This paper proposed the H_∞ state feedback and H_∞ output feedback design methods for unstable plants, which improved the original H_∞ state feedback and H_∞ output feedback. For the H_∞ state feedback design of unstable plants, it presents the complete robustness constraint which is based on solving Riccati equation and Bode integral. For the H_∞ output feedback design of unstable plants, the medium‐frequency band should be considered in particular. Besides, this paper presents the method to select weight function or coefficients in the H_∞ design, which employs Bode integral to optimize the H_∞ design. It takes a magnetic levitation system as an example. The simulation results demonstrate that the optimal performance of perturbation suppression is obtained with the design of robustness constraint. The presented method is of benefit to the general H_∞ design.

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.

Winding system is a production system commonly used in industry. Its control goal is to achieve constant tension control of the system by controlling motor torque, and to cope with various external disturbances in industrial environment. The Fractional Order PID(FOPID) controller is well known for its simplicity with better tuning flexibility as well as robustness with respect to disturbances b1b2. Thus, this paper proposes a control strategy based on FOPID to achieve the control objectives. First, a linear approximation model is established for the constant tension control system. Then, a FOPID controller is designed based on the model. Four non-linear constraints and an objective function of the controller are proposed based on the frequency domain. Five tunable parameters of FOPID are determined by combining genetic algorithm(GA) with FMINCON non-linear optimization. Finally, through the simulation of MATLAB, the performance of the controller is compared with the classic integer order PID. The results show that the FOPID controller has better control characteristics and robustness.

This paper focuses on the tracking control problem of Magnetic Levitation System (MLS). MLS is highly unstable nonlinear system, in the process of operation, MLS needs to have strong robustness and anti-interference ability. Due to disturbance and parameter uncertainties in MLS, it is complex to obtain the exact dynamics of MLS, and it is difficult to design a suitable controller. The dynamic equation of the system is established by Lagrange equation, and then we propose an adaptive Sliding Mode Control (SMC) based on Radial Basis Function Neural Network (RBFNN). Because of the parameter uncertainties and disturbance in MLS, RBFNN is used to approximate the unknown dynamics in MLS. The stability of the closed-loop system is strictly proved by using the Lyapunov stability theory, which can achieve the uniform ultimate boundedness (UUB) of the signals of the closed-loop MLS. MATLAB environment is used to verify the performance of the proposed controller. Considering disturbance, parameter change, or unmodeled dynamics in MLS, proposed controller is compared with other nonlinear controllers, simulation results verify the effectiveness of the proposed approach.

The control aim of parabolic distributed solar collector (PDSC) is to make the heat generated follow the expected reference value when the solar radiation varies unevenly. This article proposes a control strategy based on fractional-order PID (FOPID) to achieve control objectives. Four non-linear constraints and an objective function of the controller are proposed based on the frequency domain. Most studies regard the constraint measuring stability as the objective function, while the real difficulty for the control system design should be how to obtain better robustness. Therefore, this article determines five parameters of FOPID by selecting the constraint measuring robustness as the objective function based on Particle swarm optimization (PSO). Besides, three non-parametric statistical tests, including Friedman ranks, Friedman Aligned ranks and Quade ranks, have been adopted to compare PSO with other evolutionary algorithms to demonstrate its advantage. Finally, the superiority of proposed strategy to previous design is demonstrated by the simulation studies on a PDSC in terms of the control performance of step response, static gain variation, time constant variation, high frequency noise and output interference.

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.