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:
MaxIterations Automatic maximum number of differentiations ZeroTest Automatic zero test
Examplesopen allclose all
The vector-relative order of a single-output system:
The vector-relative orders of a multiple-output system:
The vector-relative orders of a StateSpaceModel:
The vector-relative order of a TransferFunctionModel:
Determine the number of states in the feedback-linearized and residual system:
The number of states in the linear subsystem:
The number of states in the residual subsystem:
In AsymptoticOutputTracker, the decay rates are assigned based on relative orders:
The first three poles correspond to the first output and the last two to the second:
The first output has slower decay rates and hence takes longer to start following the reference input:
Properties & Relations (7)
The length of the vector-relative orders is equal to the number of outputs:
Use SystemsModelDimensions to get the number of outputs:
For scalar systems with no zero dynamics, the vector-relative order equals the order:
Use SystemsModelOrder to get the order of the system:
If the vector-relative orders are finite, a linearizing feedback control can be computed:
Use FeedbackLinearize to compute the linearizing feedback:
If the sum of vector-relative orders is equal to the order, there are no zero dynamics:
The relative orders give the lengths of the integrator chains of the feedback-linearized system:
Output 1 has a single integrator and output 2 has a double integrator:
The vector-relative orders are invariant wrt a coordinate transform:
The vector-relative orders correspond to the asymptotic orders of the transfer functions:
Output 1 (first row) has asymptotic order 3, and output 2 (second row) has asymptotic order 2:
Possible Issues (3)
For the decoupling matrix to be invertible, the outputs must be independent:
There must be at least as many inputs as outputs:
If an operating point is specified, the decoupling matrix is evaluated at that point:
Although the system is feedback linearizable, there is a singularity at the operating point:
Wolfram Research (2014), SystemsModelVectorRelativeOrders, Wolfram Language function, https://reference.wolfram.com/language/ref/SystemsModelVectorRelativeOrders.html.
Wolfram Language. 2014. "SystemsModelVectorRelativeOrders." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/SystemsModelVectorRelativeOrders.html.
Wolfram Language. (2014). SystemsModelVectorRelativeOrders. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/SystemsModelVectorRelativeOrders.html