
\]
These are the basic set classes used to formulate scaled inclusion in safe control [2502.04195].

In elastic tube MPC, the scaled object is a zonotope whose generators are independently rescaled. If
\[
Z=\langle c,G\rangle=\{x\in\mathbb R^n:\ x=c+G\xi,\ \|\xi\|_\infty\le 1\},
\]
then a scaled zonotope is
\[
\langle c,G\Delta\rangle,\qquad \Delta=\operatorname{diag}(\delta),\quad \delta\in\mathbb R^D_{\ge 0}.
\]
The tube center \(c_k\) and scaling vector \(\delta_k\) are treated as optimization variables, so each generator direction can expand or contract independently [2509.19824].

A distinct but related scaling appears in contractive safe control. For a constrained-zonotope safe set
\[
\mathcal C_x=\langle c_x,G_x,A_x,b_x\rangle,
\]
the \(\lambda\)-scaled level set is defined as
\[
\mathcal C_{x\lambda}
=\left\{x: x=c_x+G_x\zeta,\ \|\zeta\|_\infty\le \lambda,\ A_x\zeta=\lambda b_x\right\}
=\langle \lambda G_x,c_x,A_x,\lambda b_x\rangle.
\]
Here \(\lambda\in(0,1)\) enforces contraction toward the interior of the safe set; the paper explicitly states that smaller \(\lambda\) means stronger contraction, \(\lambda<1\) guarantees contraction and thus invariant safety, and smaller \(\lambda\) is safer/faster but harder to satisfy [2502.04195].

Outside control, scaled inclusion can be posed as the optimization problem
\[
\max\{s>0:sZ\subseteq Q\},
\]
where \(Q\subseteq\mathbb R^d\) is a convex body and \(Z\) 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 [2605.04183].

## 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 \(\Gamma_i\) and a vector \(\gamma\) such that, for \(Z_i=\langle c_i,G_i\rangle\subset\mathbb R^d\),
\[
Z_1\oplus\cdots\oplus Z_{n-1}\subseteq Z_n
\]
holds if
\[
G_i=G_n\Gamma_i,\qquad \sum_{i=1}^{n-1} c_i = c_n + G_n\gamma,
\]
and
\[
\sum_{i=1}^{n-1}|\Gamma_i|\mathbf 1_{D_i}+|\gamma|\le \mathbf 1_{D_n}.
\]
The proof is constructive: every point in the sum is rewritten in \(Z_n\)-coordinates, and the absolute-value bound ensures that the new coefficient vector remains in the unit cube [2509.19824].

The same condition admits an equivalent linear encoding with auxiliary variables \(\Phi\). Let
\[
V=\begin{bmatrix}I_{\bar D}&-I_{\bar D}\end{bmatrix},\qquad \bar D=1+\sum_{i=1}^{n-1}D_i.
\]
Then the inclusion condition is equivalently written as
\[
G_n\Phi^\top V^\top
=
\begin{bmatrix}
-c_n+\sum_{i=1}^{n-1}c_i & G_1 & \ldots & G_{n-1}
\end{bmatrix},
\]
with
\[
\Phi\ge 0,\qquad \Phi^\top \mathbf 1_{2\bar D}\le \mathbf 1_{D_n}.
\]
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 [2509.19824].

For scaled zonotopes, each \(G_i\) is replaced by \(G_i\Delta_i\), and the certificate becomes
\[
G_n\Phi^\top V^\top
=
\begin{bmatrix}
-c_n+\sum_{i=1}^{n-1}c_i & G_1\Delta_1 & \ldots & G_{n-1}\Delta_{n-1}
\end{bmatrix},
\]
together with
\[
\Phi\ge 0,\qquad \Phi^\top \mathbf 1_{2\bar D}\le \delta_n.
\]
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 \(\delta_i\) and the certificate \(\Phi\) [2509.19824].

