# Active Vibration Control of Helicopter Maneuvering Flight Using Feedforward-Robust Hybrid Control Based on Reference Signal Reconstruction

Authors: Yifan Qin; Yang Lu; Jinchao Ma; Huiyu Yue
Journal: Shock and Vibration
 (2021)

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## Abstract

Current control laws for active control of helicopter structural vibration are designed for steady-state flight conditions, while the vibration response of maneuvering flight has not been taken into consideration yet. In order to obtain full-time vibration suppression capability, the authors propose a filtered least mean square-mixed sensitivity robust control method based on reference signal reconstruction (LMS-MSRC), driving piezoelectric stack actuators to suppress helicopter structural vibration response in maneuvering flight. When feedback controller designed byH∞ theory is implemented, active damping is added on the secondary path to weaken the adverse effects of its sudden changes in maneuvering flight state. Furthermore, a reference signal reconstruction scheme is given concerning equivalent secondary path. In addition, the reconstruction accuracy, the convergence speed, stability, and global validity of the hybrid controller are analysed. Compared with multichannel Fx-LMS, numerical simulations of LMS-MSRC for vibration suppression are undertaken with a helicopter simplified finite element model under several typical flight conditions. Further experiments of real-time free-free beam vibration control are performed, driven by a stacked piezoelectric actuator. The instantaneous overshoot of measured response is 42% less than the peak value and its attenuation reaches 85% within 2.5 s. Numerical and experimental results reveal that the proposed algorithm is practical for suppressing transient disturbance and multifrequency helicopter vibration response during maneuvering flight with faster convergence speed and better robustness.


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## Body

## 1. Introduction

Excessive helicopter structural vibration imposes discomfort on passengers and crew members both physically and mentally. Rotor aerodynamic load is one of the main sources of fuselage vibration [1]. The alternating load and moment generated in the hub center are transmitted to the fuselage through the rotor control system and the gearbox, thereby causing harmonic vibration. In order to suppress the airframe vibration, a lot of researches have been carried out on the active helicopter vibration control and the active control of structural response (ACSR) system developed at present has been successfully applied to stereotyped helicopters, such as EH-101, S-92, EC225, and UH-60M [2–5]. Various active control algorithms have shown good performance in practice.However, conventional helicopter ACSR systems are only applicable for the steady-state vibration components; little attention is paid to the active vibration suppression during maneuvering flight [2–5]. When a helicopter performs a large-overload maneuver, the alternating aerodynamic load of the rotor changes drastically in a short time and results in severe vibration [6, 7]. Furthermore, flight tests demonstrate that the rotor hub torque changes transiently so as to produce additional uncertain disturbances, which are difficult to model and bring a nonnegligible impact on the fuselage vibration level [7, 8]. Figure 1 shows measured forward flight speed and acceleration response of a CH-53G helicopter during maneuvering flight [9]. It can be seen that the vibration load of blade passing frequency changes drastically when the helicopter performs acceleration and deceleration maneuvers in 40 s–60 s. In addition, instantaneous power demand changes in maneuvering flight also causes the rotor speed fluctuating significantly in a short time. Figure 2 illustrates vibration frequency oscillation when Bell 427 performed a left-turn maneuver [10]. The real-time vibration frequency may deviate from the target value (22.4 Hz) and can be reduced up to 5%. On one hand, the control mechanism of the current ACSR systems limits the vibration suppression effects. On the other hand, the commonly used mechanical inertial actuators, such as electromagnetic actuators and circular force generators, have the shortcomings of narrow working frequency bands and slow response to instructions, respectively, which reduce the efficiency of ACSR systems severely [4, 11, 12]. For better adaptation to the working environment of helicopter maneuvering flight, research should be carried out on both actuator and control law design.Figure 1
Measured acceleration response of a CH-53G helicopter during maneuvering flight.Figure 2
Oscillation in the vibration frequency of a Bell 427 during a left-turn maneuver.In recent years, piezoelectric stack actuator (PSA) has been widely used in helicopter active vibration control due to its advantages of large driving force, small mass and volume, wide frequency bandwidth, and fast response. Straub et al. used PSA to successfully realize the active vibration suppression of the full-scale SMART rotor [13]. Heverly et al. studied the working principle and distribution of PSAs based on the elastic finite element model of AH-64 scaled airframe comprehensively [14]. However, the characteristics of PSAs and their coupling effects with fuselage will change the actual secondary path in practice, imposing a negative impact on the active vibration suppression [14]. Viswamurthy and Ganguli pointed out that the modeling error of PSA could make it hard to predict the optimal input of the actual secondary path and weaken control performance [15]. To improve the control effects of PSAs in the helicopter ACSR during maneuvering flight, it is necessary to consider robustness to actual secondary path changes in the controller design.Feedback control based on high-order harmonic control (HHC) and least mean square feedforward (Fx-LMS) control based on adaptive filtering technology are two control methods widely used in helicopter ACSR [16, 17]. Based on linear and quasistatic assumptions, HHC can effectively suppress steady-state harmonic vibration in the frequency domain. But it is inevitable to perform discrete Fourier transform and inverse transform, HHC has a larger control interval and a slower update rate, making it suitable only for the situation where the external disturbance is relatively stable or secondary path changes barely [18]. In view of the shortcomings of traditional HHC, Zhao and Song used least mean square and recursive least square algorithm, respectively, to update harmonic parameters [19, 20]. However, when there are frequency errors between the basis function and the excitations, the control system converges slowly [21]. Furthermore, the adverse impact of the secondary path mutation during maneuvering flight on vibration control is not considered.Fx-LMS is an adaptive feedforward controller with good narrowband control effects. It shows strong signal tracking ability without requirement for accurate model and can be applied to the active vibration control of various helicopter flight states theoretically. However, the performance of Fx-LMS is directly influenced by the identification accuracy of secondary path and coherence between the reference signal and vibration signal [22]. The modeling error generated by secondary path mutation will decrease the adaptive ability of Fx-LMS and may even cause the control system to fail [23, 24]. Moreover, for the vibration response with drastic changes in frequency and amplitude during maneuvering flight, with rotational speed signal used as reference signal directly, the coherence reduces so that the feedforward controller may not guarantee causal response and convergence speed will decrease [22, 25]. Considering the short duration of maneuver, it is necessary to find a new method with robustness enhancement and more accurate reference signal to enable the system to obtain stronger uncertain disturbance suppression performance.For working out the problem that reference signal is in poor consistency with the uncontrolled response and the characteristics of the secondary path time vary, this paper proposes a filtered Least Mean Square-Mixed Sensitivity Robust Controller (LMS-MSRC) based on reconstructed reference signal, which drives PSAs to realize helicopter ACSR in maneuvering flight. The reference signal reconstructed by error signal can track the real-time changes of the uncontrolled response, which is beneficial for faster convergence speed. A feedback controller specially is designed byH∞ theory, and by the introduction of appropriate cost functions and weighting functions, the time-varying property of the secondary path is transformed into a robust performance design problem with uncertain perturbation output to solve, which enhances the robustness and convergence of the system.The main contents of this paper are arranged as follows: the second section proposes specific improvement measures to establish the ACSR hybrid algorithm, where tracking accuracy of the reference signal reconstruction, system convergence and stability, and the global validity are analysed. A series of ACSR numerical simulations to verify the proposed controller in helicopter maneuvering flight status based on a simplified helicopter finite element model are presented in the third section. The fourth section carries out ACSR experimental research based on the free-free beam structure. Some conclusions are drawn from work of this paper in the last section.


## 2. Feedforward-Robust Hybrid Algorithm Based on Reference Signal Reconstruction

To better describe the proposed method, the classic Fx-LMS algorithm of helicopter ACSR is introduced and the deficiencies of its application in maneuvering flight are discussed. Furthermore, improvement measures are put forward to establish a hybrid control system and their advantages are analysed.


### 2.1. Classic Helicopter ACSR Fx-LMS Algorithm

As is shown in Figure3, the reference signal xn is a multifrequency signal related to the rotor excitation force, usually provided by the rotor speed. Pz is the primary path that represents the physical transfer function between the rotor excitation and the error sensor at the controlled point (e.g., the pilot’s position). dn denotes the narrowband desired response. d0n is the transient multifrequency disturbance that is undetectable or hard to model. Sz indicates the actual secondary path transfer function, including D/A, driving amplifier, actuator, dynamic characteristics between actuator and error sensor, filter, and A/D converter. x′n is the filtered reference signal after xn filtered by the secondary path offline estimation S^z. en shows the error response at the controlled point. Wz is an FIR filter of order I, which outputs the active control signal yfk.Figure 3
Block diagram of the classic helicopter ACSR Fx-LMS algorithm.The error signalen is expressed as follows:(1)en=dn+d0n−yn,where yn=Sz∗Yfn, Yfn=yfn⋯yfn−J+1T.According to the steepest descent principle, the weight coefficient vectorwn of Wz is updated to adapt to real-time changes:(2)wn+1=wn−2μenX′n,where X′n=x′n⋯x′n−I+1T, x′n=S^z∗Xn, and μ is the step factor of control process. From equations (1)-(2), it is revealed that the reference signal xn and the actual secondary path Sz play an important role in updating wn. In helicopter maneuvering flight, xn hardly provides effective information about external excitation changes, leading to a decrease in system convergence. Moreover, the system stability problem caused by time-varying secondary path Sz is not settled in maneuvering flight.


### 2.2. Proposed Algorithm

To overcome the above disadvantages, a feedforward-robust hybrid algorithm based on reference signal reconstruction is raised. The algorithm diagram is shown in Figure4. For the purpose of robustness enhancement to the uncertainty of the secondary path, a feedback controller is designed based on the H∞ robust theory, which increases damping to improve stability and obtains overall vibration attenuation at the same time. Moreover, for the constructed equivalent secondary path, the reference signal is reconstructed based on the error signal, and the real-time excitation changes are fed back effectively to improve convergence performance. Detailed descriptions of the mixed sensitivity robust controller and reference signal reconstruction are as follows.Figure 4
Sketch diagram of LMS-MSRC algorithm based on reference signal reconstruction. The red box describes the equivalent secondary path, and the purple one shows reconstruction of reference signal.


