On the Non-optimality of Linear Quadratic Gaussian Balanced Truncation for Constrained Order Controller Design

For model based control designs like $\boldsymbol{H}_{2}$ and $\boldsymbol{H}_{\boldsymbol{\infty}}$ optimal control, the controller order is typically equal to the model order. However, the synthesis and real-time implementation of these controllers may be hampered for models of large order due to computational limitations. Several closed-loop relevant order reduction techniques have been developed in order to overcome this problem. One technique is Linear Quadratic Gaussian (LQG) balanced truncation, which is closely related to LQG and $\boldsymbol{H}_{2}$ optimal controller design. This paper demonstrates the non-optimality of this technique for constrained order $\boldsymbol{H}_{2}$ optimal controller design. Furthermore, improvements to reduce the loss of performance as a result of truncation are proposed. These improvements can be applied to a large class of controller designs and order reduction techniques as well.