Choosing the variables

In general, there are several ways of choosing the variables in a complex system. Corresponding to each choice, a system of equations can be written and each system of equations can be represented in a graph. This formulation of the equations becomes direct and automatic if one has at his disposal techniques which permit the drawing of a graph directly from the schematic diagram of the system under study. The structure of the graphs thus obtained is related in a simple manner to the topology of the schematic diagram, and it becomes unnecessary to consider the equations, even implicitly, to obtain the graph. In some cases, one has simply to imagine the flow graph in the schematic diagram and the desired answers can be obtained without even drawing the flow graph.

— Robichaud^[11]

Non-uniqueness

Robichaud et al. wrote: "The signal flow graph contains the same information as the equations from which it is derived; but there does not exist a one-to-one correspondence between the graph and the system of equations. One system will give different graphs according to the order in which the equations are used to define the variable written on the left-hand side."^[9] If all equations relate all dependent variables, then there are n! possible SFGs to choose from.^[12]

Basic components

Elements and constructs of a signal flow graph.

A linear signal flow graph is related to a system of linear equations^[16] of the following form:

${\displaystyle {\begin{aligned}x_{\mathrm {j} }&=\sum _{\mathrm {k} =1}^{\mathrm {N} }t_{\mathrm {jk} }x_{\mathrm {k} }\end{aligned}}}$

where ${\displaystyle t_{jk}}$ = transmittance (or gain) from ${\displaystyle x_{k}}$ to ${\displaystyle x_{j}}$.

The figure to the right depicts various elements and constructs of a signal flow graph (SFG).^[17]

Exhibit (a) is a node. In this case, the node is labeled ${\displaystyle x}$. A node is a vertex representing a variable or signal.

A source node has only outgoing branches (represents an independent variable). As a special case, an input node is characterized by having one or more attached arrows pointing away from the node and no arrows pointing into the node. Any open, complete SFG will have at least one input node.

An output or sink node has only incoming branches (represents a dependent variable). Although any node can be an output, explicit output nodes are often used to provide clarity. Explicit output nodes are characterized by having one or more attached arrows pointing into the node and no arrows pointing away from the node. Explicit output nodes are not required.

A mixed node has both incoming and outgoing branches.

Exhibit (b) is a branch with a multiplicative gain of ${\displaystyle m}$. The meaning is that the output, at the tip of the arrow, is ${\displaystyle m}$ times the input at the tail of the arrow. The gain can be a simple constant or a function (for example: a function of some transform variable such as ${\displaystyle s}$, ${\displaystyle \omega }$, or ${\displaystyle z}$, for Laplace, Fourier or Z-transform relationships).

Exhibit (c) is a branch with a multiplicative gain of one. When the gain is omitted, it is assumed to be unity.

Exhibit (d) ${\displaystyle V_{in}}$ is an input node. In this case, ${\displaystyle V_{in}}$ is multiplied by the gain ${\displaystyle m}$.

Exhibit (e) ${\displaystyle I_{out}}$ is an explicit output node; the incoming edge has a gain of ${\displaystyle m}$.

Exhibit (f) depicts addition. When two or more arrows point into a node, the signals carried by the edges are added.

Exhibit (g) depicts a simple loop. The loop gain is ${\displaystyle A\times m}$.

Exhibit (h) depicts the expression ${\displaystyle Z=aX+bY}$.

Terms used in linear SFG theory also include:^[17]

• Path. A path is a continuous set of branches traversed in the direction indicated by the branch arrows.
  • Open path. If no node is re-visited, the path is open.
  • Forward path. A path from an input node (source) to an output node (sink) that does not re-visit any node.
• Path gain: the product of the gains of all the branches in the path.
• Loop. A closed path. (it originates and ends on the same node, and no node is touched more than once).
• Loop gain: the product of the gains of all the branches in the loop.
• Non-touching loops. Non-touching loops have no common nodes.
• Graph reduction. Removal of one or more nodes from a graph using graph transformations.
  • Residual node. In any contemplated process of graph reduction, the nodes to be retained in the new graph are called residual nodes.^[2]
• Splitting a node. Splitting a node corresponds to splitting a node into two half nodes, one being a sink and the other a source.^[18]
• Index: The index of a graph is the minimum number of nodes which have to be split in order to remove all the loops in a graph.
  • Index node. The nodes that are split to determine the index of a graph are called index nodes, and in general they are not unique.

