On common quadratic Lyapunov functions for pairs of stable LTI systems whose system matrices are in companion form

response of the frequency estimation loop and simplified its design. The estimates were unbiased and ripple-free when the signal contained no noise and the parameters of the signal were constant. A modified version of the algorithm provided improvements for situations in which the fundamental component of the signal could become small, or vanish for some periods of time. In this case, information from all components of the signal was used in the fundamental frequency estimation. Multiple signals with the same fundamental frequency were also combined to yield consistent estimation results despite changes in signal characteristics. In consequence, an advantage of the modified algorithm over the basic algorithm is that it is not necessary to know a priori which component is the most suitable to base the frequency estimation on. The algorithms were designed with real-time tracking applications in mind. They were simple in design and implementation, and effective in tracking time-varying parameters. The linear time-invariant approximations gave useful information about the dynamic behavior of the system, the tradeoff between convergence speed and noise sensitivity, and the selection of the design parameters. A stable and efficient adaptive notch filter for direct frequency estimation, " IEEE Trans. A minimal parameter adaptive notch filter with constrained poles and zeros, " IEEE Trans. A magnitude/phase-locked loop approach to parameter estimation of periodic signals, " in Proc. [19] , " Frequency estimation using multiple sources and multiple harmonic components, " in Proc. Abstract—In this note, the problem of determining necessary and sufficient conditions for the existence of a common quadratic Lyapunov function for a pair of stable linear time-invariant systems whose system matrices and are in companion form is considered. It is shown that a necessary and sufficient condition for the existence of such a function is that the matrix product does not have an eigenvalue that is real and negative. Examples are presented to illustrate the result.