Bode’s Integral Theorem and Flight Control

In this paper, we discuss an extension of the original Bode integral theorem to multi-input–multi-output (MIMO) continuous-time systems, without relative degree assumption. This result was recently derived by the first author. Literature search revealed that a similar formulation was developed prior and published as a draft paper in online archives. That formula was similar but not the same as presented in this paper. Nevertheless, both expressions show that the Bode sensitivity integral for a MIMO system can be written as a difference between the plant open-loop and closed-loop eigenvalues, irrespective of the system loop gain relative degree. The revised expression for the Bode integral gives further insights into sensitivity shaping and presents an alternative expression when compared to the previously known extensions. In that regard, we present derivations of the revised Bode integral formula for MIMO systems and demonstrate Bode theorem implications for flight control. Specifically, we design, analyze, and simulate a dynamic output feedback baseline controller with direct adaptive augmentation using the problem motivation from the Inaugural IEEE Bode Lecture “Respect the Unstable,” presented by Dr. Gunter Stein in 1989. For numerical illustration purposes, we will use the X-29A aircraft models and subsystems data from a NASA technical report.