Nyquist stability criterion and positive realness of sampled-data systems

We first derive a Nyquist stability criterion of sampled-data systems in such a form that reflects the treatment of sampled-data systems with their intersample behavior taken into account. We next analyze the positive realness of sampled-data systems with D₁₁ matrix (i.e., the continuous-time direct feedthrough matrix from the disturbance input w to the controlled output z) being 0, and show that such sampled-data systems cannot be made positive real, unless the relative degree of the system from w to z is 1. We further show that adding some appropriate D₁₁ leads to positive real sampled-data systems, and give a method to find such a D₁₁ that is as small as possible. All these analyses, as well as the small-gain analysis, are compared from the viewpoint of the gain margin analysis