Int. J. Fuzzy Syst. DOI 10.1007/s40815-017-0383-1

Backing Up a Truck on Gaussian and Non-Gaussian Impulsive Noise with Extended Kalman Filter and Fuzzy Controller Junying Zhang1

•

Yuting Zhang2 • Cong Xu2

Received: 30 March 2017 / Revised: 23 July 2017 / Accepted: 4 September 2017  Taiwan Fuzzy Systems Association and Springer-Verlag GmbH Germany 2017

Abstract Truck backing-up problem is a typical test bed for fuzzy control system. The control performance affects the safety of the truck well, but has not been studied when location of the truck is given by GPS which introduces sensing noises into the system. In this paper, we study the impact of noise on control performance of the system, and we propose an extended Kalman filter which claims to adapt to only Gaussian noise for improving control performance in Gaussian and non-Gaussian impulsive noise situation. To implement the filter, we propose screening the input to get the output of the fuzzy controller such that the partial derivative of the input–output function of the controller required by the extended Kalman filter is computationally available. Our simulation results of the truck system with and without noise, the noise being Gaussian and non-Gaussian impulsive, and the system with and without the extended Kalman filter, indicate that the average performance of the system with the filter is much better than that without the filter no matter the noise is Gaussian or impulsive, the great power of the extended Kalman filter in dealing with even non-Gaussian impulsive noises for fuzzy truck control, while the great deviation

& Junying Zhang [email protected] Yuting Zhang [email protected] Cong Xu [email protected] 1

School of Computer Science and Technology, Xidian University, No. 2 Taibai Road, Xi’an 710071, People’s Republic of China

2

School of Information Engineering, Xijing University, Chang’an District, Xi’an 710123, People’s Republic of China

from the average performance makes an urgent call for non-Gaussian version of the extended Kalman filter to adapt to more general non-Gaussian impulsive noise situation. Keywords Impulsive noise  Extended Kalman filter  Fuzzy controller  Truck backing-up system

1 Introduction Truck backer-upper problem has been an excellent test bed for fuzzy control systems. The problem, which was derived by Nguyen and Widrow [1], has been investigated by many researchers in the field of computational intelligence, especially fuzzy logic and fuzzy control. A lot of studies demonstrate that fuzzy control of the truck backer-upper system is superior to neural network [2], and these two control methods have been compared [3]. Kosko et al. proposed an adaptive fuzzy system for backing up a truck and trailer [4]. Riid et al. presented a fuzzy supervisory control system over the PID controller to reduce the complexity of the control problem and enhance the control performance [5]. Fuzzy controller, formulated on the basis of human understanding of the process, can be regarded as an emulator of human operator. Recently, the study of the problem is more advanced. Examples include the fuzzy truck control system for obstacle avoidance, using newly designed 33 fuzzy inference rules for steering control and 13 rules for speed control [6], clustering approach to determine the number of fuzzy rules with neural network used to train the parameters of the constructed fuzzy model [2], evolutionary algorithm-based approach to optimize fuzzy control of truck backer-upper system [7], decomposition approach

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and hierarchical approach which reduce the classical complex control to smaller number of simpler rules for fuzzy control [8, 9], robust stabilization analysis in the sense of Lyapunov with fuzzy controller which guarantees stability of the control system under a condition designed [10], and a suboptimal solution to the problem of automatically back driving a truck using the natural parabolic paths as the shortest moving distance requirement [11]. Current focus is only on how to design a fuzzy controller to control truck for a good performance, where an implicit assumption is no sensing noise in the control system. To our knowledge, there is no work on the impact of sensing noise on control performance when the truck is automatically controlled based on the location sensed by GPS which always introduces sensing and environmental noise to the system. By referring sensing and environmental conditions as GPS sensing noise such as weather and buildings beside the truck which influence the estimation precision of the truck location, it is known that the noise in GPS coordinate time series follows a power-law noise model with different components (white noise, flicker noise, and random walk) [13, 14]. We believe that the GPS sensing noise must have impact on the control performance of the truck. Studying the impact is significant in that the safety of the truck depends on the robustness of the control system to the sensed noises. In this paper, we study the impact of GPS localization and environment noise on control performance of a truck backing-up fuzzy control system, which has not even been studied before. Fuzzy control has been acknowledged to be robust to noise in the sense that it is based on an experienced control expert who knows how to deal with noises in the system. However, we find through our simulation study that its robustness is still very limited, especially when the noise is non-Gaussian impulsive. By considering that a fuzzy control system is always nonlinear, we propose the extended Kalman filter which claims suboptimal for Gaussian noise situation to improve the robustness of the system. We then study the robustness of the system with the extended Kalman filter for both Gaussian and nonGaussian impulsive noise. The study here is significant for understanding the extended Kalman filter if and to what degree it can adapt to a fuzzy control system contaminated with non-Gaussian impulsive noises. We take the control system of backing up a truck as an example for the robustness study. The main contributions of the paper are twofolds: (1) We propose a screening approach such that the extended Kalman filter is technically and computationally available for the backing-up control system of the truck, (2) We study the impact of GPS non-Gaussian noise and the effect of the extended Kalman filter on control performance of a back-up truck

