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Int. J. Vehicle Design, Vol. 73, No. 4, 2017
A novel fuzzy-sliding automatic speed control of intelligent vehicles with adaptive boundary layer Jinghua Guo* Department of Mechanical and Electrical Engineering, Xiamen University, Xiamen 361005, China Email:
[email protected] *Corresponding author
Yugong Luo and Keqiang Li State Key Laboratory of Automotive Safety and Energy, Tsinghua University, Beijing 100084, China Email:
[email protected] Email:
[email protected]
Rongben Wang College of Traffic, Jilin University, Changchun 130012, China Email:
[email protected] Abstract: Since the vehicle longitudinal dynamic system inherently has non-linear and time-varying behaviour, it is a challenging work to design an effective automatic speed control strategy to handle the unknown nonlinearities and uncertain parameters. In this paper, a novel adaptive fuzzy sliding automatic speed controller with variable boundary layer for intelligent vehicles is proposed; the control gains and the boundary layer thicknesses of the sliding mode speed controller are adaptively tuned by fuzzy logic simultaneously. Additionally, the stability of the proposed closed-loop automatic speed control system is guaranteed by the Lyapunov theory, and the switching criterion between the throttle and brake actuators is introduced. Finally, the performance of the proposed control scheme is evaluated by simulation and experimental tests, and the results illustrate that the proposed scheme can achieve good tracking performance. Keywords: intelligent vehicles; fuzzy-sliding control; adaptive boundary layer; speed control; longitudinal dynamics. Reference to this paper should be made as follows: Guo, J., Luo, Y., Li, K. and Wang, R. (2017) ‘A novel fuzzy-sliding automatic speed control of intelligent vehicles with adaptive boundary layer’, Int. J. Vehicle Design, Vol. 73, No. 4, pp.300–318.
Copyright © 2017 Inderscience Enterprises Ltd.
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Biographical notes: Jinghua Guo received his PhD from Dalian University of Technology, China, in 2012. From 2012 to 2015, he finished his postdoctoral research in Tsinghua University. He is currently an Assistant Professor with Xiamen University. He has authored more than 20 journal papers. He has engaged in more than five sponsored projects. His current research interests include intelligent vehicles, vision system, control theory and applications. Yugong Luo received his BE and ME from Chongqing University, Chongqing, China, in 1996 and 1999, respectively, and the PhD from Tsinghua University, Tsinghua, China, in 2003. He is currently an Associate Professor with the Department of Automotive Engineer, Tsinghua University. His research interests include vehicle dynamics and control, vehicle noise and vibration. Keqiang Li received his ME and PhD from Chongqing University, Chongqing, China, in 1988 and 1995, respectively, is a Professor with the Department of Automotive Engineering, Tsinghua University, he is awarded as ‘Changjiang Scholar’ by the Ministry of Education of the People’s Republic of China. His research interests include vehicle dynamics and control, driver-assistance systems and hybrid electrical vehicles. Rongben Wang received his BE, ME and PhD from Jilin University of technology, China, in 1970, 1991 and 1995, respectively. He is a Professor in College of Traffic, Jilin University; China. His research interests include automatic guided vehicles, computer vision systems and image processing.
