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Abstract

In this paper, an enhanced integral force feedback controller is proposed for implementing active damping. Compared with the classical integral force feedback, the pure integrator is replaced with a combination of a first-order low-pass filter and a proportional term. The control performance of the proposed controller is examined on a single-degree-of-freedom host system. In order to better understand the physics behind, a full equivalent mechanical model is derived. It is found that the first-order low-pass filter is mechanically equivalent to a spring and a damper connected in parallel. The proportional term mechanically represents a spring which is connected in series with the inherent actuator spring. The optimal control parameters are numerically obtained with the ℋ_∞ optimisation criterion. The analysis shows that the optimal feedback gain can be practically taken the same as that of the classical integral force feedback controller in the sense that the difference between the resultant control performance is negligible. The numerical study is also experimentally validated. The obtained results are found to correspond well with the theoretical developments.

Keywords

H-infinity optimisation force feedback active damping

Introduction

From the standpoint of energy conservation as well as ultra-high performance, there is a prevailing inclination towards lightweight structures and integrated designs in contemporary technological and industrial developments.^1,2 However, lightweight materials are more prone to vibrations, which may cause many issues including impairment of the precision instruments and equipment, decreased accuracy in mechanical processing, detrimental effect on comfort, and even structural fatigue damage. Therefore, vibration reduction techniques should be considered during the design phase. Typical solutions such as damping and isolation are often found.³ Vibration isolation is to limit the propagation of disturbances to sensitive parts of the systems such that these key parts are less influenced.^4,5 It is often used for the cases where excessive response of the sensitive systems to the external excitations is concerned, but yield limited control effectiveness for mitigating the structural resonance peaks of the sensitive systems themselves.

In such scenarios, vibration damping should be implemented. Damping can be achieved passively, with viscoelastic materials, viscous fluids or eddy-currents, or by transferring kinetic energy to tuned mass dampers (TMDs).⁶ A TMD typically comprises a mass block, spring, and energy dissipation damping system as an auxiliary device.⁷ The damping effectiveness is known to ultimately depend on the weight of its proof mass, where better control performance comes with a heavier proof mass. However, the added mass may be penalising in lightweight applications. Piezoelectrical material thus becomes more attractive for implementing vibration damping. The mechanism is to bond piezoelectrical material, which is shunted with electrical networks, to the structures. When they deform, vibration energy is transformed into electrical energy and dissipated in electrical networks, or stored (energy harvesting). Piezoelectric transducers are also found to be equipped with switched electrical networks, which can be then featured as semi-passive dampers.^8,9

When high performance is needed, active damping can be used, which involves a set of sensors (strain, acceleration, velocity, and force), a set of actuators (force, inertial, and strain), and a control algorithm (feedback or feedforward). By active damping, the primary objective is to relocate the negative real parts of the closed-loop system poles such that the peak response of the system is suppressed. This strategy often requires relatively little control effort since only the control loop gain around the resonance frequency needs to be considered. Active damping can be implemented, for example, by directly feeding back velocity signals, or passing displacement signals through a second-order filter, or integrating acceleration signals, to drive control actuators.^10–13 However, in practice, flexibilities between actuators and sensors may introduce unwanted phase delays, which would limit the achieved damping performance.¹⁴

Integral force feedback (IFF) may exhibit advantages in this regard as collocated control plants (zeros and poles are interlacing) are more easily achieved with IFF control.³ By collocation, it means that each phase lag induced by the poles will be compensated by the phase lead of the same amount introduced by the adjacent zeros. One important design concern with an IFF control system is how to calculate the optimal feedback gain. Two kinds of optimisation criterion, the maximum damping criterion and the ℋ_∞ optimisation criterion, were adopted for designing an IFF controller.^15,16 The optimal feedback gain obtained with the maximum damping criterion is sought to obtain the most achievable damping for one specified structural mode, while that with the ℋ_∞ optimisation criterion is set to minimise the maximum steady state response of the structure. In order to further improve the control performance, several modified IFF controllers were proposed, where the single integrator is superimposed with a second-order integrator or fully replaced by a resonator.^17–19 However, more control parameters need to be configured which may jeopardise the robustness of the control system.

