Design of a denoising hybrid fuzzy-pid controller for active suspension systems of heavy vehicles based on model adaptive wheelbase preview strategy

2.1. Half-vehicle model

There are two types of heavy vehicle suspension models; they are unconnected suspensions and connected suspensions. As most of the commercial vehicles use the unconnected suspension, this research mainly focuses on the unconnected model. In this section, an unconnected half vehicle model is established to illustrate a thorough study on the basis of the modeling and control of a hydraulic-actuated active suspension system.

The half vehicle model in Fig. 1 depicts that the vehicle is set to move from right to left at a constant speed $v$. The road input for the front wheel $Z_{01}\left(t\right)$ and the rear wheel $Z_{02}\left(t\right)$ are assumed to have the same magnitude but different phases:

The static equilibrium position is used as the origin of both displacements of the center of mass and angular rotation of the vehicle body. The equations of motion for the front and rear un-sprung masses are, respectively:

where $U_f$ and $U_r$ are the front and rear active forces respectively which can be provided by hydraulic actuators.

Then, Eqs. (2) and (3) can be reorganized as:

In Eq. (5), the pitch angle $\theta$ is assumed to be very small, and then it can be used as:

where $\theta$ and $Z_c$ can be obtained by:

Eqs. (8) and (9) are then brought into Eqs. (4) and (5), allowing the acceleration of the front and rear sprung masses to be reorganized as follows:

With Eqs. (10) and (11), the equations of motion for the half-car model can be constructed.

2.2. Road roughness profile

To simulate the real running process of the vehicle properly, the road input should be defined first. There are many ways describing the road input analytically, which can be classified as shock and vibration. The International Organization for Standardization (ISO) has proposed a series of standards of road roughness classification using power spectrum density (PSD) values (ISO 1982). The standard rough road input equation can be obtained as:

where $f_0$ is the low cutoff frequency, $G_0$ is the road roughness coefficient and $w_0$ is a Gaussian white noise.

In this work, the excitation profiles of shock and vibration pavement are both built. The shock road excitation is a discrete event of relatively high intensity, which is applicable as the road input to the performance evaluation with different control strategies and performance evaluation of model adaptive wheelbase preview. Meanwhile, the vibration road excitation is characterized by continuous excitations, which can be reasonable used in the performance evaluation of wavelet denoising filter.

Fig. 2Construction and signal flow of the proposed control method and simulation model

3.1. Design of hybrid fuzzy-PID controller

The fuzzy and PID control can be used in two different situations, which are determined by a threshold. The threshold was set to be 0.5 through a trial and error method. If the signal of rear sprung mass acceleration is smaller than 0.5, (i.e. the low deviations from the reference point), the PID controller is used, otherwise the fuzzy controller takes control action. This control strategy can provide a fast system response and reduce the system overshoot. In fact, these two control strategies are operated independently at any time instance.

Conventional PID control can be described by using a transfer function as:

where $U\left(s\right)$ represents the controller output force which is produced by a hydraulic actuator, $E\left(s\right)$ is the position error, $K_p$, $T_i$, and $T_d$ denote the proportional gain, integral gain, and derivative gain, respectively; $\alpha \left(s\right)$ is a transfer function between the active force and the control signal for the hydraulic actuator.

The main function of fuzzy logic control is to deal with uncertain and imprecise problems, and the multiple-control objectives are considered in this work. Linguistic variables (small (S), medium (M), large (L), etc.) are used to represent the domain knowledge, with their membership values lying between 0 and 1. Basically, the proposed fuzzy controller has the following components: a) fuzzification module, which is used to scale and map the measured variables to suitable linguistic variables; b) knowledge-based linguistic control rules, via which the input linguistic variables can trigger some actions with the fuzzy control rules. With this idea, multiple-control objectives can be achieved; c) decision making logic, by which the final fuzzy control action can be produced; and d) defuzzification module, which is used to scale and map the linguistic control variables to a non-fuzzy output to the hydraulic actuator.

