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Nowadays almost in the all branches of control even intelligent control strictly positive real transfer functions are used to stabilize closed loop control systems. These transfer functions in a closed loop control system can help to generate a lyapunov function for guarantee the stability of closed loop system. Nevertheless, the necessary and sufficient conditions for strictly positive real transfer function matrices have been different in the literature for many years. Recently, it is shown that there are inconsistencies between definition and theorems of strict positive realness. In this paper, we present and prove an important difference between positive real and strictly positive real transfer function matrices. Moreover, we introduce more complete and simpler necessary and sufficient conditions for strictly positive real transfer function matrices. An important advantage of present work is usage of Markov parameters of multivariable systems. We know these parameters are easily calculated in both frequency and time domains, so using these parameters can make a similar framework in both frequency and time domains. Presenting some examples, we show the advantages of our new results in comparison to before works.

Keywords: Matrix Transfer Functions, Strictly Positive Real (SPR) Transfer Functions, Markov Parameter of Systems, Laurent Series Expansion.

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