Proceeding of the 11th World Congress on Intelligent Control and Automation Shenyang, China, June 29 - July 4 2014
Model Free Adaptive Control for Automatic Car Parking Systems DONG Hang-rui, JIN Shang-tai, HOU Zhong-sheng Advanced Control Systems Laboratory, School of Electronic and Information Engineering Beijing Jiaotong University, Beijing, China
[email protected],
[email protected],
[email protected] Abstract—In this paper, a Model Free Adaptive Control (MFAC) scheme is proposed for automatic car parking systems. The scheme consists of a control algorithm, a parameter estimation algorithm and a parameter reset algorithm. The design of the proposed scheme only uses the input and output (I/O) data, and does not involve any model information of the controlled car. Therefore, the MFAC based automatic parking system is applicable for different kinds of car. The simulation comparisons among MFAC scheme and PID control scheme are given for different kinds of car with different parking speed. The simulation results show that the proposed MFAC scheme has smaller tracking errors in the orientation angle of the car, the x axis and y axis.
drivers. Therefore, automatic parallel parking system has become the research focus [3] and parallel parking will be studied in the paper.
Keywords—Model free adaptive control; Automatic parking; PID
It is possible to ensure the parking along the set path only in the condition of controlling the wheel angle and speed and ensuring keeping the direction of the orientation angle of car and the trajectory tangent the same. The vehicle dynamic model can be converted into the nonholonomic chained system [5] on which the corresponding control algorithm can be designed. However, due to the features of automatic parking such as nonlinearity, time-varying and lots variables, it is difficult to set an accurate model of the nonholonomic chained system. Even if the model is established, with the complex corresponding control algorithm and large amount of calculation, it is hard to popularize the application. Therefore, the PID control algorithm without concise mechanism model and its improved form [6-8], fuzzy control [9] and neural network control [10-12] are used to solve the control problem in the process of automatic parking. But the cars on the market are different with each other in many aspects, such as type, size and performance. For different cars, it is hard to adjust the PID controller parameters and to unify the fuzzy control rules. As for the neutral network control, there are also some difficulties, such as large amount of calculation and hard training. So the automatic parking system based on the above algorithm has poor portability.
Usually, the automatic parking process includes three steps: detecting parking space, path planning and path tracking [4]. After the Park Assist is started, the system begins to search the available parking space when the driver needs to park. A feasible path is calculated if the parking space is available, and then with the start of reversing, the car will stop at the exact target point along the set path. Finally, a little move of the vehicle body and the direction of the front wheels are needed to complete the whole parking process.
I. INTRODUCTION With more and more car ownership, the roads, parking lots, residential areas are quite crowed and the available parking space is less than before. Besides, varied problems caused by inexperienced new drivers have also greatly increased. The research of Transportation Research Institute at the University of Michigan showed that [1], the car accidents caused by parking account for 44% of different kinds of accidents. About 1/2 to 3/4 automatic crash accidents are brought about by reversing, while manual parking is the major reason for that problem. The automatic parking system can avoid parking accident. With the technology of sensor, computer and automatic control, the automatic parking system can perceive the parking environment exactly, and project an optimal parking path, then track the path automatically and finally stop safely and accurately [2]. There are three basic parking types, referring to parallel parking, vertical parking and oblique parking. Among them, the parallel parking is the most common one. However, without the help and aid of others, it is difficult to operate this process for most people especially for new drivers and women
The Model Free Adaptive Control [13-15] scheme’s design only relies on the I/O data of the controlled system, not including any model information of the controlled system, so it can realize parametric adaptive and structure adaptive control of the unknown controlled system. At present, the Model Free Adaptive Control has been successfully applied in many fields, such as mold, motor [16,17], chemical industry, temperature, pressure [18,19], freeway control [20],
This work is supported by National Natural Science Foundation of China (61120106009), and the Fundamental Research Funds for the Central Universities (2014JBM005) 1. Advanced Control Systems Laboratory, School of Electronic and Information Engineering, Beijing Jiaotong University, Beijing 100044
978-1-4799-5825-2/14/$31.00 ©2014 IEEE
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engineering structure vibration damping [21] and sheet metal forming [22]. According to the theory, simulation study and practical application, the MFAC can solve the controlling problems of the unknown and nonlinear time-varying system with many advantages such as simpleness, practicability, less calculation and strong robustness.
