Abstract
A variableadaptivelaw control algorithm for application to common problems like multiobjective control, actuator output constraints, and suboptimal adaptive laws is proposed in this paper. The multiobjective control problem of a nonlinear suspension is converted to the constrained stability problem of a sprung mass using a quarter nonlinearsuspension model. A variableadaptivelaw controller is then used, along with feedback from the output error, and considering the constraints of the actuator output. The controller modifies the adaptive law to reduce the active control force and restores it to the unsaturated zone. This ensures that the suspension system is always in a controlled state when the output saturation occurs. The controller was simulated for the following two cases: (i) a bump road and (ii) a Cgrade road. The analysis is verified by experiments in the end.
1. Introduction
The suspension system of a vehicle is a nonlinear timevarying system with uncertainties. The excitation from the road inputs energy into the suspension system. The damper consumes this energy, thereby stabilizing the whole system. In a passive suspension, the adopted value of the damping coefficient is a compromise for a variety of road conditions. In a semiactive suspension, the damping coefficient is adjustable, which enables the suspension to adapt to a variety of paved surfaces [1]. An active suspension is stabilized using an actuator, which can provide the required active force [2].
Magnetorheological damper (MRD) is a damper that uses the magnetorheological fluid (MRF). The shear stresses in the MRF can change with a varying magnetic field [3]. This property can be utilized for adjusting the damping coefficient of the MRD using a varying current that gives rise to a varying magnetic field. MRDs are widely used in semiactive suspensions. There are several studies that have used MRDs quite effectively. Both a magnetorheological damper and an adaptive slidingmode algorithm were used in a bicycle in [4]. The optimal control force could be computed by tracking the idealized model using a controller designed in [5, 6]. A special MR damper was developed in [7], and the skyhook algorithm was used to reduce the vibration acceleration of the entire vehicle. An MR damper was also used in a train in [8], and the Hinfinity algorithm served to control the vibration. Another study [9] describes a controller that was built to compensate for the uncertainties of the quarter suspension model and MR damper to improve the robustness of the system. Variable damping aperture control is another type of semiactive suspension that has been studied [10]. Active suspensions are more widely studied compared to semiactive suspensions. In [11], a combination of linearquadratic regulator (LQR) and neuralnetwork controls improve the stability of the camera. Some researchers [12, 13] studied the use of an inertia container as the active output mechanism to reduce the acceleration of the body. Suspension systems that use fuzzy control was used in an electromagnetic suspension system in [14]. A disturbance observer was established based on the slidingmode controller in [15]and verified by experiments. In [16], grey predictive control and fuzzy controllers were combined. A hybrid control algorithm was also used in [17]. An active electromagnetic suspension with good control effect in the high frequency range was designed and verified in [18], while [19] used a datadriven method to adjust a fixed order controller. Active suspension systems have also been controlled using a linear feedback controller designed especially for nonlinear systems [20] and an LQR controller combined with a risksensitive controller [21]. In the literature mentioned above, regardless of the nature of the damper, whether MRD, inertia or electromagnetic actuator, the main objective was to reduce the sprungmass acceleration, and this objective was achieved to a considerable degree as verified by experiments in several cases. However, safety and the suspension travel limit should also be considered along with the increase in ride comfort. In other words, it is not enough to reduce the sprungmass acceleration alone, and active suspension control should be treated as a multiobjective control problem.
According to [22, 23], multiobjective control of the suspension can be approximated as a control problem in which minimization of the sprungmass acceleration is the primary objective, while the dynamic tire load and suspension space limit are the constraints. Multiobjective control was realized by assigning different weights for the sprungmass acceleration, space requirements of the suspension, and dynamic tire load [24, 25]. Additionally, in realworld applications, the output force of the actuator is limited, presenting a new problem: the output force of the actuator acts as a constraint on the system [26]. To model this, an actuator saturationcontrol problem was formulated and solved [27, 28]. Active fault tolerance control was studied in [2931] and the uncertainties in output were taken into consideration [32]. A significant number of adaptivecontrol algorithms were designed to make the controlled suspensions more stable. However, there is still no satisfactory solution to the problem of finding an ideal adaptive law, which is typically chosen based on experience. In addition, there is usually no connection between the adaptive law and the output constraints on a system. Hence, there is need for further improvement of suspension control systems.
