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SystemsModelVectorRelativeOrders[sys]

gives the vector-relative orders of the systems model sys.

Details and Options
Details and Options Details and Options
Examples  
Basic Examples  
Scope  
Applications  
Properties & Relations  
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SystemsModelVectorRelativeOrders

SystemsModelVectorRelativeOrders[sys]

gives the vector-relative orders of the systems model sys.

Details and Options

  • SystemsModelVectorRelativeOrders is also known as vector-relative degrees.
  • The relative order of an output of sys is essentially the number of times it must be differentiated for the input of sys to appear explicitly.
  • Output has relative order if after differentiations of the output, .
  • The vector-relative orders are the list of the relative orders for each output.
  • The matrix is called the decoupling matrix and needs to have rank .
  • The system sys can be a TransferFunctionModel, StateSpaceModel, or AffineStateSpaceModel.
  • The following options can be given:
  • MaxIterationsAutomaticmaximum number of differentiations
    ZeroTestAutomaticzero test

Examples

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Basic Examples  (1)

Compute the vector-relative orders of a system:

Wolfram Language code: SystemsModelVectorRelativeOrders[AffineStateSpaceModel[{{-Subscript[x, 2], Subscript[x, 1]* Subscript[x, 3], Subscript[x, 1]}, {{0, 1}, {1, 0}, {0, Subscript[x, 3]}}, {Subscript[x, 2], Subscript[x, 3]}, {{0, 0}, {0, 0}}}, {Subscript[x, 1], Subscript[x, 2], Subscript[x, 3]}, {Subscript[, 1], Subscript[, 2]}, {Automatic, Automatic}, Automatic, SamplingPeriod -> None]]

Scope  (4)

The vector-relative order of a single-output system:

Wolfram Language code: SystemsModelVectorRelativeOrders[AffineStateSpaceModel[{{-Subscript[x, 2], Subscript[x, 1]* Subscript[x, 2]}, {{0, 0}, {Cos[Subscript[x, 1]], 1}}, {Subscript[x, 1]}, {{0, 0}}}, {Subscript[x, 1], Subscript[x, 2]}, {Subscript[, 1], Subscript[, 2]}, {Automatic}, Automatic, SamplingPeriod -> None]]

The vector-relative orders of a multiple-output system:

Wolfram Language code: SystemsModelVectorRelativeOrders[AffineStateSpaceModel[{{Sin[Subscript[x, 1]], -Subscript[x, 1], Subscript[x, 1]*Subscript[x, 2]*Subscript[x, 3]}, {{0, 0}, {1, 0}, {0, E^Subscript[x, 1]}}, {Subscript[x, 2], Subscript[x, 3]}, {{0, 0}, {0, 0}}}, {Subscript[x, 1], Subscript[x, 2], Subscript[x, 3]}, {Subscript[, 1], Subscript[, 2]}, {Automatic, Automatic}, Automatic, SamplingPeriod -> None]]

The vector-relative orders of a StateSpaceModel:

Wolfram Language code: SystemsModelVectorRelativeOrders[StateSpaceModel[{{{2, 2, 0, 0}, {1, 2, 2, 2}, {2, 3, 2, -2}, {3, 2, -3, -3}}, {{1, 0}, {0, 1}, {0, 0}, {0, 0}}, {{1, 0, 0, 0}, {0, 0, 1, 0}}, {{0, 0}, {0, 0}}}, SamplingPeriod -> None, SystemsModelLabels -> None]]

The vector-relative order of a TransferFunctionModel:

Wolfram Language code: SystemsModelVectorRelativeOrders[TransferFunctionModel[{{{1, -4 + s}}, {{1 + 10*s, s*(1 + 10*s)}}}, s]]

Applications  (2)

Determine the number of states in the feedback-linearized and residual system:

Wolfram Language code: asys = AffineStateSpaceModel[{{-Subscript[x, 1]^3 + Subscript[x, 1]* Subscript[x, 2], Subscript[x, 1], -Subscript[x, 3], Subscript[x, 1]^2 + Subscript[x, 2]}, {{0}, {2 + 2*Subscript[x, 3]}, {1}, {0}}, {Subscript[x, 4]}, {{0}}}, {Subscript[x, 1], Subscript[x, 2], Subscript[x, 3], Subscript[x, 4]}, {u}, {Automatic}, Automatic, SamplingPeriod -> None]; vro = SystemsModelVectorRelativeOrders[asys]