1
Introduction
Intelligent vehicles integrate many kinds of advanced new technologies, including advanced communication and sensor techniques, to prevent road traffic injuries. Longitudinal control is an important part and has strong influences on the automatic driving performances of intelligent vehicles. The assignment of longitudinal control for intelligent vehicles is to track the desired velocity or the desired inter-vehicle spacing while increasing the traffic capacity and improving the safety, stability and riding comfort of vehicles. In recent years, the development of automatic speed control strategy has attracted more attention, especially in the USA, Europe and China. An H∞ methodology is dedicated to control the vehicle longitudinal oscillation by Lefebvre et al. (2003), and two tuning parameters are designed to manage the compromise between performance and robustness. Toulotte et al. (2008) presents a robust fuzzy control strategy with pole placement in linear matrix inequation (LMI) region, which is applied to the spacing policy control of an automated hybrid electric vehicle prototype; simulation and real-time experiment results show that the control law is robust enough to reject the modelling uncertainties and experimental condition variations. An adaptive backstepping longitudinal control system based on intelligent control theory is introduced by Peng (2010), in which an adaptive output recurrent cerebellar model articulation controller (ORCMAC) is used to mimic an ideal backstepping control and a robust controller is designed to attenuate the effects caused by the lumped uncertainty term. Wang et al. (2013) proposes an intelligent human-imitating longitudinal control system consisting of adaptive cruise and forward collision functions, and the driver characteristics are identified by the self-learning
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algorithm based on the recursive least-squares method with a forgetting factor. In Kim et al. (2015), an approach of timing-varying parameter adaptive throttle and brake control for vehicle speed tracking is presented, and a linearised vehicle model with lumped time-varying parameters has been used for the development of throttle/ brake control algorithms. Model predictive control (MPC) is an effective way to deal with the longitudinal control of vehicles, and Li et al. (2011) developed a model predictive controller and employed a constraint-softening method to avoid computational infeasibility. The cost function and constraints consisted of longitudinal riding comfort, fuel economy and tracking capability. The sliding mode control (SMC) is an effective and popular method used in the vehicle automatic longitudinal control applications due to its inherent capabilities to deal with the uncertainties and disturbances (Ping et al., 2013; Gerdes and Hedrick, 1997; Liang et al., 2003). Gerdes and Hedrick (1997) proposes a combined engine and brake controller for automated highway vehicles based on the ideas of multiple-surface sliding control, and experimental results show that the vehicle under the proposed control method is capable of tracking velocity profiles within 0.1 m/s. Liang et al. (2003) presents a modified sliding with variable control parameter to reduce the large change of pressure feedback in the brake control of highway vehicles. In Nouveliere and Mammar (2007), a second-order sliding mode technique is applied for longitudinal control of vehicle at low speed, and the results are given with different scenarios such as carfollowing control, stop-and-go and stopping at obstacles using stereovision detection. Ferrara and Vecchio (2009) apply SMC method to design a longitudinal driver assistance system for vehicles capable of keeping the desired inter-vehicular spacing; simulation evidence demonstrates the feasibility of the proposed approach. The sliding mode parameters can be adjusted to get faster error convergence, however, this will, in turn, increase the control gain, which may cause serve chattering on the sliding surface and degrade the system performance. Therefore, sliding mode controller has limited usage in practice since it requires fast switching on the input which may result in chattering phenomenon. The boundary layer is an effective approach to eliminate or alleviate the chattering around the switching surface in Vrabel (2012), however, as the boundary layer thickness increases, the tracking performance is degraded. To deal with the chattering reduction problem, integration of fuzzy logic control and SMC has been proposed, and this integration has been proved to be an effective control strategy for a system with strong nonlinearities, uncertainties and disturbances (Cheng et al., 2009; Guo et al., 2013). In this paper, to improve the system performance and reduce the chattering phenomenon, an adaptive fuzzy SMC scheme with variable boundary layer is presented. The main contributions of this paper are that the proposed control algorithm can effectively overcome the effects of nonlinearities, disturbances and parameter uncertainties, and the chattering phenomenon that frequently appears in the conventional SMC control systems is also alleviated without deteriorating the system robustness. The rest of this paper is organised as follows: In Section 2, a longitudinal dynamics model of intelligent vehicles is developed. In Section 3, an adaptive fuzzy-sliding control scheme with adaptive boundary layer used for vehicle automatic longitudinal control is presented. Simulation and experimental results of the proposed strategy under various operating conditions are shown in Section 4. Finally, conclusions are drawn in Section 5.
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System descriptions
The vehicle system used in the analysis is a front-wheel-driven and a front-wheel-steered system, under the following assumptions: •
the engine and brake actuator are approximated as a first-order linear model
•
torque converter is locked
•
discount the slip between the tyres and the ground.
