# Discrete Varying State Space

Discrete-time state-space model with varying matrix values

Libraries:
Control System Toolbox / Linear Parameter Varying

## Description

Use this block to implement a discrete-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:

`$\begin{array}{c}{x}_{k+1}=A{x}_{k}+B{u}_{k}\\ {y}_{k}=C{x}_{k}+D{u}_{k},\end{array}$`

where uk is the system input, yk is the system output, xk is the current system state, and xk+1 is the system state at the next time step.

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

`$\begin{array}{c}{x}_{k+1}=d{x}_{0k}+A\left({x}_{k}-{x}_{0k}\right)+B\left({u}_{k}-{u}_{0k}\right)\\ {y}_{k}={y}_{0k}+C\left({x}_{k}-{x}_{0k}\right)+D\left({u}_{k}-{u}_{0k}\right),\end{array}$`

where u0k is the input offset, y0k is the output offset, and x0k and dx0k 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 yk. If you have such dependence, the resulting state-space equation yk = C(yk)xk + D(yk)uk creates an algebraic loop, because computing the output value yk 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 uk, and the state outputs xk.

For similar reasons, avoid making A and B depend on the xk+1 output. Note that it is safe for A and B to depend on yk when yk is a fixed combination of states and inputs (in other words, when yk = Cxk + Duk, 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)

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)

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)

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. (since R2024a)

Since R2024b

Initial conditions to use with the local model to start the simulation, specified as a vector of length equal to the number of model states.

#### Dependencies

To enable this port, set Initial condition source to `external`.

### Output

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System output signal.

Current state values.

#### Dependencies

To enable this port, select the Output states parameter.

State values at next time step.

#### Dependencies

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

## Parameters

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### Main Tab

Since R2024b

Specify the initial condition source.

• `internal` — Specify the initial conditions of the states using the Initial condition parameter.

• `external` — Specify the initial conditions of the states at the IC input port of the block.

#### Programmatic Use

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

 Parameter: `InitialConditionSource` Values: `internal` (default) | `external`

Initial state values, specified as a vector of length equal to the number of model states.

#### Dependencies

To enable this parameter, set Initial condition source to `internal`.

#### Programmatic Use

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

 Parameter: `InitialCondition` Values: `"0"` (default) | initial state values in quotes

Example: ```set_param(gcb,"InitialCondition","[0 0.1]")```

Block sample time, specified as either –1 (inherited sample time) or a positive scalar value.

Select to enable the state values output port, x.

Select to enable the state updates output port, xk+1.

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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