Research of multi-point adaptive control strategy based on electromagnetic active vibration absorber

2.1. Design of structure

Electromagnetic active vibration absorber is a typical active inertial actuator, the structure is as shown in Fig. 1.

In the mechanical structure, electromagnetic actuators are used to change the output force, including passive device consisted of springs, weights and actuating elements [15]. The coil is used as stator and the soft iron with permanent magnets on the upper and lower end is used as mover that clearance fit with middle pillar. The spring is multifunctional part in the absorber. First, it can provide sufficient axial elasticity for the mover of vibration absorber; second it can ensure sufficient radial stiffness when the mover is reciprocating. The resonant frequency of the electromagnetic actuator ${\omega }_p$ is determined by the weight of counterweight and the mover mass of actuating element [16], which in turn makes influence on the peak output and operating frequency of electromagnetic actuator.

Fig. 1Electromagnetic active vibration absorber

Fig. 2Model of magnetic circuit

2.2. Magnetic circuit analysis and calculation of output force

Model of magnetic circuit is as shown in Fig. 2.

In the model of magnetic circuit, the magnetic potential of upper permanent magnet increases, but decreases in the end permanent magnet. Thus. the soft iron will be in upward force on the support column along the vertical direction, the force in horizontal is canceled by each other. The meanings and symbols of related parameters are shown in Table 1.

Equivalent magnetic circuit method is used to analyze the magnetic circuit model, as shown in Fig. 3.

Fig. 3Equivalent magnetic circuit

The magnetic potential $U_{m1}$ and magnetic resistance $R_{m1}$ are the equivalent magnetic circuit of permanent magnet on the upper end of the soft iron. The magnetic potential $U_{m2}$ and magnetic resistance $R_{m2}$ are the equivalent magnetic circuit of permanent magnet on the lower end of the soft iron. The magnetic potential of control coil is $U$, here $U=iN$, magnetic resistance of air gap is $R$. Permanent magnet’s materials are NdFeB (neodymium iron boron), so the demagnetized curve is basically a straight line. The formula is as follows [17]:

Magnetic resistance of air gap can be expressed as $R=y/\left({\mu }_{0}S\right)$, $S$ is used as the magnetic area of a permanent magnet, ${\mu }_0$ is used as air permeanility. If the magnetic resistance, leakage of the magnetic yoke and iron core are ignored, the magnetic fulx of iron core magnetic route can be calculated as [18]:

The electromagnetic force on iron core is:

The electromagnetic active vibration absorber designed in this article can provide electromagnetic force in both directions through the electromagnets in upper and lower of soft iron, without applying the bias current. Because of this unique design, the relationship between input current and nonlinear electromagnetic actuator output can be approximately linear.

3.1. Influence of identification error on control

The error signal will be influenced by secondary path, which in turn affects the convergence criterion. The adjustment of control vector according to control algorithm may not be in the steepest direction of the gradient. The adaptive control process is affected or even failed. So, the accurate identification model is the basis of control. Parameter identification is essential to get the transfer function of secondary path. Considered by the vibration absorption platform and good performance of actuator’s output force, the transversal filter is acted as identification model. The transfer function of secondary path is approached by updating the weight coefficients of identification filter.

Fig. 4Schematic diagram of secondary path identification

There is always be the identification error between the estimated model $\widehat{S}\left(Z\right)$ and the actual transfer function $S\left(Z\right)$. S. Snyder and C. Hansen found that when the reference signal is sinusoidal signal, the LMS (Least Mean Square) algorithm converges with a sufficiently small step size, so long as the phase difference between the secondary path identification model and actual transfer function is not more than 90 degrees. The following is the analysis of identification error from the control effect, stability of algorithm and convergence.

The objective function can be expressed as [19]:

where $R$ is the autocorrelation matrix of the filtered reference signal, $R=E\left[X\right(n\left)X^T\right(n\left)\right]$. $P$ is the correlation matrix of the desired signal and the filtered reference signal, $P=E\left[d\right(n\left)X^T\right(n\left)\right]$.