turning right the steering wheel and reversing, turning back the steering wheel and straight reversing, turning left the steering wheel and reversing to the end. As shown in Fig. 2, there are three rectangular parking spaces among which two sides are occupied. Wc stands for the vehicle width, Lc for the vehicle length, L p for the parking space length, L for the wheelbase, ΔS for the safe distance between two vehicles and d for the distance between the middle line of the body and vehicles parked on the street .We establish a rectangular coordinate as is shown in Fig. 2 in which the x axis stands for the middle line of the parking space and ΔS is the distance between y axis and the head of the back car. The vehicle trajectory can be fully embodied by the movement process of the rear wheels, therefore, the rear wheel axis center coordinates is made as the coordinate of body movement, which means the movement path of the midpoint of the rear wheel axis is the parking path.
Aiming at parking system, this paper proposed the Model Free Adaptive Control scheme which includes control algorithm, parameters estimation algorithm and reset algorithm. This control scheme’s design only use the I/O data of the automatic parking system, not including any vehicle model information, therefore, the Model Free Adaptive Control can be applied in the automatic parking system for different types of car. In the last part of this paper, a simulation comparison between MFAC and PID control algorithm is made in terms of different cars and different speed. The simulation results show that the proposed MFAC scheme has smaller tracking errors in the orientation angle of the car, the x axis and y axis. Therefore, it proves the superiority and effectively of MFAC.
Automatic parking process starts at point P5, and ends at point O. The parking paths of four stages are line segment P5P4, arc segment P4P2, line segment P2P0 and arc segment P0O. Arc P0O is tangent to the x axis at point O, and the radius R1 should not be smaller than the minimum turning radius. In order to keep safe with the front car, draw a circle arc in radius R2 = 0.5Wc + ΔS with left-rear corner of the front car as the circle center. Draw the common tangent line of these two arcs, and two arc tangent points are P0 and P1 respectively. The common tangent line intersects line P5P4 at P3. Arc P4P2 which is tangent to P3P0 at P2 and P5P3 at P4 is used to connect line segment P1P0 and P5P4 commonly. The length of radius R3 depends on the length of segment P4P3 with limit R3>R1.
This paper is organized as follows: in Section 2: the control of the automatic parking system; in Section 3: the Model Free Adaptive Control scheme; in Section 4: simulation between MFAC and PID; in Section 5: conclusion. II.
DESCRIPTION OF THE AUTOMATIC PARKING PROCESS
A. Parking Space Detection When the driver prepares to stop, he can start the automatic parking system which will scan the space information with the ultrasonic probe set on the side of the car. We can see from the diagram of parking space detection (Fig.1), the ultrasonic probe can detect the distance information. If there are vehicles parked by the side of the road, the system would get the d1 (m), otherwise d2 (m). If there is parking space available, the pulse in dotted line can be obtained, and then the length of the space will be get according to the speed and the pulse duration. The space will be available if it is longer than Lp min (listed in the next section). If the space is not available, you can move on until find one. After get the available space, the system will enter the stage of path planning.
Fig.2 Path planning of automatic parking
Based on the above analysis, if O is the origin of coordinates, we can give the coordinates of P0, P2 and P4 respectively as follows
P0 : ( R1 sin α , R1 (1 − cos α ) ) ,
(1)
P2 : ( X p 4 − L34 (1 + cos α ) , Yp 4 − L34 sin α ) ,
(2)
P4 : ( X p 4 , Yp 4 ) ,
(3)
R1 − R1 + d + 0.5Wc cos α where X p 4 = , Yp 4 = d + 0.5Wc . L34 is tan α the length of the line P3P4， α is the included angel between line P3P0 and the positive direction of X axis which can be obtained through the following equation:
Fig.1 Diagram of parking space detection
B. Path Planning According to the driver’s experience, the Parallel parking process includes four steps, referring to straight reversing,
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Wc + R2 cos α − R1 (1 − cos α ) tan α = 2 . L p − R1 sin α − R2 sin α − ΔS
tracking will be fully ensured. So we can conclude that reducing the tracking error with the car’s target orientation angle is quite important for improving automatic parking accuracy. Therefore, the control target of the automatic parking system is to maintain a constant speed, then control the orientation angle of the car by controlling the angle of the front wheel which aims to make equal to the target orientation angle of the car so as to ensure that the coordinates of the car falls on the target trajectory constantly. Given that in the process of reversing, if we only focus on tracking the target trajectory while ignoring the orientation angle errors of the car, vehicle collisions are easy to occur. That is why we put the orientation angle of the car as the control target.
