Adaptive NN control based on Butterworth low-pass filter for quarter active suspension systems with actuator failure

1.   Introduction

In the last twenty years, with the rapid development of industry, automobile has become a necessary tool for people's daily travel. The vehicle can run smoothly on the road, which is achieved by the cooperation of various subsystems of the vehicle. The suspension system, as one of the main subsystems of the vehicle, plays an indispensable role. Vehicle suspension system is mainly divided into passive suspension system, semi-active suspension system and active suspension system (ASS). Today, many vehicles are equipped with only passive or semi-active suspension ^[1,2,3,4]. However, their spring stiffness and damping are not adjustable, according to the random vibration theory, it can only guarantee to achieve results in specific road conditions. Compared with them, the active suspension system has an actuator which can adjust the force, and greatly increases the controllability of the vehicle in the driving process. Therefore, the control problem of the active suspension systems ^[5] has been widely concerned by lots of scholars. Many scholars have proposed excellent control algorithms ^[6,7,8] for active suspension systems.

According to the established vehicle model, the active suspension control research is divided into three categories: quarter vehicle model ^[9], half-car model ^[10] and full vehicle model ^[11]. There are many excellent control algorithms applied to active suspension systems, but in practice, many uncertain factors are ignored in nonlinear suspension systems. In order to deal with the uncertain factors, NN control methods ^{[12,13,14,15,16,17,18,19]} and fuzzy control methods ^{[20,21,22,23]} have been proposed in many articles. In addition, some intelligent decision-making ^[24] and nonlinear analysis methods ^[25] are proposed. Many systems also use NN control such as stochastic systems ^[26,27], pure-feedback systems ^[28], and nonlinear delayed systems ^[29]. At the same time, NN control is also used in a lot of practical systems. At a result, ASS has become a hot issue in recent years, and many different control strategies have been reported. In ^[30] sliding mode control is developed for quarter active suspension systems. The finite-time stability is proposed in ^[31], and the problem of actuator failure in semi vehicle active suspension system has been raised in ^[32]. As the quarter ASS is most in line with the real situation, it is representative to study some control strategies such as ^[33,34]. However, the general quarter ASS ^[35,36] is a second-order system, which needs zero dynamics analysis. This may lead to the loss of information. Therefore, it is proposed in ^[37] by using Butterworth low-pass filter (LPF) to convert the non-strict feedback system to strict feedback system. Based on the Butterworth LPF, it makes the quarter ASS expressed from a second-order system to a fourth-order system.

Generally speaking, due to the influence of structural, environmental and other problems, many practical systems have different constraints. In view of this, many scholars have achieved a lot of works. For example, in ^[38], the robotic system with full-state constraints is considered. Constraint methods are proposed for switched systems in ^[39,40]. In ^[41], the delayed systems with time-varying full-state constraints is considered. An effective finite-frequency fault detection method is proposed for descriptor system in ^[42]. An input constrained control method is proposed in ^[43]. In addition, many excellent constraint control methods have been applied to many practical systems. Although a large number of scholars have carried out a series of studies on ASS, there are also many difficult problems in the research on active suspension control. The problem of actuator failure can be not ignored. Actuator failure is equally important for vehicle stability.

In the actual production and life, the actuator failure of machines cannot be ignored. of course, the actuator failure of active suspension system is quite important as well. This kind of fault may lead to the instability of the control systems and even lead to catastrophic accidents. Therefore, it is necessary to study fault-tolerant control (FTC) strategy ^[44,45]. In order to deal with the problem of actuator failure, some control strategies have been proposed ^[46,47,48]. FTC is also very important for ASS. In ^[49,50,51], some fault tolerant control strategies are proposed for quarter ASSs or half-car ASSs. Meanwhile, the corresponding control strategy is proposed for the vehicle ASS in ^[52]. Through the intelligent FTC of the active suspension, the control reliability of the suspension system can be effectively improved. Therefore, it is very meaningful to propose an FTC strategy for a quarter ASS.

An adaptive NN control based on Butterworth low-pass filter for quarter active suspension systems with actuator failure is proposed in this paper. The main contributions of the proposed control are as follows.

1) Different from the previous works ^[35,36], by using Butterworth LPF, the second-order quarter active suspension system is changed into a fourth-order system. Considering all states in the quarter suspension system is more general.

2) The actuator failure of quarter ASS is considered. An adaptive NN control is put forward in this paper, and the FTC solves the problem of actuator failure in the process of vehicle driving, so that the vehicle can quickly return to normal driving.

