4th Order System Research Articles

Barrier Function-Based Integral Sliding Mode Controller Design for a Single-Link Rotary Flexible Joint Robot

Abstract This paper proposes and evaluates a novel control approach for trajectory tracking, stability enhancement, and vibration reduction of a flexible joint robot (FJR). The FJR is a 2-degree-of-freedom underactuated nonlinear system that is challenging to control due to vibration, underactuation, uncertainties, and external disturbances. The control objectives are high trajectory tracking performance together with suppressing vibration. The proposed control approach uses integral sliding mode control (ISMC) combined with a barrier function based on back-stepping. This ensures robust and smooth performance by eliminating the reaching phase where sliding mode control (SMC) is typically not robust. Robustness is thus guaranteed from the start. The FJR is modeled as a 4th-order system using Lagrangian mechanics and decomposed into two 2nd-order subsystems for control design. Sliding variables are defined for each subsystem. Using these variables, the proposed control achieves robust trajectory tracking, stability, and vibration reduction. Numerical simulations in MATLAB validate the superior performance of the proposed ISMC-barrier function control compared to conventional ISMC for the FJR. This novel control approach addresses the challenges of controlling this FJR.

Установившееся вращательное движение спутника Фотон М-2

At the end of the flight, the attitude motion of the Photon M-2 satellite (it was in orbit 2005.05.31-2005.06.16) can be described by a generalized conservative system of differential equations. The secular change in the own kinetic moment of this satellite is described by the so-called evolutionary equations of Beletsky, which also form a generalized conservative system. The preprint examines the relationship between these systems. The satellite motion equations are reduced to equations of the 4th order describing the motion of the satellite axis of symmetry. Beletsky's equations are of the second order and describe the secular motion of the ort of the satellite's own kinetic moment. The solutions of these systems of equations corresponding to the real movements of the satellite are, respectively, conditionally periodic and periodic. The solutions of the 4th-order system are dominated by two frequencies – high and low ones. The spectral analysis showed that the low frequency coincides with the frequency of solutions of Beletsky’s equations. And the solutions of these equations coincide with the low-frequency component in the solution of the 4th-order system with respect to the variables that determine the direction of the axis of symmetry of the satellite.

Active vibration control of lightweight floor systems

Long-span and lightweight floors are often prone to structural vibrations due to their low resonance frequency and inherent damping. They are easily excited by human walking or running and present a serviceability problem for persons working or living in these buildings. Tuned mass dampers can be used to improve the dynamic behaviour of such structures. The drawback of these passive damping systems is that they require high inertial damper masses and that the damping efficiency in transient phase is low. Especially for systems with resonance frequencies that change over time (due to an addition mass or a change of the structure or temperature dependant phenomena) the damping efficiency drops rapidly.Active controlled tuned mass damper can be used to reduce the inertial mass and improve the damping efficiency for a broad band of frequencies in transient phase. Several constraints, like the stability in closed loop, robustness and performance criteria and the maximum stroke and saturation of the actuator have to be considered when designing a controller for an active vibration control device. Different control strategies are analysed in this work and simulated using a 2nd and a 4th order system in order to analyse advantages and drawbacks.

双分散多孔介质圆形和圆环形通道内高速流动分析

Based on the two-velocity Brinkman-extended Darcy flow model, the characteristics of high speed flow in circular and annular ducts occupied by bidisperse porous media were analyzed. The flow fields of the fracture (f) and porous (p) phases were inherently governed by the 4th-order system of coupled differential equations. The original governing equations were simplified to a 2nd-order system of decoupled differential equations with the normal mode reduction method. Furthermore, the analytical solutions of velocity distributions were readily derived for the f- and p-phases. Results from both the circular and the annular ducts show that an increase in the Darcy number leads to a reduction in not only the flow velocities of the two phases but their difference. However, the flow velocities of the two phases exhibit an opposite trend with the increase of the momentum transfer between the two phases, resulting in a decrease in the velocity difference.

