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On-off nonlinear active control of floor vibrations
ARTICLE IN PRESS Mechanical Systems and Signal Processing 24 (2010) 1711–1726
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Mechanical Systems and Signal Processing journal homepage: www.elsevier.com/locate/jnlabr/ymssp
On-off nonlinear active control of floor vibrations Iva´n M. Dı´az a,b,, Paul Reynolds b a b
Escuela Te´cnica Superior de Ingenieros Industriales, Universidad de Castilla-La Mancha, Edificio Polite´cnica, Av. Camilo Jose´ Cela s/n, 13071 Ciudad Real, Spain Department of Civil and Structural Engineering, The University of Sheffield, Sir Frederick Mappin Building, Mappin Street, Sheffield S1 3JD, United Kingdom
a r t i c l e in fo
abstract
Article history: Received 17 September 2008 Received in revised form 9 December 2009 Accepted 28 February 2010 Available online 4 March 2010
Human-induced floor vibrations can be mitigated by means of active control via an electromagnetic proof-mass actuator. Previous researchers have developed a system for floor vibration comprising linear velocity feedback control (LVFC) with a command limiter (saturation in the command signal to avoid actuator overloading). The performance of this control is highly dependent on the linear gain utilised, which has to be designed for a particular excitation and might not be optimum for other excitations. This work explores the use of on-off nonlinear velocity feedback control (NLVFC) as the natural evolution of LVFC when high gains and/or significant vibration level are present together with saturation in the control law. Firstly, the describing function tool is employed to analyse the stability properties of: (1) LVFC with saturation, (2) on-off NLVFC with a dead zone and (3) on-off NLVFC with a switching-off function. Particular emphasis is paid to the resulting limit cycle behaviour and the design of appropriate dead zone and switching-off levels to avoid it. Secondly, experimental trials using the three control laws are conducted on a laboratory test floor. The results corroborate the analytical stability predictions. The pros of on-off NLVFC are that no gain has to be chosen and maximum actuator energy is delivered to cancel the vibration. In contrast, the requirement to select a dead zone or switching-off function provides a drawback in its application. & 2010 Elsevier Ltd. All rights reserved.
Keywords: Active control Structural control Floor vibrations Nonlinear velocity feedback Human-induced vibrations
1. 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 [2–4] 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 [5–8]. 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.
Corresponding author at: Escuela Te´cnica Superior de Ingenieros Industriales, Universidad de Castilla-La Mancha, Edificio Polite´cnica, Av. Camilo Jose´ Cela s/n, 13071 Ciudad Real, Spain. Tel.: + 34 926295238; fax: + 34 926295300. E-mail addresses: [email protected], I.M.Diaz@sheffield.ac.uk (I.M. Dı´az).
0888-3270/$ - see front matter & 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.ymssp.2010.02.011
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I.M. Dı´az, P. Reynolds / Mechanical Systems and Signal Processing 24 (2010) 1711–1726
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 HN control [13–15] 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 modelbased, 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. 2. 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. 2.1. Control strategies Fig. 1 shows a block diagram of a general velocity feedback control particularised for a control system in which the objective is to achieve zero vibration (reference command r(t) =0) and the considered output of the floor structure (the plant) is the acceleration. In this figure, GA(s) is the transfer function of the proof-mass actuator, G(s) is of the floor
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Fig. 1. General velocity feedback control scheme.
_ structure, GS(s) is of the sensor and f ðyðtÞÞ is the control law. Three particular cases of Fig. 1 are assessed throughout this paper. The first one corresponds to LVFC with saturation in which the control law to generate the control voltage V0(t) can be written as follows: ( _ _ jyðtÞj rVs =Kc Kc yðtÞ _ , ð1Þ f ðyðtÞÞ ¼ _ _ Vs signðyðtÞÞ jyðtÞj 4 Vs =Kc where Kc is the control gain and Vs is the maximum allowable control voltage to the actuator (saturation level). This command limiter is introduced to avoid force and stroke limitation [21]. The second control law corresponds to on-off NLVFC with a dead zone in which the control voltage is obtained as ( _ 0 jyðtÞj r ddead _ , ð2Þ f ðyðtÞÞ ¼ _ _ Vs signðyðtÞÞ jyðtÞj 4 ddead where ddead is the velocity dead zone level. The third one corresponds to on-off NLVFC with a switching-off function in which the control voltage is calculated as ( 0 jy_ RMS,T ðtÞj r dRMS _ , ð3Þ f ðyðtÞÞ ¼ _ Vs signðyðtÞÞ jy_ RMS,T ðtÞj 4 dRMS where y_ RMS,T is the running root mean square (RMS) velocity computed using T-seconds and dRMS is the disconnection level in terms of y_ RMS,T .
2.2. Experimental setup The experimental floor used in this work consists of a simply supported slab strip made of in-situ cast post-tensioned concrete [22]. It has a span of 10.8 m and a total length of 11.2 m, including 0.2 m overhangs over each support. It has a width of 2.0 m, a thickness of 0.275 m and it weighs approximately 15 tonnes. Fig. 2 shows photographs of the structure, in which a general view (Fig. 2a) and a detail of one of the supports (Fig. 2b) are observed. A single collocated actuator/sensor is placed at mid span, where the first vibration mode shape has its maximum value. The actuator is a proof-mass electromagnetic shaker (APS Dynamics Model 113 electrodynamic shaker operated in inertial mode) (Fig. 3a) and the sensor is a piezoelectric accelerometer (Endevco 7754A-1000) mounted on a levelled baseplate (Fig. 3b). The peak harmonic force given by the actuator is 133 N (at 10 Hz) and the maximum stroke is 0.158 m. The saturation voltage has to consider both limitations. However, in the case of human-induced vibrations, in which lowfrequency components are mainly excited, the saturation voltage is mainly due to stroke limit and is taken as Vs =0.5 V in this work. The controller hardware completes the experimental setup. It comprises of a PC with a National Instruments PCI-6030E DAQ card installed. It is important to note that since the system response is measured in terms of acceleration, the velocity is obtained digitally together with the computation of the control law (see Fig. 1).
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Fig. 2. Photograph of the experimental slab strip. (a) General view. (b) Simple support and overhang.
Accelerometer Active mass Rubber bands Baseplate
Fig. 3. (a) Proof-mass electromagnetic shaker (APS Dynamics Model 113 electrodynamic shaker). (b) Piezoelectric accelerometer (Endevco 7754-1000) mounted on a levelled baseplate (top view).
2.3. Floor dynamics If the aforementioned collocated case between acceleration (output) and the force (input) is considered, the transfer function of the floor dynamics can be represented as an infinite sum of second-order systems as follows [23]: GðsÞ ¼
1 X
ai s2 , oi s þ o2i
s2 þ 2zi i¼1
ð4Þ
where s= jo, o being the frequency, ai Z0, zi and oi are the damping ratio and natural frequency associated to the ith mode, respectively. It is common practice to truncate the model for the frequency bandwidth of interest, such that only N vibration modes are considered. The transfer function G(s) is thus approximated by ~ GðsÞ ¼
N X i¼1
ai s2 : 2 i oi s þ oi
s2 þ2z
ð5Þ
The test floor structure was designed in such a way that its first mode (approximately at 4.4 Hz) was prone to be excited by the second and third harmonics of walking excitation [24]. The frequency response function (FRF) corresponding with the mid span (collocated control location) using a frequency span of 0–50 Hz was obtained experimentally and a posteriori parameter identification of model (5) (N= 2) was undertaken, giving ~ GðsÞ ¼
1:39 104 s2 1:31 104 s2 þ : s2 þ0:385s þ 755:64 s2 þ 6:509sþ 54046
ð6Þ
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0
Phase (deg)
−40 Magnitude (dB)
1715
−60 −80
−50 −100 −150
−100 10
20
30
40
50
−200
10
Frequency (Hz)
20
30
40
50
Frequency (Hz)
~ Fig. 4. Transfer function of the floor structure GðsÞ: (– –) experimental; (—) model. (a) Magnitude in dB referenced to 1(m/s2)/N. (b) Phase in degrees.