#### 2.2.1. Mixed Sensitivity Robust Controller

The secondary path that changes drastically in a short time poses a threat to the stability of Fx-LMS during helicopter maneuvering flight. As is shown in Figure5, the secondary path mutation ΔStz during maneuvering flight is regarded as uncertain parameter perturbation, including modeling errors caused by helicopter dynamic changes and unmodeled dynamics in maneuvering flight. The design of mixed sensitivity robust controller is to convert the controller design problem of system perturbation into the robust performance design problem of uncertain parameter system. The robust stabilization problem is solved by selecting appropriate performance and weighting functions so as to obtain a suitable controller, reducing the sensitivity to changes in secondary path and increasing the system damping simultaneously.Figure 5
Hybrid sensitivityH∞ feedback control system based on multiplicative uncertainty.Multiplicative output uncertainty is used to represent the actual secondary pathSz. Snomz denotes the nominal secondary path when ΔStz=0, which is generally identified by experiment. Kz is the designed controller. en, ybn, and dn+d0n are error, control signal, and disturbance, respectively. z1n and z2n are the observation output. Therefore, the actual secondary path is(3)Sz=Snomz×1+ΔStz.According to [26], the control system is robust if it maintains stability and complies with certain performance criteria in the presence of uncertainty. As we can see from Figure 5, the transfer function from disturbance signal dn+d0n to en is(4)Hedz=−1+SzKz−1.If disturbance suppression performance of the closed-loop system is required to be as good as possible to suppress the response perturbation ofdn+d0n to en caused by secondary path change ΔStz, Kz needs to meet the following requirements [27]:(5)ΔStz×1+SnomzKz−1∞≤1.Or in a strengthened form:(6)1+SnomzKz−1∞≤1ΔStz∞.In the sense of ∞-norm, if it is necessary to maintain sufficient disturbance suppression performance for the largest possible perturbationsΔStz, we need to solve the following minimization problem to find Kz:(7)minKstabilizing1+SnomzKz−1∞.In many cases, we may have a priori knowledge of the perturbation, and the perturbation block can be exhibited as follows:(8)ΔStz=Δ˜StzW1z,where Δ˜Stz is the unit norm perturbation set and weighting function. W1z is designed to satisfy disturbance suppression demands. At this time, the suppression target is changed from en to z1n. Correspondingly, the robust performance condition becomes(9)W1z1+SnomzKz−1∞≤1Δ˜Stz∞.The optimization problem turns into(10)minKstabilizingW1z1+SnomzKz−1∞.In engineering practice, a combination of various cost functions should be included in a reasonable design. For example, disturbance suppression and limiting control signal energy are meaningful. As is illustrated in Figure5, the above two goals are considered at the same time through mixed sensitivity controller design.(11)minKstabilizingW1z×HedzW2z×Hydz∞=W1z1+SnomzKz−1W2zKz1+SnomzKz−1∞,where Hydz=Kz×1+SzKz−1 represents the transfer function from disturbance signal dn+d0n to control signal ybn.


#### 2.2.2. Reference Signal Reconstruction

To solve the problem that the speed signal is difficult to reflect the real-time changes of disturbance amplitude, it is a better choice to directly use the feedforward component of the uncontrolled responsedn+d0n to update Wz. However, since the actuator works continuously, it is hard to obtain the dn+d0n from the control system in real time. At this moment, the feedback neutralization method can be used to obtain the reconstructed reference signal based on the error signal [28]. The reconstruction process is shown in Figure 4.(12)Seqz=Sz1+SzKz,S^eqz is a FIR filter of the equivalent secondary path Seqz, which is identified offline, and its order is J. Through en and Yfn obtained in real time, the reference signal reconstruction process is(13)d^eqn=en+S^eqn∗Yfn,where d^eqn is the reconstructed reference signal, representing the component that needs to be suppressed by feedforward control, that is, the vibration response only under the action of the robust feedback controller Kz.The proposed hybrid control law can be summarized as follows:(14)yfn=wn∗D^eqn,ycn=yfn+ybn,en=dn+d0n−sn∗Ycn,d^eqn=en+s^eqn∗Yfn,x′n=s^eqn∗D^eqn,wn+1=wn+2μenD^eqn.


### 2.3. Performance Analysis

To satisfy the requirements of helicopter ACSR in maneuvering flight, the advantages of the proposed hybrid system in terms of reference signal reconstruction accuracy, control system convergence and stability, and global effectiveness are analysed.


#### 2.3.1. Reference Signal Reconstruction Accuracy

It can be seen from equation (13) that the reconstruction error of d^eqn mainly depends on the modeling error between the offline identification of the equivalent secondary path and the actual. When Sz changes during maneuvering flight, the equivalent secondary path Seqn is expressed as equation (12). Assuming the offline identification of Sz is S^z=Snomz, the error compared to Sz is(15)ΔS^z=Sz−S^z=SnomzΔStz.Converting equation (15) in the form of sensitivity, we get(16)ΔS^zS^z=ΔStz.WhenStz≠0, the reconstruction error of the reference signal increases, and the hybrid control may diverge. With the robust feedback controller Kz introduced, the increment of equivalent secondary path Seqn estimated offline is as follows:(17)ΔS^eqz=Snomz1+ΔStz1+Snomz1+ΔStzKz−Snomz1+SnomzKz.Converting equation (17) into the form of sensitivity, we get(18)ΔS^eqzS^eqz=ΔStzGz,where(19)Gz=11+S^eqnKz.WhenS^eqzKz≫1, then Gz≪1. Compared with the case without Kz, the influence of the model error ΔS^eqz on S^eqz declines at this time. In terms of the time-varying secondary path, the reference signal reconstruction error is reduced, which is beneficial to effectively track the changes of the external disturbance response, and the stability of the hybrid control system is enhanced.


#### 2.3.2. Convergence and Stability Analysis

As is illustrated in equation (14), according to the reconstructed reference signal d^eqn, the recursive formula of the filter coefficient wn turns into(20)wn+1=wn+2μenD^eqn.Compared to speed signal,d^eqn enhances the coherence between reference signal and uncontrolled responses, which means that wn needs to pay less adjustment cost to adapt to drastically changing response during maneuvering flight, leading to faster convergence of the control system.Furthermore, according to Diniz [29], the stability condition of Fx-LMS algorithm is(21)0<μ<1trR,where R is the autocorrelation matrix of filtered reference signal and trR is its trace, which satisfies(22)R=ED^eqnD^eqTn=d^eq2nd^eqndeq′n−1⋯d^eqnd^eqn−I+1d^eqn−1deq′nd^eq2n−1⋯d^eqn−1d^eqn−I+1⋮⋮⋱⋮d^eqn−I+1deqnd^eqn−I+1deqn−1⋯d^eq2n−I+1,(23)trR=∑i=0I−1d^eq2i.According to equations (21)–(23), if Sz dynamics is low-damped and the impulse response of x′n is large with long duration, trR will increase and the upper limit of μ will decrease, making system convergence and stability reduce. However, after the robust controller Kz is introduced, S^z turns into the offline estimation S^eqz of the equivalent secondary path Sz/1+SzKz. Owing to 1+SzKz≫1, S^eqz≪S^z, then the impulse response of d^eqn is small and short, contributing to small trR and little μ. In fact, Seqz provides higher damping than Sz, which is beneficial for convergence and stability enhancement.


#### 2.3.3. Global Validity

From Figure4, the vibration response after control is(24)Ez=11+SzKzDz+D0z+SzWz1+SzKzD^eqz,where Ez, Dz, D0z, and D^eqz denote the z-transforms of en, dn, d0n, and d^eqn. If designed Kz can ensure system stability and satisfy 1+SzKz≫1 in the target frequency band, for the drastically changing desired signal dn+d0k under maneuvering flight conditions, the feedback loop can still control it to a small level (but not to 0), even if Wz=0. If the filter is adjusted to Wz=−S−1zDz+D0zD^eq−1z, global vibration reduction can be achieved theoretically. Therefore, the hybrid control system indicates advantages over either independent feedback or feedforward control when there is transient disturbance: the former is responsible for overall response reduction, while the latter helps to offset the vibration residuals. The complementarity is effective for realizing vibration control of maneuvering flying helicopters.


## 2.1. Classic Helicopter ACSR Fx-LMS Algorithm

As is shown in Figure3, the reference signal xn is a multifrequency signal related to the rotor excitation force, usually provided by the rotor speed. Pz is the primary path that represents the physical transfer function between the rotor excitation and the error sensor at the controlled point (e.g., the pilot’s position). dn denotes the narrowband desired response. d0n is the transient multifrequency disturbance that is undetectable or hard to model. Sz indicates the actual secondary path transfer function, including D/A, driving amplifier, actuator, dynamic characteristics between actuator and error sensor, filter, and A/D converter. x′n is the filtered reference signal after xn filtered by the secondary path offline estimation S^z. en shows the error response at the controlled point. Wz is an FIR filter of order I, which outputs the active control signal yfk.Figure 3
Block diagram of the classic helicopter ACSR Fx-LMS algorithm.The error signalen is expressed as follows:(1)en=dn+d0n−yn,where yn=Sz∗Yfn, Yfn=yfn⋯yfn−J+1T.According to the steepest descent principle, the weight coefficient vectorwn of Wz is updated to adapt to real-time changes:(2)wn+1=wn−2μenX′n,where X′n=x′n⋯x′n−I+1T, x′n=S^z∗Xn, and μ is the step factor of control process. From equations (1)-(2), it is revealed that the reference signal xn and the actual secondary path Sz play an important role in updating wn. In helicopter maneuvering flight, xn hardly provides effective information about external excitation changes, leading to a decrease in system convergence. Moreover, the system stability problem caused by time-varying secondary path Sz is not settled in maneuvering flight.


## 2.2. Proposed Algorithm

To overcome the above disadvantages, a feedforward-robust hybrid algorithm based on reference signal reconstruction is raised. The algorithm diagram is shown in Figure4. For the purpose of robustness enhancement to the uncertainty of the secondary path, a feedback controller is designed based on the H∞ robust theory, which increases damping to improve stability and obtains overall vibration attenuation at the same time. Moreover, for the constructed equivalent secondary path, the reference signal is reconstructed based on the error signal, and the real-time excitation changes are fed back effectively to improve convergence performance. Detailed descriptions of the mixed sensitivity robust controller and reference signal reconstruction are as follows.Figure 4
Sketch diagram of LMS-MSRC algorithm based on reference signal reconstruction. The red box describes the equivalent secondary path, and the purple one shows reconstruction of reference signal.


