Vehicle chaos identification and intelligent suppression under combined excitation of speed bump and engine

: For non-linear suspension, vehicle passes through a continuous speed bump, the chaos that may occur under the combined excitation of the speed bump and the engine. This study takes the five-degree-of-freedom vehicle model as the object of research, through the vehicle body poincaré section and the maximum Lyapunov index to identify the chaos produced by the vehicle under joint excitation, and utilises the feedback control of the optimal feedback gain coefficient based on the particle swarm optimisation (PSO) algorithm to suppress vehicle chaos. The results indicate that the vehicle is in a chaotic state in all speed range. Under low and medium speeds, the route to the chaos of the vehicle is the system coupling vibration under multi- frequency excitation, whereas in the high-speed condition, the vehicle approaches the chaos through the bifurcation. The chaos of the vehicle can be effectively suppressed by feedback control with the global optimal feedback gain searched by PSO. This study reveals the chaotic characteristics of non-linear suspension vehicles under combined excitation, which provides a new method for intelligent suppression of chaos.


Introduction
The continuous speed bump is set up on some special sections of expressways and its principle is to create vibration for passing vehicles and prompt drivers to decelerate [1]. Under the periodic excitation of the speed bump, the non-linear suspension vehicle may generate chaos. Chaos may cause adverse effects on the vehicle. It is of great significance to identify and suppress vehicle chaos.
Researches on non-linear characteristics of the vehicle have been made by scholars of the whole world. Litak and Borowiec [2] used Melnikov process to analyse the way of single-degree-offreedom vehicle model entering chaos under white noise road excitation. Zhu and Ishitobi [3] analysed the chaotic state of vehicle excited by a sinusoidal road surface and found that the vehicle was chaotic in the excitation frequency range. Naik and Singru [4] established single-degree-of-freedom vehicle model including time-delay and speed feedback, which found that a proper delay time can effectively eliminate vehicle chaos. Niu and Wu [5] established the non-linear eight-degree-of-freedom vehicle model and obtained the dynamic response of the system under sinusoidal road excitation and found that the system entered the evolutionary process of chaotic motion through the period and quasi-periodic. Liang et al. [6,7] studied the chaotic vibration characteristics of two-degree-of-freedom 1/4 vehicle non-linear suspension model by simulation and experiment, used the frequency bifurcation diagram and the phase trajectory map, which reveals the possibility of the vehicle to generate chaos under the continuous road surface excitation. The direct variable feedback control is introduced to transform the chaotic motion of the vehicle into periodic motion.
Although scholars have done a lot of researches on vehicle chaos, there are still some areas for improvement in related research. In vehicle modelling, the effect of the engine on vehicle chaos is not considered. In the aspect of recognising vehicle chaos, the chaotic state of the vehicle can only be qualitatively analysed by the frequency bifurcations and phase trajectories, which also have a certain subjectivity in the identification of the vehicle. In the aspect of vehicle chaos suppression, the feedback gain coefficient has an obvious impact on the effect of the vehicle chaos suppression when the feedback control is used to suppress the vehicle chaos. It is difficult to determine the best feedback gain coefficient by the traditional method. Therefore, the five-degree-offreedom vehicle model considering engine excitation is established in this paper, using the maximum Lyapunov index of the vehicle system to measure the chaos of the vehicle. The particle swarm optimisation (PSO) algorithm is introduced to search the optimal feedback gain coefficient of feedback control to improve the search efficiency of the optimal feedback gain coefficient and to optimise the effect of the feedback control on the vehicle chaos.

Road excitation model of a continuous speed bump
A half sine function is used to establish the speed bump model. The height of the speed bump is h, the width of the single speed bump is w 1 , and the intervals between the speed bumps are w 2 and w 1 = w 2 = w. The static model of the speed bump is shown in Fig. 1. The height of the speed bump is h = 0.005 m, the width of the speed bump is w 1 = 0.5 m, and the deceleration interval is w 2 = 0.5 m.
The vehicle passes the speed bump at the speed of v, the time required to pass a single speed bump is t 1 , t 1 = w 1 /v, the time required to pass the spacing of the speed bump is t 2 , t 2 = w 2 /v, and w 1 = w 2 , so t 1 = t 2 . The time of the vehicle passes through a speed bump and a speed bump spacing at the speed of v, that is, the dynamic model of a continuous sinusoidal speed bump is T and based on formula (1), the angular velocity ω of the dynamic model is Thus, the dynamic model of continuous sinusoidal speed bump can be obtained as