Fig. 4 shows the magnitude and phase responses of the model and experimental floor. As can be observed, the model captures the floor dynamics with sufficient accuracy. 2.4. Proof-mass actuator dynamics The proof-mass actuator can be described as a linear second-order system with the following transfer function [23]: GA ðsÞ ¼
KA s2 , s2 þ 2zA oA s þ o2A
ð7Þ
in which KA is a gain which relates the released inertial force FA(t) (see Fig. 1) with the input voltage V(t), and zA and oA are, respectively, the damping ratio and natural frequency which take into account the suspension system and internal damping. The actuator model was determined to be GA ðsÞ ¼
150s2 , s2 þ 4:021s þ 101:1
ð8Þ
where the natural frequency of the shaker is estimated as oA = 10.05 rad/s (1.60 Hz). 2.5. Sensor dynamics As was mentioned before, the velocity was obtained digitally from the acceleration signal. The employed procedure used the trapezoidal method and a DC offset correction. Again, its dynamics were identified and closely modelled by the following filter: GS ðsÞ ¼
s s ¼ 2 , s þ 5s þ 25 s2 þ 2zS oS s þ o2S
ð9Þ
in which oS and zS are the natural frequency and damping ratio of the filter. 3. 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 is usually used to discover the existence of limit cycles and determine their stability. In order to use the DF tool in its basic form, the following conditions have to be satisfied [25]: (1) there is only a single nonlinear component, (2) the nonlinearity is time-invariant, (3) the nonlinearity is odd and (4) when the input is sinusoidal, only the fundamental component of the Fourier series of the output (of the nonlinear part) has to be considered. The first condition implies that if there are several nonlinearities, these have to be merged into one. The second condition is needed due to the fact that the DF tool is based on the Nyquist criterion, which is applied only to linear time-invariant systems. The third condition simplifies the Fourier expansion since the static term can be neglected. Finally, the fourth condition is actually an assumption and is usually known as the fundamental assumption of the DF tool. This assumption represents an approximation since the output of a nonlinear element due to a sinusoidal input usually contains higher
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V (t)
r (t) = 0 +
GT (s)
. y(t)
– V0 (t)
N (A, ω)
Fig. 5. Stability analysis by the DF tool.
30 200
F
100
Imag Axis
Imag Axis
20
0 −100
↑Kc
10
A S
0 −10 −20
−200 −40
−30
−20 −10 Real Axis
0
10
−30
−30
−20
−10 Real Axis
0
10
~ oÞGS ðsÞ. (b) Zoom of the origin: A, actuator, S, sensor and F, floor. Fig. 6. (a) Root locus of the total transfer function GT ðjoÞ ¼ GA ðjoÞGðj
harmonics than the fundamental one. This implies that the dynamic response of a limit-cycling system is governed mainly by a single harmonic, i.e. the output should be approximately sinusoidal. For the stability analysis by the DF tool, the control system of Fig. 1 is transformed into the scheme of Fig. 5, in which the total transfer function of the linear part is represented by GT(jo) and the nonlinear element (i.e., the control law) is substituted by its DF, denoted by N(A,o). Thus, the Nyquist criterion can be applied in its extended form by just substituting a control gain by the DF. The outline of the stability analysis carried out herein is as follows: Step 1: Analyse the root locus of the total open-loop transfer function (the transfer function of the linear part). In this case, this is (see Fig. 1) ~ oÞGS ðjoÞ, GT ðjoÞ ¼ GA ðjoÞGðj
ð10Þ
assuming the truncated model (5) for the floor structure. Step 2: Obtain the DF, which in general depends on the amplitude A and the frequency o of the sinusoidal input, corresponding to the significant hard nonlinearities present in the control system. Step 3: Apply the extended Nyquist criterion using the DF N(A,o) GT ðjoÞ ¼ 1=NðA,oÞ:
ð11Þ
Each solution of Eq. (11) predicts a limit cycle behaviour. Step 4: Use the criterion of the stability of limit cycles to decide about their nature.
3.1. Root locus study In Step 1, the root locus maps the roots of the closed-loop transfer function for control gains from zero (open loop) to infinity. The root locus of GT(jo) (Eq. (10)) using (6), (8) and (9) is equivalent to analysis of the stability when the control _ ¼ Kc y_ with Kc 2 ½0,1Þ (equivalent to Eq. (1) without saturation). This is also equivalent to a constant DF equal to law is f ðyÞ the control gain N(A,o) =Kc in Fig. 5. Fig. 6 shows the root locus. It can be observed that the controlled system becomes unstable for high gains due to the fact that the actuator dynamics branches go to the right-half plane. This clearly illustrates the fact that the inclusion of sensor and actuator dynamics makes LVFC conditionally stable. For this particular case, the limit gain is found to be Kc,limit = 1000.4 V s/m. Increasing values of Kc increases the damping of the floor vibration modes but decreases the damping of the actuator up to its instability (see Fig. 6b).
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x 10−4 -1×10-3 = -1/Kc,limit
1 5
Imag Axis
N (A)/Kc
GT (j)
Linearity range
0.8 0.6 0.4
0
-∞
-1/N (A) -1/Kc
−5
0.2 0 0
2
4
6
8
A/a
10
−10
−8
−6 −4 Real Axis
−2
0
2 x 10−4
Fig. 7. (a) DF for the saturation nonlinearity. (b) Nyquist diagram of GT(jo) and 1/N(A).
3.2. Linear velocity feedback with saturation Steps 2, 3 and 4 were carried out on the control law of Eq. (1). As has been explained previously, similar to the concept of FRF, the DF is the ratio between the fundamental component of the nonlinear element output and a sinusoidal input given by xðtÞ ¼ A sinðotÞ. If the nonlinearity is hard, odd and single-valued (the case of the nonlinearities analysed in this work), the DF depends only on the input amplitude N(A,o)=N(A), i.e., it is a real function. The DF for a saturation nonlinearity is [25] 8 Kc A ra > > 2 sffiffiffiffiffiffiffiffiffiffiffiffiffi3 > < a a a2 NðAÞ ¼ 2Kc 4 , ð12Þ arcsin 1 2 5 A4 a > > p A A A > : with a=Vs/Kc (see Eq. (1)). The normalized DF N(A)/Kc (12) is plotted in Fig. 7a as a function of A/a. As can be seen, N(A)=Kc if the input amplitude is in the linear range (Ara) and N(A) decreases as the input amplitude increases when A4a. That is, saturation does not occur for small signals and it reduces the ratio of the output to input as the input increases. Once the DF of the nonlinearity has been identified (Step 2), the extended Nyquist criterion (11) is applied (Step 3). Fig. 7b shows a plot of GT(jo) (varying o) and the negative inverse DF 1/N(A) (varying A) in the complex plane. If the two curves intersect, limit cycles are then predicted, and their properties (amplitudes and frequencies) are obtained from the solution of (11). Focusing on 1/N(A), it can be seen that it starts at 1/Kc and remains there while the linear range. Then, 1/N(A) goes to N as A increases. Examining next the plot GT(jo), it is observed that it can intersect with 1/N(A) depending on the value of Kc. The conclusion is that: if Kc oKc,limit, the system is asymptotically stable and goes to zero vibration (no intersection); otherwise, a limit cycle is predicted (intersection). Such a limit cycle is deduced to be stable by using the limit cycle stability criterion [25]. This criterion says that if points of 1/N(A) approaching the intersection with increasing values of A are not encircled by the curve GT(jo), then the corresponding limit cycle is stable; otherwise, it is unstable. The frequency of the limit cycle oLC is obtained from GT(jo) at the intersecting point and the amplitude ALC is obtained from Eq. (11) particularised to the same point 1 p ffii , ¼ h a a qffiffiffiffiffiffiffiffiffiffiffi 2 Kc,limit 2Kc arcsin A þ A 1 Aa2
ð13Þ
with Kc 4Kc,limit and |GT(joLC)| =1/Kc,limit. In this case, the frequency of the limit cycle was oLC =11.40 rad/s (or 1.81 Hz), which is close to the estimated natural frequency of the actuator (Eq. (8)), and the amplitude was obtained by the solution of Eq. (13). If, for instance, Kc =1400 V s/m, the solution is a/A=0.599, and hence the predicted amplitude is ALC =6.0 10 4 m/s, in which Vs =0.5 V has been assumed. 3.3. On-off nonlinear velocity feedback with a dead zone The same steps as before were carried out for the control law of Eq. (2), on-off NLVFC with a dead zone. Following the standard procedure to obtain DFs [25], it is quite straightforward to derive the DF for an on-off nonlinearity including a dead zone which can be written as follows: 8 0 A r ddead > > sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi < 2 , ð14Þ NðAÞ ¼ 4Vs ddead > A 4 ddead > : pA 1 A2
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in which ddead is the dead zone level in velocity-units (see Eq. (2)). The normalized DF N(A)/Vs for ddead =1 (14) is plotted in Fig. 8a. Next 1/N(A) is plotted for increasing values of A (see Fig. 8b). It starts at N, then goes to 1/N(A)max and finally it goes back to N. The parameter N(A)max is the maximum value of the DF. Note that if 1/N(A) intersects with GT(jo), there will be two intersections. Actually, two limit cycles are predicted; the first one is unstable (with smaller amplitude) and the second one is stable (with higher amplitude). For the first intersection, increasing values of A are encircled by GT(jo) whereas, for the second intersection, increasing values of A are not encircled. This means that: if NðAÞmax o Kc,limit , the system is asymptotically stable and goes to zero vibration (no intersection); otherwise, the system is stable and goes to the stable pffiffiffilimit cycle (intersection). Parameter N(A)max can be derived from Eq. (14). From dN(A)/dA= 0, it is obtained that A ¼ 2ddead . Then, the maximum value of the DF is NðAÞmax ¼
2Vs
pddead
,
ð15Þ
and then, a zero vibration rule can be deduced Vs
ddead
o
p 2
Kc,limit :
ð16Þ
The amplitude of the limit cycles ALC is obtained from the solution of Eq. (11) particularised to the intersecting point 1 ¼ Kc,limit
pA qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : d2 4Vs 1 Adead 2
ð17Þ
Considering in Eq. (17) that ddead 5A, then the amplitude of the stable limit cycle can be predicted directly as ALC ¼
4Vs
pKc,limit
:
ð18Þ
x 10−4 -1×10-3 = -1/Kc,limit 0.6
5
N (A)max
Imag Axis
0 -∞
-1/N (A)
0.2
-1/N (A)max
−5
Dead zone 0 0
2
4
6
8
10
−10
−8
−6
A
−4
−2
Real Axis
Fig. 8. (a) DF for the on-off nonlinearity with dead zone. (b) Nyquist diagram of GT(jo) and 1/N(A).
N (A)max 1
N (A)/Vs
N (A)/Vs
Vs 0.4
GT (j)
Vs
0.5
Disconnection 0 0
2
4
6
8
10
A Fig. 9. DF for the on-off nonlinearity with a switching-off function.
0
2 x 10−4
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For the structure and controller utilised here, the predicted amplitude of the limit cycle was ALC = 6.4 10 4 m/s. It is also possible to predict the minimum value of the dead zone level to avoid limit cycle behaviour. Using Eq. (16), it was obtained that ddead,limit =3.2 10 4 m/s. 3.4. On-off nonlinear velocity feedback with a switching-off function Next, on-off NLVFC with a switching-off function was analysed (see Eq. (3)). The DF for an on-off nonlinearity including a switching-off function is as follows: 8 A r ddis < 0 ð19Þ , NðAÞ ¼ 4Vs : A 4 ddis pA
Settling time ts (s)
25 20 15
Bifurcation point
10 5 0 0
200
400 600 800 Control gain Kc (Vs/m)
1000
Fig. 10. Gain selection using a heel-drop excitation. (– –) Settling time for 5%, (– –) for 1%, (—) for 0.5%.
0.02
0.4 Velocity (m/s)
Acceleration (m/s2)
0.6
0.2 0 −0.2
0.01 0 −0.01
−0.4 −0.02 2
4
6 Time (s)
8
10
0.6
150
0.4
100 Control force (N)
Control voltage (V)
0
0.2 0 −0.2
0
2
4
6 Time (s)
0
2
4 6 Time (s)
8
10
50 0 −50 −100
−0.4
−150 0
2
4 6 Time (s)
8
10
8
10
Fig. 11. Experiments LVFC using Kc = 500 V s/m. Heel-drop excitation. a) (– –) Open-loop acceleration, (—) closed-loop acceleration. b) (– –) Open-loop velocity, (—) closed-loop velocity. c) Control voltage. d) Control force.
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where ddis is a prescribed value to disconnect the control system in velocity-units. The normalized DF N(A)/Vs for ddis =1 (19) is plotted in Fig. 9. A similar behaviour as in the case of the dead zone is predicted (see Fig. 8b). If NðAÞmax oKc,limit , the system is asymptotically stable and goes to zero vibration (no intersection). Otherwise, the system is stable and goes to the stable limit cycle (intersection). However, the maximum value of the DF is different from that predicted for the dead zone NðAÞmax ¼
4Vs
pddis
:
ð20Þ
The following zero vibration rule can be deduced: Vs
ddis
o
p 4
Kc,limit :
ð21Þ
0.01
5 Velocity (m/s)
Acceleration (m/s2)
x 10−4
0.005 0 −0.005
0
−5
−0.01 −0.015 2
4 6 Time (s)
8
10
0.6
150
0.4
100 Control force (N)
Control voltage (V)
0
0.2 0 -0.2 -0.4
0
2
0
2
4 6 Time (s)
8
10
50 0 −50 −100
0
2
4 6 Time (s)
8
10
−150
4
6
8
10
Time (s)
Fig. 12. Experiments LVFC using Kc =1400 V s/m. Ambient excitation. (a) Closed-loop acceleration. (b) Closed-loop velocity. (c) Control voltage. (d) Control force.
Settling time ts (s)
20
Bifurcation point
15 10 5 0 0
0.5
1 Dead zone level (m/s)
1.5
2 x 10−3
Fig. 13. Dead zone selection using a heel-drop excitation. (– –) Settling time for 5%, (– –) for 1%, (—) for 0.5%.
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Again, the amplitude ALC is obtained 1 pA 4Vs ¼ -ALC ¼ : pKc,limit Kc,limit 4Vs
ð22Þ
Thus, the predicted amplitude of the stable limit cycle was ALC = 6.4 10 4 m/s and the minimum value of the disconnection level to avoid limit cycle behaviour was ddis,limit =6.4 10 4 m/s, using Eq. (21). This value can be considered equivalent to dRMS,limit = 4.5 10 4 m/s RMS-velocity (dRMS in Eq. (3)). It is notable that the DF tool suggests that the switching-off level ddis,limit must be twice the dead zone level ddead,limit in order to avoid limit cycle behaviour.
4. 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 disturbance force considered was a heel-drop excitation, which is an impact excitation that is useful in evaluating transient response of floor structures and is also appropriate for studying system stability properties. The heel-drop is the force created by a person standing on their toes and suddenly dropping to their heels to hit the floor. The results for each control law are presented following this sequence: (1) controller parameter design using numerical simulations; (2) experimental trial under zero-vibration condition; and (3) experimental trial under limit cycle condition. Points 1–2 were carried out using heel-drop excitation and ambient excitation was used in Point 3. To design each control law, the settling time (ts), defined as the time taken for the response to fall within and remain within some specified percentage of the maximum peak value of the acceleration response, was considered. That is, ts was obtained from numerical simulations in which a control parameter was modified: the control gain Kc for LVFC with saturation, the dead zone value ddead for on-off NLVFC with a dead zone and the disconnection level dRMS for on-off NLVFC
0.02
0.4 0.01 Velocity (m/s)
Acceleration (m/s2)
0.6
0.2 0 −0.2
0 −0.01
−0.4 −0.02 0
2
4
6
8
10
0
2
4
6
8
10
6
8
10
Time (s)
0.6
150
0.4
100 Control force (N)
Control voltage (V)
Time (s)
0.2 0 −0.2
50 0 −50 −100
−0.4
−150 0
2
4
6 Time (s)
8
10
0
2
4 Time (s)
Fig. 14. Experiments on-off NLVFC with dead zone using ddead =4.5 10 4 m/s. Heel-drop excitation. (a) Closed-loop acceleration. (b) Closed-loop velocity. (c) Control voltage. (d) Control force.
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with a switching-off function. The simulations were undertaken using a standard idealisation of a heel drop consisting of a 2670N decreasing ramp with 50 ms duration [26].
4.1. Linear velocity feedback with saturation Taking the control scheme of Fig. 1 using the control law (1) with Vs =0.5 V, a suitable gain Kc was obtained. To this end, the control scheme for Kc 2 ½0,Kc,limit Þ was simulated under a heel-drop excitation and the settling time was computed considering an acceleration threshold of 5%, 1% and 0.5% of the peak acceleration. Fig. 10 shows the settling time for the three considered acceleration thresholds. The limit cycle behaviour was obtained for Kc,limit = 1005 V s/m. This point is identified as bifurcation point in the figure. The control system changes its stability nature from an under-damped system to a self-sustained vibrating system. Additionally, it is observed that ts is drastically reduced as Kc is increased for small values of it, whereas increasing values of Kc from Kc = 500 V s/m do not reduce ts. Therefore, this value is selected to be used since higher values reduce the actuator damping and the relative stability (Fig. 6). Several heel drops were carried out using Kc =500 V s/m and Fig. 11 shows one of them. The open-loop response, closed-loop response, control voltage and control force are shown in this figure. Observe that the control voltage generated (Fig. 11c) is mainly the result of the saturation range and that the linear range is used only when the vibration level is small. To study the limit cycle behaviour, Kc = 1400 V s/m was used and the system was subjected to ambient vibrations. Fig. 12 shows the results in this case. Depending on the noise level at the particular moment in which the experiment is recorded, either just ambient noise or a small impact is enough to initiate limit cycle behaviour. Observe that the system is involved in a self-sustained vibration that is mainly governed by a single harmonic. This demonstrates that the main approximation of the DF analysis is valid. Additionally, it was observed that the system started to be unstable for gains greater that 900 V s/m and exhibited limit cycle behaviour for gains greater than 1000 V s/m. The measured limit cycle frequency obtained from Fig. 12d was oLC = 13.20 rad/s (2.1 Hz) and the amplitude was ALC = 6.5 10 4 m/s.
1
x 10−3
Velocity (m/s)
Acceleration (m/s2)
0.02 0.01 0 −0.01
0.5 0 −0.5
−0.02 −0.03
−1 0
2
4
6
8
10
0
2
4
6
8
10
10
12
Time (s)
0.6
150
0.4
100 Control force (N)
Control voltage (V)
Time (s)
0.2 0 −0.2
50 0 −50 −100
−0.4
−150 0
2
4
6 Time (s)
8
10
0
2
4
6 Time (s)
8
Fig. 15. Experiments on-off NLVFC with dead zone using ddead = 1 10 4 m/s. Ambient excitation. (a) Closed-loop acceleration response. (b) Closed-loop velocity. (c) Control voltage. (d) Control force.
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4.2. Nonlinear velocity feedback with a dead zone Control law (2) with Vs = 0.5 V was considered. The settling time computed using 5%, 1% and 0.5% of the peak acceleration versus the dead zone level ddead 2 ðddead,min ,20 104 is plotted in Fig. 13. The minimum values of the settling time were found for ddead 2 ð4 104 ,5 104 Þ and the minimum value of the dead zone to avoid limit cycle behaviour was obtained to be ddead,limit =3.2 10 4 m/s. This point is identified as bifurcation point in the figure. That is, smaller values of the dead zone lead to limit cycle behaviour. The system response, control voltage and control force were obtained for ddead = 4.5 10 4 m/s (zero-vibration condition) and ddead = 1 10 4 m/s (limit cycle condition). Fig. 14 shows the simulated results for ddead = 4.5 10 4 m/s. Observe that the effect of the dead zone becomes apparent only at the end of the control voltage (Fig. 14c), where lower
Bifurcation point
Settling time ts (s)
20 15 10 5 0 0
0.5
1
1.5
Switching−off level (m/s)
2 x 10−3
Fig. 16. Switching-off selection using a heel-drop excitation. Settling time for T= 0.22 s (one cycle of the fundamental frequency), (– –) 5%, (– –) for 1%, (—) for 0.5%.
0.01
Velocity (m/s)
Acceleration (m/s2)
0.5
0
0.005 0 −0.005 −0.01 −0.015
−0.5 0
2
4
6
8
10
0
2
4
6
8
10
Time (s)
0.6
150
0.4
100
Control force (N)
Control voltage (V)
Time (s)
0.2 0 −0.2
50 0 −50 −100
−0.4
−150 0
2
4 6 Time (s)
8
10
0
2
4 6 Time (s)
8
10
Fig. 17. Experiments on-off NLVFC with switching-off using T= 0.22 s and dRMS = 8.6 10 4 m/s. Heel-drop excitation. (a) Closed-loop acceleration. (b) Closed-loop velocity. (c) Control voltage. (d) Control force.
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voltages were generated by LVFC. Fig. 15 shows the results for ddead = 1 10 4 m/s, which was not sufficiently high to avoid limit cycle behaviour. Experimentally, it was observed that the system became unstable for ddead,min ¼ 23 104 m=s. The measured limit cycle frequency was oLC =13.8 rad/s and the amplitude was ALC =6.5 10 4 m/s.
4.3. Nonlinear velocity feedback with a switching-off function The control law given by (3) using T= 0.22 s (the period of the first floor frequency) was considered. The settling time computed using 5%, 1% and 0.5% of the peak acceleration versus the switching-off level dRMS 2 ðdRMS,min ,20 104 is shown in Fig. 16. The bifurcation point was found at dRMS,limit = 3.7 10 4 m/s and the minimum values of the settling time were obtained for dRMS 2 ð8 104 ,8:7 104 Þ. Fig. 17 shows the system response, control voltage and control force for dRMS =8.6 10 4 m/s (zero-vibration condition) and Fig. 18 shows the results for dRMS =1 10 4 m/s which led to limit cycle behaviour. This limit cycle again was governed by a single harmonic. The system became unstable for dRMS,min ¼ 23 104 m=s. The measured limit cycle frequency was oLC =13.80 rad/s and the amplitude was ALC = 7.0 10 4 m/s.
5. 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 dramatic detrimental effects on the control system hardware and/or annoying continuous vibration. As shown Table 1, a good agreement between analytical predictions and experimental results was observed. The analytical studies carried out using the DF tool have shown that the stability properties of on-off NLVFC are similar to those of LVFC when high gains are required since both control schemes are involved in limit cycle behaviour. It has also
1
0.02 Velocity (m/s)
Acceleration (m/s2)
x 10−3
0.01 0 −0.01
0.5 0 −0.5
−0.02 −1
−0.03 0
2
4
6
8
10
0
2
4 6 Time (s)
8
0
2
4 6 Time (s)
8
0.6
150
0.4
100 Control force (N)
Control voltage (V)
Time (s)
0.2 0 −0.2
10
50 0 −50 −100
−0.4
−150 0
2
4 Time (s)
6
8
10
10
Fig. 18. Experiments on-off NLVFC with switching-off using T= 0.22 s and dRMS =1 10 4 m/s. Ambient excitation. (a) Closed-loop acceleration. (b) Closed-loop velocity. (c) Control voltage. (d) Control force.
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Table 1 Limit cycle behaviour. Comparison between the analytical studies and experimental trials.
LVFC + saturation
Parameter
Analyticala
Experimentalb
Kc,limit(V s/m)
1000 11.40 6.0 10 4
900–1000 13.20 6.5 10 4
3.2 10 4 11.40 6.4 10 4
2.5–3.5 10 4 13.80 6.5 10 4
4.5 10 4 11.40 6.4 10 4
3.0–4.0 10 4 13.80 7.0 10 4
oLC(rad/s) ALC(m/s) NLVFC + dead zone
ddead,limit(m/s)
oLC(rad/s) ALC(m/s) NLVFC + switching-off
dRMS,limit(m/s)
oLC(rad/s) ALC(m/s) a b
Analytical results obtained through the use of the DF tool (see Section 3). Experimental results conducted on the test floor.
been demonstrated that the inclusion of a dead zone or switching-off function into on-off nonlinear control can be utilised to avoid limit cycle behaviour. These studies have been validated by experimental trials on a laboratory test floor. Finally, on-off nonlinear control based-on velocity feedback has been shown to be a feasible alternative to linear velocity feedback. The advantages are that no gain has to be designed and the maximum control force is always applied to the structure. Future research will consider the use of on-off NLVFC together with feed-through term inclusion which has shown to improve stability margins of LVFC [27]. Acknowledgments The authors would like to acknowledge the financial support of Conserjerı´a de Educacio´n 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. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22]
H. Bachmann, Case studies of structures with man-induced vibrations, Journal of Structural Engineering 118 (1992) 631–647. ISO10137, Basis for the design of structures—serviceability of building against vibration, International Standards Organization, 1992, pp. 41–43. T.M. Murray, D.E. Allen, E.E. Ungar, Floor vibrations due to human activity, AISC Steel Design Guide 11, CISC, Chicago, IL, 1997. A. Pavic, M. Willford, Vibration serviceability in post-tensioned floors, Appendix G in Post-Tensioned Concrete Floors Design Handbook, Technical Report 43, 2005. M. Setareh, R.D. Hanson, Tuned mass damper to control floor vibration from humans, Journal of Structural Engineering 118 (1992) 741–762. M. Setareh, Floor vibration control using semi-active tuned mass dampers, Canadian Journal of Civil Engineering 29 (2002) 76–84. J.H. Koo, M. Ahmadian, M. Setareh, T.M. Murray, In search of suitable control methods for semi-active tuned vibration absorbers, Journal of Vibration Control 10 (2004) 163–174. Q.P. Ha, N.M. Kwok, M.T. Nguyen, J. Li, B. Samali, Mitigation of seismic responses on building structures using MR dampers with Lyapunov-based control, Structural Control and Health Monitoring 15 (2008) 604–621. L.M. Hanagan, T.M. Murray, K. Premaratne, Controlling floor vibration with active and passive devices, The Shock and Vibration Digest 35 (2003) 347–365. D. Nyawako, P. Reynolds, Technologies for mitigation of human-induced vibration in civil engineering structures, The Shock and Vibration Digest 36 (2007) 465–493. A. Ebrahimpour, R.L. Sack, A review of vibration serviceability criteria for floor structures, Computers and Structures 83 (2005) 2488–2494. B.F. Spencer Jr., S. Nagarajaiah, State of the art of structural control, ASCE Journal of Structural Engineering 129 (2003) 845–856. W.K. Gawronski, Advanced Structural Dynamics and Active Control of Structures, Springer-Verlag, New York, USA, 2004. H. Du, N. Zhang, H. Nguyen, Mixed H2/HN control of tall buildings with reduced-order modelling technique, Structural Control and Health Monitoring 15 (2008) 64–89. L. Huo, G. Song, H. Li, K. Grigoriadis, HN robust control design of active structural vibration suppression using an active mass damper, Smart Materials and Structures 17 (2008) 1–10. L.M. Hanagan, T.M. Murray, Active control for reducing floor vibrations, Journal of Structural Engineering 123 (1997) 1497–1505. L.M. Hanagan, T.M. Murray, Experimental implementation of active control to reduce annoying floor vibration, Engineering Journal 35 (1998) 123–127. M.J. Balas, Direct velocity feedback control of large space structures, Journal of Guidance and Control 2 (1979) 252–253. D. Nyawako, P. Reynolds, Response-dependent velocity feedback control for mitigation of human-induced floor vibrations, Smart Materials and Structures 18 (2009) 075002. I.M. Diaz, D. Nywako, P. Reynolds, On-off nonlinear velocity feedback control for cancelling floor vibrations, in: Proceedings of the 4th European Conference on Structural Control, St. Petersburg, pp. 175–182. K.D. Lindner, G.A. Zvonar, D. Borojevic, Nonlinear control of proof-mass actuator, AIAA Journal of Guidance, Control, and Dynamics 20 (1997) 464–470. P. Reynolds, A. Pavic, Effects of false floor on vibration serviceability of building floors. I: modal properties,, ASCE Journal of Performance of Constructed Facilities 17 (2003) 75–86.
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[23] A. Preumont, Vibration Control of Active Structures: An introduction, Kluwer Academic, Dordrecht, The Netherlands, 1997. [24] H. Bachmann, W. Ammann, Vibrations in structures—induced by man and machines, Structural Engineering Documents, vol. 3e, International Association of Bridge and Structural Engineering, Zurich, 1987. [25] J.J. Slotine, W. Li, Applied Nonlinear Control, Prentice-Hall, 1991 (Chapter 5). [26] T.M. Murray, Design practice to prevent floor vibrations, AISC Engineering Journal 12 (1975) 82–87. [27] I.M. Diaz, P. Reynolds, Robust saturated control of human-induced floor vibrations via a proof-mass actuator, Smart Materials and Structures 18 (2009) 1250024.
Contents lists available at ScienceDirect
Mechanical Systems and Signal Processing journal homepage: www.elsevier.com/locate/jnlabr/ymssp
On-off nonlinear active control of floor vibrations Iva´n M. Dı´az a,b,, Paul Reynolds b a b
Escuela Te´cnica Superior de Ingenieros Industriales, Universidad de Castilla-La Mancha, Edificio Polite´cnica, Av. Camilo Jose´ Cela s/n, 13071 Ciudad Real, Spain Department of Civil and Structural Engineering, The University of Sheffield, Sir Frederick Mappin Building, Mappin Street, Sheffield S1 3JD, United Kingdom
a r t i c l e in fo
abstract
Article history: Received 17 September 2008 Received in revised form 9 December 2009 Accepted 28 February 2010 Available online 4 March 2010
Human-induced floor vibrations can be mitigated by means of active control via an electromagnetic proof-mass actuator. Previous researchers have developed a system for floor vibration comprising linear velocity feedback control (LVFC) with a command limiter (saturation in the command signal to avoid actuator overloading). The performance of this control is highly dependent on the linear gain utilised, which has to be designed for a particular excitation and might not be optimum for other excitations. This work explores the use of on-off nonlinear velocity feedback control (NLVFC) as the natural evolution of LVFC when high gains and/or significant vibration level are present together with saturation in the control law. Firstly, the describing function tool is employed to analyse the stability properties of: (1) LVFC with saturation, (2) on-off NLVFC with a dead zone and (3) on-off NLVFC with a switching-off function. Particular emphasis is paid to the resulting limit cycle behaviour and the design of appropriate dead zone and switching-off levels to avoid it. Secondly, experimental trials using the three control laws are conducted on a laboratory test floor. The results corroborate the analytical stability predictions. The pros of on-off NLVFC are that no gain has to be chosen and maximum actuator energy is delivered to cancel the vibration. In contrast, the requirement to select a dead zone or switching-off function provides a drawback in its application. & 2010 Elsevier Ltd. All rights reserved.
Keywords: Active control Structural control Floor vibrations Nonlinear velocity feedback Human-induced vibrations
1. 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 [2–4] 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 [5–8]. 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.
Corresponding author at: Escuela Te´cnica Superior de Ingenieros Industriales, Universidad de Castilla-La Mancha, Edificio Polite´cnica, Av. Camilo Jose´ Cela s/n, 13071 Ciudad Real, Spain. Tel.: + 34 926295238; fax: + 34 926295300. E-mail addresses: [email protected], I.M.Diaz@sheffield.ac.uk (I.M. Dı´az).
0888-3270/$ - see front matter & 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.ymssp.2010.02.011
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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 HN control [13–15] 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 modelbased, 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. 2. 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. 2.1. Control strategies Fig. 1 shows a block diagram of a general velocity feedback control particularised for a control system in which the objective is to achieve zero vibration (reference command r(t) =0) and the considered output of the floor structure (the plant) is the acceleration. In this figure, GA(s) is the transfer function of the proof-mass actuator, G(s) is of the floor
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Fig. 1. General velocity feedback control scheme.
_ structure, GS(s) is of the sensor and f ðyðtÞÞ is the control law. Three particular cases of Fig. 1 are assessed throughout this paper. The first one corresponds to LVFC with saturation in which the control law to generate the control voltage V0(t) can be written as follows: ( _ _ jyðtÞj rVs =Kc Kc yðtÞ _ , ð1Þ f ðyðtÞÞ ¼ _ _ Vs signðyðtÞÞ jyðtÞj 4 Vs =Kc where Kc is the control gain and Vs is the maximum allowable control voltage to the actuator (saturation level). This command limiter is introduced to avoid force and stroke limitation [21]. The second control law corresponds to on-off NLVFC with a dead zone in which the control voltage is obtained as ( _ 0 jyðtÞj r ddead _ , ð2Þ f ðyðtÞÞ ¼ _ _ Vs signðyðtÞÞ jyðtÞj 4 ddead where ddead is the velocity dead zone level. The third one corresponds to on-off NLVFC with a switching-off function in which the control voltage is calculated as ( 0 jy_ RMS,T ðtÞj r dRMS _ , ð3Þ f ðyðtÞÞ ¼ _ Vs signðyðtÞÞ jy_ RMS,T ðtÞj 4 dRMS where y_ RMS,T is the running root mean square (RMS) velocity computed using T-seconds and dRMS is the disconnection level in terms of y_ RMS,T .
2.2. Experimental setup The experimental floor used in this work consists of a simply supported slab strip made of in-situ cast post-tensioned concrete [22]. It has a span of 10.8 m and a total length of 11.2 m, including 0.2 m overhangs over each support. It has a width of 2.0 m, a thickness of 0.275 m and it weighs approximately 15 tonnes. Fig. 2 shows photographs of the structure, in which a general view (Fig. 2a) and a detail of one of the supports (Fig. 2b) are observed. A single collocated actuator/sensor is placed at mid span, where the first vibration mode shape has its maximum value. The actuator is a proof-mass electromagnetic shaker (APS Dynamics Model 113 electrodynamic shaker operated in inertial mode) (Fig. 3a) and the sensor is a piezoelectric accelerometer (Endevco 7754A-1000) mounted on a levelled baseplate (Fig. 3b). The peak harmonic force given by the actuator is 133 N (at 10 Hz) and the maximum stroke is 0.158 m. The saturation voltage has to consider both limitations. However, in the case of human-induced vibrations, in which lowfrequency components are mainly excited, the saturation voltage is mainly due to stroke limit and is taken as Vs =0.5 V in this work. The controller hardware completes the experimental setup. It comprises of a PC with a National Instruments PCI-6030E DAQ card installed. It is important to note that since the system response is measured in terms of acceleration, the velocity is obtained digitally together with the computation of the control law (see Fig. 1).
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Fig. 2. Photograph of the experimental slab strip. (a) General view. (b) Simple support and overhang.
Accelerometer Active mass Rubber bands Baseplate
Fig. 3. (a) Proof-mass electromagnetic shaker (APS Dynamics Model 113 electrodynamic shaker). (b) Piezoelectric accelerometer (Endevco 7754-1000) mounted on a levelled baseplate (top view).
2.3. Floor dynamics If the aforementioned collocated case between acceleration (output) and the force (input) is considered, the transfer function of the floor dynamics can be represented as an infinite sum of second-order systems as follows [23]: GðsÞ ¼
1 X
ai s2 , oi s þ o2i
s2 þ 2zi i¼1
ð4Þ
where s= jo, o being the frequency, ai Z0, zi and oi are the damping ratio and natural frequency associated to the ith mode, respectively. It is common practice to truncate the model for the frequency bandwidth of interest, such that only N vibration modes are considered. The transfer function G(s) is thus approximated by ~ GðsÞ ¼
N X i¼1
ai s2 : 2 i oi s þ oi
s2 þ2z
ð5Þ
The test floor structure was designed in such a way that its first mode (approximately at 4.4 Hz) was prone to be excited by the second and third harmonics of walking excitation [24]. The frequency response function (FRF) corresponding with the mid span (collocated control location) using a frequency span of 0–50 Hz was obtained experimentally and a posteriori parameter identification of model (5) (N= 2) was undertaken, giving ~ GðsÞ ¼
1:39 104 s2 1:31 104 s2 þ : s2 þ0:385s þ 755:64 s2 þ 6:509sþ 54046
ð6Þ
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0
Phase (deg)
−40 Magnitude (dB)
1715
−60 −80
−50 −100 −150
−100 10
20
30
40
50
−200
10
Frequency (Hz)
20
30
40
50
Frequency (Hz)
~ Fig. 4. Transfer function of the floor structure GðsÞ: (– –) experimental; (—) model. (a) Magnitude in dB referenced to 1(m/s2)/N. (b) Phase in degrees.
Fig. 4 shows the magnitude and phase responses of the model and experimental floor. As can be observed, the model captures the floor dynamics with sufficient accuracy. 2.4. Proof-mass actuator dynamics The proof-mass actuator can be described as a linear second-order system with the following transfer function [23]: GA ðsÞ ¼
KA s2 , s2 þ 2zA oA s þ o2A
ð7Þ
in which KA is a gain which relates the released inertial force FA(t) (see Fig. 1) with the input voltage V(t), and zA and oA are, respectively, the damping ratio and natural frequency which take into account the suspension system and internal damping. The actuator model was determined to be GA ðsÞ ¼
150s2 , s2 þ 4:021s þ 101:1
ð8Þ
where the natural frequency of the shaker is estimated as oA = 10.05 rad/s (1.60 Hz). 2.5. Sensor dynamics As was mentioned before, the velocity was obtained digitally from the acceleration signal. The employed procedure used the trapezoidal method and a DC offset correction. Again, its dynamics were identified and closely modelled by the following filter: GS ðsÞ ¼
s s ¼ 2 , s þ 5s þ 25 s2 þ 2zS oS s þ o2S
ð9Þ
in which oS and zS are the natural frequency and damping ratio of the filter. 3. 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 is usually used to discover the existence of limit cycles and determine their stability. In order to use the DF tool in its basic form, the following conditions have to be satisfied [25]: (1) there is only a single nonlinear component, (2) the nonlinearity is time-invariant, (3) the nonlinearity is odd and (4) when the input is sinusoidal, only the fundamental component of the Fourier series of the output (of the nonlinear part) has to be considered. The first condition implies that if there are several nonlinearities, these have to be merged into one. The second condition is needed due to the fact that the DF tool is based on the Nyquist criterion, which is applied only to linear time-invariant systems. The third condition simplifies the Fourier expansion since the static term can be neglected. Finally, the fourth condition is actually an assumption and is usually known as the fundamental assumption of the DF tool. This assumption represents an approximation since the output of a nonlinear element due to a sinusoidal input usually contains higher
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V (t)
r (t) = 0 +
GT (s)
. y(t)
– V0 (t)
N (A, ω)
Fig. 5. Stability analysis by the DF tool.
30 200
F
100
Imag Axis
Imag Axis
20
0 −100
↑Kc
10
A S
0 −10 −20
−200 −40
−30
−20 −10 Real Axis
0
10
−30
−30
−20
−10 Real Axis
0
10
~ oÞGS ðsÞ. (b) Zoom of the origin: A, actuator, S, sensor and F, floor. Fig. 6. (a) Root locus of the total transfer function GT ðjoÞ ¼ GA ðjoÞGðj
harmonics than the fundamental one. This implies that the dynamic response of a limit-cycling system is governed mainly by a single harmonic, i.e. the output should be approximately sinusoidal. For the stability analysis by the DF tool, the control system of Fig. 1 is transformed into the scheme of Fig. 5, in which the total transfer function of the linear part is represented by GT(jo) and the nonlinear element (i.e., the control law) is substituted by its DF, denoted by N(A,o). Thus, the Nyquist criterion can be applied in its extended form by just substituting a control gain by the DF. The outline of the stability analysis carried out herein is as follows: Step 1: Analyse the root locus of the total open-loop transfer function (the transfer function of the linear part). In this case, this is (see Fig. 1) ~ oÞGS ðjoÞ, GT ðjoÞ ¼ GA ðjoÞGðj
ð10Þ
assuming the truncated model (5) for the floor structure. Step 2: Obtain the DF, which in general depends on the amplitude A and the frequency o of the sinusoidal input, corresponding to the significant hard nonlinearities present in the control system. Step 3: Apply the extended Nyquist criterion using the DF N(A,o) GT ðjoÞ ¼ 1=NðA,oÞ:
ð11Þ
Each solution of Eq. (11) predicts a limit cycle behaviour. Step 4: Use the criterion of the stability of limit cycles to decide about their nature.
3.1. Root locus study In Step 1, the root locus maps the roots of the closed-loop transfer function for control gains from zero (open loop) to infinity. The root locus of GT(jo) (Eq. (10)) using (6), (8) and (9) is equivalent to analysis of the stability when the control _ ¼ Kc y_ with Kc 2 ½0,1Þ (equivalent to Eq. (1) without saturation). This is also equivalent to a constant DF equal to law is f ðyÞ the control gain N(A,o) =Kc in Fig. 5. Fig. 6 shows the root locus. It can be observed that the controlled system becomes unstable for high gains due to the fact that the actuator dynamics branches go to the right-half plane. This clearly illustrates the fact that the inclusion of sensor and actuator dynamics makes LVFC conditionally stable. For this particular case, the limit gain is found to be Kc,limit = 1000.4 V s/m. Increasing values of Kc increases the damping of the floor vibration modes but decreases the damping of the actuator up to its instability (see Fig. 6b).
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x 10−4 -1×10-3 = -1/Kc,limit
1 5
Imag Axis
N (A)/Kc
GT (j)
Linearity range
0.8 0.6 0.4
0
-∞
-1/N (A) -1/Kc
−5
0.2 0 0
2
4
6
8
A/a
10
−10
−8
−6 −4 Real Axis
−2
0
2 x 10−4
Fig. 7. (a) DF for the saturation nonlinearity. (b) Nyquist diagram of GT(jo) and 1/N(A).
3.2. Linear velocity feedback with saturation Steps 2, 3 and 4 were carried out on the control law of Eq. (1). As has been explained previously, similar to the concept of FRF, the DF is the ratio between the fundamental component of the nonlinear element output and a sinusoidal input given by xðtÞ ¼ A sinðotÞ. If the nonlinearity is hard, odd and single-valued (the case of the nonlinearities analysed in this work), the DF depends only on the input amplitude N(A,o)=N(A), i.e., it is a real function. The DF for a saturation nonlinearity is [25] 8 Kc A ra > > 2 sffiffiffiffiffiffiffiffiffiffiffiffiffi3 > < a a a2 NðAÞ ¼ 2Kc 4 , ð12Þ arcsin 1 2 5 A4 a > > p A A A > : with a=Vs/Kc (see Eq. (1)). The normalized DF N(A)/Kc (12) is plotted in Fig. 7a as a function of A/a. As can be seen, N(A)=Kc if the input amplitude is in the linear range (Ara) and N(A) decreases as the input amplitude increases when A4a. That is, saturation does not occur for small signals and it reduces the ratio of the output to input as the input increases. Once the DF of the nonlinearity has been identified (Step 2), the extended Nyquist criterion (11) is applied (Step 3). Fig. 7b shows a plot of GT(jo) (varying o) and the negative inverse DF 1/N(A) (varying A) in the complex plane. If the two curves intersect, limit cycles are then predicted, and their properties (amplitudes and frequencies) are obtained from the solution of (11). Focusing on 1/N(A), it can be seen that it starts at 1/Kc and remains there while the linear range. Then, 1/N(A) goes to N as A increases. Examining next the plot GT(jo), it is observed that it can intersect with 1/N(A) depending on the value of Kc. The conclusion is that: if Kc oKc,limit, the system is asymptotically stable and goes to zero vibration (no intersection); otherwise, a limit cycle is predicted (intersection). Such a limit cycle is deduced to be stable by using the limit cycle stability criterion [25]. This criterion says that if points of 1/N(A) approaching the intersection with increasing values of A are not encircled by the curve GT(jo), then the corresponding limit cycle is stable; otherwise, it is unstable. The frequency of the limit cycle oLC is obtained from GT(jo) at the intersecting point and the amplitude ALC is obtained from Eq. (11) particularised to the same point 1 p ffii , ¼ h a a qffiffiffiffiffiffiffiffiffiffiffi 2 Kc,limit 2Kc arcsin A þ A 1 Aa2
ð13Þ
with Kc 4Kc,limit and |GT(joLC)| =1/Kc,limit. In this case, the frequency of the limit cycle was oLC =11.40 rad/s (or 1.81 Hz), which is close to the estimated natural frequency of the actuator (Eq. (8)), and the amplitude was obtained by the solution of Eq. (13). If, for instance, Kc =1400 V s/m, the solution is a/A=0.599, and hence the predicted amplitude is ALC =6.0 10 4 m/s, in which Vs =0.5 V has been assumed. 3.3. On-off nonlinear velocity feedback with a dead zone The same steps as before were carried out for the control law of Eq. (2), on-off NLVFC with a dead zone. Following the standard procedure to obtain DFs [25], it is quite straightforward to derive the DF for an on-off nonlinearity including a dead zone which can be written as follows: 8 0 A r ddead > > sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi < 2 , ð14Þ NðAÞ ¼ 4Vs ddead > A 4 ddead > : pA 1 A2
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in which ddead is the dead zone level in velocity-units (see Eq. (2)). The normalized DF N(A)/Vs for ddead =1 (14) is plotted in Fig. 8a. Next 1/N(A) is plotted for increasing values of A (see Fig. 8b). It starts at N, then goes to 1/N(A)max and finally it goes back to N. The parameter N(A)max is the maximum value of the DF. Note that if 1/N(A) intersects with GT(jo), there will be two intersections. Actually, two limit cycles are predicted; the first one is unstable (with smaller amplitude) and the second one is stable (with higher amplitude). For the first intersection, increasing values of A are encircled by GT(jo) whereas, for the second intersection, increasing values of A are not encircled. This means that: if NðAÞmax o Kc,limit , the system is asymptotically stable and goes to zero vibration (no intersection); otherwise, the system is stable and goes to the stable pffiffiffilimit cycle (intersection). Parameter N(A)max can be derived from Eq. (14). From dN(A)/dA= 0, it is obtained that A ¼ 2ddead . Then, the maximum value of the DF is NðAÞmax ¼
2Vs
pddead
,
ð15Þ
and then, a zero vibration rule can be deduced Vs
ddead
o
p 2
Kc,limit :
ð16Þ
The amplitude of the limit cycles ALC is obtained from the solution of Eq. (11) particularised to the intersecting point 1 ¼ Kc,limit
pA qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : d2 4Vs 1 Adead 2
ð17Þ
Considering in Eq. (17) that ddead 5A, then the amplitude of the stable limit cycle can be predicted directly as ALC ¼
4Vs
pKc,limit
:
ð18Þ
x 10−4 -1×10-3 = -1/Kc,limit 0.6
5
N (A)max
Imag Axis
0 -∞
-1/N (A)
0.2
-1/N (A)max
−5
Dead zone 0 0
2
4
6
8
10
−10
−8
−6
A
−4
−2
Real Axis
Fig. 8. (a) DF for the on-off nonlinearity with dead zone. (b) Nyquist diagram of GT(jo) and 1/N(A).
N (A)max 1
N (A)/Vs
N (A)/Vs
Vs 0.4
GT (j)
Vs
0.5
Disconnection 0 0
2
4
6
8
10
A Fig. 9. DF for the on-off nonlinearity with a switching-off function.
0
2 x 10−4
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For the structure and controller utilised here, the predicted amplitude of the limit cycle was ALC = 6.4 10 4 m/s. It is also possible to predict the minimum value of the dead zone level to avoid limit cycle behaviour. Using Eq. (16), it was obtained that ddead,limit =3.2 10 4 m/s. 3.4. On-off nonlinear velocity feedback with a switching-off function Next, on-off NLVFC with a switching-off function was analysed (see Eq. (3)). The DF for an on-off nonlinearity including a switching-off function is as follows: 8 A r ddis < 0 ð19Þ , NðAÞ ¼ 4Vs : A 4 ddis pA
Settling time ts (s)
25 20 15
Bifurcation point
10 5 0 0
200
400 600 800 Control gain Kc (Vs/m)
1000
Fig. 10. Gain selection using a heel-drop excitation. (– –) Settling time for 5%, (– –) for 1%, (—) for 0.5%.
0.02
0.4 Velocity (m/s)
Acceleration (m/s2)
0.6
0.2 0 −0.2
0.01 0 −0.01
−0.4 −0.02 2
4
6 Time (s)
8
10
0.6
150
0.4
100 Control force (N)
Control voltage (V)
0
0.2 0 −0.2
0
2
4
6 Time (s)
0
2
4 6 Time (s)
8
10
50 0 −50 −100
−0.4
−150 0
2
4 6 Time (s)
8
10
8
10
Fig. 11. Experiments LVFC using Kc = 500 V s/m. Heel-drop excitation. a) (– –) Open-loop acceleration, (—) closed-loop acceleration. b) (– –) Open-loop velocity, (—) closed-loop velocity. c) Control voltage. d) Control force.
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where ddis is a prescribed value to disconnect the control system in velocity-units. The normalized DF N(A)/Vs for ddis =1 (19) is plotted in Fig. 9. A similar behaviour as in the case of the dead zone is predicted (see Fig. 8b). If NðAÞmax oKc,limit , the system is asymptotically stable and goes to zero vibration (no intersection). Otherwise, the system is stable and goes to the stable limit cycle (intersection). However, the maximum value of the DF is different from that predicted for the dead zone NðAÞmax ¼
4Vs
pddis
:
ð20Þ
The following zero vibration rule can be deduced: Vs
ddis
o
p 4
Kc,limit :
ð21Þ
0.01
5 Velocity (m/s)
Acceleration (m/s2)
x 10−4
0.005 0 −0.005
0
−5
−0.01 −0.015 2
4 6 Time (s)
8
10
0.6
150
0.4
100 Control force (N)
Control voltage (V)
0
0.2 0 -0.2 -0.4
0
2
0
2
4 6 Time (s)
8
10
50 0 −50 −100
0
2
4 6 Time (s)
8
10
−150
4
6
8
10
Time (s)
Fig. 12. Experiments LVFC using Kc =1400 V s/m. Ambient excitation. (a) Closed-loop acceleration. (b) Closed-loop velocity. (c) Control voltage. (d) Control force.
Settling time ts (s)
20
Bifurcation point
15 10 5 0 0
0.5
1 Dead zone level (m/s)
1.5
2 x 10−3
Fig. 13. Dead zone selection using a heel-drop excitation. (– –) Settling time for 5%, (– –) for 1%, (—) for 0.5%.
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Again, the amplitude ALC is obtained 1 pA 4Vs ¼ -ALC ¼ : pKc,limit Kc,limit 4Vs
ð22Þ
Thus, the predicted amplitude of the stable limit cycle was ALC = 6.4 10 4 m/s and the minimum value of the disconnection level to avoid limit cycle behaviour was ddis,limit =6.4 10 4 m/s, using Eq. (21). This value can be considered equivalent to dRMS,limit = 4.5 10 4 m/s RMS-velocity (dRMS in Eq. (3)). It is notable that the DF tool suggests that the switching-off level ddis,limit must be twice the dead zone level ddead,limit in order to avoid limit cycle behaviour.
4. 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 disturbance force considered was a heel-drop excitation, which is an impact excitation that is useful in evaluating transient response of floor structures and is also appropriate for studying system stability properties. The heel-drop is the force created by a person standing on their toes and suddenly dropping to their heels to hit the floor. The results for each control law are presented following this sequence: (1) controller parameter design using numerical simulations; (2) experimental trial under zero-vibration condition; and (3) experimental trial under limit cycle condition. Points 1–2 were carried out using heel-drop excitation and ambient excitation was used in Point 3. To design each control law, the settling time (ts), defined as the time taken for the response to fall within and remain within some specified percentage of the maximum peak value of the acceleration response, was considered. That is, ts was obtained from numerical simulations in which a control parameter was modified: the control gain Kc for LVFC with saturation, the dead zone value ddead for on-off NLVFC with a dead zone and the disconnection level dRMS for on-off NLVFC
0.02
0.4 0.01 Velocity (m/s)
Acceleration (m/s2)
0.6
0.2 0 −0.2
0 −0.01
−0.4 −0.02 0
2
4
6
8
10
0
2
4
6
8
10
6
8
10
Time (s)
0.6
150
0.4
100 Control force (N)
Control voltage (V)
Time (s)
0.2 0 −0.2
50 0 −50 −100
−0.4
−150 0
2
4
6 Time (s)
8
10
0
2
4 Time (s)
Fig. 14. Experiments on-off NLVFC with dead zone using ddead =4.5 10 4 m/s. Heel-drop excitation. (a) Closed-loop acceleration. (b) Closed-loop velocity. (c) Control voltage. (d) Control force.