control system. The first contribution makes a computation tool which leads to the second contribution available. In designing the extended Kalman filter, we meet the problem of how to find partial derivative of the fuzzy controller due to no explicit expression of the controller. To tackle it, the input–output function of the fuzzy controller is obtained by screening the input in related range to get the output of the function from the fuzzy rules of the controller such that the partial derivative of the function is computationally available. Our simulation results of the truck backing-up system with and without noise, the noise being Gaussian and impulsive, and the system with and without the extended Kalman filter, indicate that the average performance of the system with the extended Kalman filter is much better than that without the filter no matter the noise is Gaussian or impulsive: It adapts to impulsive noises well and is not seriously sensitive to noise type and level, the great power of the extended Kalman filter in dealing with non-Gaussian impulsive noises for fuzzy truck control; and the deviation of the performance from the average is large and even inacceptable, especially for high noise level showing the drawbacks of the extended Kalman filter for non-Gaussian impulsive noise situation. We believe that such situation also holds for other fuzzy control system. To further improve the robustness of the system, it is anticipated that the nonGaussian version of the extended Kalman filter which especially adapts to non-Gaussian impulsive noise should be theoretically studied and designed in the first place.

2 Control System Backing a truck to the loading dock or parking spot is a difficult task even for a skilled truck driver. Most truck drivers have to back up the truck and go forward several times before they finally back it to the desired position. In some circumstances, the drivers have to back the truck directly to the loading dock due to the space limitation. The truck and the loading zone of ½xmin ; xmax  9 ½ymin ; ymax  are plotted in Fig. 1. The state of the truck is

Fig. 1 Truck backing-up system

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J. Zhang et al.: Backing Up a Truck on Gaussian and Non-Gaussian Impulsive Noise…

2.1 Truck Model

Definition of membership functions

m(x)

1 0.5 0 0

where ðxk ; yk Þ is the x- and y-coordinate and uk is the angle of the truck at stage k, and hk is the steering angle which moves the truck r meters backward from ðxk1 ; yk1 Þ to ðxk ; yk Þ. The three-dimensional vector ðx; y; uÞ is considered the state variables of the truck at stage k. It is assumed that both the location and the angle of the truck are sensed by GPS, which introduces noises in observation of the truck state. For simplicity, only the x-coordinate x and orientation u of the truck are used as inputs and the steering angle h is the output of a fuzzy controller. 2.2 Fuzzy Controller For the fuzzy control of the truck, the state variables are respectively assigned into 5, 7, and 7 fuzzy sets [12]. The detailed membership functions of the variables associated with the fuzzy subsets are given in Fig. 2. Triangle shapes are used here for simple computation and better description of the problem depending on expert. The total number of control rules is 5 9 7 = 35, which is summarized in Table 1. 2.3 Control System Fuzzy control has been acknowledged to be robust to noise in the sense that it is based on an experienced control

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Movement of the truck can be modeled by the three-state equation 8 < xk ¼ xk1 þ r cosðuk1 þ hk Þ ð1Þ y ¼ yk1 þ r sinðuk1 þ hk Þ : : k uk ¼ uk1 þ hk

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represented by three state variables x, y, and u, where u is the angle of the truck with respect to the horizontal line, and the pair ðx; yÞ specifies the center location of the truck. Control variable is the steering angle h, the angle of the front wheel with the truck. In this study, the range of x and y is [0, 100], that of u is [-90, 270], and that of h is [-30, 30]. The truck is initialized at the state of ðx0 ; y0 ; u0 Þ, where ðx0 ; y0 Þ is an arbitrary location in the zone, and u0 is an arbitrary angle in the range of ½umin ; umax . The goal is to make the truck stop at a loading dock ðxf ; yf Þ with the final angle uf . For simplicity, we assume enough clearance between the truck and the loading dock. In this paper, only backing up is considered. The truck moves in a uniform speed, i.e., r meters in each stage, under the control of h until when the goal is reached.

