Robust iterative learning control design for linear systems with time-varying delays and packet dropouts

Abstract

This paper proposes the robust iterative learning control (ILC) design for uncertain linear systems with time-varying delays and random packet dropouts. The packet dropout is modeled by an arbitrary stochastic sequence satisfying the Bernoulli binary distribution, which renders the ILC system to be stochastic instead of a deterministic one. The main idea of this paper is to transform the ILC design into robust stability for a two-dimensional (2D) stochastic system described by the Roesser model with a delay varying in a range. A delay-dependent stability condition, which can guarantee mean-square asymptotic stability of such a 2D stochastic system, is derived in terms of linear matrix inequalities (LMIs), and formulas can be given for the ILC law design. An example for the injection molding is given to demonstrate the effectiveness of the proposed ILC method.

Xuhui et al. Advances in Dierence Equations (2017) 2017:84 DOI 10.1186/s13662-017-1140-3

Robust iterative learning control design for linear systems with time-varying delays and packet dropouts

http://crossmark.crossref.org/dialog/?doi=10.1186/s13662-017-1140-3&domain=pdf

Web End = Bu Xuhui1*, Hou Zhanwei2, Hou Zhongsheng3 and Yang Junqi1

*Correspondence: mailto:[email protected]

Web End [email protected]

1School of Electrical Engineering & Automation, Henan Polytechnic University, Jiaozuo, ChinaFull list of author information is available at the end of the article

1 Introduction

Iterative learning control (ILC) is an eective technique for systems that could perform the same task over a nite time interval repetitively. It updates the control input signal only depending on I/O data of previous iteration, and the tracking performance can become better and better. Due to its simplicity and eectiveness, ILC has been widely applied in many practical systems such as robotics, chemical batch processes, hard disk drives and urban trac systems [].

Time delay, which is a source of instability and poor performance, often appears in practical systems due to the nite speed of signal transmission and information processing. As a result, studies on batch process control have attracted considerable attention []. Recently, ILC has been introduced to systems with time delays to improve the tracking performance. In [], a new ILC method is proposed for a class of linear systems with time delay using a holding mechanism. In [], an ILC algorithm integrated with Smith predictor for batch processes with xed time delay is proposed and analyzed in the frequency domain; itcan obtain perfect tracking performance under certain conditions. In [], an ILC scheme is proposed for systems with time delay and model uncertainties based on the internal model control principle. In [], a two-dimensional (D) model based on ILC

The Author(s) 2017. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Xuhui et al. Advances in Dierence Equations (2017) 2017:84 Page 2 of 17

methods for systems both with state delays and with input delays is presented; necessary and sucient conditions for the stability of ILC are also provided. In [], the problem of ILC design for time-delay systems in the presence of initial shifts is considered, and the D system theory is employed to develop a convergence condition for both asymptotic stability and monotonic convergence of ILC.

In these existing studies [], the time delay considered is known and xed constants. As we know, many practical systems suer from time-varying delays, which are less conservative than constant delays. It is a challenge to design ILC for systems with time-varying delays. There have already been a few results on this issue. In [], a robust state feedback integrated with ILC scheme is proposed for batch processes with interval time-varying delay. The design is considered using a D Roesser model based on the general D system theory []. In [], a robust closed-loop ILC scheme is proposed for batch processes with state delay and time-varying uncertainties, and the batch processes are described as a D FM model. In [], a robust output feedback incorporated with ILC scheme is proposed for a kind of batch process with uncertainties and interval time-varying delay. The batch process is transformed into a D FM model with a delay varying in a range, and the design is cast as a robust H control for uncertain D systems. It is noticed that almost all available results on ILC systems with time delays are based on an implicit assumption that sensor output measurement is perfect. However, the assumption is often not true in most cases in practice. The main reason is that sensors may suer from probabilistic signal missing especially in a networked environment [].

Actually, the problem of ILC for networked control systems (NCSs) with packet dropouts has received some attention in the research eld. The stability of ILC for linear and nonlinear systems with intermittent measurement is investigated in [, ]. Some robust ILC designs are proposed for NCSs to suppress the eect of data dropouts in [ ]. However, to our best knowledge, no work considering ILC systems with data dropouts and time delay simultaneously has been done up to now.

