Sliding mode based-variable selective control for robust tracking of uncertain Internet-based systems

ISA Transactions xxx (xxxx) xxx

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Research article

Sliding mode based-variable selective control for robust tracking of uncertain Internet-based systems Behrooz Rahmani Control Research Lab., Department of Mechanical Engineering, Yasouj University, 75914-353 Yasouj, Iran

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Article history: Received 24 April 2018 Received in revised form 17 August 2019 Accepted 17 August 2019 Available online xxxx Keywords: Sliding mode control Networked control system Mismatched uncertainty Robust control Tracking

a b s t r a c t In this research, the robust control of uncertain linear Internet-based systems has been studied. In this way, an event-triggered strategy has been used to reduce the amount of communication between the plant and the remote controller. The proposed control strategy also ensures stability, robustness and output tracking against the model uncertainties and communication network negative effects. For this purpose, a new sliding surface is designed for linear Internet-based systems by solving a set of linear matrix inequalities which is achieved using the Lyapunov’s stability theorem. Accordingly, corresponding to a range of expected time delays, a discrete-time sliding mode control method is used to design some control signals. The proposed method can cope with relatively large network delays and packet dropouts. The asymptotic stability and robustness of the closed-loop system are also evaluated. Simulation studies on four well-known benchmark problems demonstrate the effectiveness of the proposed method with respect to the model uncertainty and also band-limited communication networks. © 2019 ISA. Published by Elsevier Ltd. All rights reserved.

1. Introduction With the quick development of communication network technologies (such as the Internet), their implementation in distributed control systems has dramatically increased. This results in the so-called Internet-based control systems (IBCSs) or networked control systems (NCSs). In such systems, different control system components, i.e., sensors, controllers and actuators, are spatially distributed. They are connected through a band-limited communication network, e.g., wired or wireless Internet. This configuration leads to a lower cost and weight, easier maintenance, higher reliability, and systems flexibility compared with the traditional point-to-point wiring closed-loop systems. Consequently, IBCSs have been found applications in many areas such as remote surgeries, network-based traffic control systems, robots, automobiles, unmanned aerial and underwater vehicles, sensor networks for environmental monitoring, industrial automation systems and Internet-based home automation systems [1–4]. But, the insertion of Internet into the closed-loop control system has resulted in some challenging and also interesting issues, such as packet dropout, time-varying time delay and packet disordering phenomenon [1,5]. Consequently, straight implementation of many traditional control strategies for IBCSs may seriously degrade the performance and also stability of the networked E-mail address: [email protected].

control system. For this purpose, recently, remarkable researches have been performed to compensate these negative effects. Most of the available results in the literature, employ a timetriggered control scheme, by considering a fixed sampling interval for different sensors. Zhang et al. [6] presented an observerbased method for proper output tracking of NCSs. Lu et al. [7] proposed a model predictive tracking control strategy for NCSs with packet loss and uncertainty. For this purpose, a new system model was constructed by combining state variables and tracking errors, and based on it, a controller was then designed. Halder et al. [8] extended a linear matrix inequality-based control method for a networked plant. Souza et al. [9] used a networked proportional–derivative–integral controller for second-order delayed processes. In this way, linear matrix inequalities (LMIs) are used to design the controller. Seitz et al. [10] proposed a method to stabilize a perturbed linear NCS robustly. In this way, they studied stability bounds on the uncertainties of the continuoustime system. Wu et al. [11] considered a linear discrete-time networked system with the quantization effect and discussed on the optimal tracking performance for the closed-loop system. Niu and Li [12] used an adaptive backstepping control strategy for a switched multi-input multi-output (MIMO) nonlinear system with input delay. Zhang et al. [13] considered a proportional– integral controller for an NCS and used a discontinuous augmented Lyapunov–Krasovskii functional to study the stability conditions. Yu et al. [14] studied an uncertain linear networked system with actuator saturation, stochastic packet dropout and unmeasurable state vector. In this way, a Bernoulli distributed

https://doi.org/10.1016/j.isatra.2019.08.030 0019-0578/© 2019 ISA. Published by Elsevier Ltd. All rights reserved.

Please cite this article as: B. Rahmani, Sliding mode based-variable selective control for robust tracking of uncertain Internet-based systems. ISA Transactions (2019), https://doi.org/10.1016/j.isatra.2019.08.030.

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Nomenclature

Rn Rm×n 0 I P >0 PT DT CT IBCS SMVSC MIMO QoS eLMIs TDM ZOH

τv x(t) y(t) u(t) ∆A(t), ∆B(t) xm ym

Θk ∆d Ai , ∆d Bi Φi ∆Φi Γ0 σk Ki , Ki′ dki dˆ ki vkeq vkrb α tk

τk τksc τkca τβv τ1v β Lp

β Lp Lc N Lc

n-dimensional Euclidean space m × n real matrix R Zero matrix of appropriate dimension Identity matrix of appropriate dimension Matrix P is symmetric and positive definite (PD) Transpose of matrix P Discrete-time Continuous-time Internet-based control system Sliding mode-based variable selective control Multiple-input multiple-output Quality of service Linear matrix inequalities Time delay manager Zero-order hold Delay vector State vector of the physical plant Output vector of the physical plant Input vector of the physical plant Time-varying norm-bounded system uncertainties State vector of the reference model output vector of the reference model Augmented state vector DT uncertainty matrices Augmented DT state matrix for delay τi Uncertainty matrix for Φi Augmented DT input matrix Switching sliding variable Sliding surface parameters Unknown term which represents model uncertainties Predicted value for dki Equivalent control term Robust control term Design parameter kth sampling instant Total control loop delay for kth sampling instant Sensor-to-controller time delay for kth sampling instant Controller-to-actuator time delay for kth sampling instant Upper-bound of network time delay Lower-bound of network time delay Size of the delay vector Maximum allowable packet length of the network Size of the delay vector Maximum allowable packet length of the network Data size required for encoding a control signal Consequent data packet dropouts Data size required for encoding a control signal

