Documentation

# linearize

Linear approximation of Simulink model or subsystem

## Syntax

``linsys = linearize(mdl,io)``
``linsys = linearize(mdl,io,op)``
``linsys = linearize(mdl,io,param)``
``linsys = linearize(mdl,io,blocksub)``
``linsys = linearize(mdl,io,options)``
``linsys = linearize(mdl,io,op,param,blocksub,options)``
``linsys = linearize(mdl,blockpath)``
``linsys = linearize(mdl,blockpath,op)``
``linsys = linearize(mdl,blockpath,param)``
``linsys = linearize(mdl,blockpath,blocksub)``
``linsys = linearize(mdl,blockpath,options)``
``linsys = linearize(mdl,blockpath,op,param,blocksub,options)``
``linsys = linearize(___,'StateOrder',stateorder)``
``[linsys,linop] = linearize(___)``
``[linsys,linop,info] = linearize(___)``

## Description

example

````linsys = linearize(mdl,io)` returns a linear approximation of the nonlinear Simulink® model `mdl` at the model operating point using the analysis points specified in `io`. If you omit `io`, then `linearize` uses the root level inports and outports of the model as analysis points.```

example

````linsys = linearize(mdl,io,op)` linearizes the model at operating point `op`.```

example

````linsys = linearize(mdl,io,param)` linearizes the model using the parameter value variations specified in `param`. You can vary any model parameter with a value given by a variable in the model workspace, the MATLAB® workspace, or a data dictionary.```

example

````linsys = linearize(mdl,io,blocksub)` linearizes the model using the substitute block or subsystem linearizations specified in `blocksub`.```

example

````linsys = linearize(mdl,io,options)` linearizes the model using additional linearization options.```
````linsys = linearize(mdl,io,op,param,blocksub,options)` linearizes the model using any combination of `op`, `param`, `blocksub`, and `options` in any order.```

example

````linsys = linearize(mdl,blockpath)` returns a linear approximation of a block or subsystem in model `mdl`, specified by `blockpath`, at the model operating point. The software isolates the block from the rest of the model before linearization.```

example

````linsys = linearize(mdl,blockpath,op)` linearizes the block or subsystem at operating point `op`.```
````linsys = linearize(mdl,blockpath,param)` linearizes the block or subsystem using the parameter value variations specified in `param`. You can vary any model parameter with a value given by a variable in the model workspace, the MATLAB workspace, or a data dictionary.```
````linsys = linearize(mdl,blockpath,blocksub)` linearizes the block or subsystem using the substitute block or subsystem linearizations specified in `blocksub`.```
````linsys = linearize(mdl,blockpath,options)` linearizes the block or subsystem using additional linearization options.```
````linsys = linearize(mdl,blockpath,op,param,blocksub,options)` linearizes the block or subsystem using any combination of `op`, `param`, `blocksub`, and `options` in any order.```

example

````linsys = linearize(___,'StateOrder',stateorder)` specifies the order of the states in the linearized model for any of the previous syntaxes.```

example

````[linsys,linop] = linearize(___)` returns the operating point at which the model was linearized. Use this syntax when linearizing at simulation snapshots or when varying parameters during linearization.```

example

````[linsys,linop,info] = linearize(___)` returns additional linearization information. To select the linearization information to return in `info`, enable the corresponding option in `options`.```

## Examples

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```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.

```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); ```

```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); ```

```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); ```

```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. ```

```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`.

```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); ```

```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); ```

```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'} ```