The focus of this study is thus to develop a modified force feedback controller which allows to further boost the control performance, but with less control parameters to be optimised. The proposed controller is built upon our previous developments.^16,18 A proportional term is added to the original controller which allows to adjust the effective stiffness of the actuator as in.^20,21 On top of this, a simple first-order low-pass filter is used to replace the pure integrator in the original controller which helps limit the saturation problems for example caused by the low-frequency drift of the instrument in practical applications. The ℋ_∞ optimisation criterion is utilised to examine the influence of the cut-off frequency of the first-order low-pass filter on the control performance. The mechanical equivalence of the components in the proposed controller is also derived in order to better understand the physics behind. Experimental validations are provided which verify the theoretical analysis. The principle contributions are as follows: (a) the development of the equivalent mechanical models which enables a straightforward interpretation of the physics behind the proposed controller, (b) the investigation of the influence of the cut-off frequency of the first-order low-pass filter on the control performance, and (c) the experimental validation of the theoretical analysis.

The paper is structured into four sections. In the second section, the mathematical models are presented, based on which the optimal feedback gains for implementing the controller are derived. In the third section, experimental results are presented for the validation of the numerical study. The conclusions are drawn in the last section.

Mathematical modelling and parameter optimisation

Mathematical modelling

The system under investigation is shown in Figure 1(a). It represents an undamped single-degree-of-freedom (SDOF) system which is equipped with a massless actuator whose stiffness is denoted by $k_{a}$ . The SDOF system is defined through the mass $m_{1}$ and the suspension stiffness $k_{1}$ . It is excited by a disturbance force $F$ . A collocated force sensor which measures the transmission force $F_{s}$ is installed between the actuator and the primary structure. The control loop is implemented by feeding the output of the force sensor $F_{s}$ through a controller $u (F_{s})$ to drive the actuator.

Figure 1.

(a) The schematics of the system under investigation and (b) the equivalent mechanical model.

The governing equations of the coupled system can be written as:

m_{1} \ddot{x} (t) + k_{1} x (t) = F (t) + F_{s} (t)

(1)

F_{s} (t) = u (F_{s} (t)) - k_{a} x (t)

(2)

The proposed controller $u (F_{s} (t))$ reads:

u (F_{s} (t)) = - g_{s} L_{1} (F_{s} (t)) - g_{0} F_{s} (t)

(3)

where

L_{1} (•)

represents a first-order low-pass filter,

g_{s}

, and

g_{0}

are the control gains.

Transforming equation (3) into Laplace domain, one obtains:

U (F_{s} (s)) = - g_{s} \frac{F_{s} (s)}{s + a} - g_{0} F_{s} (s)

(4)

where

a

is the corner frequency of the low-pass filters and

s

represents the Laplace variable.

In order to better understand the physics behind the proposed controller, the pure mechanical presentation of the coupled system, that is, the mechanical system coupled with the electrical system is derived, which is shown in Figure 1(b). The proof is provided in Appendix A.

As can be also seen in Figure 1(b), the use of the first-order low-pass filter introduces a spring $k_{d}$ in addition to the desired dashpot $d_{a}$ . The proportional term $g_{0} F_{s} (s)$ is equivalent to placing a spring in series with the inherent spring of the actuator so that the effective stiffness of actuator can be tuned. The effective stiffness of the actuator, denoted by $k_{a}^{'}$ , can be thus derived:

k_{a}^{'} = \frac{k_{a}}{1 + g_{0}}

(5)

When the gain $g_{0}$ is positive, the equivalent stiffness $k_{a}^{'}$ is reduced as two positive springs are connected in series. On the other hand, $k_{a}^{'}$ is increased assuming $- 1 < g_{0} < 0$ , which actually favours the vibration mitigation performance as demonstrated in.¹⁶ The underlying idea is similar to the concept of shunting negative capacitors to piezoelectric elements, thanks to which the generalised electromechanical coupling factor can be boosted leading to a better vibration mitigation performance.^9,22,23