In this study, the tire deflection ($Z_{02}$-$Z_2$), suspension deflection ($Z_2$-$Z_r$), the acceleration of the rear sprung mass ${\ddot{Z}}_r$ and its derivative ${\stackrel{⃛}{Z}}_r$ are the four input variables to the fuzzy controller while the control voltage $V_r$ producing active force $U_r$ is its output. Triangular membership functions are used to represent the different linguistic variables for ($Z_{02}$-$Z_2$), ($Z_2$-$Z_r$), ${\ddot{Z}}_r$, ${\stackrel{⃛}{Z}}_r$ and $V_r$ because of its popularity. The linguistic variable sets for ${\ddot{Z}}_r$, ${\stackrel{⃛}{Z}}_r$ and $V_r$ are all (S, M, B) where S, M, B stand for Small, Medium and Big respectively. Four scaling factors $G_{Z_{02}-Z_2}$, $G_{Z_2-Z_r}$, $G_{{\ddot{Z}}_r}$ and $G_{{\stackrel{⃛}{Z}}_r}$ are used for ${\ddot{Z}}_r$ and ${\stackrel{⃛}{Z}}_r$ respectively to map them to the respective universes of discourses. The fuzzy control voltage $V_{r-f}$ is determined by using Eq. (14):

In Eq. (14), $F$ denotes the fuzzy relation defined in the fuzzy rule base, and $S$ is the transfer function between the active force and the control signal. The conventional max-min composition rule of inference is used for achieving the appropriate fuzzy control voltage, $V_{r-f}$. The centroid method is used for defuzzification. In this method, the centroid of each output membership function for each rule is evaluated first and the final control voltage, $V_r$, is calculated by multiplying the gain $G_{U_r}$ with the average of the individual centroid, weighted by their heights as Eq. (15):

3.2. Design of model adaptive wheelbase preview

The model adaptive wheelbase preview approach is composed of an adaptive front quarter vehicle model and a wheelbase preview algorithm. Before the integration of the adaptive quarter vehicle model and the wheelbase preview algorithm, a big concern coming upon is how to ensure that the parameters from the adaptive quarter vehicle model can be effectively transferred to the wheelbase preview controller without any uncertain disturbance. To address this problem, a classical disturbance observer (DOB) is employed inside the wheelbase preview approach. The main function of the DOB is to overcome the disturbance of the signals from the vehicle model and then delivers the necessary signals to the wheelbase preview controller.

In the framework of the adaptive quarter car model, the most important parameters for model update are the real sprung mass acceleration and active force of the front part of the vehicle. To obtain the above two parameters, an accelerometer is placed on the front part of the sprung mass, while a load cell is mounted on the top of the actuator. Then, the two real sensed parameters are sent to a front quarter vehicle model to update the current status of the vehicle timely, and the corresponding updated parameters are then used to construct the adaptive quarter vehicle model. By this way, the relations between the adaptive quarter vehicle model and the DOB can be represented as:

Regarding the wheelbase preview algorithm, the stable signals from DOB are first received by a nominal quarter vehicle model $G_n$. The role of the wheelbase preview algorithm is to reduce the sprung mass acceleration of the rear part of the vehicle, so $G_n$ has the same structure but different parameters with that of the front quarter vehicle model. Moreover, DOB consists of an inverse transfer function $G_n^{-1}$ and a filter $Q\left(s\right)$. From the nominal quarter vehicle model, it can be observed that the hysteresis term $X_{01}$ is the source of the system nonlinearity. $X_{01}$ is considered to be an uncertainty or external disturbance to the system. If $X_{01}$ can be suppressed, the nominal quarter vehicle model will be equivalent to a second-order linear system. In this work, the hysteresis effect $X_{01}$ is compensated by using the DOB. Thus, as a linear model, the transfer function between the sprung mass acceleration ${\ddot{X}}_2$ and the active force $U$ of the nominal quarter vehicle model is:

and this means the input is the active suspension force $U$, and the output is acceleration ${\ddot{X}}_2$.

So, the inverse of the transfer function of Eq. (18) is:

In many cases, the inverse of the transfer function cannot be obtained because the relative degree (the order of the denominator minus the order of the numerator) is negative after applying an inverse operation, and thus cannot be computed. A filter $Q\left(s\right)$ is then adopted to make the DOB proper and positive, so that its inverse can be computed. Such an operation is possible because after the inverse of the transfer function is multiplied by $Q\left(s\right)$, the relative order of $G_n^{-1}\left(s\right)$ can be nonnegative by letting the relative degree of $Q\left(s\right)$ be equal to or greater than that of $G_n\left(s\right)$. So, the relative degree of the filter should be no less than that of the quarter-car model. Here, the relative degree of the nominal model of the quarter-car suspension $G_n\left(s\right)$ is zero, and so the filter $Q$ is able to start the relative degree from zero. For convenience, this case sets the $Q$ filter to be 1.