The vehicle longitudinal dynamic model is derived as J e ⋅ ω e = Te − Tp f e (ωe , α th ) = Te + teTe ωt Tt = τ Tp ωp τ bTb + Tb = K p Pb T T M e v = s − b − Mgf cos ϑ − Ca Aa vx2 − Mg sin ϑ + ∆E (t ), r r
(1)
where M e = M + (( J wr + J wf ) r 2 ), Me is the equivalent mass, Jwf and Jwr denote the front and rear wheel inertias, respectively. r is the radius of wheel, M is the vehicle mass, f is the rolling resistance coefficient, ϑ is the road slope angle, Ts is the driving torque, Te denotes the engine torques, Tb denotes the brake torques, Ca is the aerodynamic drag coefficient, v is the vehicle speed, ωe is the engine speed, Je is the engine/transmission inertias, Te is the engine torque, Aa is the equivalent windward area, ∆E(t) represents uncertainties and external disturbances. f e (ωe , α th ) denotes the steady-state engine map which give the engine torque as a function of throttle angle and the engine angular velocity, αth is the throttle angle, te is the time constant of throttle actuator, ωt is the rotational speed of turbine, ωp is the rotational speed of pump, Tt is the torque of turbine and Tp is the torque of pump, τ is the torque ratio coefficient. The transitive relationship between the rotational speed and the torque of units can be described as follows: ω p = ωe v = rωt Rg io , T = T ⋅ i ⋅η o o t s To = Tt Rg
(2)
where To is the torque of transmission gearbox’s output shaft, Rg is the gear ratio, io is the transmission ratio of main reducer, ηt is the drive efficiency.
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Assumption 1: The unknown non-linear function ∆E is bounded, and there exists a known continuous function dr which ensures the following inequality is satisfied. ∆E ≤ dr.
(3)
The established model has the features of non-linear structure and can accurately describe the vehicle longitudinal dynamic characteristics; it is suitable to design the automatic longitudinal controller of intelligent vehicles.
3
Automatic speed hierarchical control architecture
Since the vehicle longitudinal dynamic system inherently has unknown nonlinearities and uncertain parameters, a hierarchical control architecture consisting of an upper control layer and a lower control layer is presented, as shown in Figure 1. Figure 1
Automatic speed control architecture
3.1 Upper control layer The longitudinal speed controller has the two-layer structure, and the design requirement of upper control layer is to determine the desired acceleration while maintaining safety and riding comfort of intelligent vehicles. The tracking error is the difference between a desired velocity vdes and a measured velocity v, which can be given as ev (t ) = vdes (t ) − v(t ).
(4)
The proportion-integration-differentiation (PID) upper controller is designed as follows: u (t ) = kp ev (t ) + kI ∫ ev (t )dt
(5)
where u denotes the control output of the upper controller, kp and kI are the proportional and integral gains, respectively. Larger accelerations would reduce the safety and riding comfort during the vehicle travelling, therefore, the desired acceleration is limited within –1.5 m/s2 to 1 m/s2. The diagram of upper control layer is shown in Figure 2, and the corresponding saturation function is designed as
A novel fuzzy-sliding automatic speed control of intelligent vehicles
ades
1 m/s 2 , (u ≥ 1 m/s 2 ) = u, (−1.5 m/s 2 ≤ u ≤ 1 m/s 2 ). 2 2 −1.5 m/s , (u ≤ −1.5 m/s )
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(6)
The parameters of upper controller are obtained by the trial and error method, in this section, kp = 2, kI = 0.3. As shown in Figure 3, the upper controller can generate the smooth desired acceleration in real-time, which satisfies the requirement of lower control layer. The simulation results manifest the effectiveness and reliability of the proposed upper controller. Figure 2
Block diagram of upper control layer
Figure 3
Desired velocity and acceleration
3.2 Lower control layer The task of lower control layer is achieved by regulating the vehicle acceleration precisely. Because the vehicle longitudinal dynamic systems have the properties of strong coupling, nonlinearities and parameter uncertainties, the robustness is the core issue for the design of lower controller. Sliding mode control technique has strong robustness to deal with the nonlinearities, parameter uncertainties and external disturbances (Nekoukar and Erfanian, 2011), it is suitable for widespread use in the design of vehicle longitudinal controller. But it has the disadvantage of high frequency chattering near the sliding hyperplane. According to the properties of strong coupling, nonlinearities and parameter uncertainties of intelligent vehicles, in this section, an adaptive fuzzy SMC scheme for automatic speed control which can overcome the effects of nonlinearities, disturbances and parameter uncertainties is proposed. To prevent the chattering phenomena and improve the performance of system, the control gains and the thicknesses of boundary layer are adaptively adjusted by fuzzy logic. As shown in Figure 4, first, the control gains of SMC law are adaptively adjusted by fuzzy logic, secondly, a boundary layer is combined with the conventional quasi-SMC, and the fuzzy controller is used to
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dynamically manage the thicknesses of boundary layer, the proposed lower controller algorithm can effectively overcome the parameter uncertainties and unmodelled dynamic characteristics, and guarantee the stability of control system. Figure 4
Block diagram of lower control layer
The objective of lower controller is to achieve the accurate tracking of desired acceleration via regulating the throttle and brake pressure. The desired acceleration produced by the upper controller is ades, the actual acceleration is a, then, the deviation between the desired and actual accelerations can be obtained as e = ades − a.
