Adaptive control of a nonlinear suspension with timedelay compensation
Jun Yao^{1} , Jin Qiu Zhang^{2} , Ming Mei Zhao^{3} , Xin Li^{4}
^{1, 2, 3, 4}Academy of Armored Force Institute, Beijing, China
^{1}Corresponding author
Journal of Vibroengineering, Vol. 21, Issue 3, 2019, p. 684695.
https://doi.org/10.21595/jve.2019.20077
Received 14 July 2018; received in revised form 3 January 2019; accepted 14 January 2019; published 15 May 2019
JVE Conferences
This paper addresses the challenge of predictive control of a quartercar nonlinear suspension and low controllerprecision. This is done by designing and implementing an adaptive controller with timedelay compensation. First, a realtime control model is created. Then, timedelay compensation is realized and both frequencydomain and timedomain simulation of the controller performance are conducted. According to the simulation results, the sprungmass acceleration of the suspension controlled by an adaptive controller with timedelay compensation is superior to that without timedelay compensation. Both the period to settle down and the peak of vibration acceleration are smaller. This means the proposed controller is capable of dealing with problems including variable time delay, nonlinear vibration and predictive control.
 The structure of this algorithm can deal with nonlinear problems, so it can solve the nonlinear items of suspension system, such as nonlinear coefficient or nonlinear stiffness.
 For the full frequency band, a satisfactory control effect can be achieved as well as significantly reduced sprungmass acceleration.
 With the proposed controller, the problem of time delay of the front suspension can be solved by a compensator.
Keywords: timedelay compensation, tracking error, adaptive controller, nonlinear suspension, active control.
1. Introduction
Much research has been conducted on the timedelay problem in system control processes. Examples include the semiactive control of singleDOF oscillators [1], the tracking control of reference models [2], the stability analysis of systems with timedelay effect [3], and the reduction of timedelay through improved hardware processing [4]. The timedelay of any given system consists of both variable delay and fixed delay.
For a structurally complex system, system delay is usually variable during the control process. Researchers [5] analyzed the effect of timedelay on overall stability for highorder nonlinear systems with feedforward control. A similar case is the global output feedback control of an uncertain nonlinear system with unknown timedelay [6]. A neural network system is another complex system, and studies of its timedelay include systemstate estimation that considers the effect of variable timedelay [7], stability conditions estimation [8], and its improvements [9]. Others have studied the effect of both timedelay and nonlinear perturbation on the finitetime stability of discrete systems [10] as well as the effect of variable timedelay on the masterslave synchronization conditions of Lur’e chaotic systems [11].
When the structure of a system is relatively simple, its timedelay is usually a constant value such as the timedelay of a car suspension with active control. To reduce the effect of timedelay, one research group [12] used Smith compensation. Another group [13] performed a coordinate transformation to convert a timedelay system into a timedelayfree system. Yet another group [14, 15] converted the timedelay into an error of the secondary path and reduced the effect of timedelay using error control. One study [16] conducted indepth research of timedelay in magnetorheological dampers. In addition, both ${H}_{\mathrm{\infty}}$ controllers [1719] and sliding mode controllers [20] have been used extensively with linear systems to reduce the effects of timedelay. The above studies aim to improve systemcontrol precision by reducing the effect of timedelay. Another feasible way of achieving this aim is to use predictive control.
Two studies [21, 22] investigated predictive control to optimize linear systems. However, the results are inapplicable with respect to the control of nonlinear carsuspensions. The key for predictive control of nonlinear cars lies in the collection of road information using the front tires, which enables predictive control of the rear tires [2328]. In this case, there is no predictive control of the front suspension, so the control effect of the entire car is affected. The above studies helped to eliminate the effect of timedelay but there is room for improvement.