The rest of this paper is organized as follows: Section Ⅱ introduces the quarter active suspension system based on LPF and the fault model used in this paper. An adaptive controller with actuator failure and stability analysis are considered in Section Ⅲ. In Section Ⅳ, a simulation example is given to demonstrate the effectiveness of the proposed method, and the conclusion is given in Section Ⅴ.

2.   Problem analysis

The quarter active suspension system is a representative model in vehicle system, which is displayed in Figure 1. It is often used in the design and analysis of automotive auxiliary system. According to the dynamic characteristics of the active suspension system, the force analysis is carried out as

where the control input of the quarter active suspension system is $u$, the sprung mass and the unsprung mass are represented by ${{m}_{s}}$, ${{m}_{u}}$. ${{z}_{s}}$ and ${{z}_{w}}$ represent the displacement of the sprung and unsprung mass, ${{z}_{r}}$ is the disturbance of the road surface, ${{k}_{a}}$, ${{k}_{t}}$, ${{l}_{a}}$ and ${{l}_{t}}$ stand for the stiffness coefficients and the damping coefficients.

According to the stress analysis, these forces are expressed as

Define the following state variables ${{x}_{1}} = {{z}_{s}}$, ${{x}_{2}} = {{\dot{z}}_{s}}$, ${{x}_{3}} = {{z}_{w}}$, ${{x}_{4}} = {{\dot{z}}_{w}}$. It easy to know that the state space form of quarter car active suspension systems is

Remark 1: Previous works ^[37,38] considered the equation of states with respect to ${{x}_{1}}$ and ${{x}_{2}}$, which may result in the information loss of the state variables ${{x}_{3}}$ and ${{x}_{4}}$. In this work, by using Butterworth low-pass filter, all states are considered when designing the controller.

Assumption 1: The desired tracking trajectory ${{y}_{d}}$ and its time derivative ${{\dot{y}}_{d}}$ are assumed to be bounded with $\left| {{y}_{d}} \right|\le A$.

Assumption 2: There is only one failure of the quarter active suspension system at the same time.

Assumption 3: Because the load of the vehicle is limited, the car-body mass is limited by ${{m}_{s\min }}\le {{m}_{s}}\le {{m}_{s\max }}$, where ${{m}_{s\min }}$ and ${{m}_{s\max }}$ are constants.

Assumption 4: ^[37] The control signal can be expressed as $u = {{u}_{f}}+\Delta u$, where $\Delta u$ is an error and ${{u}_{f}}$ is a filtered signal defined as:

where ${{H}_{L}}\left(s \right)$ is a Butterworth low-pass filter.

The corresponding filter parameters of Butterworth filters with the cutoff frequency are shown in Table 1, in which it holds that ${{\omega}_{c}}$ = 1rad/s. for different values of $n$.

In this way, it will not only lead to algebraic cycles, but also is unable to implement directly. In order to avoid the above problems, the filtered signal is used to avoid the algebraic loop problem effectively. According to the approach proposed in ^[37], it is worth noting that, due to the low pass characteristics of most actuators, the following replacement is reasonable.

where ${{f}_{2}} = {{F}_{s}}+{{F}_{d}}$ and ${{f}_{4}} = -{{F}_{s}}-{{F}_{d}}+{{F}_{t}}+{{F}_{b}}$ are unknown functions.

In this paper, the actuator faults include deviation faults and ineffective faults, which are given as ^[53]

where $0\le \rho \le 1$ is the effectiveness factor of the fault model, $\zeta$ is an unknown constant and ${{\zeta }_{\max }} = {{v}_{m}}$. Let ${{t}_{v}}$ and ${{t}_{e}}$ be the time instants when the actuator takes place and ends. The actuator fault model includes the following situations:

1) when $\rho \text{ = }1$ and $\zeta\text{ = }0$, it means that there is not any actuator fault.

2) when $0 < \underset{ -}{\rho }\le \rho < \bar{\rho }$ and $\zeta\text{ = }0$, it shows partial actuator failure, where $0 < \underset{ -}{\rho }\le 1$, and $0 < \bar{\rho }\le 1$ are constant.

3) when $\rho \text{ = }0$ and $\zeta \text{ = }1$, it implies that the signal ${{u}_{c}}$ can no longer be influenced by the control inputs $u$.

Then, (2.5) becomes

In the next section, the actuator failure problem will be solved for the ASS.

3.   Adaptive FTC controller design

3.1.   Adaptive FTC controller design

The controller design is implemented by backstepping technologies. Define the errors variable as follows:

where ${{y}_{d}}$ is the expected trajectory of ${{x}_{1}}$ and ${\alpha _{i - 1}}$ are the virtual controls which will be given in the later.

