# linearize

Linear approximation of Simulink model or subsystem

## Syntax

## Description

returns a linear approximation of the nonlinear Simulink`linsys`

= linearize(`model`

,`io`

)^{®} model `model`

at the model operating point
using the analysis points specified in `io`

. Using
`io`

, you can specify individual analysis points or you
can specify a block or subsystem to linearize. If you omit
`io`

, then `linearize`

uses the
root-level inports and outports of the model as analysis points.

specifies the order of the states in the linearized model for any of the
previous syntaxes.`linsys`

= linearize(___,'StateOrder',`stateorder`

)

## Examples

### Linearize Model Using Specified I/O Set

Open the Simulink model.

```
mdl = 'watertank';
open_system(mdl)
```

Specify a linearization input at the output of the PID Controller block, which is the input signal for the Water-Tank System block.

io(1) = linio('watertank/PID Controller',1,'input');

Specify a linearization output point at the output of the Water-Tank System block. Specifying the output point as open-loop removes the effects of the feedback signal on the linearization without changing the model operating point.

io(2) = linio('watertank/Water-Tank System',1,'openoutput');

Linearize the model using the specified I/O set.

linsys = linearize(mdl,io);

`linsys`

is the linear approximation of the plant at the model operating point.

### Linearize Model at Specified Operating Point

Open the Simulink model.

```
mdl = 'magball';
open_system(mdl)
```

Find a steady-state operating point at which the ball height is `0.05`

. Create a default operating point specification, and set the height state to a known value.

opspec = operspec(mdl); opspec.States(5).Known = 1; opspec.States(5).x = 0.05;

Trim the model to find the operating point.

options = findopOptions('DisplayReport','off'); op = findop(mdl,opspec,options);

Specify linearization input and output signals to compute the closed-loop transfer function.

io(1) = linio('magball/Desired Height',1,'input'); io(2) = linio('magball/Magnetic Ball Plant',1,'output');

Linearize the model at the specified operating point using the specified I/O set.

linsys = linearize(mdl,io,op);

### Linearize Model at Simulation Snapshot Time

Open the Simulink model.

```
mdl = 'watertank';
open_system(mdl)
```

To compute the closed-loop transfer function, first specify the linearization input and output signals.

io(1) = linio('watertank/PID Controller',1,'input'); io(2) = linio('watertank/Water-Tank System',1,'output');

Simulate `sys`

for `10`

seconds and linearize the model.

linsys = linearize(mdl,io,10);

### Batch Linearize Model for Parameter Variations

Open the Simulink model.

```
mdl = 'scdcascade';
open_system(mdl)
```

Specify parameter variations for the outer-loop controller gains, `Kp1`

and `Ki1`

. Create parameter grids for each gain value.

Kp1_range = linspace(Kp1*0.8,Kp1*1.2,6); Ki1_range = linspace(Ki1*0.8,Ki1*1.2,4); [Kp1_grid,Ki1_grid] = ndgrid(Kp1_range,Ki1_range);

Create a parameter value structure with fields `Name`

and `Value`

.

params(1).Name = 'Kp1'; params(1).Value = Kp1_grid; params(2).Name = 'Ki1'; params(2).Value = Ki1_grid;

`params`

is a 6-by-4 parameter value grid, where each grid point corresponds to a unique combination of `Kp1`

and `Ki1`

values.

Define linearization input and output points for computing the closed-loop response of the system.

io(1) = linio('scdcascade/setpoint',1,'input'); io(2) = linio('scdcascade/Sum',1,'output');

Linearize the model at the model operating point using the specified parameter values.

linsys = linearize(mdl,io,params);

### Specify Substitute Block Linearization and Linearize Model

Open the Simulink model.

```
mdl = 'scdpwm';
open_system(mdl)
```

Extract linearization input and output from the model.

io = getlinio(mdl);

Linearize the model at the model operating point.

linsys = linearize(mdl,io)

linsys = D = Step Plant Model 0 Static gain.

