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Sliding-Mode Control Design for Nonlinear Systems Using Probability Density Function Shaping Yu Liu, Hong Wang, Senior Member, IEEE, and Chaohuan Hou, Fellow, IEEE

Abstract— In this paper, we propose a sliding-mode-based stochastic distribution control algorithm for nonlinear systems, where the sliding-mode controller is designed to stabilize the stochastic system and stochastic distribution control tries to shape the sliding surface as close as possible to the desired probability density function. Kullback–Leibler divergence is introduced to the stochastic distribution control, and the parameter of the stochastic distribution controller is updated at each sample interval rather than using a batch mode. It is shown that the estimated weight vector will converge to its ideal value and the system will be asymptotically stable under the rank-condition, which is much weaker than the persistent excitation condition. The effectiveness of the proposed algorithm is illustrated by simulation. Index Terms— Kullback–Leibler divergence, probability density function, sliding-mode control, stochastic distribution control.

I. I NTRODUCTION

O

VER the past decades, there have been increasing demands for controller designs for stochastic systems. This is largely due to the fact that most practical systems are subjected to either random inputs or noise [1]. When dealing with these stochastic systems, one of the important practical issues in the controller design is the minimization of the randomness in the closed-loop system. Traditional stochastic control was focused on the output mean and variance under the assumption that the system variables are of Gaussian types. Linear quadratic Gaussian (LQG) [2] control is one of the most fundamental optimal control methods in stochastic control theory, where linear systems and additive white Gaussian noise are assumed. The LQG problem can be solved by using the separation principle. Recent works include predictive stochastic control [3], adaptive nonlinear stochastic control [4] and robust fuzzy control for uncertain Markovian stochastic systems [5]. However, most of these approaches are unsuitable for nonlinear systems with non-Gaussian noise. Originated by Wang [6], stochastic distribution control focuses on controlling Manuscript received September 26, 2012; revised February 18, 2013 and May 3, 2013; accepted July 20, 2013. Date of publication August 15, 2013; date of current version January 10, 2014. This work was supported by the Natural Science Foundation of China under Grant 61290323, Grant 61333007, Grant 61134006, and the Director Foundation of IACAS under Grant Y154221511. Y. Liu and C. Hou are with the Institute of Acoustics, Chinese Academy of Science, Beijing 100190, China (e-mail: [email protected]; [email protected]). H. Wang is with the Control Systems Centre, School of Electrical and Electronic Engineering, The University of Manchester, Manchester M60 1QD, U.K. (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TNNLS.2013.2275531

the shape of the output probability density function (pdf) for non-Gaussian and dynamic stochastic systems. The objective of stochastic distribution control is to find a control input that makes the shape of the measured output pdf follow a given pdf. In stochastic distribution control, there is no assumption on the Gaussian noise and the stochastic system can be subjected to arbitrary random input [7]. There are different methods to implement stochastic distribution control. Wang and Yue [8], [9] presented a pdf tracking control strategy for stochastic parameter systems based on a B-spline model for the output pdf. The tracking problem of pdf was transferred to the tracking of given weights which corresponded to the given pdf by using the B-spline approximation. In recent years, the minimum error entropy (MEE) criterion was developed to deal with non-Gaussian and nonlinear systems. The MEE algorithm measures the average uncertainty contained in a given pdf, and uses stochastic information gradient (SIG) method [10]–[13] to achieve the optimum. On the other hand, sliding-mode control (SMC) is considered as an important method in modern control theory, which can be used to deal with control systems subjected to uncertainties and disturbances [14]–[16]. It is well known for its insensitivity to parameter variations and matched uncertainties once the system reaches the sliding surface [17]. In spite of its robustness and accuracy, the key problem of SMC is its high (theoretically infinite) frequency control switching which leads to the so-called chattering effect [16]. Indeed, how the chattering effect can be reduced remains a hotspot of recent research on SMC. Saturation and sigmoid functions are used as “filters” of a discontinuous signal in order to obtain a continuous control. However such methods can also reduce the control accuracy [14]. The implementation of radial basis function (RBF) networks is another choice to reduce the chattering effect in the sliding mode, where the sliding gain is represented by an RBF network and adjusted adaptively using the least mean-squared (LMS) criterion [18]. Levant [19], [20] introduced a homogeneity-based higher order SMC theory that could produce continuous control signal while retaining the accuracy of a normal SMC. To the best of our knowledge, there are few sliding-mode controllers that consider shaping the pdf of their tracking error. In this paper, we introduce a control method to achieve highprecision tracking performance for nonlinear systems with uncertainties and disturbances. The main purpose is to present a hybrid control scheme that incorporates SMC with pdf shaping. The SMC part deals with the nonrepeatable uncertainties and unmodeled dynamics, while stochastic control part, which

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LIU et al.: SMC DESIGN FOR NONLINEAR SYSTEMS USING PDF SHAPING

pd

DKL(ps,pd)

PDF part

Neural network

noise updf

xr +

e -

C

S

Sliding Mode Controller

usmc + u +

Plant P

x

SMC part

333

relationship between the Renyi’s entropy [26] and information divergence. The Parzen window method [27], which is one of the widely used kernel estimator, is a nonparametric method to estimate the pdf of a random process. Indeed, the Parzen window method is just a biased estimator of pdf based on the tradeoff between low bias and low variance. An appropriate window size is of paramount importance to fulfill the need of the estimation. When we utilize the Parzen window method to estimate the pdf of f (x) from the samples, we have N 1  K σ (x − x i ) fˆ(x) = N i=1

Fig. 1.

Schematic of SMC-based PDF control.

is also called pdf control, shapes the pdf of the tracking error to the desired function. Considering the assumption that we have little a priori knowledge about the noise and the fact that RBF networks have the ability to approximate any continuous function with arbitrary accuracy [21]–[24], we introduce an RBF network to represent the pdf shaping control to compensate for the loss caused by all the uncertainties, nonlinearities, and non-Gaussian inputs. As is shown in Fig. 1, the control input to plant P is divided into two parts: u smc comes from a normal discrete sliding-mode controller and u pdf is the output of an RBF network whose weights are tuned by the Kullback–Leibler (K–L) divergence between the pdf of the sliding surface ps and a desired pdf pd . The rest of this paper is organized as follows. Section II gives some backgrounds and notations about pdf estimation and information divergence measure. Section III presents the details of SMC-based PDF shaping control design using K–L divergence. The convergence analysis of the proposed controller is given in Section IV. Some simulations are provided in Section V to show the advantages of the proposed method. Finally, concluding remarks are given in Section V. II. P RELIMINARIES In this section, some basic terminologies and notations of pdf estimation are described. Information divergence measure and SMC will be used for this paper. In the following, matrices are assumed to have appropriate dimensions except when specially pointed out. For two vectors v 1 and v 2 , the notation v 1 ≥ v 2 denotes that every element of v 1 is no less than the corresponding one in v 2 , while max(v 1 ) denotes a map Rn → Rn with the function max(.) operating on every element of the vector. In represents the identity matrix with n × n dimensions, N depicts the set of positive integers, and P T is the transpose of the matrix P.