Non-linear vehicle suspension model
The vehicle structure is complex and needs to be simplified according to certain rules. Considering that the vehicle is basically the same on both sides of the road when the vehicle passes through the speed bump, there is a phase difference between the front and rear wheels, and the vehicle is regarded as the left and right symmetry structures, and the five-degree-of-freedom 1/2 vehicle model considering the engine excitation is set up, as shown in Fig. 2.
In Fig. 2, z e is the vertical displacement of the engine; z b is the vertical displacement of the body; z f , z r are the vertical displacements of the front and rear suspensions, respectively; q f and q r are vertical road excitations for front and rear tyres, respectively. m e is the engine quality; m b is the body quality; m f and m r are the front and rear tyre qualities. a and b are the distances between the front and rear suspensions to the centre of mass, respectively, and d is the distance from the engine mounting device to the centre of mass. k e is the spring stiffness of engine mounting device; k f2 and k r2 are spring stiffness of front and rear suspensions, respectively; k f1 and k r1 are elastic stiffness of front and rear tyres, respectively. c e is the damping coefficient of the engine mounting device; c f2 and c r2 are the damping coefficients of the front and rear suspensions, respectively; c f1 and c r1 are the damping coefficients of the front and rear tyres, respectively. The symbol: e denotes the engine, f indicates front, r represents rear, 1 is the wheel, and the 2 is the suspension.
The equilibrium position of the vehicle at rest is the origin of coordinates in each generalised coordinate system, and the motion equation of a vehicle can be expressed as The elastic force of a non-linear spring can be expressed as In formula (5), F k is the elastic force; k is the elastic coefficient; the Δz represents the elastic variable; sgn(·) represents the positive and negative sign functions. When n = 1, the spring is linear and when n ≠ 1, the spring is non-linear. The damping force of the suspension dampers and the tyres can be expressed as c u is compression stroke damping and c d is stretching stroke damping in the formula.

Analysis of chaotic characteristics of image method
The speed range of a general vehicle through a continuous speed bump is 10-20 m/s. Taking the vehicle speed as the abscissa and taking the poincaré section point of the body displacement at each speed as the vertical coordinate, the speed bifurcation diagram of the body displacement is made, as shown in Fig. 3. The bifurcation diagram cannot effectively distinguish the quasi-periodic motion and the non-periodic motion at each speed of the vehicle. The chaotic state of the vehicle system and the approach to chaos need to be further analysed by using the poincaré section diagram. For the poincaré section, a phase point on the plane is taken as the starting point, and the phase trajectory goes back to the plane after a periodic motion, and the intersection point of the phase trajectory and the plane is the point of the poincaré section. After several cycles, the number of points on the section reflects the motion state of the system. When the poincaré section has only one fixed point and a few discrete points, it can be determined that the motion is periodic; when the poincaré section  is a closed curve, it can be determined that the motion is quasiperiodic; when the poincaré section is a densely packed point and there is a hierarchical structure, it can be determined that the motion is in a chaotic state. Three speeds 10, 15, and 20 m/s are selected, and the poincaré section diagrams of the vehicle body corresponding to the vehicle speed are made, which are, respectively, shown in Figs. 4-6. The chaotic state of the vehicle and the ways of a vehicle entering chaos are analysed at low speed, medium speed, and high speed. Fig. 4 shows that the poincaré section point of the vehicle body consists of two separate limit cycles, indicating that the vehicle is in a non-periodic motion, and the way that the vehicle enters the chaos is the coupling of the vibration of the system under the multi-excitation frequency.
The two separate limit cycles of the poincaré section of the vehicle body of Fig. 5 fuse into a limit cycle and have a broken trend. It shows that the vehicle is in a non-periodic motion, and the way that the vehicle enters the chaos is still coupled with the vibration of the system under the multi-excitation frequency.
In Fig. 6, the vehicle phase trajectories are a group of spiral lines. The limit cycle of the poincaré section diagram is completely broken, indicating that the vehicle is in a non-periodic motion, the limit cycle rupture, and the vehicle enters chaos by the way of bifurcation.

