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SystemsConnectionsModel[{sys1,sys2,…},conxs,ins,outs]

gives a model with inputs ins and outputs outs obtained by connecting the systems models sysi using connections conxs.

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SystemsConnectionsModel

SystemsConnectionsModel[{sys1,sys2,…},conxs,ins,outs]

gives a model with inputs ins and outputs outs obtained by connecting the systems models sysi using connections conxs.

Details and Options

Examples

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Basic Examples  (3)

A series connection:

Wolfram Language code: {Subscript[sys, 1], Subscript[sys, 2]} = {StateSpaceModel[{{{0, 1}, {-Subscript[α, 0], -Subscript[α, 1]}}, {{0}, {1}}, {{Subscript[β, 0], Subscript[β, 1]}}, {{0}}}, SamplingPeriod -> None, SystemsModelLabels -> None], StateSpaceModel[{{{0, 1, 0}, {0, 0, 1}, {-Subscript[a, 0], -Subscript[a, 1], -Subscript[a, 2]}}, {{0}, {0}, {1}}, {{Subscript[b, 0], Subscript[b, 1], Subscript[b, 2]}}, {{0}}}, SamplingPeriod -> None, SystemsModelLabels -> None]};
Wolfram Language code: sys = SystemsConnectionsModel[{Subscript[sys, 1], Subscript[sys, 2]}, {{1, 1} -> {2, 1}}, {{1, 1}}, {{2, 1}}]

A feedback connection:

Wolfram Language code: {cp, Subscript[sys, 1], Subscript[sys, 2]} = {StateSpaceModel[{{}, {}, {}, {{1, -1}}}, SamplingPeriod -> None, SystemsModelLabels -> None], StateSpaceModel[{{{0, 1}, {-Subscript[α, 0], -Subscript[α, 1]}}, {{0}, {1}}, {{Subscript[β, 0], Subscript[β, 1]}}, {{0}}}, SamplingPeriod -> None, SystemsModelLabels -> None], StateSpaceModel[{{{0, 1, 0}, {0, 0, 1}, {-Subscript[a, 0], -Subscript[a, 1], -Subscript[a, 2]}}, {{0}, {0}, {1}}, {{Subscript[b, 0], Subscript[b, 1], Subscript[b, 2]}}, {{0}}}, SamplingPeriod -> None, SystemsModelLabels -> None]};
Wolfram Language code: sys = SystemsConnectionsModel[{cp, Subscript[sys, 1], Subscript[sys, 2]}, {{1, 1} -> {2, 1}, {2, 1} -> {3, 1}, {3, 1} -> {1, 2}}, {{1, 1}}, {{2, 1}}]

A nested connection:

Wolfram Language code: {cp, Subscript[sys, 11], Subscript[sys, 12], Subscript[sys, 2]} = {TransferFunctionModel[{{{1, -1}}, 1}, ], TransferFunctionModel[{{{s + Subscript[β, 0]}}, s + Subscript[α, 0]}, s], TransferFunctionModel[{{{b + s}}, a + s}, s], TransferFunctionModel[{{{1}}, s*α}, s]};

The inner feedback system:

Wolfram Language code: Subscript[sys, 1] = SystemsConnectionsModel[{cp, Subscript[sys, 11], Subscript[sys, 12]}, {{1, 1} -> {2, 1}, {2, 1} -> {3, 1}, {3, 1} -> {1, 2}}, {{1, 1}}, {{2, 1}}]

The complete system:

Wolfram Language code: sys = SystemsConnectionsModel[{Subscript[sys, 1], Subscript[sys, 2]}, {{1, 1} -> {2, 1}}, {{1, 1}}, {{2, 1}}]

Scope  (6)

Connect systems of the same kind:

Wolfram Language code: {Subscript[sys, 1], Subscript[sys, 2]} = {TransferFunctionModel[{{{1}}, s*α}, s], TransferFunctionModel[{{{b + s}}, a + s}, s]};
Wolfram Language code: sys = SystemsConnectionsModel[{Subscript[sys, 1], Subscript[sys, 2]}, {{1, 1} -> {2, 1}}, {{1, 1}}, {{2, 1}}]

Connect systems of different kinds:

Wolfram Language code: {Subscript[sys, 1], Subscript[sys, 2], Subscript[sys, 3]} = {StateSpaceModel[{{{Subscript[a, 11], Subscript[a, 12]}, {Subscript[a, 21], Subscript[a, 22]}}, {{Subscript[b, 11]}, {Subscript[b, 21]}}, {{Subscript[c, 11], Subscript[c, 12]}}, {{0}}}, SamplingPeriod -> None, SystemsModelLabels -> None], TransferFunctionModel[{{{k}}, s + α}, s], AffineStateSpaceModel[{{Subscript[x, 1]^2, -Subscript[x, 1] + Subscript[x, 2]}, {{0}, {1}}, {Subscript[x, 1]*Subscript[x, 2]}, {{0}}}, {Subscript[x, 1], Subscript[x, 2]}, {u}, {Automatic}, Automatic, SamplingPeriod -> None]};
Wolfram Language code: sys = SystemsConnectionsModel[{Subscript[sys, 1], Subscript[sys, 2], Subscript[sys, 3]}, {{1, 1} -> {2, 1}, {2, 1} -> {3, 1}}, {{1, 1}}, {{3, 1}}]

Connect discrete-time systems:

Wolfram Language code: {Subscript[sys, 1], Subscript[sys, 2], Subscript[sys, 3]} = {TransferFunctionModel[{{{b + z}}, a + z}, z, SamplingPeriod -> T], TransferFunctionModel[{{{B + z}}, A + z}, z, SamplingPeriod -> T], TransferFunctionModel[{{{z + β}}, z + α}, z, SamplingPeriod -> T]};
Wolfram Language code: sys = SystemsConnectionsModel[{Subscript[sys, 1], Subscript[sys, 2], Subscript[sys, 3]}, {{1, 1} -> {2, 1}, {2, 1} -> {3, 1}}, {{1, 1}}, {{3, 1}}]

Multivariable systems:

Wolfram Language code: {Subscript[sys, 1], Subscript[sys, 2], Subscript[sys, 3]} = {TransferFunctionModel[{{{Subscript[α, 11], Subscript[α, 12]}, {Subscript[α, 21], Subscript[α, 22]}}, {{s + Subscript[β, 11], s + Subscript[β, 12]}, {s + Subscript[β, 21], s + Subscript[β, 22]}}}, s], StateSpaceModel[{{{Subscript[a, 11], Subscript[a, 12]}, {Subscript[a, 21], Subscript[a, 22]}}, {{Subscript[b, 11]}, {Subscript[b, 21]}}, {{Subscript[c, 11], Subscript[c, 12]}}, {{0}}}, SamplingPeriod -> None, SystemsModelLabels -> None], NonlinearStateSpaceModel[{{f[x, u]}, {g[x, u]}}, {x}, {u}, {Automatic}, Automatic, SamplingPeriod -> None]};
Wolfram Language code: sys = SystemsConnectionsModel[{Subscript[sys, 1], Subscript[sys, 2], Subscript[sys, 3]}, {{1, 1} -> {2, 1}, {1, 1} -> {3, 1}}, {{1, 1}, {1, 2}}, {{2, 1}, {3, 1}}]

A system with an internal signal that is also an output:

Wolfram Language code: {Subscript[sys, 1], Subscript[sys, 2]} = {TransferFunctionModel[{{{1}}, s*α}, s], TransferFunctionModel[{{{b + s}}, a + s}, s]};
Wolfram Language code: sys = SystemsConnectionsModel[{Subscript[sys, 1], Subscript[sys, 2]}, {{1, 1} -> {2, 1}}, {{1, 1}}, {{2, 1}, {1, 1}}]

