[Paper Review] Exact Volume of Zonotopes Generated by a Matrix Pair

Mingwang Zhao|arXiv (Cornell University)|Apr 12, 2020

Point processes and geometric inequalities13 references4 citations

TL;DR

This paper presents a recursive, analytical method to compute the exact volume of zonotopes generated by a matrix pair {A, B} in discrete-time linear systems, with applications to finite- and infinite-time controllable and reachable regions. The approach leverages eigenvalue structure—particularly real, distinct, and non-negative eigenvalues—to achieve significant reductions in computational complexity, validated through numerical experiments showing exact volume agreement and orders-of-magnitude speedups via optimized recursion.

ABSTRACT

In this article, we define a class of special zonotopes generated by a matrix pair with finite-interval parameters. We discuss the relationship between the volume of these zonotopes and the controllability of one aspect (the volume of the controllable region) of the dynamic systems. We present a corollary and develop an effective recursive method to compute the volume of the special zonotopes. Furthermore, we develop two recursive and analytical volume-computation methods for the finite- and infinite-time controllable regions with real eigenvalues. We conduct numerical experiments to demonstrate the effectiveness of these new volume-computation methods for zonotopes and regions.

Motivation & Objective

• To define and analyze a class of zonotopes generated by matrix pairs {A,B} with finite-interval parameters in discrete-time linear systems.
• To establish a connection between the volume of these zonotopes and the controllability of dynamic systems, particularly the size of the reachable and controllable regions.
• To develop efficient, recursive algorithms for computing the exact volume of zonotopes associated with finite- and infinite-time controllable and reachable regions.
• To reduce computational complexity in volume computation by exploiting eigenvalue structure, especially for systems with real, distinct, and non-negative eigenvalues.
• To validate the proposed methods through numerical experiments demonstrating accuracy and efficiency gains over standard determinant-based computation.

Proposed method

• Proposes a zonotope model E_q(P_N) defined as the Minkowski sum of vectors from the controllability matrix P_N = [B, AB, ..., A^{N-1}B], with parameters c_i ∈ [0,1].
• Introduces a recursive algorithm based on the rank q of P_N and the eigenvalue distribution of A, particularly for cases with n distinct real eigenvalues λ_i ≥ 0 or λ_i ∈ [0,1).
• Derives Corollary 1 and Theorem 2 to compute finite-time reachable region volumes using recursive determinant updates, reducing the number of required n×n determinant computations.
• Develops Theorem 3 for infinite-time controllable region volume computation by analyzing the limit of the reachable region as N→∞, assuming spectral radius ρ(A) < 1.
• Employs recursive equations (19) and (20) to compute volumes without recalculating full determinants at each step, significantly lowering computational cost.
• Applies the method to zonotopes with generators from the controllability matrix, enabling exact volume computation for geometric analysis of controllability.
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Experimental results

Research questions

• RQ1 How can the volume of zonotopes generated by a matrix pair {A,B} be computed exactly and efficiently in discrete-time linear systems with bounded inputs?
• RQ2 What is the relationship between the volume of the reachable/controllable region and the controllability of a linear time-invariant system?
• RQ3 Can recursive algorithms be designed to reduce the computational complexity of volume computation for zonotopes arising from controllability matrices?
• RQ4 How do different eigenvalue distributions of matrix A—particularly real, distinct, and non-negative eigenvalues—affect the efficiency and structure of volume computation algorithms?
• RQ5 What is the performance gain of the proposed recursive methods compared to standard determinant-based volume computation in terms of time and operations?

Key findings

• The proposed recursive methods (Corollary 1 and Theorem 2) compute the exact volume of finite-time reachable regions with identical results to the standard determinant-based method (Theorem 1), confirming correctness.
• For N=800, the number of required n×n determinant computations was reduced from 8.501×10^7 (Theorem 1) to 3.188×10^5 (Corollary 1), a reduction of over 99%.
• The computational time for N=800 dropped from 3.585×10^2 seconds (Theorem 1) to 1.692 seconds (Corollary 1), a speedup of nearly 212x.
• The infinite-time controllable region volume was computed in just 1.328×10^{-3} seconds with only 26 multiplications (Theorem 3), demonstrating extreme efficiency for the limit case.
• The method scales well: for N=500, the recursive method (Theorem 2) required only 1.243×10^5 determinant computations and 0.6586 seconds, compared to 2.071×10^7 and 85.19 seconds for the standard method.
• The results confirm that the recursive approach maintains numerical accuracy while drastically reducing computational complexity, especially for large N and high-dimensional systems.