# 1D Observer Form [A(v),B(v),C(v),F(v),H(v)]

Implement gain-scheduled state-space controller in observer form depending on one scheduling parameter

GNC/Control

## Description

The 1D Observer Form [A(v),B(v),C(v),F(v),H(v)] block implements a gain-scheduled state-space controller defined in the following observer form:

`$\begin{array}{l}\stackrel{˙}{x}=\left(A\left(v\right)+H\left(v\right)C\left(v\right)\right)x+B\left(v\right){u}_{meas}+H\left(v\right)\left(y-{y}_{dem}\right)\\ {u}_{dem}=F\left(v\right)x\end{array}$`

The main application of this block is to implement a controller designed using H-infinity loop-shaping, one of the design methods supported by Robust Control Toolbox.

## Parameters

A-matrix(v)

A-matrix of the state-space implementation. The A-matrix should have three dimensions, the last one corresponding to the scheduling variable v. Hence, for example, if the A-matrix corresponding to the first entry of v is the identity matrix, then `A(:,:,1) = [1 0;0 1];`.

B-matrix(v)

B-matrix of the state-space implementation. The B-matrix should have three dimensions, the last one corresponding to the scheduling variable v. Hence, for example, if the B-matrix corresponding to the first entry of v is the identity matrix, then `B(:,:,1) = [1 0;0 1];`.

C-matrix(v)

C-matrix of the state-space implementation. The C-matrix should have three dimensions, the last one corresponding to the scheduling variable v. Hence, for example, if the C-matrix corresponding to the first entry of v is the identity matrix, then `C(:,:,1) = [1 0;0 1];`.

F-matrix(v)

State-feedback matrix. The F-matrix should have three dimensions, the last one corresponding to the scheduling variable v. Hence, for example, if the F-matrix corresponding to the first entry of v is the identity matrix, then `F(:,:,1) = [1 0;0 1];`.

H-matrix(v)

Observer (output injection) matrix. The H-matrix should have three dimensions, the last one corresponding to the scheduling variable v. Hence, for example, if the H-matrix corresponding to the first entry of v is the identity matrix, then `H(:,:,1) = [1 0;0 1];`.

Scheduling variable breakpoints

Vector of the breakpoints for the scheduling variable. The length of v should be same as the size of the third dimension of A, B, C, F, and H.

Initial state, x_initial

Vector of initial states for the controller, i.e., initial values for the state vector, x. It should have length equal to the size of the first dimension of A.

## Inputs and Outputs

InputDimension TypeDescription

First

Contains the set-point error.

Second

Contains the scheduling variable.

Third

Contains the measured actuator position.

OutputDimension TypeDescription

First

Contains the actuator demands.

## Assumptions and Limitations

If the scheduling parameter inputs to the block go out of range, then they are clipped; i.e., the state-space matrices are not interpolated out of range.

## Examples

See H-Infinity Controller (1 Dimensional Scheduling) in `aeroblk_lib_HL20` for an example of this block.

## Reference

Hyde, R. A., "H-infinity Aerospace Control Design - A VSTOL Flight Application," Springer Verlag, Advances in Industrial Control Series, 1995. ISBN 3-540-19960-8. See Chapter 6.