2. Governing equation for a controlled suspension
As shown in Fig. 1, the body mass of the car and the passenger mass are taken together as the sprung mass ${m}_{s}$, which varies with the load over a small range. The mass of the wheel, brake system, and other connections are lumped together as the unsprung mass ${m}_{u}$. The sprung and unsprung masses are connected by a spring and a damper ${c}_{s}$. The spring in the model is a combination of a linear spring ${k}_{s}$ and a nonlinear spring ${\delta}_{s}$. The unsprung mass is supported on the ground by an equivalent linear spring ${k}_{u}$ and a linear damper ${c}_{u}$. In an active suspension, the actuator generates an active force$u$, whose function is to improve the ride comfort.
Fig. 1A nonlinear quarter carsuspension with an active controller
According to Newton’s second law, the governing equations are:
${m}_{u}{\ddot{z}}_{u}{f}_{cs}\left({\dot{z}}_{s}{\dot{z}}_{u}\right){f}_{ks}\left({z}_{s}{z}_{u}\right)+{f}_{cu}\left({\dot{z}}_{u}\dot{q}\right)+{f}_{ku}\left({z}_{u}q\right)=sa{t}_{a,b}\left(u\right),$
where:
${f}_{cu}\left({\dot{z}}_{u}\dot{q}\right)={c}_{u}\left({\dot{z}}_{u}\dot{q}\right),{f}_{ku}\left({z}_{u}q\right)={k}_{u}\left({z}_{u}q\right),$
which are the restoring forces at the corresponding dampers and springs.
We define the state variables ${x}_{1}={z}_{s}$, ${x}_{2}={\dot{z}}_{s}$, ${x}_{3}={z}_{u}$, ${x}_{4}={\dot{z}}_{u}$, and Eq. (1) can be modified as:
${\dot{x}}_{2}=\theta \left({f}_{cs}\left({\dot{z}}_{s}{\dot{z}}_{u}\right){f}_{ks}\left({z}_{s}{z}_{u}\right)+sa{t}_{a,b}\left(u\right)\right),$
${\dot{x}}_{3}={x}_{4},$
${\dot{x}}_{4}=\frac{1}{{m}_{u}}\left({f}_{cs}\left({\dot{z}}_{s}{\dot{z}}_{u}\right)+{f}_{ks}\left({z}_{s}{z}_{u}\right){f}_{cu}\left({\dot{z}}_{u}\dot{q}\right){f}_{ku}\left({z}_{u}q\right)sa{t}_{a,b}\left(u\right)\right),$
where $sa{t}_{a,b}\left(u\right)$ represents the saturation function of the output force. $\theta =1/{m}_{s}$ is a variable parameter, which changes with the load and satisfies the equation:
In addition, since the actual actuator output has the maximum value, the active controlforce saturation function is defined as:
Vehicle suspension control can be expressed as a multiobjective control problem, whose objective is the maximization of ride comfort while ensuring the safety of the vehicle and keeping within the limited space available for the suspension system. These conditions can be expressed as:
$\left{z}_{s}{z}_{u}\right\le {z}_{\mathrm{m}\mathrm{a}\mathrm{x}},$
$\underset{x\to T}{\mathrm{l}\mathrm{i}\mathrm{m}}{z}_{s}=0,$
where ${m}_{s\mathrm{m}\mathrm{i}\mathrm{n}}$ is the minimum sprung mass. ${z}_{\mathrm{m}\mathrm{a}\mathrm{x}}$ is the maximum space available for the suspension, and $T$ is a finite time.
2.1. Controller design with an ideal actuator
We define the tracking error as ${e}_{1}={x}_{1}{x}_{1r}$ where ${x}_{1r}$ is the reference signal. It is continuous and differentiable and satisfies the condition $\left{x}_{1r}\right\le {\epsilon}_{0}$, where ${\epsilon}_{0}$ is a positive constant. In the differential form:
The ideal function of the virtual control input ${x}_{2}$ is ${x}_{2d}$ and we define the error:
Substituting Eq. (7) into (6) yields the following equation:
We use the Lyapunov candidate function:
Differentiating ${v}_{1}$ yields the following equation:
We then choose an ideal function:
where ${k}_{1}$ is a positive constant. Substituting Eq. (11) into (10) yields:
From Eq. (12) we can see that, if ${e}_{2}=0$, then ${\dot{v}}_{1}={k}_{1}{{e}_{1}}^{2}\le 0$, ensuring that ${e}_{1}$ converges to zero.
Differentiating Eq. (7) yields the following equation:
We define the term:
where $\widehat{\theta}$ is an estimate of $\theta $. The Lyapunov candidate function now considered is:
where $r$ is a positive constant. Differentiating Eq. (15) yields:
We choose:
and substitute Eq. (17) into Eq. (16) to yield the following equation:
We then define the adaptive law as follows:
Eq. (19) implies that when the rate of variation of the adaptive law rate is positive, it eventually reaches the maximum and when the rate of variation in the adaptive law is negative, the value eventually reaches the minimum. When the rate of variation of the adaptive law is set to zero, i.e., there is no variation, the value remains stable within a range.