where x 1 , . . . , x N are samples and K σ is the kernel function with bandwidth σ . A kernel is a nonnegative real-valued integrable function satisfying the following requirements:  +∞ 1) −∞ K (x)d x = 1; 2) K (−x) = K (x), ∀x. The first requirement ensures that the Parzen window method results in a pdf, while the second one guarantees that the expectation of the process is kept unchanged. A commonly used kernel is Gaussian kernel, which has the following form:   1 x2 exp − 2 . K σ (x) = √ 2σ 2πσ Remark 1: The bandwidth of the kernel σ is a free parameter which exhibits a strong influence on the resulting estimate. Intuitively, one wants to choose σ as small as the data allows. However, there is always a tradeoff between the bias of the estimator and its variance. The most common optimality criterion used to select this parameter is the mean integrated squared error  MISE(σ ) = E ( fˆ(x) − f (x))2 d x. On the other hand, one can just simply choose σ = 0.2σˆ , where σˆ is the standard deviation of the underestimated random process [25]. A better way of selecting this parameter is to implement an adaptive kernel size method [10] where the bandwidth is optimized with the minimum entropy criterion. B. Kullback–Leibler Divergence Information divergence is a kind of measure of the distance between two distributions. Considering two pdfs p(x) and q(x), the K–L divergence [10] or relative entropy is defined as    p(x) p(x) d x = E P ln (1) p(x) ln DKL ( p||q) = q(x) q(x)

A. Probability Density Estimation

or, in the discrete form    p(x) p(x) DKL ( p||q) = = E P ln . p(x) ln q(x) q(x)

There are numerous methods to estimate the probability density, such as histograms, kernel estimator, the nearest neighbor method, orthogonal series estimator, and maximum penalized likelihood estimator [25]. In this paper, we focus on the kernel estimator because of its wide applicability and its

Although it is an effective dissimilarity measure, K–L divergence is not actually a true metric or distance. First of all, it is asymmetric, i.e., the K–L divergence from p(x) to q(x) is not equal to that from q(x) to p(x), which means DKL (q|| p) = DKL ( p|| p). Moreover, it does not satisfy the

(2)

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IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS, VOL. 25, NO. 2, FEBRUARY 2014

triangle inequality. On the other hand, the K–L divergence is always nonnegative and equal to zero if and only if p(x) = q(x) almost everywhere, which makes it a useful tool to measure the “distance” between distributions. In fact, Kullback and Leibler themselves introduced a symmetric divergence as D( p||q) = DKL ( p||q) + DKL (q|| p). There are numerous symmetrized definitions of divergence, such as Jensen–Shannon divergence [28], Jeffreys divergence [29], and the λ divergence [30]. Since we are considering an optimization problem of pdf, the exact quantity of divergence is less important. Therefore, we use the K–L divergence for the sake of simplicity. For this purpose, two of the most important properties of the K–L divergence that will be used in this paper are described as follows. Lemma 1: DKL ( p||q) ≥ 0, with equality if and only p(x) = q(x), ∀x. Lemma 2: DKL ( p||q) is a convex function. Remark 2: Generally speaking, K–L divergence is convex and has a minimum if and only if p(x) = q(x). We can design a pdf shaping scheme to make the error pdf as close as possible to a desired one. This is the main idea of the proposed method.

Until recently, most researches on SMC were based on continuous systems. Perhaps the most interesting thing about SMC is, even though the actual control u is discontinuous, the switching across the sliding-mode forces the system to act as if it were driven by a continuous signal. However, in practice most control systems are in fact computer controlled. Because of the finite sampling frequency, the actual closed-loop system cannot be driven into a true sliding mode, but into a quasisiding mode which has a limited switching frequency along the sliding manifold [14], [15], [31], [32]. Therefore, a perfect sliding mode is possible only in the case of perfect model knowledge and in the absence of disturbances. Otherwise, discrete sliding-mode control (DSMC) must be considered. According to Gao [32], the closed-loop DSMC system should possess the following properties.

S2

S3

Wherever the initial state is, the trajectory will move monotonically toward the sliding manifold and cross it within finite time. Once the trajectory crosses the sliding manifold, it will cross the manifold again and again in every successive sampling interval, resulting in a zigzag motion. The size of the zigzagging step is nonincreasing and within a specified band.

In this case, the discrete-time controllable system is expressed as x(k + 1) = Ax(k) + Bu(k) + f (x, k)

s(k) = C x(k).

(4)

Assuming that B is a full column matrix, there must exist a matrix T1 ∈ Rm×n such that T1 B = Im . Then, defining the transformation     T2 x¯1 (k) = x(k) = T x(k) (5) x¯2 (k) T1 where the rows of T2 span the nullspace of T1 , the following system representation can be obtained: (6) x¯1 (k + 1) = A¯ 11 x¯1 (k) + A¯ 12 x¯2 (k) + f u (x, k) x¯2 (k + 1) = A¯ 21 x¯1 (k) + A¯ 22 x¯2 (k) + u(k) + f m (x, k) (7) (8) s(k) = C1 x¯1 (k) + C2 x¯2 (k) where T AT −1 =



A¯ 11 A¯ 21

   A¯ 12 0 −1 = [C , C ], TB = , CT . 1 2 Im A¯ 22

f u (x, k) and f m (x, k) are unmatched and matched disturbances, respectively, and they are both bounded because f (x, k) is bounded. Assuming that C is selected such that the matrix product CB is nonsingular, then the matrix C2 is also nonsingular because |C2 Im | = |CT−1 TB| = |CB| = 0.

C. Sliding-Mode Contol

S1

are parameter matrices of the appropriate size. The discrete switching function is defined by

(3)

where x ∈ Rn and u ∈ R are the state and input, respectively. f ∈ Rn is uniformly bounded, and matrices A and B

On the sliding mode s(k) = 0, (8) yields x¯2 (k) = F x¯1 (k) −C2−1 C1 .

where F = Thus x¯2 (k) is seen to be linearly related to x¯1 (k) on the sliding manifold. Then x¯1 (k) can be described as x¯1 (k + 1) = ( A¯ 11 + A¯ 12 F)x¯1 (k) + f u (x, k).