```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]); ```

```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

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

Analysis point set that contains inputs, outputs, and openings, specified as a linearization I/O object or a vector of linearization I/O objects. To create `io`:

• Define the inputs, outputs, and openings using `linio`.

• If the inputs, outputs, and openings are specified in the Simulink model, extract these points from the model using `getlinio`.

Each linearization I/O object in `io` must correspond to the Simulink model `mdl` 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.

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

• Operating point object, created using:

• Array of operating point objects, specifying multiple operating points. To create an array of operating point objects, you can:

• 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.

Block or subsystem to linearize, specified as a character vector or string that contains its full block path.

The software treats the inports and outports of the specified block as open-loop inputs and outputs, which isolates it from the rest of the model before linearization.

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.

Each substitute linearization structure has the following fields:

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

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 nu-by-ny array of gain values, where nu is the number of inputs and ny is the number of outputs.

• 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:

FieldDescription
`Specification`

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

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.

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

• Structure — Vary the value of a single parameter by specifying `param` 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, to use the first element of vector `V` as a parameter, use:

`param.Name = 'V(1)';`
• `Value` — Parameter sample values, specified as a double array.

For example, vary the value of parameter `A` in the 10% range:

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

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`.

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 `mdl` that have states, or any named states in `mdl`.

You do not have to specify every block and state from `mdl` in `stateorder`. The states you specify appear first in `linsys`, followed by the remaining states in their default order.

Linearization algorithm options, specified as a `linearizeOptions` option set.

## Output Arguments

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Linearization result, returned as a state-space model or an array of state-space models. 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 VariationBlock SubstitutionLinearize At...Resulting `linsys` Dimensions
No parameter variationNo block substitutionModel operating pointSingle state-space model
Single operating point, specified as an operating point object or snapshot time using `op`
N1-by-`...`-by-Nm array of operating point objects, specified by `op`N1-by-`...`-by-Nm
Ns snapshots, specified as a vector of snapshot times using `op`Column vector of length Ns
N1-by-`...`-by-Nm model array for at least one block, specified by `blocksub.Value`Model operating pointN1-by-`...`-by-Nm
Single operating point, specified as an operating point object or snapshot time using `op`
N1-by-`...`-by-Nm array of operating points, specified as an array of operating point objects using `op`
Ns snapshots, specified as a vector of snapshot times using `op`Ns-by-N1-by-`...`-by-Nm
N1-by-`...`-by-Nm parameter grid, specified by `param`Either no block substitution or an N1-by-`...`-by-Nm model array for at least one block, specified by `blocksub.Value`Model operating pointN1-by-`...`-by-Nm
Single operating point, specified as an operating point object or snapshot time using `op`
N1-by-`...`-by-Nm array of operating point objects, specified by `op`
Ns snapshots, specified as a vector of snapshot times using `op`Ns-by-N1-by-`...`-by-Nm

For example, suppose:

• `op` is a 4-by-3 array of operating point 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 operating point 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 (Control System Toolbox).

Operating point at which the model was linearized, returned as an operating point object or an array of operating point 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 operating point object or an array of operating point 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`.

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

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:

FieldDescription
`x`State offsets used for linearization, returned as a column vector of length nx, where nx is the number of states in `linsys`.
`y`Output offsets used for linearization, returned as a column vector of length ny, where ny is the number of outputs in `linsys`.
`u`Input offsets used for linearization, returned as a column vector of length nu, where nu is the number of inputs in `linsys`.
`dx`Derivative offsets for continuous time systems or updated state values for discrete-time systems, returned as a column vector of length nx.
`StateName`State names, returned as a cell array that contains nx elements that match the names in `linsys.StateName`.
`InputName`Input names, returned as a cell array that contains nu elements that match the names in `linsys.InputName`.
`OutputName`Output names, returned as a cell array that contains ny elements that match the names in `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 Approximating Nonlinear Behavior Using an Array of LTI Systems.

Linearization diagnostic information, returned as `[]` if `options.StoreAdvisor` is `false`. Otherwise, `Advisor` is returned as one of the following:

`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.

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### 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:

FieldDescription
`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 `Inputs` has the following fields:

FieldDescription
`BlockName`Full block path of the block whose output connects to the corresponding block input.
`PortIndex`Output port of the block specified by `BlockName` that connects to the corresponding block input.
`Values`Value of the signal specified by `BlockName` and `PortIndex`. If this signal is a vector signal, then `Values` is a vector with the same dimension.
`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

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### Model Properties for Linearization

By default, `linearize` automatically sets the following Simulink model properties:

• `BufferReuse = 'off'`

• `RTWInlineParameters = 'on'`

• `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 requests the Jacobian for these levels from each block.

For some blocks, the software cannot compute an analytical linearization. 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 ${10}^{-5}\left(1+|x|\right)$, where 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.

### 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.

## Alternatives

As an alternative to the `linearize` function, you can linearize models using one of the following methods:

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. 