The finite dimensional attractor for a 4th order system of Cahn-Hilliard type with a supercritical nonlinearity

This article is devoted to the study of the long-time behavior of solutions of the following 4th order parabolic system in a bounded smooth domain $\Omega\subset\subset{\mathbb R}^n$: $$ b\partial_t u=-{\Delta_x} (a{\Delta_x} u-\alpha\partial_t u-f(u)+\tilde g )\,, \tag*{(1)} $$ where $u=(u^1,\cdots,u^k)$ is an unknown vector-valued function, $a$ and $b$ are given constant matrices such that $a+a^*>0$, $b=b^*>0$, $\alpha>0$ is a positive number, and $f$ and $g$ are given functions. Note that the nonlinearity $f$ is not assumed to be subordinated to the Laplacian. The existence of a finite dimensional global attractor for system (1) is proved under some natural assumptions on the nonlinear term $f$.

Optimal multirate control design for hard disk drive servo systems

This paper presents a multirate digital controller design for HDD servo systems via direct parameter optimization approach. In order to find a satisfactory multirate controller with good transient and steady state performance, a linear quadratic performance index with stability constraints is introduced. The controller parameters are selected based on the HDD servo system transient response data instead of a detailed plant model. The proposed method can improve the performance of LQG multirate servo designs without knowing the exact plant parameters or noise model. The effect of controller order on system performance is also studied. For a 4th order system studied, it is found that a 4th order controller is sufficient to achieve the optimal performance.

Design of robust excitation controllers for synchronous generators using the spectral and control-canonical-form methods with output feedback

The linearized modelling of a synchronous generator with exciter, along with a powerful model order-reduction method and two efficient pole-placing algebraic control methods (i.e. the spectral and control-canonical-form (CCF) using state and/or output feedback) have been presented in systematic and concise form. They are used to design a robust excitation controller of an 87.5 kVA synchronous generator with a conventional excitation system supplying power to the electric utility system through a transformer and a transmission line. The excitation controller design is based on the obtained reduced 4th-order system model of the developed single-input multiple-output (SIMO) 8th-order, linearized (about a nominal operating point), time-invariant state-space open-loop system model. It was tested at several operating points in the over- and under-excitation operating regions of the generator. All the designed systems with the above excitation controller (closed-loop system models) were tested and displayed significantly improved dynamic stability characteristics and overall system performance with reference to those of the open-loop system models. The proposed procedure for designing implementable robust excitation controllers is relatively simple to apply, as amply verified by the example given and relevant system simulations.

Approximations of the 3-particle toda lattice

We study numerically Hamiltonians derived from the 3-particle Toda lattice by truncation at orders i ≤ 10. The third order system is the Hénon - Heiles (HH) Hamiltonian. The behavior of the systems of order 5, 7 and 9 is similar but their degree of stochasticity is smaller. In contrast with the HH system, they have appreciable ordered regions for energies larger than the escape energy. In all cases of odd order systems do not have an escape energy. The even order systems do not have an escape energy. The 4th order system seems integrable but all other systems show transition to a large degree of stochasticity. As the energy increases beyond E crit the stochastic regions increase, first abruptly and then more slowly. As the order i increases the systems tend to be integrable Toda lattice. The topology on the surface of section x = 0 is studied in detail. The changes of topology are related to changes in the stability character of the various periodic orbits.

Ballooning instabilities in the solar corona: Conditions for stability

A method for analysing the stability of cylindrical loops and arcades to ballooning modes is presented. Using a WKB method, simple tests for instability (or stability to localised modes) are obtained for the two commonly used, line tying, conditions. In addition, when instabilities are present, the physical growth rate and the radial structure of the least stable can be computed from the solution of a 4th order system of ordinary differential equations. The detailed analysis for a particular field is presented in an Appendix.

Analysis of Control Systems with Time Delay

The stability and sensitivity of a fourth-order feedback control system with time delay have been studied. The condition for the stability has been found out. The permissible loop gain and time delay for the stability of the system have been analytically evaluated. The closed-loop transfer function sensitivities w.r.t. both loop gain and time delay have been derived. The results obtained for a practical 4th-order system are discussed and a design example is presented.

Dynamic-system simplification and an application to power-system-stability studies

A method of system simplification having application to power-system dynamic-stability problems is discussed in the paper. The method is based on the geometric properties of Lyapunov functions, and is suitable for modelling higher-order systems that are expressed in state-variable form. Techniques are given for modelling both the free and forced responses. Two examples illustrating the application of this method are given. In the first example, simplification of a 4th-order system by this method is considered and the response of the resulting model is compared with the response of the model obtained by the eigenvalue-grouping method. In the second example, lower-order dynamic equivalents for an 11th-order differential equation describing the performance of a synchronous machine are derived. Advantages of this method over existing methods are discussed.