Filtered reference signal is $X\left(n\right)=\sum _{m=0}^{M-1}{s}_{m}x(n-m)$. In order to get the minimum value of the objective function, take the derivative of the objective function:

The optimal coefficients of adaptive control filter can be calculated as follows, substituted into objective function to get the minimum value:

The optimal coefficients are substituted into objective function, the minimum value is:

where $R$ is positive definite quadratic matrix, so $P^T{R}^{-1}P>$ 0. It shows that the total power of error signal after convergence is lower than that of desired signal.

However, the actual transfer function of secondary path is unknown and difficult to be calculated. So, in the actual control system, the estimation model $\widehat{S}\left(Z\right)$ acted as actual transfer function is used to update the weight coefficients of control filter. At this time, the optimal coefficients is:

The new optimal coefficients are substituted into objective function and the minimum value of it is:

In the Eq. (9), if the identification error is big, it can be inferred that even if the algorithm converges, the total power of error signal may still be higher than that of desired signal. The control fail.

Because of the identification error, the convergence step of LMS algorithm is as follows:

The error signal is directly influenced by estimation model, and the range of iteration step is also affected. Above all, it can be concluded that the accurate estimation model of secondary model is the prerequisite for the convergence of control algorithm. The inaccurate estimation model will lead to the control system in the risk of divergence.

3.2. Identification algorithm of secondary path

The vibration absorption platform is the research object in this paper. The characteristic frequency of platform is not changed greatly, and the output force of absorber is nearly linear, so transversal filter is selected as the estimation model to approach the actual transfer function of secondary path. The transversal filter is a finite impulse response filter, which does not need feedback and past related output. It is non-recursive and commonly used. The nest is the design of identification algorithm, which is expected to improve the identification accuracy as much as possible and reduce the influence of identification error on control system.

The least squares algorithm is introduced in this paper. Convergence speed and steady-state error are influenced by iterative step to some extent in the algorithm. But these two indexes requirements for iterative step is contradictory. When the iterative step is large, the convergence speed is fast, but the steady-state error is large, and vice versa. Variable step method is introduced in this paper. In the beginning of identification, large step is used to ensure fast convergence speed. After the algorithm tends to stabilize, small step is changed to obtain small steady-state error. Update of weight coefficients are as follows:

The key of this algorithm is the step-change mechanism, this to say, the selection of $\mu \left(n\right)$. According to the existing research, the step-change mechanism is roughly divided into two directions: First, Build the relationship between iterative step and error signal of transversal filter. Second, build the relationship between iterative step and gradient vector. In this paper, the change of iterative step is based on identification error.

The frequently-used normalized variable step algorithm is as follows:

where, ${\mu }_0$ is initial step, $eps$ is the tiny positive, which prevents the system diverging. Because the denominator may become zero without the tiny positive. While, the influence of identification error on algorithm is not reflected in Eq. (12). So, a new improved variable step identification algorithm is proposed in this paper. Combined with the above requirements, the change mechanism is changed as:

In the Eq. (13), $E\left(n\right)$ is the power of identification error, $E\left(n\right)=\lambda E(n-1)+(1-\lambda )e^2\left(n\right)$. It can be seen from the Eq. (13) that identification error is large at the beginning of the identification, so the value of variable step is large to get the fast convergence speed. But there is a necessary condition, the initial step must meet the requirement for convergence. And then, the identification error is getting smaller with the updating of weight coefficients, small variable step is changed into identification to obtain small steady-state error. The improved variable step identification algorithm not only ensures the convergence speed of identification, but solves the problem of identification accuracy. The variable step may be too large duo to the pulse impulsion in the process of identification, it should be limited in the convergence range. If $\mu \left(n\right)>{\mu }_{max}$, then $\mu \left(n\right)={\mu }_{max}$. Here, ${\mu }_{max}$ is the upper limit of convergence step.

The force diagram of vibration absorption platform is as shown in Fig. 5.

