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# 1D Controller [A(v),B(v),C(v),D(v)]

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

GNC/Control

## Description

The 1D Controller [A(v),B(v),C(v),D(v)] block implements a gain-scheduled state-space controller as defined by the equations

`$\begin{array}{l}\stackrel{˙}{x}=A\left(v\right)x+B\left(v\right)y\\ u=C\left(v\right)x+D\left(v\right)y\end{array}$`

where v is a parameter over which A, B, C, and D are defined. This type of controller scheduling assumes that the matrices A, B, C, and D vary smoothly as a function of v, which is often the case in aerospace applications.

## Parameters

A-matrix(v)

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

D-matrix(v)

D-matrix of the state-space implementation. In the case of 1-D scheduling, the D-matrix should have three dimensions, the last one corresponding to the scheduling variable v. For example, if the D-matrix corresponding to the first entry of v is the identity matrix, then `D(:,:,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, and D.

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

AnyContains the measurements.

Second

Contains the scheduling variable conforming to the dimensions of the state-space matrices.

OutputDimension TypeDescription

First

AnyContains 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.