Furthermore, the above studies indicate that a highprecision controller like the ${H}_{\mathrm{\infty}}$ controller is effective in a linear system but tends to be less effectivein a nonlinear system. With a variable timedelay in a nonlinear system, the controllers without a predictator reported so far are inadequate in decreasing the vibration of car body. The predictive control cannot solve time delay problem of the front suspension. In other words, the timedelay is not completely solved. Inspired by researches done by Wu Deng, et al. [29, 30], this paper introduces an adaptive controller with the aim to improve the active control of quartercar nonlinear suspensions, while considering the timedelay effect. The proposed method addresses the problems of the inefficient of a controller used in a nonlinear system, the inadequate of a controller used to eliminate the variable time delay and the low precision of a controller used in front suspension.
This paper is organized as follows: Chapter 2 describes a control model for quartercar nonlinear active suspensions and an adaptive controller with timedelay compensation. Chapter 3 focuses on the simulation of the designed controller. This is followed by an experimental verification in Chapter 4, while Chapter 5 summarizes the conclusions of this paper.
2. Mathematical model for quartercar nonlinear active suspensions
Shown in Fig. 1 is the quartercar suspension model, where body mass and passenger mass are modeled as a single sprung mass ${m}_{s}$ (which varies with the load within a small range, and it has both upper and lower limits). Tire mass, brake system, and other link masses are modeled as unsprung mass ${m}_{u}$. Sprung mass and unsprung mass are connected via a spring and damper ${c}_{s}$. The spring is modeled as linear spring ${k}_{s}$ and nonlinear spring ${\delta}_{s}$. The unsprung mass is supported by the ground via an equivalent linear spring ${k}_{u}$ and linear damper ${c}_{u}$. For the active suspension, the actuator generates an active control force $u$ to improve comfort. $q$ represents the unevenness of the road.
Fig. 1. Nonlinear quartercar suspension with active controller
According to Newton’s second law, the dynamic equation of the quartercar suspension model is:
where:
${f}_{ks}\left({z}_{s}\right(t){z}_{u}(t\left)\right)={k}_{s}\left[{z}_{s}\right(t){z}_{u}(t\left)\right]+{\delta}_{s}[{z}_{s}\left(t\right){z}_{u}\left(t\right){]}^{3},$
${f}_{cu}\left({\dot{z}}_{u}\left(t\right)\dot{q}\left(t\right)\right)={c}_{u}\left[{\dot{z}}_{u}\left(t\right)\dot{q}\left(t\right)\right],\mathrm{}\mathrm{}\mathrm{}{f}_{ku}\left({z}_{u}\left(t\right)q\left(t\right)\right)={k}_{u}\left[{z}_{u}\left(t\right)q\left(t\right)\right].$
After defining the state variables ${x}_{1}={z}_{s}\left(t\right)$, ${x}_{2}={\dot{z}}_{s}\left(t\right)$, ${x}_{3}={z}_{u}\left(t\right)$ and ${x}_{4}={\dot{z}}_{u}\left(t\right)$, Eq. (1) can be rewritten as:
where, $f(z,t)={f}_{cs}\left({\dot{z}}_{s}\right(t){\dot{z}}_{u}(t\left)\right){f}_{ks}\left({z}_{s}\right(t){z}_{u}(t\left)\right)$. $\theta =1/{m}_{s}$ is a parameter that varies with load and meets the condition:
Car suspension control is a multiobjective control problem, through which we can not only improve ride comfort but also guarantee a safe performance and to stay within the suspension limits. These conditions can be expressed as:
where, ${z}_{\mathrm{m}\mathrm{a}\mathrm{x}}$ represents the maximum working space of the suspension; $T$ represents a finite period of time.
The tracking error is defined as ${e}_{1}={x}_{1}{x}_{1r}$; ${x}_{1r}$ represents a reference signal thatis continuous and differentiable, and $\left{x}_{1r}\right\le {\epsilon}_{0}$. Its differential form is:
${x}_{2}$ is selected as the virtual control input of the above formula, and its ideal function is ${x}_{2d}$. At the same time, a timedelay is unavoidable in a realistic control system, so the control effect of the controlled suspension is affected. At this point, a compensation is needed. The error is defined as:
After substituting Eq. (6) into Eq. (5), we obtain:
After taking the derivative of Eq. (6), we can write:
We then select the following ideal function:
where, ${k}_{1}$ represents a positive constant.
We define:
where, $\widehat{\theta}$ represents the estimate of $\theta $.
We define the adaptive rate as:
where, $r$ is a positive constant.
And set the active control force as:
We then consider the Lyapunov candidate function:
For any $\zeta >$ 0, a set $A$ is always a compact set:
In a compact set $A$, $\Vert {\int}_{t\tau}^{t}u\left(s\right)ds\Vert $ has a maximum value, and, given that $u\left(t\right)$ is continuous, $\Vert u\left(t\right)u(t\tau )\Vert $ also has a maximum value. In addition, considering the perfect nature of the square, we can write:
After taking the derivative of $v$ and substituting it into Eqs. (6)(9), (11) and (12), we obtain:
where, $\alpha =\frac{1}{4}\left[u\right(t)u(t\tau ){]}^{2}$. As can be seen from Eq. (16), if:
We can formulate:
to obtain:
According to Eq. (19), when $t\to \mathrm{\infty}$, ${e}_{1}$, ${e}_{2}$, ${\int}_{t\tau}^{t}u\left(s\right)ds$ has a boundary.
A schematic of the controller with timedelay compensation is shown in Fig. 2.
Fig. 2. Adaptive controller with timedelay compensation
3. Numerical simulation
The parameters for the quartercar suspension model are shown in Table 1.
The controller parameters are shown in Table 2.
Table 1. Parameters used in the quartercar suspension model
Parameter

