Guidance Matrix Projection (GMP) Method

• The Guidance Matrix Projection (GMP) Method is a two-stage approach that projects initial opinions onto the consensus subspace T_P and then applies standard DeGroot iteration to ensure convergence.
• It introduces an orthogonal projector constructed from the system’s Laplacian, yielding a regularized power limit operator that guarantees a rank-1 consensus mapping.
• The method generalizes the DeGroot model by recovering consensus in scenarios where the typical influence matrix regularity conditions fail, offering robust solutions for multi-agent dynamics.

The Guidance Matrix Projection (GMP) Method, also referred to as the “Projection Method” for consensus, is a two-stage algorithmic framework designed to ensure consensus in multi-agent discrete-time systems modeled by the DeGroot process, especially when the standard consensus conditions on the influence matrix are not met. GMP facilitates convergence to consensus by orthogonally projecting initial opinions onto a specific subspace of consensus-convergent states and then applying the standard DeGroot iteration. This approach enables the recovery of consensus in scenarios where the standard DeGroot dynamics would otherwise fail, and naturally leads to the definition of a “regularized power limit” of the influence matrix (Agaev et al., 2011).

1. Fundamentals of the DeGroot Consensus Model

The DeGroot model describes the dynamics of $n$ agents, each holding an opinion $x_i^{(t)}$ at time $t$, with the vector of opinions denoted $x^{(t)} \in \mathbb{R}^n$. The evolution is governed by a row-stochastic influence matrix $P \in \mathbb{R}^{n\times n}$. The update rule is

$x^{(t+1)} = P x^{(t)}, \quad t=0,1,2,\ldots$

where $P_{ij} \geq 0$ and $\sum_j P_{ij} = 1$ for all $i$. Consensus is achieved if, for every initial $x^{(0)}$, the sequence $\{x^{(t)}\}$ converges to a constant vector $s \mathbf{1}$, with $\mathbf{1}=(1,\ldots,1)^T$.

A necessary and sufficient condition for consensus under all initial conditions is that $P$ is regular (stochastic, indecomposable, and aperiodic; SIA): this ensures $\lim_{t\to\infty}P^t = P^\infty$ exists and all rows of $P^\infty$ coincide.

2. Consensus-Convergence Subspace $T_P$

When $P$ fails to be regular, consensus may still be attainable for a subset of initial conditions. The subspace $T_P \subseteq \mathbb{R}^n$ is defined as

$$T_P = \{ x \in \mathbb{R}^n : P^\infty x = a\mathbf{1}\ \text{for some}\ a \in \mathbb{R} \}$$

Theorem 1 states that $$T_P = R(L) \oplus \operatorname{span}\{\mathbf{1}\}$$ where $L = I - P$. Alternatively,

$$T_P = N(P^\infty) + \operatorname{span}\{\mathbf{1}\} = R(U)$$

for any full-column-rank matrix $U$ constructed by omitting one column in $L$ for each final strongly-connected class of the communication graph and appending $\mathbf{1}$ as an extra column.

3. Construction of the Orthogonal Projector onto $T_P$

The orthogonal projection of an arbitrary initial opinion vector $x^{(0)}$ onto $T_P$ is realized via

$$\operatorname{Proj}_{T_P}(x^{(0)}) = U (U^T U)^{-1} U^T x^{(0)}$$

where $U\in\mathbb{R}^{n\times m}$ with $m = \dim T_P$ is a basis for $T_P$. The projector $\operatorname{Proj}_{T_P}$ is symmetric and idempotent, and its range is exactly $T_P$.

The procedure for constructing $U$ involves, for each final strongly connected component in the digraph of $P$, deleting a corresponding column from $L$ and then appending the all-ones vector as an additional column.