This paper proposes a robust ILC design scheme for uncertain linear systems with time-varying delays and random packet dropouts. Here the considered systems are implemented in a network environment, where data packet may be missed during transmission. For convenience, only the measurement packet dropout is taken into account. The packet dropout is modeled by an arbitrary stochastic sequence satisfying the Bernoulli binary distribution, which renders the ILC system to be stochastic instead of a deterministic one. Then, a D stochastic Roesser model with a delay varying in a range is established to describe the entire dynamics. Based on D system theory, the ILC law is designed to guarantee mean-square asymptotic stability of the considered D stochastic systems. Afterwards, a delay-dependent stability condition is derived in terms of linear matrix inequalities (LMIs), and formulas can be given for the ILC law design. Finally, an example for the injection molding is given to demonstrate the eectiveness of the proposed ILC method.

Throughout the present paper, the following notations are used. The superscript T denotes the matrix transposition, I denotes the identity matrix, denotes the zero vector or matrix with the required dimensions. diag{} denotes the standard (block) diagonal matrix whose o-diagonal elements are zero. In symmetric block matrices, an asterisk

is used to denote the term that is induced by symmetry. The notation refers to the Euclidean vector norm, E{x}, E{x|y} mean the expectation of x and the expectation of

Xuhui et al. Advances in Dierence Equations (2017) 2017:84 Page 3 of 17

x conditional on y, respectively. Matrices, if the dimensions are not explicitly stated, are assumed to be compatible for algebraic operations.

2 Problem formulation and 2D system representation

Consider the following linear discrete time system with time-varying delay:

x(t + , k) = (A + A)x(t, k) + (Ad + Ad)x(t d(t), k) + Bu(t, k),

y(t, k) = Cx(t, k), x(, k) = xk,

()

where k denotes iteration, t denotes discrete time. x(t, k), u(t, k), y(t, k) are state, input and output variables, A, Ad, B, C are the matrices with appropriate dimensions describing the system in the state space. xk stands for the initial condition of the process in the kth iteration. d(t) is time-varying delay satisfying

dm d(t) dM,

where dm, dM denote the lower and upper delay bounds. A, Ad denote admissible uncertain perturbations of matrices A and Ad, which can be represented as

A = E F, Ad = E F,

where E, F, F are known real constant matrices characterizing the structures of uncertain perturbations, and is an uncertain perturbation of the system that satises T I.

For system (), design the following ILC update law:

u(t, k + ) = u(t, k) + u(t, k), ()

where u(t, k) is the ILC update law to be designed.

Denote e(t, k) = yd(t) y(t, k), (t, k) = x(t , k + ) x(t , k). From () and (), we can obtain

(t + , k) e(t, k + )

= A

(t, k) e(t, k)

+ Ad

(t d(t), k) e(t, k h(k))

+ B u(t , k), ()

where

A = + , Ad = (d + d),

=

A CA I

,d =

Ad

CAd

, B =

B

CB

,

= F, d = F,

=

E

CE

, F = [F ], F = [F ],

Xuhui et al. Advances in Dierence Equations (2017) 2017:84 Page 4 of 17

h(k) is a new delay variable satisfying hm h(k) hM with hm, hM denoting the lower and upper delay bounds. The new vector e(t, k h(k)) is introduced here as is also done in []

so that system () can be modeled as a normal D system with interval time delay.Consider the following ILC update laws:

u(t , k) = K

, ()

where the stochastic parameter (t, k) is a random Bernoulli variable taking the values of and with

Prob (t, k) = = E (t, k) = ,

E (t, k) = ( ),

in which satisfying is a known constant.From () and (), it can be derived

(t + , k) e(t, k + )

x(t , k + ) x(t , k) e(t, k)

, ()

where K = [K, K] is a gain matrix to be designed. ILC law () contains two parts: a P-type ILC law and a state feedback control law. This control scheme has the advantages of feedback loop such as robustness and meanwhile enjoys the extra performance improvement from ILC.