### 2.2.1. Mixed Sensitivity Robust Controller

The secondary path that changes drastically in a short time poses a threat to the stability of Fx-LMS during helicopter maneuvering flight. As is shown in Figure5, the secondary path mutation ΔStz during maneuvering flight is regarded as uncertain parameter perturbation, including modeling errors caused by helicopter dynamic changes and unmodeled dynamics in maneuvering flight. The design of mixed sensitivity robust controller is to convert the controller design problem of system perturbation into the robust performance design problem of uncertain parameter system. The robust stabilization problem is solved by selecting appropriate performance and weighting functions so as to obtain a suitable controller, reducing the sensitivity to changes in secondary path and increasing the system damping simultaneously.Figure 5
Hybrid sensitivityH∞ feedback control system based on multiplicative uncertainty.Multiplicative output uncertainty is used to represent the actual secondary pathSz. Snomz denotes the nominal secondary path when ΔStz=0, which is generally identified by experiment. Kz is the designed controller. en, ybn, and dn+d0n are error, control signal, and disturbance, respectively. z1n and z2n are the observation output. Therefore, the actual secondary path is(3)Sz=Snomz×1+ΔStz.According to [26], the control system is robust if it maintains stability and complies with certain performance criteria in the presence of uncertainty. As we can see from Figure 5, the transfer function from disturbance signal dn+d0n to en is(4)Hedz=−1+SzKz−1.If disturbance suppression performance of the closed-loop system is required to be as good as possible to suppress the response perturbation ofdn+d0n to en caused by secondary path change ΔStz, Kz needs to meet the following requirements [27]:(5)ΔStz×1+SnomzKz−1∞≤1.Or in a strengthened form:(6)1+SnomzKz−1∞≤1ΔStz∞.In the sense of ∞-norm, if it is necessary to maintain sufficient disturbance suppression performance for the largest possible perturbationsΔStz, we need to solve the following minimization problem to find Kz:(7)minKstabilizing1+SnomzKz−1∞.In many cases, we may have a priori knowledge of the perturbation, and the perturbation block can be exhibited as follows:(8)ΔStz=Δ˜StzW1z,where Δ˜Stz is the unit norm perturbation set and weighting function. W1z is designed to satisfy disturbance suppression demands. At this time, the suppression target is changed from en to z1n. Correspondingly, the robust performance condition becomes(9)W1z1+SnomzKz−1∞≤1Δ˜Stz∞.The optimization problem turns into(10)minKstabilizingW1z1+SnomzKz−1∞.In engineering practice, a combination of various cost functions should be included in a reasonable design. For example, disturbance suppression and limiting control signal energy are meaningful. As is illustrated in Figure5, the above two goals are considered at the same time through mixed sensitivity controller design.(11)minKstabilizingW1z×HedzW2z×Hydz∞=W1z1+SnomzKz−1W2zKz1+SnomzKz−1∞,where Hydz=Kz×1+SzKz−1 represents the transfer function from disturbance signal dn+d0n to control signal ybn.


### 2.2.2. Reference Signal Reconstruction

To solve the problem that the speed signal is difficult to reflect the real-time changes of disturbance amplitude, it is a better choice to directly use the feedforward component of the uncontrolled responsedn+d0n to update Wz. However, since the actuator works continuously, it is hard to obtain the dn+d0n from the control system in real time. At this moment, the feedback neutralization method can be used to obtain the reconstructed reference signal based on the error signal [28]. The reconstruction process is shown in Figure 4.(12)Seqz=Sz1+SzKz,S^eqz is a FIR filter of the equivalent secondary path Seqz, which is identified offline, and its order is J. Through en and Yfn obtained in real time, the reference signal reconstruction process is(13)d^eqn=en+S^eqn∗Yfn,where d^eqn is the reconstructed reference signal, representing the component that needs to be suppressed by feedforward control, that is, the vibration response only under the action of the robust feedback controller Kz.The proposed hybrid control law can be summarized as follows:(14)yfn=wn∗D^eqn,ycn=yfn+ybn,en=dn+d0n−sn∗Ycn,d^eqn=en+s^eqn∗Yfn,x′n=s^eqn∗D^eqn,wn+1=wn+2μenD^eqn.


## 2.2.1. Mixed Sensitivity Robust Controller

The secondary path that changes drastically in a short time poses a threat to the stability of Fx-LMS during helicopter maneuvering flight. As is shown in Figure5, the secondary path mutation ΔStz during maneuvering flight is regarded as uncertain parameter perturbation, including modeling errors caused by helicopter dynamic changes and unmodeled dynamics in maneuvering flight. The design of mixed sensitivity robust controller is to convert the controller design problem of system perturbation into the robust performance design problem of uncertain parameter system. The robust stabilization problem is solved by selecting appropriate performance and weighting functions so as to obtain a suitable controller, reducing the sensitivity to changes in secondary path and increasing the system damping simultaneously.Figure 5
Hybrid sensitivityH∞ feedback control system based on multiplicative uncertainty.Multiplicative output uncertainty is used to represent the actual secondary pathSz. Snomz denotes the nominal secondary path when ΔStz=0, which is generally identified by experiment. Kz is the designed controller. en, ybn, and dn+d0n are error, control signal, and disturbance, respectively. z1n and z2n are the observation output. Therefore, the actual secondary path is(3)Sz=Snomz×1+ΔStz.According to [26], the control system is robust if it maintains stability and complies with certain performance criteria in the presence of uncertainty. As we can see from Figure 5, the transfer function from disturbance signal dn+d0n to en is(4)Hedz=−1+SzKz−1.If disturbance suppression performance of the closed-loop system is required to be as good as possible to suppress the response perturbation ofdn+d0n to en caused by secondary path change ΔStz, Kz needs to meet the following requirements [27]:(5)ΔStz×1+SnomzKz−1∞≤1.Or in a strengthened form:(6)1+SnomzKz−1∞≤1ΔStz∞.In the sense of ∞-norm, if it is necessary to maintain sufficient disturbance suppression performance for the largest possible perturbationsΔStz, we need to solve the following minimization problem to find Kz:(7)minKstabilizing1+SnomzKz−1∞.In many cases, we may have a priori knowledge of the perturbation, and the perturbation block can be exhibited as follows:(8)ΔStz=Δ˜StzW1z,where Δ˜Stz is the unit norm perturbation set and weighting function. W1z is designed to satisfy disturbance suppression demands. At this time, the suppression target is changed from en to z1n. Correspondingly, the robust performance condition becomes(9)W1z1+SnomzKz−1∞≤1Δ˜Stz∞.The optimization problem turns into(10)minKstabilizingW1z1+SnomzKz−1∞.In engineering practice, a combination of various cost functions should be included in a reasonable design. For example, disturbance suppression and limiting control signal energy are meaningful. As is illustrated in Figure5, the above two goals are considered at the same time through mixed sensitivity controller design.(11)minKstabilizingW1z×HedzW2z×Hydz∞=W1z1+SnomzKz−1W2zKz1+SnomzKz−1∞,where Hydz=Kz×1+SzKz−1 represents the transfer function from disturbance signal dn+d0n to control signal ybn.


## 2.2.2. Reference Signal Reconstruction

To solve the problem that the speed signal is difficult to reflect the real-time changes of disturbance amplitude, it is a better choice to directly use the feedforward component of the uncontrolled responsedn+d0n to update Wz. However, since the actuator works continuously, it is hard to obtain the dn+d0n from the control system in real time. At this moment, the feedback neutralization method can be used to obtain the reconstructed reference signal based on the error signal [28]. The reconstruction process is shown in Figure 4.(12)Seqz=Sz1+SzKz,S^eqz is a FIR filter of the equivalent secondary path Seqz, which is identified offline, and its order is J. Through en and Yfn obtained in real time, the reference signal reconstruction process is(13)d^eqn=en+S^eqn∗Yfn,where d^eqn is the reconstructed reference signal, representing the component that needs to be suppressed by feedforward control, that is, the vibration response only under the action of the robust feedback controller Kz.The proposed hybrid control law can be summarized as follows:(14)yfn=wn∗D^eqn,ycn=yfn+ybn,en=dn+d0n−sn∗Ycn,d^eqn=en+s^eqn∗Yfn,x′n=s^eqn∗D^eqn,wn+1=wn+2μenD^eqn.


## 2.3. Performance Analysis

To satisfy the requirements of helicopter ACSR in maneuvering flight, the advantages of the proposed hybrid system in terms of reference signal reconstruction accuracy, control system convergence and stability, and global effectiveness are analysed.


### 2.3.1. Reference Signal Reconstruction Accuracy

It can be seen from equation (13) that the reconstruction error of d^eqn mainly depends on the modeling error between the offline identification of the equivalent secondary path and the actual. When Sz changes during maneuvering flight, the equivalent secondary path Seqn is expressed as equation (12). Assuming the offline identification of Sz is S^z=Snomz, the error compared to Sz is(15)ΔS^z=Sz−S^z=SnomzΔStz.Converting equation (15) in the form of sensitivity, we get(16)ΔS^zS^z=ΔStz.WhenStz≠0, the reconstruction error of the reference signal increases, and the hybrid control may diverge. With the robust feedback controller Kz introduced, the increment of equivalent secondary path Seqn estimated offline is as follows:(17)ΔS^eqz=Snomz1+ΔStz1+Snomz1+ΔStzKz−Snomz1+SnomzKz.Converting equation (17) into the form of sensitivity, we get(18)ΔS^eqzS^eqz=ΔStzGz,where(19)Gz=11+S^eqnKz.WhenS^eqzKz≫1, then Gz≪1. Compared with the case without Kz, the influence of the model error ΔS^eqz on S^eqz declines at this time. In terms of the time-varying secondary path, the reference signal reconstruction error is reduced, which is beneficial to effectively track the changes of the external disturbance response, and the stability of the hybrid control system is enhanced.