(9)

Under the assumption that (A, B) is controllable, controllability of ( A¯ 11 , A¯ 12 ) is guaranteed [16] and appropriate F, C1 , and C2 can be selected to stabilize (9). On the other hand, the discrete sliding-mode reaching law [32] can be described as ⎤ ⎡ α1 sign(s1 (k)) ⎢ α2 sign(s2 (k)) ⎥ ⎥ ⎢ (10) s(k + 1) = s(k) − ⎢ ⎥ .. ⎦ ⎣ . αm sign(sm (k)) for some diagonal matrix  ∈ Rm×m with its elements satisfying 0 ≤ i,i < 1 ∀i = 1, . . . , m, and αi > 0 ∀i = 1, . . . , m. The following theorem can be stated [31]. Theorem 1: The closed-loop system (6)–(8) satisfies properties S1, S2, and S3 if the controller is chosen as follows: ⎤ ⎡ α1 sign(s1 (k)) ⎢ α2 sign(s2 (k)) ⎥ ⎥ ⎢ u(k) = ( − A¯ 22 )s(k) − A¯ 21 x¯1 (k) − ⎢ ⎥ (11) .. ⎦ ⎣ . αm sign(sm (k))

1 + i,i ˆ f m,i αi > 1 − i,i where fˆm = max(| f m (x, k)|).

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LIU et al.: SMC DESIGN FOR NONLINEAR SYSTEMS USING PDF SHAPING

Remark 3: There are two concepts that may cause confusion here: the stable dynamic system and the stationary stochastic process. “Stable dynamic system” is a concept that is usually used in the control field to describe the convergent property of the closed-loop system, whereas “stationary stochastic process” is often used in signal processing areas to mean that the pdf of a signal (or a variable) does not change with time. It is worth noticing that, even for a nonstationary stochastic process, it has its pdf at each sample time instant, except that such a pdf is time-variant. As for the proposed method, which will be seen in later sections, since asymptotic stability is guaranteed, the sliding surface and the parameter estimation errors will exponentially converge to zero. During the parameter-updating phase, the parameters in the control law (i.e., the pdf controller) change with time, which makes the closed-loop system a nonstationary stochastic process from the signal processing point of view. However, after the adaptive process settles down (at least approximately settles down), the parameters of the feedback control will be kept unchanged, and the closed-loop system can be seen as a stationary stochastic process thereafter. All the above indicate that, during the convergence phase of the close-loop system, the stochastic process will gradually turn out to be stationary from the signal processing view point.

Now we will introduce a novel control scheme to achieve high-precision tracking performance for the stochastic nonlinear system with uncertainties and disturbances, which is a scheme that incorporates SMC with pdf control. It is proposed that the SMC part stabilizes the system and deals with the nonrepeatable uncertainties and unmodeled dynamics, while the pdf control part attenuates the randomness of the tracking error by shaping the pdf of the sliding surface to a narrow range. Considering the following nonlinear system: (13)

where x ∈ Rn denotes the state, u ∈ R is the input, and d(k) ∈ Rn is the unknown disturbance of the system. For simplicity, we only deal with the single input system here. A. Sliding-Mode Control Design Suppose that there exists a sliding mode s = Ce = C(x − x d ), on which the state of the system can converge to zero. By left-multiplying C on both sides of (13) we have s(k + 1) = C f (x, k) + CBu(k) + Cd(k) − C x d (k + 1). (14) When C B is invertible, one of the feasible SMC can be u smc (k) = (C B)−1 [hs(k) + C x d (k + 1) − C f (x, k)]

Substituting (15) into (14), the dynamics of the sliding mode becomes s(k + 1) = hs(k) + Cd(k). (17) Therefore, stability is ensured for this first-order system. However, this is not enough, as we are trying to track the desired trajectory with high precision. Considering the intrinsic characteristic of DSMC, the sliding mode will inevitably fall into a quasi-sliding manifold by the impact of the uncertain term Cd(k) as mentioned above. Also, the inaccurate model parameter will enlarge this quasi-sliding band. Our next step is to use a pdf control scheme to compensate this quasi-sliding manifold to obtain better tracking performance. We now add a pdf control part to the SMC part to make the whole controller as follows: u = u SMC + u pdf .

(18)

According to Fig. 1, (15), and (17), we obtain s(k + 1) = hs(k) + Cd(k) + CBupdf .

(19)

Our pdf controller design will be based on this first-order stochastic system. To make the design procedure easier to follow, we define  T (20) Z k = x 1 (k), . . . , x n (k) s(k) s(k) and use an RBF-based network as

III. SMC-BASED PDF C ONTROL A LGORITHM

x(k + 1) = f (x, k) + B(x, k)u(k) + d(k)

335

u pdf (k) = −W (k)T H (Z k )

(21)

Rm

where W (k) ∈ denotes weight vector of the RBF network and the RBFs are represented by   ||Z − ci ||2 , i = 1, . . . , m. Hi (Z ) = exp − 2 2σrbf In the following, we will focus on how the weight parameter W (k) can be tuned using pdf shaping techniques. Remark 4: The selection of the optimal matrix C is a complicated issue, which is out of the scope of this paper. Interested readers may refer to [14] and [34]–[36]. B. Pseudo-Error We will try to compensate the uncertainties by representing them as a weighted combination of nonlinear functions, and then use a weight update law to achieve the purpose. We hope that the adaptive weights can converge to their ideal values and exhibit significant improvement in the tracking performance. Rewriting the sliding-mode dynamics with g = C B,

(k) = g −1 Cd(k)

(22)

we obtain the following first-order system: s(k + 1) = hs(k) + g[(k) + u pdf (k)], k = 1, 2, . . . . (23)

(15)

where h is a constant satisfying 0 < h < 1. According to [33], the closed-loop system will be driven into a quasi-sliding band δ with 1 ˆ δ= (16) f m and fˆm = max{|Cd(k)|}. 1−h

Since (23) holds for all k ∈ N, we are to predict the sliding manifold, and the term hs(i ) + g(i ) is what we will deal with, where i ∈ [k − N +1, k] and N is the number of samples involved in the adaption. Letting h s(i ) + (i ) = W ∗T H (Z i ) ∀i ∈ [k − N + 1, k] g

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and substituting u pdf (i ) = −W T (i )H (Z i ) and W˜ (i ) = W (i )− W ∗T into (23), we have s(i + 1) = −g W˜ T (i )H (Z i ) ∀i = [k − N + 1, k].

(25)

The gradient of each J (e j ) with respect to Wk has the form

Therefore, we can define the pseudo-error ei (k) as follows: =

ei (k) = −g W˜ T (k)H (Z i ) ∗

TH

= −g[W (k) − W + W (i ) − W (i )] (Z i ) = −g[W˜ (i ) + W (k) − W (i )]TH (Z i ) = s(i + 1) − g[W (k) − W (i )]TH (Z i ) i = [k − N + 1, k].