It is assumed that the ILC law is implemented in a networked control system as shown in Figure , where the network only exists on the side from plant to controller for convenience. The data x(t, k +), x(t, k), e(t, k) are transferred as one whole packet from the plant to the controller. It is further supposed that the controller can detect whether the packet is dropped or not. If the data packet is missed, then u(t, k) = , that is, u(t, k + ) = u(t, k). If the data packet is not missed, the term u(t, k) is calculated as (). In this case, the ILC update law () can be described as

u(t , k) = (t, k)K

x(t , k + ) x(t , k) e(t, k)

()

= A + (t, k) BK

(t, k) e(t, k)

+ Ad

(t d(t), k) e(t, k h(k))

. ()

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Denote (t, k) = xh(i, j), e(t, k) = xv(i, j), system () can be rewritten as the following typical

D Roesser system:

xh(i + , j) xv(i, j + )

= A + (i, j) BK

xh(i, j)

xv(i, j)

+ Ad

xh(i d(i), j)

xv(i, j h(j))

, ()

where the bounded conditions are dened by

xh(i, j) = ij, j < r, dM i < , xv(i, j) = ij, i < r, hM j < ,

= ,

()

where r < and r < are positive integers, ij and ij are given vectors.

Remark It is worth pointing out that D system () is a stochastic system due to the introduction of the stochastic variable (i, j). It diers from the deterministic D or ILC

systems with time delay in recent works such as []. Here, the ILC design should be

discussed under the framework of stochastic stability. To this end, we need to give the

following denition of stochastic stability for D systems.

Denition ([]) For all initial bound conditions in (), D system () is said to be mean-square asymptotically stable if the following is satised:

lim

i+j E

x(i, j) = .

3 Stability analysis and controller design3.1 Stability analysis

In this section, we focus on the problem of robust stability and robust stabilization for D

stochastic system () using an LMI technique. The following lemma is needed in the proof

of our main result.

Lemma ([]) For any vector (t)

Rn, two positive integers , and matrix < R

Rnn, the following inequality holds:

( + )

t=

T(t)R(t)

t=

T(t)R

t=

(t).

Now, we can give our main result.

Theorem Given positive integers dm, dM, hm, hM, D system () is mean-square asymptotically stable if there exists a positive denite symmetric matrix P = P

h

Pv , Q =

Qh

Qv ,

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M = M

h

Mv , and G =

Gh

Gv such that the following matrix inequality holds:

G ( A + BK)TP ( BK)TP (( A + BK)T I)G ( BK)TGQ ATdP ATdG

M G P

P

HG HG

< ,

()

where

= P + M + DQ + Q G, D =

(dM dm)I

(hM hm)I

,

H =

dMI

hMI

, = ( ).

i,j = (i, j) , it is obvious that

E{ i,j} = , E{ i,j

i,j} = ( ).

Denote

xd(i, j) =

xh(i d(i), j)

xv(i, j h(j))

Proof Dene

, xM(i, j) =

xh(i dM, j)

xv(i, j hM)

,

h(r, j) = xh(r + , j) xh(r, j), v(i, t) = xv(i, t + ) xv(i, t),

(i, j) = xT(i, j) xTd(i, j) xTM(i, j) T,

(i, j) = xh(i, j), xh(i , j), . . . , xh i d(i), j , xv(i, j), xv(i, j ), . . . , xv i, j h(j) .

Consider the following Lyapunov function candidate:

V x(i, j) = Vh xh(i, j) + Vv xv(i, j) ,

where

Vh xh(i, j) =

k=

Vhk xh(i, j) , Vh xh(i, j) = xhT(i, j)Phxh(i, j),

Vh xh(i, j) =

i

r=id(i)

xhT(r, j)Qhxh(r, j), Vh xh(i, j) =

i

r=idM

xhT(r, j)Mhxh(r, j),

Vh xh(i, j) =

dm

s=dM

i

r=i+s

xhT(r, j)Qhxh(r, j),

Vh xh(i, j) = dM

s=dM

i

r=i+s

hT(r, j)Ghh(r, j), ()

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Vv xv(i, j) =

k=

Vvk xv(i, j) , Vv xv(i, j) = xvT(i, j)Pvxv(i, j),

Vv xv(i, j) =

j

t=jh(j)

xvT(i, t)Qvxv(i, t), Vv xv(i, j) =

j

t=jhM

xvT(i, t)Mvxv(i, t),

Vv xv(i, j) =

hm

s=hM

j

t=j+s

xvT(i, t)Qvxv(i, t),

Vv xv(i, j) = hM

s=hM

j

t=j+s

vT(i, t)Gvv(i, t),

and Ph, Pv, Qh, Qv, Mh, Mv, Gh, Gv are positive denite matrices to be determined.Dene the following index:

J = E V x(i, j) | (i, j)

= E

Vh(xh(i + , j)) Vh(xh(i, j)) +Vv(xv(i, j + )) Vv(xv(i, j))

(i, j)

= E

Vhk xh(i, j) +

k=

Vvk xv(i, j)