Systematic reduction to sources and sinks

A signal-flow graph may be simplified by graph transformation rules.^[19][20][21] These simplification rules are also referred to as signal-flow graph algebra.^[22] The purpose of this reduction is to relate the dependent variables of interest (residual nodes, sinks) to its independent variables (sources).

The systematic reduction of a linear signal-flow graph is a graphical method equivalent to the Gauss-Jordan elimination method for solving linear equations.^[23]

The rules presented below may be applied over and over until the signal flow graph is reduced to its "minimal residual form". Further reduction can require loop elimination or the use of a "reduction formula" with the goal to directly connect sink nodes representing the dependent variables to the source nodes representing the independent variables. By these means, any signal-flow graph can be simplified by successively removing internal nodes until only the input and output and index nodes remain.^[24][25] Robichaud described this process of systematic flow-graph reduction:

The reduction of a graph proceeds by the elimination of certain nodes to obtain a residual graph showing only the variables of interest. This elimination of nodes is called "node absorption". This method is close to the familiar process of successive eliminations of undesired variables in a system of equations. One can eliminate a variable by removing the corresponding node in the graph. If one reduces the graph sufficiently, it is possible to obtain the solution for any variable and this is the objective which will be kept in mind in this description of the different methods of reduction of the graph. In practice, however, the techniques of reduction will be used solely to transform the graph to a residual graph expressing some fundamental relationships. Complete solutions will be more easily obtained by application of Mason's rule.^[26] The graph itself programs the reduction process. Indeed a simple inspection of the graph readily suggests the different steps of the reduction which are carried out by elementary transformations, by loop elimination, or by the use of a reduction formula.^[26]

— Robichaud, Signal flow graphs and applications, 1962

For digitally reducing a flow graph using an algorithm, Robichaud extends the notion of a simple flow graph to a generalized flow graph:

Before describing the process of reduction...the correspondence between the graph and a system of linear equations ... must be generalized...The generalized graphs will represent some operational relationships between groups of variables...To each branch of the generalized graph is associated a matrix giving the relationships between the variables represented by the nodes at the extremities of that branch...^[27] The elementary transformations [defined by Robichaud in his Figure 7.2, p. 184] and the loop reduction permit the elimination of any node j of the graph by the reduction formula:[described in Robichaud's Equation 7-1]. With the reduction formula, it is always possible to reduce a graph of any order... [After reduction] the final graph will be a cascade graph in which the variables of the sink nodes are explicitly expressed as functions of the sources. This is the only method for reducing the generalized graph since Mason's rule is obviously inapplicable.^[28]

— Robichaud, Signal flow graphs and applications, 1962

The definition of an elementary transformation varies from author to author:

• Some authors only consider as elementary transformations the summation of parallel-edge gains and the multiplication of series-edge gains, but not the elimination of self-loops^[23][29]
• Other authors consider the elimination of a self-loop as an elementary transformation^[30]

Parallel edges. Replace parallel edges with a single edge having a gain equal to the sum of original gains.

The graph on the left has parallel edges between nodes. On the right, these parallel edges have been replaced with a single edge having a gain equal to the sum of the gains on each original edge.

The equations corresponding to the reduction between N and node I₁ are:

${\displaystyle {\begin{aligned}N&=I_{\mathrm {1} }f_{\mathrm {1} }+I_{\mathrm {1} }f_{\mathrm {2} }+I_{\mathrm {1} }f_{\mathrm {3} }+...\\N&=I_{\mathrm {1} }(f_{\mathrm {1} }+f_{\mathrm {2} }+f_{\mathrm {3} })+...\\\end{aligned}}}$

Outflowing edges. Replace outflowing edges with edges directly flowing from the node's sources.