(26)

C. PDF Controller Design For simplicity, x k , sk , e j , Wk , and Hk are used to represent x(k), s(k), e j (k), W (k), and H (k), respectively. At time interval k, the pdf of the latest N pseudo-errors ei (k), i = k − N + 1, . . . , k, can be estimated to give 1 N

k 

K σ (y − ei )

(28)

i=k−N+1

where K σ is the kernel function with the parameter σ . √We choose Gaussian kernel here, i.e., K σ (y) = 1/( 2πσ ) exp (−y 2 /(2σ 2 )). As mentioned earlier, the reason we use pseudo-error ei (k) rather than the sliding mode s(i ) is based on the fact that they share the same weight estimation W (k). The desired pdf can be of any form, and here we use the following form:   1 y2 (29) exp − 2 pd (y) = √ 2σd 2πσd where σd is the desired parameter. Then using the mean instead of expectation operator, the information divergence between the estimated pdf and the desired one can be written as ˆ ps || pd ) = 1 D( N

k  j =k−N+1

ln

ps (e j ) 1 = pd (e j ) N

k  j =k−N+1

k  i=k−N+1 k 

g g e j H j −1 = 2 2 σ σd g − 2 e j H j −1 σd

where

(27)

Remark 5: In the traditional LMS method, only the current error is involved in the adaptive law, and thus the pseudo-error concept is unnecessary. However, in the pdf-based criterion, a set of past sliding errors s(i ), i ∈ [k − N + 1, k] are used at each time interval k, and thus each of them depends linearly on different weight vectors W (i ). This will give rise to a difficulty in the adaptive tuning process. To solve this problem, we introduce pseudo-errors ei (k), i ∈ [k − N + 1, k] at time interval k, which are calculated as ei (k) = s(i +1)−g[W (k)− W (i )]TH (Z i ) according to (26). As a result, these pseudoerrors are linearly dependent on the same weight vector W (k), and can be used in the adaptive tuning process.

ps (y) =

g i=k−N+1 σ2

−

At time interval k, W (i ) is already known and can be taken into feedback control because i ≤ k. Therefore we have ek (k) = s(k + 1).

∂ ∂ 1 1 ps (e j ) − pd (e j ) ps (e j ) ∂ W pd (e j ) ∂ W     k  (e −e )2 exp − j2σ 2i (e j − ei )(H j −1 − Hi−1 )

∇ J (e j ) =

J (e j ). (30)

 exp

αji =

k  j =k−N+1

  (e j −ei )2 exp − 2σ 2   α j i (e j − ei )(H j −1 − Hi−1 )

i=k−N+1

(e −e )2 − j2σ 2i

(31) 

  , αji > 0, (e j −ei )2 exp − 2σ 2



α j i = 1.

i

(32) By summing up ∇ J (e j ) and using the deepest descent method to update the weight vector, we can obtain ˆ p S || pd ) = Wk−1 − Wk = Wk−1 − μ∇ D(

μ N

k 

∇ J (e j ).

j =k−N+1

(33) The above algorithm can be summarized as follows. 1) Design a proper sliding-mode controller to stabilize the system. 2) Choose the kernel bandwidth σ , desired pdf parameter σd , and other parameters of the RBF network. 3) Record W j , H j , and s j at each sample interval j . 4) Use recorded information to calculate e j (k − 1) = s j − T (Wk−1 − W j )TH j , for each j = k − N, . . . , k − 1. 5) Form the estimated pdf ps (x) by using (28). 6) Calculate Wk using (31) and (33) by a properly chosen μ. 7) Construct the pdf part of the controller TH u pdf (k) = −Wk k , and apply the whole control signal u(k) = u smc (k) + u pdf (k) to the control system (13). 8) Go back to 3). IV. C ONVERGENCE A NALYSIS The following lemmas will be useful in the convergence analysis, whose proof can be found in the Appendix. Lemma 3 (Chebyshev’s Sum Inequality): If a1 ≥ a2 ≥ · · · ≥ an ≥ 0, and b1 ≥ b2 ≥ · · · ≥ bn ≥ 0, then ⎛ ⎞⎛ ⎞ n n n    1 1 1 ajbj ≥ ⎝ aj⎠ ⎝ bj⎠ (34) n n n j =1 j =1 j =1 ⎛ ⎞⎛ ⎞ n n n   1 1 1 a j bn− j +1 ≥ ⎝ aj⎠ ⎝ bj⎠ (35) n n n j =1

j =1

j =1

where the equality signs hold if and only if a j s or b j s are equal.

LIU et al.: SMC DESIGN FOR NONLINEAR SYSTEMS USING PDF SHAPING

Lemma 4: Let A be a symmetric matrix and B = B T > 0 is a positive definite matrix, and assume that

337

T and W˜ k−1 on M, and using Left- and right-multiplying W˜ k−1 (26) we have

y T Ay ≥ y T By ≥ 0 Rn .

(36) y T (I − μA)T (I − μA)y < y T y. As mentioned earlier, the K–L divergence is a convex function. Therefore, given a proper μ, the deepest descent method should reach the optimum. However, there is a problem here: we substitute the expectation with average as we cannot actually obtain the expectation. It seems that the best result we can have is the convergence in probability. However, we will show that, under a weak condition, which can be seen in [37, Cond. 1], the proposed method guarantees the convergence of the weight estimation to its ideal value. Condition 1 (Rank-Condition): If the history stack of the recorded data contains as many linearly independent elements H (Z i ) ∈ Rm as the dimension of the basis function of the uncertainties, the recorded data is sufficiently rich. In other words, if = [H (x 1), H (x 2), . . . , H (x p )] represents the history stack, then it is sufficiently rich if rank( ) = m. Theorem 2: Consider the first-order system given by (23). The pseudo-error given by (26), and the pdf controller given by (21) and the weight update law of (31)–(33); then the weight error dynamics W˜ k = Wk − W ∗ and the system state sk will globally and uniformly exponentially converge to zero if the followings conditions are met. 1) Condition 1 (Rank-condition) is satisfied. 2) Parameters σ and σd for the pdf are chosen such that σ > 2σd . Proof: Using W˜ k−1 = Wk−1 − W ∗ , (31) and (33) can be changed into g ∇ J (e j ) = 2 H j −1 H j −1T W˜ k−1 σd k    g αji (H j −1 − Hi−1 )(H j −1 − Hi−1 )T W˜ k−1 − 2 σ i=k−N+1

(37) and μ W˜ k = W˜ k−1 − N

k 

∇ J (e j )

j =k−N+1

 μ μ  = W˜ k−1 − MW˜ k−1 = I − M W˜ k−1 N N

(38)

where M=

g2 σd2 −

k 

[H j −1 H j −1T ]

k 



=

k 

1 σd2 −

e2j

j =k−N+1

1 σ2

k 



k 

 αji (e j − ei )2 .

j =k−N+1 i=k−N+1

(40) Using Lemma 4, substituting a j , b j with αji and (e j − ei )2 , respectively, and noticing that in the numerator part of (32), exp(−x 2 ) is a decreasing function on x 2 , we have 

k  i=k−N+1

 1 αji (e j − ei )2 ≤ N



k 

 (e j − ei )2 .