Substituting Eq. (19) into (18) yields:
Integrating Eq. (20) yields the following:
From Eqs. (20) and (21), we can see that ${e}_{1}$, ${e}_{2}$, and $\stackrel{~}{\theta}$ are finite, and ${e}_{1}$, ${e}_{2}\to 0$ as $t\to \infty $.
The structure of the controller is shown in Fig. 2.
Fig. 2Schematic diagram of an adaptive controller
2.2. Controller design with an actuator having output constraints
In reality, the output of the actuator has the maximum value. Therefore, when the active control force exceeds the maximum value, the actuator cannot track it further. In this situation, the error ${e}_{2}$ should be compensated:
where ${k}_{3}$ is a positive constant, ${\theta}_{0}$ is the initial value of the uncertain parameter $\theta $, and $\mathrm{\Delta}u=usa{t}_{a,b}\left(u\right)$ represents the difference in the output. Substituting Eq. (22a) into Eq. (6) yields the following equation:
Substituting Eq. (23) into Eq. (10) yields:
We choose the ideal function:
Substituting Eq. (25) into Eq. (24) yields:
From Eq. (26) we can see that, if ${e}_{2}=0$, then ${\dot{v}}_{1}={k}_{1}{{e}_{1}}^{2}\le 0$, ensuring that ${e}_{1}$ converges to zero.
The active control force is given in Eq. (17). When output saturation occurs, the variable adaptive law stated below is used:
where $\tau $ is a positive constant. This particular form of the variable adaptive law, shown in Eq. (27), is chosen because:
(a) When output saturation occurs, the variation in the adaptive law is increased to reduce the active control force and restore it to the unsaturated region. This ensures that the suspension system is always in a controlled state.
(b) When output saturation occurs, $\theta $ is modified by changing the adaptive law.
We consider a Lyapunov candidate function:
Here, the set $A$ is a compact set for any $\zeta >0$:
In a compact set $A$, $\Vert \mathrm{\Delta}u\Vert $ has the maximum value. In addition, the following can be obtained according to the properties of a perfect square:
$\eta \mathrm{\Delta}u\le {\eta}^{2}+\frac{1}{4}\mathrm{\Delta}{u}^{2},$
${e}_{2}\mathrm{\Delta}u\le {{e}_{2}}^{2}+\frac{1}{4}\mathrm{\Delta}{u}^{2},$
$\stackrel{~}{\theta}\tau \le {\stackrel{~}{\theta}}^{2}+\frac{1}{4}{\tau}^{2}.$
Differentiating Eq. (28), we get:
$={k}_{1}{{e}_{1}}^{2}{k}_{2}{{e}_{2}}^{2}+(\eta {e}_{2})\dot{\eta}+{r}^{1}\stackrel{~}{\theta}\tau $
$\le {k}_{1}{{e}_{1}}^{2}\left({k}_{2}{k}_{3}{\theta}_{0}\right){{e}_{2}}^{2}\left(\frac{3}{4}{k}_{3}{\theta}_{0}\right){\eta}^{2}+{r}^{1}{\stackrel{~}{\theta}}^{2}+\alpha ,$
where, $\alpha =\frac{1}{2}{\theta}_{0}\mathrm{\Delta}{u}^{2}+\frac{1}{4}{r}^{1}{\tau}^{2}$. From Eq. (31) we can see that if the following condition:
is satisfied, we obtain:
The following equation can then be obtained from Eq. (33):
where $e$ is the exponential index. From Eq. (34) we can see that ${e}_{1}$, ${e}_{2}$, $\eta $ are finite when $t\to \infty $.
The mathematical model of the controller with output constraints is shown in Fig. 3.
Fig. 3Schematic diagram of adaptive controller with output constraints
The zero dynamic stability of the system can be proved using the following equations. When ${e}_{1}=0$, ${\dot{e}}_{1}=0$, which indicates that the output does not reach the constraint ($\eta =0$), we obtain:
After replacing $sa{t}_{a,b}\left(u\right)$ in Eq. (2d) with that in Eq. (35), the zerodynamic system obtained can be represented as:
where:
Since matrix $B$ is a Hurwitz matrix, the zerodynamic system is stable.
3. Numerical simulation
The quarter suspension model parameters are shown in Table 1.
