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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.

As it is known, the equal variables method can be used to create and solve difficult symmetric inequalities in nonnegative variables involving the expressions ... is strictly convex. In this paper, we extend the equal variables method to real variables.

The control algorithm incorporates a process feedback path of pure proportional type (only for integral-type or unstable processes), a process model of second order plus time delay and a realizable second-order internal controller which has not a tuning filter time constant as usual in the classical IMC design, but a tuning gain with standard value 1, that can be used by the human operator to get a strong or weak control action. The model parameters (steady-state gain, time delay and transient time) can be easily experimentally determined and can be online verified and corrected. The algorithm is practical and quasi-universal because it is easily tunable, has a unique form and can be applied to almost all process types: stable proportional processes (with or without time delay, with or without overshoot, of minimum or nonminimum phase), integral processes and even some unstable processes. The proposed algorithm is better than the proportional-integral-derivative (PID) algorithm (which is also quasi-universal and practical) due to its robustness, high control performance (especially for processes with time delay) and simple experimental procedure for determining the controller parameters. Some applications are presented to highlight the main features of the algorithm and the tuning procedure for all process types.

Let f6(x, y, z) be a symmetric homogeneous polynomial of degree six. Based on cancelling the high coefficient of f6(x, y, z), we give some practical sufficient conditions to have f6(x, y, z) ≥ 0 for any nonnegative real variables x, y, z. Some applications are given in order to emphasize the effectiveness of the proposed method.

Let fn(x,y,z) be a symmetric homogeneous polynomial of degree n . In this paper, we give the necessary and sufficient conditions to have fn(x,y,z) 0 for n 6 and any nonnegative real numbers x,y,z . In addition, we extend some results to n = 7 and n = 8 , and then apply the proposed method to prove several elegant symmetric homogeneous polynomial inequalities of degree n , 4 n 8 . Mathematics subject classification (2010): 26D05.

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.

The main aim of this paper is to give a complete proof to the open inequality with power-exponential functions a ea + b eb ‚ a eb + b ea ; which holds for all positive real numbers a and b. Notice that this inequality was proved in (1) for only a ‚ b ‚ 1 and 1 ‚ a ‚ b. In addition, other two open inequalities with power-exponential functions are proved, and three new conjectures are presented.