The normalized variable step algorithm and improved variable step identification algorithm are respectively used in the identification of secondary path. The length of transversal filter is 300. The estimation models are as shown in Fig. 6, each line represents an iterative process of a particular filter coefficient.

Fig. 5Force diagram of vibration absorption platform

Fig. 6Identification of secondary path with different algorithm

a) Identification with normalized variable step

b) Identification with improved algorithm

In Fig. 6, the convergence speed with improved algorithm is faster than that with normalized variable step. It is shown that the improved variable step algorithm could be well applied in the identification of secondary path.

3.3. Multi-channel control algorithm

In addition to implement of control system, multi-channel vibration absorber algorithm has no essential difference compared with single-channel algorithm. First, the calculation is greatly complicated with the increase in the quantity of error sensors and vibration absorbers in the multi-channel control system. Second, each error signal collected by sensors is the effect of all vibration absorbers not a single one. So serious coupling between different channels will decrease the stability of control algorithm in the actual control system. The multi-channel control system discussed in this article includes one reference sensor, $I$ electromagnetic active absorbers and $J$ error sensors. FIR filters with length $M$ work as response function of error channel, the quantity of filters is $I*J$. In order to expound the principle of control system clearly, one reference sensor, two electromagnetic active absorbers and two error sensors are included in the algorithm flow graph.

Fig. 7Multi-channel algorithm flow graph

The reference signal becomes a diagonal matrix:

Here, every vector $X\left(n\right)=\left[x\right(n),x(n-1)\cdots x(n-L+1){]}^T$.

Desired signals $D\left(n\right)=X\left(n\right)H\left(Z\right)=\left[d_1\right(n),d_2(n)\cdots d_j(n){]}^T\text{.}$$H\left(Z\right)$work as transfer function of primary path.

Weight coefficients matrix of control filter shown as follow:

where the single weight coefficients $w_i\left(n\right)={\left[w_{i1}\left(n\right),w_{i2}\left(n\right)\cdots w_{iL}\left(n\right)\right]}^T$.

The output matrix of controllers is:

The output of a single controller at a certain moment work as $y_i\left(n\right)=w_i^T\left(n\right)x\left(n\right)$, and each output signal is in the form of vector, in other word $y_i\left(n\right)=\left[y_i\right(n),y_i(n-1)\cdots y_i(n-M+1\left)\right]$.

$S\left(Z\right)$ acted as transfer function matrix of secondary path, its dimension is $I*J$:

where $S_{ij}\left(Z\right)$ represents the transfer function from the $i$ control signal to the $j$ error sensor. Then the error signal is supposed to be:

The objective function here can be expressed as:

According to convergence rule, calculate the gradient-descent direction:

The weight coefficients of multi-channel controller update as follow formula:

In the Eq. (11), $\widehat{X}\left(n\right)$ is the result of filtering where the reference signals are filtrated by secondary path matrix, the details are as follows:

The estimation model of secondary path can be expressed as:

The secondary path estimation model matrix must be spilt for the realization of the distributed multi-channel adaptive control algorithm. For example, the secondary path estimation model of dual-channel control system is as follows:

Estimation model matrix must be put into respective channel for the realization of algorithm, so the estimation model of each channel can be expressed as follows:

${\widehat{S}}_1\left(Z\right)$ and ${\widehat{S}}_2\left(Z\right)$ are compensation for channel 1 and channel 2 in the actual control system. At the same time, excessive output of some controllers will result in damage to the vibration absorber, So the leakage factor$\lambda$ is introduced to limit the output of the controllers in the actual control system.

4.1. Output force characteristics of electromagnetic active vibration absorber

In this article, electromagnetic active vibration absorber is designed, simulated and experimental analyzed. Specific implementation method is included. The force sensors are used to collect vibration signals, the excitation current with different frequency and amplitude. The output force can be calculated according to the signals. The main experimental equipment are as follows in Fig. 8.