(i, j)

. ()

Calculating () along the solutions of system (), we can obtain

E Vh

xh(i, j) | (i, j)

= E xhT(i + , j)Phxh(i + , j)| (i, j) xhT(i, j)Phxh(i, j), ()

E Vh

xh(i, j) | (i, j)

= E

r=i+d(i+)

xhT(r, j)Qhxh(r, j)

i

i

r=id(i)

xhT(r, j)Qhxh(r, j)

(i, j)

E

xhT(i, j)Qhxh(i, j) xhTd(i, j)Qhxhd(i, j)

+

idm

r=i+dM

xhT(r, j)Qhxh(r, j)

(i, j)

, ()

E Vh

xh(i, j) | (i, j)

= E xhT(i, j)Mhxh(i, j) xhTM(i, j)MhxhM(i, j) | (i, j) , ()

E Vh

xh(i, j) | (i, j)

= E

(dM dm)xhT(i, j)Qhxh(i, j)

idm

r=i+dM

xhT(r, j)Qhxh(r, j)

(i, j)

, ()

xh(i, j) | (i, j)

= E

E Vh

dMhT(i, j)Ghh(i, j) dM

i

r=idM

hT(r, j)Ghh(r, j)

(i, j)

, ()

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and

E Vv

xv(i, j) | (i, j)

= E xvT(i, j + )Pvxv(i, j + )| (i, j) xvT(i, j)Pvxv(i, j), ()

E Vv

xv(i, j) | (i, j)

= E

t=j+d(j+)

xvT(i, t)Qvxv(i, t)

j

j

t=jh(j)

xvT(i, t)Qvxv(i, t)

(i, j)

E

xvT(i, j)Qvxv(i, j)xvT xvTd(i, j)Qvxvd(i, j)

+

jhm

t=j+hM

xvT(i, t)Qvxv(i, t)

(i, j)

, ()

E Vv

xv(i, j) | (i, j)

= E xvT(i, j)Mvxv(i, j) xvTM(i, j)MvxvM(i, j) | (i, j) , ()

E Vv

xv(i, j) | (i, j)

= E

(hM hm)xvT(i, j)Qvxv(i, j)

jhm

t=j+hM

xvT(i, t)Qvxv(i, t)

(i, j)

, ()

xv(i, j) | (i, j)

= E

E Vv

hMvT(i, j)Gvv(i, j) hM

j

t=jdM

vT(i, t)Gvv(i, t)

(i, j)

. ()

For () and (), using Lemma yields

E Vh

xh(i, j) | (i, j)

E

dMhT(i, j)Ghh(i, j)

i

r=idM

hT(r, j)Gh

i

r=idM

h(r, j)

(i, j)

= E

dM(xh(i + , j) xh(i, j))TGh(xh(i + , j) xh(i, j))

(xh(i, j) xhM(i, j))TGh(xh(i, j) xhM(i, j))

(i, j)

, ()

and

E Vv

xv(i, j) | (i, j)

E

hMvT(i, j)Gvv(i, j)

j

t=jdM

vT(i, t)Gv

j

t=jdM

v(i, t)

(i, j)

= E

hM(xv(i, j + ) xv(i, j))TGv(xv(i, j + ) xv(i, j))

(xv(i, j) xvM(i, j))TMv(xv(i, j) xvM(i, j))

(i, j)

. ()

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Since

i,j is independent with x(i, j) and xd(i, j), we have

E xhT(i + , j)Phxh(i + , j) | (i, j) + E xvT(i, j + )Pvxv(i, j + ) | (i, j)

= E

xh(i + , j) xv(i, j + )

T

P

xh(i + , j) xv(i, j + )

(i, j)

= E

x(i, j) xd(i, j)

T

( A + BK)T ( BK)T

ATd

P

P

( A + BK) Ad BK

x(i, j) xd(i, j)

(i, j)

. ()

From ()-() and using the result in (), we can obtain

J E T(i, j)(i, j)| (i, j) ,

where

=

G

Q

M G

+ T

P P

HG

HG

+ T

, ()

and

=

( A + BK) Ad BK

, =

( A + BK) I Ad BK

.

