Balanced parametrization of classes of linear systems
SIAM Journal on Control and Optimization
Robust and optimal control
Multidimensional Systems and Signal Processing
Approximation of Large-Scale Dynamical Systems (Advances in Design and Control) (Advances in Design and Control)
Analysis of Second-Order Modes of Linear Continuous-Time Systems under Positive-Real Transformations
IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences
Gramian-Preserving Frequency Transformation for Linear Discrete-Time State-Space Systems
IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences
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This paper discusses the behavior of the second-order modes (Hankel singular values) of linear discrete-time systems under bounded-real transformations, where the transformations are given by arbitrary transfer functions with magnitude bounded by unity. Our main result reveals that the values of the second-order modes are decreased under any of the above-mentioned transformations. This result is the generalization of the theory of Mullis and Roberts, who proved that the second-order modes are invariant under any allpass transformation, i.e. any lossless bounded-real transformation. We derive our main result by describing the controllability/observability Gramians of transformed systems with the help of the discrete-time bounded-real lemma.