white sequence with a known conditional probability distribution was used to model the random packet dropout and an observerbased H∞ output feedback controller was then implemented. Zhang et al. [15] considered an uncertain networked system and used a Smith predictor to estimate its future states. Based on this estimation, a feedback control law was then designed. On the other hand, some researchers used event-triggered control scheme. In this way, Li et al. [16] considered an NCS with communication time delays and external disturbances and used L∞ control co-design strategy to improve the performance of the closed-loop dynamical system. Li and Chen [17] implemented a networked predictive control scheme to compensate external disturbances and communication constraints. For this purpose, a discrete-time sliding-mode control strategy was utilized. Gu et al. [18] used a dynamic output feedback controller for an NCS. For this purpose, the Lyapunov–Krasovskii functional method was implemented to design controller gains and also event-triggered sampling parameters. Yan et al. [19] proposed an H∞ output tracking control problem. They firstly suggested an adaptive method for adjusting network data transmission and then used the state-feedback controller for output tracking. Lian et al. [20] considered an NCS with packet disordering and used a Markov chain to predict the probability of the happening of this phenomenon. They then implemented sliding mode control to stabilize the resulting Markovian jump system. Li et al. [21] used the systems output to design an event-triggered scheme and then studied strategies to reduce the conservatism of the stability analysis. However, the stability of the closed-loop system is guaranteed by these researches, only for relatively small packet dropout rates and time delays. Recently, in order to solve these difficulties, variable selective control ( VSC ) method has been proposed by Rahmani and Markazi [1,22,23]. The general idea of the VSC method is based on the implementation of eventdriven sensors. However, the VSC suffers from three important drawbacks in real-time applications: 1. The accurate mathematical model of the physical plant is needed to design the controller. Unfortunately, industrial processes are complex and nonlinear, such that obtaining their accurate dynamical model using first principles of physics and also identification methods are challenging. Moreover, structural and parametric uncertainties commonly exist in practical systems; 2. A close distance between sensors and actuators is needed; i.e., they must be connected through point-to-point wiring communication technologies with negligible delay. Since, in IBCSs, the sensors and the actuators are spatially distributed, this assumption may not be acceptable; 3. The robust tracking and model following issues are not studied. Given these, new methods are still needed to improve the performance of the IBCSs with relatively large time delays, packet dropouts and uncertainties. Motivated by the above discussions, this research attempts to propose a sliding mode-based variable selective control (SMVSC ) for model following and robust tracking of uncertain linear MIMO systems with matched and mismatched uncertainties. In this way, when new control inputs are received by the actuators, the sensors are triggered to measure the outputs of the plant. Accordingly, time delay of the control loop will be equal to the sampling period, at each sampling instant. Based on this triggering scheme, the IBCS is modeled as a switched linear uncertain system. Consequently, a set of LMIs is solved to design a new sliding surface for such an uncertain linear system. Control inputs are then designed such that the acceptable tracking

Please cite this article as: B. Rahmani, Sliding mode based-variable selective control for robust tracking of uncertain Internet-based systems. ISA Transactions (2019), https://doi.org/10.1016/j.isatra.2019.08.030.

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performance is achieved, in the presence of parameter uncertainties, external disturbances, time delays, packet losses, and also packet disorders. For this purpose, by presenting four theorems, stability, robustness and acceptable tracking performance of such dynamical systems are evaluated. Numerical simulations of wellknown benchmark problems demonstrate the effectiveness of the proposed algorithm. The main contributions of this research can be summarized as follows: 1. A new framework for model following and robust tracking of linear systems with time delay, jitter, packet loss, packet disordering and parameter uncertainty is established. 2. A new sliding surface is designed for uncertain linear IBCSs by solving a set of LMIs. Accordingly, using SMVSC, the robustness against unknown matched and mismatched parameter uncertainties are guaranteed. 3. Using SMVSC, sensors, controllers and actuators can be connected through local area networks with considerable time delays. 2. Problem description Here, a MIMO, uncertain, linear, dynamical system described by the following state-space model is considered as the physical plant for the closed-loop control system shown in Fig. 1,

n

m

(1) l

In this model, x(t) ∈ R , u(t) ∈ R , y(t) ∈ R , denote state, input and output vectors, respectively. A ∈ Rn×n , B ∈ Rn×m and C ∈ Rl×n are matrices associated to the state-space model. On the other hand, ∆A(t) and ∆B(t) represent norm-bounded system uncertainties. By considering F (t) as a time-varying matrix with Lebesgue-measurable elements which satisfies F T (t)F (t) ≤ I, Gc , EA , and EB as known constant matrices of appropriate dimensions, the uncertainty matrices can be written as follows:

[∆A(t) ∆B(t)] = Gc F (t)[EA EB ].

(2)

In the following, tk , τ and τ represent kth time instant, sensor-to-controller and controller-to-actuator time delays of kth packet, respectively. Consider Fig. 1. Evidently, in such an Internet-based control system, based on the plant data, the control input signals are generated in a discrete-time (DT ) controller and transmitted back to the plant through the wired or wireless Internet. sc k

delay occurs during transmission of each control packet. In the light of the proposed scheme, the sampled-data model of this Internet-based closed-loop system can be written as x˙ (t) = [A + ∆A(t)]x(t) + [B + ∆B(t)]u(tk − τk−1 ), tk ≤ t ≤ tk+1 , y(t) = Cx(t). (3) Consequently, DT, ZOH-equivalent, state-space model of Eq. (3), for a time delay τk = Tk = τi , can be described by xk+1 = (Ai + ∆d Ai )xk + (Bi + ∆d Bi )uk−1 , yk = Cxk , where, Ai = eAτi , Bi =

ca k

Remark 1. For the sake of simplicity of presentation, Suppose firstly that, packet dropout is temporarily ignored. In Remark 3, a method is proposed to overcome its negative effects. The plants states are sampled using an adaptive method [1, 22,23]. According to Figure 3 of [1], assume that at time instant tk , new control input signal uk−1 is received by the actuator. The event-driven sensors are now triggered to measure the states xk of the plant. Based on these, the kth plant-side packet Xk = [xk ; uk−1 ; tk ] will be constructed and transmitted to the controller. Suppose this packet is received at time tk + τksc by the controller. As a result, associated with some values for possible time delays, control input signals are designed and the specified control-side packet Uk is constructed and sent back to the plant-side through Internet. If this packet arrives at time tk + τksc + τkca , the total control loop delay will be τk = τksc + τkca . In the light of the implemented event triggering scheme for sensors and controllers, it can be easily shown that k + 1th sampling instant will be such that Tk = tk+1 − tk = τk . In other words, only one step time

∫ tk +τi tk

(4)

eA(τi −λ) dλB. In this formula, ∆d Ai

and ∆ Bi are constant matrices with appropriate dimensions. For i = 1, 2, . . . , β , they denote the corresponding DT, normbounded, uncertainty and are supposed to be estimated by the information of the norm-bounded uncertainties in Eq. (2) (see Remark 2.2 of [23]). On the other hand, xk+1 = x(tk+1 ), xk = x(tk ) and uk−1 = u(tk−1 ) = u(tk − Tk−1 ) = u(tk − τk−1 ). By defining a ]T [ , the state-space model (4) new state vector Θk = xTk uTk−1 is described in the following augmented form d

Θk+1 =

[

xk+1 uk

[ +

x˙ (t) = [A + ∆A(t)]x(t) + [B + ∆B(t)]u(t), y(t) = Cx(t).