By using Schur complements, it is obvious that () is equivalent to (). Since < , it is obvious that

E Vh xh(i + , j) Vh xh(i, j) + Vv xv(i, j + ) Vv xv(i, j) | (i, j) < . ()

Taking mathematical expectation on both sides of () leads to

E Vh xh(i + , j) + Vv xv(i, j + ) E V

h xh(i, j) + Vv xv(i, j) . ()

Summing up both sides of () from N to with respect to i and to N with respect to j, for any nonnegative integer N and considering boundary conditions (), we have

E

Vh(xh(, N)) + Vv(xv(, N + )) + Vh(xh(, N ))

+Vv(xv(, N)) + + Vh(xh(N + , )) + Vv(xv(N, ))

=

i+j=N+

E Vh xh(i, j) +

E Vv xv(i, j)

i+j=N+

=

i+j=N+

E

xh(i, j)

V xv(i, j)

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E

xh(, N) xv(, N)

+ + V

xh(N, )

xv(N, )

=

i+j=N

E

xh(i, j)

V xv(i, j)

. ()

It is clear that the sum of the Lyapunov functional value decreases along the state trajectories. Then, from [], we can conclude that

lim

i+j E

x(i, j) = ,

which implies that system () is asymptotically stable.This completes the proof of Theorem .

Remark Theorem provides a sucient condition for D uncertain systems with a delay varying in a range and random packet dropouts. If the communication link existing between the plant and the controller is perfect, that is, there is no packet dropout during their transmission, then = and = . In this case, the condition in Theorem becomes the condition obtained in [, ] for a D deterministic system with time delay. From this point of view, Theorem can be seen as an extension to D time-delay systems with packet dropout.

3.2 Controller design

Theorem gives a mean-square asymptotic stability condition where the controller gain matrix K is known. However, our eventual purpose is to determine a suitable K by system matrices A, Ad, B, E, F, F and parameter .

Lemma ([]) Let U, V, and W be real matrices of appropriate dimensions with W satisfying W = WT, then for all T I,

W + U V + VT TUT < ,

if and only if there exists > such that

W + UUT + VTV < .

Now, we can give the following result.

Theorem Given positive integers dm, dM, hm, hM, D system () is mean-square asymptotically stable if there exist a positive denite symmetric matrix L = L

h

Lv , S =

Sh

Sv ,

Xv matrix Y and positive scalar such that the following matrix inequality holds:

M =

Mh

Mv

, M =

Mh

Mv

, X = X

h

< ,

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where

=

X

S

M X

, =

L AT + YT BT YT BT

L ATd

,

=

L AT + YT BT L YT BT

L ATd

, =

L FT

L FT

,

=

T L

T L

, =

T HX

T HX

,

= I,

= L + M + DS + S LXL.

In this case, a suitable ILC law can be selected as K = YL.

Proof Pre- and post-multiply inequality () by diag{L, L, G, L, L, G, G}, by using

Schur complements and letting L = P, the following LMI can be obtained:

L L AT + YT BT YT BT L AT + YT BT L YT BTS L ATd L ATd

M X

L L

HG HG

< ,

()

that is,

L LT + YT BT YT BT LT + YT BT L YT BT

S LTd LTd

M X

L L

HG HG

T

L FT

L FT

T

L FT

L FT

T

+

T

+

T

< . ()

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By Lemma , () holds if and only if there exists a scalar > such that

L LT + YT BT YT BT LT + YT BT L YT BT S LTd LTd

M X T L

T L

T HG

T HG

FL FL < . ()

Using Schur complements, condition () implies that the condition in Theorem holds.

Remark Note that the condition in Theorem is no longer LMIs owing to the term LXL in

+

L FT L FT

. The available LMI tools cannot be used directly to obtain a feasible solution. However, we can use the idea of iterative algorithms in combination with LMI convex optimization problems as is done in [, ]. Then an available control law can be obtained.

Remark It is noticed that the data dropout may occur on both system output and control input sides in NCSs. In this paper, we only consider output measurement missing for the sake of convenience, as is also done in most existing works. However, the result in this paper can be extended to the control input signal dropouts.

Remark Since the system performs the same task repetitively, computation complexity is an important issue for ILC systems. The large number of iterations leads to high accuracy but heavy computational burden. How to keep the balance between iteration number and tracking accuracy is an important problem for practical ILC systems. Some eorts can be made to address this issue. Firstly, we can pre-calculate an acceptable tracking error, which satises the accuracy requirement of the control objective. If the tracking error reaches the given value, then the iteration process is stopped. Secondly, we can design

Xuhui et al. Advances in Dierence Equations (2017) 2017:84 Page 13 of 17

optimal ILC with monotonic convergence and fast convergence speed such that learning

transient behavior is reduced.

