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

GNC/Control

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

$$\begin{array}{l}\dot{x}=A(v)x+B(v)y\\ u=C(v)x+D(v)y\end{array}$$

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.

**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*.

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

First | Any | Contains the measurements. |

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

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

First | Any | 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`

for
an example of this block.

1D Controller Blend u=(1-L).K1.y+L.K2.y

1D Observer Form [A(v),B(v),C(v),F(v),H(v)]

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

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