The discontinuities in the Voltage to PWM block cause the model to linearize to zero. To treat this block as a unit gain during linearization, specify a substitute linearization for this block.

```
blocksub.Name = 'scdpwm/Voltage to PWM';
blocksub.Value = 1;
```

Linearize the model using the specified block substitution.

linsys = linearize(mdl,blocksub,io)

linsys = A = State Space( State Space( State Space( 0.9999 -0.0001 State Space( 0.0001 1 B = Step State Space( 0.0001 State Space( 5e-09 C = State Space( State Space( Plant Model 0 1 D = Step Plant Model 0 Sample time: 0.0001 seconds Discrete-time state-space model.

### Specify Sample Time of Linearized Model

Open the Simulink model.

```
mdl = 'watertank';
open_system(mdl)
```

To linearize the Water-Tank System block, specify a linearization input and output.

io(1) = linio('watertank/PID Controller',1,'input'); io(2) = linio('watertank/Water-Tank System',1,'openoutput');

Create a linearization option set, and specify the sample time for the linearized model.

```
options = linearizeOptions('SampleTime',0.1);
```

Linearize the plant using the specified options.

linsys = linearize(mdl,io,options)

linsys = A = H H 0.995 B = PID Controll H 0.02494 C = H Water-Tank S 1 D = PID Controll Water-Tank S 0 Sample time: 0.1 seconds Discrete-time state-space model.

The linearized plant is a discrete-time state-space model with a sample time of `0.1`

.

### Linearize Block or Subsystem at Model Operating Point

Open the Simulink model.

```
mdl = 'watertank';
open_system(mdl)
```

Specify the full block path for the block you want to linearize.

```
blockpath = 'watertank/Water-Tank System';
```

Linearize the specified block at the model operating point.

linsys = linearize(mdl,blockpath);

### Linearize Block or Subsystem at Trimmed Operating Point

Open Simulink model.

```
mdl = 'magball';
open_system(mdl)
```

Find a steady-state operating point at which the ball height is `0.05`

. Create a default operating point specification, and set the height state to a known value.

opspec = operspec(mdl); opspec.States(5).Known = 1; opspec.States(5).x = 0.05;

options = findopOptions('DisplayReport','off'); op = findop(mdl,opspec,options);

Specify the block path for the block you want to linearize.

```
blockpath = 'magball/Magnetic Ball Plant';
```

Linearize the specified block at the specified operating point.

linsys = linearize(mdl,blockpath,op);

### Specify State Order in Linearized Model

Open the Simulink model.

```
mdl = 'magball';
open_system(mdl)
```

Linearize the plant at the model operating point.

```
blockpath = 'magball/Magnetic Ball Plant';
linsys = linearize(mdl,blockpath);
```

View the default state order for the linearized plant.

linsys.StateName

ans = 3x1 cell array {'height' } {'Current'} {'dhdt' }

Linearize the plant and reorder the states in the linearized model. Set the rate of change of the height as the second state.

stateorder = {'magball/Magnetic Ball Plant/height';... 'magball/Magnetic Ball Plant/dhdt';... 'magball/Magnetic Ball Plant/Current'}; linsys = linearize(mdl,blockpath,'StateOrder',stateorder);

View the new state order.

linsys.StateName

ans = 3x1 cell array {'height' } {'dhdt' } {'Current'}

### Linearize Model at Multiple Snapshot Times

Open the Simulink model.

```
mdl = 'watertank';
open_system(mdl)
```

To compute the closed-loop transfer function, first specify the linearization input and output signals.

io(1) = linio('watertank/PID Controller',1,'input'); io(2) = linio('watertank/Water-Tank System',1,'output');

Simulate `sys`

and linearize the model at `0`

and `10`

seconds. Return the operating points that correspond to these snapshot times; that is, the operating points at which the model was linearized.

