WOLFRAM SYSTEMMODELER LINK PACKAGE SYMBOL

WSMLinearize

gives a linearized state-space to the model

at an equilibrium.

linearizes at state values

and input values

.

MORE INFORMATION

linearizes a continuous-time system designed in Wolfram SystemModeler.

returns a StateSpaceModel object.

allows the following symbolic values for spec:

	"EquilibriumValues"	uses WSMFindEquilibrium
	"InitialValues"	uses WSMModelData

uses spec to add missing values in vals.

The default spec is taken to be .

The resulting has states , inputs , and outputs as defined in .

The list of states, inputs, and outputs can be found from WSMModelData:

	"StateVariables"	state variables
	"InputVariables"	input variables
	"OutputVariables"	output variables

reduces a system of DAEs to a system of ODEs and linearizes the resulting ODEs.

A system of ODEs with state equations and output equations is linearized at a point and .

The linearized system has state , input , and output , with state equations and output equation . The matrices are given by , , , and , all evaluated at and .

The returned StateSpaceModel corresponds to the system , .

The following options can be given:

Method

"NumericDerivative"

methods for linearization algorithm

The option Method has the following possible settings:

	"NumericDerivative"	uses SystemModeler linearization
	"SymbolicDerivative"	uses StateSpaceModel linearization

EXAMPLES

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Basic Examples (3)

Load Wolfram SystemModeler Link:

Linearize a DC-motor model around an equilibrium:

Linearize a mixing tank model around equilibrium with given state and input constraints:

Load Wolfram SystemModeler Link:

In[1]:=

Linearize a DC-motor model around an equilibrium:

In[1]:=

Out[1]=

Linearize a mixing tank model around equilibrium with given state and input constraints:

In[1]:=

In[2]:=

Out[2]=

Scope (12)

Linearize a textual RLC circuit model:

Linearize an RLC circuit block diagram model:

Linearize an acausal RLC circuit:

Linearize a DAE model:

Linearize a DAE model symbolically:

Linearize a model without inputs:

Linearize a model without states:

Linearize a model without outputs:

Linearize around an equilibrium:

Linearize around initial values:

Linearize around equilibrium with given state and input constraints:

Linearize around given partial states and inputs, using initial values for remaining values:

Generalizations & Extensions (2)

Linearize a model by using the variables from WSMModelData:

Hide labels in the resulting StateSpaceModel:

Options (6)

By default, the

method is used:

The method

uses StateSpaceModel to linearize system equations:

Linearize symbolically, keeping some parameters symbolic:

Linearize symbolically, using a partially symbolic linearization point:

Use

variable names:

Use

variable names:

Applications (11)

Compare responses from a model and its linearization at an equilibrium point:

Linearize around the equilibrium point:

Compare the stationary output response with a nonlinear model:

Compare the y

output:

Test the stability of a linearized system from eigenvalues of the system matrix:

Since there is an eigenvalue with a positive real part, the system is unstable:

Plotting the output response also indicates an unstable system:

Test the stability of a linearized system from poles of the transfer function:

Since there is a pole with a positive real part, the system is unstable:

Do a frequency analysis using a linear model:

By plotting

for the linearized transfer function

:

Verify the result using Fourier on simulated data:

Compute

from

:

Alternatively, the imaginary parts of the eigenvalues give the resonance peaks:

Stability may depend on the linearization point:

Linearize at a symbolic angle

:

Compute the eigenvalues:

Analyze stability for different angles:

Linearization takes place at time 0:

Linearize with the switch connecting at time 0:

If the switch is not connected at time 0, the result is different:

Design a PID controller:

Linearize the model:

Define a PID controller and closed-loop transfer function:

Select PID parameters for appropriate step response:

Design a lead-based controller for a DC motor based on its linearization:

Define a PI-lead controller transfer function:

Open-loop transfer function:

Select controller parameters:

Use selected parameters and close the loop with the PI-lead controller:

Design a controller using pole placement:

Place the closed-loop poles:

Compute the closed-loop state-space model:

Show the step response:

Design an LQ controller:

Define state and input weight matrices:

Define LQ controller gain:

Closed-loop state-space model:

Closed-loop step response:

Design a state estimator:

Compute estimator gains and the estimator state-space model:

The state and output response to a unit step on the inputs:

Observer state response:

Compare each state and its estimate:

Properties & Relations (8)

Linearize around initial values using WSMModelData:

Compare results:

Linearize around equilibrium using WSMFindEquilibrium:

Compare results:

Compare responses from a model and its linearization at an equilibrium point:

Linearize around the equilibrium point:

Compare the stationary output response with a nonlinear model:

Compare the first output:

Compare responses from a model and its linearization at a non-equilibrium point:

Linearize around the given point:

Compare the stationary output response with a nonlinear model:

Get the output equations:

Compute the stationary output:

Compare the first output:

Use TransferFunctionModel to convert to a transfer function representation:

Use ToDiscreteTimeModel to discretize a linearized model:

Discretize using sample time

:

The linearized state-space model is not unique:

Change the order in which the variables x1 and x2 are declared:

The models are equivalent and have identical transfer functions:

StateSpaceModel can linearize systems of ordinary differential equations:

Using approximate numeric parameter values:

Using SystemModeler to linearize a model of the same system:

Possible Issues (1)

Some models cannot be linearized symbolically:

Use

to linearize numerically: