## 1 Introduction  
Non-linear passive vibration isolators have been proven to be advantageous to overcome the inherent drawback of linear isolators in different applications [1]. Thus, the effects of non-linear stiffness or/and damping on an isolator's performance have drawn a lot of attention; see, for example, [2; 3; 4; 5; 6]. It is known that a non-linear isolator performs well over larger frequency ranges compared to a corresponding linear isolator. The soft non-linearity of its stiffness reduces the vibration transmissibility in the resonant frequency [7] and the influence of non-linearity on the characteristics of an isolator may also include shifts in resonance frequency, internal resonance, and chaotic response [1; 8; 9; 10; 11; 7]. Recently, a type of non-linear passive vibration isolator with quasi-zero stiffness (QZS) at the equilibrium position has received increasing attention [10; 11; 12; 13; 14; 15; 16; 17; 18]. This isolator possesses high-static-low-dynamic stiffness so that it can maintain the load-bearing capacity with a small static displacement and achieve a lower natural frequency to enlarge frequency ranges of vibration isolation. But, it should be noted that isolators with QZS are very sensitive facilities and if something occurs to cause deviation of its characteristic parameters (e.g., mismatch of the load and negative stiffness parameters) and/or change in the non-linearity properties of its forces (e.g., destruction of the symmetry), then the isolation performance will change dramatically [19; 20; 21; 22]. This situation is particularly true when the amplitude of excitation is not relatively small, and the isolator is exposed to a sustained excitation. Therefore, they need careful tuning and fine precision during manufacturing. Besides, a non-linear passive isolator cannot perform well over all the frequencies that a system may be confronted with, because it is designed and fabricated with predetermined system parameters. A better scheme to overcome these shortcomings is to introduce semi-active or active devices which cooperate with the passive isolators. Such an isolator is referred to as a hybrid isolator. Due to the limitations of the scope of the discussion in this article, we only focused our attention on an active type of hybrid isolator with QZS. A simple linear time-delayed displacement feedback control strategy was introduced to improve the robustness and transmissibility performance of an isolator with QZS under both force and base excitation in Reference [23]. This control strategy can be used to gain better isolation performance in a resonant region. It was also shown that undesirable bifurcation and chaotic behaviors can be avoided or greatly suppressed when the controlled stiffness and especially the introduced time delay are appropriately chosen. But the performance gets worse in the high-frequency band. In Reference [24], the cubic displacement feedback with time delay was proposed, which can effectively suppress the transmissibility in resonant region and reduce the resonant frequency with the performance in higher frequency range unaffected with appropriate feedback parameters. In Reference [25], the time-delayed cubic velocity feedback control strategy was investigated. It turned out that a better isolation performance in the high-frequency band where isolation is required can also be achieved, but the 1/3 subharmonic resonance can occur for a smaller excitation amplitude and increasing the feedback gain cannot eliminate the 1/3 subharmonic resonance. In these studies, a single-degree-of-freedom (SDOF) oscillator with cubic non-linearity was analyzed by using some approximate analytical methods, e.g., the multiple scales method and the harmonic balance method, and the amplitude-frequency responses were obtained. These control analyses of hybrid isolators with QZS were conducted based upon good knowledge of mathematical representations of the systems.