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:

$\begin{matrix} d x = A x + B u \\ y = C x + D u, \end{matrix}$

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:

$\begin{matrix} d x = d x_{0} + A (x - x_{0}) + B (u - u_{0}) \\ y = y_{0} + C (x - x_{0}) + D (u - u_{0}), \end{matrix}$

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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u — System input
scalar | vector

System input signal.

A — State matrix
matrix

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

B — Input matrix
matrix

Input matrix of dimensions N_x-by-N_u, where N_u is the number of system inputs.

C — Output matrix
matrix

Output matrix N_y-by-N_x, where N_y is the number of system outputs.

D — Feedforward matrix
matrix

Feedforward matrix of dimensions N_y-by-N_u.

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

dx0 — State derivative offsets
matrix

Since R2024a

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

Dependencies

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

x0 — State offsets
matrix

Since R2024a

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

Dependencies

To enable this port, select Enable state offsets x0.

u0 — Input offsets
matrix

Since R2024a

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

Dependencies

To enable this port, select Enable input offsets u0.

y0 — Output offsets
matrix

Since R2024a

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

Dependencies

To enable this port, select Enable output offsets y0.

IC — Initial condition
vector

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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y — System output
scalar | vector

System output signal.

x — Current state vector
vector

Current state values.

Dependencies

To enable this port, select the Output states parameter.

dx — State derivatives
vector

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 condition source — Option to set initial condition using external signal
`internal` (default) | `external`

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 conditions — System initial conditions
0 (default) | scalar | vector

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]")

State names — System state names
`''` (default) | character vector | cell array

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.

Output states — Provide state output
on (default) | off

Select to enable the state values output port, x.

Output state derivatives — Provide state derivatives
on (default) | off

Select to enable the state derivatives output port, dx.

Enable feedthrough — Add input port for feedthrough matrix
on (default) | off

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 state offsets x0 — Add input port for state offsets
off (default) | on

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 state derivative offsets dx0 — Add input port for state derivative offsets
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 input offsets u0 — Add input port for input offsets
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 output offsets y0 — Add input port for output offsets
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"`

Extended Capabilities

C/C++ Code Generation
Generate C and C++ code using Simulink® Coder™.

Version History

Introduced in R2017b

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R2024b: Specify initial condition externally as block input

The block now allows you to specify initial state values at the block input port IC. To enable this port, set the Initial condition source parameter to external.

R2024b: Initial condition parameter `x0` renamed to `InitialCondition`

The Initial state parameter x0 for programmatic use is now called InitialCondition. If your code sets this parameter programmatically using, for example set_param(blockPath,"x0","[0 0.1]"), update your code to set_param(blockPath,"InitialCondition","[0 0.1]").

R2024a: Support for specifying offsets

Use the Offsets tab of the block parameters to enable the input ports for specifying instantaneous values of the varying offsets.

R2023a: Enable feedthrough checkbox added

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

Varying State Space

Description

Ports

Input

u — System input scalar | vector

A — State matrix matrix

B — Input matrix matrix

C — Output matrix matrix

D — Feedforward matrix matrix

dx0 — State derivative offsets matrix

Dependencies

x0 — State offsets matrix

Dependencies

u0 — Input offsets matrix

Dependencies

y0 — Output offsets matrix

Dependencies

IC — Initial condition vector

Dependencies

Output

y — System output scalar | vector

x — Current state vector vector

Dependencies

dx — State derivatives vector

Dependencies

Parameters

Main Tab

Initial condition source — Option to set initial condition using external signal internal (default) | external

Programmatic Use

Initial conditions — System initial conditions 0 (default) | scalar | vector

Dependencies

Programmatic Use

State names — System state names '' (default) | character vector | cell array

Output states — Provide state output on (default) | off

Output state derivatives — Provide state derivatives on (default) | off

Enable feedthrough — Add input port for feedthrough matrix on (default) | off

Offsets Tab

Enable state offsets x0 — Add input port for state offsets off (default) | on

Programmatic Use

Enable state derivative offsets dx0 — Add input port for state derivative offsets off (default) | on

Programmatic Use

Enable input offsets u0 — Add input port for input offsets off (default) | on

Programmatic Use

Enable output offsets y0 — Add input port for output offsets off (default) | on

Programmatic Use

Extended Capabilities

C/C++ Code Generation Generate C and C++ code using Simulink® Coder™.

Version History

R2024b: Specify initial condition externally as block input

R2024b: Initial condition parameter x0 renamed to InitialCondition

R2024a: Support for specifying offsets

R2023a: Enable feedthrough checkbox added

See Also

Topics

u — System input
scalar | vector

A — State matrix
matrix

B — Input matrix
matrix

C — Output matrix
matrix

D — Feedforward matrix
matrix

dx0 — State derivative offsets
matrix

x0 — State offsets
matrix

u0 — Input offsets
matrix

y0 — Output offsets
matrix

IC — Initial condition
vector

y — System output
scalar | vector

x — Current state vector
vector

dx — State derivatives
vector

Initial condition source — Option to set initial condition using external signal
`internal` (default) | `external`

Initial conditions — System initial conditions
0 (default) | scalar | vector

State names — System state names
`''` (default) | character vector | cell array

Output states — Provide state output
on (default) | off

Output state derivatives — Provide state derivatives
on (default) | off

Enable feedthrough — Add input port for feedthrough matrix
on (default) | off

Enable state offsets x0 — Add input port for state offsets
off (default) | on

Enable state derivative offsets dx0 — Add input port for state derivative offsets
off (default) | on

Enable input offsets u0 — Add input port for input offsets
off (default) | on

Enable output offsets y0 — Add input port for output offsets
off (default) | on

C/C++ Code Generation
Generate C and C++ code using Simulink® Coder™.

R2024b: Initial condition parameter `x0` renamed to `InitialCondition`