(4)
According to the path planning, we can get the shortest ava ilable parking space Lpmin when the line P0P1 equals Lc, L p min = ( R1 + R2 ) sin α + Lc cos α .
(5)
And α can be obtained through the following equation
Lc sin α = ( R1 + R2 ) cos α + Wc / 2 − R1 .
(6)
III. MFAC SCHEME Fig. 4 shows the basic control block diagram of the automatic parking system. In this system, the vehicle could stop at the designated place accurately as long as the controller can make θ fully tracking θ * . The orientation angle of the car in real time can be measured by sensors like gyroscope, while the controller revising the angle of steering wheel through the Steering-by-Wire System based on the error between the target orientation angle of the car and current orientation angle.
C. Path Tracking 1) Vehicle dynamics model The diagram of the vehicle dynamics model [9, 23] which is generally accepted is shown as Fig. 3, and the model as follows ⎧ x = ν cos θ , ⎪ ⎨ y = ν sin θ , ⎪θ = ν tan β / L. ⎩
(7)
Of which, X, Y is the abscissa and ordinate of the midpoint of rear wheel axle, ν is the speed; L is the wheelbase, and θ is the included angle between vehicle body and X axis; β is the front wheel turning angle.
Fig.4 Block diagram of automatic parking control system
In the practical application, the signals are processed into digital ones by the MCU, therefore, model (7) usually discrete into
⎧θ ( k + 1) = θ ( k ) + Tν tan β ( k ) / L, ⎪ ⎨ x ( k + 1) = x ( k ) + Tν cos θ ( k ) , ⎪ y ( k + 1) = y ( k ) + Tν sin θ ( k ) , ⎩
The MFAC scheme only depends on the I/O data of the controlled system without modeling vehicles, so according to the target orientation angle of the car provided by the path planning, we can directly propose the MFAC scheme [13] aiming at automatic parking system. First, we need to transfer the nonlinear system of the car automatic parking into the dynamic linearization data model just as follows,
(8)
where T is the sampling time.
Δθ ( k + 1) = φ1 ( k ) Δθ ( k ) + φ2 ( k ) Δβ ( k ) ,
(9)
where β is the rotating angle of the front wheel, φ1 ( k ) 、
φ2 ( k ) are two pseudo-partial derivatives estimated on line. Δθ ( k +1) = θ ( k + 1) −θ ( k ) , Δβ ( k ) = β ( k ) − β ( k −1) .
We used the following control input index function to design the control law: J ( β ( k ) ) = ⎡⎣θ * ( k + 1) − θ ( k + 1) ⎤⎦ + λΔβ ( k ) , 2
2
(10)
where λ > 0 is a weighting constant.
Fig.3 Vehicle model
Besides, the front wheel is limited by the maximum steering angle β max as β ( k ) ≤ β max .
Substituting (9) into (10) ， differentiating (10) w.r.t.
β ( k ) , and letting it be zero gives
2) Control Target If the orientation angle of the car can keep consistent with the slope of the trajectory at every moment, the coordinates of
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This algorithm is easy to perform with little adjustment parameter and calculation, so it’s suitable for different vehicles to realize the adaptive control of the automatic parking system.
β ( k ) = β ( k − 1) +
ρφ2 ( k ) (θ * ( k + 1) − θ ( k ) − φ1 ( k ) Δθ ( k ) ) λ + φ (k ) 2 2
(11)
.