### 2.3.2. Convergence and Stability Analysis

As is illustrated in equation (14), according to the reconstructed reference signal d^eqn, the recursive formula of the filter coefficient wn turns into(20)wn+1=wn+2μenD^eqn.Compared to speed signal,d^eqn enhances the coherence between reference signal and uncontrolled responses, which means that wn needs to pay less adjustment cost to adapt to drastically changing response during maneuvering flight, leading to faster convergence of the control system.Furthermore, according to Diniz [29], the stability condition of Fx-LMS algorithm is(21)0<μ<1trR,where R is the autocorrelation matrix of filtered reference signal and trR is its trace, which satisfies(22)R=ED^eqnD^eqTn=d^eq2nd^eqndeq′n−1⋯d^eqnd^eqn−I+1d^eqn−1deq′nd^eq2n−1⋯d^eqn−1d^eqn−I+1⋮⋮⋱⋮d^eqn−I+1deqnd^eqn−I+1deqn−1⋯d^eq2n−I+1,(23)trR=∑i=0I−1d^eq2i.According to equations (21)–(23), if Sz dynamics is low-damped and the impulse response of x′n is large with long duration, trR will increase and the upper limit of μ will decrease, making system convergence and stability reduce. However, after the robust controller Kz is introduced, S^z turns into the offline estimation S^eqz of the equivalent secondary path Sz/1+SzKz. Owing to 1+SzKz≫1, S^eqz≪S^z, then the impulse response of d^eqn is small and short, contributing to small trR and little μ. In fact, Seqz provides higher damping than Sz, which is beneficial for convergence and stability enhancement.


### 2.3.3. Global Validity

From Figure4, the vibration response after control is(24)Ez=11+SzKzDz+D0z+SzWz1+SzKzD^eqz,where Ez, Dz, D0z, and D^eqz denote the z-transforms of en, dn, d0n, and d^eqn. If designed Kz can ensure system stability and satisfy 1+SzKz≫1 in the target frequency band, for the drastically changing desired signal dn+d0k under maneuvering flight conditions, the feedback loop can still control it to a small level (but not to 0), even if Wz=0. If the filter is adjusted to Wz=−S−1zDz+D0zD^eq−1z, global vibration reduction can be achieved theoretically. Therefore, the hybrid control system indicates advantages over either independent feedback or feedforward control when there is transient disturbance: the former is responsible for overall response reduction, while the latter helps to offset the vibration residuals. The complementarity is effective for realizing vibration control of maneuvering flying helicopters.


## 2.3.1. Reference Signal Reconstruction Accuracy

It can be seen from equation (13) that the reconstruction error of d^eqn mainly depends on the modeling error between the offline identification of the equivalent secondary path and the actual. When Sz changes during maneuvering flight, the equivalent secondary path Seqn is expressed as equation (12). Assuming the offline identification of Sz is S^z=Snomz, the error compared to Sz is(15)ΔS^z=Sz−S^z=SnomzΔStz.Converting equation (15) in the form of sensitivity, we get(16)ΔS^zS^z=ΔStz.WhenStz≠0, the reconstruction error of the reference signal increases, and the hybrid control may diverge. With the robust feedback controller Kz introduced, the increment of equivalent secondary path Seqn estimated offline is as follows:(17)ΔS^eqz=Snomz1+ΔStz1+Snomz1+ΔStzKz−Snomz1+SnomzKz.Converting equation (17) into the form of sensitivity, we get(18)ΔS^eqzS^eqz=ΔStzGz,where(19)Gz=11+S^eqnKz.WhenS^eqzKz≫1, then Gz≪1. Compared with the case without Kz, the influence of the model error ΔS^eqz on S^eqz declines at this time. In terms of the time-varying secondary path, the reference signal reconstruction error is reduced, which is beneficial to effectively track the changes of the external disturbance response, and the stability of the hybrid control system is enhanced.


## 2.3.2. Convergence and Stability Analysis

As is illustrated in equation (14), according to the reconstructed reference signal d^eqn, the recursive formula of the filter coefficient wn turns into(20)wn+1=wn+2μenD^eqn.Compared to speed signal,d^eqn enhances the coherence between reference signal and uncontrolled responses, which means that wn needs to pay less adjustment cost to adapt to drastically changing response during maneuvering flight, leading to faster convergence of the control system.Furthermore, according to Diniz [29], the stability condition of Fx-LMS algorithm is(21)0<μ<1trR,where R is the autocorrelation matrix of filtered reference signal and trR is its trace, which satisfies(22)R=ED^eqnD^eqTn=d^eq2nd^eqndeq′n−1⋯d^eqnd^eqn−I+1d^eqn−1deq′nd^eq2n−1⋯d^eqn−1d^eqn−I+1⋮⋮⋱⋮d^eqn−I+1deqnd^eqn−I+1deqn−1⋯d^eq2n−I+1,(23)trR=∑i=0I−1d^eq2i.According to equations (21)–(23), if Sz dynamics is low-damped and the impulse response of x′n is large with long duration, trR will increase and the upper limit of μ will decrease, making system convergence and stability reduce. However, after the robust controller Kz is introduced, S^z turns into the offline estimation S^eqz of the equivalent secondary path Sz/1+SzKz. Owing to 1+SzKz≫1, S^eqz≪S^z, then the impulse response of d^eqn is small and short, contributing to small trR and little μ. In fact, Seqz provides higher damping than Sz, which is beneficial for convergence and stability enhancement.


## 2.3.3. Global Validity

From Figure4, the vibration response after control is(24)Ez=11+SzKzDz+D0z+SzWz1+SzKzD^eqz,where Ez, Dz, D0z, and D^eqz denote the z-transforms of en, dn, d0n, and d^eqn. If designed Kz can ensure system stability and satisfy 1+SzKz≫1 in the target frequency band, for the drastically changing desired signal dn+d0k under maneuvering flight conditions, the feedback loop can still control it to a small level (but not to 0), even if Wz=0. If the filter is adjusted to Wz=−S−1zDz+D0zD^eq−1z, global vibration reduction can be achieved theoretically. Therefore, the hybrid control system indicates advantages over either independent feedback or feedforward control when there is transient disturbance: the former is responsible for overall response reduction, while the latter helps to offset the vibration residuals. The complementarity is effective for realizing vibration control of maneuvering flying helicopters.


## 3. Simulation Results and Analysis

To verify the effectiveness of the LMS-MSRC control law based on reference signal reconstruction, a single-input single-output control system is constructed to carry out ACSR simulation research of the helicopter maneuver flight, which is based on the secondary path identified offline of the simplified helicopter finite element model. The simulations of conventional ACSR methods are carried out simultaneously for comparison so as to discuss the advantages of the proposed method in control effect, stability, and robustness with complex external disturbances and structural parameter changes during maneuvering flight.


### 3.1. Simulation Model

Currently, the elastic beam structure model has been widely used in helicopter vibration response analysis [30]. As is shown in Figure 6, a free-free finite element beam model built by ANSYS is regarded as the simplified helicopter fuselage [31]. The structure parameters are as follows: m=6.1×103kg, ρ=7.8×103kg/m3, E=210Gpa, and μ=0.3. Among them, m, ρ, E, and μ represent mass, density modulus of elasticity, and Poisson’s ratio, respectively. The first seven modes are analysed as 8.2, 11.2, 16.7, 26.9, 33.2, 50.4, and 56.9 Hz. Figure 6(b) displays a partial enlargement of front part of the fuselage. External disturbance is applied at node 94 to simulate excitation force from the hub, and secondary control force is introduced at node 109. The 37th node represents the driver’s seat, and its acceleration response is used to evaluate the global effectiveness of the active control system.Figure 6
Simplified finite element model of the helicopter fuselage. (a) The whole fuselage. (b) Front part of the fuselage.
(a)(b)Figure7 indicates the measured acceleration response power spectrum of a typical helicopter [19]. To simulate the amplitude and frequency characteristics of the helicopter vibration response, the blade passing frequency is set to 19.5 Hz with second-order harmonic set to 39 Hz. By adjusting disturbance amplitude, the vibration response of 19.5 Hz is about 12 dB larger than the second-order harmonic component at node 37. Gaussian white noise with 0 mean value is added to the multifrequency signal.Figure 7
Measured acceleration response power spectrum of a typical helicopter.


### 3.2. Design of the Robust Feedback Controller

The performance functionW1z is related to the excitation load by helicopter, and it needs a large amplitude at the disturbance frequencies. Taking into consideration time-varying transfer functions during maneuvering, the design margin of the controller can be increased and the parameter perturbation range of the nominal secondary path can be set to 20%. Moreover, for the sake of multifrequency characteristics of disturbance and avoiding system natural frequencies to reduce complexity, W1z is designed as a narrowband pass filter with excitation frequencies, as shown in equation (25). λ, fwi, and σi reveal adjustment coefficient, disturbance frequency, and system damping ratio separately. W2z is used to limit input amplitude of the actuator and can be set as a constant gain 1.(25)W1s=λ∏i=1nfwi2s2+2fwiσis+fwi2.With proper weighting functions and MATLAB robust control toolbox, the robust feedback controller is designed as a 10-orderz-domain controller.


### 3.3. Numerical Results and Analysis

It is worth noting that the environment of helicopter maneuvering flight is very complicated, which is caused by time-varying secondary path, drastic changes of external disturbance, and so on. Firstly, the accuracy of the reference signal reconstruction after secondary path mutation is analysed. Furthermore, the robust feedback controller’s ability to suppress transient external disturbance and secondary channel mutation is verified. In addition, a simulation is carried out for time-varying disturbance amplitude and frequency. At last, all the above factors are considered simultaneously to check the performance of simulated helicopter maneuvering flight.


#### 3.3.1. Reference Signal Reconstruction

To confirm the accuracy of the reconstructed reference signal, the original and reconstructed reference signals after secondary path mutation are exhibited in Figure8. The amplitude-frequency characteristic of the secondary path is magnified by a factor of 1.3, and the damping is slightly reduced. Extra disturbance is applied at 15 s, and all responses are normalized by the peak value of the original reference signal before interference. As is shown in Figure 8, the reconstructed reference signal shows good consistency with the original, and the overall error is less than 10.2% after mutation. Although the error oscillates after 15s, it does not show a tendency to diverge, which proves the reconstructed signal good tracking abilities to change to the original. It is found from the subsequent simulation that the error to this extent is acceptable and the hybrid control system works well.Figure 8
The original and reconstructed reference signal after secondary path mutation.


