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# Self-Conditioned [A,B,C,D]

Implement state-space controller in self-conditioned form

GNC/Controls

## Description

The Self-Conditioned [A,B,C,D] block can be used to implement the state-space controller defined by

$\left[\begin{array}{l}\stackrel{˙}{x}=Ax+Be\\ u=Cx+De\end{array}\right]$

in the self-conditioned form

$\begin{array}{l}\stackrel{˙}{z}=\left(A-HC\right)z+\left(B-HD\right)e+H{u}_{meas}\\ {u}_{dem}=Cz+De\end{array}$

The input ${u}_{meas}$ is a vector of the achieved actuator positions, and the output ${u}_{dem}$ is the vector of controller actuator demands. In the case that the actuators are not limited, then ${u}_{meas}={u}_{dem}$ and substituting the output equation into the state equation returns the nominal controller. In the case that they are not equal, the dynamics of the controller are set by the poles of A-HC.

Hence H must be chosen to make the poles sufficiently fast to track ${u}_{meas}$ but at the same time not so fast that noise on e is propagated to ${u}_{dem}$. The matrix H is designed by a callback to the Control System Toolbox™ command place to place the poles at defined locations.

## Dialog Box

A-matrix

A-matrix of the state-space implementation.

B-matrix

B-matrix of the state-space implementation.

C-matrix

C-matrix of the state-space implementation.

D-matrix

D-matrix of the state-space implementation.

Initial state, x_initial

This is a vector of initial states for the controller, i.e., initial values for the state vector, z. It should have length equal to the size of the first dimension of A.

Poles of A-H*C

This is a vector of the desired poles of A-H*C. Hence the number of pole locations defined should be equal to the dimension of the A-matrix.

## Inputs and Outputs

InputDimension TypeDescription

First

Contains the control error.

Second

Contains the measured actuator position.

OutputDimension TypeDescription

First

Contains the actuator demands.

## Assumptions and Limitations

 Note:   This block requires the Control System Toolbox product.

## Examples

This Simulink® model shows a state-space controller implemented in both self-conditioned and standard state-space forms. The actuator authority limits of ±0.5 units are modeled by the saturation block.

Notice that the A-matrix has a zero in the 1,1 element, indicating integral action.

The top trace shows the conventional state-space implementation. The output of the controller winds up well past the actuator upper authority limit of +0.5. The lower trace shows that the self-conditioned form results in an actuator demand that tracks the upper authority limit, which means that when the sign of the control error, e, is reversed, the actuator demand responds immediately.

## 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.