(7)
The sliding hyperplane in the space of tracking deviation is defined as t
S = e + λ ∫ e dt. 0
(8)
The time-derivative of the sliding hyperplane defined by equation (8) can be given as
dS = ades − a + λ (ades − a ) = 0. dt
(9)
The control target is satisfied when S = 0 and S = 0, and the equivalent control ueq is derived as the solution of the problem s(t ) = 0, the equivalent control term is valid only on the sliding hyperplane, so an additional term ur should be designed to deal with the uncertainties and external disturbances.
3.2.1 Throttle control strategy Based on the equilibrium relationship between driving force and running resistance, from equation (1), the vehicle longitudinal dynamic model can be expressed as follows: v =
1 Ts − Mgf cos ϑ − Cd Aa v 2 − Mg sin ϑ + ∆E1 . M e r
(10)
Assumption 2: The unknown non-linear function ∆E1 is bounded, there exists a known continuous function dr1 which ensures the following inequality can be satisfied.
A novel fuzzy-sliding automatic speed control of intelligent vehicles ∆E1 ≤ dr1 .
307 (11)
Using the SMC technique, substituting equation (10) into equation (9), thus e + λ ades +
λ Mer
⋅ ( Mgf cos ϑ + Cd Aa v 2 + Mg sin ϑ ) =
λ Mer
Ts ,eq .
(12)
From equation (12), the equivalent control of driving torque is found on the sliding hyperplane, as follows: Ts ,eq =
M er
λ
e + M e rades + r ⋅ ( Mgf cos ϑ + Cd Aa v 2 + Mg sin ϑ ).
(13)
Defining the sliding hyperplane S11 as S11 = S .
(14)
To deal with the uncertainties and external disturbances of the vehicle longitudinal dynamics system, the variable structure control law is designed as Tvs = K11 sgn( S11 ),
where K11 ≥ defined as
Mer
λ
(15)
dr1 . The sliding mode controller of desired driving torque Ts is
Ts ,des = Teq + Tvs .
(16)
From the equations Ts = To ioηt and To = Tt Rg , the desired turbine torque is derived as Tt ,des =
1
ηt Rg io
⋅ Ts ,des .
(17)
Assuming the rotate speeds of engine and bump are equal as ωp = ωe , then
ωe,des = Tpt−1 (Tt ,des , ωt ).
(18)
The engine model is J eω e = Te − Tp .
(19)
Defining the derivation between the desired and actual rotate speeds of engine as e12 = ωe ,des − ωe .
(20)
Based on equation (20), the sliding hyperplane S12 is given as S12 = e12 .
(21)
The time-derivative of the sliding hyperplane defined by equation (21) can be given as 1 S12 = ω e.des − ω e = ω e ,des − (Te − Tp ) . Je
(22)
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To improve the dynamic performance of the whole system, the reaching law is selected as follows: S12 = − K12 sgn( S12 ).
(23)
From equations (22) and (23), the desired engine torque is defined as Te ,des = J eω e ,des + TP (ωe , ωt ) + J e K12 sgn( S12 ).
(24)
The linear first-order model of engine is defined as
τ eTe + Te = f (ωe , α th ).
(25)
The derivation between the desired and actual output torques of engine is given as e13 = Te ,des − Te .
(26)
Defining the sliding hyperplane S13 as S13 = e13 .
(27)
The time-derivative of the sliding hyperplane defined by equation (27) can be obtained as 1 S13 = Te ,des − Te = Te ,des − ( f (ωe , α th ) − Te ).