[linsys,linop] = linearize(mdl,io,[0,10]);

### Batch Linearize Plant Model and Obtain Linearization Offsets

Open the Simulink model.

```
mdl = 'watertank';
open_system(mdl)
```

Vary parameters `A`

and `b`

within 10% of their nominal values.

```
[A_grid,b_grid] = ndgrid(linspace(0.9*A,1.1*A,3),...
linspace(0.9*b,1.1*b,4));
```

Create a parameter structure array, specifying the name and grid points for each parameter.

params(1).Name = 'A'; params(1).Value = A_grid; params(2).Name = 'b'; params(2).Value = b_grid;

Create a default operating point specification for the model.

opspec = operspec(mdl);

Trim the model using the specified operating point specification, parameter grid. Suppress the display of the operating point search report.

opt = findopOptions('DisplayReport','off'); [op,opreport] = findop(mdl,opspec,params,opt);

`op`

is a 3-by-4 array of operating point objects that correspond to the specified parameter grid points.

Specify the block path for the plant model.

```
blockpath = 'watertank/Desired Water Level';
```

To store offsets during linearization, create a linearization option set and set `StoreOffsets`

to `true`

.

```
options = linearizeOptions('StoreOffsets',true);
```

Batch linearize the plant at the trimmed operating points, using the specified I/O points and parameter variations.

[linsys,linop,info] = linearize(mdl,blockpath,op,params,options);

You can use the offsets in `info.Offsets`

when configuring an LPV System block.

info.Offsets

ans = 3x4 struct array with fields: x dx u y StateName InputName OutputName Ts

## Input Arguments

`model`

— Simulink model name

character vector | string

Simulink model name, specified as a character vector or string. The model must be in the current working folder or on the MATLAB path.

`io`

— Analysis points

linearization I/O object | vector of linearization I/O objects | string | character vector

Analysis points for linearizing model, specified as one of the following:

Linearization I/O object or a vector of linearization I/O objects — Specify a list of one or more inputs, outputs, and loop openings. To create the list of analysis points:

String or character vector — Specify the full path of a block or subsystem to linearize. The software treats the inports and outports of the specified block as open-loop inputs and outputs, which isolates the block from the rest of the model before linearization.

The analysis points defined in `io`

must correspond to
the Simulink model `model`

or some normal-mode model
reference in the model hierarchy.

If you omit `io`

, then `linearize`

uses the root-level inports and outports of the model as analysis
points.

For more information on specifying linearization inputs, outputs, and openings, see Specify Portion of Model to Linearize.

`op`

— Operating point

`OperatingPoint`

object | array of `OperatingPoint`

objects | vector of positive scalars

Operating point for linearization, specified as one of the following:

`OperatingPoint`

object, created using:Array of

`OperatingPoint`

objects, specifying multiple operating points. To create an array of`OperatingPoint`

objects, you can:Extract operating points at multiple snapshot times using

`findop`

.Batch trim your model using multiple operating point specifications. For more information, see Batch Compute Steady-State Operating Points for Multiple Specifications.

Batch trim your model using parameter variations. For more information, see Batch Compute Steady-State Operating Points for Parameter Variation.

Vector of positive scalars representing one or more simulation snapshot times. The software simulates

`sys`

and linearizes the model at the specified snapshot times.If you also specify parameter variations using

`param`

, the software simulates the model for each snapshot time and parameter grid point combination. This operation can be computationally expensive.

If you specify parameter variations using `param`

,
and the parameters:

Affect the model operating point, then specify

`op`

as an array of operating points with the same dimensions as the parameter value grid. To obtain the operating points that correspond to the parameter value combinations, batch trim your model using`param`

before linearization. For more information, see Batch Linearize Model at Multiple Operating Points Derived from Parameter Variations.Do not affect the model operating point, then specify

`op`

as a single operating point.

`blocksub`

— Substitute linearizations for blocks and subsystems

structure | structure array

Substitute linearizations for blocks and subsystems, specified
as a structure or an *n*-by-1 structure array, where *n* is
the number of blocks for which you want to specify a linearization.
Use `blocksub`

to specify a custom linearization
for a block or subsystem. For example, you can specify linearizations
for blocks that do not have analytic linearizations, such as blocks
with discontinuities or triggered subsystems.