(41)

i=k−N+1

Then it can be seen that T MW˜ k−1 W˜ k−1 ≥

1 σd2

k 

e2j −

j =k−N+1

⎧ 1 ⎨1 = 2 σ ⎩N

1 σ2N 

k 

  (e j − ei )2

k  i, j =k−N+1





(e j + ei )2 +

i, j =k−N+1

σ2 −4 σd2



k 

e2j

j =k−N+1

⎫ ⎬ ⎭

T

W˜ k−1 ≥ W˜ k−1

(42) where 1

= 2 σ



σ2 −4 σd2



k

k 

H j H jT .

j =k−N+1

T Noting that σ > 2σd and j =k−N+1 H j H j > 0 due to Condition 1, we can conclude that > 0. Furthermore, the equality in (42) holds if and only if e j = 0 for all j = k − N + 1, . . . , k. Now consider the Lyapunov function given by

1 ˜T ˜ W Wk 2 k and note that V (0) = 0 and V (W˜ ) > 0 for all W˜ = 0. Based on Lemma 5, while A and B are substituted by 1/NM and 1/N , respectively, there exists a μ0 = 2N(λmin ( ))/(λmax (M TM )) such for all 0 < μ < μ0 , the following holds:  μ T  μ  T T I− M I − M W˜ k−1 < W˜ k−1 W˜ k−1 W˜ k−1 . N N According to (38), we have V (W˜ k ) =

T W˜ kT W˜ k < W˜ k−1 W˜ k−1 .

j =k−N+1

g2 σ2

∇ J (e j )

j =k−N+1

and y ∈ Then for some vector y, where A, B ∈ there exists a μ0 > 0 such that for all 0 < μ < μ0 , the following inequality holds: Rn×n ,

k 

T T W˜ k−1 MW˜ k−1 = W˜ k−1

 αji (H j −1 − Hi−1 )(H j −1 − Hi−1 )T .

i, j =k−N+1

(39)

Hence, if we choose σ > 2σd , Lyanpunov stability of the solution W˜ = 0 can be established. Therefore, we have sk = T H −g W˜ k−1 k−1 → 0 as k → ∞. Moreover, since the sliding manifold (23) is a first-order linear system under bounded

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uncertainty which is BIBO, the whole adaptive control system will be bounded and thus asymptotically converge to zero. Remark 6: It is well known that the traditional gradientbased adaptive law may fall into local minima and the convergence of the parameters is not guaranteed when only instantaneous data is used except when the persistent excitation (PE) condition is satisfied. However, the PE condition is restrictive and often infeasible to be implemented online. With the rank-condition, which is much weaker than PE, our proposed method can guarantee global convergence of the weight vector together with the K–L divergence criterion. Remark 7: Obviously, an entire set of data is not necessary to guarantee the convergence of the proposed method. However, if more previous data samples are involved, a larger step size μ can be used, and thus faster convergence rate can be obtained at the cost of more computational time. This can be seen from the choice of the step size μ < μ0 = 2N(λmin ( ))/(λmax (M T M)), where N is the number of data involved in the pdf estimation.

Fig. 2.

Sliding surface s with SMC, SMC + PDF, and SMC + ASG.

Suppose that yi,d = 0; then ei (k) = yi (k), and the switch surface is designed as s(k) = 2e1 (k) + e2 (k).

V. I LLUSTRATIVE E XAMPLES

(46)

Let h = 0.96; then the sliding mode can be derived as

In order to show the applicability of the proposed method, two examples are illustrated as follows.

s(k + 1) = 0.96s(k)

(47)

which is stable. The SMC controller is denoted as u smc (k) = 100[0.96s(k) − 2e1 (k) − 1.02e2 (k)].

A. Example 1 Consider the nonlinear stochastic system       x (k) 0 x 1 (k + 1) =A 1 + u(k) x 2 (k + 1) x 2 (k) b + b     0 f 1 (k) + + f 2 (k) w(k) y1 (k) = x 1 (k) + d1 (k)

(43)

(44)

y2 (k) = x 2 (k) + d2 (k) (45)   1 0.01 where A = , b = 0.01; b = 0.005x 22(k) is the 0 1 parameter uncertainty; w(k) is a β-distribution noise with w(k) ∼ 0.01 × β(0.2, 1) f i (k), i = 1, 2 represents unmodeled uncertainties and disturbances with f1 (k) = 0.002 sin(0.2πk) + 0.005x 1(k)x 2 (k)2 + x 1 (k)2 and 3 f 2 (k) = 0.01 sin(0.02πk) + 0.01 cos(x 1 (k)) d1 and d2 are uniformly distributed measurement noise subject to di (k) ∼ U (−0.01, 0.01). In fact, this example comes from a robot arm subjected to nonlinear perturbations and non-Gaussian distributed noise, with the sample period T = 0.01. We do not mention the discretization procedures because we are unwilling to get into the discretization problem of a nonlinear system, which is too complicated to be mentioned here.

(48)

Parameters of the PDF controller are chosen as L = 50, σ = 0.04, and σd = 0.01, which means that the target pdf is defined as   1 x2 exp − 2 . pd (x) = √ 2σd 2πσd The updating step μ and the number of RBF basis N are selected as 0.025 and 30, respectively. In this section, the proposed method is also compared with the method presented in [18] (simplified as SMC + ASG), where the sliding gain is represented by an RBF network and adjusted adaptively using the LMS criterion in order to reduce the chattering effect. Parameters of the RBF network in SMC + ASG are the same as those of the proposed method. The dynamics of the sliding surface s(k) are shown in Fig. 2. The sliding mode (dotted line) has a bias because of two factors: first, there is a β-distributed noise in the system which is not of zero mean. Second, the impact of the nonlinearity, such as cos(.) and x 2 which are odd functions, makes the output biased. Notice that our pdf control starts from the 400th sample point, the time before which is used to gather data and wait for the normal sliding system to settle down. It takes about 150 points for the pdf controller to reach its optimum, and after that the bias and the periodic dynamics of the sliding mode s(k) are almost cancelled by the pdf controller. Fig. 3 shows the pdf of sliding mode s(k) with different control methods. It can be seen that, by using the proposed method, the pdf of s(k) is shaped from a double-peaked form—caused by sinusoidal disturbance—to a Gaussian-like one. However, we cannot expect an arbitrarily narrow pdf. Instead, we obtain

LIU et al.: SMC DESIGN FOR NONLINEAR SYSTEMS USING PDF SHAPING

Fig. 3.

Probability density function of s.