For constrained zonotopes, inclusion is encoded by a different certificate. If
\[
\mathcal C_1=\langle c_1,G_1,A_1,b_1\rangle,\qquad
\mathcal C_2=\langle c_2,G_2,A_2,b_2\rangle,
\]
then \(\mathcal C_1\subseteq\mathcal C_2\) holds if there exist \(\Gamma\), \(L\), and \(P\) such that
\[
c_2-c_1=G_2L,\qquad
G_1=G_2\Gamma,
\]
\[
PA_1=A_2\Gamma,\qquad
Pb_1=b_2+A_2L,
\]
\[
|\Gamma|\bar 1+|L|\le \bar 1.
\]
This certificate aligns the center, generators, and equality constraints while simultaneously preserving the unit-box admissibility of latent coefficients [2502.04195].

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 \(\mathcal C_{cl}\subseteq \mathcal C_{x\lambda}\) 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
\[
x_{k+1}=Ax_k+Bu_k+\omega_k,
\]
with disturbance
\[
\omega_k\in\mathcal W=\langle c_W,G_W\rangle,
\]
and feedback law
\[
u_k=\bar u_k+K(x_k-\bar x_k).
\]
The tube cross-section at time \(k\) is parameterized as
\[
x_k\in \bar x_k\oplus \langle c_k,G\Delta_k\rangle.
\]
The one-step reachability condition is imposed by requiring
\[
\left\langle A\bar x_k+B\bar u_k+A_Kc_k,\ A_KG\Delta_k \right\rangle\oplus \langle c_W,G_W\rangle
\subseteq
\langle \bar x_{k+1}+c_{k+1},G\Delta_{k+1}\rangle,
\]
where \(A_K=A+BK\). The online certificate is then written as the linear constraints
\[
G\Phi_k^\top V^\top =
\begin{bmatrix}
-c_{k+1}-\bar x_{k+1} + A\bar x_k + B\bar u_k + A_Kc_k + c_\omega
&
A_KG\Delta_k
&
G_\omega
\end{bmatrix},
\]
with
\[
\Phi_k\ge 0,\qquad \Phi_k^\top \mathbf 1_{2\bar D}\le \delta_{k+1}.
\]
This is the zonotopic-elastic realization of scaled-zonotope containment in the MPC loop [2509.19824].

A second formulation precomputes a fixed certificate \(\Phi_0\). If \(\Phi_0\) satisfies the unscaled inclusion condition, then a sufficient condition for the scaled case is
\[
\Phi_0^\top \left( \begin{bmatrix}1\end{bmatrix}\otimes \begin{bmatrix} 1 & \delta_1^\top & \cdots & \delta_{n-1}^\top \end{bmatrix}^\top \right)\le \delta_n.
\]
The derivation uses
\[
T=\operatorname{diag}(1,\delta_1,\ldots,\delta_{n-1}),
\]
rewrites the certificate with
\[
\Phi=(I_2\otimes T)\Phi_0\Delta_n^{-1},
\]
and then enforces \(\Phi^\top\mathbf 1\le \mathbf 1\). The practical interpretation given in the paper is that \(\Phi_0\) 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 \(\Phi_0\) can make some otherwise feasible scaling choices infeasible [2509.19824].

The same inclusion machinery is used to compute a zonotopic approximation of the robust positively invariant set for the error dynamics
\[
z_{k+1}=A_K z_k+\omega_k.
\]
With
\[
\bar G=\begin{bmatrix}I & A_K & \cdots & A_K^s\end{bmatrix}G_\omega,
\]
the proposed LP is
\[
\max_{c,\Phi,\delta}\ \delta
\]
subject to
\[
\bar G\Phi^\top V^\top=
\begin{bmatrix}
-c+A_Kc+c_\omega & A_K\bar G\Delta & G_\omega
\end{bmatrix},
\qquad
\Phi\ge 0,\quad \Phi^\top\mathbf 1_{2\bar D}\le \delta.
\]
The result is a zonotopic RPI set \(\langle c,\bar G\Delta\rangle\) [2509.19824].

