Note on the root locus method

A new approach to the calculation of the points at which the root locus crosses the imaginary axis is proposed and the corresponding parameters are given. Further, this method to analyze polynomial convexity is used. Examples are given for illustration. It is shown that this approach is simple and useful to determine the Hurwitz stable polynomial.

设计串联校正装置的一种解析方法

根据时域指标设计串联校正装置,一般使用根轨迹法(图解法)。此类方法无法将静态指标(如位置误差系数 K_p 或速度误差系数 K_r)与根轨迹作图有机结合,使设计计算复杂化。此处提出的解析法可免于这一问题,使设计计算更合理更简单,且物理概念清楚明了。

Delay-dependent stability and added damping of SDOF real-time dynamic hybrid testing

It is well-recognized that a transfer system response delay that reduces the test stability inevitably exists in real-time dynamic hybrid testing （RTDHT）. This paper focuses on the delay-dependent stability and added damping of SDOF systems in RTDHT. The exponential delay term is transferred into a rational fraction by the Pad6 approximation, and the delay-dependent stability conditions and instability mechanism of SDOF RTDHT systems are investigated by the root locus technique. First, the stability conditions are discussed separately for the cases of stiffness, mass, and damping experimental substructure. The use of root locus plots shows that the added damping effect and instability mechanism for mass are different from those for stiffness. For the stiffness experimental substructure case, the instability results from the inherent mode because of an obvious negative damping effect of the delay. For the mass case, the delay introduces an equivalent positive damping into the inherent mode, and instability occurs at an added high frequency mode. Then, the compound stability condition is investigated for a general case and the results show that the mass ratio may have both upper and lower limits to remain stable. Finally, a high-emulational virtual shaking table model is built to validate the stability conclusions.

Stability of an explicit time-integration algorithm for hybrid tests, considering stiffness hardening behavior

An explicit unconditionally stable algorithm for hybrid tests,which is developed from the traditional HHT-α algorithm,is proposed.The unconditional stability is first proven by the spectral radius method for a linear system.If the value of α is selected within [-0.5,0],then the algorithm is shown to be unconditionally stable.Next,the root locus method for a discrete dynamic system is applied to analyze the stability of a nonlinear system.The results show that the proposed method is conditionally stable for dynamic systems with stiffness hardening.To improve the stability of the proposed method,the structure stiffness is then identified and updated.Both numerical and pseudo-dynamic tests on a structure with the collision effect prove that the stiffness updating method can effectively improve stability.

Theoretical and experimental studies on critical time delay of multi-DOF real-time hybrid simulation

This paper aims to investigate the critical stability of a multi-degree-of-freedom(multi-DOF)real-time hybrid simulation(RTHS).First,the critical time-delay analysis models are developed using the continuous-and discrete-time root locus(RL)techniques,respectively.A bilinear transform is introduced into the first-order Padéapproximation while conducting the discrete RL analysis.Based on this technique,the time delay can be explicitly used as the gain factor and thus the instability mechanism of the multi-DOF RTHS system can be analyzed.Subsequently,the critical time delays calculated by the continuous-and discrete-time RL techniques,respectively,are compared for a 2-DOF RTHS system.It is shown that assuming the RTHS system to be a continuous-time system will result in overestimating the critical time delay.Finally,theoretically calculated critical delays are demonstrated and validated by numerical simulation and a set of RTHS experiments.Parametric analysis provides a glimpse of the effects of time step,frequency and damping ratio in a performing partitioning scheme.The constructed analysis model proves to be useful for evaluating the critical time delay to predict stability and performance,therefore facilitating successful RTHS.

Use of genotype-environment interactions to elucidate the pattern of maize root plasticity to nitrogen deficiency

Maize（Zea mays L.） root morphology exhibits a high degree of phenotypic plasticity to nitrogen（N） de ficiency,but the underlying genetic architecture remains to be investigated Using an advanced BC_4F_3 population,we investigated the root growth plasticity under two contrasted N levels and identi fied the quantitative trait loci（QTLs） with QTL-environment（Q×E）interaction effects. Principal components analysis（PCA） on changes of root traits to N de ficiency（D LN-HN） showed that root length and biomass contributed for 45.8% in the same magnitude and direction on the first PC,while root traits scattered highly on PC_2 and PC_3. Hierarchical cluster analysis on traits for D LN-HN further assigned the BC_4F_3 lines into six groups,in which the special phenotypic responses to N de ficiency was presented These results revealed the complicated root plasticity of maize in response to N de ficiency that can be caused by genotype environment（G×E） interactions. Furthermore,QTL mapping using a multi-environment analysis identi fied 35 QTLs for root traits. Nine of these QTLs exhibited signi ficant Q×E interaction effects. Taken together,our findings contribute to understanding the phenotypic and genotypic pattern of root plasticity to N de ficiency,which will be useful for developing maize tolerance cultivars to N de ficiency.

Adaptive Equivalent-input-disturbance Approach to Improving Disturbance-rejection Performance

This paper presents an adaptive equivalent-input-disturbance(AEID)approach that contains a new adjustable gain to improve disturbance-rejection performance.A linear matrix inequality is derived to design the parameters of a control system.An adaptive law for the adjustable gain is presented based on the combination of the root locus method and Lyapunov stability theory to guarantee the stability of the AEID-based system.The adjustable gain is limited in an allowable range and the information for adjusting is obtained from the state of the system.Simulation results show that the method is effective and robust.A comparison with the conventional EID approach demonstrates the validity and superiority of the method.