x-position NB

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expert who knows how to deal with noises in the system. However, we find through our simulation study (see Sect. 4) that its robustness is still very limited, especially when the noise is non-Gaussian impulsive. By considering that a fuzzy control system is always nonlinear, we propose the extended Kalman filter to improve the robustness of the system. The study here is significant for understanding the extended Kalman filter, which adapts to only Gaussian noise, if and to what degree it can adapt to a fuzzy control system contaminated with non-Gaussian impulsive noises. By developing an extended Kalman filter for the estimate of the state of the truck when the state variables are sensed by GPS which introduces noises on them, the structure of the control system is shown in Fig. 3. 2.4 Noise Model Though it is known that the noise in GPS coordinate time series follows a power-law noise model with different components (white noise, flicker noise, and random walk) [13, 14], no one studies the impact of the noises on the fuzzy control system of a truck even in Gaussian noise

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X k-1 ϕk-1

VK Fuzzy Controller

θK

Truck with the state of

(x k,y k,ϕk)

(xk-1,yk-1,ϕκ−1)

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ˆk Z-1

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the characteristic function the pdf of alpha-stable variables with different a and the fixed dispersion c ¼ 1, given in Fig. 4c. Seen from the figure is that the tail of the pdf for a\2 is heavier than that of Gaussian (the situation of a ¼ 2). It is such heavier tail of the pdf that leads to the impulsiveness of the noise. And the smaller the a is, the heavier the tail of the pdf is, leading to more impulsiveness of the noise.

Operang point (x k-1,θ k)

Fig. 3 Structure of the truck fuzzy control system with extended Kalman filter

situation. Here, alpha-stable noise is adopted as the model of the noise under study, within which Gaussian noise is only its typical and significant subfamily. Mandelbrot popularized the model in the early 1960s [17, 18], and nowadays a large number of works have been done, including domain of attraction [15, 16], computations of stable pdf, cdf, and its tail properties [19–23], stability test [21, 22] [24, 25], and a vast amount of work relating to applications, mainly in finance and then spread to a variety of ranges in different areas due to its ability of modeling a class of phenomena characterized by impulses or spikes. Due to the nonexistence of an analytical expression for the alpha-stable probability density function, this family is usually expressed by means of its characteristic function, which is given by  jcxja ½1isignðxÞb tanðpa=2Þþilx /ðxÞ¼ ejcxj½1þisignðxÞð2=pÞb logðjxjÞþilx ða 6¼ 1Þ ð2Þ e ða ¼ 1Þ It has four parameters: a (0\a  2) is the characteristic exponent which sets the level of impulsiveness, b is the skewness parameter, b 2 ½1; 1 (b ¼ 0 for symmetric distributions), c [ 0 is the dispersion, a scale parameter, and l is the location parameter. As a noise model, generally the location l and the skewness b are both set zeros. We simply denote such stable variable by Sða; cÞ. Gaussian distribution Nð0; r2 Þ is a specific family of alpha-stable distribution Sða; cÞ in that the characteristic exponent is a = 2 and r2 ¼ 2c2 . Thus, we model the noise to be Gaussian, say Sð2; cÞ, and non-Gaussian stable, say Sða; cÞ, a\2. The impulsiveness of the noise following stable distribution can be seen from both the samples drawn from the probability density function (pdf) and the pdf of stable variables. To see it, 1000 independent samples drawn from Gaussian Sð2; cÞ and 1000 ones drawn from Sð1:5; cÞ with c ¼ 1 are shown in Fig. 4a, b, respectively. It indicates the non-Gaussian stable variable is much more impulsive than Gaussian when a ¼ 1:5. We computationally estimate from

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3 Extended Kalman Filter for Truck Fuzzy Control Since the state of the truck is sensed by GPS which introduces sensing and environmental noises into the truck control (non-linear) system, we develop an extended Kalman filter for the estimate of the state of the truck. 3.1 Extended Kalman Filter For a system with the state and observation equation  xk ¼ fðxk1 ; uk Þ þ wk ð3Þ zk ¼ hðxk Þ þ vk where wk and vk are Gaussian noises with covariance matrices being Qk and Rk respectively, the extended Kalman filter consists of two steps: prediction and updating. In the prediction step, the filter produces estimates of the current state variables along with their uncertainties. Once the outcome of the next measurement is observed, these estimates are updated using a weighted average, with more weight being given to estimates with more certainty. The two steps are below:

Predict Predicted state estimate Predicted covariance estimate

x^kjk1 ¼ fð^ xk1jk1 ; uk Þ Pkjk1 ¼

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(4.2)

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y~k ¼ zk  hð^ xkjk1 Þ

(4.3)

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Sk ¼ Hk Pkjk1 HTk þ Rk

(4.4)

Near-optimal Kalman gain

Kk ¼ Pkjk1 HTk S1 k

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x^kjk ¼ x^kjk1 þ Kk y~k

(4.6)

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Pkjk ¼ ðI  Kk Hk ÞPkjk1

(4.7)

where the state transition and observation matrices are following Jacobians:

J. Zhang et al.: Backing Up a Truck on Gaussian and Non-Gaussian Impulsive Noise…

Fig. 4 Impulsiveness of stable noise a 1000 samples drawn from the Gaussian Sð2; 1Þ, b 1000 samples drawn from the stable Sð1:5; 1Þ, and c the pdf of stable variables Sða; 1Þ for a ¼ 0:5; 0:75; 1; 1:25; 1:5

 of  Fk1 ¼  ox x^k1jk1 ;uk  oh Hk1 ¼  ox x^kjk1

ð4:8Þ ð4:9Þ

3.2 Truck System Equations In the truck control system, the state equation is given in Eq. (1), or in vector form of xk ¼ fðxk1 ; hk Þ

ð5Þ

where the state vector is xk ¼ ½xk ; yk ; uk T , fðxk1 ; hk Þ ¼ ½f1 ðxk1 ; hk Þ; f2 ðxk1 ; hk Þ; f3 ðxk1 ; hk ÞT , which

and in

f1 ðxk1 ; hk Þ ¼ xk1 þ r cosðuk1 þ hk Þ

ð6:1Þ

f2 ðxk1 ; hk Þ ¼ xk1 þ r sinðuk1 þ hk Þ

ð6:2Þ

f3 ðxk1 ; hk Þ ¼ uk1 þ hk

ð6:3Þ

and the steering control hk is given by fuzzy controller: hk ¼ gðxk1 ; uk1 Þ:

ð7Þ

Thus, we have Qk ¼ 0. Assuming that all the states are measurable and can be sensed (determined) by GPS introducing noises, we have zk ¼ xk þvk ;

ð8Þ

indicating that Hk ¼ I leads to no necessity for computing the partial derivative in Eq. (4.9). An acceptable assumption is that the GPS noise on the x-coordinate and y-coordinate and orientation of a truck are independent and stationary. Thus, we have Rk being a fixed diagonal matrix with positive diagonal elements relating to noise levels of the three measured states. 3.3 Derivative of the Filter To apply extended Kalman filter to the truck control system, we need to compute the state transition Jacobi matrix

 of  Fk1 ¼ ox x^

k1jk1 ;uk

, the matrix computed on the operating

point of x^k1=k1 ; uk , where in this study, we simply denote them by x^k1 ; hk . From Eqs. (6.1), (6.2), and (6.3), we have 8 of1 og > > ¼ 1  r sinðuk1 þ hk Þ > > > ox ox k1 > < ofk1 1 ¼0 oyk1 > >   > > of1 og > > ¼ r sinðu þ h Þ 1 þ : k k1 ouk1 ouk1 8 of2 og > > ¼ r cosðuk1 þ hk Þ > > > ox ox k1 k1 > < of 2 ¼1 oyk1 >   > > > of2 = og > > ¼ r cosðuk1 þ hk Þ 1 þ : ouk1 ouk1 8 of3 og > > ¼ > > > oxk1 ox > < ofk1 3 ¼0 > oyk1 > > > of3 og > > ¼1þ : ouk1 ouk1 To represent the above equations with a matrix form, we notice the property that an elementary row/column transform of a matrix is equivalent to the matrix left/right multiplied by an identity matrix performed a same transform. The obtained Jacobi matrix is then 2

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6 7 6 Fk1 ¼ 4 0 1 0 5þ4 0 0 0 0 0 2 32 og=oxk1 1 0 1 6 76  4 1 0 1 54 0 1 0

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ð9Þ Now, we meet the problem of how to find the partial derivatives og=oxk1 and og=ouk1 due to the strong nonlinearity and no explicit expression of the controller

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hk ¼ gðxk1 ; uk1 Þ. To tackle the problem, the input–output function of the fuzzy controller is obtained by discretizing related range of each input into l equally spaced points, i.e., xðiÞ ¼ xmin þ ði  1Þðxmax  xmin Þ=ðl  1Þ, i ¼ 1; 2; . . .; l and uðjÞ ¼ xmin þ ðj  1Þðumax  umin Þ=ðl  1Þ, j ¼ 1; 2; . . .; l, and scanning all the respective x and u to get the output h of the function with the fuzzy rules of the controller such that the partial derivative of the function is computationally available. The larger l provides the more precise input–output function of the controller, and thus the more preciseness of its partial derivatives. The obtained input–output function of the controller is shown in Fig. 5. In our experiment, we set l = 5001. To estimate the derivative of the control function at the operating point x^k1 ; hk , for the estimate x^k1 , we simply find the closest point of x, denoted as xclose , and its left and right adjacent points among fxðiÞ; i ¼ 1; 2; . . .; lg, respectively, denoted as xlarge and xsmall . Similarly, for the esti^k1 , we find uclose , ularge , and usmall , respectively. mate u Then, the partial derivative is og=oxk1 jx^k1 ;hk ¼

hðxlarge ; uclose Þ  hðxsmall ; uclose Þ ðxmax  xmin Þ=ðl  1Þ

ð10:1Þ

og=ouk1 jx^k1 ;hk ¼

hðxclose ; ularge Þ  hðxclose ; usmall Þ ðumax  umin Þ=ðl  1Þ

ð10:2Þ

Substituting the above equations into Eq. (9), Fk1 jx^k1 ;hk then can be estimated.