#### 3.3.2. Secondary Path Mutation and Transient External Disturbance

In order to confirm the benefits of the proposed LMS-MSRC to the robustness of ACSR system in maneuvering flight, four helicopter ACSR methods are compared in Figure9. The employed ACSR methods for comparison including enhanced HHC, frequency-domain LMS, and multichannel LMS are wildly studied in the literature [19, 21, 23]. The parameters in all the methods are optimized after several attempts. The control starts to be exerted 10s after the external disturbance. In order to simulate secondary path mutation and transient disturbance that are hard to model during maneuvering, the amplitude-frequency characteristic of the secondary channel is decreased by 15% with damping slightly reduced at 40 s, and additional disturbance is applied at 70 s. The comparison of acceleration response with different methods is displayed in Figure 9.Figure 9
Comparison of acceleration response with different helicopter ACSR methods. (a) Acceleration response at the control point. (b) Control process at mutation and disturbance.
(a)(b)Considering enhanced HHC, the convergence speed after secondary path mutation is slower than that of other methods and the transient external disturbance that is difficult to model can hardly be suppressed. Secondary path mutation adaptability is improved by frequency-domain LMS, while residual response oscillation is more severe under the circumstances of injected disturbance. In contrast, control effects in multichannel LMS and proposed LMS-MSRC are relatively more effective. It takes 30 s for multichannel LMS to achieve 93% of vibration attenuation, while LMS-MSRC converges quickly, and the acceleration response is almost suppressed completely after 8 s. Both control laws can adapt to transfer function changes, but the convergence is decreased with multichannel LMS. Owing to the lack of necessary reference information, multichannel LMS shows poor adaptability to interference changes. On the contrary, the vibration contributed by transient disturbance is attenuated over 90% by LMS-MSRC. It proves that the robust feedback loop can compensate secondary path and perform a stronger suppression performance on system parameter perturbation and uncertain transient disturbance, which is beneficial for robustness and effectiveness improvement of helicopter maneuvering ACSR system. Subsequent simulations and experiments are carried out with multichannel LMS and LMS-MSRC for consideration of application potential.


#### 3.3.3. Time-Varying Excitation Amplitude and Frequency

The system is excited by 19.5 Hz + 39 Hz external disturbance and control begins after 10s. Taking into account actual working conditions of forward flight to speed up [8], excitation signal amplitude increases by 50% every 8 s starting from 40 s and returns to initial value at 64 s. The disturbance frequencies are reduced by 5% (turn to 18.5 Hz + 37 Hz) to reappear rotor speed reduction during maneuvering flight at 80 s. The parameter settings remain unchanged. The filter order is set to 64, and the initial value of the weight coefficients is 0. The system sampling rate is 1000 Hz. Convergence factors of multichannel LMS are μ19.5Hz=6×10−4, μ39Hz=1.5×10−3 while the value of LMS-MSRC is μ=6×10−2. The adjustment coefficients, external disturbance frequencies, and system damping ratio are designed as follows: λ=0.2, fwi=2π×19.5,39, and σi=0.06,0.025. The vibration control process is displayed in Figures 10 and 11.Figure 10
Acceleration response and filter output with multichannel LMS algorithm. (a) Acceleration response at the control point. (b) Output of the filter.
(a)(b)Figure 11
Acceleration response and filter output with LMS-MSRC algorithm. (a) Acceleration response at the control point. (b) Output of the filter.
(a)(b)The increase in excitation amplitude requires more control energy, which is reflected in the enlargement in filter output shown in Figures10(b) and 11(b). For multichannel LMS in Figure 10(b), it is observed that the transient increment of filter output is almost close to zero at the beginning of mutation, indicating that the algorithm converges slowly. Comparing Figure 10(a) with Figure 11(a), it is found that multichannel LMS tends to seek out stability at the cost of convergence in the circumstances of amplitude surge. The reasons accounting for this phenomenon are that convergence factors affect each other when dealing with multifrequency components. On the other hand, the speed signal fails to provide real-time changes in disturbance, resulting in additional costs to fit system changes. In contrast, the reconstructed reference signal in LMS-MSRC can effectively track excitation changes and the convergence speed is accelerated under the premise of stability.


#### 3.3.4. Complex Operating Conditions in Helicopter Maneuvering Flight

All the above factors are taken into account to simulate complex situation of helicopter maneuvering ACSR [8]. Vibration is suppressed before 10 s. With the magnitude-frequency characteristic of secondary path multiplied by 0.85 and damping slightly reduced, excitation amplitude increases by 50% and additional transient disturbance is applied. At the same time, excitation frequency is reduced by 5% to simulate the decrease in rotor speed. The parameter settings remain unchanged, and the control processes are shown in Figures 12 and 13, respectively.Figure 12
Acceleration response and filter output with multichannel LMS algorithm. (a) Acceleration response at the control point. (b) Output of the filter.
(a)(b)Figure 13
Acceleration response and filter output with LMS-MSRC algorithm. (a) Acceleration response at the control point. (b) Output of the filter.
(a)(b)It is difficult for multichannel LMS to adapt to complex external disturbance and secondary path changes. The vibration overshoot exceeds the peak value of uncontrolled response. The filter output regulates slowly so that residual vibration still reaches 15% after 20 s. LMS-MSRC shows good adaptability by contrast. The peak response of closed-loop system is 60% less than the uncontrolled response, and acceleration response is decreased by 95% within 3.5 s. Therefore, the proposed hybrid algorithm can suit structural parameter changes and transient response via a robust controller and accelerate convergence according to the reconstructed reference signal reflecting the real-time changes of external disturbance. The results demonstrate that LMS-MSRC is superior to multichannel LMS in terms of vibration response suppression on complex maneuvering conditions.Summarizing the above simulation results, it is not difficult to draw the conclusions: compared with multichannel LMS, LMS-MSRC based on reference signal reconstruction shows stronger adaptive ability with better robustness and faster convergence. It indicates great vibration suppression capability for complicated operating conditions of helicopter maneuvering flight.


## 3.1. Simulation Model

Currently, the elastic beam structure model has been widely used in helicopter vibration response analysis [30]. As is shown in Figure 6, a free-free finite element beam model built by ANSYS is regarded as the simplified helicopter fuselage [31]. The structure parameters are as follows: m=6.1×103kg, ρ=7.8×103kg/m3, E=210Gpa, and μ=0.3. Among them, m, ρ, E, and μ represent mass, density modulus of elasticity, and Poisson’s ratio, respectively. The first seven modes are analysed as 8.2, 11.2, 16.7, 26.9, 33.2, 50.4, and 56.9 Hz. Figure 6(b) displays a partial enlargement of front part of the fuselage. External disturbance is applied at node 94 to simulate excitation force from the hub, and secondary control force is introduced at node 109. The 37th node represents the driver’s seat, and its acceleration response is used to evaluate the global effectiveness of the active control system.Figure 6
Simplified finite element model of the helicopter fuselage. (a) The whole fuselage. (b) Front part of the fuselage.
(a)(b)Figure7 indicates the measured acceleration response power spectrum of a typical helicopter [19]. To simulate the amplitude and frequency characteristics of the helicopter vibration response, the blade passing frequency is set to 19.5 Hz with second-order harmonic set to 39 Hz. By adjusting disturbance amplitude, the vibration response of 19.5 Hz is about 12 dB larger than the second-order harmonic component at node 37. Gaussian white noise with 0 mean value is added to the multifrequency signal.Figure 7
Measured acceleration response power spectrum of a typical helicopter.


## 3.2. Design of the Robust Feedback Controller

The performance functionW1z is related to the excitation load by helicopter, and it needs a large amplitude at the disturbance frequencies. Taking into consideration time-varying transfer functions during maneuvering, the design margin of the controller can be increased and the parameter perturbation range of the nominal secondary path can be set to 20%. Moreover, for the sake of multifrequency characteristics of disturbance and avoiding system natural frequencies to reduce complexity, W1z is designed as a narrowband pass filter with excitation frequencies, as shown in equation (25). λ, fwi, and σi reveal adjustment coefficient, disturbance frequency, and system damping ratio separately. W2z is used to limit input amplitude of the actuator and can be set as a constant gain 1.(25)W1s=λ∏i=1nfwi2s2+2fwiσis+fwi2.With proper weighting functions and MATLAB robust control toolbox, the robust feedback controller is designed as a 10-orderz-domain controller.


## 3.3. Numerical Results and Analysis

It is worth noting that the environment of helicopter maneuvering flight is very complicated, which is caused by time-varying secondary path, drastic changes of external disturbance, and so on. Firstly, the accuracy of the reference signal reconstruction after secondary path mutation is analysed. Furthermore, the robust feedback controller’s ability to suppress transient external disturbance and secondary channel mutation is verified. In addition, a simulation is carried out for time-varying disturbance amplitude and frequency. At last, all the above factors are considered simultaneously to check the performance of simulated helicopter maneuvering flight.


### 3.3.1. Reference Signal Reconstruction

To confirm the accuracy of the reconstructed reference signal, the original and reconstructed reference signals after secondary path mutation are exhibited in Figure8. The amplitude-frequency characteristic of the secondary path is magnified by a factor of 1.3, and the damping is slightly reduced. Extra disturbance is applied at 15 s, and all responses are normalized by the peak value of the original reference signal before interference. As is shown in Figure 8, the reconstructed reference signal shows good consistency with the original, and the overall error is less than 10.2% after mutation. Although the error oscillates after 15s, it does not show a tendency to diverge, which proves the reconstructed signal good tracking abilities to change to the original. It is found from the subsequent simulation that the error to this extent is acceptable and the hybrid control system works well.Figure 8
The original and reconstructed reference signal after secondary path mutation.