${m}_{s}$

${m}_{u}$

${k}_{s}$

${\delta}_{s}$

${k}_{u}$

${c}_{s}$

${c}_{u}$

Unit

kg

kg

N/m

N/m

N/m

Ns/m

Ns/m

Numerical value

324

45

20,000

2,000

180,000

1,000

500

Table 2. Controller parameters
Parameter

$r$

${k}_{1}$

${k}_{2}$

${\theta}_{min}$

${\theta}_{max}$

Numerical value

0.01

10

10

1/350

1/300

A simulation analysis was conducted for the following scenarios:
(a) Adaptive control with timedelay compensation ($\tau =$ 0.01 s),
(b) Adaptive control without timedelay compensation ($\tau =$ 0.01 s),
(c) Passive state.
3.1. Frequency domain analysis
As shown in Fig. 3(a), the new controller can ensure a satisfactory ride comfort. In addition, for the full frequency range and despite delays in the control system, the sprung mass acceleration in the controlled state remains below that of the passive state. This meets the requirements for highprecision system control.
As shown in Fig. 3(b) and Fig. 3(c), within the lowfrequency range (< 20 rad/s), both the dynamic movement of the suspension and the dynamic load of the tire with adaptive control are reduced.
Fig. 3. Frequency response of: a) sprung mass acceleration, b) suspension space, c) dynamic load of tire
a)
b)
c)
For the mediumfrequency range (20 rad/s80 rad/s) both dynamic movement of the suspension and the dynamic load of the tire have increased due to the adaptive control. They exceed that of the passive suspension, and a peak value is produced within the frequency band.
For the highfrequency range (> 80 rad/s), the dynamic movement of the suspension and the dynamic load of the tire with adaptive controller are similar to the passive suspension. In addition, Fig. 3 suggests that timedelay compensation can increase the controller effect and reduce sprungmass acceleration.
3.2. Time domain analysis
A simulation was performed fortwo road types: bump road and C level road. The simulation for the bump road was conducted using the following function:
The Clevel road is the most common road type for cars, and its frequency band covers the points of resonance for both sprung mass and unsprung mass.
Fig. 4 shows the suspension response for two road types. According to Eq. (20), the duration from zero to peak value point of the bump road is 0.125 s, and the delaytime of the controller is 0.01 s. In other words, the suspension was set to a controlled state 0.01 s after excitation, and a delay of 0.01 s is maintained afterwards. Therefore, given that the delay is less than 0.125 s, the suspension is already in the controlled state before it reaches the peak value for the bump road. However, if the delaytime exceeds 0.125 s, the maximum sprungmass acceleration of the controlled suspension is equal to that of the passive suspension. In other words, controller delay should be minimized.
As seen in Fig. 4(a) and 4(c), the adaptive controller with timedelay compensation performs better thanks to a smaller error and reduced sprungmass acceleration for bump road. The same result can be obtained in C level road from Fig. 4(b) and 4(d).
Fig. 4. Sprungmass acceleration of: a) bump road, b) Clevel road; tracking error of, c) bump road, d) Clevel road
a)
b)
c)
d)
As shown in Fig. 5, due to the timedelay, the adaptive controller without timedelay compensation may even deteriorate ride comfort and increase sprungmass acceleration near the point of resonance (60 rad/s). The adaptive controller with timedelay compensation, on the other hand, can reduce sprungmass acceleration. For a realistic random road type, the higher the frequency, the lower is the power spectrum, and the lower the frequency, the higher is the power spectrum. For largeamplitude vibrations, the sprungmass acceleration of the controlled suspension is lower because of the low frequency – see Fig. 4(b). For smallamplitude vibrations, the sprungmass acceleration of the controlled suspension shows no significant decline because of the high frequency – see Fig. 5. Overall, timedelay is unavoidable and may even reduce sprungmass acceleration for high frequencies. However, the controller with timedelay compensation can reduce the effect of timedelay and improve ride comfort.
An overview of the RMS values for sprungmass acceleration for the two road types is shown in Table 3.
According to Table 3, the controller with timedelay compensation performs better than the controller without timedelay compensation. Compared to the passive suspension, the sprungmass acceleration declined by 89.15 % (with timedelay compensation), by 78.55 % (without timedelay compensation) for the bump road, and by 69.06 % (with timedelay compensation), and 54.04 % (without timedelay compensation) for the Clevel road.
Fig. 5. PSD of sprungmass acceleration in frequency domain
Table 3. RMS values for sprungmass acceleration
Road type