τe
(28)
The reaching law is selected as follows: S13 = − K13 sgn( S13 ).
(29)
From equations (28) and (29), the following equation is true, such as 1 Te ,des − ( f (ωe , α th ) − Te ) = − K13 sgn( S13 ).
τe
(30)
Therefore,
τ eTe,des + Te + K13 sgn( S13 ) = f (ωe , α th ).
(31)
The desired throttle control law is obtained as
α th ,des = f −1 (τ eTe ,des + Te + K13 sgn( S13 ), ωe ).
(32)
3.2.2 Brake control strategy From equation (1), the vehicle longitudinal dynamic model under braking condition can be derived as v =
1 Te ,min Tb − − Mgf cos ϑ − Cd Aa v 2 − Mg sin ϑ + ∆E2 , Me r r
(33)
where Te,min denotes the lowest output torque of engine when the throttle angle is zero.
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Assumption 3: The unknown non-linear function ∆E2 is bounded, there exists a known continuous function dr2, which ensures the following inequality can be satisfied. ∆E2 ≤ dr2 .
(34)
Substituting equation (33) into equation (10), then e + λ ades −
λ M er
Te ,min +
λ M er
Tb +
λ Me
( Mgf cos ϑ + Cd Aa v 2 + Mg sin ϑ ) = 0.
(35)
The equivalent control of braking torque is obtained as 1 Tb ,eq = − M e r e + α des − r ( Mgf cos ϑ + Ca Aa v 2 + Mg sin ϑ ) + Te ,min . λ
(36)
Defining the sliding hyperplane as S 21 = S ,
(37)
then, the variable structure control law is designed as Tb ,vs = K 21sgn( S21 ),
where K 21 ≥
Mer
λ
(38)
dr2 , therefore, the desired braking torque is defined as
Tb ,des = Tb,eq + Tb ,vs .
(39)
Introducing the braking model, as follows:
τ bTb + Tb = K p ⋅ Pb .
(40)
The derivation between the desired and actual braking torque is given as e22 = Tb ,des − Tb .
(41)
Defining the sliding hyperplane as S 22 = e22 .
(42)
The reaching law is selected as below S22 = Tb,des − Tb = − K 22 sgn( S 22 ).
(43)
Consequently, it can be seen that 1 Tb ,des − ( K p Pb − Tb ) = − K 22 sgn( S22 ).
τb
(44)
From equation (44), the following equation can be obtained as K p Pb = τ bTb ,des + Tb + τ b K 22 sgn( S 22 ).
(45)
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Therefore, the desired braking torque is given as Pb =
1 (τ bTb ,des + Tb + τ b K 22 sgn( S 22 )). Kp
(46)
Control gains K1i(i=1,2,3) and K2i(i=1,2) of lower control layer are selected unsuitable may result in the chattering with high frequency, which can reduce the tracking performance and destroy the stability of the system, therefore, the selection of control gain parameters is a core issue of the design of sliding mode controller (Roopaei and Jahromi, 2009). In this section, an adaptive fuzzy SMC algorithm with variable boundary layer used for longitudinal control is presented to achieve favourable tracking performance.
3.2.3 Adaptive control gains The control gain parameters K1i(i=1,2,3) and K2i(i=1,2) have a significant impact on the robustness and stability of control system, hence they should be chosen according to the magnitude of the uncertainties, external disturbances and process noises (Guo et al., 2013). However, because the magnitude of external disturbances is unmeasurable, it is difficult to select the appropriate control gains. To overcome the chattering with high frequency in the SMC system, recently, the method integrating the neural network, adaptive control and genetic algorithm with SMC is proposed via assigning and learning the uncertain region of system. Fuzzy control has strong robustness and allows a human approach to control design without the demand for knowledge of mathematical modelling. In this section, a novel fuzzy SMC strategy, in which the SMC gains are regulated by the fuzzy logic inference system, is proposed to guarantee the robustness, reduce the chattering, and overcome the parameter uncertainties. The triangular and trapezoidal membership functions are used for the fuzzification of the input variables ( Sij and Sij ) and the output variable Kij, the membership functions of the linguistic terms negative big(NB), negative medium(NM), negative small (NS), zero (ZE), positive small (PS), positive medium (PB) and positive big (PB) are assigned to the inputs ( Sij and Sij ) and very small (VS), small (S), medium (M), big (B) and very big(VB) are assigned to the output Kij. The linguistic fuzzy rules of the throttle and brake control system are expressed in Table 1. It can be seen that, if the system states are above the sliding surface and are moving away from it, thus the control gain might be VB so that the system states could be enforced fast enough to return to the sliding surface. If the system states are on the sliding surface and moving to it, thus the control gain might be VS in order to avoid overshoot. The Mamdani implication operator is adopted to carry out the fuzzy inference, and the centroid defuzzification method which is the most widely adopted strategy especially for continuous systems, is utilised to implement the defuzzifier.