4. The Guidance Matrix Projection Algorithm

The GMP method consists of two sequential steps:

1. Preequalization (Guidance Stage): Compute the orthogonal projection $$\hat{x} = \operatorname{Proj}_{T_P}(x^{(0)})\in T_P$$.
2. Iterative Consensus (DeGroot Iteration): Run the standard DeGroot iteration:

$x^{(t+1)} = P x^{(t)}$

but with initial state $\hat{x}$. As $\hat{x} \in T_P$, the resulting sequence is guaranteed to converge to a consensus vector.

A concise schematic of the GMP method is provided below:

Stage	Operation	Output
1. Project	$\hat{x} = \operatorname{Proj}_{T_P}(x^{(0)})$	$\hat{x} \in T_P$
2. Iterate	$x^{(t+1)} = P x^{(t)}$ with $x^{(0)} = \hat{x}$	Consensus vector

This process guarantees consensus regardless of the regularity of $P$, provided the projection step is performed.

5. Regularized Power Limit and Operator Structure

When the power limit $P^\infty = \lim_{t \to \infty} P^t$ exists (e.g., $P$ is proper: having no spectrum on the unit circle except a single $1$), the combined GMP map takes the form

$$x^{(0)} \mapsto \lim_{t\to\infty} x^{(t)} = P^\infty \operatorname{Proj}_{T_P} x^{(0)}$$

Defining $M := P^\infty \operatorname{Proj}_{T_P}$, the operator $M$ is rank-1, stochastic, and idempotent onto $\operatorname{span}\{\mathbf{1}\}$. Thus, $M$ acts as the unique “regularized power limit” of $P$. In the case where $P$ is already regular, $\operatorname{Proj}_{T_P} = I$ and $M = P^\infty$, recovering the standard DeGroot consensus map.

Key operator relations include:

• $P^\infty L = 0$
• $M^2 = M$ (idempotent)
• All rows of $M$ coincide

A plausible implication is that $M$ provides a canonical way to substitute $P^\infty$ by a regularized rank-one operator even when $P$ lacks regularity.

6. Exemplifying the GMP Method

For a seven-agent system with a block influence matrix

$$P = \begin{bmatrix} P_B & * \ 0 & D \end{bmatrix}, \quad L = \begin{bmatrix} L_B & * \ 0 & D \end{bmatrix}$$

where the first $b=5$ agents form two strongly connected classes and agents 6,7 are nonbasic, the following procedure is executed:

1. Construct $U$ by deleting one $L$ column per strongly connected class, appending $\mathbf{1}$.
2. Compute $S = U(U^T U)^{-1} U^T$ as the projector for $T_P$.
3. Calculate $P^\infty$ so nonbasic agent components vanish.
4. Form $M = P^\infty S$.

In this example, all rows of $M$ are identical to $a^T = (.2364, .2364, .1182, .1636, .2455, 0, 0)$, ensuring for any $x^{(0)}$, $M x^{(0)} = (a^T x^{(0)})\cdot \mathbf{1}$. Thus, a one-time preequalization followed by DeGroot iteration always yields consensus, regardless of the communication topology failing the usual arborescence condition (Agaev et al., 2011).

7. Theoretical Properties and Implications

Key theoretical findings include:

• The GMP method is fully characterized by the readily computable projection matrix $\operatorname{Proj}_{T_P}$.
• The Iterates $x^{(t)}$ converge to consensus if and only if $$x^{(0)}\in R(L)\oplus \operatorname{span}\{\mathbf{1}\}$$.
• $M=P^\infty S$ is a rank-1, stochastic, idempotent operator mapping all inputs to consensus vectors.
• The GMP method generalizes the power limit concept, extending robust consensus guarantees to systems with nonregular influence matrices.
• Spectral preconditions for the existence of $P^\infty$ require $P$ to be proper. If not, consensus cannot be guaranteed via this method.

The GMP method hence provides a systematic solution for consensus in multi-agent discrete-time systems where conventional DeGroot iterations are insufficient, offering a regularized, analytically well-defined operator that enforces consensus through a deterministic preequalization transform and standard linear iteration (Agaev et al., 2011).