The graph on the left has an intermediate node N between nodes from which it has inflows, and nodes to which it flows out. The graph on the right shows direct flows between these node sets, without transiting via N.

For the sake of simplicity, N and its inflows are not represented. The outflows from N are eliminated.

The equations corresponding to the reduction directly relating N's input signals to its output signals are:

${\displaystyle {\begin{aligned}N&=I_{\mathrm {1} }f_{\mathrm {1} }+I_{\mathrm {2} }f_{\mathrm {2} }+I_{\mathrm {3} }f_{\mathrm {3} }\\O_{\mathrm {1} }&=g_{\mathrm {1} }N\\O_{\mathrm {2} }&=g_{\mathrm {2} }N\\O_{\mathrm {3} }&=g_{\mathrm {3} }N\\O_{\mathrm {1} }&=g_{\mathrm {1} }(I_{\mathrm {1} }f_{\mathrm {1} }+I_{\mathrm {2} }f_{\mathrm {2} }+I_{\mathrm {3} }f_{\mathrm {3} })\\O_{\mathrm {2} }&=g_{\mathrm {2} }(I_{\mathrm {1} }f_{\mathrm {1} }+I_{\mathrm {2} }f_{\mathrm {2} }+I_{\mathrm {3} }f_{\mathrm {3} })\\O_{\mathrm {3} }&=g_{\mathrm {3} }(I_{\mathrm {1} }f_{\mathrm {1} }+I_{\mathrm {2} }f_{\mathrm {2} }+I_{\mathrm {3} }f_{\mathrm {3} })\\O_{\mathrm {1} }&=I_{\mathrm {1} }f_{\mathrm {1} }g_{\mathrm {1} }+I_{\mathrm {2} }f_{\mathrm {2} }g_{\mathrm {1} }+I_{\mathrm {3} }f_{\mathrm {3} }g_{\mathrm {1} }\\O_{\mathrm {2} }&=I_{\mathrm {1} }f_{\mathrm {1} }g_{\mathrm {2} }+I_{\mathrm {2} }f_{\mathrm {2} }g_{\mathrm {2} }+I_{\mathrm {3} }f_{\mathrm {3} }g_{\mathrm {2} }\\O_{\mathrm {3} }&=I_{\mathrm {1} }f_{\mathrm {1} }g_{\mathrm {3} }+I_{\mathrm {2} }f_{\mathrm {2} }g_{\mathrm {3} }+I_{\mathrm {3} }f_{\mathrm {3} }g_{\mathrm {3} }\\\end{aligned}}}$

Zero-signal nodes.

Eliminate outflowing edges from a node determined to have a value of zero.

If the value of a node is zero, its outflowing edges can be eliminated.

Nodes without outflows.

Eliminate a node without outflows.

In this case, N is not a variable of interest, and it has no outgoing edges; therefore, N, and its inflowing edges, can be eliminated.

Self-looping edge. Replace looping edges by adjusting the gains on the incoming edges.

The graph on the left has a looping edge at node N, with a gain of g. On the right, the looping edge has been eliminated, and all inflowing edges have their gain divided by (1-g).

The equations corresponding to the reduction between N and all its input signals are:

Solving linear equations

Signal flow graphs can be used to solve sets of simultaneous linear equations.^[33] The set of equations must be consistent and all equations must be linearly independent.

Putting the equations in "standard form"

Flow graph for three simultaneous equations. The edges incident on each node are colored differently just for emphasis. Rotating the figure by 120° simply permutes the indices. ${\displaystyle x_{1}=\left(c_{11}+1\right)x_{1}+c_{12}x_{2}+c_{13}x_{3}-y_{1}\ ,}$ ${\displaystyle x_{2}=c_{21}x_{1}+\left(c_{22}+1\right)x_{2}+c_{23}x_{3}-y_{2}\ ,}$ ${\displaystyle x_{3}=c_{31}x_{1}+c_{32}x_{2}+\left(c_{33}+1\right)x_{3}-y_{3}\ .}$

For M equations with N unknowns where each y_j is a known value and each x_j is an unknown value, there is equation for each known of the following form.