The corresponding equivalent damping coefficient $d_{a}$ , the stiffness $k_{d}$ , and $k_{0}$ can be linked with the control parameters according to:

\begin{array}{l} d_{a} = \frac{k_{a}}{g_{s}}, & k_{d} = \frac{a k_{a}}{g_{s}}, & k_{0} = \frac{k_{a}}{g_{0}} \end{array}

(6)

Note that the corner frequency related parameters $k_{d}$ goes to zero when the corner frequency $a$ is set to zero.

Compared with the classical integral force feedback controller as proposed in,¹⁵ the pure integrator is now replaced by a first-order low-pass filter. In this way, the saturation problems due to the undesirable amplification of the control signal at low frequencies can be limited. The proportional term is introduced mainly to be able to tune the effective stiffness of the actuator as better control performance comes with a larger actuator stiffness.

Although the saturation problems can be suppressed by the proposed controller, the optical control gain g_s needs to be revisited in the present of the corner frequency of the low-pass filter. Before proceeding with the optimisation, a more general formulation is implemented with the following parameters:

\begin{array}{l} τ = ω_{1} t, & x_{1} = x, & x_{2} = F_{s} \ k_{1}, & x_{d} = F \ k_{1}, \\ Ω_{c} = \frac{a}{ω_{1}}, & μ = \frac{k_{a}}{(1 + g_{0}) k_{1}}, & ω_{1} = \sqrt{k_{1} \ m_{1}}, & g_{s n} = \frac{g_{s}}{(1 + g_{0}) ω_{1}} \end{array}

(7)

The system governing equations (1) and (2) are thus normalised:

{\ddot{x}}_{1} + x_{1} - x_{2} = x_{d}

(8)

{\dot{x}}_{2} + (g_{s n} + Ω_{c}) x_{2} + μ ({\dot{x}}_{1} + Ω_{c} x_{1}) = 0

(9)

where the derivatives are calculated with respect to the time scale

τ

as defined in equation (7).

Parameters optimisation

The ℋ_∞ norm is employed to optimise the control parameters of the proposed controller. The magnitude of the normalised driving-point receptance $| x_{1} / x_{d} |$ is taken as the performance index.

The normalised driving-point receptance of the primary structure can be derived according to equations (8) and (9):

\frac{x_{1}}{x_{d}} = \frac{S + Ω_{c} + g_{s n}}{S^{3} + (Ω_{c} + g_{s n}) S^{2} + (μ + 1) S + Ω_{c} (μ + 1) + g_{s n}}

(10)

and its magnitude is given as:

| \frac{x_{1}}{x_{d}} | = \frac{\sqrt{Ω^{2} + {(Ω_{c} + g_{s n})}^{2}}}{\sqrt{Ω^{2} {(Ω^{2} + μ + 1)}^{2} + {(Ω_{c} μ + (Ω_{c} + g_{s n}) (1 - Ω^{2}))}^{2}}}

(11)

where

S = s / ω_{1}

is the normalised Laplace variable and

Ω = S / j

is the normalised frequency.

