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Implement gain-scheduled state-space controller in observer form depending on three scheduling parameters

GNC/Control

The 3D 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(v1,v2,v3)**-matrix of the state-space implementation. In the case of 3-D scheduling, the*A*-matrix should have five dimensions, the last three corresponding to scheduling variables*A*1,*v*2, and*v*3. Hence, for example, if the*v*-matrix corresponding to the first entry of*A*1, the first entry of*v*2, and the first entry of*v*3 is the identity matrix, then*v*`A(:,:,1,1,1) = [1 0;0 1];`

.**B-matrix(v1,v2,v3)**-matrix of the state-space implementation. In the case of 3-D scheduling, the*B*-matrix should have five dimensions, the last three corresponding to scheduling variables*B*1,*v*2, and*v*3. Hence, for example, if the*v*-matrix corresponding to the first entry of*B*1, the first entry of*v*2, and the first entry of*v*3 is the identity matrix, then*v*`B(:,:,1,1,1) = [1 0;0 1];`

.**C-matrix(v1,v2,v3)**-matrix of the state-space implementation. In the case of 3-D scheduling, the*C*-matrix should have five dimensions, the last three corresponding to scheduling variables*C*1,*v*2, and*v*3. Hence, for example, if the*v*-matrix corresponding to the first entry of*C*1, the first entry of*v*2, and the first entry of*v*3 is the identity matrix, then*v*`C(:,:,1,1,1) = [1 0;0 1];`

.**F-matrix(v1,v2,v3)**State-feedback matrix. In the case of 3-D scheduling, the

-matrix should have five dimensions, the last three corresponding to scheduling variables*F*1,*v*2, and*v*3. Hence, for example, if the*v*-matrix corresponding to the first entry of*F*1, the first entry of*v*2, and the first entry of*v*3 is the identity matrix, then*v*`F(:,:,1,1,1) = [1 0;0 1];`

.**H-matrix(v1,v2,v3)**Observer (output injection) matrix. In the case of 3-D scheduling, the

-matrix should have five dimensions, the last three corresponding to scheduling variables*H*1,*v*2, and*v*3. Hence, for example, if the*v*-matrix corresponding to the first entry of*H*1, the first entry of*v*2, and the first entry of*v*3 is the identity matrix, then*v*`H(:,:,1,1,1) = [1 0;0 1];`

.**First scheduling variable (v1) breakpoints**Vector of the breakpoints for the first scheduling variable. The length of

1 should be same as the size of the third dimension of*v*,*A*,*B*,*C*, and*F*.*H***Second scheduling variable (v2) breakpoints**Vector of the breakpoints for the second scheduling variable. The length of

2 should be same as the size of the fourth dimension of*v*,*A*,*B*,*C*, and*F*.*H***Third scheduling variable (v3) breakpoints**Vector of the breakpoints for the third scheduling variable. The length of

3 should be same as the size of the fifth dimension of*v*,*A*,*B*,*C*, and*F*.*H***Initial state, x_initial**Vector of initial states for the controller, i.e., initial values for the state vector,

. It should have length equal to the size of the first dimension of*x*.*A*

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

First | Contains the set-point error. | |

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

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

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

Fifth | 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.

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.

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

2D Observer Form [A(v),B(v),C(v),F(v),H(v)]

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