Feedback properties of multivariable systems: The role and use of the return difference matrix

For linear time-invariant multivariable feedback systems, the feedback properties of plant disturbance attenuation, sensor noise response, stability margins, and sensitivity to plant and sensor variation are quantitatively related to the Bode magnitude versus frequency plots of the singular values of the return difference matrix <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">I + L</tex> and of the associated inverse-return difference matrix <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">I + L⁻¹</tex> . Implied fundamental limits of feedback performance are quantitatively described and design tradeoffs are discussed. The penalty function in the stochastic linear quadratic Gaussian (LQG) optimal control problem is found to be a weighted-sum of the singular values, with the weights determined by the quadratic cost and noise intensity matrices. This enables systematic "tuning" of LQG cost and noise matrices so that the resulting optimal return difference and inversereturn difference meet inequality constraints derived from design specifications on feedback properties. The theory has been used to synthesize a multivariable automatic controller for the longitudinal dynamics of an advanced fighter aircraft.