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Implement 1-D vector of state-space controllers by linear interpolation of their outputs

**Library:**Aerospace Blockset / GNC / Control

The 1D Controller Blend u=(1-L).K1.y+L.K2.y block implements
an array of state-space controller designs. The model runs the controllers in parallel
and interpolates their outputs according to the current flight condition or operating
point. The advantage of this implementation approach is that the state-space matrices
*A*, *B*, *C*, and
*D* for the individual controller designs do not need to vary
smoothly from one design point to the next. The output from this block is the actuator
demand, which you can input to an actuator block.

This block requires the Control System Toolbox™ license.

The block implements

$$\begin{array}{l}{\dot{x}}_{1}={A}_{1}{x}_{1}+{B}_{1}y\\ {u}_{1}={C}_{1}{x}_{1}+{D}_{1}y\\ {\dot{x}}_{2}={A}_{2}{x}_{2}+{B}_{2}y\\ {u}_{2}={C}_{2}{x}_{2}+{D}_{2}y\\ u=(1-\lambda ){u}_{1}+\lambda {u}_{2}\\ \\ \lambda =\{\begin{array}{ccc}0& & v<{v}_{\mathrm{min}}\\ \frac{v-{v}_{\mathrm{min}}}{{v}_{\mathrm{max}}-{v}_{\mathrm{min}}}& & {v}_{\mathrm{min}}\le v\le {v}_{\mathrm{max}}\\ 1& & v>{v}_{\mathrm{max}}\end{array}\\ \end{array}$$

For example, suppose two controllers are designed at two operating points
*v*=*v*_{min} and
*v*=*v*_{max}. For longer
arrays of design points, the block only implements nearest neighbor designs. At any
given instant in time, the block updates three controller designs, reducing
computational requirements.

As the value of the scheduling parameter varies and the index of the controllers that need to be run changes, the block initializes the states of the oncoming controller using the self-conditioned form as defined for the Self-Conditioned [A,B,C,D] block.

[1] Hyde, R. A., "H-infinity Aerospace Control Design — A VSTOL Flight Application." ,
*Advances in Industrial Control Series*, Springer Verlag,
1995.

1D Controller [A(v),B(v),C(v),D(v)] | 1D Observer Form [A(v),B(v),C(v),F(v),H(v)] | 1D Self-Conditioned [A(v),B(v),C(v),D(v)] | 2D Controller Blend | Linear Second-Order Actuator | Nonlinear Second-Order Actuator | Self-Conditioned [A,B,C,D]