IV. SIMULATION RESULTS Aiming at two types of vehicle, this paper gives the simulation comparison results between two control scheme MFAC (11)、 (13)、 (14) and PID(15) at different parking speed ( υ = 1.44 km / h and υ = 2.88km / h ) to verify the superiority of MFAC. The PID controller is in positional form as follows
Define index function for the unknown parameters φ1 ( k )
and φ2 ( k ) estimation as
⎛ ⎡φ ( k ) ⎤ ⎞ J ⎜⎢ 1 = θ ( k ) − θ ( k − 1) ⎜ φ2 ( k ) ⎥ ⎟⎟ ⎦⎠ ⎝⎣ ⎡φ ( k ) ⎤ −⎢ 1 ⎥ ⎡⎣ Δθ ( k − 1) , Δβ ( k − 1) ⎤⎦ ⎣φ2 ( k ) ⎦
2
k
(12)
β ( k ) = K p e ( k ) + K i ∑e ( k ) + K d ( e ( k ) − e ( k − 1) ) , (15) j =1
2
⎡φ ( k ) ⎤ ⎡φ1 ( k − 1) ⎤ +μ ⎢ 1 ⎥−⎢ ⎥ , ⎣φ2 ( k ) ⎦ ⎣φ2 ( k − 1) ⎦
where e ( k ) = θ * ( k ) − θ ( k ) is the error of the orientation angle of the car, and Kp, Ki and Kd stands for proportion, integration and differential parameter respectively.
where μ > 0 is a weighting factor.
PID parameters are a set of better ones which is obtained through several simulations and comparisons with considering two types of vehicle under the guide of the principle of quickness and small overshoot. The parameter settings of the three algorithms are shown in Table Ⅰ.
With the matrix inversion lemma and the optimal conditions, we can estimate the algorithm of φ1 ( k ) , φ2 ( k ) as following ⎡φl1 ( k ) ⎤ ⎡φl1 ( k − 1) ⎤ ⎡Δθ ( k − 1) ⎤ ⎢ ⎥=⎢ ⎥ +η ⎢ ⎥ l l ⎣ Δβ ( k − 1) ⎦ ⎣⎢φ2 ( k ) ⎦⎥ ⎣⎢φ2 ( k − 1) ⎦⎥
TABLE Ⅰ ALGORITHM PARAMETERS
⎡Δθ ( k − 1) ⎤ Δθ ( k ) − ⎡φl1 ( k − 1) , φl2 ( k − 1) ⎤ ⎢ ⎣ ⎦ Δβ ( k − 1) ⎥ ⎣ ⎦. × ⎡Δθ ( k − 1) ⎤ μ + ⎣⎡ Δθ ( k − 1) , Δβ ( k − 1) ⎦⎤ ⎢ ⎥ ⎣ Δβ ( k − 1) ⎦
(13)
MFAC
PID
φl (1) = 2.6
K p = 21.5
φl2 (1) = 0.4
K i = 0.18
ρ = 7.6
K d = 0.08
1
ε = 10−4 μ = 0.01 η = 0.01 λ = 0.06
In order to make the control algorithm more flexible, the step factor η ∈ (0, 2] is added. The parameter reset algorithm is given in order to ensure that the parameter estimation algorithm has good ability to track the time-varying parameters.
The simulation includes two parts. The first part is the automatic parking simulation of the FAW-VW CC (2012); the second part is the simulation of the Audi A6L.
⎡φl1 ( k ) ⎤ ⎡φl1 (1) ⎤ ⎡φl ( k ) ⎤ ⎢ ⎥=⎢ ⎥ , if ⎡φl1 ( k ) , φl2 ( k ) ⎤ ⎢ 1 ⎥≤ε ⎣ ⎦ ⎢φl k ⎥ ⎢⎣φl2 ( k ) ⎥⎦ ⎢⎣φl2 (1) ⎥⎦ ⎣ 2 ( )⎦ ⎡Δθ ( k − 1) ⎤ or ⎡⎣ Δθ ( k − 1) , Δβ ( k − 1) ⎤⎦ ⎢ ⎥ ≤ ε (14) ⎣ Δβ ( k − 1) ⎦ or sign φl ( k ) ≠ sign φl (1) ,
A. The simulation result of automatic parking of the FAW-VW CC (2012) The basic sizes of the FAW-VW CC (2012) and relevant information about parking spaces are shown in Table Ⅱ.
