Dynamic Quotients and Additive-State Realization

Conceptual map

III-06.01Accessible states
III-06.02Transported target spans
III-06.03Finite closure algorithms
III-06.04Additive frames
III-06.05The Gorman connection

1. Future responses define the state quotient

A report can preserve a current target and lose a future target reached after admissible transitions. Dynamic sufficiency closes the target rows under every transition sequence that the policy experiment permits.

Let $x_{t+1}=A_{u_{t}}x_{t}$ for controls $u_{t}$ in a finite menu $$U$$ , and let the terminal target be $Cx_{H}$ . For a word $w=(u_{0},\ldots,u_{h-1})$ , write $A_{w}=A_{u_{h-1}}\cdots A_{u_{0}}$ . The response row space through horizon $$H$$ is

\[S_{H}=\operatorname{span}\{\operatorname{row}(CA_{w}): |w|\le H\}.\](1)

Definition 1 · Dynamic response equivalence

States $$x$$ and $x^{\prime}$ are equivalent through horizon $$H$$ when $r(x-x^{\prime})=0$ for every row $r\in S_{H}$ . The quotient removes exactly the joint kernel of all tested future responses.

2. Finite closure algorithm

Proposition 1 · Minimal linear response state

Initialize $S_{0}=\operatorname{row}(C)$ and iterate

\[S_{j+1}=S_{j}+\operatorname{span}\{rA_{u}: r\in S_{j}, u\in U\}.\](2)

The sequence stabilizes after at most $$n$$ strict dimension increases. If $$R$$ is a full-row-rank matrix whose rows span the stable space, $$z=Rx$$ reconstructs every target $CA_{w}x$ . Every linear representation with that property has dimension at least $\operatorname{rank}(R)$ .

Proof. The spaces form a nested sequence of subspaces of $\mathbb{R}^{n}$ . A strict inclusion raises dimension, so closure occurs after at most $$n$$ increases. Each tested response row belongs to the stable span and therefore factors through $$R$$ . If a representation $$Bx$$ reconstructs all responses, $\operatorname{ker}(B)$ lies in the joint kernel of $$S$$ ; hence $\operatorname{row}(R)$ lies in $\operatorname{row}(B)$ and $\operatorname{rank}(B)\ge \operatorname{rank}(R)$ . ∎

3. A three-state closure calculation

Take one transition matrix and one terminal row:

\[A=\begin{bmatrix}1 & 1 & 0 \\ 0 & 1 & 1 \\ 0 & 0 & 1\end{bmatrix}, \quad C=\begin{bmatrix}1 & 0 & 0\end{bmatrix}.\](3)

The first three response rows are $$C$$ =[1,0,0], $$CA$$ =[1,1,0], and $CA^{2}$ =[1,2,1]. They are linearly independent, so the dynamic response state has dimension three. The current target $x_{1}$ is one-dimensional; two additional coordinates become relevant as hidden components propagate forward.

Changing the target to $$C$$ =[0,0,1] gives $$CA=C$$ , so the closure dimension is one. The transition law alone does not determine the needed state; the target row enters the construction.

4. Additive realization is a separate condition

Suppose the aggregate state is built from micro states $x^{i}$ . The quotient coordinate $R\sum _{i}x^{i}$ is additive when the underlying state itself aggregates linearly and the transition and target maps respect that aggregation. A low-dimensional response quotient for one aggregate system does not establish a representative-agent interpretation or behavioral aggregation across heterogeneous agents.

Failure case · Dimension without aggregation

A nonlinear coordinate can separate all response-equivalence classes while failing $z(x^{1}+x^{2})=z(x^{1})+z(x^{2})$ . The quotient dimension then describes information compression, while additive implementation remains unresolved.

5. Implementation, exercises, and sources

Store an orthonormal row basis for $S_{j}$ . At each iteration multiply the current basis by every transition matrix, append the candidate rows, and use a rank-revealing QR or SVD with a declared tolerance. Stop when the numerical rank stabilizes and verify the residual of every discarded row.

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Exercises

Verify that the three rows following (3) are independent and that the next row adds no dimension.
Add a second switching matrix and compute the stable span by hand.
Construct a two-agent model with a two-dimensional response quotient that lacks an additive realization.

Partial solutions

1. Stacking $$C$$ , $$CA$$ , and $CA^{2}$ gives an upper-triangular row matrix with determinant one. The next row [1,3,3] cannot raise rank beyond the state dimension and is a linear combination of the first three. 3. A nonlinear one-to-one reparameterization of a two-dimensional quotient preserves response information and generally breaks additivity across agents, supplying the requested separation.

R. E. Kalman (1960), “On the General Theory of Control Systems,” Proceedings of the First IFAC Congress.Reachability and observability subspaces.
Thomas Kailath, Linear Systems, Chapters 6–7.Minimal realizations and invariant subspaces.
William Gorman (1953), “Community Preference Fields,” Econometrica 21, 63–80.Additive aggregation restrictions.

6. Audit checkpoint

State the control menu, horizon, transition ordering, target rows, closure tolerance, stable dimension, decoder, and aggregation operation. A finite response quotient and an additive economic state are distinct conclusions.

7. Scope boundary

The chapter covers finite-dimensional linear switching systems and exact row-space closure. Nonlinear accessibility, infinite-dimensional realization, and full aggregation theory require additional results.