To study the effects of varying the linearization of a block on the model dynamics, you can batch linearize your model by specifying multiple substitute linearizations for a block.

If you substitute a linearization with a sample time that differs from that of the
original block or subsystem, it is best practice to set the overall linearization sample
time (`options`

`.SampleTime`

) to a nondefault
value.

Each substitute linearization structure has the following fields.

`Name`

— Block path

character vector | string

Block path of the block for which you want to specify the linearization, specified as a character vector or string.

`Value`

— Substitute linearization

double | double array | LTI model | model array | structure

Substitute linearization for the block, specified as one of the following:

Double — Specify the linearization of a SISO block as a gain.

Array of doubles — Specify the linearization of a MIMO block as an

*n*-by-_{u}*n*array of gain values, where_{y}*n*is the number of inputs and_{u}*n*is the number of outputs._{y}LTI model, uncertain state-space model, or uncertain real object — The I/O configuration of the specified model must match the configuration of the block specified by

`Name`

. Using an uncertain model requires Robust Control Toolbox™ software.Array of LTI models, uncertain state-space models, or uncertain real objects — Batch linearize the model using multiple block substitutions. The I/O configuration of each model in the array must match the configuration of the block for which you are specifying a custom linearization. If you:

Vary model parameters using

`param`

and specify`Value`

as a model array, the dimensions of`Value`

must match the parameter grid size.Specify

`op`

as an array of operating points and`Value`

as a model array, the dimensions of`Value`

must match the size of`op`

.Define block substitutions for multiple blocks, and specify

`Value`

as an array of LTI models for one or more of these blocks, the dimensions of the arrays must match.

Structure with the following fields.

Field Description `Specification`

Block linearization, specified as a character vector that contains one of the following:

MATLAB expression

Name of a Custom Linearization Function in your current working folder or on the MATLAB path

The specified expression or function must return one of the following:

Linear model in the form of a D-matrix

Control System Toolbox™ LTI model object

Uncertain state-space model or uncertain real object (requires Robust Control Toolbox software)

The I/O configuration of the returned model must match the configuration of the block specified by

`Name`

.`Type`

Specification type, specified as one of the following:

`'Expression'`

`'Function'`

`ParameterNames`

Linearization function parameter names, specified as a cell array of character vectors. Specify

`ParameterNames`

only when`Type = 'Function'`

and your block linearization function requires input parameters. These parameters only impact the linearization of the specified block.You must also specify the corresponding

`blocksub.Value.ParameterValues`

field.`ParameterValues`

Linearization function parameter values, specified as a vector of doubles. The order of parameter values must correspond to the order of parameter names in

`blocksub.Value.ParameterNames`

. Specify`ParameterValues`

only when`Type = 'Function'`

and your block linearization function requires input parameters.

`param`

— Parameter samples

structure | structure array

Parameter samples for linearization, specified as one of the following:

Structure — Vary the value of a single parameter by specifying parameters as a structure with the following fields.

`Name`

— Parameter name, specified as a character vector or string. You can specify any model parameter that is a variable in the model workspace, the MATLAB workspace, or a data dictionary. If the variable used by the model is not a scalar variable, specify the parameter name as an expression that resolves to a numeric scalar value. For example, use the first element of vector`V`

as a parameter.`parameters.Name = 'V(1)';`

`Value`

— Parameter sample values, specified as a double array.

For example, vary the value of parameter

`A`

in the 10% range.`parameters.Name = 'A'; parameters.Value = linspace(0.9*A,1.1*A,3);`

Structure array — Vary the value of multiple parameters. For example, vary the values of parameters

`A`

and`b`

in the 10% range.[A_grid,b_grid] = ndgrid(linspace(0.9*A,1.1*A,3),... linspace(0.9*b,1.1*b,3)); parameters(1).Name = 'A'; parameters(1).Value = A_grid; parameters(2).Name = 'b'; parameters(2).Value = b_grid;

For more information, see Specify Parameter Samples for Batch Linearization.