The optimisation parameters could in principle be any of the three, that is, $Ω_{c}$ , $g_{s n}$ , and $μ$ in equation (11), but the stiffness ratio $μ$ is often constrained by the requirements on the change of the structure’s dynamics due to the installation of the active damping device and is thus considered as fixed. On the other hand, the role of the cut-off frequency $Ω_{c}$ , as shown in Figure 1(b), is to introduce an additional spring parallel to the damper. This spring $k_{d}$ is series with the inherent actuator spring which generally decreases the effective stiffness of the actuator, and thus degrades the control effectiveness. Therefore, the value of the cut-off frequency $Ω_{c}$ is suggested to be set relatively small compared with the resonance frequency to be damped. As such, neither $μ$ nor $Ω_{c}$ is considered a parameter to be optimised, and the remaining optimisation parameter is left with the normalised feedback gain $g_{s n}$ . However, an explicit expression for the optimal $g_{s n}$ cannot be obtained since the resultant equations to be resolved are hexic polynomials. A numerical ℋ_∞ optimisation (a gradient-based method implemented in MATLAB function fminsearch) is performed instead where optimal values of $g_{s n}$ is sought to minimise the resonance peaks for a pair of fixed stiffness ratio $μ$ and cut-off frequency $Ω_{c}$ . This procedure is repeated for a large variation of $μ$ and $Ω_{c}$ ( $\begin{array}{l} 0 < μ \leq 1; & 0 < Ω_{c} \leq 0.65 \end{array}$ ). Before showing the results, the optimal control gain under the $H_{\infty}$ criterion is revisited when the pure integrator is used, as in our previous work.¹⁶ The optimal control gain and the corresponding minimal maximum response can be derived as

g_{s n}^{o p t} = \sqrt{1 + \frac{μ}{2}}

(12)

{| \frac{x}{x_{d}} |}_{o p t} = \frac{2}{μ}

(13)

The optimal gain $g_{s n}$ obtained during t numerical ℋ_∞ optimisation is compared with that corresponding to the benchmark where a pure integrator is used. Figure 2(a) and (b) compare the optimal control gain ratio and the performance index ratio, respectively. It can be seen that visible difference only occurs when the stiffness ratio and the cut-off frequency are chosen relatively large. For the extreme case when $μ$ and $Ω_{c}$ approach 1 and 0.65, the estimated optimal control gain assuming no high-pass filters are used is less than the true optimal gain by 15%, but the difference in terms of the performance index is only around 1% which means that the estimated optimal gain is good enough in the sense of the control performance for most scenarios.

Figure 2.

The performance index against normalised frequency for different cut-off frequencies (left) and feedback gains (right) where the stiffness ratio $μ = 0.3$ is chosen.

In order to better understand how the proportional gain $g_{0}$ influences the control effectiveness, the control plant, defined as the transfer function between the output of the force sensor and the input to the actuator, is derived. $G (j Ω)$ represents its normalised frequency response function, which is expressed as:

G (j Ω) = \frac{1 - Ω^{2}}{(1 - Ω^{2} + μ) (1 + g_{0})}

(14)

Note that there exists a pair of complex zeros and a pair of complex poles as in the case of classical integral force feedback control. The distance between the location of zeros and poles is often used as a performance index to characterise the control effectiveness. When the proposed controller is used, the stiffness ratio $μ$ can be effectively adjusted by changing the proportional feedback gain $g_{0}$ according to equation (7). For example, a large value of $μ$ can be obtained when $g_{0}$ falls in the range $(- 1, 0]$ where $μ$ increases as $g_{0}$ approaches −1.

Since the proposed active damping device can be fully represented by a mechanical network (given idealised force sensors and actuators are employed and $g_{0} > - 1$ ), the gain margin of system is infinite. In order to investigate the phase margin, the open-loop gain $L = G (j Ω) H (j Ω)$ is derived, where $H (j Ω)$ represent the normalised frequency response controller. It is expressed as:

H (j Ω) = \frac{g_{s} (1 + g_{0})}{j Ω + Ω_{c}}

(15)

The phase margin is calculated as:

P M = π - L (j Ω_{u})

(16)

where

Ω_{u}

denotes the unit gain frequency at which the amplitude of the loop gain

L (j Ω)

equals to unity.

Figure 3 plots the phase margin of the coupled system against the stiffness ratio $μ$ when the optimal control parameters are applied. It can be seen that the phase margin increases with an increase in $μ$ . For a large range of the stiffness ratio $μ$ , the phase margin is greater than 80 degrees, which would be sufficient for most practical applications.

Figure 3.

Phase margin of the coupled system when the optical settings are applied.