3.2.4 Adaptive variable boundary layer A boundary layer is introduced to replace the sign function to further alleviate the chattering and improve the tracking performance, the control law is obtained as
A novel fuzzy-sliding automatic speed control of intelligent vehicles Sij u = ueq + K ( Sij , Sij )sat ∆ ij
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,
(47)
where Sij sat ∆ ij
+1, Sij > ∆ ij = Sij / ∆ ij , Sij ≤ ∆ ij . −1, Sij < −∆ ij
(48)
Although the introduced saturation function can reduce the chattering, the robustness and steady performance are damaged by the unreasonable saturated parameters. A large/small thickness of the boundary layer more/less effectively alleviates the chattering phenomenon, but leads to less/more accurate control results. Table 1
Rule-base of fuzzy logic control gains
Kij Sij
Sij PB
PM
PS
ZE
NS
NM
NB
PB
VB
VB
B
B
M
S
VS
PM
VB
B
B
M
S
VS
PS
PS
B
B
M
S
VS
VS
M
ZE
B
M
S
VS
S
M
B
NS
M
S
VS
S
M
B
B
NM
S
VS
S
M
B
B
VB
NB
VS
S
M
B
B
VB
VB
To solve the above shortages and get better tracking performance, the time-varying boundary layer approach, which needs a strategy to automatically adjust the thickness of saturation function, has been given more attentions. In this section, an adaptive variable boundary layer based on fuzzy logic is introduced, and the boundary layer thicknesses are adjusted by a specified fuzzy control rule base. The fuzzy system adopts the absolute values of sliding hyperplane Sij and thickness ∆ij as the input and output variables, respectively. The membership functions of the linguistic terms small (SM), medium (NM), big (B) are assigned to the inputs Sij and wide (W), medium(S), narrow (N) are assigned to the outputs ∆ij. The triangular membership functions are used for the fuzzification of the input and output variables, as shown in Figure 5. Figure 5
Membership functions
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The linguistic fuzzy rules are expressed as the following procedure: R1: If S ij is big, then ∆ij is narrow R2: If S ij is small, then ∆ij is wide R3: If S ij is medium, then ∆ij is medium. The proposed adaptive boundary layer strategy follows the principle that the thickness is reduced gradually when the system tends towards steady state. The fuzzy control rule R1 narrows the thickness to achieve the aim of reaching the sliding surface quickly, which indicates the narrowed thickness drives the trajectory into the boundary layer as the surface outside the thickness of the boundary layer. The fuzzy control rule R2 broadens the thickness to achieve the aim of alleviating the chattering phenomena. The minimum–maximum reasoning method and centroid defuzzification method are also adopted here.
3.2.5 Stability analysis The stability of closed-loop throttle and brake control system is analysed by the Lyapunov theory to ensure the convergence of system states. Here, a two-step process is required: first, a positive define Lyapunov function is programmed, then, the time derivative of the Lyapunov function along the trajectory of control system is determined. Theorem 1: The stability of closed-loop vehicle longitudinal control system can be guaranteed by the proposed throttle and brake control laws.