${\displaystyle {\begin{aligned}\sum _{\mathrm {k} =1}^{\mathrm {N} }c_{\mathrm {jk} }x_{\mathrm {k} }&=y_{\mathrm {j} }\end{aligned}}}$ ; the usual form for simultaneous linear equations with 1 ≤ j ≤ M

Although it is feasible, particularly for simple cases, to establish a signal flow graph using the equations in this form, some rearrangement allows a general procedure that works easily for any set of equations, as now is presented. To proceed, first the equations are rewritten as

${\displaystyle {\begin{aligned}\sum _{\mathrm {k} =1}^{\mathrm {N} }c_{\mathrm {jk} }x_{\mathrm {k} }-y_{\mathrm {j} }&=0\end{aligned}}}$

and further rewritten as

${\displaystyle {\begin{aligned}\sum _{\mathrm {k=1} }^{\mathrm {N} }c_{\mathrm {jk} }x_{\mathrm {k} }+x_{\mathrm {j} }-y_{\mathrm {j} }&=x_{\mathrm {j} }\end{aligned}}}$

and finally rewritten as

${\displaystyle {\begin{aligned}\sum _{\mathrm {k=1} }^{\mathrm {N} }(c_{\mathrm {jk} }+\delta _{\mathrm {jk} })x_{\mathrm {k} }-y_{\mathrm {j} }&=x_{\mathrm {j} }\end{aligned}}}$  ; form suitable to be expressed as a signal flow graph.

where δ_kj = Kronecker delta

The signal-flow graph is now arranged by selecting one of these equations and addressing the node on the right-hand side. This is the node for which the node connects to itself with the branch of weight including a '+1', making a self-loop in the flow graph. The other terms in that equation connect this node first to the source in this equation and then to all the other branches incident on this node. Every equation is treated this way, and then each incident branch is joined to its respective emanating node. For example, the case of three variables is shown in the figure, and the first equation is:

${\displaystyle x_{1}=\left(c_{11}+1\right)x_{1}+c_{12}x_{2}+c_{13}x_{3}-y_{1}\ ,}$

where the right side of this equation is the sum of the weighted arrows incident on node x₁.

As there is a basic symmetry in the treatment of every node, a simple starting point is an arrangement of nodes with each node at one vertex of a regular polygon. When expressed using the general coefficients {c_in}, the environment of each node is then just like all the rest apart from a permutation of indices. Such an implementation for a set of three simultaneous equations is seen in the figure.^[34]

Often the known values, y_j are taken as the primary causes and the unknowns values, x_j to be effects, but regardless of this interpretation, the last form for the set of equations can be represented as a signal-flow graph. This point is discussed further in the subsection Interpreting 'causality'.

Signal-flow graphs for dynamic systems analysis

When building a model of a dynamic system, a list of steps is provided by Dorf & Bishop:^[47]

• Define the system and its components.
• Formulate the mathematical model and list the needed assumptions.
• Write the differential equations describing the model.
• Solve the equations for the desired output variables.
• Examine the solutions and the assumptions.
• If needed, reanalyze or redesign the system.

—RC Dorf and RH Bishop, Modern Control Systems, Chapter 2, p. 2

In this workflow, equations of the physical system's mathematical model are used to derive the signal-flow graph equations.

Signal-flow graphs for design synthesis

Signal-flow graphs have been used in Design Space Exploration (DSE), as an intermediate representation towards a physical implementation. The DSE process seeks a suitable solution among different alternatives. In contrast with the typical analysis workflow, where a system of interest is first modeled with the physical equations of its components, the specification for synthesizing a design could be a desired transfer function. For example, different strategies would create different signal-flow graphs, from which implementations are derived.^[48] Another example uses an annotated SFG as an expression of the continuous-time behavior, as input to an architecture generator^[49]

Simple voltage amplifier

Figure 1: SFG of a simple amplifier

The amplification of a signal V₁ by an amplifier with gain a₁₂ is described mathematically by

${\displaystyle V_{2}=a_{12}V_{1}\,.}$

This relationship represented by the signal-flow graph of Figure 1. is that V₂ is dependent on V₁ but it implies no dependency of V₁ on V₂. See Kou page 57.^[53]

Ideal negative feedback amplifier

Figure 3: A possible signal-flow graph for the asymptotic gain model

Figure 4: A different signal-flow graph for the asymptotic gain model

A signal flow graph for a nonideal negative feedback amplifier based upon a control variable P relating two internal variables: x_j=Px_i. Patterned after D.Amico et al.^[54]

A possible SFG for the asymptotic gain model for a negative feedback amplifier is shown in Figure 3, and leads to the equation for the gain of this amplifier as

${\displaystyle G={\frac {y_{2}}{x_{1}}}}$ ${\displaystyle =G_{\infty }\left({\frac {T}{T+1}}\right)+G_{0}\left({\frac {1}{T+1}}\right)\ .}$

The interpretation of the parameters is as follows: T = return ratio, G_∞ = direct amplifier gain, G₀ = feedforward (indicating the possible bilateral nature of the feedback, possibly deliberate as in the case of feedforward compensation). Figure 3 has the interesting aspect that it resembles Figure 2 for the two-port network with the addition of the extra feedback relation x₂ = T y₁.

From this gain expression an interpretation of the parameters G₀ and G_∞ is evident, namely:

${\displaystyle G_{\infty }=\lim _{T\to \infty }G\$ ;\ G_{0}=\lim _{T\to 0}G\ .}

There are many possible SFG's associated with any particular gain relation. Figure 4 shows another SFG for the asymptotic gain model that can be easier to interpret in terms of a circuit. In this graph, parameter β is interpreted as a feedback factor and A as a "control parameter", possibly related to a dependent source in the circuit. Using this graph, the gain is

${\displaystyle G={\frac {y_{2}}{x_{1}}}}$ ${\displaystyle =G_{0}+{\frac {A}{1-\beta A}}\ .}$

To connect to the asymptotic gain model, parameters A and β cannot be arbitrary circuit parameters, but must relate to the return ratio T by:

${\displaystyle T=-\beta A\ ,}$

and to the asymptotic gain as:

${\displaystyle G_{\infty }=\lim _{T\to \infty }G=G_{0}-{\frac {1}{\beta }}\ .}$

Substituting these results into the gain expression,

${\displaystyle G=G_{0}+{\frac {1}{\beta }}{\frac {-T}{1+T}}}$

${\displaystyle =G_{0}+(G_{0}-G_{\infty }){\frac {-T}{1+T}}}$

${\displaystyle =G_{\infty }{\frac {T}{1+T}}+G_{0}{\frac {1}{1+T}}\ ,}$

which is the formula of the asymptotic gain model.

Electrical circuit containing a two-port network

Signal flow graph of a circuit containing a two port. The forward path from input to output is shown in a different color. The dotted line rectangle encloses the portion of the SFG that constitutes the two-port.

The figure to the right depicts a circuit that contains a y-parameter two-port network. V_in is the input of the circuit and V₂ is the output. The two-port equations impose a set of linear constraints between its port voltages and currents. The terminal equations impose other constraints. All these constraints are represented in the SFG (Signal Flow Graph) below the circuit. There is only one path from input to output which is shown in a different color and has a (voltage) gain of -R_Ly₂₁. There are also three loops: -R_iny₁₁, -R_Ly₂₂, R_iny₂₁R_Ly₁₂. Sometimes a loop indicates intentional feedback but it can also indicate a constraint on the relationship of two variables. For example, the equation that describes a resistor says that the ratio of the voltage across the resistor to the current through the resistor is a constant which is called the resistance. This can be interpreted as the voltage is the input and the current is the output, or the current is the input and the voltage is the output, or merely that the voltage and current have a linear relationship. Virtually all passive two terminal devices in a circuit will show up in the SFG as a loop.