4 Simulations and Results In experiment, the loading zone is the area in a 100 by 100 m2. The truck is initially arbitrarily located at any position of the square with an arbitrary angle, i.e., x0 2 ½0; 100, y0 2 ½0; 100, and u0 2 ½90; 270. The unit for x and y is

Truck-to-dock error ¼ jxend  xf j

ð11:2Þ

In fact, angle variation represents the seriousness of the swing of the truck, hence symbolizes the safety of the truck. Since the initial state is ðx0 ; y0 ; u0 Þ ¼ ð10; 0; 90Þ and the destination is ðxf ; yf ; uf Þ ¼ ð50; 100; 90Þ, the truck is generally backward first in about horizontal and then turns to about vertical direction, finally reaching the location of about ðxend ; 100; 90Þ. Thus, we focus on only horizontal error jxend  xf j in measuring the truck-to-dock error. We did a lot of experiments for various initial state of the truck for conclusions. For the limitation of the space, we only demonstrate our results for the case that initial state of the truck is ðx0 ; y0 ; u0 Þ ¼ ð10; 0; 90Þ. 4.1 No Noise Situation In the case of no any noise on sensing the state of the truck, the trajectory of the truck controlled by the fuzzy controller is shown in Fig. 6. Seen from the figure is a very smooth trajectory of the truck from the initial location to the loading dock, indicating that the fuzzy rules in the controller is excellent in controlling the truck.

Input-output function of the fuzzy controller

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meter, and for u is degree. The truck is supposed to be 6 m long and 2 m wide. The truck under the control steps backward r ¼ 2 meters in each stage under the control of the fuzzy controller. Noise model for sensing the state of the truck by GPS is supposed to be respective Gaussian and (impulsive) nonGaussian stable variable for study. And two situations are in study: Initial state of the truck is precisely known, and is not precisely known. Comparisons are on the performance of the control system with no GPS sensing noise (no noise), with GPS sensing noise (noise), and with extended Kalman filter (Kalman). Two aspects reflect the control performance of the truck: angle variation over stages, and truck-to-dock error, which are respectively defined below: 1X Angle variation ¼ jukþ1  uk j ð11:1Þ N k

4.2 Impact of Noise When Initial State of the Truck is Precisely Known

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Error covariance matrix Pk measures preciseness of the state of the truck at stage k. The error covariance matrix P0 is a 3 9 3 zero matrix to indicate that the initial state of the truck is precisely known. We did a lot of experiments on truck control with Gaussian and stable GPS sensing noise, respectively. Figure 7a–c shows the trajectory of the truck when there is no sensing noise, when the sensing noise is

J. Zhang et al.: Backing Up a Truck on Gaussian and Non-Gaussian Impulsive Noise…

Fig. 6 Trajectory of the truck controlled by the fuzzy controller when the state of the truck can be sensed precisely without any sensing noise

Gaussian Nð0; 4Þ, when it is stable Sð1:01; 10Þ, and when the noise is filtered with extended Kalman filter, respectively. Though the controller is satisfactory in the situation of no sensing noise, which can be seen from the trajectory given in Fig. 6, based on the satisfactory rules in the controller however, control performance of the fuzzy controller degrades with respect to noise, which can be clearly seen from the swing of the trajectory of the truck. The magnitude of the swing is such large that it even challenges the safety of the truck in such not so large noise situation of Nð0; 4Þ, while the truck with the sensing noise of Sð1:01; 10Þ is imporssible to be on road.