In practice, the proposed controller is implemented in a cascade way. The control plant $G$ in equation (14) actually represents the transfer function between the output of the force sensor and the input to the actuator when the control loop with the proportional term is closed, whereas H in equation (15) stands for a first-order low-pass filter left in the cascaded controller. A first-order low-pass filter can be also known as a lag controller where the corresponding zero and pole are at infinite and the corner frequency $Ω_{c}$ , respectively. Its function is to introduce a phase lag of $π / 2$ after the corner frequency, which brings the phase of the loop gain in Figure 4(b) down away from $π$ .

Figure 4.

The frequency response function of (a) the control plant, (b) the open-loop gain, and (c) magnitude of the performance index for different values of $g_{0}$ .

In practical applications, the effective stiffness ratio $μ$ can be estimated according to equation (14) as:

μ = {(\frac{ω_{p}}{ω_{z}})}^{2} - 1 = {Ω_{p}}^{2} - 1

(17)

where

ω_{p}

and

ω_{z}

correspond to the frequencies of the pole and the zero of the control plant respectively, and

Ω_{p}

is the normalised frequency of the pole.

In the following, the controller effectiveness of the proportional term is illustrated. Figure 4(a) compares the control plant of the considered system when the gain $g_{0}$ is set to 0, −0.5 and −0.786, respectively. The effective stiffness ratio $μ$ for these values of $g_{0}$ can be calculated based on equation (7) and equals 0.3, 0.6, and 1.41. As shown, a pair of complex zeros and a pair of complex poles are presented for each scenario. The location of the zeros is invariant with respect to the gain $g_{0}$ since it is solely determined by the resonance frequency of the primary structure when the actuator is removed. On the other hand, the location of the poles shifts rightwards when $g_{0}$ moves towards to −1. In addition, the magnitude of the control plant is amplified by a factor of $1 / (1 + g_{0})$ .

In Ref. 21 a proportional controller is considered in order to boost the control performance of integral force feedback controllers. It is implemented as a feedforward term, while the proposed controller takes an opposite architecture where the proportional term is introduced in the feedback path. In order to compare the two controllers, the performance index introduced in this manuscript, that is, the magnitude of the normalised driving-point receptance $| x_{1} / x_{d} |$ is used.

With the controller in Ref. 21 the performance index is written as:

{| \frac{x_{1}}{x_{d}} |}_{F F} = \sqrt{\frac{{(β g + g)}^{2} + Ω^{2}}{{(- β g Ω^{2} + β μ_{0} g - g Ω^{2} + β g + g)}^{2} + {(- Ω^{3} + μ_{0} Ω + Ω)}^{2}}}

(18)

where the parameters

β

and

g

represent the gains associated with the proportional and the integral terms, respectively, and

μ_{0} = k_{a} / k_{1}

denotes the stiffness ratio between the actuator and the host structure.

Under the $H_{\infty}$ optimisation criterion, the optimal gain associated with the integral term, that is, g can be derived:

g_{o p t} = \frac{\sqrt{2 β (μ_{0} + 1) + μ_{0} + 2}}{(β + 1) \sqrt{2 (β + 1)}}

(19)

and the resultant optimal performance index is written as:

{| \frac{x_{1}}{x_{d}} |}_{F F_{o p t}} = \frac{2 β + 2}{μ_{0}}

(20)

The performance index decreases when the proportional gain $β$ is set to negative values. Comparing equations (20) and (13), it can be seen that the optimal performance index associated with the proposed controller in this manuscript is exactly the same to that associated with the controller in Ref. 21 given the corner frequency is equal to zero and $g_{0} = β$ . Note that the parameter $μ = μ_{0} / (1 + g_{0})$ in equation (13) denotes the effective stiffness ratio when the proportional feedback controller is applied.

