The Hamilton–Jacobi–Issacs (HJI) inequality is the most basic relation in nonlinear H_∞ design, to which no effective analytical solution is currently available. The sum of squares (SOS) method can numerically solve nonlinear problems that are not easy to solve analytically, but it still cannot solve HJI inequalities directly. In this paper, an HJI inequality suitable for SOS is firstly derived to solve the problem of nonconvex optimization. Then, the problems of SOS in nonlinear H_∞ design are analyzed in detail. Finally, a two‐step iterative design method for solving nonlinear H_∞ control is presented. The first step is to design an adjustable nonlinear state feedback of the gain array of the system using SOS. The second step is to solve the L₂ gain of the system; the optimization problem is solved by a graphical analytical method. In the iterative design, a diagonally dominant design idea is proposed to reduce the numerical error of SOS. The nonlinear H_∞ control design of a polynomial system for large satellite attitude maneuvers is taken as our example. Simulation results show that the SOS method is comparable to the LMI method used for linear systems, and it is expected to find a broad range of applications in the analysis and design of nonlinear systems.

We present a sum of squares (SOS) method for the synthesis of nonlinear polynomial control systems. As an emerging numerical solution method in recent years, SOS targets polynomials as the research object. It guarantees that the polynomial we solve for is always nonnegative. In this paper, we give a generalized S-procedure to solve the SOS problem. As an illustration of how the SOS method can be used, the region of attraction (ROA) in a nonlinear polynomial system is analyzed in detail. The method of determining decision variables is given in the SOS problem. We discuss the determination and solution of set-containment constraints and the conservatism problem in solving the SOS problem. SOS provides a convenient numerical method to solve nonlinear problems that are not easy to solve analytically.

User location prediction in location‐based social networks can predict the density of people flow well in terms of intelligent transportation, which can make corresponding adjustments in time to make traffic smooth, reduce fuel consumption, reduce greenhouse gas emissions, and help build a green cycle low‐carbon transportation green system. This paper proposes a Markov chain position prediction model based on multidimensional correction (MDC‐MCM). Firstly, extract corresponding information from the user’s historical check‐in position sequence as a position‐position conversion map. Secondly, the influence of check‐in period, space distance, and other factors on the position prediction is linearly weighted and merged with the position prediction of the n‐order Markov chain to construct MDC‐MCM. Finally, we conduct a comprehensive performance evaluation of MDC‐MCM using the dataset collected from Brightkite. Experimental results show that compared with other advanced location prediction technologies, MDC‐MCM achieves better location prediction results.

Linearized model of the system is often used in control design. It is generally believed that we can obtain the linearized model as long as the Taylor expansion method is used for the nonlinear model. This paper points out that the Taylor expansion method is only applicable to the linearization of the original nonlinear function. If the Taylor expansion is used for the derived nonlinear equation, wrong results are often obtained. Taking the linearization model of the maglev system as an example, it is shown that the linearization should be carried out with the process of equation derivation. The model is verified by nonlinear system simulation in Simulink. The method in this paper is helpful to write the linearized equation of the control system correctly.

Flexible solar panels play an essential role in the field of aerospace. However, many difficulties appear in the control design due to the existence of a weakly damped resonance module. The design for flexible systems often causes an unstable controller so that the systems after design still have trouble in putting into practice. We adopt H_∞ loop-shaping design and put forward a directive method for selecting the weighting function. The simulation results indicate that system bandwidth is optimized based on the stable controller. In this way, the control precision and response speed of the system are improved. In the meantime, the system is easy to put into use.