If `param`

specifies tunable parameters only,
the software batch linearizes the model using a single model compilation.

To compute the offsets required by the LPV
System block, specify `param`

, and set `options.StoreOffsets`

to `true`

.
You can then return additional linearization information in `info`

,
and extract the offsets using `getOffsetsForLPV`

.

`stateorder`

— State order in linearization results

cell array of character vectors

State order in linearization results, specified as a cell array of block paths or state names.
The order of the block paths and states in `stateorder`

indicates the order of the states in `linsys`

.

You can specify block paths for any blocks in `model`

that have states, or
any named states in `model`

.

You do not have to specify every block and state from `model`

in
`stateorder`

. The states you specify appear first in
`linsys`

, followed by the remaining states in their
default order.

`options`

— Linearization algorithm options

`linearizeOptions`

option set

Linearization algorithm options, specified as a `linearizeOptions`

option set.

## Output Arguments

`linsys`

— Linearization result

state-space model | array of state-space models

Linearization result, returned as a state-space model or an array of state-space models.

For most models, `linsys`

is returned as an `ss`

object or an array of
`ss`

objects. However, if `model`

contains one of the following blocks in the linearization path defined by
`io`

, then `linsys`

returns the
specified type of state-space model.

Block | `linsys` Type |
---|---|

Block with a substitution specified as a
`genss` object or tunable model
object | `genss` |

Block with a substitution specified as an uncertain
model, such as `uss` | `uss` (Robust Control Toolbox) |

Sparse Second Order block | `mechss` |

Descriptor State-Space block configured to linearize to a sparse model | `sparss` |

The dimensions of `linsys`

depend on the specified
parameter variations and block substitutions, and the operating points at
which you linearize the model.

**Note**

If you specify more than one of `op`

,
`param`

, or `blocksub.Value`

as
an array, then their dimensions must match.

Parameter Variation | Block Substitution | Linearize At... | Resulting `linsys`
Dimensions |
---|---|---|---|

No parameter variation | No block substitution | Model operating point | Single state-space model |

Single operating point, specified as an
`OperatingPoint` object or snapshot time
using `op` | |||

N-by-_{1}`...` -by-N
array of _{m}`OperatingPoint` objects, specified
by `op` | N-by-_{1}`...` -by-N_{m} | ||

N snapshots,
specified as a vector of snapshot times using
_{s}`op` | Column vector of length
N_{s} | ||

N-by-_{1}`...` -by-N
model array for at least one block, specified by
_{m}`blocksub.Value` | Model operating point | N-by-_{1}`...` -by-N_{m} | |

Single operating point, specified as an
`OperatingPoint` object or snapshot time
using `op` | |||

N-by-_{1}`...` -by-N
array of operating points, specified as an array of
_{m}`OperatingPoint` objects using
`op` | |||

N snapshots,
specified as a vector of snapshot times using
_{s}`op` | N-by-_{s}N-by-_{1}`...` -by-N_{m} | ||

N-by-_{1}`...` -by-N
parameter grid, specified by
_{m}`param` | Either no block substitution or an
N-by-_{1}`...` -by-N
model array for at least one block, specified by
_{m}`blocksub.Value` | Model operating point | N-by-_{1}`...` -by-N_{m} |

Single operating point, specified as an
`OperatingPoint` object or snapshot time
using `op` | |||

N-by-_{1}`...` -by-N
array of _{m}`OperatingPoint` objects, specified
by `op` | |||

N snapshots,
specified as a vector of snapshot times using
_{s}`op` | N-by-_{s}N-by-_{1}`...` -by-N_{m} |

For example, suppose:

`op`

is a 4-by-3 array of`OperatingPoint`

objects and you do not specify parameter variations or block substitutions. In this case,`linsys`

is a 4-by-3 model array.`op`

is a single`OperatingPoint`

object and`param`

specifies a 3-by-4-by-2 parameter grid. In this case,`linsys`

is a 3-by-4-by-2 model array.`op`

is a row vector of positive scalars with two elements and you do not specify`param`

. In this case,`linsys`

is a column vector with two elements.`op`

is a column vector of positive scalars with three elements and`param`

specifies a 5-by-6 parameter grid. In this case,`linsys`

is a 3-by-5-by-6 model array.`op`

is a single operating point object, you do not specify parameter variations, and`blocksub.Value`

is a 2-by-3 model array for one block in the model. In this case,`linsys`

is a 2-by-3 model array.`op`

is a column vector of positive scalars with four elements, you do not specify parameter variations, and`blocksub.Value`

is a 1-by-2 model array for one block in the model. In this case,`linsys`

is a 4-by-1-by-2 model array.

For more information on model arrays, see Model Arrays.

`linop`

— Operating point

`OperatingPoint`

object | array of `OperatingPoint`

objects

Operating point at which the model was linearized, returned as an
`OperatingPoint`

object or an array of
`OperatingPoint`

objects with the same dimensions as
`linsys`

. Each element of `linop`

is the operating point at which the corresponding
`linsys`

model was obtained.

If you specify `op`

as a single `OperatingPoint`

object or
an array of `OperatingPoint`

objects, then
`linop`

is a copy of `op`

. If you
specify `op`

as a single operating point object and also
specify parameter variations using `param`

, then
`linop`

is an array with the same dimensions as the
parameter grid. In this case, the elements of `linop`

are
scalar expanded copies of `op`

.

To determine whether the model was linearized at a reasonable
operating point, view the states and inputs in `linop`

.

`info`

— Linearization information

structure

Linearization information, returned as a structure with the following fields:

`Offsets`

— Linearization offsets

`[]`

(default) | structure | structure array

Linearization offsets that correspond to the operating point
at which the model was linearized, returned as `[]`

if `options.StoreOffsets`

is `false`

.
Otherwise, `Offsets`

is returned as one of the following:

If

`linsys`

is a single state-space model, then`Offsets`

is a structure.If

`linsys`

is an array of state-space models, then`Offsets`

is a structure array with the same dimensions as`linsys`

.

Each offset structure has the following fields:

Field | Description |
---|---|

`x` | State offsets used for linearization, returned as a column vector of length
n, where
_{x}n is the number of states in
_{x}`linsys` . |

`y` | Output offsets used for linearization, returned as a column vector of length
n, where
_{y}n is the number of outputs in
_{y}`linsys` . |

`u` | Input offsets used for linearization, returned as a column vector of length
n, where
_{u}n is the number of inputs in
_{u}`linsys` . |

`dx` | Derivative offsets for continuous time systems or updated state values for discrete-time
systems, returned as a column vector of length
n._{x} |

`StateName` | State names, returned as a cell array that contains
n elements that match the names
in _{x}`linsys.StateName` . |

`InputName` | Input names, returned as a cell array that contains
n elements that match the names
in _{u}`linsys.InputName` . |

`OutputName` | Output names, returned as a cell array that contains
n elements that match the names
in _{y}`linsys.OutputName` . |

`Ts` | Sample time of the linearized system, returned as a scalar that matches the sample time in
`linsys.Ts` . For continuous-time systems,
`Ts` is `0` . |

If `Offsets`

is a structure array, you can
configure an LPV System block using
the offsets. To do so, first convert them to the required format using `getOffsetsForLPV`

. For an example, see Approximate Nonlinear Behavior Using Array of LTI Systems.

`Advisor`

— Linearization diagnostic information

`[]`

(default) | `LinearizationAdvisor`

object | array of `LinearizationAdvisor`

objects

Linearization diagnostic information, returned as
`[]`

if
`options.StoreAdvisor`

is
`false`

. Otherwise,
`Advisor`

is returned as one of the
following:

If

`linsys`

is a single state-space model,`Advisor`

is a`LinearizationAdvisor`

object.If

`linsys`

is an array of state-space models,`Advisor`

is an array of`LinearizationAdvisor`

objects with the same dimensions as`linsys`

.

