Sum of imaginary parts and performance limitation of the LQG problem for SIMO systems

The sum of roots, initially introduced as an indicator of `average stability´, has been proven useful in expressing achievable performance levels for some ℋ₂ control problems. The drawback of this quantity is that it misses information on the oscillation property. To complement the quantity, another notion of the sum of imaginary parts is proposed in this paper. Further an algebraic method to compute this new quantity is suggested. This paper also shows that the performance limitation of the LQG problem in the SIMO plant case can be characterized in terms of the sum of roots. It is demonstrated by way of a design example that two contributions make it possible to minimize the achievable performance over plant parameters under a constraint on the oscillation of the closed-loop system.