fractional delay compensated discrete time smc for networked control system

In this paper, a novel idea of compensating the fractional time varying communication delay in the sliding surface is presented.. The fractional time delay in the sensor to controller an

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Fractional delay compensated discrete-time SMC for networked control

D.H Shaha,⁎, A.J Mehtab

a Gujarat Technological University, Gujarat, India

b Institute of Infrastructure Technology Research and Management, Gujarat, India

A R T I C L E I N F O

Keywords:

Discrete-time sliding mode control

Networked control

Network delay

A B S T R A C T Time-varying network induced delay in the communication channel severely affects the performance of closed loop network control systems In this paper, a novel idea of compensating the fractional time varying communication delay in the sliding surface is presented The fractional time delay in the sensor to controller and controller to actuator channel is approximated using the Thiran approximation technique to design the sliding surface A discrete-time sliding mode control law is derived using the proposed surface that compensates fractional time delay in sensor to controller and controller to actuator channels for uncertain network control systems The sufficient condition for closed loop stability of the system is derived using the Lyapunov function The efficacy of the proposed strategy is supported by the simulation results

1 Introduction

The Networked Control System (NCS) is one of the frontier areas in

thefield of control, both in terms of research and application The main

cause of attraction is due to its lower cost, simpler installation, easier

maintenance and resource sharing features Any feedback control

systems, closed through some communication medium (such as CAN,

Ethernet, Profibus, Profinet, DeviceNet, etc.) are classified as

Networked Control Systems (NCSs) [16] Various issues such as

bandwidth sharing, resource allocation, time delay, packet loss,

sche-duling, etc arise due to the presence of the communication medium

[17] This degrades the performance of the closed loop system and even

leads to instability

Recently, researchers have proposed various control algorithms

addressing the time delay issue Cac, Hung and Khang[1]used a pole

placement method for compensating the time delay in the continuous

time domain The algorithm was designed for the CAN type

determi-nistic networked medium Yi, Kim and Choi[2]solved the time delay

problem by using the Smith predictor algorithm The method was

verified over wireless sensor networks (WSN) connected between the

controller output and plant input Hikichi et al [3] worked on

continuous time delay compensation using predictors and disturbance

observer for designing a PID controller Cuellar et al.[4]proposed an

observer based predictor using the Pade approximation technique for

time lag processes Vallabhan et al [5] have used the analytical

framework approach for compensation of random time delay and packet loss Ono et al.[6]designed a state feedback controller based

on a modified Smith predictor which stabilized the plant in the presence of dead time Recently, Khanesar et al.[8] used the Pade approximation technique for time delay compensation in a continuous time system Hu et al [23] designed a sliding mode intermittent controller for bidirectional associative memory (BAM) using neural networks with delays Saravanakumar et al.[24] proved the stability using a Markovian Jump approach for neural networks having varying time interval delays

Unlike the continuous time domain, very few researchers have tried

to focus their work on the discrete-time domain Jacovitti and Scarano [7]proposed various time delay estimation techniques for discrete-time systems Yue, Han and Lam[15]provided the model of NCSs with networked induced delay in the discrete domain Hespanha, Naghshtabrizi and Xu [10] designed a Luenberger output feedback observer based state feedback controller to deal with time delay Niu and Ho[9]designed a sliding mode control in the discrete domain in order to deal with network non-idealities such as time delay and packet loss Recently, Li et al.[11]designed a sliding mode predictive control for compensation of delay in a networked control system using a Kalman Predictor They considered networked delays in an integer form Shah and Mehta [19] have designed output feedback based discrete time sliding mode control[18]to deal with delay problems in NCSs They compensated the time varying delay using a zero-order

http://dx.doi.org/10.1016/j.dcan.2016.09.006

Received 18 February 2016; Received in revised form 22 September 2016; Accepted 30 September 2016

Peer review under responsibility of Chongqing University of Posts and Telecommunication.

⁎ Corresponding author.

E-mail address: dipeshshah.ic@gmail.com (D.H Shah).

2352-8648/ © 2016 Chongqing University of Posts and Telecommuniocations Production and hosting by Elsevier B.V This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by/4.0/).