Fig. 4.

Output x1 with SMC, SMC + PDF, and SMC + ASG.

the one that is the nearest to the desired pdf in the sense of K–L divergence. From Figs. 2 and 3, one can also observe that the proposed method outperforms the SMC + ASG method in terms of bias and pdf. Figs. 4 and 5 illustrate the system states x 1 and x 2 with normal SMC, SMC + PDF, and SMC + ASG. It can be seen that, since the sliding surface achieves a better performance after introducing the pdf controller, so do the states. Moreover, it is seen from Fig. 6 that, compared with SMC + ASG, the proposed method improves the tracking performance without leading to the chattering effect in the control signal, which is just what we would expect. Finally, the 3-D mesh representation of pdf of the sliding surface is further illustrated. Fig. 7 shows the dynamic transitional changes of the pdf of s(k) with time, where the pdf at each time sample is estimated based on 50 points before and after that underestimated sample point. We also observe that the pdf converges to a narrow Gaussian-like shape. B. Example 2 Consider a coupled system consisting of two cylindrical tanks, as shown in Fig. 8. The objective is to control the liquid level in the second tank by manipulating the flow rate

339

Fig. 5.

Output x2 with SMC, SMC + PDF, and SMC + ASG.

Fig. 6.

Whole control input u with SMC, SMC + PDF, and SMC + ASG.

Fig. 7.

3-D-mesh of the sliding surface.

of the liquid into the first tank. The liquid used in the plant is assumed to be steady, nonviscous, and incompressible, which leads to the use of Bernoulli’s equation to obtain the following equations: A1

$ $ dH1 = Q 1 − β1 H1 − β12 |H1 − H2 |sign(H1 − H2 ) dt (49)

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Pump

Q2 (disturbance)

Q1 (input)

Tank 1

Tank 2 H1 H2 (output)

Valve 1 Fig. 8.

Valve 2

Schematic diagram of the coupled-tank system. TABLE I

designing the DSMC, we can turn to an alternative—emulation design [39]. Emulation design is widely used in sampleddata design, where the controller is first designed in the continuous-time domain and then is implemented in discrete time by applying sample-and-hold to the controller. It has been proved that, when sampled sufficiently fast, the emulation design method preserves the property of the continuous-time system in a semiglobal practical sense, although it may cause performance degradation [39], [40]. A detailed discussion of this topic is obviously out of the scope of this paper, and interested readers may refer to relevant material. The tracking error of the coupled-tank system is defined as the difference between the measured and the desired height of H2 e = Hr − h 2 (56) where Hr is the set point of H2. In continuous time domain, the sliding surface is depicted as

PARAMETERS OF THE C OUPLED -TANK S YSTEM

s(t) = e˙ + λe

A2

$ $ dH2 = Q 2 − β2 H2 + β12 |H1 − H2|sign(H1 − H2 ) dt (50) h 1 = H1 + d 1

(51)

and h 2 = H2 + d 2 .

(52)

In the above equations, Hi and Q i , i = 1, 2 are the liquid level of and flow rate into the i th tank, where the term Q 1 is the control input which will be depicted as u later, and the term Q 2 is the disturbance (not known a priori), defined as where w is the system noise subjected to w ∼ 0.5N(0.1, 0.02) + 0.5N(−0.1, 0.02). h 1 and h 2 stand for the measured outputs of H1 and H2, respectively. d1 and d2 are uniformly distributed measurement noises subject to di ∼ U (−0.5, 0.5), i = 1, 2.

(53)

A1 and A2 represent the cross-sectional areas of corresponding tanks. Parameters β1 , β2 , and β12 are constant and dependent on the coefficients of discharge, gravitational constant, and the cross-sectional area of each outlet. Each parameter is composed of its nominal value and an uncertain part Ai = A¯ i (1 + Ai ), βk = β¯k (1 + βk ),

where λ is a positive constant; we choose λ = 5 for our example. An SMC reaching law [32] is implemented to give s˙ = −ks − αsign(s).

i = 1, 2

(54)

k = {1, 2, 12}.

(55)

All these parameters of the coupled-tank plant are given in Table I, where the nominal ones are taken from [38]. The exact discrete-time model of a nonlinear system is not always computable since we need to solve nonlinear initial value problems, as is the case in this example. Therefore, it is difficult to design a DSMC for (49)–(52) directly. However, instead of using an exact discrete-time model and then

(58)

Consequently, after some deduction and substituting the saturation function for the sign function, the continuous SMC can be obtained as   $ β¯2 u(t) = −ks − αsat(10s) − √ |h 1 − h 2 | 2 h2 +  ¯ 2 A1 1 (59) − β¯12 (d1 − d2 ) 2 β¯12 where

β¯1 $ H1 − A¯ 1 β¯2 $ d2 = − H2 + A¯ 2

d1 = −

Q 2 (t) = 2 sin(0.15t) + w

(57)

β¯12 $ |H1 − H2|sign(H1 − H2 ) A¯ 1 β¯12 $ |H1 − H2 |sign(H1 − H2) A¯ 2

k = 5, α = 2, and is an arbitrarily small positive constant to ensure the nonsingularity of the control input (59). All the parameters in the control law are used with their nominal values and, as a result, the unmatched parameter uncertainties and disturbances (in the H2 channel) will lead to a biased tracking error, as will be seen later. Considering the sampling period T = 2s and set point Hr = 10 in this example, the DSMC is obtained by applying sample-and-hold method and using (59) into the nonlinear system. As for the pdf controller, the parameters are chosen as L = 40, σ = 0.1, and σd = 0.04. The updating step size μ and the number of neurons in the RBF neural network are selected as μ = 1 and N = 15, respectively. The SMC is used in the initial 400 steps, and then the pdf controller is introduced. Seen from Fig. 9, the SMC leads to a biased tracking, which is caused by the incapability of the SMC in dealing with unmatched uncertainties and disturbances. However, the pdf controller results in a zero-mean error and a narrow Gaussianalike pdf, as is shown in Fig. 10. The 3-D mesh representation is shown in Fig. 11 to illustrate the changing process of the pdf of the sliding surface.

LIU et al.: SMC DESIGN FOR NONLINEAR SYSTEMS USING PDF SHAPING

Fig. 9.

Output x2 with SMC, SMC + PDF, and SMC + ASG.

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MSE when it encounters non-Gaussian variables, but it only concentrates on the dispersion aspect of the error distribution rather than the mean error and the detailed information of the shape of the pdf of the error. Using K–L divergence, we can shape the pdf of the tracking error as close as possible to a desired pdf, which prevails over MSE and MEE. The second advantage is that we can tune parameters of the pdf controller in every sample interval rather than use it as a batch mode. Most previous pdf control schemes were implemented in batch mode [41]. The term “batch” is referred to a period of time when the process repeats itself. In the batch mode scheme, control parameters remain fixed during each batch and are updated between any two adjacent batches, so that the closed-loop tracking error is made to decrease batch by batch. Our proposed method updates the weight vector of the RBF network at each sample interval, so that a better convergence rate may be obtained. Further studies will be focused on the reduction of the computational load on each sample interval, and the using of K–L divergence to more generalized nonlinear stochastic systems. A PPENDIX P ROOF OF S OME L EMMAS

Fig. 10.