The number of states in the linear subsystem:

Wolfram Language code: Total[vro]

Feedback linearize the system:

Wolfram Language code: ℱ = FeedbackLinearize[asys]

The linearized subsystem:

Wolfram Language code: ℱ["LinearSystem"]

The number of states in the residual subsystem:

Wolfram Language code: SystemsModelOrder[asys] - Total[vro]

The residual subsystem:

Wolfram Language code: ℱ["ResidualSystem"]

In AsymptoticOutputTracker, the decay rates are assigned based on relative orders:

Wolfram Language code: asys = AffineStateSpaceModel[{{Subscript[x, 2][t] + Subscript[x, 2][t]^2, Subscript[x, 3][t] - Subscript[x, 1][t]*Subscript[x, 4][t] + Subscript[x, 4][t]*Subscript[x, 5][t], Subscript[x, 2][t]*Subscript[x, 4][t] + Subscript[x, 1][t]*Subscript[x, 5][t] - Subscript[x, 5][t]^2, Subscript[x, 5][t], Subscript[x, 2][t]^2}, {{0, 1}, {0, 0}, {Cos[Subscript[x, 1][t] - Subscript[x, 5][t]], 1}, {0, 0}, {0, 1}}, {Subscript[x, 1][t] - Subscript[x, 5][t], Subscript[x, 4][t]}, {{0, 0}, {0, 0}}}, {{Subscript[x, 1][t], 0.1}, {Subscript[x, 2][t], 0.3}, {Subscript[x, 3][t], 0}, {Subscript[x, 4][t], 0.4}, {Subscript[x, 5][t], 1}}, {Subscript[, 1][t], Subscript[, 2][t]}, {Automatic, Automatic}, t, SamplingPeriod -> None];
Wolfram Language code: SystemsModelVectorRelativeOrders[asys]

The first three poles correspond to the first output and the last two to the second:

Wolfram Language code: ref = {Sin[t], Cos[t]}; fb = AsymptoticOutputTracker[asys, ref, {-2, -3, -4, -10, -12}]

The first output has slower decay rates and hence takes longer to start following the reference input:

Wolfram Language code: OutputResponse[SystemsModelStateFeedbackConnect[asys, fb], {0, 0}, {t, 0, 2.5}];
Wolfram Language code: Plot[Evaluate[{%, ref}//Flatten], {t, 0, 2.5}, IconizedObject[«plotOpts»]]

Properties & Relations  (7)

The length of the vector-relative orders is equal to the number of outputs:

Wolfram Language code: asys = AffineStateSpaceModel[{{(-Sin[Subscript[x, 1]])*Subscript[x, 2], (-Subscript[x, 2])*Subscript[x, 3], 0}, {{0, 0, 1, 0}, {0, 1, 0, 1 + Subscript[x, 2]}, {1, 0, 1 + Subscript[x, 2], 0}}}, {Subscript[x, 1], Subscript[x, 2], Subscript[x, 3]}, {Subscript[, 1], Subscript[, 2], Subscript[, 3], Subscript[, 4]}, {Automatic, Automatic, Automatic}, Automatic, SamplingPeriod -> None];

Use SystemsModelDimensions to get the number of outputs:

Wolfram Language code: Length[SystemsModelVectorRelativeOrders[asys]] == Last[SystemsModelDimensions[asys]]

For scalar systems with no zero dynamics, the vector-relative order equals the order:

Wolfram Language code: asys = AffineStateSpaceModel[{{-Subscript[x, 2], -Subscript[x, 3], (-Subscript[x, 1])*Subscript[x, 2]}, {{0}, {0}, {1 - Subscript[x, 1]}}, {Subscript[x, 1]}, {{0}}}, {Subscript[x, 1], Subscript[x, 2], Subscript[x, 3]}, {u}, {Automatic}, Automatic, SamplingPeriod -> None];