(
2
)
(
2
TABLE Ⅱ BASIC PARAMETERS OF FAW-VW CC
)
where ε is a small positive constant。 φl1 ( k ) and φl2 ( k ) are the
Parameter
Value
Lc
4.799 m 1.855 m 2.712 m 5.600 m
Wc
L
estimation value of φ1 ( k ) and φ2 ( k ) respectively, and only this two parameters need to be adjusted on line which means has smaller amount of calculation compared with the traditional adaptive algorithm, and the controller is simple but practical.
Lp
β max L34
ΔS
So far this paper proposes the model free adaptive control scheme (11) 、 (13) 、 (14) for the automatic parking system.
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42.00 ° 1.000 m 0.500 m
computing method of the radius R1 is in line with (14) and the algorithm parameters are same as those of Table Ⅲ.
In order to protect the vehicle performance, we need to avoid keeping the steering wheel at maximal joint angle, so the radius R1 takes R1 =
L ⎛β ⎞ tan ⎜ max ⎟ ⎝ 1.1 ⎠
TABLE Ⅲ BASIC PARAMETERS OF AUDI A6L Parameter
(16)
.
MFAC PID
2
error of θ(°)
0 -2 1
0
5
10
20 t(s)
15
error of x axis(cm)
0 -1 -2 0 5
5
10
20 t(s)
15
0
-5
0
5
10
20 t(s)
15
Lc
5.015 m
Wc L
1.874 m
β max
42.00 °
3.012 m
The automatic parking simulation result of Audi A6L is shown in Fig. 7-8. Fig. 7 shows the parking errors under the speed of υ = 1.44km / h . Obviously MFAC scheme performs better than PID scheme in terms of errors in x axis, y axis and the orientation angle of car during various stages of the parking. As we can see from Fig. 8, MFAC also tracks better than PID even though the parking speed improves. Therefore, MFAC scheme competes better under different parking speed during the process of automatic parking for Audi A6L.
error of y axis(cm)
error of y axis(cm)
error of x axis(cm)
error of θ(°)
The automatic parking simulation result of the FAW-VW CC (2012) is shown in Fig. 5-6. Fig. 5 shows the parking errors under the speed of υ = 1.44km / h . Obviously MFAC scheme performs better than PID scheme in terms of errors of x axis, y axis and the orientation angle of car during various stages of the parking. As we can see from Fig. 6, MFAC can also tracking better than PID even though the parking speed increases. Therefore, MFAC scheme competes better under different parking speed during the process of automatic parking for FAW-VW CC.
Value
Fig.5 Parking error comparisons of FAW-VW ( υ = 1.44km / h )
MFAC PID
2 0 -2 0
5
10
15
20 t(s)
-4 0 5
5
10
15
20 t(s)
5
10
15
20 t(s)
2 0 -2
0
-5
0
error of θ(°)
MFAC PID
2 0 -2 0
2
4
6
8
10 t(s) error of x axis(cm)
1 0 -1
-2 0 5
2
4
6
8
10 t(s)
error of y axis(cm)
error of y axis(cm)
error of x axis(cm)
error of θ(°)
Fig.7 Parking error comparisons of Audi A6L ( υ = 1.44km / h )
0
-5
0
2
4
6
8
10 t(s)
Fig.6 Parking error comparisons of FAW-VW ( υ = 2.88km / h )
MFAC PID
2 0 -2 0
2
4
6
8
10 t(s)
-4 0 5
2
4
6
8
10 t(s)
2
4
6
8
10 t(s)
2 0 -2
0
-5
0
Fig.8 Parking error comparisons of Audi A6L ( υ = 2.88km / h )
B. The automatic parking simulation result of Audi A6L The basic sizes of Audi A6L are shown in Table 4 and other dimension parameters are same as those in Table 3. The
According to the simulation results of two types of car, we can see the MFAC scheme can fulfill automatic parking
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process better than PID scheme for different cars with different speed.
[9]
V. CONCLUSION Aiming at the automatic parking system, this paper proposes the MFAC scheme which only depends on the I/O data without any model information. We also give automatic parking simulations of different types of vehicle in different speed and the results show that the MFAC scheme has higher accuracy and more quickly response than the PID scheme in terms of tracking the orientation angle of car and coordinates trajectory of x and y. And it can adapt to the automatic parking process under different speed.