Proof. First, the convergence of throttle control system is analysed, with regard to the sliding surface S11, defining the Lyapunov function as V11 =
1 2 S11 . 2
(49)
The time-derivative of equation (49) can be obtained as V11 = S11 ⋅ S11 lim
x →∞
= S11 ⋅ (e + λ e) = S11 ⋅ ( ades − a + λ ⋅ (ades − a )) 1 = S11 ⋅ ades − a + λ ⋅ ades − Me
(50) T ⋅ s − Mgf cos ϑ − Cd Aa v 2 − Mg sin ϑ + ∆E1 . r
Substituting equations (16) into equation (50), and K11 ≥ λ V11 ≤ − K11sgn( S11 ) + dr1 ⋅ S11 < 0. M er
Mer
λ
dr1 , yields
(51)
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Therefore, the reaching condition is satisfied, the desired torque can ensure the state variables of system asymptotically converge to the sliding surface S11 = 0, and even if the disturbances and uncertainties exist, the sliding surface will be invariant sets. With regard to the sliding surface S12, defining the Lyapunov function as V12 =
1 2 S12 . 2
(52)
The time-derivative of equation (52) can be obtained as V12 = S12 ⋅ S12 = S12 ⋅ (ω e,des − ω e )
(53)
1 = S12 ⋅ ω e,des − ⋅ (Te − Tp ) . Je
Substituting equation (24) into equation (53), yields V12 < 0.
(54)
Thus, the reaching condition is satisfied, the desired engine torque can ensure the state variables of system asymptotically converge to sliding surface S12 = 0. With regard to the sliding surface S13 , defining the Lyapunov function as V13 =
1 2 S13 . 2
(55)
The time-derivative of equation (55) can be obtained as V13 = S13 ⋅ S13 = S13 ⋅ (Ted − Te )
(56)
1 = S13 ⋅ Ted − ⋅ ( f (ωe , α th ) − Te ) . τ e
Substituting equation (33) into equation (56), yields
V13 < 0.
(57)
Therefore, the reaching condition is satisfied, the desired throttle input can ensure the state variables of system asymptotically converge to the sliding surface S13 = 0. Based on the Lyapunov theory, it can be seen that throttle control law can make the system asymptotically converge to the sliding surface in limited time; therefore, the proposed throttle controller can ensure the stability of the system. Secondly, the convergence of brake control system is analysed, with regard to the sliding surface S 21 , defining the Lyapunov function as V21 =
1 2 S21 . 2
The time-derivative of equation (58) can be obtained as
(58)
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V21 = S 21 ⋅ S21 = S21 ⋅ (e + λ e) = S21 ⋅ ( ades − a + λ ⋅ (ades − a )) 1 Te , min Tb = S21 ⋅ ades − a + λ ⋅ ades − − − Mgf cos ϑ − Cd Aa v 2 − Mg sin ϑ + ∆E2 . Me r r
(59) Substituting equations (39) into (59), and K 21 ≥ ( M e r λ ) ⋅ dr2 , yields λ V21 ≤ − K 21sgn( S21 ) + dr2 ⋅ S 21 < 0. M er
(60)
The reaching condition is satisfied V21 < 0, the desired braking pressure ensure the state variables of system asymptotically converge to the sliding surface S 21 = 0, and the trajectory remains in the sliding surface when the sliding surfaced is reached. With regard to the sliding surface S22 , defining the Lyapunov function as V22 =
1 2 S 22 . 2
(61)
The time-derivative of equation (61) can be obtained as V22 = S 22 ⋅ S22 = S 22 ⋅ (Tbd − Tb ) = S 22 ⋅ (Tbd −
1
τb
(62)
⋅ ( K p Pb − Tb )).
Substituting equation (46) into equation (62), yields V22 < 0.
(63)
The reaching condition is satisfied, the desired torque braking Pb,des can ensure that the state variables of system asymptotically converge to the sliding surface S22 = 0. Consequently, the brake control strategy can guarantee that the system states asymptotically converge to the sliding surface S21 = 0 and S22 = 0, and the convergence of system is ensured by brake control law. The stability of closed-loop vehicle longitudinal dynamic control system is proved.