In order to better understand the physics behind, the normalised control plant associated with Ref. 21 is derived:

G (j Ω) = \frac{1 + \frac{β}{1 + β} μ - Ω^{2}}{1 + μ - Ω^{2}}

(21)

As can be seen, the zeros of the control plant move leftwards when the parameter $β$ is set to a negative value between −1 and 0, while the location of the poles is kept fixed. For the proposed controller, the proportional term is added in a feedback path. This leads to the opposite phenomena that the zero of the control plant is fixed but the pole moves with the change of g₀. Although the control architectures and the optimal feedback gains are different, the optimal performance index keeps unchanged which lies in the fact that both controllers can succeed to increase the stiffness ratio between the actuator and the host structure.

Figure 4(b) plots the open-loop gain of the system for the three values of $g_{0}$ . As can be seen, the unit gain frequency increases with a decrease of $g_{0}$ . This leads to a greater phase margin which is in consistent with the phase margin shown in Figure 3. Figure 4(c) depicts the performance index for the same variation of $g_{0}$ . The peak values indeed decrease as $g_{0}$ approaches −1. This also demonstrates the validity of the optimal formulae in equation (12) for a large range of applications.

Figure 5 compares the resultant root loci of the system when applying different feedback gains of the proportional term. In consistence of the control plant as shown in Figure 4(a), the location of the poles moves away from that of the zeros when g₀ approaches to −1. As such that the most achieved damping also increases with a decrease of g₀ in the negative direction. When impulse excitations are concerned, it helps reduce the setting time of the transient response of the system. Note that the optimal gains for achieving the maximum damping and the minimal maximum steady state response of the structure are different as illustrated in Refs. 15,16.

Figure 5.

The root locus of the system for different g₀.

Experimental validation

Figure 6 shows the test bench used for experimental examination. The primary structure is a cantilever aluminium beam with the dimensions 45 cm*3 cm*0.3 cm (length*width*thickness). Two voice coil actuators (AVM24-10) are mounted at the free end of the cantilever beam. One of them is used to introduce the disturbance force and the other one is used to deliver the control force. According to the Lorentz law, the force delivered by an ideal voice coil actuator (its mass and damping are neglected) can be regulated by³:

f = - 2 π n r B i

(22)

where

i

is the input current,

B

the magnetic flux,

2 π r n

is the length exposed to the magnetic flux.

Figure 6.

The experimental test set-up: (1) cantilever beam, (2) eddy-current sensor (ECL101), (3) force sensor (PCB 221B02), (4) voice coil actuator (AVM24-10), (5) MicroLabBox, (6) current amplifier (ADD-45N), and (7) ICP conditioner (PCB 482C05).

In the theoretical analysis, the transfer function between the displacement and the disturbance force was chosen as the performance index. Therefore, an eddy-current sensor (ECL101) was used to measure the tip displacement of the beam, which is installed close to the voice coil actuators. In this study, only the first bending mode of the beam is considered such that the single-mode beam can dynamically represent a linear SDOF system. The block diagram of the control scheme is depicted in Figure 7. Underneath the control actuator, a force sensor (PCB 221B02) is installed in order to measure the transmission force and provide the feedback signal. However, additional damping is induced by the voice coil actuators (the eddy-current effect and air viscous damping effect), which violates the low damping assumption of the primary structure. Thus, a negative damping control loop was implemented in order to eliminate the total inherent damping of the system. This was achieved by positively feeding the tip velocity signal back to drive the voice coil actuators. The velocity signal is obtained by passing the displacement signal measured by the eddy-current sensor through a first-order high-pass filter with a corner frequency of 500 Hz. The high-pass filter is used in order to avoid the over-injection of the sensor noise in the high-frequency range. The feedback gain of this negative damping loop is denoted as $g_{n e g}$ . Note that the force sensor is placed underneath the magnet of the voice coil actuator which is different to the proposed configuration in Figure 1(a). This is done mainly to avoid adding too much weight to the beam. Nevertheless, the actuator actually delivers a pair of reactive force which on one hand acts on the beam and on the other hand on the ground. Therefore, the force sensor in the current configuration measures the same transmission force as it would be in the configuration as indicated in Figure 1(a) given the weight of the actuator is negligible compared to that of the host structure. For the considered test bench, this assumption makes sense since only the coil part of the actuator is attached to the beam.