Analysis of chaotic characteristics of the numerical method
The maximum Lyapunov index of the vehicle system is used to identify the chaotic state of the vehicle system. The Lyapunov index is the most reliable method of identifying chaos [6], which represents the numerical characteristics of the average exponential divergence rate of the adjacent trajectories in the phase space, reflecting the sensitivity of the chaotic system to the initial value.
A set of time series is obtained by solving formula (4) by fourorder five-level Runge-Kutta algorithm, and a set of onedimensional time series is formed by the first and end connection of the time series. Phase space reconstruction of the time series by using the coordinate delay method In (8), Y k is the m-dimensional phase space, k = 1, 2, …, σ − (m − 1) , is the delay time, and m is the embedding dimension. To ensure the fidelity of the phase space reconstruction and the accuracy of the calculation of the Lyapunov exponent, the selection of these two parameters is very important. The mutual information method is used to obtain the relationship between the delay time and the information entropy I( ) [10]. So the conclusion is that the optimal delay time is 14 sampling points backward, and the sampling compensation T = 0.001 means that the delay time is = 0.014 s. Using the artificial neighbouring point method to determine the embedding dimension, using MATLAB to calculate the relationship between the embedding dimension and the proportion of false neighbour points, it can be concluded that when the embedding dimension is m = 29, if the embedding dimension will continue to expand, the proportion of false neighbour points no longer change. Therefore, when the road excitation frequency is 10 Hz, the embedding dimension of the vehicle system time series should be 29 [10]. Then, the wolf method is used to calculate the maximum Lyapunov index of time series after phase space reconstruction. The maximum Lyapunov index of vehicle system at different speeds is shown in Fig. 7. The maximum Lyapunov index of the vehicle is greater than zero at all speeds, and the vehicle is in chaos. There is a great difference between the pavement excitation frequency and the engine excitation frequency at low speed, and under multifrequency excitation, the vehicle chaos is more obvious. The difference between the excitation frequency of the road surface and the engine frequency is reduced at medium speed, the limit cycle of vehicle gradually fuses, and the vehicle chaos degree decreases. At high speed, the limit cycle of the vehicle is broken, and the vehicle enters into chaos through the bifurcation route, and the degree of chaos increases dramatically. It is noted that at 15.69 m/s, the maximum Lyapunov index of the vehicle system increases sharply, which is due to the complete fusion of the limit ring near the speed of the vehicle, and the way that the vehicle enters the chaos is transformed from multi-frequency excitation to bifurcation, and the vehicle is chaotic at this time.

Feedback control
Direct variable feedback control [7] can achieve chaos suppression by adjusting the feedback gain coefficient on the premise that the unsteady period orbit of the system target is unknown. Using the piecewise quadratic function x|x| as the form of feedback control, this control method is easy to implement and has little effect on the system [11]. The controller of the design system is in the form of θ˙= z 4 , z e = z 5 , z˙e = z 6 , z f = z 7 , z˙f = z 8 , z˙r = z 9 , and z˙r = z 10 , then the state space equations of the first four terms of the system can be rewritten according to (4)

Intelligent searches for optimal feedback control coefficient
The feedback gain coefficient determines the effect of the chaos suppression. An improved PSO algorithm is introduced to minimise the maximum Lyapunov index of the vehicle system as the control target and search for the best feedback gain coefficient [k 1 , k 2 , k 3 , k 4 ]. Different from the traditional feedback control, the feedback gain coefficient in this paper keeps changing in the control process and expands the search domain of the PSO algorithm to obtain better feedback control effect. At the same time, in order to avoid the 'premature' problem commonly found in PSO, it is necessary to judge the early maturing of the particle swarm, and use chaos search to do precocious processing for 'early-maturing' particle swarms to avoid the computation falling into a local optimal solution.
The steps to search for the global optimal feedback coefficient [k 1 , k 2 , k 3 , k 4 ] are as follows: (i) Initialising M × 4 × σ random vectors to form a (P × 4) × σ matrix as the initial solution, generating an isomorphic random vector as the 'migration' velocity of the initial particle group. (ii) The maximum Lyapunov index of the vehicle system is used as the fitness of the corresponding particle swarm after feedback control, and the fitness of the ith generation particle swarm is Lya i , Lya ig is the smallest maximum Lyapunov index in the ith generation particle swarm, and the corresponding feedback coefficient is the local optimal feedback coefficient k ig . Lya q is the smallest and largest Lyapunov index ever searched so far, and the corresponding feedback coefficient is the global optimal feedback coefficient k q . (iii) If the maximum Lyapunov index of the vehicle after feedback control is non-positive or reaches the limit generation, then the calculation is ended and the position of the particle swarm is output as the global optimal solution; if this condition is not satisfied, the precocious judging is performed. (iv) The variance 2 of population fitness of particle swarm reflects the degree of 'aggregation' of particle swarm, and 2 is the prematurity criterion of the particle swarm. Its definition is as follows: In this formula, m is the particle group generation, Lya avg is the average fitness of the particle swarm so far, and L is the normalisation calibration factor, which limits the variance of the population fitness variance. The formula is When 2 < C, the particle swarm is considered to be in a precocious state and precociously treated; when 2 > C, the particle swarm is assumed to be in a non-premature state, a next-generation particle swarm is generated, and the resulting next-generation particle swarm is returned to step (iii) to continue the particle swarm calculation. This constant C is set to 0.7 in this paper.
(v) The chaos optimisation algorithm makes use of the random, ergodicity, and regularity characteristics of chaotic variables to optimise the search in the solution space and it is easy to jump out of the local optimal solution [12]. Selecting the logistic map to generate the chaotic variable y′ q where μ is the control parameter. When μ = 4, the logistic system is in a completely chaotic state. Utilising the sensitivity of chaos to the initial condition, we choose n initial values y q with a slight difference and get n chaotic variable y′ q by formula (12). According to (13), it generates n new feedback gain coefficients k ′ igq These n variables form the solution space of the chaos optimisation algorithm, and these n new feedback gain coefficients are brought into (13) to calculate the maximum Lyapunov index of the vehicle system. This solution space makes the feedback gain coefficient of the maximum Lyapunov index minimum of the vehicle system as k′ ig , update k ig with k′ ig , and return to step (iii) to continue particle swarm calculation.
(vi) Generating the ith generation particle group and its 'migration' speed according to the equations below: where i = 1, 2, 3, 4; d = 1, 2, …, M; a is the constraint coefficient, which controls the weight of the 'migration' speed of the particle swarm; ω ≥ 0 is the inertia factor; c 1 , c 2 ≥ 0 are learning factors; and r 1 , r 2 are random numbers between random factors [0,1].
The process is shown in Fig. 8.