Step 1: From above definition, ${{\dot{z}}_{1}}$ is given as

The Lyapunov function ${{V}_{1}}$ is selected as

The time derivative of ${{V}_{1}}$ is became

The virtual control input ${{\alpha }_{1}}$is selected as

where ${{c}_{1}} > 0$ is a positive constant. The time derivative of ${{V}_{1}}$ is written as

Step 2: The time derivative of ${{z}_{2}}$ as defined in above is

It is easy to get

In order to deal with the influence of unknown functions, the neural network is introduced to approximate it as

where $w = {{\left[{{w}_{1}}, {{w}_{2}}, \cdots {{w}_{n}} \right]}^{T}}\in R$ is the weight vector of NN, $\phi \left(X \right)$ is the basis functions, $X = {{\left[{{x}_{1}}, {{x}_{2}}, {{x}_{3}}, {{x}_{4}} \right]}^{T}}$ is the input of the basis functions, $\kappa$ represents the approximation error and $\bar{\kappa }$ is a constant with $\kappa \le \bar{\kappa }$.

In addition, define ${{H}_{2}}\left({{X}_{2}} \right)\text{ = }w_{2}^{T}{{\phi }_{2}}\left({{X}_{2}} \right)+{{\kappa }_{2}}$, where ${{H}_{2}}\left({{X}_{2}} \right)$ denotes an unknown function, given as

The error weight vector is defined as ${{\tilde{w}}_{2}} = {{w}_{2}}-{{\hat{w}}_{2}}$, ${{\hat{w}}_{2}}$ represents the estimate of ${{w}_{2}}$.

The Lyapunov function ${{V}_{2}}$ is selected as

where ${{\Lambda }_{i}}\left(i = 2, 3, 4 \right)$ is a positive definite symmetric matrix. The time derivative of ${{V}_{2}}$ is regarded as

Substituting (3.7) and (3.9) into (3.11), one obtains

Lemma 1: ^[54,55] For any $x, y\in {{R}^{n}}$, there are following inequalities where $a > 0, p > 1, q > 1$ and $\left(p-1 \right)\left(q-1 \right) = 1$.

Then, the time derivative of ${{V}_{2}}$ is

It is easy to know

The virtual control input and parameter adaptive law are designed as

Now, substituting (3.16) into (3.15) gives

where ${{c}_{2}} > 0$ and ${{\gamma }_{1}} > 0$ are designed constants.

Step 3: The Lyapunov function ${{V}_{3}}$ is defined as

The error weight vector is defined as ${{\tilde{w}}_{3}} = {{w}_{3}}-{{\hat{w}}_{3}}$, ${{\hat{w}}_{3}}$ represents the estimate of ${{w}_{3}}$.

The time derivative of ${{V}_{3}}$ is regarded as

Define

Then, the time derivative of ${{V}_{3}}$ becomes

The virtual control input of active suspension system and parameter adaptive law are given as

From (3.22) ${{\dot{V}}_{3}}$ becomes

where ${{c}_{3}} > 0$ and ${{\gamma }_{2}} > 0$ are designed constants.

Step 4: The time derivative of ${{z}_{4}}$ is as follows

Then, it gets

Let

The Lyapunov function ${{V}_{4}}$ is defined as

The error weight vector is defined as ${{\tilde{w}}_{4}} = {{w}_{4}}-{{\hat{w}}_{4}}$, ${{\hat{w}}_{4}}$ represents the estimate of ${{w}_{4}}$.

The time derivative of ${{V}_{4}}$ is regarded as

The controller of active suspension system is designed as

The parameter adaptive law is given as

Then, from (3.29) and (3.30), it leads to

where ${{c}_{4}} > 0$ and ${{\gamma }_{3}} > 0$ are designed constants.

3.2.   Stability analysis

Theorem 1: The quarter active suspension systems with actuator faults based on Butterworth low-pass filter is considered (2.7). Through the design of virtual controllers ${{\alpha }_{1}}$, ${{\alpha }_{2}}$ and ${{\alpha }_{3}}$, actual controller ${{u}_{c}}$ and adaptive laws ${{\hat{w}}_{2}}$, ${{\hat{w}}_{3}}$ and ${{\hat{w}}_{4}}$, all the signals in the closed-loop systems are bounded.