3.3. Design of wavelet denoising filter

The main target of the wavelet denoising filter is to suppress the noisy part of the measured signals. So, the discrete wavelet transform (DWT) is used in this work to discriminate the noisy signal of the state variable ${\ddot{Z}}_r$. The general wavelet noise filtering procedure can be described as follows [30]: a) Apply wavelet transform to the noisy signal to produce the noisy wavelet coefficients to a suitable level. b) Select appropriate threshold limit at each level and threshold method to remove the noise. c) Conduct inverse wavelet transform of the threshold wavelet coefficients to obtain a filtered signal. Firstly, the DWT can be represented as:

where $\mathbf{w}$ is a vector with regard to the wavelet transform coefficient, $\mathbf{W}$ is the matrix of the wavelet filter coefficients and $f$ is the discrete signal. In order to effectively and efficiently filter the noise in the state variable, “mother wavelet” must be selected carefully. The mother wavelet $\mathbf{\psi }$ can be obtained as:

where $p$ is a scaling variable, $q$ is a translation variable and $t$ is time. The choice of mother wavelet in this work is based on the Pearson’s product-moment coefficient of the original signal and the filtered signal over many tests. Based on the Pearson’s product-moment coefficient, the Daubechies wavelet Db5 is used as the “mother wavelet” in this work.

With selected mother wavelet, the decomposition and reconstruction of the signal can be achieved by Mallat method [31] as:

where $s$ is the scaling variable and $l$ is the location index. Then, by constraining the $\mathbf{\psi }$, $s$ and $l$, an orthogonal decomposition can be achieved. In this work, both the threshold limits and the level of decomposition were set by many simulation trials. The final threshold limits are 5 times of the standard deviation of the wavelet-transformed signal, and the level of decomposition is set to be 3. As a result, the final filtered signal $f$ can then be obtained as:

4.1. Analysis of different control strategies on shock excitation

With the road input of shock excitation in Fig. 3, the numerical simulation with different control strategies are conducted based on the system parameters in Tables 1 and 2. Upon completion of the numerical simulation, the root mean square (RMS) values of the performance indices with different control methods are obtained in Figs. 4-7, in which a smaller RMS value can be regarded as a better performance. To further reveal the performance of different methods, a more detailed IAE and MO of the four performance indices with different control strategies are summarized in Table 3.

Fig. 4 shows the comparison of the sprung mass acceleration with different control strategies. It can be seen that the proposed FPMWW method obtains the best performance in both the front and rear sprung mass acceleration, whereas the fuzzy method performs the worst in both parts. Moreover, by examining Table 3, the performance of the proposed FPMWW is superior to that of the PMWW method, which proves the advantages of the integration of the fuzzy and PID control methods. Furthermore, the PMWW method exceeds the SMC method in reducing the sprung mass acceleration. The above evidences verify the model adaptive wheelbase preview method and wavelet denoising filter can effectively improve the ride comfort.

Fig. 3Road input of shock excitation

Fig. 5 demonstrates the comparison of the tire deflection with different control strategies. It can be concluded from Fig. 5 that all the control methods hold nearly the same performance in front tire deflection, whereas fuzzy control method behaves the worse road holding capacity in the rear suspension. By examining Table 3, it can be also found that the PMWW method outperforms the other methods in both front and rear tire deflections, but not very obvious as compared with the proposed FPMWW method. In addition, the proposed FPMWW method obtains the same MO as the PMWW method, which also proves the effectiveness of the proposed method.

Fig. 4RMS values of sprung mass acceleration with different controllers under shock road excitation

a) Front sprung mass acceleration

b) Rear sprung mass acceleration

Fig. 5RMS values of tire deflection with different controllers under shock road excitation

a) Front tire deflection

b) Rear tire deflection

Fig. 6 illustrates the pitch acceleration of the vehicle body with different control strategies. It is manifest that the proposed FPMWW and SMC methods are acceptable in maintaining small pitch acceleration, while the PMWW and the fuzzy control methods behave rather worse. Table 3 also shows that the IAE value and MO of the pitch acceleration with the proposed FPMWW method outperforms the other control methods.

Fig. 7 depicts the suspension deflections with different control strategies. With the summarized data in Table 3, it can be found that the PMWW method is the best in terms of reduction of the front and rear suspension deflections, with an improvement of 30.84 % and 78.19 % respectively as compared with the fuzzy control method, but it does not surpass the proposed FPMWW method too much. Besides, the fuzzy control method performs the worse in the rear suspension deflection, which is far from satisfaction for a heavy vehicle. Overall, the above results show that the proposed FPMWW method is effective under shock excitation.