3

yk

]

Ai + ∆d Ai 0m×n

[ =

]

0n,m Im,m

][

xk uk−1

uk ≡ (Φi + ∆Φi )Θk + Γ0 uk ,

[

=

Bi + ∆d Bi 0m×m

C 0l×m

]

] (5)

Θk ,

where,

Φi =

[

∆Φi =

Ai

Bi

0m×n

0m×m

[

∆d Ai ∆d Bi 0m×n 0m×m

] ]

, Γ0 =

[

0n,m Im,m

] and

.

But, at tk , the time delay τk , is random. As a result, it may not be accurately predicted beforehand by the controller. Therefore, it is not an easy task to design a stabilizing control signal uk for the system with governing Eq. (3). As shown in Fig. 1, in order to solve this problem, here, a time delay manager (TDM) is used in the plant-side. For this purpose, according to Remark 7, τβv = τmax is selected as the upper-bound for time delay. Based on this, a delay vector with size β over the range [0, τβv ] is constructed in TDM and controller as τ v = [ τ1v τ2v · · · τβv ]. Suppose that at time tk′ , the controller receives a plant-side packet Xk = [xk ; uk−1 ; tk ]. After that, according to Remark 2, the controllers buffer is updated. By the assumption of the newness of this packet, a control-side packet Uk = [uk1 , uk2 , . . . , ukβ ; tk ] is designed and sent back to the plant-side. Here, for i = 1, 2, . . . , β , uki denotes the control input associated with time delay τiv (see Section 3). Preceding the plant, the TDM receives this packet. If at time tk′′ , the plant-side receives a control-side packet Uk = [uk1 , uk2 , . . . , ukβ ; tk ], the procedure illustrated by Remark 2 is utilized to update the TDM buffer. In this way, when a latest data packet is received by TDM, the time difference τe = tk′′ − tk is evaluated which represents the Internet-induced time delay. Based on this, the control-side packet is used to select the proper control signal. However, usually, elements of τ v are not exactly equal to τe . To solve this difficulty, an index ks ∈ {1, 2, . . . , β} is selected such that τkvs −1 < τe ≤ τkvs and τ0v = 0. It then waits for τkvs −τe s, and consequently, the time delay of the control loop will be τkvs . The appropriate control input signals uk = ukks are now selected from the packet of the TDM buffer and implemented to the actuators through a zero-order hold (ZOH) element. An event detector module then sends a triggering signal to sensors through a local area network to transmit a new plant-side packet.

Please cite this article as: B. Rahmani, Sliding mode based-variable selective control for robust tracking of uncertain Internet-based systems. ISA Transactions (2019), https://doi.org/10.1016/j.isatra.2019.08.030.

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Fig. 1. Sliding-mode based variable selective control structure.

As a result, at different time instants, τk and then Tk are switched between elements of τ v . The Internet-based control system (5) is therefore described by the following switching linear system,

Θk+1 = yk =

∑β

i=1

θi (k)

(

(Φi + ∆Φi )Θk + Γ0 uk C1 Θk ,

)

,

(6)

C 0l×m and θ (k) = θ1 (k) θ2 (k) . . . θβ (k) where, C1 = demonstrates that at the time instant tk which switching model is activated. It is shown with the only entry of θ (k) which is unity. Here, the aim is to propose a sliding mode-based control strategy which can enable an uncertain linear switched system described by Eq. (6) for tracking output of model described by (7) for the whole range of Internet-induced time delay which is smaller than τβv ,

[

]

x˙ m (t) = Am xm (t), ym (t) = Cm xm (t).

{

}

(7)

In this formula, models state and output vector are denoted by xm (t) ∈ Rnm and ym (t) ∈ Rl , respectively, Am ∈ Rnm ×nm and v Cm ∈ Rl×nm . By assuming Aimd = eAm τi and sampling period as v v τi ∈ τ , DT, ZOH-equivalent, state-space model of Eq. (7) can be described by xmk+1 = Aimd xmk , ymk = Cm xmk ,

(8)

for each i = 1, 2, . . . , β . According to the implemented sampling scheme, this model can be considered as the following switching linear system, xmk+1 = ymk =

∑β

i=1

θi (k)Aimd xmk , Cm xmk .

(9)

Assumptions of this research can be summarized as follows: (1) The plant is controllable; (2) State vector can be measured; (3) Maximum values for the number of consecutive packet dropouts and Internet-induced time delays are known.

Remark 2. Assume that in the controller and the TDM, two different buffers are allocated, named by Bc and BT , respectively. According to Section 2, Xk1 := [xk1 ; uk1 −1 ; tk1 ] can be the content of Bc at time instant t1 . It is the plant-side packet constructed at time tk1 . Suppose a new control-side packet Xk2 := [xk2 ; uk2 −1 ; tk2 ] sent at time tk2 is received by the controller at t1 . In this way, firstly, tk1 is compared with tk2 . If the condition tk2 > tk1 is satisfied, Xk2 will be allocated in Bc , and by Xk1 is deleted. Otherwise, if such a condition is not satisfied, Xk2 is ignored. A similar process is performed in BT , to select new control-side packets. Remark 3. In order to solve the negative effects of packet dropout, network data senders are considered in every controller and sensor node. In this way, each new plant-side or control-side packet is transmitted N times. Here, it is assumed the number of consecutive packet dropouts is less than N [1]. 3. Main idea As described in Section 2, the mathematical model of the IBCS with the proposed sampling strategy (see Fig. 1) can be represented as a DT switched system described by Eq. (6). In this research, a SMVSC is used to guarantee that output yk of the system described by Eq. (6) follows the output ymk of the model of Eq. (7) with a small tracking error. In this way, a perquisite is the existence of matrices Gi ∈ R(n+m)×nm and Hi ∈ Rm×nm satisfying the following matrix equation, for i = 1, 2, . . . , β and each k, β ∑

θi (k)

{[

i=1

Φi Γ0

][

C1 0l×m

Gi Hi

]

[ =

Gi Aimd Cm

]}

.