The SFG and the schematic depict the same circuit, but the schematic also suggests the circuit's purpose. Compared to the schematic, the SFG is awkward but it does have the advantage that the input to output gain can be written down by inspection using Mason's rule.

Mechatronics : Position servo with multi-loop feedback

Angular position servo and signal flow graph. θ_C = desired angle command, θ_L = actual load angle, K_P = position loop gain, V_ωC = velocity command, V_ωM = motor velocity sense voltage, K_V = velocity loop gain, V_IC = current command, V_IM = current sense voltage, K_C = current loop gain, V_A = power amplifier output voltage, L_M = motor inductance, V_M = voltage across motor inductance, I_M = motor current, R_M = motor resistance, R_S = current sense resistance, K_M = motor torque constant (Nm/amp), T = torque, M = moment of inertia of all rotating components α = angular acceleration, ω = angular velocity, β = mechanical damping, G_M = motor back EMF constant, G_T = tachometer conversion gain constant,. There is one forward path (shown in a different color) and six feedback loops. The drive shaft assumed to be stiff enough to not treat as a spring. Constants are shown in black and variables in purple.

This example is representative of a SFG (signal-flow graph) used to represent a servo control system and illustrates several features of SFGs. Some of the loops (loop 3, loop 4 and loop 5) are extrinsic intentionally designed feedback loops. These are shown with dotted lines. There are also intrinsic loops (loop 0, loop1, loop2) that are not intentional feedback loops, although they can be analyzed as though they were. These loops are shown with solid lines. Loop 3 and loop 4 are also known as minor loops because they are inside a larger loop.

• The forward path begins with θ_C, the desired position command. This is multiplied by K_P which could be a constant or a function of frequency. K_P incorporates the conversion gain of the DAC and any filtering on the DAC output. The output of K_P is the velocity command V_ωC which is multiplied by K_V which can be a constant or a function of frequency. The output of K_V is the current command, V_IC which is multiplied by K_C which can be a constant or a function of frequency. The output of K_C is the amplifier output voltage, V_A. The current, I_M, though the motor winding is the integral of the voltage applied to the inductance. The motor produces a torque, T, proportional to I_M. Permanent magnet motors tend to have a linear current to torque function. The conversion constant of current to torque is K_M. The torque, T, divided by the load moment of inertia, M, is the acceleration, α, which is integrated to give the load velocity ω which is integrated to produce the load position, θ_LC.
• The forward path of loop 0 asserts that acceleration is proportional to torque and the velocity is the time integral of acceleration. The backward path says that as the speed increases there is a friction or drag that counteracts the torque. Torque on the load decreases proportionately to the load velocity until the point is reached that all the torque is used to overcome friction and the acceleration drops to zero. Loop 0 is intrinsic.
• Loop1 represents the interaction of an inductor's current with its internal and external series resistance. The current through an inductance is the time integral of the voltage across the inductance. When a voltage is first applied, all of it appears across the inductor. This is shown by the forward path through ${\displaystyle {\frac {1}{s\mathrm {L} _{\mathrm {M} }}}\,}$. As the current increases, voltage is dropped across the inductor internal resistance R_M and the external resistance R_S. This reduces the voltage across the inductor and is represented by the feedback path -(R_M + R_S). The current continues to increase but at a steadily decreasing rate until the current reaches the point at which all the voltage is dropped across (R_M + R_S). Loop 1 is intrinsic.
• Loop2 expresses the effect of the motor back EMF. Whenever a permanent magnet motor rotates, it acts like a generator and produces a voltage in its windings. It does not matter whether the rotation is caused by a torque applied to the drive shaft or by current applied to the windings. This voltage is referred to as back EMF. The conversion gain of rotational velocity to back EMF is G_M. The polarity of the back EMF is such that it diminishes the voltage across the winding inductance. Loop 2 is intrinsic.
• Loop 3 is extrinsic. The current in the motor winding passes through a sense resister. The voltage, V_IM, developed across the sense resister is fed back to the negative terminal of the power amplifier K_C. This feedback causes the voltage amplifier to act like a voltage controlled current source. Since the motor torque is proportional to motor current, the sub-system V_IC to the output torque acts like a voltage controlled torque source. This sub-system may be referred to as the "current loop" or "torque loop". Loop 3 effectively diminishes the effects of loop 1 and loop 2.
• Loop 4 is extrinsic. A tachometer (actually a low power dc generator) produces an output voltage V_ωM that is proportional to is angular velocity. This voltage is fed to the negative input of K_V. This feedback causes the sub-system from V_ωC to the load angular velocity to act like a voltage to velocity source. This sub-system may be referred to as the "velocity loop". Loop 4 effectively diminishes the effects of loop 0 and loop 3.
• Loop 5 is extrinsic. This is the overall position feedback loop. The feedback comes from an angle encoder that produces a digital output. The output position is subtracted from the desired position by digital hardware which drives a DAC which drives K_P. In the SFG, the conversion gain of the DAC is incorporated into K_P.