Use SystemsModelOrder to get the order of the system:

Wolfram Language code: SystemsModelVectorRelativeOrders[asys] == {SystemsModelOrder[asys]}

If the vector-relative orders are finite, a linearizing feedback control can be computed:

Wolfram Language code: SystemsModelVectorRelativeOrders[asys = AffineStateSpaceModel[{{-Subscript[x, 2], Subscript[x, 1]* Subscript[x, 3], -Subscript[x, 1] + Subscript[x, 3]}, {{1, 0}, {0, 1}, {0, 1 + Subscript[x, 3]}}, {Subscript[x, 1]}, {{0, 0}}}, {Subscript[x, 1], Subscript[x, 2], Subscript[x, 3]}, {Subscript[u, 1], Subscript[u, 2]}, {Automatic}, Automatic, SamplingPeriod -> None]]

Use FeedbackLinearize to compute the linearizing feedback:

Wolfram Language code: FeedbackLinearize[asys]

If the sum of vector-relative orders is equal to the order, there are no zero dynamics:

Wolfram Language code: asys = AffineStateSpaceModel[ {{Subscript[x, 3] + (Subscript[x, 2]* (1 + Subscript[x, 3]^2))/(1 + Subscript[x, 3]), Subscript[x, 3]/(1 + Subscript[x, 3]), Subscript[x, 1]*Subscript[x, 2]}, {{2*Subscript[x, 2]*((1 + Subscript[x, 2])/ (1 + Subscript[x, 3]))}, {-(1 + Subscript[x, 2])/ (1 + Subscript[x, 3])}, {1}}, {Subscript[x, 1] + Subscript[x, 2]^2}, {{0}}}, {Subscript[x, 1], Subscript[x, 2], Subscript[x, 3]}, {u1}, {Automatic}, Automatic, SamplingPeriod -> None];
Wolfram Language code: Total[SystemsModelVectorRelativeOrders[asys]] == SystemsModelOrder[asys]
Wolfram Language code: FeedbackLinearize[asys, Automatic, "ZeroDynamics"]

The relative orders give the lengths of the integrator chains of the feedback-linearized system:

Wolfram Language code: asys = AffineStateSpaceModel[{{0, -Subscript[x, 2] - Subscript[x, 3], Subscript[x, 2] + Subscript[x, 3]}, {{Subscript[x, 1], -Subscript[x, 2]}, {Subscript[x, 1], 0}, {-Subscript[x, 1], Subscript[x, 2]}}, {Subscript[x, 1] + Subscript[x, 2] + Subscript[x, 3], Subscript[x, 1] + Subscript[x, 3]}, {{0, 0}, {0, 0}}}, {Subscript[x, 1], Subscript[x, 2], Subscript[x, 3]}, {Subscript[, 1], Subscript[, 2]}, {Automatic, Automatic}, Automatic, SamplingPeriod -> None];

Output 1 has a single integrator and output 2 has a double integrator:

Wolfram Language code: SystemsModelVectorRelativeOrders[asys]
Wolfram Language code: TransferFunctionModel[FeedbackLinearize[asys, Automatic, "LinearSystem"]]

The vector-relative orders are invariant wrt a coordinate transform:

Wolfram Language code: asys = AffineStateSpaceModel[{{3*Subscript[x, 2], Subscript[x, 1]* Subscript[x, 2], -Subscript[x, 3]^3}, {{-1, 0}, {1 + Subscript[x, 2], 0}, {0, 1 + Subscript[x, 2]}}, {Subscript[x, 1] + Subscript[x, 2], Subscript[x, 3]}, {{0, 0}, {0, 0}}}, {Subscript[x, 1], Subscript[x, 2], Subscript[x, 3]}, {Subscript[, 1], Subscript[, 2]}, {Automatic, Automatic}, Automatic, SamplingPeriod -> None];
Wolfram Language code: asys1 = StateSpaceTransform[asys, {{Subscript[x, 1] -> Subscript[z, 1] + Subscript[z, 3] Subscript[z, 2], Subscript[x, 2] -> Subscript[z, 2] - Subscript[z, 1], Subscript[x, 3] -> Subscript[z, 3]}, {Subscript[z, 1], Subscript[z, 2], Subscript[z, 3]}}];
Wolfram Language code: SystemsModelVectorRelativeOrders /@ {asys, asys1}