[10]
[11]
[12] [13] [14]
REFERENCES [1]
[2] [3]
[4]
[5]
[6]
[7]
[8]
[15]
Green P. Parking Crashes and Parking Assistance System Design: Evidence from Crash Databases, the Literature and Insurance Agent Interviews [C]. Proc. of SAE World Congress, 2006, 1685-1700. Song J.Z. Research on the Key Technology for Auto-parking system[D]. Hunan: National University of Defense Technolegy,2009. Hanafy M., Gomaa M.M., Taher M. and Wahba A.M. Path generation and tracking for car automatic parking employing swarm algorithm[C].Proc. of the 2011 International Conference on Computer Engineering & Systems (ICCES).2011:99-104. Jung H.G., Lee Y.H. and Kim J.H. Uniform User Interface for Semiautomatic Parking Slot Marking Recognition [J], IEEE Transactions on Vehicular Technology.2010, 59(2):616-626. Liu K.Z., Horii M. An experimental comparison of nonholonomic control methods: Automatic parking benchmark[C]. Proc. of SICE, 2007 Annual Conference. 2007: 1712-1717. Hsu T.H., Liu J.F., Yu P.N., Lee W.S. and Hsu J.S. Development of an automatic parking system for vehicle[C]. Vehicle Power and Propulsion Conference. 2008:1-6. Chen C.H., Hsu C.W. and Yao C.C. A Novel Design for Full Automatic Parking System[C]. Proc. of 2012 12th International Conference on ITS Telecommunications, 2012: 175-179. Liang Z., Zheng G.Q. and Li J.S. Application of Fuzzy Control Strategy in Automatic Parking Path Planning[C]. Proc. of 2012 4th International
[16]
[17]
[18] [19] [20] [21]
[22]
[23]
1774
Conference on Intelligent Human-Machine Systems and Cybernetics. 2012, 2 :132–135. Baturone I., Moreno-Velo F.J., Sánchez-Solano S. and Ollero A. Automatic design of fuzzy controllers for car-like autonomous robots [J]. IEEE Transactions on Fuzzy Systems. 2004, 12(4):447-465. Gorinevsky D., Kapitanovsky A. and Goldenberg A. Neural network architecture for trajectory generation and control of automated car parking[C]. Proc. of Control Systems Technology.1996, 4(1):50–56. Daxwanger W.A., Schmidt G.K. Skill-based visual parking control using neural and fuzzy networks[C]. Proc. of System, Man, and Cybernetics. 1995, 2:1659–1664. Nguyen D.H., Widrow B. Neural networks for self-learning control systems [J]. Control Systems Magazine, IEEE. 1990, 10(3):18-23. Hou Z.S, Jin S.T. Model Free Adaptive Control: Theory and Application [M].Beijing: Science Press, 2013. Hou Z.S, Huang W.H. The model-free learning adaptive control of a class of SISO nonlinear systems[C]. Proc. of American Control Conference. Albuquerque, 1997: 343-344. Hou Z.S. Nonparametric model and its adaptive control theory [M].Beijing: Science Press, 1999. Tan K.K., Lee T.H., Huang S.N. and Leu F.M. Adaptive predictive PI control of a class of SISO systems[C]. Proc. of American Control Conference. California, 1999:3848-3852. Tan K.K., Lee T.H., Huang S.N. and Leu F.M.Adaptive predictive control of a class of SISO nonlinear systems[J]. Dynamics and Control,2001,11(2):151-174． Han Z.G. The application of model free controller [J].Control Engineering of China, 2002, 9(4):22-25. Han Z.G. Model free control law in the prediction of chemical fertilizer [J].Control Theory & Application, 2004,21(6):858-863. Wang C.W. Studies on the model-free adaptive control based local ramp metering strategies [D]. Beijing: Beijing Jiaotong University, 2005. Zhou Q, Qu W.L. Model-free learning adaptive semi-active control of structure with MR dampers [J]. Earthquake Engineering and Engineering Vibration, 2004, 24(4):127-132. Liu C.G, Sui Z, Fu W.Z, et al. Nonparametric model and adaptive control for multi-point forming process [J]. Control Engineering of China, 2004, 11(4):306-308. Muller B., Deutscher J. and Grodde S. Continuous Curvature Trajectory Design and Feedforward Control for Parking a Car [J]. IEEE Transactions on Control Systems Technology. 2007, 15(3):541-553.
