# 1D Self-Conditioned [A(v),B(v),C(v),D(v)]

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

GNC/Control

## Description

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

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

in the self-conditioned form

`$\begin{array}{l}\stackrel{˙}{z}=\left(A\left(v\right)-H\left(v\right)C\left(v\right)\right)z+\left(B\left(v\right)-H\left(v\right)D\left(V\right)\right)e+H\left(v\right){u}_{meas}\\ {u}_{dem}=C\left(v\right)z+D\left(v\right)e\end{array}$`

For the rationale behind this self-conditioned implementation, refer to the Self-Conditioned [A,B,C,D] block reference. This block implements a gain-scheduled version of the Self-Conditioned [A,B,C,D] block, v being the 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. 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];`.

D-matrix(v)

D-matrix of the state-space implementation. The D-matrix should have three dimensions, the last one corresponding to the scheduling variable v. Hence, 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 first 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.

Poles of A(v)-H(v)*C(v)

Vector of the desired poles of A-HC. Note that the poles are assigned to the same locations for all values of the scheduling parameter v. Hence the number of pole locations defined should be equal to the length of the first dimension of the A-matrix.

## Inputs and Outputs

InputDimension TypeDescription

First

AnyContains the measurements.

Second

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

Third

Contains the measured actuator position.

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.

 Note:   This block requires the Control System Toolbox™ product.

## Reference

The algorithm used to determine the matrix H is defined in Kautsky, Nichols, and Van Dooren, "Robust Pole Assignment in Linear State Feedback," International Journal of Control, Vol. 41, No. 5, pages 1129-1155, 1985.