(10)

Since by the proposed method, at each time step k, the switch between different subsystems happens randomly, the following matrix equation must be solved for i = 1, 2, . . . , β ,

[

Φi Γ0 C1 0l×m

][

Gi Hi

]

[ =

Gi Aimd Cm

]

.

(11)

To solve this equation, the procedure proposed in [24] is used. A different output matrix C or reference model must be selected,

Please cite this article as: B. Rahmani, Sliding mode based-variable selective control for robust tracking of uncertain Internet-based systems. ISA Transactions (2019), https://doi.org/10.1016/j.isatra.2019.08.030.

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if Eq. (10) does not have a solution [25]. After that, for reaching an acceptable tracking and model following performance, a DT sliding mode controller is used. For this purpose, a novel sliding surface is firstly defined. The well-known Lyapunov theory for mismatched uncertain systems is then implemented to design this surface. σk is here considered as the sliding variable and σk = 0 is assumed to be the sliding surface for the Internet-based control system. Remark 4. Continuous-time ( CT ) and sampled-data closed-loop systems have different sliding-mode characteristics. The states of a CT control system can easily be kept on the sliding surface. But, keeping the states of an uncertain sampled-data system completely on σk = 0 is with difficulties. In such cases, states remain within the specified band with respect to the sliding surface. This case is named by the quasi-sliding mode. It occurs because that by considering the ZOH actuators in Internet-based control systems, DT control signals are generated only at certain sampling instants and are held constant during the entire sampling period [26].

σk is chosen as a switching sliding variable which is defined as follows: σk = SZk − εk ,

(12)

∑β

εk = i=1 θi (k − 1)[εk−1 + S(Φi + Γ0 Ki )Zk−1 + S Γ0 Ki′ εk−1 ], ε0 = 0 , (13) gains Ki ∈ Rm×(n+m) and Ki′ ∈ Rm×m are the sliding surface parameters to be designed later and S ∈ Rm×(n+m) is selected such that S Γ0 is not singular. Here, Zk is an auxiliary variable defined as β ∑

θi (k)Gi xmk ≡ Θk − Gxmk ,

Using Eqs. (6), (9), (10), (14) and (15), a forward expression for the state vector of the auxiliary systems is written as Zk+1 = Θk+1 −

(14)

uk =

θi (k)Hi xmk + vk =

i=1

i=1

θi (k)(Hi xmk + vki ) =

θi (k + 1)Gi xmk+1

∑β = Θk+1 − i=1 θi (k + 1)Gi xmk+1 + i=1 θi (k)Gi xmk+1 ∑β − i=1 θi (k)Gi xmk+1 ∑β = Θk+1 − i=1 θi (k)Gi xmk+1 + µkm ∑β ∑β = i=1 θi (k)(Φi + ∆Φi )Θk + i=1 θi (k)Γ0 uki ∑β ∑β − i=1 j=1 θi (k)θj (k)Gi Ajmd xmk + µkm ∑β ∑β = i=1 j=1 θi (k)θj (k)[(Φi + ∆Φi )Θk + Γ0 Hi xmk

β ∑

(12), (13) and (18):

εk = SZk .

(20)

In this surface, it is also assumed that vki = v Eq. (20), Eq. (19) can be rewritten as

σk+1 =

∑β

i=1

eq ki ;

θi (k)uki ,

θi (k)[S ∆Φi Θk + S Γ0 vkeqi − SZk

(21) eq

According to Remark 5, equivalent control term vk will be dei termined using the fact that σk+1 = 0 at the sliding surface; eq therefore, vk can be estimated as follows, using Eq. (21):

v

i=1

Remark 5. In order to stabilize the quasi-sliding mode, the eq control input term vk = vk + vkrb is used, here. In this way, the eq following equivalent control term vk will be determined such that σk+1 = 0 at the sliding surface σk = 0, assuming a certain dynamical model for the plant,

v

=

eq ki

= (S Γ0 )−1 (S Γ0 Ki Zk + S Γ0 Ki′ εk + SZk − S ∆Φi Θk − S µkm ) = [Ki + (S Γ0 )−1 S ]Zk + Ki′ εk − (S Γ0 )−1 dki .

eq ki

θi (k)v .

(16)

For i = 1, 2, β , in the sequel, a strategy will be proposed to determine vkrb . In order to compensate unknown dynamics, i parameter uncertainties, and also external disturbances, a robust control law vkrb is designed. In this regard, the quasi-sliding mode must be reached and maintained thereafter. For this purpose, vkrb is considered as follows: β ∑

(22) Here, dki = S ∆Φi Θk + S µkm represents model uncertainties and therefore is an unknown term. But, as mentioned in Remark 5, a eq certain dynamical model must be used to calculate vk . As a result, i eq vk is determined as i

vkeqi = [Ki + (S Γ0 )−1 S ]Zk + Ki′ εk .

(23)

As the next step, a robust control term v is calculated such that in the presence of model uncertainties and external disturbances, quasi-sliding mode is reached and maintained thereafter. A necessary and sufficient condition for this statement can be written as follows [27]: rb ki

i=1

vkrb =

therefore, using

i

where, uki = Hi xmk + vki .

β ∑

(19)

In the sliding surface, σk = 0 is defined; therefore, Eq. (12) leads to

(15)

eq k

(18)

+ Γ0 vki − Φj Gj xmk − Γ0 Hj xmk + µkm ] ∑β = i=1 θi (k)[(Φi + ∆Φi )Θk + Γ0 Hi xmk + Γ0 vki − Φi Gi xmk − Γ0 Hi xmk + µkm ] ∑β = i=1 θi (k)(Φi Zk + ∆Φi Θk + Γ0 vki + µkm ), ∑β ∑β where, µkm = i=1 θi (k)Gi xmk+1 − i=1 θi (k + 1)Gi xmk+1 . A forward expression for σk is written as follows, using Eqs.

− S Γ0 Ki Zk − S Γ0 Ki′ εk + S µkm ].

and G = i=1 θi (k)Gi . In order to have an acceptable tracking and model following performance, the following control input is now considered, β ∑

i=1

∑β

i=1

∑β

β ∑

∑β

σk+1 = SZk+1 − εk+1 ∑β = i=1 θi (k)[S Φi Zk + S ∆Φi Θk + S Γ0 vki − εk − S Φi Zk − S Γ0 Ki Zk − S Γ0 Ki′ εk + S µkm ].

where,

Zk = Θk −

5

|σk+1 | < |σk |.

