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

GNC/Control

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:

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

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.

**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*.

Input | Dimension Type | Description |
---|---|---|

First | Contains the set-point error. | |

Second | Contains the scheduling variable. | |

Third | Contains the measured actuator position. |

Output | Dimension Type | Description |
---|---|---|

First | Contains the actuator demands. |

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.

See H-Infinity Controller (1 Dimensional Scheduling) in `aeroblk_lib_HL20`

`aeroblk_lib_HL20`

for
an example of this block.

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.

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