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Sparse Polynomial Zonotope

Definition​

PZ≔{∑i=1h(∏k=1pαkE(k,i))G(⋅,i)D+∑j=1qβjG(⋅,j)I ∣ αk,βj∈[−1,1]}{\cal PZ} \coloneqq \left\{ \sum_{i=1}^{h} ( \prod_{k=1}^{p} \alpha_{k}^{E_{(k,i)}} ) G_{(\cdot, i)}^{D} + \sum_{j=1}^{q} \beta_{j} G_{(\cdot, j)}^{I} \ \big| \ \alpha_{k}, \beta_{j} \in [-1,1] \right\}

shorthand as PZ=<GD,GI,E,U>PZ∈Rn{\cal PZ} = \left< G^{D}, G^{I},E,\mathrm{U}\right>_{\cal PZ} \in \R^{n}, where U∈N>01×p\mathrm{U} \in \N_{> 0}^{1 \times p} introduced as a row vector storing all unambiguous integer identifier for each dependent factor αk\alpha_{k}.

Example​

PZ=<[42124022],[10],[01030011],[01]>PZ{\cal PZ}=\left< \begin{bmatrix} 4 & 2 & 1 & 2 \\ 4 & 0 & 2 & 2 \end{bmatrix}, \begin{bmatrix} 1 \\ 0 \end{bmatrix}, \begin{bmatrix} 0 & 1 & 0 & 3 \\ 0 & 0 & 1 & 1 \end{bmatrix}, \begin{bmatrix} 0 & 1 \end{bmatrix} \right>_{\cal PZ}

defines the set

PZ={[44]+[20]α0+[12]α1+[22]α03α1+[10]β0 ∣ α0,α1,β0∈[−1,1]}{\cal PZ}=\left\{ \begin{bmatrix} 4 \\ 4 \end{bmatrix} + \begin{bmatrix} 2 \\ 0 \end{bmatrix} \alpha_{0} + \begin{bmatrix} 1 \\ 2 \end{bmatrix} \alpha_{1} + \begin{bmatrix} 2 \\ 2 \end{bmatrix} \alpha_{0}^{3} \alpha_{1} + \begin{bmatrix} 1 \\ 0 \end{bmatrix} \beta_{0} \ \bigg| \ \alpha_{0},\alpha_{1}, \beta_{0} \in [-1,1] \right\}
from pyrat.geometry import SPZono
from pyrat.util.visualization import plot

# TODO

Operations​

Exact Addition​

Cartesian Product​

Quadratic Map​

Convex Hull​

Order Reduction​

Restructure​

Arithmetic​

addition or '+'
  • with another spzono
    PZ0⊕PZ1=<[G0DG1D],[G0IG1I],[E00p0,h10p1,h0E1],U(p0+p1)>PZ{\cal PZ}_{0} \oplus {\cal PZ}_{1} = \left< \begin{bmatrix} G_{0}^{D} & G_{1}^{D} \end{bmatrix}, \begin{bmatrix} G_{0}^{I} & G_{1}^{I} \end{bmatrix}, \begin{bmatrix} E_{0} & \bold{0}^{p_{0},h_{1}} \\ \bold{0}^{p_{1},h_{0}} & E_{1} \end{bmatrix}, \mathrm{U}(p_{0}+p_{1}) \right>_{\cal PZ}
  • with another zonotope
    PZ⊕Z=<[czGD],[GI,0Gz],[0E0],U>PZ{\cal PZ} \oplus {\cal Z} = \left< \begin{bmatrix} c_{z} & G_{D} \end{bmatrix}, \begin{bmatrix} G_{I,0} & G_{z} \end{bmatrix}, \begin{bmatrix} \bold{0} & E_{0} \end{bmatrix},\mathrm{U} \right>_{\cal PZ}
Matrix multiplication or '@'
M⊗PZ=<M⊗G,M⊗GI,E,U>PZM \otimes {\cal PZ} = \left< M \otimes G,M \otimes G_{I},E,\mathrm{U} \right>_{PZ}

References​

[1]: Kochdumper, N., & Althoff, M. (2020). Sparse polynomial zonotopes: A novel set representation for reachability analysis. IEEE Transactions on Automatic Control, 66(9), 4043-4058.