The controller parameters are shown in Table 2.
Table 1Quarter suspension model parameters
Parameters  ${m}_{s}$  ${m}_{u}$  ${k}_{s}$  ${\delta}_{s}$  ${k}_{u}$  ${c}_{s}$  ${c}_{u}$ 
Unit  kg  kg  N/m  N/m  N/m  Ns/m  Ns/m 
Value  324  45  20000  2000  180000  1000  500 
Table 2Controller parameters
Parameters  $r$  ${k}_{1}$  ${k}_{2}$  ${k}_{3}$  ${\theta}_{0}$  ${\theta}_{min}$  ${\theta}_{max}$  $\tau $  $a$  $b$ 
Value  0.01  10  10  10  1/300  1/350  1/300  0.5  1250  700 
Simulations were carried out for the following three states:
(a) Variableadaptivelaw controller with output constraint,
(b) Fixedadaptivelaw controller with output constraint,
(c) Passive states.
3.1. Frequency response
As shown in Fig. (4a), due to the output constraint of the actuator, the sprungmass acceleration of the nonlinear suspension with a fixedadaptivelaw controller does not differ much from that of a passive suspension. However, it has a slightly smaller value than the sprung mass acceleration of the passive suspension. In other words, given the output constraint, the equilibrium state of the adaptivecontrol system is destroyed, and the controller performance degrades. However, the sprungmass acceleration of the nonlinear suspension with a variableadaptivelaw controller is much smaller than the passive one, and the controller performance is very good thereby validating its efficiency. As can be seen from Fig. (4b), the dynamic stroke of the active suspension is increased at an excitation frequency greater than 25 rad/s, which ensures good controller performance. The effect on the dynamic load of the tire is small and becomes progressively smaller for lower frequencies.
Fig. 4Frequency response of: a) sprung mass acceleration; b) suspension space; c) dynamic load of the tire
a)
b)
c)
3.2. Timedomain analysis
The simulation is carried out for a bump road and C level road. The bumproad surface is generated according to the following function:
3.2.1. Comparison with a bump road
Fig. 5 shows the response of the suspension to a bump road. The bump road can be treated as a case of transient unbalanced excitation, which can reflect the performance of a controller with an asymmetrical output constraint. In Fig. (5a), the sprungmass acceleration of the active suspension with a variableadaptivelaw controller drops to zero rapidly after a slight increase. It then remains stable. For the fixedadaptivelaw controller, it will continue to increase in the reverse direction, and then gradually decrease to zero. This shows the superiority of the variableadaptivelaw controller. To maintain the small sprungmass acceleration for as long as possible, the output force with avariableadaptivelaw controller decreases to zero for longer periods. This results in a longer time to stabilize for tracking error and suspension travel – see Figs. (5b), (5c) and (5d). In Fig. (5e), the dynamic load of the tire increased significantly after a sudden impact, but these temporary changes do not affect vehicle safety.
Fig. 5Performance comparison with a bump road: a) sprung mass acceleration; b) tracking error; c) output force; d) suspension space; e) dynamic load of tire
a)
b)
c)
d)
e)
3.2.2. Comparison with the Clevel road
Fig. 6 shows the response of the suspension for a Clevel road when the vehicle speed is 20 m/s. As shown in Fig. 6, for a random road, when the calculated active force is less than the actuator output constraint, the performance of the variableadaptivelaw controller is identical to the fixed adaptivelaw controller, and the sprung mass acceleration is zero. When the driving force exceeds the actuator output constraint, the system balance is disturbed. Until this point, the variableadaptivelaw controller is superior to the fixed adaptivelaw controller with a smaller sprungmass acceleration (Fig. (6a)), smaller tracking error (Fig. (6b)), and smaller suspension travel (Fig. (6d)). However, it can be seen from Fig. (6e) that the controller cannot reduce the dynamic tire load when the output is constrained. In cases where the road excitation is very strong, even the controller with no output constraints imposed on the system approaches its limit. If the output control force is restricted at this moment, the control effect on dynamic tire load will also necessarily be reduced.
Fig. 6Performance comparison with the Clevel road: a) sprung mass acceleration; b) tracking error; c) output force; d) suspension space; e) dynamic load of tire
a)
b)
c)
d)
e)
Fig. 7 shows that, for the lowfrequency range, the sprungmass acceleration of the active suspension is smaller than the passive suspension. However, when the frequency is greater than 15 rad/s, the effectiveness of the fixedadaptivelaw controller decreases significantly because of the output constraint on the actuator. The sprungmass acceleration with this controller is only slightly less than that for the passive suspension. However, the performance of the variableadaptivelaw controller is still acceptable.