However, when the Kalman filter is utilized, it is found from Fig. 7c that the trajectory nearly recovers its original one: It is nearly the same as that of the truck without any noise. This indicates that Kalman filter is very effective in filtering noises for the control of the truck when the initial state of the truck is precisely known. What surprises us is that the trajectory of the truck when Kalman filter is utilized is exactly the same, not influenced at all by GPS noise, no matter the noise is Gaussian or stable, and no matter what parameter of Gaussian and stable is. Even when the variance of Gaussian or the dispersion of the stable is greatly large, or even when the characteristic exponent is very small indicating that the noise is highly impulsive, the trajectory of the truck is exactly the same and especially satisfactory. Inspecting the Kalman filter, since Q is a zero matrix, from Eqs. (4.2) and (4.5), we get P ¼ 0, and hence K ¼ 0. Thus, we have x^kjk ¼ x^kjk1 þ Kk y~k ¼ fð^ xk1jk1 ; uk Þ. This means that when initial state of the truck is precisely known, theoretically the trajectory of the truck is totally determined by the initial x^0 , since in truck control system, uk is a function of x^k1jk1 (i.e., hk is the function of ðxk1 ; uk1 Þ). Our experiments match the theoretical analysis perfectly. 4.3 Impact of Noise When Initial State of the Truck is Roughly Known In this experiment, P0 is set to be a 3 9 3 diagonal nonzero matrix. When the sensing noise is Gaussian, we set the diagonal elements of P0 being 2 m/2 degrees, which is the same as the width of the truck, reflecting the situation of the high quality of current GPS; when the sensing noise is stable, we set them being the square of dispersion of the stable no matter what the characteristic exponent a of the stable is. Two examples of controlling results when noise is Gaussian and stable, respectively, is shown in Fig. 8. It

Fig. 7 Trajectory of the truck controlled by the fuzzy controller when the state of the truck is sensed with the sensing noise of a Gaussian Nð0; 4Þ, b non-Gaussian stable Sð1:01; 10Þ, and c the noise is filtered by Kalman filter

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indicates that noises really have a serious impact on the trajectory of the truck. We did a lot of experiments on Gaussian noise situation with standard deviation r from 0.1 to 5, on stable noise situation with fixed characteristic exponent a ¼ 1:5 and various dispersion c from 0.1 to 5, and on stable noise situation with fixed dispersion c ¼ 2 and various characteristic exponent a from 1.02 to 2, where we refer to the r, c, and a as noise level, respectively, where the larger r, c and the smaller a is, the higher the noise level is. Each experiment is performed 30 times, and we measure average and standard deviation of their corresponding control performance: angle variation, and truck-to-dock error, as well as trajectory length of the truck defined by ffi X qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Trajectory length ¼ ðxkþ1  xk Þ2 þ ðykþ1  yk Þ2 : k

ð12Þ The result is shown in Fig. 9. For easy comparison, we made vertical axis to be the same in each row panel of the figure. Compared with the trajectory of the truck with sensing noise but no filter, what can be seen from Fig. 9 are smoother trajectory, smaller angle variation, and smaller truck-to-dock error when the extended Kalman filter is applied. The trajectory length of the truck without filter is generally a little bit shorter in average than that with Kalman filter (seen from the upper row panel of the figure), while its angle variation is much larger in average than that with Kalman (seen from the middle row panel of the figure), reaching to the average truck-to-dock error much larger than that with Kalman filter. This indicates that the Kalman filter functions in smoothing the trajectory with smaller angle variation of the truck toward the loading dock with smaller truck-to-dock error. Performance deterioration with respect to noise level can be seen from all the subfigures in the figure, including

the mean performance and the variance of the performance. For a same specific noise level, the deterioration without filter is more serious than that with the extended Kalman filter. This indicates that the extended Kalman filter really improves the control performance of the truck. Impulsiveness of sensing noise really has impact on the control performance of the truck. This can be seen by comparing the left and middle column panel of Fig. 9, where the left and middle column panel is the result of sensing noise with a ¼ 2 and a ¼ 1:5, respectively. It is seen that the smaller a or equally the more impulsiveness of the noise deteriorates control performance more seriously. This can also be seen from the right panel: The smaller a is, the larger angle variation and higher truckto-dock error are, no matter if Kalman filter is utilized. However, what can also be seen from the figure is that Kalman filter alleviates such deterioration and improves control performance well. On the other hand, with the increase of noise level, the variance of both angle variation and truck-to-dock error becomes serious, while such serious variation is alleviated greatly by Kalman filter. The improvement in average performance by Kalman filter is about irrelevant to noise level, especially for high noise level situation, which can be seen from Fig. 10, which provides the ratio of average performance without filter to that with Kalman filter. In the figure, the ratio on angle variation/truck-to-dock error is the ratio of the mean angle variation/truck-to-dock error without a filter to that with Kalman filter. The ratio value less than one indicates real improvement by Kalman filter, and the smaller the value is, the more improvement is gained by Kalman filter. Figure 10 indicates that in average, Kalman filter improves control performance well, compared with no filter case. An interesting thing is that even in the worst case of impulsiveness in our experiment, i.e., a = 1.02 (and

Fig. 8 Trajectory of the truck controlled by fuzzy controller, when a the sensing noise is Gaussian Nð0; 4Þ, and b is filtered with the extended Kalman filter; and when c the sensing noise is stable Sð1:5; 4Þ, and d is filtered with the extended Kalman filter

