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:

$$\begin{array}{c}dx=d{x}_0+A(x-x_0)+B(u-u_0)\\ y=y_0+C(x-x_0)+D(u-u_0),\end{array}$$

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

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.

Output

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

Main Tab

Initial conditions — System initial conditions

0 (default) | scalar | vector

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

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

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.