Proof: Based on (3.28), (3.29), and (3.30), (3.31) is rewritten as

where $N = \sum\limits_{i = 2}^{4}{\frac{1}{2}\bar{\kappa }_{i}^{2}}+\frac{1}{2}{{\left(\Delta u \right)}^{2}}+\sum\limits_{i = 1}^{3}{\frac{1}{2}{{\gamma }_{i}}{{\left\| {{w}_{i+1}} \right\|}^{2}}}$, $M = \min\left\{2{{c}_{1}}, {2{{a}_{2}}}/{{{m}_{s\max}}, }\; \right.2{{a}_{3}}, {2{{a}_{4}}}/{{{m}_{b\max }}, }\; {{\gamma }_{i}}{{\Lambda }_{i+1}}\ i = 2, \left. 3, 4 \right\}$, ${{a}_{2}} = {{c}_{2}}+1$, ${{a}_{3}} = {{c}_{3}}+0.5$, and ${{a}_{4}} = {{c}_{4}}-0.5$.

Then, (3.32) can be further written as

From (3.33), it is clear that ${{z}_{i}}$, ${{x}_{i}}$, $u$ are bounded.

Meanwhile, the errors satisfy

From (3.34), it clear that the errors of quarter active suspension system are within bounds. ${{z}_{1}}$ is bounded as $\left| {{z}_{1}} \right|\le B$ and the desired trajectory $\left| {{y}_{d}} \right|\le A$ is also bounded. At the same time, $\left| {{x}_{1}} \right| = \left| {{z}_{1}} \right|+\left| {{y}_{d}} \right|\le A+B$. We can get ${{x}_{2}}, {{x}_{3}},$ ${{x}_{4}}$are bound similar. It implies that all the signals of the quarter active suspension closed-loop system with actuator failure are bounded.

4.   Simulation example

In this section, an example of quarter active suspension system is provided to prove the effectiveness of proposed method. The quarter-car parameters are given.

The initial state values are selected as: ${{x}_{1}}\left(0 \right) = 0.03$, ${{x}_{2}}\left(0 \right) = 0.03$, ${{x}_{3}}\left(0 \right)\text{ = }0.03$, ${{x}_{4}}\left(0 \right) = 0.03$, ${{x}_{5}}\left(0 \right) = 0.0\text{3}$, ${{x}_{6}}\left(0 \right) = 0.03$ and ${{x}_{7}}\left(0 \right) = 0.0\text{3}$. The desired tracking trajectories ${{y}_{d}}$ is given as ${{y}_{d}} = 0$, then ${{\dot{y}}_{d}} = 0$. The road excitation is represented as: ${{z}_{r}}\text{ = }0.\text{002}\sin (\text{0}\text{.001}\pi t)$. In addition, the designed parameters are as follows ${{c}_{1}}{3}$, ${{c}_{2}} = 3.\text{2}$, ${{c}_{3}} = 1$, ${{c}_{4}} = \text{64}\text{.5}$ ${{\gamma }_{1}}\text{ = }100$, ${{\gamma }_{2}}\text{ = }100$, and ${{\gamma }_{3}}\text{ = }100$. The problem of actuator failure is considered in this paper. So, the actuator faults deviation coefficient $\rho$ and unknown constant $\zeta$ are designed as $\rho \text{ = }0.8$, $\zeta \text{ = }0.5$. The parameters of the vehicle is described by Table 2.

Figures 2 and 4 show that the vertical displacement of the car-body converges to a small neighborhood of zero. Figures 3 and 5 show that the vertical speed converges to the neighborhood of zero. At the same time, ${{z}_{1}}$, ${{z}_{2}}$, ${{z}_{3}}$ and ${{z}_{4}}$ are given in Figures 6 and 7. Figure 8 shows that the suspension space is small enough around zero. The adaptive laws are plotted in Figure 9. It is clear that all the adaptive laws converge to a small neighborhood of zero. In Figure 10, we can see that the control input $u$ is stable.

5.   Conclusion

An adaptive fault tolerant controller is put forward for a quarter ASS in this paper. By using Butterworth low-pass filter, the second order ASS is converted to a fourth order system, which includes all the vehicle variables in the quarter suspension system. At the same time, the fault of vehicle actuator is also considered. NN control is used to approximate unknown functions in quarter suspension system. On account of backstepping method, the corresponding adaptive controller is designed. Through stability analysis, it can be concluded that all signals in the quarter active suspension system are bounded. Finally, the simulation results show that the method is effective. In the future, more complex situations can be considered in the controller design of the fault-tolerant control, such as the performance constraints, saturated or dead-zone, etc. The control studies will be further enriched in active suspension systems.

Acknowledgments

This work is supported in part by the National Natural Science Foundation of China under Grants 61973147, 61803189, in part by the LiaoNing Revitalization Talents Program under Grant XLYC1907050.

Conflict of interest

All authors declare no conflicts of interest.