(24)

The condition (24) is attained if a design parameter 0 < α < 1 exists such that

σk+1 = ασk .

θi (k)vkrbi ,

(17)

i=1

where, for i = 1, 2, β , vkrb will be determined in the sequel. i

In this regard, v

(25) rb ki

is designed as follows:

vkrbi = (S Γ0 )−1 ασk − (S Γ0 )−1 σk − (S Γ0 )−1 dˆ ki .

(26)

Please cite this article as: B. Rahmani, Sliding mode based-variable selective control for robust tracking of uncertain Internet-based systems. ISA Transactions (2019), https://doi.org/10.1016/j.isatra.2019.08.030.

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Accordingly, using Eqs. (23) and (26), the control law of Eq. (15) is then modified as uk =

∑β

=

∑β

i=1

θi (k)uki =

∑β

i=1

θi (k)(Hi xmk + vkeqi + vkrbi )

θi (k)[Hi xmk + (Ki + (S Γ0 )−1 S)Zk + Ki′ εk − (S Γ0 )−1 dˆ k + (S Γ0 )−1 ασk − (S Γ0 )−1 σk ]. i=1

(27)

−1

Kia = Mis (X s )

i

Here, dˆ ki denotes the predicted value for the unknown term dki , which can be estimated through the following equation, dˆ ki = dˆ k−1i + (σk − ασk−1 ).

where, G′d and EΦi are known constant matrices with appropriate i dimensions. Fd′ is a time dependent matrix which has Lebesguei measurable elements satisfying Fd′T Fd′ ≤ I. In this way, the gain i i Kia = [Ki Ki′ ] is designed as follows:

(28)

As mentioned in Remark 5, v tries to draw the trajectories of the eq states towards the vicinity of σk = 0, since vk tries to maintain i it at the neighborhood of σk = 0. The problem is now to design the sliding surface parameters, i.e. Ki and Ki′ , for i = 1, 2, . . . , β . This causes the trajectories of the auxiliary system slide along the vicinity of σk = 0 towards the direction of the origin. In this way, it is shown that the implementation of the control input of Eq. (27) results in four important properties: (a) the auxiliary system described by Eq. (18) will be bounded-input bounded-output ( BIBO) stable (see Theorem 1); (b) the quasi-sliding mode will be reached and maintained thereafter (see Theorem 2); (c) a tracking error will be ultimately bounded and an acceptable tracking performance will be achieved (see Theorem 3); (d) the Internet-based system with auxiliary dynamics of Eq. (18) will be robustly stable in the quasi-sliding mode (see Theorem 4).

.

(33)

Proof. Using Eqs. (18) and (27), a forward expression for auxiliary state vector Zk can be described by Zk+1 =

rb ki

∑β

i=1

θi (k)[Φi Zk + ∆Φi Θk + µkm + Γ0 vki ]

∑β

θi (k)[Φi Zk + ∆Φi Θk + Γ0 Ki Zk + Γ0 Ki′ εk + Γ0 (S Γ0 )−1 SZk − Γ0 (S Γ0 )−1 dˆ ki + Γ0 (S Γ0 )−1 ασk − Γ0 (S Γ0 )−1 σk + µkm ] ∑β = i=1 θi (k)[(Φi + Γ0 (S Γ0 )−1 S)Zk + ∆Φi Zk + ∆Φi Gi xm + Γ0 Ki Zk + Γ0 K ′ εk − Γ0 (S Γ0 )−1 dˆ k =

i=1

i

k

i

+ Γ0 (S Γ0 )−1 (α − 1)SZk − Γ0 (S Γ0 )−1 (α − 1)εk + µkm ]. (34) The errors for estimation of S ∆Φi Θk and S µkm are assumed to be e S ∆Φie Θk and S ∆Φm xmk , respectively. It can then be shown that e dˆ ki = S(∆Φi Θk + ∆Φie Θk ) + S(µkm + ∆Φm xmk ) 1 1 ≡ S ∆Φi Θk + S ∆Φm xmk .

(35)

Consequently, Eq. (34) can be rewritten as follows: Remark 6. Michaletzky and Gerencser [28] showed that a DT linear switched system is BIBO stable, if and only if it is exponentially stable. On the other hand, Liu and Zhao showed that such systems are exponentially stable if they are asymptotically stable (See Lemma 1 of [29]). Therefore, for the DT linear switched systems, the asymptotic stability guarantees BIBO stability. Lemma 1 ([30]). Given a symmetric constant matrix S, constant matrices D and E with appropriate dimensions, the following inequality holds: S + DFE + E T F T DT < 0,

S+

[

γ −1 E T

γD

]

[

R 0

0 I

][

γ E γ DT −1

]

< 0.

Theorem 1. With the implementation of the control input of Eq. (27), the closed-loop, uncertain, auxiliary system described by Eq. (18) is BIBO stable, if there exist matrices X s > 0, Mis and some scalar γi , such that

⎡

−X s ⎢ Φi′ X s + Γ0′ Mis ⎢ ⎣ EΦi X s 0

⋆ −X s 0 γi2 G′ Tdi

⋆ ⋆ −γi2 I

⎤ ⋆ ⋆ ⎥ ⎥ ⋆ ⎦ < 0, 2 −γi I

0

(29)

(30) and the uncertainty matrix is as follows:

∆Φi =

[

∆Φi − Γ0 (S Γ0 )−1 S ∆Φi1 0 0

0

]

.

(31)

It can be described by

∆Φi′ = G′di Fd′ i EΦi ,

According to Eq. (13), a forward expression for εk can be written as

∑β

i=1

θi (k)[εk + S Φi Zk + S Γ0 Ki Zk + S Γ0 Ki′ εk ].

