On-off nonlinear active control of floor vibrations
Introduction
Advancements in structural technology and modern trends in building layouts have resulted in light, slender, open plan floor structures that are more susceptible to vibration under human excitations [1]. Such vibrations can cause a serviceability problem in terms of disturbing the building occupants, but they rarely affect the fatigue behaviour or safety of structures.
Several general guidelines [2], [3], [4] are available to consider human-induced vibrations. These guidelines take into account the usual human activities (normal living and business activities or dancing and aerobic exercises) and dynamic properties (mass, stiffness and damping ratios of structural and non-structural elements). Nevertheless, floor structures can still experience excessive vibration levels that are not accepted by their occupants. Improvement of these floors is usually complicated and involves significant structural and non-structural changes and severe disruptions of occupation. An alternative procedure is the use of passive and semi-active devices [5], [6], [7], [8]. However, due to their passive nature, the vibration cancellation is often of limited effectiveness and they often have to be tuned to damp a single vibration mode. As a consequence, when either more effective vibration cancellation is required, multiple vibration modes need to be damped or the floor dynamics change substantially, these passive devices do not perform well. In this case, an active control approach rather than passive or semi-active systems can be useful [9]. A state-of-the-art review of technologies (passive, semi-active and active) for mitigation of human-induced vibration can be found in [10]. Furthermore, techniques to cancel floor vibrations (especially passive and semi-active techniques) are reviewed in [11].
Active control has been implemented successfully in a number of civil engineering structures using active mass dampers [12]. Approaches such as LQR, LQG, H2 and H∞ control [13], [14], [15] are commonly found in research works and they are usually focused on cancelling hazardous vibrations due to earthquakes or wind. All these techniques, which are model-based, usually require complex design methodologies and full state feedback which result in high-order controllers and possible poor stability margins.
With regards to active control for floor vibrations induced by humans, Hanagan and Murray [16], [17] have studied analytically and implemented experimentally linear velocity feedback control (LVFC), i.e., the velocity output is multiplied by a constant gain and feeds back to a collocated force actuator. The merits of this method are its robustness to spillover effects due to high-order unmodelled dynamics and that it is unconditionally stable in the absence of actuator and sensor dynamics [18]. However, when such dynamics are considered, it is observed that a couple of branches in the root locus of the closed-loop system go to the right-half plane and the stability for high gains is no longer guaranteed. The control law used is completed by a command limiter (i.e., a saturation nonlinearity in the command signal) with the following objectives: (a) to avoid actuator force overloading; (b) to avoid actuator stroke saturation; and (c) to level off the system performance in the case of unstable behaviour, which can be due to a non-adequate choice of the control gain (uncertainties can make this task difficult) or changes in the system dynamics that might modify importantly the predicted stability conditions. Unstable performance is thus avoided, but the closed-loop system can exhibit stable limit cycle behaviour, which is non-desirable since it could result in dramatic adverse effects on the control system performance and its components. This behaviour was observed in [16], but no more explanation of the phenomenon was provided. One of the drawbacks of LVFC is that its performance is highly dependent on the control gain used. Such a gain has to be designed according to a specific excitation (for instance, heel-drop excitation). Consequently, optimal or acceptable performance for a different excitation (such as walking excitation) is then not guaranteed. An attempt to avoid the dependence on the gain choice has been recently presented in [19], in which the gain is selected automatically from the velocity output. Limit cycle behaviour was also observed, but no analytical explanation was provided.
This paper addresses mainly the two following issues: firstly, an analytical study of LVFC with saturation is carried out in order to demonstrate the existence of limit cycle behaviour for high gains and establish the conditions necessary for it to appear; secondly, on-off nonlinear control based-on velocity feedback is studied as an alternative to LVFC with saturation. Some preliminary results were presented in [20], motivating this paper. When high gains are used and/or significant vibration levels are reached, LVFC with saturation is essentially working in the saturation range and it can then be approximated by on-off nonlinear velocity feedback control (NLVFC). Its main advantage is that no gain has to be designed and the actuator always imparts maximum energy to the floor system. However, it is shown here that on-off NLVFC exhibits similar stability properties as LVFC with high gains, i.e., the controlled system is involved in stable limit cycle behaviour. Furthermore, it is demonstrated that this behaviour can be avoided if either a dead zone or switching-off function (disconnection rule) is included in the control law. It is analytically demonstrated that the condition to cause limit cycle behaviour is given by the ratio between the saturation level and the dead zone or switching-off level, depending on the solution adopted. Hence, by prediction of this ratio, one can easily design on-off NLVFC without limit cycle behaviour. In this paper, it is shown that this ratio can be predicted with sufficient accuracy by using the describing function (DF) tool in its basic form. All the analytical predictions have been corroborated by experimental trials on a test floor which consists of a simply supported post-tensioned concrete slab strip.
The remainder of the paper is organized as follows. The general velocity feedback strategy and its particularisation to LVFC with saturation and on-off NLVFC with a dead zone and with a switching-off function are presented in the following section. This section also contains the experimental setup used in this work and the dynamics involved. Section 3 considers in detail the stability properties of the above-mentioned control strategies by means of the DF tool. Experimental results are conducted on a laboratory experimental test floor in Section 4. Finally, some conclusions and suggestions for future work are given in Section 5.
Section snippets
Control system description
This section presents the control strategies followed and the experimental setup. Additionally, the dynamics of the control system components are briefly described.
Stability analysis
Stability is the primary concern in any active control system applied to civil engineering structures mainly due to safety reasons. This section analyses the stability properties of LVFC with saturation (1) and on-off NLVFC with a dead zone (2) and a switching-off function (3) by means of the DF tool in its basic form. Note that saturation, on-off nonlinearity, dead zone and switching-off function are hard nonlinearities, which are especially easy to treat by means of the DF tool. Such a tool
Experimental trials on a test floor
The results of experimental trials on the laboratory test floor described in Section 2 are presented in this section. The three control laws described in Section 2 were utilised and assessed. The two main objectives of this section are (a) to corroborate the stability property predictions obtained from the analytical study of Section 3 and (b) to assess whether on-off NLVFC with either a dead zone or a switching-off function could be an alternative to LVFC with saturation. To this end, the
Conclusions
On-off nonlinear control based-on velocity feedback has been presented and compared with its linear counterpart in the context of cancellation of floor vibrations. The paper focuses on the stability properties of LVFC with saturation and on-off NLVFC with a dead zone and with a switching-off function. The stability properties of the three control laws have been obtained analytically and experimentally paying special attention to the prediction of limit cycle behaviour, which can result in
Acknowledgments
The authors would like to acknowledge the financial support of Conserjería de Educación y Ciencia of Junta de Comunidades de Castilla-La Mancha, the European Social Fund and UK Engineering and Physical Sciences Research Council (Ref: EP/G061130/1). The authors are also grateful to Emiliano Pereira from Automatic Engineering Department of Universidad de Castilla-La Mancha for his valuable suggestions.
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