3.2.6 Switching criteria One of the assumptions in the design of the throttle and brake SMC law is that the actuators can be instantaneously switched from one value to another. However, switching criteria between throttle and brake actuators at a very fast rate will result in the chattering problem and destroy the stability of the system, therefore, the switching criteria need to ensure the response of vehicle speed smoothly and rapidly. As shown in Figure 6, the proposed switching criteria are designed as follows: when both Te(t) and Tb(t) are positive, the throttle control is activated; when both Te(t) and Tb(t) are negative,
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as well as the absolute value of velocity error greater than threshold value, the brake control is activated; in other cases, the zero control is activated. The proposed switching strategy can effectively implement the combined control of throttle and brake actuators. Figure 6
4
Schematic diagram of switching criteria (see online version for colours)
Simulation and experimental tests
Firstly, a numerical Matlab–Adams co-simulation test is carried out to verify the feasibility of the proposed control strategy, and the behaviours of the proposed control strategy are compared with the conventional SMC system, which is a popular method. The vehicle parameters are M = 1770 kg, Jwf = Jwr = 1.8 kg·m2, Kp = 0.84, i0 = 4.71, Je = 0.15 kg·m2, τe = 0.3, h = 0.55 m, r = 0.28 m, ηt = 0.99, τb = 0.5, Rg = 0.27, Ca = 0.4 kg/m, respectively. The external disturbances and uncertainties are the random numbers within the region [–0.5 0.5]. Figure 7(a) shows the desired velocity of intelligent vehicle: at first, the vehicle runs at a constant velocity 54 km/h, then, it begins to accelerate for 8 s, in the final stage, it returns to run at a uniform speed of 40 km/h. The corresponding desired acceleration is the time-derivative of the desired velocity, and the initial speed error is set at –0.8 m/s. The response results of velocity error controlled by the proposed control strategy and conventional SMC method are shown in Figure 7(b), the steady-state velocity error of the proposed control strategy can converge to zero in finite time, but the velocity error controlled by the conventional SMC method oscillates within ±0.2 m/s, which indicates that, compared with the conventional strategy, the proposed control method not only improves the tracking precision, but also has less overshoot and smaller oscillation. Figure 7(c) and (d) shows the dynamic responses of control inputs including throttle angle and brake pressure. It can be seen that the switching between throttle and brake controllers of the proposed control method is much smoother than SMC, in addition, the chattering phenomenon is effectively attenuated by the proposed control strategy. The simulation results manifest that the proposed control strategy is feasible and effective, and has strong robustness to high uncertainties and disturbances.
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J. Guo et al. The response results of simulation test: (a) desired velocity; (b) velocity vector; (c) throttle angle and (d) brake pressure (see online version for colours)
To further appraise the performances of the automatic speed control scheme, the proposed control strategy which was implemented on the experimental vehicle has been tested. The desired velocity is shown in Figure 8(a), and the expected acceleration is shown in Figure 8(b). Figure 8(c) shows the response results of velocity error, it indicates that the actual velocity can converge to the desired velocity quickly, and the maximum velocity error which occurs during the periods of 14–17 s, is within 0.6 m/s, it is caused by the immediate changes of acceleration. Figure 8(d) shows the comparison diagram between the desired and actual accelerations. It is interesting to note that the proposed control scheme can ensure the intelligent vehicles track the desired velocity and acceleration trajectory smoothly and quickly. Figure 8(e) shows the response of throttle control input, it can be seen that the variations of throttle can be divided into three stages, the throttle control remains a constant value during the first and third stages, and the throttle control is inactivated at the second stage. The results of brake pressure are shown in Figure 8(f) the opposite of throttle control, the brake control is activated at the deceleration stage. Figure 8(e) and (f) indicates that the responses of both throttle and brake control inputs have no sharp fluctuations, and the proposed switching strategy can guarantee a smooth transition between throttle and brake actuators, which can improve the riding comfort and safety of intelligent vehicles.
A novel fuzzy-sliding automatic speed control of intelligent vehicles Figure 8
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The response results of experimental test: (a) desired velocity; (b) desired acceleration; (c) velocity; (d) acceleration; (e) throttle and (f) brake pressure (see online version for colours)
Conclusions
This paper presents a novel automatic speed control strategy for intelligent vehicles, which is aimed to improve the robustness to the high nonlinearities and parameter uncertainties. The proposed control scheme is developed based on fuzzy technique with robust sliding model control theory, and the control gains and the boundary layer thicknesses of robust sliding mode controller are adaptively adjusted by fuzzy logic simultaneously. The stability of the proposed control system is proved by the Lyapunov theory. Simulation and experimental results demonstrate the effectiveness and feasibility of the proposed control schemes.
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