(32)

(37)

Now, by considering a new augmented state vector of the form Zka = [ZkT , εkT ]T , the closed-loop dynamics for such an auxiliary system is illustrated as given in Box I, where, wk′ = ∑β ∑β a a ′ a i=1 θi (k)Ki Zk and i=1 θi (k)[Ki Ki ]Zk ≡ dmk =

β { ∑

θi (k)

[

∆Φi Gi − Γ0 (S Γ0 )−1 S(∆Φi1 Gi + ∆Φm1 )

− [θi (k + 1) − θi (k)]

]

0

i=1

for i = 1, 2, . . . , β . Here, Γ0′ = [Γ0 ; S Γ0 ], [ ] Φi + Γ0 (S Γ0 )−1 S + Γ0 (S Γ0 )−1 (α − 1)S −Γ0 (S Γ0 )−1 (α − 1) Φi′ = , S Φi I

′

i=1

(36)

εk+1 =

where F satisfies F T F ≤ R, if and only if for some γ > 0

∑β

θi (k)[(Φi + Γ0 (S Γ0 )−1 S)Zk + ∆Φi Zk + ∆Φi Gi xmk + Γ0 Ki Zk + Γ0 Ki′ εk − Γ0 (S Γ0 )−1 S ∆Φi1 Zk − Γ0 (S Γ0 )−1 S ∆Φi1 Gi xmk − Γ0 (S Γ0 )−1 S ∆Φm1 xmk + Γ0 (S Γ0 )−1 (α − 1)SZk − Γ0 (S Γ0 )−1 (α − 1)εk ] ∑β − i=1 [θi (k + 1) − θi (k)]Gi Aimd xmk .

Zk+1 =

[

Gi Aimd 0

]}

xmk .

(39)

As demonstrated in Eq. (31), S , ∆Φi , Γ0 , Kia and α affect ∆Φi′ ; therefore, ∆Φi′ is assumed to be a matrix with norm-bounded uncertainty, for i = 1, 2, . . . , β . Accordingly, it is illustrated by Eq. (32). The term dmk can be considered as a bounded disturbance input. This is due the fact that ∥xmk ∥ ≤ ρ for some positive constant ρ and each sampling instant k. As illustrated in Remark 6, the asymptotic stability of the auxiliary system described by Eq. (38) without term dmk guarantees its BIBO stability. In this way, dmk is assumed to be zero for k = 0, 1, . . . , ∞, for the purpose of asymptotic stability analysis. As a result, the closed-loop dynamics for the new auxiliary system is described by Zka+1 =

∑β

i=1

θi (k)[(Φi′ + ∆Φi′ )Zka + Γ0′ w ′ k ].

(40)

Please cite this article as: B. Rahmani, Sliding mode based-variable selective control for robust tracking of uncertain Internet-based systems. ISA Transactions (2019), https://doi.org/10.1016/j.isatra.2019.08.030.

B. Rahmani / ISA Transactions xxx (xxxx) xxx

Zka+1 =

∑β

i=1 θi (k)

∑β

7

Φi + Γ0 (S Γ0 )−1 S + Γ0 (S Γ0 )−1 (α − 1)S + Γ0 Ki Γ0 Ki′ − Γ0 (S Γ0 )−1 (α − 1) S Φi + S Γ0 Ki S Γ0 Ki′ + I [ ] ∆Φi − Γ0 (S Γ0 )−1 S ∆Φi1 0 a

[

]

Zka

θi (k) Zk 0 0 { [ ] [ ]} ∑β Gi Aimd ∆Φi Gi − Γ0 (S Γ0 )−1 S(∆Φi1 Gi + ∆Φm1 ) + i=1 θi (k) − [θi (k + 1) − θi (k)] xmk +

i=1

0

≡

(38)

0

∑β

′ a ′ a ′ ′ i=1 θi (k)[Φi Zk + ∆Φi Zk + Γ0 w k ] + dmk ,

Box I.

Designing a switched state-feedback controller for system described by Eq. (40) is now studied. In this regard, the following controller is considered,

wk′ =

∑β

i=1

θi (k)Kia Zka ,

(41)

where, = 1, 2, . . . , β . Suppose V (k) = function. The asymptotically stability of the system with Eq. (40) and control input (41) is guaranteed as long as ∆V (k) < 0, for all k = 0, 1, . . . , ∞, Kia is the stabilizing gain, for i T Zka PZka as a Lyapunov candidate

T

T

∆V (k) = V (k + 1) − V (k) = Zka+1 PZka+1 − Zka PZka ∑β T = i=1 θi (k)Zka (Φi′ + ∆Φi′ + Γ0′ Kia )T P T

× (Φi′ + ∆Φi′ + Γ0′ Kia )Zka − Zka PZka ∑β T [ = i=1 θi (k)Zka (Φi′ + ∆Φi′ + Γ0′ Kia )T ] × P(Φi′ + ∆Φi′ + Γ0′ Kia ) − P Zka .

(42)

The last equality in Eq. (42) is negative definite, when the following inequality is satisfied, for i = 1, 2, . . . , β , (Φi′ + ∆Φi′ + Γ0′ Kia )T P(Φi′ + ∆Φi′ + Γ0′ Kia ) − P < 0.

(43)

Since ∆Φi′ is a norm-bounded uncertainty of the form of Eq. (32), by the implementation of the Schur complement, LMI (43) is rewritten as

−P ⋆ Φi′ + ∆Φi′ + Γ0′ Kia −P −1 ] ] [ [ −P ⋆ 0 ∆Φi′ T + = Φ ′ + Γ0′ Kia −P −1 ∆Φi′ 0 [ i ] [ ] ] [ 0 −P ⋆ EΦi 0 = + G′ Fd′ ′ ′ a −1 i Φ +Γ K −P di [ i T ]0 i [ ] EΦi T + Fd′T 0 G′ di < 0. i

]

[

(44)

the auxiliary system of Eq. (38) with control input (41) will be asymptotically stable. dmk is a bounded disturbance input, while it is nonzero and bounded. Based on Remark 6, the closed loop auxiliary systems described by Eq. (38) and therefore Eq. (18) are BIBO stable. This completes the proof. As it will be proved in the following theorem, the control law of Eq. (27) will steer the systems trajectories to arrive at the quasi-sliding mode and to maintain thereafter. Theorem 2. Consider the auxiliary system with the state equation of Eq. (18). Assume that with the sliding surface defined by Eq. (12), the control law of Eq. (27) and the uncertainty estimator of Eq. (28), there exist matrices Mis , X s > 0 and scalar γi satisfying the LMI (29), for i = 1, 2, . . . , β . The state vector Zk will be then driven close to the domain of the quasi-sliding mode band. In this way, such a band can be established as limk→∞ σk <

−P ⋆ + Φi + Γ0′ Kia −P −1 [ ] EΦi 0 × < 0. ′T

[

′

0

T

EΦi 0

0 G′d

][

γi I −2

0

i

0

|d(k+1)i − dki | < δ.