The complexity motivation is explicit. Compared with polyhedral elastic tubes, the zonotopic parameterization uses only the number of generators \(D\) as scaling factors, whereas a polyhedral elastic representation may require \(2\binom{D}{n-1}\) scaling parameters after converting a zonotope to half-space form via Buck’s formula. Among the zonotopic variants, the \(\Phi\)-form is the most flexible but introduces many auxiliary variables scaling roughly with \(D\bar D\); the \(\Gamma\) and \(\Phi_0\) 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 \(\Phi\) variant yields the largest domain of attraction but the highest runtime, while the \(\Phi_0\) variant is slightly more conservative but considerably cheaper [2509.19824].

## 4. \(\lambda\)-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
\[
x(t+1)=A^*x(t)+B^*u(t)+w(t),
\]
with additive disturbance \(w(t)\in\mathcal Z_w\), and the controller is linear state feedback,
\[
u(t)=Kx(t).
\]
Rather than proving invariance directly, the safe set is made \(\lambda\)-contractive for some \(\lambda\in(0,1)\), meaning
\[
x(t)\in \mathcal P \implies x(t+1)\in \lambda \mathcal P
\quad \forall w(t)\in \mathcal Z_w.
\]
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 \(\lambda\) [2502.04195].

Prior knowledge enters through a matrix zonotope restriction on the unknown parameter matrix
\[
\theta^*=[A^*\ \ B^*],\qquad \theta^*\in \mathcal M_{\text{prior}}=\langle G_\theta,C_\theta\rangle.
\]
From data \(X_0,U_0,X_1\), the construction uses
\[
D_0=\begin{bmatrix}X_0 & U_0\end{bmatrix},
\]
and the right-inverse parameterization
\[
K=U_0G_K,\qquad
D_0G_K=\begin{bmatrix}I\\ K\end{bmatrix}.
\]
The set of all closed-loop systems consistent with data and prior knowledge is
\[
\Sigma^K_{X,U_0}=\left\{ A_K: A_K=A+BK,\ [A\ B]\in\Sigma_{X,U_0} \right\},
\]
and the key theorem gives its exact constrained-matrix-zonotope representation,
\[
\mathcal M_{cl}
=
\Big\langle [G_w\circ G_K\quad 0],\ (X_1-C_w)G_K,\ A_C,\ B_C \Big\rangle,
\]
with
\[
A_C=[G_w\ \ G_\theta] \begin{bmatrix} D_0 & I \end{bmatrix}^{\bot},
\qquad
B_C= \begin{bmatrix} X_1-C_w & C_\theta \end{bmatrix} \begin{bmatrix} D_0 & I \end{bmatrix}^{\bot}.
\]
The equality conformity constraint involving \(\begin{bmatrix}D_0 & I\end{bmatrix}^{\bot}\) is the mechanism that filters out models that cannot simultaneously explain the data and prior knowledge [2502.04195].