Figure 7.

The configuration scheme of the experimental set-up.

A dSpace MicroLabBox system was used both for the data acquisition and the control implementation. The control scheme was implemented in the Matlab Simulink environment and then downloaded to the processor unit of the MicroLabBox system. The control scheme was updated at a sampling frequency of 10 kHz, and the measured data were recorded at the same sampling frequency. A current amplifier (ADD-45N) was used to drive the voice coil actuators.

For the considered set-up or future applications, the optimal value of the control gain can be calculated with the knowledge of the effective stiffness ratio between the actuator and the host structure. Practically, the stiffness ratio can be directly estimated by performing a measurement of the control plant. Based on the measured control plant, the locations of the zeros and the poles as well as the DC gain can be extracted. Before further proceeding, one should also pay attenuation to the modal interactions between the targeted mode and the other adjacent modes. For the considered set-up, the first mode of the cantilever beam is chosen for the experimental validation. In such a case, the contributions from the higher modes can be characterised as a quasi-static gain,²⁴ which needs to be taken into account when estimating the effective stiffness ratio. More precisely, the DC gain defined as the response of the control plant at 0 Hz, that is, $G (0)$ is only determined by the parameters $μ$ and $g_{0}$ given in equation (14) for a SDOF system. While for the equivalent SDOF system, the DC gain is also influenced by the contributions from the higher modes. Therefore, this difference referring to as interaction correction factor (ICF) needs to be taken into account. The derived optimal gain as in equation (12) needs to be calibrated by multiplying with the ICF in order to eliminate the influence of the mode interactions.

Finally, the identified stiffness ratio $μ$ , the ICF, and the frequency of the zero (corresponding to the resonance of the primary structure when the actuator is removed, that is, $ω_{1}$ in equation (7)) are substituted into equation (12), to obtain the optimal feedback gain $g_{s}$ for implementing the proposed controller defined in equation (4).

The optimal control parameters for different values of $g_{0}$ are experimental examined. A white noise signal was applied to excite the beam in the vicinity of its first bending mode. The duration of the measurement was set to 200 seconds.

Figure 8(a) plots the frequency response of the control plant when $g_{0}$ is set to 0, −0.5, and −0.786, respectively. The stiffness ratio, ICF, and the optimal control parameters are summarised in Table 1. Note that the coherence of the control plant is relatively low when $g_{0}$ is set to −0.786. This is because the magnitude of the control plant is amplified by a factor of $1 / (1 + g_{0})$ , and the input disturbance force has to be limited to avoid an overwhelming response around the resonance frequency. As a consequence, the signal-to-noise ratio is low at other frequencies leading to a low coherence. Nevertheless, the optimal control parameters can still be accurately estimated. Figure 8(b) depicts the driving-point receptance (the transfer function between the disturbance force and the measured tip displacement). As expected, a better control effectiveness is achieved when $g_{0}$ approaches −1. The comparison between Figures 8 and 4 shows that the experimental results are in a good accordance with the theoretical analysis.

Figure 8.

The frequency response of (a) the control plant and (b) magnitude of the performance index for different values of $g_{0}$ .

Table 1.

Control parameters.

Parameters	$g_{0} = 0$	$g_{0} = - 0.492$	$g_{0} = - 0.786$
$μ$	0.307	0.604	1.425
ICF	0.82	0.814	0.778
$g_{d}$	3652.1	2128.1	1181.6
$a$	6.28	6.28	6.28
$g_{n e g}$	850	850	850