Bump road

C level road

Passive (m/s^{2})

0.7868

1.1139

Adaptive active (with timedelay compensation) (m/s^{2})

0.0854

0.3446

Adaptive active (without timedelay compensation) (m/s^{2})

0.1688

0.512

4. Experimental verification
To better understand the effectiveness of the controller with timedelay compensation, a bench test was carried out – see Fig. 6. The actuator used in the test consists of a rack and a pinion.
Fig. 6. Experimental setup for the bench test
The base excitation is provided by a hydraulic cylinder. A spring, which functions as a tire, is placed between the hydraulic cylinder and the unsprung mass. The sprung mass and the unsprung mass are connected via a damper and an actuator. The parameters of the system are shown in Table 1. The controller proposed in this work runs on an embedded system.
Because four different sensors were installed for the test, we can measure the sprungmass displacement, the sprungmass acceleration, the unsprungmass acceleration, and the relative displacement between sprung mass and unsprung mass, respectively. After differentiating the displacement, we can derive the velocity. These signals are used as inputs of the control system. The actuator represents the output of the system.
Fig. 7 shows the sprung mass acceleration results for both the bump road and the C level road. The delaytime $\tau =$ 0.01 s and other parameters are identical to those shown in Table 1 and Table 2. According to Fig. 7(a) and Fig. 7(b), the performance of the adaptive controller with timedelay compensation is clearly superior to that of the adaptive controller without timedelay compensation. Both the period to settle down and the peak of vibration acceleration are the least for controller with compensation, shown in Fig. 7(a). The performance remains the same in regardless of the road type. This can be concluded from the result depicted in Fig. 7(b), in which the average acceleration under the controller with compensation is the smallest. The power spectrum chart in Fig. 7(c) confirms this result. Both the time domain result and the frequency domain result demonstrate that the proposed method can solve the problems mentioned above.
Fig. 7. Sprungmass acceleration for: a) bump road, b) C level road, c) PSD of sprung mass acceleration
a)
b)
c)
5. Conclusions
It is challenging to realize effective predictive control for a quartercar nonlinear suspension system. One approach this challenge is to transform the problem into a problem of control with timedelay. This paper proposes an adaptive controller with improved precision for nonlinear systems and timedelay compensation. The results are compared with a controller without timedelay compensation:
1) With the proposed controller, the problem of time delay of the front suspension can be solved. This is done by a compensator. Unlike the linear algorithm, this algorithm does not need to establish the state equation of the linear system. The structure of this algorithm can deal with nonlinear problems, so it can solve the nonlinear items of suspension system, such as nonlinear coefficient or nonlinear stiffness.
2) Using the controller designed in this study, both dynamic travel and dynamic load of the tire with the suspension increased within the mediumfrequency band. However, for the full frequency band, a satisfactory control effect can be achieved as well as significantly reduced sprungmass acceleration. The bench test results are consistent with the results of the theoretical analysis, which confirms the improved effectiveness of the new controller.
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