linmod

Extract continuous-time linear state-space model around operating point using block by block linearization algorithm

Syntax

[A,B,C,D] = linmod(mdl)

[A,B,C,D] = linmod(mdl,x,u)

[A,B,C,D] = linmod(mdl,x,u,opts)

[A,B,C,D] = linmod(mdl,x,u,-v5)

[A,B,C,D] = linmod(mdl,x,u,opts,-v5)

[A,B,C,D] = linmod(mdl,x,u,opts,xpert,upert,-v5)

[n,d] = linmod(___)

sys = linmod(___)

Description

[A,B,C,D] = linmod(mdl) computes the linear state-space model of the system of ordinary differential equations represented in the model mdl by linearizing each block one by one. Inport and Outport blocks in the model represent the system inputs and outputs.

The block-by-block analytic algorithm uses preprogrammed analytic block Jacobians for most blocks, which typically results in more accurate linearization compared to the perturbation algorithm. The Simulink^® Control Design™ documentation contains a list of blocks that have preprogrammed analytic Jacobians and a discussion of the block-by-block analytic algorithm for linearization.

The block-by-block analytical algorithm also allows for special treatment of problematic blocks such as the Transport Delay block and the Quantizer block.

Note

The linmod function provides only basic linearization capabilities. For full linearization functionality, use Simulink Control Design software. For more information, see Choose Linearization Tools (Simulink Control Design).

[A,B,C,D] = linmod(mdl,x,u) computes the linear state-space model around the operating point defined by the state values x and the input values u.

[A,B,C,D] = linmod(mdl,x,u,opts) computes the linear state-space model around the specified operating point using options opts.

[A,B,C,D] = linmod(mdl,x,u,-v5) uses the perturbation algorithm from versions 5.3 and earlier to compute the linear state-space model of the system of ordinary differential equations represented in the model mdl at the operating point specified as the state x and inputs u. Using the perturbation algorithm is equivalent to using the linmodv5 function.

[A,B,C,D] = linmod(mdl,x,u,opts,-v5) computes the linear state-space model using the perturbation algorithm with options opts.

[A,B,C,D] = linmod(mdl,x,u,opts,xpert,upert,-v5) computes the linear state-space model using the perturbation algorithm with options opts using state perturbation xpert and input perturbation upert.

[n,d] = linmod(___) returns the transfer function representation of the linearized model.

sys = linmod(___) returns a structure that contains the linearized model, state names, input and output names, and information about the operating point.

Examples

Linearization with Referenced Models

You can use the linmod function to extract a linear model from a model reference hierarchy.

Open the model mdlref_f14 and the model mdlref_dynamics which is referenced by the Model block named Aircraft Dynamics Model in the model mdlref_f14.

topmdl = "mdlref_f14";
refmdl = "mdlref_dynamics";

open_system(topmdl)
open_system(refmdl)

For best results when linearizing model reference hierarchies, configure the top and all referenced models to simulate in normal mode. Linearization has limitations for multirate referenced models, and the linmod function uses a different algorithm to linearize referenced models that simulate in accelerator mode.

Set the simulation mode for the top model and the Model block to Normal.

set_param(topmdl,SimulationMode="Normal")
mdlblk = topmdl + "/Aircraft Dynamics Model";
set_param(mdlblk,SimulationMode="Normal")

Linearize the model mdlref_f14 using the linmod function.

[A,B,C,D] = linmod(topmdl);

The state-space model that the linmod function returns corresponds to the complete model, including the referenced model.

Input Arguments

mdl — Name of model to linearize

string | character vector

Name of model to linearize, specified as a string or a character vector.

Data Types: char | string

x — Model states of operating point at which to linearize model

0 (default) | structure | vector

Model states of operating point at which to linearize model, specified as a structure or a vector. The operating point at which to linearize the model is specified as a combination of the model states and the input values.

To extract the state of the model as a structure, use the Simulink.BlockDiagram.getInitialState function. You can edit the values of the model state by modifying the values field of the signals substructure. For example, use these commands to access the values of the state of the model named mdl.

mdlState = Simulink.BlockDiagram.getInitialState("mdl");
stateVals = mdlState.signals.values;

You must use the structure format to specify the state at which to linearize the model when:

• The model is a model reference hierarchy.

• The model has states that have different data types.

• The model has states that have a data type other than double.

u — Input values of operating point at which to linearize model

0 (default) | vector

Input values of operating point at which to linearize model, specified as a vector. The operating point at which to linearize the model is specified as a combination of the model states and the input values.

Data Types: double

opts — Linearization options

vector

Linearization options, specified as a vector with the elements and values described in the table.