Fig. 6RMS values of pitch acceleration with different controllers under shock road excitation

Fig. 7RMS values of suspension deflection with different controllers under shock road excitation

a) Front suspension deflection

b) Rear suspension deflection

4.2. Analysis of model adaptive wheelbase preview method

This test mainly exams the effectiveness of the wheelbase preview method with the consideration of updatable vehicle model. The model adaptive method is accomplished by receiving the variation of the updated load signal that is sensed by the load cell. In the test, the mass of the vehicle body is increased from 7200 kg to 13000 kg (see Table 1), and then the front sprung mass is changed from 3500 kg to 5720 kg (see Table 2) which is detected by the load cell. The value of 5720 kg in Table 2 implies that the updated model information is introduced in the process of wheelbase preview. In this test, the road input is also the shock excitation and the test period is 10 seconds. The parameters of the front quarter vehicle with model update can be found in Table 2.

Figs. 8-11 concisely indicate the performance of the proposed controller with and without model adaptive wheelbase preview consideration. It is distinct that the four indices used for quantifying the vehicle performance are improved at a certain degree.

Fig. 8Sprung mass acceleration with and without model adaptive wheelbase preview

a) Front sprung mass acceleration

b) Rear sprung mass acceleration

Table 4 reveals that the sprung mass acceleration, the tire deflections, the pitch acceleration and the suspension deflections in both the front and rear parts of the vehicle can be improved by the model adaptive wheelbase preview method. With this method, the IAE values of the tire and suspension deflections in the front and rear parts of the vehicle are significantly decreased by 16.21 %, 22.49 %, 14.17 % and 28.03 %, respectively. In addition, the IAE values of the front and rear sprung mass acceleration, as well as the pitch acceleration are reduced by 5.31 %, 8.99 % and 9.84 % respectively, which are also in a satisfactory range. However, the maximum overshoot of sprung mass acceleration and pitch acceleration with model adaptive wheelbase preview are increased as compared with that without model adaptive wheelbase preview. Nevertheless, the increase of the aforesaid maximum overshoot is very small and viewed as a random phenomenon, which is insignificant to the general performance of vehicle. Overall, the aforesaid results demonstrate that the proposed model adaptive wheelbase preview method is effective.

Fig. 9Tire deflections with and without model adaptive wheelbase preview

a) Front tire deflection

b) Rear tire deflection

Fig. 10Pitch acceleration with and without model adaptive wheelbase preview

Fig. 11Suspension deflections with and without model adaptive wheelbase preview

a) Front suspension deflection

b) Rear suspension deflection

4.3. Analysis of wavelet denoising filter

To examine the effectiveness of the wavelet denoising filter, a test was conducted based on the proposed controller with and without the wavelet denoising filter simultaneously. Apart from the noise disturbance from the vibration excitation of the road input (Fig. 12), an additional noise is also introduced (with 0.005 m/s² Hz) so that the denoising performance of the proposed wavelet method can be greatly validated. Figs. 13-16 reveal the four performance indices with and without wavelet denoising filter. It can be seen clearly from Fig. 13 that the ride comforts of the front and rear suspensions are both improved and the noise disturbance are dramatically reduced. At the same time, it can also be found in Figs. 14 and 15 that the proposed controller with wavelet denoising filter can reduce the noise disturbance and improve the corresponding vehicle performances to some extents. In addition, the control performance with wavelet denoising filter in suspension deflection seems the same as that of without wavelet denoising filter, as shown in Fig. 16.

Fig. 12Road input of vibration excitation

The according data of the performance evaluation of the proposed controller with and without wavelet denoising filter is summarized in Table 5. As compared with the system without wavelet denoising filter, the sprung mass acceleration and tire deflection of the front part of the vehicle are reduced by 2.98 and 2.29 %, respectively. It is also noted that the sprung mass acceleration and tire deflection of the rear part of the vehicle are remarkably reduced by 9.52 % and 15.55 %, respectively. Moreover, the pitch acceleration and the front and rear suspension deflections are all reduced slightly by 0.84 %, 0.14 % and 0.46 %, respectively. In a nutshell, the promising results show that the wavelet denoising filter is effective.

Fig. 13Sprung mass acceleration with and without wavelet denoising filter

a) Front sprung mass acceleration

b) Rear sprung mass acceleration