Proof. To check validity of Eq. (25), σk+1 is calculated firstly, by the combination of Eqs. (12), (18) and (27), as follows:

σk+1 = SZk+1 − εk+1 ∑β = i=1 θi (k)[S Φi Zk + S ∆Φi Θk + S Γ0 vki + S µkm ] − εk+1 ∑β = i=1 θi (k)[S Φi Zk + dki + S Γ0 Ki Zk + S Γ0 Ki′ εk + SZk − dˆ k + ασk − σk

i=1

γ

2 i I

0

⋆ −P −1 0 T G′ di

0

⋆ ⎥ ⋆ ⎥ < 0. ⎦ ⋆ −γi−2 I

i

As a result, it can be shown that

∑β

i=1

θi (k − 1)[d(k−1)i − dˆ (k−1)i + ασk−1 ].

(50)

Substituting Eq. (28) into Eq. (50), it can be shown that (45)

⋆ ⋆ −γi2 I

i

(49)

σk =

]

G di

−P ⎢ Φi′ + Γ0′ Kia ⎢ ⎣ EΦi

(48)

i

Using Schur complement for LMI (45), it can be shown that

⎡

(47)

− εk − S Φi Zk − S Γ0 Ki Zk − S Γ0 Ki′ εk ] ∑β = θi (k)[dk − dˆ k + ασk ].

According to the assumption of Fd′T Fd′ ≤ I and Lemma 1, LMI (44) i i holds if and only if some γi > 0 exists such that

]

.

Here, the maximum change of dki in one sampling period is denoted by δ ; i.e.,

0

[

δ 1−α

⎤

(46)

By pre- and post-multiplying of both sides of LMI (46) with diag {P −1 , I , I , γi2 I }, and defining Kia = Mis P and X s = P −1 , LMI (29) is achieved. As a result, if the term dmk is ignored,

dˆ ki = d(k−1)i .

(51)

By the assumption of arbitrary bounded values for dˆ 0i and d0i , using Eqs. (50) and (51), it is straightforward to show that ∑β ˆ σ1 = i=1 θi (0)(d0i − d0i + ασ0 ), ∑β σ2 = i=1 θi (1)(d1i − d0i + ασ1 ), ∑β ∑β σ3 = θ (2) [ d − d1i + j=1 θj (1)α (d1j − d0j ) + α 2 σ1 ], 2i i =1 i ∑k−2 ∑β σk = j=0 ij =1 θij (k − j − 1)α j [d(k−2−j+1)ij − d(k−2−j)ij ] + α k−1 σ1 . (52)

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8

B. Rahmani / ISA Transactions xxx (xxxx) xxx

Since there are matrices X s > 0 and Mis and scalers γi satisfying the LMI (29), gains Ki and Ki′ can be found using Theorem 1 to guarantee BIBO stability of the system illustrated by Eq. (38), for i = 1, 2, . . . , β . On the other hand, dki , is a function of Zk and xmk . Therefore, dki is bounded and the inequality (48) is satisfied for different values of k. Consequently, |σk | is bounded as follows:

|σk | < =

∑k−2 ∑β

ij =1

j=0

∑k−2 j=0

θij (k − j − 1)α j δ + α k−1 σ1

α j δ + α k−1 σ1 .

(53)

Since 0 < α < 1, it can be proved that limk→∞ σk <

δ 1−α

.

(54)

Therefore, the control law of Eq. (27) drives the state vector δ Zk arbitrarily close to the quasi-sliding mode band 1−α . This completes the proof. Theorem 3. The problem of tracking and model following for the Internet-based control system described by Eq. (3) is studied here. If with the sliding surface defined as Eq. (12), the control law of Eq. (27) and the uncertainty estimator of Eq. (28), there exist matrices X s > 0, Mis and scalar γi satisfying LMI (29), for i = 1, 2, . . . , β , then the tracking error ek is ultimately bounded and the desirable tracking performance will be achieved. Proof. From Theorems 1 and 2, it can be seen that the quasisliding mode of auxiliary system described by Eq. (18) with the control law of Eq. (27) can be achieved. It is also demonstrated that the closed-loop auxiliary system with Eq. (38) in the quasi-sliding mode is stable, and the auxiliary state vector Zk is ultimately bounded. On the other hand, from Eq. (10), it can be shown that C1 G = Cm . Consequently, the relationship between tracking error ek and the auxiliary state Zk is described by ek = yk − ymk = C1 Θk − Cm xmk = C1 (Θk − Gxmk ) = C1 Zk .

(55)

As a result, Eq. (55) yields that

∥ek ∥ ≤ ∥C1 ∥ ∥Zk ∥.

(56)

In the following theorem, the asymptotic stability and robustness of the Internet-based system with auxiliary dynamics of Eq. (38) in the quasi-sliding mode are evaluated. Theorem 4. Consider the Internet-based control system described by Eq. (38) which is in the quasi-sliding mode. If with the sliding surface of Eq. (12), the control law of Eq. (27) and the uncertainty estimator of Eq. (28), there exist matrices Mis , X s > 0 and scalar γi which satisfy LMI (29), for i = 1, 2, . . . , β , the closed-loop system in the quasi-sliding mode is stable, and Zka is bounded as lim ∥Zka ∥ ≤ ξ ,

(57)

ν , and λA is the maximum norm for where, ξ = ∥T ∥ ∥T −1 ∥ 1−λ A all eigenvalues of the system matrices of the closed-loop system described by Eq. (38) without dmk , for i = 1, 2, . . . , β . Here, T is a similarity transformation matrix and ν = max ∥dmj ∥, for j = 1, 2, . . . , k − 1. Moreover, the auxiliary system defined by Eq. (38) and also the main Internet-based control system of Eq. (6) in the quasi-sliding mode will be asymptotically stable, if

lim ∥dmk ∥ → 0,

k→∞

(58)

as lim ∥xmk ∥ → 0.

k→∞

Zka = TJ k T −1 Z a (0) + T

k−1 ∑

J j T −1 dmk−j−1 .

(59)

(60)

j=0

Assume that λA is the maximum of norm of all eigenvalues of the system matrices of the closed-loop system described by Eq. (38) without dmk , for i = 1, 2, . . . , β , and ν = max ∥dmj ∥, for j = 1, 2, . . . , k − 1. From Eq. (60), it can be written that lim ∥Zka ∥ ≤ lim ∥T ∥

k→∞

k→∞

≤ lim ∥T ∥ k→∞

k−1 ∑

∥J ∥j ∥T −1 ∥∥dmk−j−1 ∥

j=0 k−1 ∑

λjA ∥T −1 ∥ν.

