# Varying State Space

State-space model with varying matrix values

Libraries:
Control System Toolbox / Linear Parameter Varying

## Description

Use this block to implement a continuous-time state-space model with varying matrices. Feed the instantaneous values of the state matrix A, input matrix B, output matrix C, and feedforward matrix D to the corresponding input ports. The system response is given by:

`$\begin{array}{c}dx=Ax+Bu\\ y=Cx+Du,\end{array}$`

where u is the system input, y is the system output, and x and dx are the state vector and state derivatives, respectively.

The block also lets you specify varying offsets (since R2024a). Then, the system response is given by:

`$\begin{array}{c}dx=d{x}_{0}+A\left(x-{x}_{0}\right)+B\left(u-{u}_{0}\right)\\ y={y}_{0}+C\left(x-{x}_{0}\right)+D\left(u-{u}_{0}\right),\end{array}$`

where u0 is the input offset, y0 is the output offset, and x0 and dx0 are the state offset and state derivative offset, respectively.

Use this block and the other blocks in the Linear Parameter Varying library to implement common control elements with variable parameters or coefficients. For more information, see Model Gain-Scheduled Control Systems in Simulink.

Caution

Avoid making the C and D matrices depend on the system output y. If you have such dependence, the resulting state-space equation y = C(y)x + D(y)u creates an algebraic loop, because computing the output value y requires knowing the output value. This algebraic loop is prone to instability and divergence. Instead, try expressing C and D in terms of the time t, the block input u, and the state outputs x.

For similar reasons, avoid making A and B depend on the dx output. Note that it is safe for A and B to depend on y when y is a fixed combination of states and inputs (in other words, when y = Cx + Du, where C and D are constant matrices).

## Ports

### Input

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System input signal.

State matrix of dimensions Nx-by-Nx, where Nx is the number of system states.

Input matrix of dimensions Nx-by-Nu, where Nu is the number of system inputs.

Output matrix Ny-by-Nx, where Ny is the number of system outputs.

Feedforward matrix of dimensions Ny-by-Nu.

For a zero-feedthrough state-space model, clear the Enable feedthrough checkbox. Clearing this checkbox disables the D input port. (since R2023a)

Since R2024a

State derivative offsets of dimensions Nx-by-1, where Nx is the number of system states.

#### Dependencies

To enable this port, select Enable state derivative offsets dx0.

Since R2024a

State offsets of dimensions Nx-by-1, where Nx is the number of system states.

#### Dependencies

To enable this port, select Enable state offsets x0.

Since R2024a

Input offsets of dimensions Nu-by-1, where Nu is the number of system inputs.

#### Dependencies

To enable this port, select Enable input offsets u0.

Since R2024a

Output offsets of dimensions Ny-by-1, where Ny is the number of system outputs.

#### Dependencies

To enable this port, select Enable output offsets y0.

### Output

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System output signal.

Current state values.

#### Dependencies

To enable this port, select the Output states parameter.

Current derivatives of the corresponding states in x.

#### Dependencies

To enable this port, select the Output state derivatives parameter.

## Parameters

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### Main Tab

Initial state values, specified as a scalar or a vector whose length is the number of system states.

To identify system states, specify state names as a:

• character vector, for a one-state plant.

• Cell array of character vectors, for a multistate plant.

Select to enable the state values output port, x.

Select to enable the state derivatives output port, dx.

Since R2023a

Enable the D input port for a state-space model with direct feedthrough. For a zero-feedthrough model, clear this checkbox. Disabling the port for zero-feedthrough models is numerically more reliable than feeding a zero-constant into the port.

### Offsets Tab

Enable the x0 input port for specifying varying state offsets.

#### Programmatic Use

To set the block parameter value programmatically, use the `set_param` (Simulink) function.

 Parameter: `InputX0` Values: `"off"` (default) | `"on"`

Enable the dx0 input port for specifying varying state derivative offsets.

#### Programmatic Use

To set the block parameter value programmatically, use the `set_param` (Simulink) function.

 Parameter: `InputDX0` Values: `"off"` (default) | `"on"`

Enable the u0 input port for specifying varying input offsets.

#### Programmatic Use

To set the block parameter value programmatically, use the `set_param` (Simulink) function.

 Parameter: `InputU0` Values: `"off"` (default) | `"on"`

Enable the y0 input port for specifying varying output offsets.

#### Programmatic Use

To set the block parameter value programmatically, use the `set_param` (Simulink) function.

 Parameter: `InputY0` Values: `"off"` (default) | `"on"`

## Version History

Introduced in R2017b

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