Behavior of a Signal Flow Graph

definition example

The behavior of a Signal Flow Graph $g:m\to n$ over a Rig $R$ is the set of input/output pairs

$$B(g):=\{(x,S(g)(x))\mid x\in R^m\}\subseteq R^m\times R^n,$$

the graph of the linear map given by the matrix $S(g)$ (Eq. 5.75). For the copy icon, $B=\{(x,(x,x))\}$. The mirror image $g^{\mathrm{op}}:n\to m$ of an icon has the transposed relation $B(g^{\mathrm{op}}):=\{(S(g)(x),x)\}$ (Eq. 5.76). Behaviours compose as relations: $B_1;B_2=\{(x,z)\mid \exists y.(x,y)\in B_1,(y,z)\in B_2\}$ (Eq. 5.78), giving a prop functor $B:{\mathbf{SFG}}_R^{+}\to {\mathbf{Rel}}_R$ on non-simplified signal flow graphs (Eq. 5.81).

Sources: 7 Sketches §5.4.3 (“The behavioral approach”, “Mirror image of an icon”, “Combining directions”), Exercises 5.77, 5.80, 5.82–5.85, Theorem 5.87; Willems’ behavioural approach [Wil07]; Kittenlab Lecture 14 (a model as an exclusion law: a subset of a universum).

• Reversed add: $\{(x,(y,z))\mid x=y+z\}$; reversed copy: $\{((y,z),x)\mid x=y=z\}$ (7S Exercise 5.77).
• $B(g;h^{\mathrm{op}})=\{(x,y)\mid S(g)x=S(h)y\}$ and $B(g^{\mathrm{op}};h)=\{(S(g)x,S(h)x)\}$ (7S Exercise 5.82, 7S Exercise 5.83): systems of linear equations and parametrized subspaces.
• Behaviours are linear relations; kernel = compose with reversed zeros, image = compose with reversed discards (7S Exercise 5.84).
• Cup and cap behaviours $\{(0,(x,x))\}$, $\{((x,x),0)\}$ make ${\mathbf{Rel}}_R$ compact closed (Theorem 5.87). See Graphical Linear Algebra.