[Paper Review] Analyzing Controllability of Bilinear Systems on Symmetric Groups: Mapping Lie Brackets to Permutations

Wei Zhang, Jr-Shin Li|arXiv (Cornell University)|Aug 7, 2017

Mitochondrial Function and Pathology27 references3 citations

TL;DR

This paper proposes a novel algebraic framework that maps Lie bracket operations of vector fields in bilinear systems to permutations on symmetric groups, enabling efficient controllability analysis via permutation cycle structures. The key contribution is a necessary and sufficient condition for controllability on SO(n) based on cycle length, with extensions to graph-based systems like multi-agent networks and Markov chains through graph connectivity.

ABSTRACT

Bilinear systems emerge in a wide variety of fields as natural models for dynamical systems ranging from robotics to quantum dots. Analyzing controllability of such systems is of fundamental and practical importance, for example, for the design of optimal control laws, stabilization of unstable systems, and minimal realization of input-output relations. Tools from Lie theory have been adopted to establish controllability conditions for bilinear systems, and the most notable development was the Lie algebra rank condition (LARC). However, the application of the LARC may be computationally expensive for high-dimensional systems. In this paper, we present an alternative and effective algebraic approach to investigate controllability of bilinear systems. The central idea is to map Lie bracket operations of the vector fields governing the system dynamics to permutation multiplications on a symmetric group, so that controllability and controllable submanifolds can be characterized by permutation cycles. The method is further applicable to characterize controllability of systems defined on undirected graphs, such as multi-agent systems with controlled couplings between agents and Markov chains with tunable transition rates between states, which in turn reveals a graph representation of controllability through the graph connectivity.

Motivation & Objective

• Address the computational intractability of the Lie algebra rank condition (LARC) for high-dimensional bilinear systems.
• Develop an alternative, algebraic approach to controllability analysis that avoids costly iterative Lie bracket computations.
• Establish a correspondence between Lie bracket operations on so(n) and permutation multiplications on the symmetric group S_n.
• Characterize controllability and controllable submanifolds using permutation cycle structures.
• Extend the framework to systems defined on undirected graphs, such as multi-agent systems and Markov chains, revealing a graph-theoretic representation of controllability via connectivity.

Proposed method

• Map vector fields of right-invariant bilinear systems on SO(n) to permutations in the symmetric group S_n via a Lie bracket-to-permutation correspondence.
• Define a binary operation * on S_n that models the Lie bracket operation, preserving cycle structure and orbit behavior.
• Use cycle decomposition and orbit union to determine the order and type of resulting permutations after * operations.
• Introduce an equivalence relation ~ on S_n based on cycle type (conjugacy class), enabling abstraction over permutation details.
• Establish that the * operation is commutative and associative modulo ~, allowing consistent algebraic manipulation.
• Derive a necessary and sufficient controllability condition: the system is controllable if and only if the generated permutations span a single cycle of maximal length n.

Experimental results

Research questions

• RQ1 Can Lie bracket operations in bilinear systems be systematically mapped to permutation operations on symmetric groups to simplify controllability analysis?
• RQ2 What algebraic properties of the permutation operation * ensure consistency and correctness in characterizing controllability?
• RQ3 How can the cycle structure of permutations derived from Lie brackets determine the controllability of systems on SO(n)?
• RQ4 To what extent can this framework be generalized to systems defined on undirected graphs, such as multi-agent systems and Markov chains?
• RQ5 What is the relationship between graph connectivity and controllability in systems modeled via permutation cycles?

Key findings

• The system on SO(n) is controllable if and only if the permutation cycles generated by the * operation span a single cycle of length n.
• The * operation on S_n preserves cycle structure and orbit union, enabling the construction of higher-order cycles from lower-order ones.
• The equivalence relation ~ based on cycle type ensures that the * operation is well-defined and commutative modulo ~.
• The associativity of * modulo ~ is proven through case analysis on overlapping and disjoint orbits of cycles.
• The framework reveals that controllability of graph-based systems is equivalent to the connectivity of the underlying graph, linking algebraic structure to network topology.
• The method provides a computationally efficient alternative to the LARC, avoiding iterative Lie bracket computation by leveraging permutation algebra.