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J. Zhang et al.: Backing Up a Truck on Gaussian and Non-Gaussian Impulsive Noise… Stable noise situation (α =1.5)

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3 2 1 0

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(a)

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5

(b)

(c)

Fig. 9 Comparison on impact of noise type and noise level on control performance of the truck, where the GPS noise is a Gaussian, b stable with a¼ 1:5 and c ¼ 0:1  5, and c stable with c ¼ 2 and a ¼ 1:02  2

Gaussian noise situation

Gaussian noise situation

0.6 0.4 0.2

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Fig. 10 Ratio of control performance in noise case to that with Kalman filter showing the improvement in performance by the extended Kalman filter. The upper panel is the ratio on angle variation, and the lower panel is the ratio on truck-to-dock error

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c ¼ 2 m), which corresponds to extremely serious impulsiveness of sensing noise, both the angle variation and the truck-to-dock error in the extended Kalman filter situation are about the same level as that when the impulsiveness of the noise a ¼ 1:5 is not so serious (and the dispersion is about only c ¼ 3 m), and about the same level as that when the noise is Gaussian with the standard deviation of even r ¼ 10 m. This indicates that the extended Kalman filter, though suboptimal in adapting Gaussian noise, is also effective in adapting non-Gaussian impulsive noise. Unlike its linear counterpart, the extended Kalman filter in general is not an optimal estimator. Why the extended Kalman filter in truck control system works so well in average? The reason is believed to be the well-designed fuzzy control rules which is strongly nonlinear and the well-estimated partial derivative (or the linearization of the control system). In this study, though the controller is fuzzy which is strongly nonlinear, and so is the truck model given in Eq. (1), the input–output function of the controller is well estimated by screening all available inputs (i.e., discretization of the input range is subtle), which makes the computation of the derivative be effective for a good estimate of the derivative of the function. On the other hand, as seen from the input–output function curve of the controller (Fig. 5), the function appears to be multiple flat steps in input regions with sharp jumps from region to region. In each flat region, the derivative is almost independent to operating point, making our technique be some precise in derivative estimation, while in the jump area, the function is seriously dependent on operating point, which makes the technique only a rough estimate of the derivative. Fortunately, the truck control system works mostly in the flat region, the reason why the truck is with a good control performance. In fact, the precision of the estimated state is given by the updated covariance of the estimate. To understand the precision of the estimated state of the truck equipped with Kalman filter with respect to control function gð; Þ, we investigate error covariance, which is given in Eqs. (4.4), (4.5), and (4.7). Noticing Hk ¼ I, we have h i Pk=k ¼ I  Pk=k1 ðPk=k1 þ Rk Þ1 Pk=k1 ð13:1Þ By noting that h i I  Pk=k1 ðPk=k1 þ Rk Þ1 ðPk=k1 þ Rk Þ ¼ Pk=k1 þ Rk  Pk=k1 ¼ Rk

ð13:2Þ

and 1 1 1 P1 k=k1 ðPk=k1 þ Rk ÞRk ¼ Rk þ Pk=k1

123

ð13:3Þ

we have h i Pk=k ¼ I  Pk=k1 ðPk=k1 þ Rk Þ1 Pk=k1 ¼ Rk ðPk=k1 þ Rk Þ1 Pk=k1 1 ¼ ðR1 k þ Pk=k1 Þ

ð13:4Þ

1

Substituting Eq. (4.2) into the above equation, we obtain h i1 1 T Pk=k ¼ R1 ð13:5Þ k þ ðFk1 Pk1=k1 Fk1 Þ where Fk1 is the derivative of the input–output function of the controller. Therefore, the larger derivative of the control function gð; Þ at an operating point xk1 results in the larger updated (a posteriori) estimate covariance, indicating that the estimate is more roughly estimated by the Kalman filter. Seen from Fig. 5, the truck input–output function of the fuzzy controller is largely flat in most regions of the loading zone. This might be the reason why the Kalman filter works well in most cases. One thing must be noticed. Not only average but also standard deviation can tell the performance of an approach. Inspecting Fig. 9, standard deviation of the performance for impulsive noise (a\2) is much larger than that for Gaussian noise even though Kalman filter is utilized. This is because the extended Kalman filter is originally developed to deal with only Gaussian noise, without ability of robustness to impulsive noise theoretically. Thus, to further improve the robustness of the system, the non-Gaussian version of the extended Kalman filter to adapt to more general non-Gaussian impulsive noises should be theoretically studied and designed.