### 3.3.2. Secondary Path Mutation and Transient External Disturbance

In order to confirm the benefits of the proposed LMS-MSRC to the robustness of ACSR system in maneuvering flight, four helicopter ACSR methods are compared in Figure9. The employed ACSR methods for comparison including enhanced HHC, frequency-domain LMS, and multichannel LMS are wildly studied in the literature [19, 21, 23]. The parameters in all the methods are optimized after several attempts. The control starts to be exerted 10s after the external disturbance. In order to simulate secondary path mutation and transient disturbance that are hard to model during maneuvering, the amplitude-frequency characteristic of the secondary channel is decreased by 15% with damping slightly reduced at 40 s, and additional disturbance is applied at 70 s. The comparison of acceleration response with different methods is displayed in Figure 9.Figure 9
Comparison of acceleration response with different helicopter ACSR methods. (a) Acceleration response at the control point. (b) Control process at mutation and disturbance.
(a)(b)Considering enhanced HHC, the convergence speed after secondary path mutation is slower than that of other methods and the transient external disturbance that is difficult to model can hardly be suppressed. Secondary path mutation adaptability is improved by frequency-domain LMS, while residual response oscillation is more severe under the circumstances of injected disturbance. In contrast, control effects in multichannel LMS and proposed LMS-MSRC are relatively more effective. It takes 30 s for multichannel LMS to achieve 93% of vibration attenuation, while LMS-MSRC converges quickly, and the acceleration response is almost suppressed completely after 8 s. Both control laws can adapt to transfer function changes, but the convergence is decreased with multichannel LMS. Owing to the lack of necessary reference information, multichannel LMS shows poor adaptability to interference changes. On the contrary, the vibration contributed by transient disturbance is attenuated over 90% by LMS-MSRC. It proves that the robust feedback loop can compensate secondary path and perform a stronger suppression performance on system parameter perturbation and uncertain transient disturbance, which is beneficial for robustness and effectiveness improvement of helicopter maneuvering ACSR system. Subsequent simulations and experiments are carried out with multichannel LMS and LMS-MSRC for consideration of application potential.


### 3.3.3. Time-Varying Excitation Amplitude and Frequency

The system is excited by 19.5 Hz + 39 Hz external disturbance and control begins after 10s. Taking into account actual working conditions of forward flight to speed up [8], excitation signal amplitude increases by 50% every 8 s starting from 40 s and returns to initial value at 64 s. The disturbance frequencies are reduced by 5% (turn to 18.5 Hz + 37 Hz) to reappear rotor speed reduction during maneuvering flight at 80 s. The parameter settings remain unchanged. The filter order is set to 64, and the initial value of the weight coefficients is 0. The system sampling rate is 1000 Hz. Convergence factors of multichannel LMS are μ19.5Hz=6×10−4, μ39Hz=1.5×10−3 while the value of LMS-MSRC is μ=6×10−2. The adjustment coefficients, external disturbance frequencies, and system damping ratio are designed as follows: λ=0.2, fwi=2π×19.5,39, and σi=0.06,0.025. The vibration control process is displayed in Figures 10 and 11.Figure 10
Acceleration response and filter output with multichannel LMS algorithm. (a) Acceleration response at the control point. (b) Output of the filter.
(a)(b)Figure 11
Acceleration response and filter output with LMS-MSRC algorithm. (a) Acceleration response at the control point. (b) Output of the filter.
(a)(b)The increase in excitation amplitude requires more control energy, which is reflected in the enlargement in filter output shown in Figures10(b) and 11(b). For multichannel LMS in Figure 10(b), it is observed that the transient increment of filter output is almost close to zero at the beginning of mutation, indicating that the algorithm converges slowly. Comparing Figure 10(a) with Figure 11(a), it is found that multichannel LMS tends to seek out stability at the cost of convergence in the circumstances of amplitude surge. The reasons accounting for this phenomenon are that convergence factors affect each other when dealing with multifrequency components. On the other hand, the speed signal fails to provide real-time changes in disturbance, resulting in additional costs to fit system changes. In contrast, the reconstructed reference signal in LMS-MSRC can effectively track excitation changes and the convergence speed is accelerated under the premise of stability.


### 3.3.4. Complex Operating Conditions in Helicopter Maneuvering Flight

All the above factors are taken into account to simulate complex situation of helicopter maneuvering ACSR [8]. Vibration is suppressed before 10 s. With the magnitude-frequency characteristic of secondary path multiplied by 0.85 and damping slightly reduced, excitation amplitude increases by 50% and additional transient disturbance is applied. At the same time, excitation frequency is reduced by 5% to simulate the decrease in rotor speed. The parameter settings remain unchanged, and the control processes are shown in Figures 12 and 13, respectively.Figure 12
Acceleration response and filter output with multichannel LMS algorithm. (a) Acceleration response at the control point. (b) Output of the filter.
(a)(b)Figure 13
Acceleration response and filter output with LMS-MSRC algorithm. (a) Acceleration response at the control point. (b) Output of the filter.
(a)(b)It is difficult for multichannel LMS to adapt to complex external disturbance and secondary path changes. The vibration overshoot exceeds the peak value of uncontrolled response. The filter output regulates slowly so that residual vibration still reaches 15% after 20 s. LMS-MSRC shows good adaptability by contrast. The peak response of closed-loop system is 60% less than the uncontrolled response, and acceleration response is decreased by 95% within 3.5 s. Therefore, the proposed hybrid algorithm can suit structural parameter changes and transient response via a robust controller and accelerate convergence according to the reconstructed reference signal reflecting the real-time changes of external disturbance. The results demonstrate that LMS-MSRC is superior to multichannel LMS in terms of vibration response suppression on complex maneuvering conditions.Summarizing the above simulation results, it is not difficult to draw the conclusions: compared with multichannel LMS, LMS-MSRC based on reference signal reconstruction shows stronger adaptive ability with better robustness and faster convergence. It indicates great vibration suppression capability for complicated operating conditions of helicopter maneuvering flight.


## 3.3.1. Reference Signal Reconstruction

To confirm the accuracy of the reconstructed reference signal, the original and reconstructed reference signals after secondary path mutation are exhibited in Figure8. The amplitude-frequency characteristic of the secondary path is magnified by a factor of 1.3, and the damping is slightly reduced. Extra disturbance is applied at 15 s, and all responses are normalized by the peak value of the original reference signal before interference. As is shown in Figure 8, the reconstructed reference signal shows good consistency with the original, and the overall error is less than 10.2% after mutation. Although the error oscillates after 15s, it does not show a tendency to diverge, which proves the reconstructed signal good tracking abilities to change to the original. It is found from the subsequent simulation that the error to this extent is acceptable and the hybrid control system works well.Figure 8
The original and reconstructed reference signal after secondary path mutation.


## 3.3.2. Secondary Path Mutation and Transient External Disturbance

In order to confirm the benefits of the proposed LMS-MSRC to the robustness of ACSR system in maneuvering flight, four helicopter ACSR methods are compared in Figure9. The employed ACSR methods for comparison including enhanced HHC, frequency-domain LMS, and multichannel LMS are wildly studied in the literature [19, 21, 23]. The parameters in all the methods are optimized after several attempts. The control starts to be exerted 10s after the external disturbance. In order to simulate secondary path mutation and transient disturbance that are hard to model during maneuvering, the amplitude-frequency characteristic of the secondary channel is decreased by 15% with damping slightly reduced at 40 s, and additional disturbance is applied at 70 s. The comparison of acceleration response with different methods is displayed in Figure 9.Figure 9
Comparison of acceleration response with different helicopter ACSR methods. (a) Acceleration response at the control point. (b) Control process at mutation and disturbance.
(a)(b)Considering enhanced HHC, the convergence speed after secondary path mutation is slower than that of other methods and the transient external disturbance that is difficult to model can hardly be suppressed. Secondary path mutation adaptability is improved by frequency-domain LMS, while residual response oscillation is more severe under the circumstances of injected disturbance. In contrast, control effects in multichannel LMS and proposed LMS-MSRC are relatively more effective. It takes 30 s for multichannel LMS to achieve 93% of vibration attenuation, while LMS-MSRC converges quickly, and the acceleration response is almost suppressed completely after 8 s. Both control laws can adapt to transfer function changes, but the convergence is decreased with multichannel LMS. Owing to the lack of necessary reference information, multichannel LMS shows poor adaptability to interference changes. On the contrary, the vibration contributed by transient disturbance is attenuated over 90% by LMS-MSRC. It proves that the robust feedback loop can compensate secondary path and perform a stronger suppression performance on system parameter perturbation and uncertain transient disturbance, which is beneficial for robustness and effectiveness improvement of helicopter maneuvering ACSR system. Subsequent simulations and experiments are carried out with multichannel LMS and LMS-MSRC for consideration of application potential.


## 3.3.3. Time-Varying Excitation Amplitude and Frequency

The system is excited by 19.5 Hz + 39 Hz external disturbance and control begins after 10s. Taking into account actual working conditions of forward flight to speed up [8], excitation signal amplitude increases by 50% every 8 s starting from 40 s and returns to initial value at 64 s. The disturbance frequencies are reduced by 5% (turn to 18.5 Hz + 37 Hz) to reappear rotor speed reduction during maneuvering flight at 80 s. The parameter settings remain unchanged. The filter order is set to 64, and the initial value of the weight coefficients is 0. The system sampling rate is 1000 Hz. Convergence factors of multichannel LMS are μ19.5Hz=6×10−4, μ39Hz=1.5×10−3 while the value of LMS-MSRC is μ=6×10−2. The adjustment coefficients, external disturbance frequencies, and system damping ratio are designed as follows: λ=0.2, fwi=2π×19.5,39, and σi=0.06,0.025. The vibration control process is displayed in Figures 10 and 11.Figure 10
Acceleration response and filter output with multichannel LMS algorithm. (a) Acceleration response at the control point. (b) Output of the filter.
(a)(b)Figure 11
Acceleration response and filter output with LMS-MSRC algorithm. (a) Acceleration response at the control point. (b) Output of the filter.
(a)(b)The increase in excitation amplitude requires more control energy, which is reflected in the enlargement in filter output shown in Figures10(b) and 11(b). For multichannel LMS in Figure 10(b), it is observed that the transient increment of filter output is almost close to zero at the beginning of mutation, indicating that the algorithm converges slowly. Comparing Figure 10(a) with Figure 11(a), it is found that multichannel LMS tends to seek out stability at the cost of convergence in the circumstances of amplitude surge. The reasons accounting for this phenomenon are that convergence factors affect each other when dealing with multifrequency components. On the other hand, the speed signal fails to provide real-time changes in disturbance, resulting in additional costs to fit system changes. In contrast, the reconstructed reference signal in LMS-MSRC can effectively track excitation changes and the convergence speed is accelerated under the premise of stability.