A system with no internal connections:

Wolfram Language code: {Subscript[sys, 1], Subscript[sys, 2], Subscript[sys, 3]} = {TransferFunctionModel[{{{b + z}}, a + z}, z, SamplingPeriod -> T], TransferFunctionModel[{{{B + z}}, A + z}, z, SamplingPeriod -> T], TransferFunctionModel[{{{z + β}}, z + α}, z, SamplingPeriod -> T]};
Wolfram Language code: sys = SystemsConnectionsModel[{Subscript[sys, 1], Subscript[sys, 2], Subscript[sys, 3]}, {}, {{1, 1}, {2, 1}, {3, 1}}, {{1, 1}, {2, 1}, {3, 1}}]

Applications  (2)

Reduce a block diagram:

The subsystems:

Wolfram Language code: {Subscript[sys, 1], Subscript[sys, 2], Subscript[sys, 3], comp} = {StateSpaceModel[{{{Subscript[a, 1], 0}, {0, Subscript[a, 2]}}, {{Subscript[b, 1], 0}, {0, Subscript[b, 2]}}, {{Subscript[c, 1], 0}, {0, Subscript[c, 2]}}, {{Subscript[d, 1], 0}, {0, Subscript[d, 2]}}}, SamplingPeriod -> None, SystemsModelLabels -> None], StateSpaceModel[{{{Subscript[α, 11], Subscript[α, 12]}, {Subscript[α, 21], Subscript[α, 22]}}, {{Subscript[β, 1], 0}, {0, Subscript[β, 2]}}, {{Subscript[γ, 11], Subscript[γ, 12]}}, {{Subscript[δ, 11], Subscript[δ, 12]}}}, SamplingPeriod -> None, SystemsModelLabels -> None], TransferFunctionModel[{{{s}}, a + s}, s], StateSpaceModel[{{}, {}, {}, {{1, -1}}}, SamplingPeriod -> None, SystemsModelLabels -> None]};

The connections model:

Wolfram Language code: SystemsConnectionsModel[{Subscript[sys, 1], Subscript[sys, 2], Subscript[sys, 3], comp}, {{1, 2}{4, 1}, {4, 1}{2, 1}, {2, 1}{3, 1}, {3, 1}{4, 2}}, {{1, 1}, {1, 2}, {2, 2}}, {{1, 1}, {2, 1}}]

Reduce the system:

Wolfram Language code: SystemsModelMerge[%]

A systems model with three subsystems running on an Arduino Uno microcontroller board:

The first subsystem is a moving-average filter:

Wolfram Language code: ma = StateSpaceModel[y[t] == (u[t] + u[t - 1] + u[t - 2]/3), y[t], u[t], y[t], t, SamplingPeriod -> 0.2]//N

The second subsystem is a ButterworthFilterModel:

Wolfram Language code: bf = ToDiscreteTimeModel[ButterworthFilterModel[2], 0.2]

The third subsystem compares the outputs of the earlier subsystems:

Wolfram Language code: cf = NonlinearStateSpaceModel[{{}, Which[u1 > u2, 5, True, 0]}, {}, {u1, u2}, SamplingPeriod -> 0.2]

The complete system:

Wolfram Language code: sys = SystemsConnectionsModel[{ma, bf, cf}, {{1, 1} -> {3, 1}, {2, 1} -> {3, 2}}, {{1, 1}, {2, 1}}, {{3, 1}}]

Embed it in the microcontroller board:

Wolfram Language code: Needs["MicrocontrollerKit`"]
Wolfram Language code: MicrocontrollerEmbedCode[sys, <|"Target" -> "ArduinoUno", "Inputs" -> {"A0" -> "Analog", "A1" -> "Analog"}, "Outputs" -> {"4" -> "Digital"}|>, "/dev/tty.usbmodem1411"]

Properties & Relations  (4)

SystemsModelMerge will try to merge a SystemsConnectionsModel:

Wolfram Language code: sys = SystemsConnectionsModel[{TransferFunctionModel[{{{1}}, s*α}, s], TransferFunctionModel[{{{b + s}}, a + s}, s]}, {DirectedEdge[{1, 1}, {2, 1}]}, {{1, 1}}, {{2, 1}, {1, 1}}];
Wolfram Language code: SystemsModelMerge[sys]

SystemsModelSeriesConnect is a special case:

Wolfram Language code: {Subscript[sys, 1], Subscript[sys, 2]} = {TransferFunctionModel[{{{1}}, s*α}, s], TransferFunctionModel[{{{b + s}}, a + s}, s]};
Wolfram Language code: SystemsModelSeriesConnect[Subscript[sys, 1], Subscript[sys, 2]]//TransferFunctionExpand
Wolfram Language code: SystemsModelMerge[SystemsConnectionsModel[{Subscript[sys, 1], Subscript[sys, 2]}, {{1, 1} -> {2, 1}}, {{1, 1}}, {{2, 1}}]]//TransferFunctionExpand

SystemsModelParallelConnect is a special case:

Wolfram Language code: {Subscript[sys, 1], Subscript[sys, 2], com, sum} = {TransferFunctionModel[{{{1}}, s*α}, s], TransferFunctionModel[{{{b + s}}, a + s}, s], TransferFunctionModel[{{{1}}, 1}, s], TransferFunctionModel[{{{1, 1}}, 1}, s]};
Wolfram Language code: SystemsModelParallelConnect[Subscript[sys, 1], Subscript[sys, 2]]//TransferFunctionExpand
Wolfram Language code: SystemsModelMerge[SystemsConnectionsModel[{com, Subscript[sys, 1], Subscript[sys, 2], sum}, {{1, 1} -> {2, 1}, {1, 1} -> {3, 1}, {2, 1} -> {4, 1}, {3, 1} -> {4, 2}}, {{1, 1}}, {{4, 1}}]]//TransferFunctionExpand

SystemsModelFeedbackConnect is a special case:

Wolfram Language code: {Subscript[sys, 1], Subscript[sys, 2], cp} = {TransferFunctionModel[{{{1}}, s*α}, s], TransferFunctionModel[{{{b + s}}, a + s}, s], TransferFunctionModel[{{{1, -1}}, 1}, s]};
Wolfram Language code: SystemsModelFeedbackConnect[Subscript[sys, 1], Subscript[sys, 2]]
Wolfram Language code: SystemsModelMerge[SystemsConnectionsModel[{cp, Subscript[sys, 1], Subscript[sys, 2]}, {{1, 1} -> {2, 1}, {2, 1} -> {3, 1}, {3, 1} -> {1, 2}}, {{1, 1}}, {{2, 1}}]]//TransferFunctionExpand
Wolfram Research (2019), SystemsConnectionsModel, Wolfram Language function, https://reference.wolfram.com/language/ref/SystemsConnectionsModel.html.

Text

Wolfram Research (2019), SystemsConnectionsModel, Wolfram Language function, https://reference.wolfram.com/language/ref/SystemsConnectionsModel.html.

CMS

Wolfram Language. 2019. "SystemsConnectionsModel." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/SystemsConnectionsModel.html.

APA

Wolfram Language. (2019). SystemsConnectionsModel. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/SystemsConnectionsModel.html

BibTeX

@misc{reference.wolfram_2026_systemsconnectionsmodel, author="Wolfram Research", title="{SystemsConnectionsModel}", year="2019", howpublished="\url{https://reference.wolfram.com/language/ref/SystemsConnectionsModel.html}", note=[Accessed: 13-June-2026]}

BibLaTeX

@online{reference.wolfram_2026_systemsconnectionsmodel, organization={Wolfram Research}, title={SystemsConnectionsModel}, year={2019}, url={https://reference.wolfram.com/language/ref/SystemsConnectionsModel.html}, note=[Accessed: 13-June-2026]}