`LinearizationAdvisor`

objects store
linearization diagnostic information for individual linearized
blocks. For an example of troubleshooting linearization results
using a `LinearizationAdvisor`

object, see Troubleshoot Linearization Results at Command Line.

## Limitations

Linearization is not supported for model hierarchies that contain referenced models configured to use a local solver.

Linearization is not supported for Simscape™ networks configured to use a local solver.

## More About

### Custom Linearization Function

You can specify a substitute linearization for a block or subsystem in your Simulink model using a custom function on the MATLAB path.

Your custom linearization function must have one `BlockData`

input
argument, which is a structure that the software creates and passes
to the function. `BlockData`

has the following fields:

Field | Description | ||||||||
---|---|---|---|---|---|---|---|---|---|

`BlockName` | Name of the block for which you are specifying a custom linearization. | ||||||||

`Parameters` | Block parameter values, specified as a structure array with `Name` and `Value` fields. `Parameters` contains
the names and values of the parameters you specify in the `blocksub.Value.ParameterNames` and `blocksub.Value.ParameterValues` fields. | ||||||||

`Inputs` |
Input signals to the block for which you are defining a linearization,
specified as a structure array with one structure for each block input.
Each structure in
| ||||||||

`ny` | Number of output channels of the block linearization. | ||||||||

`nu` | Number of input channels of the block linearization. | ||||||||

`BlockLinearization` | Current default linearization of the block, specified as a
state-space model. You can specify a block linearization that depends
on the default linearization using `BlockLinearization` . |

Your custom function must return a model with `nu`

inputs
and `ny`

outputs. This model must be one of the following:

Linear model in the form of a D-matrix

Control System Toolbox LTI model object

Uncertain state-space model or uncertain real object (requires Robust Control Toolbox software)

For example, the following function multiplies the current default block linearization, by a
delay of `Td = 0.5`

seconds. The delay is represented by a Thiran filter
with sample time `Ts = 0.1`

. The delay and sample time are parameters
stored in `BlockData`

.

function sys = myCustomFunction(BlockData) Td = BlockData.Parameters(1).Value; Ts = BlockData.Parameters(2).Value; sys = BlockData.BlockLinearization*Thiran(Td,Ts); end

Save this function to a location on the MATLAB path.

To use this function as a custom linearization for a block or subsystem, specify the
`blocksub.Value.Specification`

and
`blocksub.Value.Type`

fields.

blocksub.Value.Specification = 'myCustomFunction'; blocksub.Value.Type = 'Function';

To set the delay and sample time parameter values, specify the `blocksub.Value.ParameterNames`

and `blocksub.Value.ParameterValues`

fields.

blocksub.Value.ParameterNames = {'Td','Ts'}; blocksub.Value.ParameterValues = [0.5 0.1];

## Algorithms

### Model Properties for Linearization

By default, `linearize`

automatically sets the following Simulink model properties.

`BufferReuse = 'off'`

`BlockReductionOpt = 'off'`

`SaveFormat = 'StructureWithTime'`

After linearization, Simulink restores the original model properties.

### Block-by-Block Linearization

Simulink Control Design™ software linearizes models using a block-by-block approach. The software individually linearizes each block in your Simulink model and produces the linearization of the overall system by combining the individual block linearizations.

The software determines the input and state levels for each block from the operating point, and obtains the Jacobian for each block at these levels.

For some blocks, the software cannot compute an analytical linearization in this manner. For example:

Some nonlinearities do not have a defined Jacobian.

Some discrete blocks, such as state charts and triggered subsystems, tend to linearize to zero.

Some blocks do not implement a Jacobian.

Custom blocks, such as S-Function blocks and MATLAB Function blocks, do not have analytical Jacobians.

