1 Matrix-weighted graphs
1.1 From scalar- to matrix-weighted graphs
1.2 Matrix-weighted graph
1.3 Notes and references
2 Matrix-weighted Laplacian
2.1 Undirected and leader-follower topologies
2.2 Directed topologies
2.2.1 General properties
2.2.2 Directed acyclic leader-follower graphs
2.2.2 Generalized balance graphs
2.2.3 Directed cycles
2.3 Notes and references
3 Physical interpretation and motivational examples
3.1 Matrix weights
3.1.1 Multi-layer and social networks
3.1.2 Networked input-output economic model
3.1.3 Formation control
3.1.4 Network localization
3.1.5 Mechanical- and electrical networks
3.2 Electrical analogies of matrix-weighted networks
3.2.1 Voltage and current flow
3.2.2 Effective resistance
3.2.3 Energy
3.3 Notes and references
4 Connectivity
4.1 Algebraic criteria for connectedness
4.2 Algebraic graph conditions for connectedness and clustering
4.3 Warshall algorithm for matrix-weighted graphs
4.4 Discussions and examples
4.5 Notes and references
5 Spanning trees
5.1 Expanded and quasi-expanded positive spanning trees
5.2 Matrix-tree theorem
5.3 Expanded positive spanning trees with minimum edges
5.4 Generating matrix-weighted graphs with prescribed number of clusters
5.5 Notes and references
6 Quantitative measures
6.1 The energy of a matrix-weighted graph
6.2 The matrix-weighted resistance distances
6.3 Vertex- and edge importance
6.4 Notes and references
7 Matrix-weighted consensus
7.1 Matrix-weighted consensus
7.1.1 Basic definitions
7.1.2 Matrix-weighted consensus algorithm
7.1.3 Leader-follower matrix-weighted consensus
7.2 Matrix-weighted consensus of double integrators
7.2.1 Generalized balance topology
7.2.2 Directed cycles
7.3 Matrix-weighted consensus of higher-order integrators
7.4 Notes and references
8 Discrete-time and randomized algorithms
8.1 Discrete-time matrix-weighted consensus
8.1.1 Leaderless topology
8.1.2 Leader-follower topology
8.2 Randomized matrix-weighted consensus
8.2.1 The expected matrix-weighted graph
8.2.2 The randomized algorithm
8.2.3 Convergence analysis
8.2.4 Application to bearing-based network localization
8.3 Notes and references
9 Accelerated algorithms
9.1 Fast convergence matrix-weighted consensus algorithms
9.1.1 Finite-time and fixed-time stability
9.1.2 Finite-time matrix-weighted consensus
9.1.3 Fixed-time matrix-weighted consensus
9.2 Accelerated matrix-weighted consensus algorithms using memories
9.2.1 Continuous-time algorithm
9.2.2 Discrete-time algorithm
9.3 Notes and references
10 Robustness
10.1 Adaptive matrix-weighted consensus algorithms
10.1.1 Single-integrators with uncertainties
10.1.2 Double-integrators with matched uncertainties
10.1.3 Higher-order integrators with matched uncertainties
10.2 Disturbance observers
10.2.1 Single-integrator agents with disturbance
10.2.2 Double-integrator agents
10.2.3 Higher-order integrator agents with mismatched disturbances
10.3 Sliding-mode
10.3.1 Single-integrator agents
10.3.2 Double-integrator agents
10.4 Notes and references
11 Tracking
11.1 The matrix-weighted consensus tracking problem
11.2 Leaders with constant velocity
11.2.1 Single-integrator follower agents: PI consensus tracking
11.2.2 Double-integrator follower agents: Consensus tracking without velocity measurement
11.3 Disturbance observer
11.3..1 Single-integrator follower agents
11.3.2 Double-integrator follower agents
11.4 Sliding-mode
11.4.1 Single-integrator follower agents
11.4.2 Double-integrator follower agents
11.5 Notes and references
12 Time-delays
12. 1 Matrix-weighted consensus with a uniform time-delay
12.2 Matrix-weighted consensus with heterogeneous constant time-delays
12.3 Adaptive algorithm for unknown constant time-delays
12.3.1 Adaptive matrix-weighted consensus algorithm
12.3.2 Matrix-weighted consensus with heterogeneous time-delays
12.4 Notes and references
13 Synchronization
13.1 The state synchronization problem
13.2 Leaderless synchronization
13.2.1 Observer-based adaptive synchronization algorithm
13.2.2 Convergence analysis
13.3 Leader-follower synchronization of homogeneous LTI agents
13.3.1 Observer-based adaptive synchronization algorithm
13.3.2 Convergence analysis
13.4 Notes and references
14 Scaling matrices
14.1 Matrix-scaled consensus over undirected matrix-weighted networks
14.1.1 Matrix-scaled consensus
14.1.2 Geometry of the matrix-scaled consensus space in R^2
14.1.3 Properties of the matrix-scaled Laplacian
14.1.4 Matrix-scaled consensus algorithm
14.2 Multi-dimensional Altafini model with scaling matrices
14.2.1 Signed matrix-weighted graphs
14.2.2 Bipartite consensus over undirected signed matrix-weighted graphs
14.2.3 Bipartite consensus over generalized balanced signed matrix-weighted graphs
14.2.4 Matrix-scaled consensus over signed undirected matrix-weighted graphs
14.3 Notes and references
15 Networked input-output economic model
15.1 Modeling
15.1.1 Perron-Frobenius theorem
15.1.2 Leontief's input-output economic model
15.1.3 The networked input-output model and problem formulation
15.2 Distributed computation of the equilibrium price structure
15.2.1 The proposed algorithm
15.2.2 The closed networked input-output model
15.2.3 The open networked input-output model
15.3 Simulation results
15.3.1 A closed input-output economic network
15.3.2 An open input-output economic network
15.4 Notes and references
16 Bearing-only formation control
16.1 Bearing vectors and bearing-only measurement based navigation
16.2 Leader-first follower formations
16.2.1 Bearing-based Henneberg construction
16.2.2 Bearing-only control of leader-first follower formation
16.3 Notes and references
A Linear algebra
B Control theory
C Proofs
Matlab & Simulink Simulations: [Download at Github]
The materials in the book can be used as the main reference for a course on Matrix-weighted graphs [Sample teaching materials].
Chapter 4, p. 65 & p. 75: The notation (union) $\union_i^n ker(A_i)$ should be understood as the subspace sum $ker(A_1)+\ldots+ker(A_n)$. The analysis is still valid, however, this notational issue that may cause confusion.
Chapter 5, p. 91: in the proof of Corollary 5.1, the term in row 1, column 2 of the determinant should be $-A_2$ instead of $-A_1$. The proof after the change is still valid.
Chapter 6, p. 110 and Appendix C, p. 348: in equation of $A_a$, the first summand should be $(A_2^{-1}+A_1^{-1})^{-1}(A_1+A_2+A_3)^{-1}A_1$.
Chapter 9, p. 174, Equation (9.26): insert $\beta_r $ before $\mathbf{x}[k-r]$. The literature review in this chapter overlooked several works on discrete-time accelerated consensus algorithms with multiple memory elements, e.g., see https://ieeexplore.ieee.org/document/10066203 and the references therein.