Control effect analysis
To analyse the effect of feedback control on chaos suppression, optimal feedback gain coefficient obtained by the PSO algorithm is brought into (9), and the vehicle body poincaré section is taken at a controlled vehicle speed of 20 m/s, as shown in Fig. 9. Comparing Fig. 6 with Fig. 9, it is found that the state of motion of the vehicle after the control is transformed from chaotic motion to periodic motion. Feedback control can effectively suppress vehicle system chaos.
Further analyse the feedback control of vehicle chaos at various vehicle speeds, the maximum Lyapunov index of the vehicle system at each vehicle speed after feedback control is calculated. The maximum Lyapunov indexes of the vehicle system before and after feedback control are shown in Fig. 10. The maximum Lyapunov index of the vehicle system is significantly reduced at each vehicle speed, indicating that feedback control can effectively suppress vehicle chaos at each vehicle speed.

Conclusion
In this paper, a five-degree-of-freedom vehicle model considering engine vibration is established. The chaotic characteristics of the vehicle under continuous speed bump and engine combined excitation are analysed, and the feedback control based on PSO is used to suppress the vehicle chaos intelligently. The following conclusions are drawn: (i) Under the joint excitation of the speed bump and the engine, the vehicle is in a chaotic state in the range of 10-20 m/s. The way of vehicle entering chaos in the middle-to-low-speed range is the coupling of system vibration under multi-frequency excitation and the vehicle enters chaos through the bifurcation approach at high speed.
(ii) According to the maximum Lyapunov index at each vehicle speed, the difference between the road excitation frequency and the engine excitation frequency at low speed is obvious, and the vehicle chaos is more obvious under multi-frequency excitation. The difference between the road excitation frequency and the engine frequency is reduced at medium-speed condition, the limit cycle of the vehicle is gradually merged, and the degree of vehicle chaos decreases. The limit cycle of the vehicle is broken at highspeed condition, the vehicle enters the chaos through the bifurcation route, and the degree of chaos increases sharply. At the speed of 15.69 m/s, the maximum Lyapunov index of the vehicle system increases dramatically, the vehicle limit ring is completely integrated near the speed, and the vehicle enters the chaotic way from multiple frequency excitation to bifurcation, which means that the vehicle undergoes paroxysmal chaos.
(iii) The feedback control is used to suppress vehicle chaos. The PSO algorithm is used to search the global optimal feedback gain coefficient of feedback control. Moreover, chaotic search is utilised to improve the PSO algorithm of avoiding 'premature' phenomenon of the particle swarm and to make the calculation jump out of the local optimal solution as soon as possible. The results show that the feedback control can effectively reduce the maximum Lyapunov index of the vehicle system at various vehicle speeds, restrain vehicle chaos, and convert the vehicle motion state from chaotic motion to periodic or quasi-periodic motion.