# Varying Observer Form

Observer-form state-space model with varying matrix values

**Libraries:**

Control System Toolbox /
Linear Parameter Varying

## Description

Use this block to implement a continuous-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:

$$\begin{array}{c}d{x}_{e}=A{x}_{e}+Bu+L\epsilon \\ u=-K{x}_{e}\\ \epsilon =y-C{x}_{e}-Du,\end{array}$$

where *u* is the plant input (control signal), *y*
is the plant output, *x _{e}* is the estimated
state, and

*ε*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*tracks the plant state, and all controllers are expressed with the same state coordinates.

_{e}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**. If you have such dependence, the resulting equation *u* =
–*K*(*u*)*x _{e}*
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**.

For similar reasons, avoid making **A** and
**B** depend on the
**dx _{e}** output.

## Ports

### Input

### Output

## Parameters

## Extended Capabilities

## Version History

**Introduced in R2017b**