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In this paper, we present a new complete and constructive characterization of all the stabilizing regions in the PID gain space. It is based on D-partition theory and the fact that for a given proportional gain $k_{p}$ the stability boundaries in the plane of integral and derivative gains are all straight lines. Actually, we present the notion of the critical $k_{p}$ point, which is defined in the sense that the topology of the D-partition of $(k_{i},k_{d})$ plane can change when the $k_{p}$ crosses a critical $k_{p}$ point. Moreover, we identify seven types of critical $k_{p}$ points and provide formulas for their computation. With the availability of all critical $k_{p}$ points, the stabilizing $k_{p}$ intervals can be exhaustively and exactly determined. By sweeping the $k_{p}$ parameter over the stabilizing $k_{p}$ intervals, the whole set of stabilizing PID controllers for a given plant is created. In addition, it allows for an efficient test of the existence of a stabilizing PID controller set without sweeping the $k_{p}$ parameter. To validate the newly presented analytical and constructive characterization of stabilizing PID controller sets and demonstrate the existence of seven types of critical $k_{p}$ points, five examples are provided.

The resultant-based frequency-sweeping approach is appealing for creating a stability map to study stability robustness against delay uncertainties. It relies on converting the system’s transcendental characteristic function into a polynomial form such that the crossing frequency set and the corresponding stability switching curves in the delay parameter plane can be exactly and exhaustively determined using only algebraic computations. In the existing literature, the equation conversion has been achieved by using Rekasius substitution, bilinear substitution, or half-angle tangent substitution, and variable eliminations are carried out by using Sylvester resultant or Dixon resultant. In this paper, we first point out that since the computation of Dixon resultant involves a polynomial division, it could be beneficial in a symbolic programming environment. Then, we provide numerical evidence to compare the computational complexity of using different equation conversion methods and the subsequent resultant elimination to obtain the critical crossing frequencies from which the exact crossing frequency set can be determined. Next, based on the resultant theory, we reveal the roles of the critical crossing frequencies that play in the characterization of the delay map. Finally, we provide an efficient procedure for constructing stability charts in the delay parameter plane for linear time-invariant systems with two delays in states. To demonstrate our contributions, explanatory examples are provided along with a complete illustrative example.

In this paper, we propose a recessed-gate tunneling field-effect transistor (TFET) to improve the on current of TFETs by increasing the tunnel area with line tunneling. We investigate the effects of the recessed-body thickness and the doping level on the device performance. For optimal device structures, our proposedn-TFET reaches 1.44 x 10-6 A/µm of on current and 3.22 x 109 ON/OFF current ratio. A minimum subthreshold swing SSmin = 28.3 mV/dec and an average swing SSavg = 59.8 mV/dec over seven orders of drain current are achieved. In addition, complementary TFET inverters showgood noisemargins of NMH = 65 mV (38.5 % VDD) and NML = 77 mV (32.5 % VDD) and also a high voltage gain even at VDD = 0.2 V. [ABSTRACT FROM AUTHOR]