Discrete Varying State Space

Discrete-time state-space model with varying matrix values

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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 u_k is the system input, y_k is the system output, x_k is the current system state, and x_k+1 is the system state at the next time step.

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_k. If you have such dependence, the resulting state-space equation y_k = C(y_k)x_k + D(y_k)u_k creates an algebraic loop, because computing the output value y_k 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_k, and the state outputs x_k.

For similar reasons, avoid making A and B depend on the x_k+1 output. Note that it is safe for A and B to depend on y_k when y_k is a fixed combination of states and inputs (in other words, when y_k = Cx_k + Du_k, where C and D are constant matrices).

Ports

Input

u_k — 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)

Output

y_k — System output

scalar | vector

System output signal.

x_k — Current state vector

vector

Current state values.

Dependencies

To enable this port, select the Output states parameter.

x_k+1 — Next state vector

vector

State values at next time step.

Dependencies

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

Parameters

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.

Sample time (-1 for inherited) — Block sample time

–1 (default) | positive scalar

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

Output states — Provide state output

on (default) | off

Select to enable the state values output port, x.

Output state derivatives — Provide state updates

on (default) | off

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

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.

Extended Capabilities

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

PLC Code Generation
Generate Structured Text code using Simulink® PLC Coder™.

Version History

Introduced in R2017b

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.