Vasile Cirtoaje

The paper presents a practical algorithm of the proportional-internal model control (P-IMC) type that can be applied to control a wide class of processes: Stable proportional processes, integral processes and some unstable processes. The P-IMC algorithm is a suitable combination between the P0-IMC algorithm and the P1-IMC algorithm, which are characterized by a too weak and a too strong impact of the tuning gain on the control action, respectively. The overall controller has five parameters: A tuning parameter K, three model parameters (steady-state gain, settling time, and time delay) and a process feedback gain used only for integral or unstable processes, to turn them into a compensated process (stable and of proportional type). For a step setpoint, the initial value of the compensated process input is approximately K times its final value. Furthermore, for $K=1$ , the compensated process input is close to a step shape (step control principle). These properties enable a human operator to check and adjust online the model parameters. Due to its control performance, robustness to modeling error, and capability to be easily tuned and applied for all industrial processes, the P-IMC algorithm could be a viable alternative to the known PID algorithm. Numerical simulations are given to highlight the performance and the flexibility of the algorithm.

We establish some strong sufficient conditions such that the inequality $f_4(x,y,z)\ge 0$ holds for any real numbers $x,y,z,$ where $f_4(x,y,z)$ is a cyclic homogeneous polynomial of degree four. In addition, we give a conjecture which states some new necessary and sufficient conditions such that the inequality $f_4(x,y,z)\ge 0$ holds for any real numbers $x,y,z.$ Several applications are given to show the effectiveness of the proposed method.