4 Illustrative example

In this section, the result is applied to an injection molding process to demonstrate the

eectiveness of the proposed ILC design approach. Injection molding process is a typical

repetitive process, where key process variables are controlled to follow certain proles

repetitively to ensure the product quality [, ]. Injection velocity is a key variable in the

lling stage, which is controlled by manipulating the opening of a hydraulic valve. A state-

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space representation of injection molding velocity control can be described as follows:

x(k, t + ) = (A + A)x(k, t) + (Ad + Ad)x(k, t d(t)) + Bu(k, t),

y(k, t) = Cx(k, t),

where matrices A = [ . .], B = [ ], C = [. .]. A = E F, Ad = E F indi

cate uncertain parameter perturbation with

E =

=

. + .t, t ,., < t , . + .(t ), < t , ., < t .

In simulation, the initial states are given as x(, k) = x(, k) = for all k, and the control input is selected as u(t, ) = for all t. The time-varying delay d(t) changes randomly with d(t) [. ]. The uncertain parameters , are assumed to vary randomly within [ ]

along with both time and iteration direction.

To perform the simulation, we consider the following two cases: ()

= , ()

= ..

Obviously, Case means that there is no packet dropout. Case means that there is percent dropout. Using Theorem , we can obtain the controller K = [. . .] for Case and K = [. . .] for Case . Figure shows the tracking error on the iteration domain for Case . It is obvious that the tracking error converges to a small steady state without packet dropouts, where the nal error is caused by unpredictable random parameter disturbances. To see the convergence procedure more clearly, system outputs at st, th and th iteration are plotted in Figure .

. .

, F =

, F =

. .

,

, || < , || < ,

where , are unknown variables. The values of time-varying delays are chosen as . d(t) and . h(k) , which is derived from [].

The desired trajectory is given as follows:

yd(t) =

Xuhui et al. Advances in Dierence Equations (2017) 2017:84 Page 15 of 17

For Case , the tracking error on the iteration domain is plotted in Figure , and system

outputs at the three dierent iterations are plotted in Figure . It can be seen that even

though the tracking performance has degraded and signicant tracking errors exist in the

start iteration due to packet dropouts, better tracking can also be achieved after some iter

ations. It is thus demonstrated that the proposed method can be used for robust tracking

against non-repeatable uncertainties along iteration, random packet dropouts and time-

varying delay. Besides, the convergence speed of Case is slower than that of Case as

shown in Figures and , which illustrates that convergence speed gets slower as dropout

rate increases.

Xuhui et al. Advances in Dierence Equations (2017) 2017:84 Page 16 of 17

5 Conclusions

The robust ILC design is considered for uncertain linear systems with time-varying delays and random packet dropouts. By modeling the packet dropout as an arbitrary stochastic sequence satisfying the Bernoulli binary distribution, the considered system can be transformed into a D stochastic system described by the Roesser model with a delay varying in a range. Then, a delay-dependent stability condition is derived in terms of linear matrix inequalities (LMIs), and formulas can be given for the ILC law design. The results on injection velocity control have illustrated the feasibility and eectiveness of the proposed design.

Competing interests

The authors declare that they have no competing interests.

Authors contributions

BXH and HZW carried out the main part of this article. HZS and YJQ corrected and revised the manuscript. All authors have read and approved the nal manuscript.

Author details

1School of Electrical Engineering & Automation, Henan Polytechnic University, Jiaozuo, China. 2College of Computer Science and Technology, Henan Polytechnic University, Jiaozuo, China. 3Advanced Control Systems Lab., School of Electronics and Information Engineering, Beijing Jiaotong University, Beijing, China.

Acknowledgements

This work was supported by the National Natural Science Foundation of China (Nos. 61573129, 61433002, 61573130), the Program for Science & Technology Innovation Talents in Universities of Henan Province (16HASTIT046), the Fundamental Research Funds for the Universities of Henan Province, the program of Key Young Teacher of Higher Education of Henan Province (2014GGJS-041), the program of Key Young Teacher of Henan Polytechnic University, the Science and Technology Innovation Talents Project of Henan Province (164100510004) and the Innovation Scientists and Technicians Troop Construction Projects of Henan Province (CXTD2016054).

Publishers Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional aliations.

Received: 5 November 2016 Accepted: 10 March 2017

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