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Parabolic Equation (PE) is one of the effective methods used to study electromagnetic wave propagation at the troposphere. Especially, at an elevation angle between 0-15°, Narrow-angle Parabolic Equation is an efficient method. Firstly, the Narrow-angle Parabolic Equation is derived step by step from wave equation. Secondly, the main errors from narrow-angle parabolic equation derivation are discussed respectively; and the quantitative expressions between relative errors and different factors are established. Thirdly, in order to guarantee the calculation precision of electromagnetic wave propagation at troposphere, the chosen rule of the calculation step length at different frequencies is also studied. Finally, the propagation losses of electromagnetic wave distribution in evaporation duct and hybrid duct are calculated and imaged by Narrow-angle Parabolic Equation respectively.

Active disturbance rejection control (ADRC) is a very popular algorithm used in many industrial plants owing to its capability of anti-uncertainties. However, there exists the steady-state error caused by the effect of actuator saturation. In this paper to overcome this issue, an anti-saturation strategy named integral-linear ADRC (I-LADRC) is proposed. A parameter tuning method for I-LADRC is also presented to achive good performance. Comparing with PID and LADRC algorithms, the proposed method achieved an excellent performance which is validated by the 300MW thermal power plant.

Dual variables system is a new systematic methodology for analysis of dynamic systems, and can be applied in plane SH wave problems. In dual variables system, uniform SH wave and inhomogeneous SH wave can be analyzed by the general methods. Corresponding to the solutions of Hamilton dual equation, there are three forms of solution for SH wave: eigenvector expansion solution, modal expansion solution and transfer form solution. Based on the eigenvector expansion solution, reflection problem, transmission problem and boundary value problem of SH waves are solved, reflection coefficients and transmission coefficients are given. In dual variables system, the analysis and calculation methods of SH wave problems obtain the same results as the traditional methods, and the methods are intuitional and simple, moreover, the intrinsic relationships among different forms of solution of SH wave can be revealed.

Based on method of electro-mechanical analogy, displacement and force variables of mechanical structure are regarded as dual variables corresponding to electrical network, thus some concepts, parameter equations, analyzing and solving methods of electrical network theory can be applied in electromechanical systems. Electro-mechanical coupling systems are considered as dual variables systems, and are analyzed by the unified methods of dual variables systems. As the typical examples, mechanical vibration model, mechanical transmission system and DC servomotor system are considered as chain dual systems with lumped-parameter, and are analyzed by the analysis methods of dual variables systems. The analytical results agree with those obtained by using traditional methods, and the solving process is intuitional and simple compared with traditional methods. The applications show that the analysis methods of dual variables systems are validity and versatility for electromechanical systems.

Dual variables system is a new systematic methodology for analysis of dynamic systems, and regarding displacement variables and the corresponding stress variables as dual variables, plane P-SV wave problems can be steered to Hamiltonian dual variables system. Based on eigenvector expansion solution of the dual variables equation, the unified analyzing and solving methods for reflection and transmission problems of P-SV waves are presented, and the methods are fit for uniform waves, inhomogeneous waves and Stoneley wave problems. As the typical applications, Zeoppritz equation for reflection coefficients and transmission coefficients of uniform waves, the equation for propagation coefficients of inhomogeneous waves, and the frequency equation of Stoneley wave are obtained by the unified methods. In dual variables system, the solving process is intuitional and simple, physical meanings are concise, and these show the validity and versatility of the analysis methods of dual variables systems for elastic waves problems.

Regarding displacement and internal force as dual variables, Euler beam problems can be steered to dual variables system. Based on eigenvalue problems of Hamiltonian dual equation of Euler beam, eigenvector expansion method and modal expansion method of Euler beam are deduced, and the unification of elastic wave problems and vibration problems is revealed. According to transfer form solution of Hamiltonian dual equation of Euler beam, equivalent stiffness and equivalent flexibility of beam end are proposed, elemental stiffness equation and the shape functions are inferred, and boundary integral equation and fundamental solutions are derived. In dual variables system, dynamic characteristics of elastic beam are more obvious, the analysis and calculation of beam system are intuitional and simple, and the intrinsic relationships among the analysis and solving methods of elastic beam are embodied profoundly.

Based on generalized variational principle of long beams and interpolation of shape functions of B-spline wavelet on the interval (BSWI), mixed FEM of BSWI based on node parameters for beam structures was proposed. By taking node parameters as interpolation coefficients, this method has many advantages, such as easy dealing with the boundary conditions and meeting the continuity of node quantities between adjacent elements. And with it, the deflection and moment functions of beam can be calculated simultaneously, the shortcoming of low accuracy in beam moment calculation with traditional FEM can be overcome. Mixed FEM of BSWI based on node parameters can be used in various kinds of analysis of beam structures, and the numerical examples indicate that the method have the advantages of high accuracy and less computation amounts.

Linear quadratic optimal control and Kalman filtering problem can be described by Hamiltonian dual equations uniformly. Based on the T-parameter solution of Hamiltonian dual equations, the recursive formulas for Riccati matrix and the equivalent action were deduced. The backward and forward recursive algorithms were presented for the nodal variables of multiple segments system, and the recursive algorithms can be applied in linear quadratic optimal control and Kalman filtering problem.

For electrical networks with equal number of port each side, its various parameter equations were given, and merging algorithm for cascaded network was expressed in terms of T-parameter equation. Resembling structural mechanics analysis and the solutions for control system response, the recurrence method based on T-parameter equations, and the finite element method based on Y-parameter equations were presented for cascaded networks system.

Regarding displacement components and the corresponding stress components as dual variables, elastodynamic problems can be steered to Hamiltonian dual variables system. For normal incidence problems of simple harmonic plane elastic waves, the decoupling of P-, SV- and SH-waves corresponds to decomposition of dual equation, and as one-dimensional wave, each decoupled wave has the corresponding dual equation with same equation forms, analyzing and solving methods. Wave properties, analyzing and solving methods of plane elastic waves can be all embodied in dual variables system. The eigenvalue problems of dual equation can solve wave number, wave velocity and wave impedance of medium. The eigenvector expansion solutions of dual equation are suited to analysis of the reflection and transmission of elastic waves, the modal expansion solutions of dual equation are suited to the modal analysis of layer structure, and stratified media have the transfer matrix methods based on the transition form solutions of dual equation.