The vector-relative orders correspond to the asymptotic orders of the transfer functions:

Wolfram Language code: sys = TransferFunctionModel[{{{1, 2}, {1, 1}}, {{2 + ^3, 3 + ^3}, {1 + ^2, 2 + ^2}}}, , SamplingPeriod -> None, SystemsModelLabels -> None];
Wolfram Language code: SystemsModelVectorRelativeOrders[sys]

Output 1 (first row) has asymptotic order 3, and output 2 (second row) has asymptotic order 2:

Wolfram Language code: Series[sys[s], {s, ∞, 3}]//MatrixForm

Possible Issues  (3)

For the decoupling matrix to be invertible, the outputs must be independent:

Wolfram Language code: SystemsModelVectorRelativeOrders[AffineStateSpaceModel[{{Subscript[x, 1], Subscript[x, 1]* Subscript[x, 2]}, {{1 + Subscript[x, 2], 0}, {1, -1}}, {Subscript[x, 1], Subscript[x, 1]}, {{0, 0}, {0, 0}}}, {Subscript[x, 1], Subscript[x, 2]}, {Subscript[, 1], Subscript[, 2]}, {Automatic, Automatic}, Automatic, SamplingPeriod -> None]]

There must be at least as many inputs as outputs:

Wolfram Language code: SystemsModelVectorRelativeOrders[AffineStateSpaceModel[{{-Subscript[x, 2], Subscript[x, 1] + Subscript[x, 2]}, {{1}, {1 + Subscript[x, 1]}}, {Subscript[x, 1], Subscript[x, 2]}, {{0}, {0}}}, {Subscript[x, 1], Subscript[x, 2]}, {Subscript[, 1]}, {Automatic, Automatic}, Automatic, SamplingPeriod -> None]]

If an operating point is specified, the decoupling matrix is evaluated at that point:

Wolfram Language code: SystemsModelVectorRelativeOrders[sys = AffineStateSpaceModel[{{-Subscript[x, 2], Subscript[x, 1] + Subscript[x, 2]}, {{0}, {Subscript[x, 1]}}, {Subscript[x, 1]}, {{0}}}, {{Subscript[x, 1], 0}, {Subscript[x, 2], 0}}, {Subscript[u, 1]}, {Automatic}, Automatic, SamplingPeriod -> None]]

Although the system is feedback linearizable, there is a singularity at the operating point:

Wolfram Language code: FeedbackLinearize[sys, Automatic, "FeedbackCompensator"]
Wolfram Research (2014), SystemsModelVectorRelativeOrders, Wolfram Language function, https://reference.wolfram.com/language/ref/SystemsModelVectorRelativeOrders.html.

Text

Wolfram Research (2014), SystemsModelVectorRelativeOrders, Wolfram Language function, https://reference.wolfram.com/language/ref/SystemsModelVectorRelativeOrders.html.

CMS

Wolfram Language. 2014. "SystemsModelVectorRelativeOrders." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/SystemsModelVectorRelativeOrders.html.

APA

Wolfram Language. (2014). SystemsModelVectorRelativeOrders. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/SystemsModelVectorRelativeOrders.html

BibTeX

@misc{reference.wolfram_2026_systemsmodelvectorrelativeorders, author="Wolfram Research", title="{SystemsModelVectorRelativeOrders}", year="2014", howpublished="\url{https://reference.wolfram.com/language/ref/SystemsModelVectorRelativeOrders.html}", note=[Accessed: 13-June-2026]}

BibLaTeX

@online{reference.wolfram_2026_systemsmodelvectorrelativeorders, organization={Wolfram Research}, title={SystemsModelVectorRelativeOrders}, year={2014}, url={https://reference.wolfram.com/language/ref/SystemsModelVectorRelativeOrders.html}, note=[Accessed: 13-June-2026]}