Element	Value	Default
opts(1)	Perturbation value of delta used to perturb the model states and inputs. The linmod function uses the value you specify only when you use the v5 perturbation algorithm. When you use the default block-by-block linearization algorithm, the linmod function ignores the value you specify and always uses the default value of 1e-5.	1e-5
opts(2)	Nonnegative time at which to evaluate blocks during linearization.	0
opts(3)	Option to remove extra states from blocks that do not have direct feedthrough. 0 — Do not remove extra states from blocks that do not have direct feedthrough. 1 — Remove extra states from blocks that do not have direct feedthrough.	0

xpert — State perturbation values

opts(1) + 1e-3*opts(1)*abs(x) (default) | structure | vector

State perturbation values, specified as a structure or a vector. You must use the structure format to specify the state perturbation values when:

• The model is a model reference hierarchy.

• The model has states that have different data types.

• The model has states that have a data type other than double.

To extract the model states as a structure, use the Simulink.BlockDiagram.getInitialState function. You can then edit the values in the structure to specify the perturbation value for each state by modifying the values field of the signals substructure. For example, use these commands to access the states of the model named mdl.

mdlState = Simulink.BlockDiagram.getInitialState("mdl");
stateVals = mdlState.signals.values;

You can specify state and input perturbation values only when you linearize the model using the perturbation algorithm from versions 5.3 and earlier by calling the linmodv5 function or by specifying the -v5 input option.

By default, the perturbation values are calculated using the delta value specified in the first element of the opts input vector as shown in this code.

xpert = opts(1) + 1e-3*opts(1)*abs(x);

upert — Input perturbation values

opts(1) + 1e-3*opts(1)*abs(u) | structure | vector

Input perturbation values, specified as a structure or a vector.

You can specify state and input perturbation values only when you linearize the model using the perturbation algorithm from versions 5.3 and earlier by calling the linmodv5 function or by specifying the -v5 input option.

By default, the perturbation values are calculated using the delta value specified in the first element of the opts input vector as shown in this code.

upert = opts(1) + 1e-3*opts(1)*abs(u);

Output Arguments

[A,B,C,D] — State-space representation of linearized model

vector

State-space representation of linearized model, returned as a vector.

[n,d] — Transfer function representation of linearized model

vector

Transfer function representation of linearized model, returned as a vector.

sys — Linearized model

structure

Linearized model, returned as a structure that contains state names, input and output names, and information about the operating point.

Limitations

• The linmod function provides only basic linearization capabilities. For full linearization functionality, use Simulink Control Design software. For more information, see Choose Linearization Tools (Simulink Control Design).

• When you linearize a model reference hierarchy, configure the top and referenced models to simulate in normal mode. Multiple limitations exist for linearizing multirate referenced models that simulate in accelerator mode.

• When you linearize a model reference hierarchy that contains referenced models configured to simulate in accelerator mode, the software uses the default algorithm to linearize the top model and the numeric perturbation algorithm to linearize the referenced model.

• Linearization is not supported for models that contain one or more referenced models configured to use a local solver. For more information, see Use Local Solvers in Referenced Models.

Tips

• By default, the system time is zero. For systems that are dependent on time, you can specify the system time using the second element of the opts input argument.

• State order is maintained in linearization such that the order of states in the linearized model matches the order of states in the nonlinear model. You can get information about the states in a model and the blocks associated with the states by using the model name as a programmatic interface to execute the sizes phase. The return argument named blks is a vector that contains the name of each block associated with a state. For more information, see Use Model Name as Programmatic Interface.

  [sys,x0,blks,st] = modelName([],[],[],'sizes');

• You can convert the state-space linearized representation of a linearized single-input, multiple-output system to another form using these functions:

  • ss2tf — Convert from state-space to transfer function form.

  • ss2zp (Signal Processing Toolbox) — Convert from state-space to zero-pole form.

• You can create a state-space model object from a linearized model using the ss (Control System Toolbox) function. You can use state-space model objects to represent a linear time invariant (LTI) system for control design. You can also combine multiple LTI state-space models to represent more complex systems.

• After creating a state-space model object, you can convert to transfer function form using the tf (Control System Toolbox) function or convert to zero-pole-gain form using the zpk (Control System Toolbox) function.

• The default block-by-block algorithm is recommended for linearizing models that contain Transport Delay or Derivative blocks. The block-by-block algorithm replaces Transport Delay and Derivative blocks with Pade approximations. Using the v5 perturbation algorithm to linearize models that contain Transport Delay or Derivative blocks can be troublesome. For more information, see Linearizing Models.

• Before linearizing a model that contains Transport Delay or Derivative blocks using the v5 perturbation algorithm, replace the Transport Delay and Derivative blocks with the specialized blocks available in the Linearization library inside the Simulink Extras library.

Version History

Introduced in R2007a