EM1643V function signal generator, adjustable frequency range is 0.2 Hz to 2000 Hz, JiangSu JiuNeng Electronic Technology limited company. LA-200 Power Amplifier, drive the vibration exciter, manufacturer is JiangSu LianNeng Electronic Technology limited company. WYK-20040K DC Power supply, output voltage: 0-200V; Current: 0-40A., JiangSu LianNeng Electronic Technology limited company. dSPACE controller, 20 input channels and 8 output channels, manufacturer is Germany dSPACE Company. PCB acceleration sensor, manufacturer is PCB Piezoelectric Sensor Technology (BeiJing) limited company. T460 Thinkpad, Lenovo, used for calculation. The absorbers used in experiment are self-developed and designed.

The frequency of exciting current is respectively 30 Hz, 80 Hz, 100 Hz, 150 Hz and 200 Hz. Variable control method is applied in the experiment. The current is limited in some certain frequency, then change the amplitude to get the output force. The details are given in Fig. 9.

Fig. 8Experimental equipment

As shown in Fig. 9(a), no matter the frequency is 30 Hz around the natural frequency or 200 Hz away from the natural frequency, the output force of absorber changes with the coil current approximate linearly. That is to say, the absorber is driven by a signal frequency current, output force will be a primary frequency, without too much other frequency caused by absorber. Fig. 9(b) shows that output force decreases with the increase of the frequency under same amplitude current. It is consistent with the simulation. In conclusion, linear output characteristic eliminate the influence of nonlinear output force to the adaptive vibration absorption system, basically meet the design requirements.

Fig. 9Characteristic analysis of output force

a) Output force characteristic of different coil current

b) Output force characteristic of different frequency

4.2. Multi-channel vibration absorption control with fixed frequency excitation

The experiment bench and installation of vibration absorbers are as shown in Fig. 10.

The sinusoidal signal send out by signal generator is seemed as the initial vibration, with frequency of 37 Hz and amplitude of 2. Nest the experimental plan is designed to verify the validity of the multi-channel control algorithm. The details are as follows: No. 1 and No. 2 vibration absorbers used as passive vibration absorber are installed in the experiment bench without active vibration absorber, as shown in Fig. 10. Error vibration signal collected by error sensor is seemed as the vibration evaluation index. Because the fixed structure of passive absorbers is different from that of active absorbers, No. 1 and No. 2 passive absorbers are replace by active vibration absorbers in the diagonal position. White noise sent out by controller is used to identify the model of secondary path, the order of FIR is 500. Identification of secondary path are fed back to the control system to carry out the adaptive vibration absorption experiment, in which, the orders of two control filter are 128 and the initial value is zero. Identification of secondary path are as shown in Fig. 11.

Error vibration signals before and after control collected by error sensors are shown in Fig. 12.

Fig. 10Experiment bench and installation of vibration absorbers

In Fig. 12, the absorption effect is not obvious when the 1 and 2 passive vibration absorbers are used in control system, because the excitation frequency is far from resonant frequency. While the 3 and 4 electromagnetic active vibration absorbers are totally different, the absorption effect is obvious. The output force can be changed to the excitation signal adaptively, ignoring the difference between excitation frequency and resonant frequency. The details of damping effect are as shown in Table 2.

Fig. 11Identification of secondary path

a) Identification of $S_{11}\left(Z\right)$

b) Identification of $S_{12}\left(Z\right)$

c) Identification of $S_{21}\left(Z\right)$

d) Identification of $S_{22}\left(Z\right)$

Fig. 12Error signals and spectrum before and after control

a) Error signal of 1 point

b) Error signal of 2 point

c) Error signal of 3 point

d) Error signal of 4 point

The phase-space reconstruction is used in the analysis of error vibration signals, the attractor has obviously changed after control, as shown in Fig. 13. The change of attractor indicates the validity of multi-channel control algorithm, and also indicates the error influence of nonlinear in the multi-channel control system. because the attractor does not have the characteristics of limit cycle. So, in the nest optimization algorithms, the nonlinear influence needs to be further compensated.

Fig. 13Phase-space reconstruction of error signals before and after control

a) Phase-space reconstruction of 3 error signal

b) Phase-space reconstruction of 4 error signal