The rootmeansquare values of the sprungmass acceleration for two kinds of road surfaces are shown in Table 3.
Fig. 7PSD (power spectral density) of sprungmass acceleration in frequency domain
Table 3The rootmeansquare value for sprungmass acceleration
Road type  Bump road  Cgrade road 
Passive (m/s^{2})  1.4018  1.5897 
Variable adaptive law (m/s^{2})  0.2603  0.3163 
Fixed adaptive law (m/s^{2})  0.2739  1.4861 
It can be seen from Table 3 that the reduction in the sprungmass acceleration is best with the variableadaptivelaw controller. Compared to the passive suspension, the sprungmass acceleration decreased by 81.43 %, 80.46 %, and 80.19 %, 6.52 %, respectively for both kinds of roads.
4. Experimental verification
The experiment detailed in this section was carried out to verify the effectiveness of the designed controller. The experimental setup is shown in Fig. 8. The actuator used in the experiment consists of a rotating motor, a rack, and a pinion.
The base excitation is generated by the hydraulic cylinder. Positioned between the hydraulic cylinder and the unsprung mass is a spring that approximates a tire. The sprung mass and the unsprung mass are connected by a damper and an actuator. The controller considered in this study operates on an embedded system.
Four sensors are used in the test to convert the analog data pertaining to physical quantities into electrical signals. The displacement sensor 1 (TWZ, 500 mm) is fixed on the test equipment with its movable rod fixed on the upper desk. It collects data regarding the displacement of the sprung mass. The acceleration sensor 1 (CSLAS02, ±10 g) is fixed at the center of the upper desk and measures the acceleration of the sprung mass. The displacement sensor 2 (TWZ, 250 mm) is fixed to the upper desk with its movable rod attached to the bottom desk in order to measure the displacement between the sprung and unsprung masses. Acceleration sensor 2 (CSLAS02, ±10 g) is attached to the center of the bottom desk and measures acceleration of the unsprung mass. DC power supply is provided for all four sensors. A data acquisition module installed on an embedded controller is responsible for collecting the signals from the four sensors. The velocity values can be obtained by differentiating the measured displacements from the sensor, and these comprise the inputs to the control system. With the data soobtained, the proposed algorithm can run on the embedded controller. The resulting data, which is the output of the control system, is used to control an actuator attached between the upper and lower desks.
During the experiment, the physical quantities are measured by the sensors and transferred to the controller. The controller generates a corresponding signal which is supplied to the actuator. The actuator generates the active force. Thus, the experiment simulates the reallife active control of the suspension system.
Fig. 8Experimental setup
Fig. 9Experimental results for: a) sprungmass acceleration; b) tracking error; c) PSD of sprungmass acceleration
a)
b)
c)
Fig. 9 shows the experimental results for the Clevel road. The parameters are the same as those in Table 1 and Table 2. From Fig. (9a) and (9c), it can be seen that the sprungmass acceleration for the variableadaptivelaw controller is smaller, and the plots of the tracking error shown in Fig. (9b) confirm this. In practice, the actuator always fluctuates between the maximum limits, when it reaches the constraints because it can maintain only a constant value in a simulated state. This means that it cannot switch instantly, which explains the variation between the experiment and the results of the theoretical analysis. In addition, the time delay and inaccurate estimate of platform parameters affect the result. However, in general, the variable adaptive law controller is superior to the fixed adaptive law controller.
5. Conclusions
A variableadaptivelaw control algorithm was studied for application to problems like multiobjective control, actuator output constraints, and suboptimal adaptivelaw choices. With an ideal actuator, the conventional adaptive control algorithm works well. However, when the output constraints of the actuator are taken into consideration, the conventional adaptivecontrol algorithm no longer performs satisfactorily. This problem is solved in the proposed controller by adjusting the adaptive law. From the experiment result, we can conclude that:
1) Control over the system can be maintained under all conditions. With a variableadaptivelaw controller, the output constraints can be met. This is critical in actual engineering applications.
2) Better riding comfort can be achieved with the variableadaptivelaw controller compared with a fixed adaptivelaw controller when output constraints are met. Smaller sprungmass accelerations and tracking errors are observed in both the simulation and experiment, confirming the outstanding performance of this controller.
3) The controller can be modified according to specific requirements. It has been proved that the controller is stable with changes like variations in the adaptive law. This means that any adaptive controller has the potential to be a better one in some extreme environment like output saturation. This is a significant improvement in the active control of suspension systems.
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