Conclusion

This paper discusses an enhanced integral force feedback controller which consists of a first-order low-pass filter and a proportional term. The physics behind is explicitly explained via an equivalent mechanical model. The first-order low-pass filter mimics the dynamic behaviour of a pure mechanical network which comprises a spring and a damper connected in parallel, while the proportional term plays the same role as a mechanical spring connected in series with the inherent actuator spring. The active damping is provided essentially by the first-order low-pass filter where the cut-off frequency is required to be smaller than the target frequency to be damped. By cascading the proportional controller, a relatively large stiffness ratio can be implemented given that the associated gain is properly tuned. Compared with the classical IFF controller, the proposed control can further improve the vibration mitigation performance and also suppress the low-frequency saturation induced by the integration operation. The numerical study, where the optimal feedback gain is calculated given a fixed pair of stiffness ratio and the cut-off frequency of the first-order low-pass filter, shows that the optimal feedback gain can be taken the same as that of the IFF controller in the sense the performance index is concerned. The obtained results were found to be in excellent agreement with the experimental tests. With these results, one can envision to develop a simple analogue electronic control system for a collocated actuator-force sensor pair such that the device would be compact enough for smart structure applications.

Footnotes

Acknowledgements

The financial supports from the Science Fund Program for Distinguished Young Scholars (RLZY20231001-10), Zhujiang Talent Plan (2021QN02Z097), the Specialized Fund for the Basic Research Operating Expenses Program of SYSU (23hytd004), and National Key R&D Program of China (Grant No. 2022YFC2204000) are gratefully acknowledged.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the Zhujiang Talent Plan (2021QN02Z097), National Key R&D Program of China (2022YFC2204000), National Natural Science Foundation of China (RLZY20231001-10), and Specialized Fund for the Basic Research Operating Expenses Program of SYSU (23hytd004).

ORCID iD

Guoying Zhao

Appendix

In this appendix, it is to prove that the systems sketched in Figure 1(a) and (b) are dynamically equivalent.

Figure A.1.

(a) the sketch of the active system and (b) its mechanical representative.

Figure A.1(b) depicts a pure mechanical system which consists of several springs and dashpots as well as an inerter. Under the excitation force denoted by $F$ , the governing equations of this system can be written as: (A.1)

F = - d_{a} ({\dot{x}}_{1} - {\dot{x}}_{2}) - k_{d} (x_{1} - x_{2})

(A.2)

F = - k_{0} (x_{2} - x_{3})

(A.3)

F = - k_{a} (x_{3} - x_{4})

Expressing the relative motion in terms of the transmission force $F$ , equations (A.1)–(A.3) can be rewritten as: (A.4)

x_{1} - x_{2} = \frac{- F}{d_{a} s + k_{d}}

(A.5)

x_{2} - x_{3} = - \frac{F}{k_{0}}

(A.6)

x_{3} - x_{4} = - \frac{F}{k_{a}}

Summing up equations (A.4)–(A.6) and multiplying both sides with $k_{a}$ , yields: (A.7)

k_{a} (x_{1} - x_{5}) = - \frac{k_{a} F}{d_{a} s + k_{d}} - \frac{k_{a} F}{k_{0}} - F

According to the control law, the governing equations of the system shown in Figure A.1(a) can be expressed as: (A.8)

F = - g_{s} \frac{F_{s} (s)}{s + a} - g_{0} F_{s} (s) - k_{a} x

where

x = x_{1} - x_{4}

represents the relative displacement across the mechanical network and

F_{s} = F

Comparing equation (A.7) with equation (A.8) and one can find the equivalence between systems shown in Figure A.1(a) and (b). The feedback gains and their corresponding mechanical components are thus related by equation (A.9): (A.9)

\begin{array}{l} d_{a} = \frac{k_{a}}{g_{s}}, & k_{d} = \frac{a k_{a}}{g_{s}}, & k_{0} = \frac{k_{a}}{g_{0}} \end{array}

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Experimental study on an enhanced integral force feedback controller for active damping

Abstract

Keywords

Introduction

Mathematical modelling and parameter optimisation

Mathematical modelling

Parameters optimisation

Experimental validation

Conclusion

Footnotes

Acknowledgements

Declaration of conflicting interests

Funding

ORCID iD

Appendix

References