Whole control input u with SMC, SMC + PDF, and SMC + ASG.

A. Lemma 2 Proof: For a convex function f (x), the perspective of f is also a convex function. That is, g(x, t) = t f (x/t), ∀t > 0 is a convex function. Substituting f (x) with − ln x, which is convex, and t with q(x), we can conclude that p(x) ln p(x)/q(x) is convex. Also integration or summation keeps the convexity, which completes our proof. B. Lemma 3 Proof: We only need to prove (34), as (35) can be similarly shown. Consider the sum with the following form: S=

n n  

(ai − a j )(bi − b j ) ≥ 0.

i=1 j =1

By opening the brackets, we deduce 2n

n  i=1

Fig. 11.

3-D mesh of the sliding surface.

VI. C ONCLUSION In this paper, a new stochastic distribution control scheme is proposed for nonlinear stochastic systems. In the proposed controller, the SMC makes the system stable, while the pdf control shapes the sliding surface to a desired pdf. There are two major differences in our pdf control part as compared to existing pdf schemes. The first one is the use of K–L divergence for the pdf control. The minimum square error (MSE) and the MEE are the commonly used criteria in pdf control schemes. However, MSE is only optimal for linear systems with Gaussian variables. MEE outperforms

ai bi − 2

n  i=1

ai

n 

bj ≥ 0

j =1

and we obtain (34). C. Lemma 4 Proof: We only need to prove that −2μy T Ay + μ2 y T A T Ay ≤ 0. Since y T Ay > y T By and A T A ≥ 0, what we need is to show μ2 y T A T Ay ≤ 2μy T By. This means that μ0 can be chosen as 2λmin (B) . μ0 = λmax (A T A) Then for any μ < μ0 , we have y T (μA T A − 2B)y < 0.

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This means that μy T A T Ay < 2y T By < 2y T Ay. Also we can further obtain −2μy T Ay + μ2 y T A T Ay ≤ 0. This ends our proof. ACKNOWLEDGMENT The authors would like to thank the editors and the reviewers for a number of helpful comments, which improved the presentation of this paper. R EFERENCES [1] H. Wang, “Minimum entropy control of non-gaussian dynamic stochastic systems,” IEEE Trans. Autom. Control, vol. 47, no. 2, pp. 398–403, Feb. 2002. [2] K. J. Astrom, Introduction to Stochastic Control Theory. New York, NY, USA: Academic, 1970. [3] N. Filatov and H. Unbehausen, “Adaptive predictive control policy for nonlinear stochastic systems,” IEEE Trans. Autom. Control, vol. 40, no. 11, pp. 1943–1949, Nov. 1995. [4] H.-B. Ji and H.-S. Xi, “Adaptive output-feedback tracking of stochastic nonlinear systems,” IEEE Trans. Autom. Control, vol. 51, no. 2, pp. 355–360, Feb. 2006. [5] H.-N. Wu and K.-Y. Cai, “Mode-independent robust stabilization for uncertain Markovian jump nonlinear systems via fuzzy control,” IEEE Trans. Syst. Man Cybern. B, Cybern., vol. 36, no. 3, pp. 509–519, Jun. 2006. [6] H. Wang, Bounded Dynamic Stochastic Systems: Modelling and Control. New York, NY, USA: Springer-Verlag, 2000. [7] L. Guo and H. Wang, Stochastic Distribution Control System Design: A Convex Optimization Approach. New York, NY, USA: Springer-Verlag, 2010. [8] H. Wang, J. Zhou, and H. Yue, “Shaping of output PDF based on the rational square-root b-spline model,” ACTA Autom. Sinica, vol. 31, no. 3, pp. 343–351, 2005. [9] H. Wang and H. Yue, “A rational spline model approximation and control of output probability density functions for dynamic stochastic systems,” Trans. Inst. Meas. Control, vol. 25, no. 2, pp. 93–105, Jun. 2003. [10] J. C. Principe, Information Theoretic Learning: Renyi’s Entropy and Kernel Perspectives. New York, NY, USA: Springer-Verlag, 2010. [11] D. Erdogmus, J. C. Principe, K. Sung-Phil, and J. C. Sanchez, “A recursive Renyi’s entropy estimator,” in Proc. 12th IEEE Workshop Neural Netw. Signal Process., Sep. 2002, pp. 209–217. [12] B. Widrow and M. E. Hoff, “Adaptive switching circuits,” Dept. Electron. Eng., Univ. Stanford, Stanford, CA, USA, Tech. Rep. IRE WESCON Conv. Rec., 4:96-104, Aug. 1960. [13] B. Widrow and S. D. Stearns, Adaptive Signal Processing. Englewood Cliffs, NJ, USA: Prentice-Hall, 1985. [14] V. I. Utkin, Sliding Modes in Control Optimization. New York, NY, USA: Springer-Verlag, 1992. [15] K. D. Young, V. I. Utkin, and U. Ozguner, “A control engineer’s guide to sliding mode control,” IEEE Trans. Control Syst. Technol, vol. 7, no. 3, pp. 328–342, May 1999. [16] C. Edwards, E. F. Colet, and L. Fridman, Advances in Variable Structure and Sliding Mode Control. Berlin, Germany: Springer-Verlag, 2006. [17] A.-M. Zou, K. D. Kumar, Z.-G. Hou, and X. Liu, “Finite-time attitude tracking control for spacecraft using terminal sliding mode and chebyshev neural network,” IEEE Trans. Syst. Man Cybern. B, Cybern., vol. 41, no. 4, pp. 950–963, Aug. 2011. [18] Z. Li, Z. Deng, and Z. Gu, “New sliding mode control of building structure using RBF neural networks,” in Proc. CCDC, May 2010, pp. 2820–2825. [19] A. Levant, “Principles of 2-sliding mode design,” Automatica, vol. 43, no. 4, pp. 576–586, Apr. 2007. [20] A. Levant, “Quasi-continuous high-order sliding-mode controllers,” IEEE Trans. Autom. Control, vol. 50, no. 11, pp. 1812–1816, Nov. 2005.