Available online xxxx

Please cite this article as: Shah, D.H., Digital Communications and Networks (2016), http://dx.doi.org/10.1016/j.dcan.2016.09.006

hold technique Recently, Argha et al designed a discrete time sliding

mode control to handle the random delays occurring in NCSs [20]

They treated delays as stochastic variables due to the random behavior

Guo et al.[21] considered the state estimation problem for wireless

NCS The sliding mode observer was designed to solve the state

estimation problem considering stochastic uncertainty and time delay

Yao et al.[22]designed a robust model predictive control (RMPC) and

state observer for a class of time varying systems under input

constraints and packet loss situations

Although lot of work is done, time delay compensation in the

discrete domain is still under investigation Most of the researchers

[8,11,12,19], have assumed the values of network delays in terms of an

integer But, in real time the delays may have non-integer type values

So, as per the authors best knowledge till date none of the researchers

in the discrete domain have tried to study the effect of fractional delay

in NCSs The compensation of fractional delays occurring within the

network is still an open research problem in NCSs in discrete domain

Apart from these, the controllers were designed based on the time delay

approximation technique without considering the effect of uncertainty

Further, the control algorithm is implemented through digital

proces-sor and the communication is also carried out in digital signal form

This motivates the authors to explore the Discrete-Time SMC algorithm

which compensates the fractional delay occurring within the network

even in the presence of system uncertainties and disturbances

In this work, the fractional time delay is compensated at the sliding

surface instead of compensating at the control law In the proposed

method, the sliding gain will change according to the networked delay

and force the system states to slide along the predetermined surface

Based on the proposed sliding surface, the control law is derived This

law provides faster convergence without increasing the amplitude of

the quasi-sliding band Once the sliding surface and the control law are

designed in SMC, the next step is to design the band which guarantees

the stability of the designed sliding surface and causes the system states

to remain within that band for afinite interval of time

The paper is organized as follows:Section 2describes the Problem

Statement The main part of the paper, design of the compensated

sliding surface is presented in Section 3 Section 4discusses about

discrete time sliding mode control for NCS with time delay Results and

discussion are enclosed in Section 5 followed by the conclusion in

Section 6

2 Problem statement

Let us consider the continuous LTI system in normal form as:

x ṫ( ) =Ax t( ) +Bu t( − ) +τ B d t d ( )), (1)

where x∈R n represents the system state vector, u∈R mrepresents the

control input, y∈R p represents the system output, A∈R n n× ,

B∈R n m× , B d∈R n m× ,C∈R p n×, D∈R p m× are the matrices of

appro-priate dimensions, d t( ) presents the matched disturbances and τ

represents the total networked induced delay

The discrete form of Eqs.(1) and (2)is given by:

l

x k( + 1) =Fx k( ) +Gu k( − ) +τ G d k d ( ), (3)

l

where F=e Ah, G=∫h e Bdh Ah

0 x k( ) ∈R n indicates the plant state,

u k( ) ∈R m defines the control input, y k( ) ∈R p represents the plant

output, d k( ) represents the matched disturbances applied at the control

input, lτis the fractional part of network induced delay occurring within

the network Mathematically it is represented as,

where τ is the total network induced delay and h is the sampling

interval

Assumption 1 The disturbance are bounded in nature which is represented as:

where d l and d udenote the lower and upper bounds of the disturbances, respectively

The network induced delay is the combination of the sensor to

controller delay (τ sc ) and controller to actuator delay (τ ca) which is represented as:

whereτ sc=lτ sc*handτ ca=lτ ca*h Remark 1 In this paper it is considered that both network delays lτ sc

and lτ caare considered as a fractional part ofτscandτcarespectively Both represent the positive scalar quantities having the same properties Thus, both the delays reach their maximum values at the same interval

Assumption 2 Network induced delay varies with time and satisfies the given condition

where τ l and τ u indicate the lower bound and upper bound of the networked delay

The objective is to design a robust DSMC algorithm which stabilizes the system (3), in the presence of a sensor controller delay τsc and

controller to actuator delay τ ca satisfying the condition (8) and matching the uncertainty satisfying condition(6)

3 Design of sliding surface for NCS Fig 1represents the schematic diagram of NCS with time delay compensator The sensor samples the data packets at regular sampling intervalh The data signals in the closed loop system will experience

sensor to controller delay (τ sc ) and controller to actuator delay (τ ca) These delayed signals must be compensated using approximation techniques to avoid the degradation of the output In this work, the Thiran approximation technique [25] is used for compensating the networked delay occurring due to the presence of the network For designing the DSMC, the sliding surface using the Thiran Approximation rule in the form ofLemma 1as under:

Lemma 1 The sliding variable for the given system(3)with sensor

to controller network delay satisfying Assumptions (1) and (2) is given as:

Fig 1 Block diagram of NCS with time delay compensation.

l l

α

τ τ

8 + 8 + 1

4 + 6 + 2

sc sc

sc sc

2

2 and C srepresent the sliding gain

Proof Let the sliding variable with network delay from sensor to

controller τ scis given by:

l

where C sindicates the sliding gain,x k( −lτ sc)indicates the delayed state

vector The value of the sliding gain C sis calculated using LQ method

with proper selection of Q and R matrices

ApplyingZtransform to Eq.(10)we get:

l

wherelτ sc=τ h sc/

Using the Thiran approximation, the discrete time delay is

repre-sented as:

l l

⎝

⎞

⎠∏

k

τ i

τ k i z

τ

i

n sc sc

k

−

=0

=0

−

sc

(12)

where n indicates the order of approximation.

Considering n = 1 and taking thefirst order approximation we have,

l l

⎝

⎞

⎠

k

τ i

τ k i z

τ

i sc sc

k

−

=0

1

=0

1

−

sc

(13) The above equation can be further expanded as:

l l

l l l

l

l l

⎣

⎝

⎞

⎠

⎧

⎝

⎞

⎠

⎧

⎩

⎫

⎭

⎤

⎦

⎥

τ

τ

τ

τ

τ

= ( − 1) 1

0

2

1 1 2

sc sc sc sc

sc

sc sc

−1

sc

(14) Further solving we get,

l

whereα = llτ lτl

τ τ

8 + 8 + 1

4 + 6 + 2

sc sc

sc sc

2

Substituting Eq.(15)in Eq.(11)we get,

further solving,

Applying the inverseZtransform we have,

This completes the proof.□

According to Eq (18), the compensated sliding variable s kc( )

depends on the difference of the present and past state variables The

past state variable is multiplied with the parameter‘α’ approximated

through the Thiran approximation rule So, the delay in the sliding

surface at each sampling instant k is compensated by past state

variables multiplied over the parameter ‘α’ which is approximated

equal to the networked delay

Now, we are ready to propose the control law using the sliding

surface(9)

4 Design of the discrete time networked sliding mode

control under time delay compensation

In this section, the derivation of the discrete time sliding mode

control law along with its stability using the compensated sliding

surface(18)is represented in the form ofTheorem 1

Theorem 1 The discrete-time sliding surface(9)is reached within a

finite time in the presence of varying time delays (8)and matched

uncertainty(6)provided the control law is designed as:

u k( ) = − (2C G s ) [−1Hx k( ) −Ix k( ) −Js k c( ) −d k c( ) +d − 2C G d k s d ( )]

1

(19) where

H= 2C F s , I=αC s , J= [1 − ( ( ))]q s k c Proof The reaching law proposed in[13]is used to derive the control law since it provides faster convergence The reaching law is given by:

s k[( + 1)] = {1 − [ ( )]} ( ) − ( ) +q s k s k d k d1, (20) where

q s k

{ [ ( )]} =κ κ s k

+ | ( )|, d k( ) represents the disturbance, d =1 d+d

2

u l, mean

value of d k ( ), d =2 d u−2d l , deviated value of d k( ) andκ is the designer's constant satisfying:

The compensated reaching law considering the network delay is given by:

s c[( + 1)] = {1 − [ ( )]} ( ) −k q s k c s k c d k c( ) +d1, (22)

where s k c( ) represents the compensated sliding surface and d k c( ) represents the compensated disturbance which is given as:

Using Eq.(18), Eq.(22)can be rewritten as:

x k C αC x k q s k s k d k d

2 ( + 1) s− s( ) = [1 − ( ( ))] ( ) −c c( ) + 1, (24)

Substituting the value of x k( + 1),

C Fx k Gu k G d k αC x k q s k s k d k d

2 [s ( ) + ( ) + d ( )] − s ( ) = [1 − ( ( ))] ( ) −c c c( ) + 1,

(25) Further simplification gives,

C Fx k C Gu k G d k αC x k q s k s k d k

d

2 ( ) + 2 ( ( ) + ( )) − ( ) = [1 − ( ( ))] ( ) − ( )