See Mason's rule for development of Mason's Gain Formula for this example.

Standards covering signal-flow graphs

• IEEE Std 155-1960, IEEE Standards on Circuits: Definitions of Terms for Linear Signal Flow Graphs, 1960.

This IEEE standard defines a signal-flow graph as a network of directed branches representing dependent and independent signals as nodes. Incoming branches carry branch signals to the dependent node signals. A dependent node signal is the algebraic sum of the incoming branch signals at that node, i.e. nodes are summative.

State transition signal-flow graph

State transition signal-flow graph. Each initial condition is considered as a source (shown in blue).

A state transition SFG or state diagram is a simulation diagram for a system of equations, including the initial conditions of the states.^[56]

Closed flowgraph

A simple RC system and its closed flowgraph. A "dummy" transmittance Z(s) is introduced to close the system.^[50]

Closed flowgraphs describe closed systems and have been utilized to provide a rigorous theoretical basis for topological techniques of circuit analysis.^[50]

• Terminology for closed flowgraph theory includes:
  • Contributive node. Summing point for two or more incoming signals resulting in only one outgoing signal.
  • Distributive node. Sampling point for two or more outgoing signals resulting from only one incoming signal.
  • Compound node. Contraction of a contributive node and a distributive node.
  • Strictly dependent & strictly independent node. A strictly independent node represent s an independent source; a strictly dependent node represents a meter.
  • Open & Closed Flowgraphs. An open flowgraph contains strictly dependent or strictly independent nodes; otherwise it is a closed flowgraph.

Examples of nonlinear branch functions

It we denote by x_j the signal at node j, the following are examples of node functions that do not pertain to a linear time-invariant system:

${\displaystyle {\begin{aligned}x_{\mathrm {j} }&=x_{\mathrm {k} }\times x_{\mathrm {l} }\\x_{\mathrm {k} }&=abs(x_{\mathrm {j} })\\x_{\mathrm {l} }&=\log(x_{\mathrm {k} })\\x_{\mathrm {m} }&=t\times x_{\mathrm {j} }{\text{ ,where }}t{\text{ represents time}}\\\end{aligned}}}$

Examples of nonlinear signal-flow graph models

• Although they generally can't be transformed between time domain and frequency domain representations for classical control theory analysis, nonlinear signal-flow graphs can be found in electrical engineering literature.^[57][58]
• Nonlinear signal-flow graphs can also be found in life sciences, for example, Dr Arthur Guyton's model of the cardiovascular system.^[59]