Given the closed-loop uncertainty \(\mathcal M_{cl}\), a current state \(x(t)\in\mathcal C_x\), and disturbance \(w(t)\in\mathcal Z_w\), the next state lies in a constrained zonotope
\[
x(t+1)\in \mathcal C_{cl}=\langle G_{cl},c_{cl},A_{cl},b_{cl}\rangle,
\]
with
\[
G_{cl}= [(G_w\circ G_K)c_x\ \ 0\ \ (X_1-C_w)G_KG_x\ \ G_f\ \ G_h],
\]
\[
c_{cl}=(X_1-C_w)G_Kc_x+c_h,
\]
\[
A_{cl}= \begin{bmatrix} \mathrm{Vec}(A_C) & 0 & 0 & 0\\ 0 & A_x & 0 & 0\\ 0 & 0 & 0 & 0 \end{bmatrix},
\qquad
b_{cl}= \begin{bmatrix} \mathrm{Vec}(B_C)\\ b_x\\ 0 \end{bmatrix}.
\]
Safety is then reduced to the inclusion problem
\[
\mathcal C_{cl}\subseteq \lambda \mathcal C_x.
\]
Using the constrained-zonotope inclusion lemma, the paper states that the safe control problem is solved if there exist \(G_K,\Gamma,L,P\) satisfying
\[
(I-(X_1-C_w)G_K)c_x-c_h=\lambda G_xL,
\]
\[
G_{cl}=\lambda G_x\Gamma,
\]
\[
PA_{cl}=A_x\Gamma,
\]
\[
Pb_{cl}=\lambda b_x+A_xL,
\]
\[
|\Gamma|\bar 1+|L|\le \bar 1.
\]
This is the exact inclusion condition for the constrained-zonotope case [2502.04195].

When the safe set is instead represented as a polytope
\[
\mathcal P(H_s,h_s)=\{x:H_sx\le h_s\},
\]
the same contractive requirement becomes
\[
\mathcal C_{cl}\subseteq \lambda \mathcal P(H_s,h_s),
\]
and the paper derives an LP:
\[
\min_{P,G_K,\rho}\ \rho
\]
subject to
\[
P h_s \le \lambda h_s - H_s c_h - \rho\, l - y,
\]
\[
P H_s = H_s(X_1-C_w)G_K,
\]
\[
\|G_K\|\le \rho,\qquad P\ge 0,
\]
where
\[
y_j=\sum_{i=1}^{s_w}\left|{H_s}_j {G_h}_i\right|,
\]
and
\[
l_j= \max_{\beta} \left( \sum_{i=1}^{s_c} \big[ |h_{sj}|\ \|{G_w}_i\|\ \ 0 \big]\beta_i \right),
\]
subject to auxiliary constrained-zonotope feasibility conditions on \(\beta\). 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 [2502.04195].

## 5. Approximate scaled containment of zonotopes in convex geometry

In convex geometry and oracle-model algorithms, the scaled-zonotope question is formulated as
\[
\max\{s>0:sZ\subseteq Q\},
\]
where the outer body \(Q\subseteq\mathbb R^d\) is described by a membership oracle and the inner body \(Z\subseteq\mathbb R^d\) is a zonotope
\[
Z(W)=\{Wx:x\in\mathbb R^n,\ \|x\|_\infty\le 1\}.
\]
The central issue is no longer an exact linear certificate but the best efficiently computable approximation factor \(s(d)\) and the ability to find a point in \(s(d)Z\setminus Q\) when containment fails [2605.04183].

The main algorithmic guarantee is a randomized polynomial-time algorithm with approximation factor
\[
O(\sqrt d).
\]
More precisely, if \(Z\nsubseteq Q\), the algorithm finds a point in
\[
O(\sqrt d)\cdot Z \setminus Q
\]
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 \(n=O(d\log d)\) generators, the core probabilistic step states that if \(Z(W)\nsubseteq Q\), then for a random \(y\in\{\pm1\}^n\),
\[
\sqrt{\frac{n}{\log n}}\,Wy \notin Q
\]
with probability at least \(1/\mathrm{poly}(n)\), so repeated sampling finds a violating point efficiently [2605.04183].

The lower-bound theory is nearly tight. The paper proves a universal oracle-model lower bound
\[
\Omega\!\left(\sqrt{\frac{d}{\log d}}\right)
\]
for all zonotopes, strengthening earlier tightness results that were known for the hypercube case \(B_\infty^d\). Under Talagrand’s conjecture, which predicts sparsification with only \(O(d/\varepsilon^2)\) generators, the approximation improves to the near-optimal factor
\[
\Theta\!\left(\sqrt{\frac{d}{\log d}}\right).
\]
For constant-\(\Delta\) modular zonotopes, the paper proves this sparsification theorem and obtains the containment approximation factor
\[
O\!\left(\Delta^2\sqrt{\frac d{\log d}}\right),
\]
which is essentially optimal up to constants when \(\Delta\) is constant [2605.04183].

