- State Scaling: Each state variable in the state-space model may have different physical units and magnitudes. State scaling involves normalizing these state variables so that they are dimensionless and of a similar order of magnitude. This can be done by multiplying the state vector by a diagonal scaling matrix, where each diagonal element corresponds to the inverse of a characteristic value (e.g., maximum value or standard deviation) of the respective state variable.
- Control Scaling: Similar to state scaling, control inputs can also vary widely in their magnitudes and units. Control scaling ensures that the control inputs used in the optimization process are of a similar order of magnitude, which can help the numerical solver work more effectively.
- Measurement Scaling: The outputs or measurements of the system are also scaled to be dimensionless and of a similar order of magnitude. This is particularly important when the measurements are used in the cost function of the optimization problem, as it ensures that no single measurement disproportionately affects the cost due to its scale.
- Identification of Scaling Factors: Determine appropriate scaling factors for the states, inputs, and outputs. These factors are typically chosen based on the ranges or standard deviations of these variables.
- Pre-multiplication by Scaling Matrices: Apply the scaling factors to the system's matrices. For a state-space system represented by ( A, B, C, D ), where ( x' = Ax + Bu ) and ( y = Cx + Du ), the matrices are transformed using diagonal scaling matrices ( S_x, S_u, S_y ) for states, inputs, and outputs, respectively.
- Transformation of the System: The original system is transformed into a scaled system by pre- and post-multiplying the appropriate matrices by their respective scaling matrices.
- Controller Synthesis: The H-infinity synthesis is performed on the scaled system. The resulting controller is designed to work well with the scaled system.
- Inverse Scaling of the Controller: Once the controller is designed for the scaled system, it needs to be transformed back so that it can be applied to the original, unscaled system. This involves inverse scaling of the controller matrices.