Eq.(26)further can be expressed in terms of control law as:

u k( ) = − (2C G s ) [−1Hx k( ) −Ix k( ) −Js k c( ) −d k c( ) +d − 2C G d k s d ( )]

1

(27) where

H= 2C F s , I=αC s , J= [1 − ( ( ))]q s k c This completes the proof.□ The next step is to prove the stability such that any trajectory of the system(3)will be driven onto the sliding surface and maintained on it within afinite interval of time So, using the sliding surface(18)and control law(27), a stability condition is derived such that the system states shall remain within the band in the presence of varying network time delay

Stability: The trajectory of the closed loop system can be driven using the sliding surface infinite time with the controller designed in (27)under varying networked time delay(8)and matched uncertainty (6)such that for any κ≥d2and γ≻0 the following condition should hold

true:

M s k s k

0⪯ ≺ ( ) ( ).c T

Proof Let the sliding surface be given by:

Let the Lyapunov function be given by:

Taking the forward difference of the above equation

V k s k s k s k s k

Δ ( ) =s c( + 1) ( + 1) − ( ) ( ),

T

T

3

Substituting the value of s k c( + 1)from Eq.(18)we get,

V k C x k αC x k C x k αC x k s k s k

Δ ( ) = [2s s ( + 1) − s ( )] [2T s ( + 1) − s ( )] − c T( ) ( ),c

(32)

Substituting the value of x k( + 1) we get,

V k C Fx k Gu k G d k αC x k

C Fx k Gu k G d k αC x k s k s k

[2 [ ( ) + ( ) + ( )] − ( )] − ( ) ( ),

T

Substituting the value of u k( ) and further solving it we have,

V k q s k s k d k d

q s k s k d k d s k s k

Δ ( ) = [[1 − ( ( ))] ( ) − ( ) + ]

*[[1 − ( ( ))] ( ) − ( ) + ] − ( ) ( ),

T c

1

We see that the term [[1 − ( ( ))] ( ) −q s k c s k c d k c( ) +d1] contains the

disturbance term inΔ ( )V k s Let

M= [[1 − ( ( ))] ( ) −q s k c s k c d k c( ) +d1] *[[1 − ( ( ))] ( ) −T q s k c s k c d k c( ) +d1]

Then we have,

V k M s k s k

Δ ( ) =s − c( ) ( )

T

The term M is tuned to zero by appropriately selecting the parameterκ

IfMis closed to zero, thens k s k c T( ) ( )c will be larger thanM Thus, for any

constant parameterγ , we have M−s k s k c T( ) ( )≺ −c γs k s k c T( ) ( )c

Therefore, by tuning the parameterκ , we have, V kΔ ( )≺ −s γs k s k c( ) ( )

T c

which guarantees the convergence ofΔ ( )V k s

This completes the proof.□

The control signal u k( ), will also experience controller to actuator

delay τ(ca) which results in the delayed control signal u k( −τlca) To

avoid the degradation of the plant response, the time delay

compensa-tion is done from the controller to actuator Using the same approach

of the Thiran approximation as discussed in the earlier section the

compensated control signal is represented as:

where

β = τ τ

τ τ

8 + 8 + 1

4 + 6 + 2

ca ca

ca ca

2

2 andlτ ca=τ ca/h

It is noted from Eq.(36)that the control signal u k a( )depends on the

difference of the present and past control signals The past control

signal is multiplied over the parameter‘β’ approximated through the

Thiran approximation This compensated control signal is further

applied to the plant

5 Results and discussion

To prove the efficiency of the proposed method an example from

[14]is simulated in the MATLAB environment

Consider the continuous LTI system as,

⎡

⎣⎢

⎤

⎦⎥

⎡

⎣⎢

⎤

⎦⎥

−0.03

Discretizing the above system parameters at the sampling interval of

h = 30 ms:

⎡

⎣⎢

⎤

⎦⎥

⎡

⎣⎢

⎤

⎦⎥

−0.001771

−0.02934 , = [1 0], = [0]

Figs 2–11 show the nature of the system under a networked

environment In order to check the robustness of the derived control

law, a slow time varying disturbance is applied to the system shown in

Fig 2.Fig 3presents the time varying network induced delay with a

range of 3 ms ≤τ≤ 35 ms under which the system shows a stable

response satisfying(8) In this work, the networked delay is considered

as the time required for the data packets to travel from sensor to

controller and controller to actuator The amount of time required for

data packets to travel from sensor to controller is1.5 ms ≤τ sc≤ 17.5 ms

and for controller to actuator is 1.5 ms ≤τ ca≤ 17.5 ms respectively

Fig 2 Time varying disturbance d(k) versus time.