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with a switching-off function. The simulations were undertaken using a standard idealisation of a heel drop consisting of a 2670N decreasing ramp with 50 ms duration [26].
4.1. Linear velocity feedback with saturation Taking the control scheme of Fig. 1 using the control law (1) with Vs =0.5 V, a suitable gain Kc was obtained. To this end, the control scheme for Kc 2 ½0,Kc,limit Þ was simulated under a heel-drop excitation and the settling time was computed considering an acceleration threshold of 5%, 1% and 0.5% of the peak acceleration. Fig. 10 shows the settling time for the three considered acceleration thresholds. The limit cycle behaviour was obtained for Kc,limit = 1005 V s/m. This point is identified as bifurcation point in the figure. The control system changes its stability nature from an under-damped system to a self-sustained vibrating system. Additionally, it is observed that ts is drastically reduced as Kc is increased for small values of it, whereas increasing values of Kc from Kc = 500 V s/m do not reduce ts. Therefore, this value is selected to be used since higher values reduce the actuator damping and the relative stability (Fig. 6). Several heel drops were carried out using Kc =500 V s/m and Fig. 11 shows one of them. The open-loop response, closed-loop response, control voltage and control force are shown in this figure. Observe that the control voltage generated (Fig. 11c) is mainly the result of the saturation range and that the linear range is used only when the vibration level is small. To study the limit cycle behaviour, Kc = 1400 V s/m was used and the system was subjected to ambient vibrations. Fig. 12 shows the results in this case. Depending on the noise level at the particular moment in which the experiment is recorded, either just ambient noise or a small impact is enough to initiate limit cycle behaviour. Observe that the system is involved in a self-sustained vibration that is mainly governed by a single harmonic. This demonstrates that the main approximation of the DF analysis is valid. Additionally, it was observed that the system started to be unstable for gains greater that 900 V s/m and exhibited limit cycle behaviour for gains greater than 1000 V s/m. The measured limit cycle frequency obtained from Fig. 12d was oLC = 13.20 rad/s (2.1 Hz) and the amplitude was ALC = 6.5 10 4 m/s.
1
x 10−3
Velocity (m/s)
Acceleration (m/s2)
0.02 0.01 0 −0.01
0.5 0 −0.5
−0.02 −0.03
−1 0
2
4
6
8
10
0
2
4
6
8
10
10
12
Time (s)
0.6
150
0.4
100 Control force (N)
Control voltage (V)
Time (s)
0.2 0 −0.2
50 0 −50 −100
−0.4
−150 0
2
4
6 Time (s)
8
10
0
2
4
6 Time (s)
8
Fig. 15. Experiments on-off NLVFC with dead zone using ddead = 1 10 4 m/s. Ambient excitation. (a) Closed-loop acceleration response. (b) Closed-loop velocity. (c) Control voltage. (d) Control force.
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4.2. Nonlinear velocity feedback with a dead zone Control law (2) with Vs = 0.5 V was considered. The settling time computed using 5%, 1% and 0.5% of the peak acceleration versus the dead zone level ddead 2 ðddead,min ,20 104 is plotted in Fig. 13. The minimum values of the settling time were found for ddead 2 ð4 104 ,5 104 Þ and the minimum value of the dead zone to avoid limit cycle behaviour was obtained to be ddead,limit =3.2 10 4 m/s. This point is identified as bifurcation point in the figure. That is, smaller values of the dead zone lead to limit cycle behaviour. The system response, control voltage and control force were obtained for ddead = 4.5 10 4 m/s (zero-vibration condition) and ddead = 1 10 4 m/s (limit cycle condition). Fig. 14 shows the simulated results for ddead = 4.5 10 4 m/s. Observe that the effect of the dead zone becomes apparent only at the end of the control voltage (Fig. 14c), where lower
Bifurcation point
Settling time ts (s)
20 15 10 5 0 0
0.5
1
1.5
Switching−off level (m/s)
2 x 10−3
Fig. 16. Switching-off selection using a heel-drop excitation. Settling time for T= 0.22 s (one cycle of the fundamental frequency), (– –) 5%, (– –) for 1%, (—) for 0.5%.
0.01
Velocity (m/s)
Acceleration (m/s2)
0.5
0
0.005 0 −0.005 −0.01 −0.015
−0.5 0
2
4
6
8
10
0
2
4
6
8
10
Time (s)
0.6
150
0.4
100
Control force (N)
Control voltage (V)
Time (s)
0.2 0 −0.2
50 0 −50 −100
−0.4
−150 0
2
4 6 Time (s)
8
10
0
2
4 6 Time (s)
8
10
Fig. 17. Experiments on-off NLVFC with switching-off using T= 0.22 s and dRMS = 8.6 10 4 m/s. Heel-drop excitation. (a) Closed-loop acceleration. (b) Closed-loop velocity. (c) Control voltage. (d) Control force.
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voltages were generated by LVFC. Fig. 15 shows the results for ddead = 1 10 4 m/s, which was not sufficiently high to avoid limit cycle behaviour. Experimentally, it was observed that the system became unstable for ddead,min ¼ 23 104 m=s. The measured limit cycle frequency was oLC =13.8 rad/s and the amplitude was ALC =6.5 10 4 m/s.
4.3. Nonlinear velocity feedback with a switching-off function The control law given by (3) using T= 0.22 s (the period of the first floor frequency) was considered. The settling time computed using 5%, 1% and 0.5% of the peak acceleration versus the switching-off level dRMS 2 ðdRMS,min ,20 104 is shown in Fig. 16. The bifurcation point was found at dRMS,limit = 3.7 10 4 m/s and the minimum values of the settling time were obtained for dRMS 2 ð8 104 ,8:7 104 Þ. Fig. 17 shows the system response, control voltage and control force for dRMS =8.6 10 4 m/s (zero-vibration condition) and Fig. 18 shows the results for dRMS =1 10 4 m/s which led to limit cycle behaviour. This limit cycle again was governed by a single harmonic. The system became unstable for dRMS,min ¼ 23 104 m=s. The measured limit cycle frequency was oLC =13.80 rad/s and the amplitude was ALC = 7.0 10 4 m/s.
5. 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 dramatic detrimental effects on the control system hardware and/or annoying continuous vibration. As shown Table 1, a good agreement between analytical predictions and experimental results was observed. The analytical studies carried out using the DF tool have shown that the stability properties of on-off NLVFC are similar to those of LVFC when high gains are required since both control schemes are involved in limit cycle behaviour. It has also
1
0.02 Velocity (m/s)
Acceleration (m/s2)
x 10−3
0.01 0 −0.01
0.5 0 −0.5
−0.02 −1
−0.03 0
2
4
6
8
10
0
2
4 6 Time (s)
8
0
2
4 6 Time (s)
8
0.6
150
0.4
100 Control force (N)
Control voltage (V)
Time (s)
0.2 0 −0.2
10
50 0 −50 −100
−0.4
−150 0
2
4 Time (s)
6
8
10
10
Fig. 18. Experiments on-off NLVFC with switching-off using T= 0.22 s and dRMS =1 10 4 m/s. Ambient excitation. (a) Closed-loop acceleration. (b) Closed-loop velocity. (c) Control voltage. (d) Control force.
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Table 1 Limit cycle behaviour. Comparison between the analytical studies and experimental trials.
LVFC + saturation
Parameter
Analyticala
Experimentalb
Kc,limit(V s/m)
1000 11.40 6.0 10 4
900–1000 13.20 6.5 10 4
3.2 10 4 11.40 6.4 10 4
2.5–3.5 10 4 13.80 6.5 10 4
4.5 10 4 11.40 6.4 10 4
3.0–4.0 10 4 13.80 7.0 10 4
oLC(rad/s) ALC(m/s) NLVFC + dead zone
ddead,limit(m/s)
oLC(rad/s) ALC(m/s) NLVFC + switching-off
dRMS,limit(m/s)
oLC(rad/s) ALC(m/s) a b
Analytical results obtained through the use of the DF tool (see Section 3). Experimental results conducted on the test floor.
been demonstrated that the inclusion of a dead zone or switching-off function into on-off nonlinear control can be utilised to avoid limit cycle behaviour. These studies have been validated by experimental trials on a laboratory test floor. Finally, on-off nonlinear control based-on velocity feedback has been shown to be a feasible alternative to linear velocity feedback. The advantages are that no gain has to be designed and the maximum control force is always applied to the structure. Future research will consider the use of on-off NLVFC together with feed-through term inclusion which has shown to improve stability margins of LVFC [27]. Acknowledgments The authors would like to acknowledge the financial support of Conserjerı´a de Educacio´n 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. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22]
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