## 3.3.4. Complex Operating Conditions in Helicopter Maneuvering Flight

All the above factors are taken into account to simulate complex situation of helicopter maneuvering ACSR [8]. Vibration is suppressed before 10 s. With the magnitude-frequency characteristic of secondary path multiplied by 0.85 and damping slightly reduced, excitation amplitude increases by 50% and additional transient disturbance is applied. At the same time, excitation frequency is reduced by 5% to simulate the decrease in rotor speed. The parameter settings remain unchanged, and the control processes are shown in Figures 12 and 13, respectively.Figure 12
Acceleration response and filter output with multichannel LMS algorithm. (a) Acceleration response at the control point. (b) Output of the filter.
(a)(b)Figure 13
Acceleration response and filter output with LMS-MSRC algorithm. (a) Acceleration response at the control point. (b) Output of the filter.
(a)(b)It is difficult for multichannel LMS to adapt to complex external disturbance and secondary path changes. The vibration overshoot exceeds the peak value of uncontrolled response. The filter output regulates slowly so that residual vibration still reaches 15% after 20 s. LMS-MSRC shows good adaptability by contrast. The peak response of closed-loop system is 60% less than the uncontrolled response, and acceleration response is decreased by 95% within 3.5 s. Therefore, the proposed hybrid algorithm can suit structural parameter changes and transient response via a robust controller and accelerate convergence according to the reconstructed reference signal reflecting the real-time changes of external disturbance. The results demonstrate that LMS-MSRC is superior to multichannel LMS in terms of vibration response suppression on complex maneuvering conditions.Summarizing the above simulation results, it is not difficult to draw the conclusions: compared with multichannel LMS, LMS-MSRC based on reference signal reconstruction shows stronger adaptive ability with better robustness and faster convergence. It indicates great vibration suppression capability for complicated operating conditions of helicopter maneuvering flight.


## 4. Active Control Tests and Results Analysis

In order to verify the effectiveness of the LMS-MSRC algorithm, considering actual demands of helicopter vibration reduction and the scale of the control system, a series of ACSR tests based on the rotor passing frequency and its second-order frequency component (19.5 Hz + 39 Hz) are carried out.


### 4.1. Experimental System Setup

The block diagram of the elastic beam ACSR test system is shown in Figure14. The beam is suspended by steel cables to realize free-free boundary conditions. Excitation signal generated by the signal generator (1000Z, DG) drives the exciter (HEA-500, SinoCera) via the power amplifier to excite the elastic beam to produce multifrequency vibration. The uncontrolled vibration is monitored by the acceleration sensor (333B30, PCB), low-pass filtered by the signal conditioner (CM3504, Centuryl), and enters into the data acquisition system (DH5902, Donghua). According to the active control law, real-time control signal calculated by DSP drives the PSA to generate the required control torque via the driving power, neutralizing the response caused by external excitation. The ACSR physical testing system is displayed in Figure 15. Specific parameters of the beam structure and the low-voltage mechanically packaged PSA used in this test are exhibited in Table 1 and Table 2, respectively.Figure 14
Schematic diagram of the elastic beam ACSR test system.Figure 15
Physical diagram of the elastic beam ACSR experimental setup.Table 1
Parameters of the beam structure.
ParametersDataLength × width × height (mm)3000 × 60 × 20First three natural frequencies (Hz)11.9, 35, 61Control point (mm)1690Actuator location (mm)1270Exciter location (mm)2090Table 2
Parameters of the piezoelectric stack.
ParametersDataPeak-to-peak displacement (μm)100Axial stiffness (N/μm)20Blocked force (N)3500First axial natural frequency (kHz)10Diameter× length (mm)15× 100Input range (V)0～100


### 4.2. Experimental Results and Analysis

The performance tests of the proposed controller are carried out as below: firstly, a dual-frequency harmonic excitation control experiment is conducted. Then, the adaptability of LMS-MSRC with time-varying phases is verified. The control experiment about simultaneous changes in excitation frequency and amplitude as well as primary/secondary path mutation is implemented. The convergence factors areμ19.5Hz=8×10−4, μ39Hz=1×10−3 for multichannel LMS and μ=2×10−3 for LMS-MSRC. The sampling rate is 1000 Hz. The filter coefficient is 64, and the initial value is set to 0. The adjustment coefficient λ, external disturbance frequency fwi, and system damping ratio σi are designed as follows: λ=0.25, fwi=2π×19.5,39, and σi=0.08,0.035.


#### 4.2.1. Dual-Frequency Harmonic Excitation

Control starts at 10s. The error response at the controlled position and the filter output are exhibited in Figure16. It can be seen that the acceleration response quickly drops below 12% after control, while it takes at least 30s for multichannel LMS to converge. After the system is stabilized, vibration of target frequencies is almost completely eliminated, which indicates the designed LMS-MSRC can suppress effective vibration level at the position of helicopter fuselage. Figure 16(d) shows the filter output with LMS-MSRC.Figure 16
Acceleration response comparison and filter output with LMS-MSRC algorithm. (a) Acceleration response with LMS-MSRC. (b) Acceleration response with multichannel LMS. (c) Comparison of LMS-MSRC response spectrum. (d) Output of LMS-MSRC filter.
(a)(b)(c)(d)


#### 4.2.2. Harmonic Disturbance with Time-Varying Phase

By adjusting the phase of the driving signal input to the exciter, harmonic excitation phase changes from 0° to +45°, −30° and 0° at 80 s, 160 s, and 240 s, respectively. The control process is shown in Figure17. The residual vibration changed slightly at 80 s, and the amplitude of the error signal fluctuates more at 160 s and 240 s with LMS-MSRC. In comparison, multichannel LMS fluctuates greatly, and it takes more time to converge under phase changes. As is shown in Figure 17(c), excitation phase change is as high as −75° at 160 s, and instantaneous change of the filter output increases, but the acceleration response is restored to the level before excitation phase fluctuation after a few seconds of adjustment. The results illustrate that LMS-MSRC hybrid controller exhibits adaptability to the disturbance phase mutation.Figure 17
Control effects after disturbance phase changes. (a) Acceleration response with LMS-MSRC. (b) Acceleration response with multichannel LMS. (c) Output of LMS-MSRC filter.
(a)(b)(c)


#### 4.2.3. Time-Varying Primary/Secondary Path with Mutation in Disturbance Frequency and Amplitude

An additional 1 kg mass is placed between the PSA installation position and the controlled point before test. At about 80s, disturbance frequency turns to 18.5 Hz + 37 Hz and amplitude increases by 50%, with the mass removed to realize primary/secondary path sudden changes of the elastic beam.As is shown in Figures18(a)-18(b), vibration attenuation is achieved before structural mutation with LMS-MSRC and multichannel LMS. However, due to the mass movement and the sharp increase in disturbance amplitude, the acceleration response increases rapidly. Figure 18(c) indicates that the instantaneous response overshoot is 42% less than the peak value, which demonstrates that additional damping is provided to effectively suppress response surge. In addition, the vibration attenuation reaches 85% within 2.5 s so as to verify the fast convergence and robustness of the algorithm. In contrast, it is observed that the response peak of multichannel LMS is the same as that of the open-loop system and the response is decreased by 83% within 20 s in Figure 18(b). The acceleration response before and after excitation frequency change with LMS-MSRC is compared in Figure 18(d). Although the 18.5 Hz and 37 Hz components are almost completely suppressed, the residual vibration increases. Analysing the response before and after the mass movement, the uncontrolled structure response caused by the movement is the main reason accounting for the increase of residual signal. The LMS-MSRC filter output in Figure 18(e) exhibits that the excitation frequencies decrease, but the uncontrolled response raises, resulting in an increase in the required control energy.Figure 18
Control effects before/after disturbance and primary/secondary path mutation. (a) Acceleration response with LMS-MSRC. (b) Acceleration response with multichannel LMS. (c) Control process at mutation with LMS-MSRC. (d) Comparison of LMS-MSRC response spectrum. (e) Comparison of LMS-MSRC filter output spectrum.
(a)(b)(c)(d)(e)From the above experimental results, compared to multichannel LMS, LMS-MSRC based on reference signal reconstruction can effectively suppress multifrequency harmonic response and show good adaptability and strong robustness to unknown environmental changes. The fast-changing control signal avoids violent fluctuation of most structural response, and the mutation response is effectively suppressed during helicopter maneuvering flight.


## 4.1. Experimental System Setup

The block diagram of the elastic beam ACSR test system is shown in Figure14. The beam is suspended by steel cables to realize free-free boundary conditions. Excitation signal generated by the signal generator (1000Z, DG) drives the exciter (HEA-500, SinoCera) via the power amplifier to excite the elastic beam to produce multifrequency vibration. The uncontrolled vibration is monitored by the acceleration sensor (333B30, PCB), low-pass filtered by the signal conditioner (CM3504, Centuryl), and enters into the data acquisition system (DH5902, Donghua). According to the active control law, real-time control signal calculated by DSP drives the PSA to generate the required control torque via the driving power, neutralizing the response caused by external excitation. The ACSR physical testing system is displayed in Figure 15. Specific parameters of the beam structure and the low-voltage mechanically packaged PSA used in this test are exhibited in Table 1 and Table 2, respectively.Figure 14
Schematic diagram of the elastic beam ACSR test system.Figure 15
Physical diagram of the elastic beam ACSR experimental setup.Table 1
Parameters of the beam structure.
ParametersDataLength × width × height (mm)3000 × 60 × 20First three natural frequencies (Hz)11.9, 35, 61Control point (mm)1690Actuator location (mm)1270Exciter location (mm)2090Table 2
Parameters of the piezoelectric stack.
ParametersDataPeak-to-peak displacement (μm)100Axial stiffness (N/μm)20Blocked force (N)3500First axial natural frequency (kHz)10Diameter× length (mm)15× 100Input range (V)0～100


## 4.2. Experimental Results and Analysis

The performance tests of the proposed controller are carried out as below: firstly, a dual-frequency harmonic excitation control experiment is conducted. Then, the adaptability of LMS-MSRC with time-varying phases is verified. The control experiment about simultaneous changes in excitation frequency and amplitude as well as primary/secondary path mutation is implemented. The convergence factors areμ19.5Hz=8×10−4, μ39Hz=1×10−3 for multichannel LMS and μ=2×10−3 for LMS-MSRC. The sampling rate is 1000 Hz. The filter coefficient is 64, and the initial value is set to 0. The adjustment coefficient λ, external disturbance frequency fwi, and system damping ratio σi are designed as follows: λ=0.25, fwi=2π×19.5,39, and σi=0.08,0.035.