Fig 3 Total networked delay τ versus time.

Fig 4 x 1 versus time with initial condition x 1 =1.

Fig 5 x 2 versus time with initial condition x 1 =1.

Figs 4 and 5 show the plant variables with the initial condition

x k( ) = [11] Both the states converge to zero from the given initial

condition in the presence of network delay The sliding gain C s is

Q=diag(1000, 1000) and R = 1 The computed values of the optimal

sliding gain are C = [ − 1.77 − 2.766] s Fig 6shows the compensated sliding surface calculated using the Thiran approximation rule We observe that the compensated sliding variable is computed from the first sampling instant in the presence of the sensor to controller fractional delay.Fig 7shows the control signal u k( ) which is computed

using the proposed compensated sliding surface s k c( ) This control signal is further applied to the plant through the network The same approach of time delay compensation is used to compute the

compen-sated control signal u k a( ) These results are shown in Fig 8 Fig 9 shows the results of stability It is observed fromFig 9that for the

given κ = 10 and d = 0.22 that guarantees the convergence of V kΔ ( )s and implies that the trajectories of system(3)will be driven on the sliding surface and maintained on it under the specified network delay and matched uncertainty The algorithm is also examined for different SNRs as shown inFig 10 It is observed fromFig 11that the system states converge to zero for different SNRs Thus, from the above results

it is justified that the Thiran approximation provides better compensa-tion within the specified band

The control law derived using Thiran approximation is more robust than [8,11,12,19] because it generates less chattering even in the presence of network delay and matched uncertainty

6 Conclusion

In this paper, a new concept for compensating the network delay having fractional behavior in sliding surface was introduced The Thiran approximation technique is used to compensate the networked delay The sliding surface is designed in such a manner that it slides on the predetermined surface according to the network delay Using this novel approach, control law for discrete-time networked sliding mode

is designed to compute the control sequences in the presence of a variable time delay and matched uncertainty Stability is checked using

Fig 6 s c (k) versus sampling instants k.

Fig 7 u(k) versus time.

Fig 8 u a (k) versus time.

Fig 9 Result of stability.

Fig 10 Response of SNR.

Fig 11 Nature of state variables for different SNR.

5

the Lyapunov function such that the system states would remain within

that band in finite interval of time even in the presence of network

delay and matched uncertainty The simulation results show that the

Thiran approximation enhanced the response under network

non-idealities The proposed algorithm is valid for deterministic or time

varying network delays It can be valid for those applications whose

bandwidths arefixed and network delays are deterministic in nature In

the future, the work can be extended for single as well as multiple

packets drop out conditions The same approach can be extended for

wireless NCS having random types of communication delays as well as

with random dropout conditions

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Dipesh Shah born in India and obtained B.E Instrumentation and Control (2007) and M.E Applied Instrumentation (2010) from Gujarat University Ahmedabad.

Currently, he is working as an Assistant Professor at Sardar Vallabhbhai Patel Institute of Technology, Vasad, Gujarat, India.

He is pursuing Ph.D from Gujarat Technological University, Ahmedabad, Gujarat, India.

His research interest is robust controllers, sliding mode control, networked control system and communication networks.

Axay Mehta obtained B.E Electrical (1996), M.Tech (2002) and Ph.D (2009) degree from Gujarat University Ahmedabad, IIT Kharagpur and IIT Mumbai, respectively.

He has worked as various faculty positions at various Institutions including Associate Faculty at Indian Institute Technology, Gandhinagar.

He also acted as Professor; Director at Gujarat Power Engineering and Research Institute, Mehsana, Gujarat, India, during 2012–2014.

Currently, he is an Associate Professor at the Institute of Infrastructure Technology Research and Management, Ahmedabad, Gujarat.

His research interest is non-linear sliding mode control and observer, sliding mode control application in electrical engineering and networked control system.

He has published 35 research papers in peer reviewed international journals and conferences of repute.

He is Senior Member IEEE, Life Member of Institution of Engineers (India), Life Member of Indian Society for Technical Education and Member of Systems Society of India.

He is conferred the Best research paper award at NSC 2002 by Systems Society of India and Pedagogical Innovation award 2014 by GTU.