NonlinearStateSpaceModel

NonlinearStateSpaceModel[{f,g},x,u]

represents the model , .

NonlinearStateSpaceModel[sys]

gives a state-space representation corresponding to the systems model sys.

NonlinearStateSpaceModel[eqns,{{x1,x10},},{{u1,u10},},{g1,},t]

gives the state-space model of the differential equations eqns with dependent variables xi, input variables ui, operating vaues xi0 and ui0, outputs gi, and independent variable t.

Details and Options

Examples

open allclose all

Basic Examples  (1)

Define the nonlinear system with output :

Plot its response to a step input with initial states at the origin:

Scope  (18)

Basic Uses  (5)

A system with 2 states , 1 input , and 1 output:

Count the number of inputs and outputs:

The order of the system:

A system with 2 inputs and 1 output:

The response to a unit step applied to each input channel while holding the other at zero:

A system with 1 input and 2 outputs:

A pure gain system:

Connect the gains in series with the original system:

Specify operating values for the states:

Obtain the StateSpaceModel using linearization about the operating values:

Explicitly specify the output and independent variables:

System Conversions  (4)

The nonlinear representation of a StateSpaceModel with specified states:

The nonlinear state-space representation of a TransferFunctionModel with default states:

The nonlinear representation of an AffineStateSpaceModel, preserving states:

The nonlinear state-space representation of a set of ODEs:

Model Manipulations  (3)

Set new state, input, and output variables:

Use state , input , and output :

Use Normal to get the model's arguments:

Operating Values  (6)

States and inputs deleted using SystemsModelDelete are set to their operating values:

So are the states and inputs not extracted by SystemsModelExtract:

StateSpaceTransform computes the operating values of the new states if the old were specified:

By default, the simulation functions assume the operating values as the initial values:

StateResponse:

OutputResponse:

StateSpaceModel linearizes about all operating values:

AffineStateSpaceModel only linearizes about operating values of input variables:

NonlinearStateSpaceModel preserves operating values specified along with ODEs:

FullInformationOutputRegulator linearizes about the operating values to compute the gains :

Compute the gains from the linearized subsystem:

Generalizations & Extensions  (1)

Variables with operating values given as are taken to be :

Options  (1)

SystemsModelLabels  (1)

Label the outputs and states of a system:

Applications  (5)

Chemical Systems  (2)

Use NonlinearStateSpaceModel to specify the equations for a two-tank system and simulate it:

Use Bernoulli's law and mass balance to derive the resulting differential equations:

Use steady-state operating values :

Define the corresponding nonlinear system with input and output :

With a higher steady-state value for flow rate, the level in the tanks settles at a higher level:

Control the level in the second tank below using a PID controller:

Obtain a PID controller using the linearized model and PIDTune:

Specify a piecewise reference value for the liquid level in the second tank:

The simulations show how the closed-loop system follows the reference:

Aerospace Systems  (2)

Find the equations of motion of an aircraft's longitudinal dynamics , where the states are and the inputs are : »

Here , , and are taken to be polynomials in α and δe:

The aerodynamic and model parameters:

Use , where is the flight path angle and the pitch angle:

Find the steady states for steady , level flight at a specific speed :

Define state variables and initial values:

The nonlinear state-space model of the aircraft with states and inputs :

The 2×2 transfer function representation of the linearized system:

The system is unstable due to the poles in the right half-plane:

It is also nonminimum phase due to the zeros in the right half-plane:

The BodePlot shows that the velocity can be affected more than the flight path angle :

The six-degree-of-freedom equations of motion of an aircraft. The rotation matrix for a rotation about an axis : »

The rotation matrices for the Euler angles , , and :

The matrix transformation of a vector from the body-fixed to the earth-fixed axes:

The inverse transformation from the earth-fixed to the body-fixed axes:

Verify they are inverses:

A figure showing the earth-fixed axes and body-fixed axes :

The Euler angle rates in terms of the angular velocity components , , and :

The kinematic equations for the Euler angles:

The kinematic equations for the flight path variables , , and :

The inertia matrix, assuming that the plane is a plane of symmetry:

The angular momentum can be computed as :

A figure showing the forces and moments acting on the aircraft:

The state equations of the angular velocity can be obtained by solving the moment equation , where is the vector of aerodynamic moments:

The dynamics of the rotational motion:

The forces acting on the aircraft are the aerodynamic forces , its weight, and the thrust :

The dynamics of the translational motion can be obtained from Newton's law :

The complete equations of motion for the aircraft:

The longitudinal dynamics are the dynamics for states , , , and :

The lateral dynamics are the dynamics for the states , , , and :

Extended Kalman Filter  (1)

Design an extended Kalman filter to track the motion of a wheeled robot:

The model of the robot:

All the measurements are assumed to be noisy, with covariance :

The gain of the filter is based on the covariance of the states:

The right-hand side of the differential equation for the estimated states with noisy measurements :

The right-hand side of the differential equation for the covariance matrix :

Assemble the estimator using NonlinearStateSpaceModel:

Simulate the robot from initial position and a set of inputs:

The noisy measurements:

Simulate the response of the filter using the inputs to the system and the noisy measurements:

Compare the actual, noisy, and filtered values of the variable :

Variable :

Variable :

Properties & Relations  (3)

NonlinearStateSpaceModel converts to AffineStateSpaceModel by linearizing inputs:

Conversion the other way is exact:

NonlinearStateSpaceModel converts to StateSpaceModel by linearizing states and inputs:

Conversion the other way is exact:

Convert to TransferFunctionModel by state and input linearization:

Conversion the other way is exact:

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

Text

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

CMS

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

APA

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

BibTeX

@misc{reference.wolfram_2023_nonlinearstatespacemodel, author="Wolfram Research", title="{NonlinearStateSpaceModel}", year="2014", howpublished="\url{https://reference.wolfram.com/language/ref/NonlinearStateSpaceModel.html}", note=[Accessed: 19-March-2024 ]}

BibLaTeX

@online{reference.wolfram_2023_nonlinearstatespacemodel, organization={Wolfram Research}, title={NonlinearStateSpaceModel}, year={2014}, url={https://reference.wolfram.com/language/ref/NonlinearStateSpaceModel.html}, note=[Accessed: 19-March-2024 ]}