### 4.2.1. Dual-Frequency Harmonic Excitation

Control starts at 10s. The error response at the controlled position and the filter output are exhibited in Figure16. It can be seen that the acceleration response quickly drops below 12% after control, while it takes at least 30s for multichannel LMS to converge. After the system is stabilized, vibration of target frequencies is almost completely eliminated, which indicates the designed LMS-MSRC can suppress effective vibration level at the position of helicopter fuselage. Figure 16(d) shows the filter output with LMS-MSRC.Figure 16
Acceleration response comparison and filter output with LMS-MSRC algorithm. (a) Acceleration response with LMS-MSRC. (b) Acceleration response with multichannel LMS. (c) Comparison of LMS-MSRC response spectrum. (d) Output of LMS-MSRC filter.
(a)(b)(c)(d)


### 4.2.2. Harmonic Disturbance with Time-Varying Phase

By adjusting the phase of the driving signal input to the exciter, harmonic excitation phase changes from 0° to +45°, −30° and 0° at 80 s, 160 s, and 240 s, respectively. The control process is shown in Figure17. The residual vibration changed slightly at 80 s, and the amplitude of the error signal fluctuates more at 160 s and 240 s with LMS-MSRC. In comparison, multichannel LMS fluctuates greatly, and it takes more time to converge under phase changes. As is shown in Figure 17(c), excitation phase change is as high as −75° at 160 s, and instantaneous change of the filter output increases, but the acceleration response is restored to the level before excitation phase fluctuation after a few seconds of adjustment. The results illustrate that LMS-MSRC hybrid controller exhibits adaptability to the disturbance phase mutation.Figure 17
Control effects after disturbance phase changes. (a) Acceleration response with LMS-MSRC. (b) Acceleration response with multichannel LMS. (c) Output of LMS-MSRC filter.
(a)(b)(c)


### 4.2.3. Time-Varying Primary/Secondary Path with Mutation in Disturbance Frequency and Amplitude

An additional 1 kg mass is placed between the PSA installation position and the controlled point before test. At about 80s, disturbance frequency turns to 18.5 Hz + 37 Hz and amplitude increases by 50%, with the mass removed to realize primary/secondary path sudden changes of the elastic beam.As is shown in Figures18(a)-18(b), vibration attenuation is achieved before structural mutation with LMS-MSRC and multichannel LMS. However, due to the mass movement and the sharp increase in disturbance amplitude, the acceleration response increases rapidly. Figure 18(c) indicates that the instantaneous response overshoot is 42% less than the peak value, which demonstrates that additional damping is provided to effectively suppress response surge. In addition, the vibration attenuation reaches 85% within 2.5 s so as to verify the fast convergence and robustness of the algorithm. In contrast, it is observed that the response peak of multichannel LMS is the same as that of the open-loop system and the response is decreased by 83% within 20 s in Figure 18(b). The acceleration response before and after excitation frequency change with LMS-MSRC is compared in Figure 18(d). Although the 18.5 Hz and 37 Hz components are almost completely suppressed, the residual vibration increases. Analysing the response before and after the mass movement, the uncontrolled structure response caused by the movement is the main reason accounting for the increase of residual signal. The LMS-MSRC filter output in Figure 18(e) exhibits that the excitation frequencies decrease, but the uncontrolled response raises, resulting in an increase in the required control energy.Figure 18
Control effects before/after disturbance and primary/secondary path mutation. (a) Acceleration response with LMS-MSRC. (b) Acceleration response with multichannel LMS. (c) Control process at mutation with LMS-MSRC. (d) Comparison of LMS-MSRC response spectrum. (e) Comparison of LMS-MSRC filter output spectrum.
(a)(b)(c)(d)(e)From the above experimental results, compared to multichannel LMS, LMS-MSRC based on reference signal reconstruction can effectively suppress multifrequency harmonic response and show good adaptability and strong robustness to unknown environmental changes. The fast-changing control signal avoids violent fluctuation of most structural response, and the mutation response is effectively suppressed during helicopter maneuvering flight.


## 4.2.1. Dual-Frequency Harmonic Excitation

Control starts at 10s. The error response at the controlled position and the filter output are exhibited in Figure16. It can be seen that the acceleration response quickly drops below 12% after control, while it takes at least 30s for multichannel LMS to converge. After the system is stabilized, vibration of target frequencies is almost completely eliminated, which indicates the designed LMS-MSRC can suppress effective vibration level at the position of helicopter fuselage. Figure 16(d) shows the filter output with LMS-MSRC.Figure 16
Acceleration response comparison and filter output with LMS-MSRC algorithm. (a) Acceleration response with LMS-MSRC. (b) Acceleration response with multichannel LMS. (c) Comparison of LMS-MSRC response spectrum. (d) Output of LMS-MSRC filter.
(a)(b)(c)(d)


## 4.2.2. Harmonic Disturbance with Time-Varying Phase

By adjusting the phase of the driving signal input to the exciter, harmonic excitation phase changes from 0° to +45°, −30° and 0° at 80 s, 160 s, and 240 s, respectively. The control process is shown in Figure17. The residual vibration changed slightly at 80 s, and the amplitude of the error signal fluctuates more at 160 s and 240 s with LMS-MSRC. In comparison, multichannel LMS fluctuates greatly, and it takes more time to converge under phase changes. As is shown in Figure 17(c), excitation phase change is as high as −75° at 160 s, and instantaneous change of the filter output increases, but the acceleration response is restored to the level before excitation phase fluctuation after a few seconds of adjustment. The results illustrate that LMS-MSRC hybrid controller exhibits adaptability to the disturbance phase mutation.Figure 17
Control effects after disturbance phase changes. (a) Acceleration response with LMS-MSRC. (b) Acceleration response with multichannel LMS. (c) Output of LMS-MSRC filter.
(a)(b)(c)


## 4.2.3. Time-Varying Primary/Secondary Path with Mutation in Disturbance Frequency and Amplitude

An additional 1 kg mass is placed between the PSA installation position and the controlled point before test. At about 80s, disturbance frequency turns to 18.5 Hz + 37 Hz and amplitude increases by 50%, with the mass removed to realize primary/secondary path sudden changes of the elastic beam.As is shown in Figures18(a)-18(b), vibration attenuation is achieved before structural mutation with LMS-MSRC and multichannel LMS. However, due to the mass movement and the sharp increase in disturbance amplitude, the acceleration response increases rapidly. Figure 18(c) indicates that the instantaneous response overshoot is 42% less than the peak value, which demonstrates that additional damping is provided to effectively suppress response surge. In addition, the vibration attenuation reaches 85% within 2.5 s so as to verify the fast convergence and robustness of the algorithm. In contrast, it is observed that the response peak of multichannel LMS is the same as that of the open-loop system and the response is decreased by 83% within 20 s in Figure 18(b). The acceleration response before and after excitation frequency change with LMS-MSRC is compared in Figure 18(d). Although the 18.5 Hz and 37 Hz components are almost completely suppressed, the residual vibration increases. Analysing the response before and after the mass movement, the uncontrolled structure response caused by the movement is the main reason accounting for the increase of residual signal. The LMS-MSRC filter output in Figure 18(e) exhibits that the excitation frequencies decrease, but the uncontrolled response raises, resulting in an increase in the required control energy.Figure 18
Control effects before/after disturbance and primary/secondary path mutation. (a) Acceleration response with LMS-MSRC. (b) Acceleration response with multichannel LMS. (c) Control process at mutation with LMS-MSRC. (d) Comparison of LMS-MSRC response spectrum. (e) Comparison of LMS-MSRC filter output spectrum.
(a)(b)(c)(d)(e)From the above experimental results, compared to multichannel LMS, LMS-MSRC based on reference signal reconstruction can effectively suppress multifrequency harmonic response and show good adaptability and strong robustness to unknown environmental changes. The fast-changing control signal avoids violent fluctuation of most structural response, and the mutation response is effectively suppressed during helicopter maneuvering flight.


## 5. Conclusions

For the purpose of helicopter ACSR during maneuvering flight, based on Fx-LMS algorithm, the reference signal reconstruction and the mixed sensitivity robust feedback controller are considered in this paper to form a feedforward-robust hybrid ACSR control law. In view of the simplified finite element helicopter model, several control simulations of typical helicopter maneuvering flight conditions are carried out. A series of ACSR tests to simulate helicopter maneuvering flight are conducted on a free-free beam. The main conclusions of the present study are summarized as follows:(1)
The designed robust feedback controller can make the hybrid control law obtain the advantages of feedforward and feedback control at the same time. That is, the feedback control can reduce the overall response and suppress unknown transient disturbance, while residual vibration related to the reference signal is cancelled by feedforward control so as to improve the global vibration attenuation during helicopter maneuvering flight. In addition, the introduction of the reconstructed reference signal can effectively track the real-time changes of the uncontrolled response, which is beneficial for the convergence and stability of the system.(2)
In the cases of complex transient external disturbance, time-varying secondary path, and changes in excitation signal amplitude and frequencies of the simulation object, LMS-MSRC based on reference signal reconstruction can effectively suppress multifrequency vibration, which shows obvious advantages in adaptive adjustment performance, robustness, and convergence speed compared with other ACSR methods. In the case of simulating the complex operating conditions of helicopter maneuvering flight, the introduction of robust feedback controller increases secondary path damping, and the response overshoot suppressing is achieved. The reconstructed reference signal that reflects the real-time change of disturbance accelerates convergence speed and enhances the robustness of the entire control system.(3)
In the ACSR tests on the free-free beam, the multifrequency vibration suppression performance and adaptability of LMS-MSRC are verified. Under the circumstances that external excitation amplitude, frequency, and primary/secondary path change simultaneously, the peak value of the closed-loop system is 42% lower than that of the open-loop system, and the vibration response is attenuated by more than 85% within 2.5 s. The experimental results demonstrate that LMS-MSRC based on reference signal reconstruction shows good adaptability and strong robustness to the unknown environmental changes with great application potential to helicopter ACSR during maneuvering flight.



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*Source: 3153531-2021-12-15.xml*