5 Conclusions Backing up a truck with fuzzy control is especially successful for no sensing noise situation, and for sensing noise situation with the initial state of the truck precisely known. However, when the state of the truck is sensed by GPS, which introduces noise into the control system, and when the initial state of the truck is only roughly known, which are real situations, the control performance of the truck deteriorates greatly with respect to noise levels no matter Gaussian or non-Gaussian impulsive noise is introduced by the GPS. To deal with the noise, we propose an extended Kalman filter to filter the noise for an estimate of the state of the truck, and perform fuzzy control based on the estimated state. Impact of noise levels for Gaussian noise and non-Gaussian impulsive noise on control performance of the truck is studied. The main contributions of the paper are twofolds: (1) we propose a screening approach such that the extended

J. Zhang et al.: Backing Up a Truck on Gaussian and Non-Gaussian Impulsive Noise…

Kalman filter is technically and computationally available for the backing-up control system of the truck; (2) we study the impact of GPS non-Gaussian noise and the effect of the extended Kalman filter on control performance of a backup truck control system. The first contribution results in a computation tool which leads to the second contribution available. By simulating the truck system with and without noise, the noise being Gaussian and non-Gaussian impulsive, and the system with and without the extended Kalman filter, we found that the performance of the system with the extended Kalman filter is much better than that without the filter no matter the noise is Gaussian or non-Gaussian impulsive. Though extended Kalman filter is suboptimal to Gaussian noise, it adapts to impulsive noise well and is not seriously sensitive to noise level in average sense, the great power of the extended Kalman filter in dealing with non-Gaussian impulsive noises for the truck control, while large deviation from the average performance makes an urgent call for the non-Gaussian version of the extended Kalman filter to adapt to more general non-Gaussian impulsive noise situation for further improving both average and deviation performance of the system. The former is for fuzzy rules, and the latter is for characteristic of noise such that a filter which adapts to the characteristic rather than the extended Kalman filter which adapts to only Gaussian noise can be designed and utilized. For improving the control performance of a fuzzy control system with noise, in our opinion, two research directions are the main focus in fuzzy control system, one is the learning of fuzzy rule and membership function, and another is learning the characteristic of noise and designing the adapted version of the extended Kalman filter to tackle the noise. In our opinion, membership function and fuzzy rule are connected rather than disconnected. Their combination (together with fuzzification and defuzzification operation) determines the performance of the control system without noise. However, the main study stream in the field is learning one with another fixed. A typical example of learning fuzzy rules is a multi-granularity approach [26], while that of learning membership function is a genetic approach [27], with the former for regression and the latter for mining fuzzy association rules from low quality data. Up to now, no study is given to fuzzy controller design by simultaneously learning fuzzy rule and membership function for backing-up control of a truck. Solving it might be an iterative learning: learning fuzzy rule to its convergence while fixing membership function, and learning membership function to its convergence while fixing the fuzzy rule learned, till when both are convergent. This is the first paper which studies the impact of noise and the effect of the extended Kalman filter on control

performance of a fuzzy control system with sensing noise. How to make a control system to adapt to noise is an important issue. Without a good uncerstanding of the characteristic of noise in the system, one can not get a good control performance of the system. The result seen from this study is the loss of control performance (especially the large deviation from the average performance) because the extended Kalman filter adopted in the study is the (sub)optimal for only Gaussian noises. This encourages the study of a fuzzy control learning system, which can learn the characteristic of the noise. As for the backing-up control system of a truck, it is anticipated that during several times of backing up of the truck, the original extended Kalman filter can evolve to adapt to the complicated real situation of the GPS noises. Acknowledgements This work was supported by the Natural Science Foundation of China under Grants 61571341 and 11401357, the National Ministry of Education Fund Projects of China (No. 20130203110017), and the Fundamental Research Funds of China for the Central Universities (Nos JBZ170301 and 20101164977).

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Junying Zhang received Ph.D. degree in signal and information processing from Xi’dian University, Xi’an, China, in 1998. She was a Visiting Scholar with the Department of Electrical Engineering and Computer Science with the Catholic University of America, Washington, D.C., USA, from 2001 to 2002, and she was a Visiting Professor with the Department of Electrical Engineering and Computer Science with Virginia Polytechnic Institute and State University, Blacksburg, VA, USA, in 2007. She is currently a Professor with the School of Computer Science and Technology, Xi’dian University, People’s Republic of China. Her current research interests include intelligent information processing and characteristics of non-Gaussian impulsive noises, including machine learning and its application to cancer related bioinformatics, cause learning, and pattern discovery. Yuting Zhang received her bachelor degree in electronic information engineering from Xijing University, Xi’an, People’s Republic of China, in 2014. She is currently a graduate student in the university. Her current research interest is intelligent information processing. Cong Xu received her M.S. degree in circuit and system from Xidian University, Xi’an, People’s Republic of China, in 2013. She is currently a teacher with the School of Information Engineering, Xijing University, People’s Republic of China. Her research interests are intelligent information processing and machine learning.