You can specify a custom linearization for any such blocks for which you know the expected linearization. If you do not specify a custom linearization, the software linearizes the model by perturbing the block inputs and states and measuring the response to these perturbations. For each input and state, the default perturbation level is:

$${10}^{-5}\left(1+\left|x\right|\right)$$ for double-precision values.

$$0.005\left(1+\left|x\right|\right)$$ for single-precision values.

Here, *x* is the value of the corresponding input or state at the
operating point. For information on how to change perturbation levels for individual blocks,
see Change Perturbation Level of Blocks Perturbed During Linearization.

For more information, see Linearize Nonlinear Models and Exact Linearization Algorithm.

### Full-Model Numerical Perturbation

You can linearize your system using full-model numerical perturbation, where the software
computes the linearization of the full model by perturbing the values of root-level inputs
and states. To do so, create a `linearizeOptions`

object and set the
`LinearizationAlgorithm`

property to one of the following:

`'numericalpert'`

— Perturb the inputs and states using forward differences; that is, by adding perturbations to the input and state values. This perturbation method is typically faster than the`'numericalpert2'`

method.`'numericalpert2'`

— Perturb the inputs and states using central differences; that is, by perturbing the input and state values in both positive and negative directions. This perturbation method is typically more accurate than the`'numericalpert'`

method.

For each input and state, the software perturbs the model and computes a linear model based on
the model response to these perturbations. You can configure the state and input
perturbation levels using the `NumericalPertRel`

linearization
options.

Block-by-block linearization has several advantages over full-model numerical perturbation:

Most Simulink blocks have a preprogrammed linearization that provides an exact linearization of the block.

You can use linear analysis points to specify a portion of the model to linearize.

You can configure blocks to use custom linearizations without affecting your model simulation.

Structurally nonminimal states are automatically removed.

You can specify linearizations that include uncertainty (requires Robust Control Toolbox software).

You can obtain detailed diagnostic information.

When linearizing multirate models, you can use different rate conversion methods. Full-model numerical perturbation can only use zero-order-hold rate conversion.

For more information, see Linearize Nonlinear Models and Exact Linearization Algorithm.

## Alternatives

As an alternative to the `linearize`

function, you can linearize models
using one of the following methods.

To interactively linearize models, use the Model Linearizer app. For an example, see Linearize Simulink Model at Model Operating Point.

To obtain multiple transfer functions without modifying the model or creating an analysis point set for each transfer function, use an

`slLinearizer`

interface. For an example, see Vary Parameter Values and Obtain Multiple Transfer Functions.

Although both Simulink
Control Design software and the Simulink
`linmod`

function perform block-by-block
linearization, Simulink
Control Design linearization functionality has a more flexible user interface and uses
Control System Toolbox numerical algorithms. For more information, see Linearization Using Simulink Control Design Versus Simulink.

## Version History

**Introduced in R2006a**

### R2020b: Linearize Simulink model to a sparse state-space model

You can linearize and obtain a sparse model from a Simulink model that contains a Sparse Second Order or Descriptor State-Space block.

`mechss`

model when you use a Sparse Second Order in your Simulink model.`sparss`

model when you use a Descriptor State-Space block and select the**Linearize to sparse model**block parameter.

For more information, see Sparse Model Basics. For an example, see Linearize Simulink Model to a Sparse Second-Order Model Object.

### R2016b: Compute operating point offsets for model inputs, outputs, states, and state derivatives during linearization

You can compute operating point offsets for model inputs, outputs, states, and state derivatives when linearizing Simulink models. Thee offsets streamline the creation of linear parameter-varying (LPV) systems.

To obtain operating point offsets, first create a `linearizeOptions`

object and set the
`StoreOffsets`

option to `true`

. Then,
linearize the model.

You can extract the offsets from the `info`

output argument and
convert them into the required format for the LPV System block using the `getOffsetsForLPV`

function.

### R2014a: Block substitution input argument

You can specify a substitute linearization for a Simulink block or subsystem using the `blocksub`

input
argument of the `linearize`

function.

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