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
\[
P=\langle c,G,E,F,\theta,R\rangle_{CPZ}
=
\left\{x \in \mathbb{R}^d :
x = c + \sum_{i=1}^{n}\prod_{k=1}^{s}\lambda(k)^{E(k,i)}G_{(:,i)},\;
\sum_{i=1}^{q}\prod_{k=1}^{s}\lambda(k)^{R(k,i)}F_{(:,i)} = \theta,\;
\|\lambda\|_\infty \le 1
\right\}.
\]
The inclusion problem \(P_1\subseteq P_2\) is certified by the existence of
\[
\gamma,\ \Gamma,\ \Pi,\ \Psi,\ \psi
\]
satisfying
\[
c_1 = c_2 + G_2 \gamma,\qquad
G_1 = G_2 \Gamma,
\]
\[
\Pi F_1 = F_2 \Psi,\qquad
\Pi \theta_1 = \theta_2 - F_2 \psi,
\]
together with the nonlinear boundedness conditions
\[
\left(E_2^\top \cdot \log\left(|\gamma| + |\Gamma| \cdot \mathbf{1}_{n_1}\right)\right) \le \mathbf{0}_{s_2 \times 1},
\]
\[
\left(R_2^\top \cdot \log\left(|\psi| + |\Psi| \cdot \mathbf{1}_{q_1}\right)\right) \le \mathbf{0}_{s_2 \times 1}.
\]
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 \(P_1\) into that of \(P_2\) [2603.26621].

When the polynomial structure is removed, the CPZ conditions reduce to the familiar constrained-zonotope form:
\[
c_1 = c_2 + G_2 \gamma,\qquad G_1 = G_2 \Gamma,
\]
\[
\Pi F_1 = F_2 \Gamma,\qquad \Pi \theta_1 = \theta_2 - F_2 \gamma,
\]
\[
|\gamma| + |\Gamma|\cdot \mathbf{1}_{n_1} \le \mathbf{1}_{s_2}.
\]
This reduction makes explicit that scaled-zonotope and constrained-zonotope inclusion tests are the linear core from which the CPZ construction generalizes [2603.26621].

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
\[
I=\langle \alpha\mid G_I\Gamma\rangle,\qquad \Gamma=\mathrm{diag}(\gamma_1,\dots,\gamma_{n_I}),\quad \gamma_i\ge 0,
\]
with fixed generator directions and optimized scaling factors. The key containment device is box inclusion:
\[
c_S-|G_S|\mathbf 1\le x\le c_S+|G_S|\mathbf 1,
\]
and for a box \(B=\{b\mid \underline b\le b\le \overline b\}\),
\[
S\subseteq B
\]
is equivalent to
\[
c_S-|G_S|\mathbf 1\ge \underline b,\qquad c_S+|G_S|\mathbf 1\le \overline b.
\]
This converts reachability and safety constraints into linear or convex programs for invariant, viable, and discriminating under-approximations [1901.01006].

For viability and discriminating-kernel computation, the same paper introduces scaled control-authority zonotopes,
\[
\mathcal P(t)=\langle 0\mid G_P\Psi(t)\rangle,\qquad \Psi(t)=\mathrm{diag}(\psi(t)),\quad \psi(t)\ge 0,
\]
and optimizes over \(\gamma\), \(\psi(t)\), 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 \(\Phi\)- or \(\lambda\)-scaled inclusion certificates, but it establishes the same basic principle that zonotope scaling can encode safety-relevant set under-approximations through convex constraints [1901.01006].

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.

Source: https://www.emergentmind.com/topics/scaled-zonotope-inclusion-conditions