[21] V. S. Elanayar and C. S. Yung, “Radial basis function neural network for approximation and estimation of nonlinear stochastic dynamic systems,” IEEE Trans. Neural Netw., vol. 5, no. 5, pp. 594–603, Jul. 1994. [22] H. Zhang, Y. Luo, and D. Liu, “Neural-network-based near-optimal control for a class of discrete-time affine nonlinear systems with control constraints,” IEEE Trans. Neural Netw., vol. 20, no. 9, pp. 1490–1503, Sep. 2009. [23] Z. Wang, Y. Liu, M. Li, and X. Liu, “Stability analysis of Markovian jumping stochastic Cohen–Grossberg neural networks with mixed time delays,” IEEE Trans. Neural Netw., vol. 17, no. 3, pp. 814–820, Mar. 2008. [24] A. K. Kostarigka and G. A. Rovithakis, “Adaptive dynamic output feedback neural network control of uncertain MIMO nonlinear systems with prescribed performance,” IEEE Trans. Neural Netw. Learn. Syst., vol. 23, no. 1, pp. 138–149, Jan. 2012. [25] B. W. Silverman, Density Estimation for Statistics and Data Analysis (Monographs on Statistics and Applied Probability). London, U.K.: Chapman & Hall, 1986. [26] A. Renyi, Probability Theory. New York, NY, USA: Dover, 2007. [27] E. Parzen, “On estimation of a probability density function and mode,” Ann. Math. Stat., vol. 33, no. 3, pp. 1065–1076, Sep. 1962. [28] J. Lin, “Divergence measures based on the Shannon entropy,” IEEE Trans. Inf. Theory, vol. 37, no. 1, pp. 145–151, Jan. 1991. [29] F. Topsoe, “Some inequalities for information divergence and related measures of discrimination,” IEEE Trans. Inf. Theory, vol. 46, no. 4, pp. 1602–1609, Jul. 2000. [30] D. Morales, L. Pardo, M. Salicr, and M. L. Menndez, “The λ-divergence and the λ-mutual information: Estimation in the stratified sampling,” J. Comput. Appl. Math., vol. 47, no. 1, pp. 1–10, Jun. 1993. [31] G. Monsees, “Discrete-time sliding mode control,” Ph.D. dissertation, Center Syst. and Control, Univ. Delft Technique, Delft, The Netherlands, 2002. [32] W. Gao, Variable Structure Control Theory. Beijing, China: Science and Technology Press, 1990. [33] A. Bartoszewicz, “Remarks on discrete-time variable structure control systems,” IEEE Trans. Ind. Electron., vol. 43, no. 1, pp. 235–238, Jan. 1996. [34] X. Yi and M. Saif, “Sliding mode observer for nonlinear uncertain systems,” IEEE Trans. Autom. Control, vol. 46, no. 12, pp. 2012–2017, Dec. 2001. [35] H. H. Choi, “LMI-based sliding surface design for integral sliding mode control of mismatched uncertain systems,” IEEE Trans. Autom. Control, vol. 52, no. 4, pp. 736–742, Apr. 2007. [36] G. Bartolini, L. Fridman, A. Pisano, and E. Usai, Modern Sliding Mode Control Theory, New Perspective and Applications. New York, NY, USA: Springer-Verlag, 2008. [37] G. V. Chowdhary, “Concurrent learning for convergence in adaptive control without persistency of excitation,” Ph.D. dissertation, Dept. Aerosp. Eng., Georgia Inst. Technol., Atlanta, GA, USA, 2010. [38] H. Abbas, S. Asghar, and S. Qamar, “Sliding mode control for coupledtank liquid level control system,” in Proc. 10th Int. Conf. FIT, Dec. 2012, pp. 325–330. [39] D. Nesic, A. R. Teel, and D. Carnevale, “Explicit computation of the sampling period in emulation of controllers for nonlinear sampled-data systems,” IEEE Trans. Autom. Control, vol. 54, no. 4, pp. 619–624, Mar. 2009. [40] D. Nesic, A. R. Teel, and P. V. Kokotovic, “Sufficient conditions for stabilization of sampled-data nonlinear systems via discrete-time approximations,” Syst. Control lett., no. 38, nos. 4–5, pp. 259–270, Dec. 1999. [41] P. Afshar, H. Wang, and T. Chai, “An ILC-based adaptive control for general stochastic systems with strictly decreasing entropy,” IEEE Trans. Neural Netw., vol. 20, no. 3, pp. 471–482, Mar. 2009.

Yu Liu was born in Tianjin, China, in 1975. He received the B.S. degree from Naikan University, Naikan, China, in 1997, and the Ph.D. degree from Institute of Acoustic, Chinese Academy of Sciences (IACAS), Beijing, China, in 2006. He has been an Associate Professor at IACAS and is currently an Academic Visitor with the University of Manchester, Manchester, U.K. His current research interests include sliding mode control, nonlinear control, and navigation.

LIU et al.: SMC DESIGN FOR NONLINEAR SYSTEMS USING PDF SHAPING

Hong Wang (M’95–SM’05) was born in Beijing, China, in 1960. He received the B.S. degree from the Huainan University of Mining Engineering, Huainan, China, in 1982, and the M.S. and Ph.D. degrees from the Huazhong University of Science and Technology, Huazhong, China, in 1984 and 1987, respectively. He was a Research Fellow with Salford, Brunel, U.K., and Southampton Universities, Southampton, U.K., from 1988 to September 1992. He then joined UMIST in 2002, and has been a Professor of process control with the Control Systems Centre, University of Manchester, Manchester, U.K., since 2002. He holds a research visiting position with the Northeastern University, Shenyang, China, Huazhong University of Science and Technology, and Institute of Automation, Chinese Academy of Sciences, Beijing, China. His current research interests include stochastic distribution control, fault detection and diagnosis, nonlinear control, and data-based modeling for complex systems. He has published 200 papers and three books. Prof. Wang was an Associate Editor of the IEEE T RANSACTIONS ON AUTOMATIC C ONTROL and serves as an Editorial Board Member for the Journal of Measurement and Control, T RANSACTIONS OF THE I NSTITUTE OF M EASUREMENT AND C ONTROL, IMechE Journal of Systems and Control Engineering, Automatica Sinica, and Journal of Control Theory and Applications.

343

Chaohuan Hou (SM’96–F’01) received the B.Sc. degree in physics from Peking University, Beijing, China, in 1958. He has been working on underwater acoustics and signal processing with the Institute of Acoustics, Chinese Academy of Sciences (IACAS), Beijing, since 1958, where he became a Professor in 1985. He was a Deputy Director at IACAS from 1993 to 1997. He has published more than 200 journal and conference papers. His current research interests include underwater acoustics, digital signal processing, arrays signal processing, VLSI signal processing, and ASIC chip design. Prof. Hou was elected as an Academician of the Chinese Academy of Sciences in 1995. He was the President of the Acoustical Society of China from 2002 to 2006, and became the Honorary President. From 2007 to 2010, he was a Board Member of the International Commission for Acoustics.