Characterization of Stability Regions in the Discrete PID Gain Space via D-Partition Approach

In the study of linear control systems, D-partition theory provides a powerful technique for controller design by dividing the controller parameter space into stability regions, within which the number of unstable characteristic roots remains invariant. By re-expressed the discrete PID controller as $G_{pid}\left ({{ z }}\right)=({k_{0}+k}_{1}z+k_{2}\left ({{ 1+z^{2} }}\right))/(z\left ({{ z-1 }}\right))$ , we revisit the problem of characterizing stability regions in the gain space $\mathbf {k} = (k_{0},k_{1},k_{2})$ using the D-partition technique. Specifically, we identify and complete seven types of critical $k_{0}$ points, which divide the $k_{0}$ axis into intervals where the topology of the D-partition of the $(k_{1},k_{2})$ -plane remains invariant. This enables the exact construction of all stability - not merely stable - regions in the three-dimensional gain space. Furthermore, we present a refined algorithm for determining all the critical $k_{0}$ points associated with a discrete PID controller for a given arbitrary-order plant. This algorithm improves upon existing methods in two key aspects: 1) two previously unrecognized types of critical $k_{0}$ points are identified, ensuring the completeness of the characterization; and 2) new mathematical expressions are derived for describing singular D-partition border lines, which eliminates the occurrence of spurious critical $k_{0}$ points. In addition, due to the finite range of the frequency variable in discrete-time systems, the proposed critical $k_{0}$ revelation algorithm is computationally well-suited for revealing all critical proportional gain values $k_{p}$ for the continuous-time PID control, through appropriate use of bilinear s-to-z and z-to-s transformations. To validate the effectiveness of the proposed method, extensive examples are provided.