Discrete Varying Observer Form

Discrete-time observer-form 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 varying state-space model in observer form. The system matrices A, B, C, and D describe the plant dynamics, and the matrices K and L specify the state-feedback and state-observer gains, respectively. Feed the instantaneous values of these matrices to the corresponding input ports. The observer form is given by:

$$\begin{array}{c}{x}_{k+1}=A{x}_k+B{u}_k+L{\epsilon }_k\\ u_k=-K{x}_k\\ {\epsilon }_k=y_k-C{x}_k-D{u}_k,\end{array}$$

where u_k is the plant input (control signal), y_k is the plant output, x_k is the estimated state, and ε_k is the innovation, the difference between the predicted and measured plant output. The observer form works well for gain scheduling of state-space controllers. In particular, the state x_k tracks the plant state, and all controllers are expressed with the same state coordinates.

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 K matrix depend on the control signal u_k. If you have such dependence, the resulting equation u_k = –K(u_k)x_k creates an algebraic loop, because computing the block output value requires knowing the block output value. This algebraic loop is prone to instability and divergence. Instead, try expressing K in terms of the time t and the block input y_k.

For similar reasons, avoid making A, B, or L depend on the u_k+1 output.

Ports

Input

y_k — Measurement signal

scalar | vector

Measured plant output signal.

A — Plant state matrix

matrix

Plant state matrix of dimensions N_x-by-N_x, where N_x is the number of plant states.

B — Plant input matrix

matrix

Plant input matrix of dimensions N_x-by-N_u, where N_u is the number of plant inputs.

C — Plant output matrix

matrix

Plant output matrix of dimensions N_y-by-N_x, where N_y is the number of plant outputs.

D — Plant feedforward matrix

matrix

Plant feedforward matrix of dimensions N_y-by-N_u.

K — State-feedback matrix

matrix

State-feedback matrix of dimensions N_u-by-N_x.

L — State-observer matrix

matrix

State-observer matrix of dimensions N_x-by-N_y.

Output

u_k — Control signal

scalar | vector

Control signal (plant input).

x_k — Estimated plant state vector

vector

Vector of estimated plant states.

Dependencies

To enable this port, select the Output states parameter.

x_k+1 — Estimated next state vector

vector

Estimated 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 plant 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 estimated states output port, x_e.

Output state derivatives — Provide state updates

on (default) | off

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

Extended Capabilities

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

Version History

Introduced in R2017b