(61)

j=0

Since λA < 1 for an asymptotically stable system, it can be shown that ∞ ∑

λjA =

j=0

1 1 − λA

.

(62)

By substituting Eq. (62) into Eq. (61), it can be shown that lim ∥Zka ∥ ≤ ξ ,

k→∞

Since ∥C1 ∥ < ∞, it follows that ek is ultimately bounded as Zk is ultimately bounded and the desirable tracking performance is therefore achieved. This completes the proof.

k→∞

Proof. According to Theorem 1, if for i = 1, 2, . . . , β , matrices Mis , X s > 0 and scaler γi exist which satisfy the LMI (29), gains Ki and Ki′ can be found to guarantee the asymptotic stability of the closed-loop system described by Eq. (38) without dmk , in the quasi-sliding mode. Consequently, all eigenvalues of the uncertain closed-loop state matrix of Eq. (38), that is (Φi′ + ∆Φi′ + ′ Γ0′ Kia ) ≡ Φi cl , lie inside the unit circle of the z-plane; i.e., |λj | < 1, j = 1, 2, . . . , 2 × m + n. Using a similarity transformation matrix ′ ′ T , this system matrix Φi cl can be expressed as Φi cl = TJT −1 , ′ where J = diag(λ1 , λ2 , . . . , λ2×m+n ). Here, the eigenvalues of Φi cl are assumed to be distinct. Then, the solution of Eq. (38) can be obtained as

(63)

ν where, ξ = ∥T ∥ ∥T −1 ∥ 1−λ . Furthermore, from Eq. (39) it can A be shown that limk→∞ ∥dmk ∥ → 0, as limk→∞ ∥xmk ∥ → 0. It demonstrates that limk→∞ ∥Zka ∥ → 0 as limk→∞ ∥xmk ∥ → 0. As a result, Zk , ϵk and Θk will converge to zero. Consequently, in this case, the uncertain Internet-based system described by Eq. (6) is asymptotically stable with the control input of Eq. (27). This completes the proof.

Remark 7. Summary of the SMVSC Method: In order to minimize the online computational burden for the real-time implementation of the SMVSC, this strategy is divided in two parts: off-line controller design and on-line control implementation. The following off-line procedure is used to design the SMVSC for a linear, uncertain, Internet-based, control system. 1. The subsequent procedure is performed to build the delay vector τv :

• Based on the quality of service (QoS) of the Internet, initial values for τβv , τ1v and the consequent data

packet dropouts N are firstly approximated. For this purpose, the methodology used in [23] can be used. • An initial value for β is selected as

β ≤ floor(Lp /Lc ) − 1, where, Lp , Lc , and floor(X ) represent largest allowable packet length of Internet, the data size which is required for encoding a single control signal and rounding X to the nearest integer towards minus infinity, respectively.

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9

Fig. 2. The proposed flowchart for selecting the delay upper-bound.

• τ v is constructed as τ v = {τ1v , (τ1v +(τβv −τ1v ) /(β − 1) ), . . . , τβv }. • The values for β , τβv and τ1v are updated according to

• ZOH elements are used to apply the selected input

τ1 , τ2 , . . . , τβv :

• Xk = [xk ; uk−1 ; tk ] is sent to the control-side through

• Step-invariant DT plant models of Eq. (6) is con-

2. At every time instant tk , when a new plant-side packet is received by the control-side, the following procedure is done:

Remark 8.

2. The following steps are done for each delay

v

v

structed, based on CT linear model of Eq. (1). • Matrix equation (10) is solved to find Gi and Hi . If no acceptable solution is achieved, then a different reference model will be chosen. 3. (S Γ0 ) is required for calculating the control signal. In this way, a nonsingular matrix S is selected as the sliding surface parameter. 4. Eq. (33) is used to calculate stabilizing gains of the sliding surface, where, Mis and X s are calculated by solving LMI (29). 5. α , which is defined as the design parameter for the reaching law, is chosen in the interval (0, 1). −1

The on-line (real-time) procedure for the SMVSC are then summarized as follows: 1. When a new control-side packet Uk is received by the plant-side, at every time instant tk , the following steps are performed:

• BT is updated according to Remark 2. • According to Section 2, τk = tk+1 − tk is determined and based on this, the associated entries ukks are selected from the TDM buffer.

signals to the plant.

• A triggering signal is sent to sensors by the event detector module to transmit new plant-side packet. Internet, according to Remark 3.

• Bc is updated according to Remark 2. • The unknown term dˆ k is estimated using Eq. (28). • For i = 1, 2, β , the tracking control signals uki are calculated using control law of Eq. (27) which is described by, uki =

β ∑

θi (k)[Hi xmk + (Ki + (S Γ0 )−1 S)Zk + Ki′ εk

i=1

− (S Γ0 )−1 dˆ ki + (S Γ0 )−1 ασk − (S Γ0 )−1 σk ].

(64)

• The control-side packet Uk = [uk1 , uk2 , . . . , ukβ ; tk ] is

constructed, and transmitted back to the plant-side according to Remark 3.

Remark 8. The feasibility of LMI (29) depends on some parameters, such as τ1v , τβv and β . Based on Remark 7, some initial values for these parameters are firstly estimated. If there is no feasible solution for this LMI, the procedure described in Fig. 2 can be performed. Here, solution 1 represents selecting a larger value for

Please cite this article as: B. Rahmani, Sliding mode based-variable selective control for robust tracking of uncertain Internet-based systems. ISA Transactions (2019), https://doi.org/10.1016/j.isatra.2019.08.030.

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Fig. 3. Internet-induced time delay (s) vs. time (s) for Example 1.

Fig. 4. Control input designed by SMVSC for Example 1.

τ1v with an increment of dτ1v in each step; evidently the largest value for τ1v must be lower than τβv . Solution 2 means choosing a smaller value for β with a decrement of 1 in each step; evidently the smallest value for β is 1. Solution 3 represents choosing a smaller value for τβv with a decrement of dτβv in each step.

DT uncertainty matrices ∆Φi′ are considered as follows for i = 1, 2, . . . , β , using Remark 2.2 of [23]:

R MATLAB⃝ 2014a (The MathWorks, Inc., Natick, MA, USA) is used here, to evaluate the effectiveness of the SMVSC strategy. In this way, four benchmark problems are simulated. The LMIs are R solved by LMI MATLAB⃝ toolbox, and the dynamical systems are R simulated by SIMULINK⃝ environment.

Example 1. A continuous-time uncertain system with the following state-space model is firstly considered, 0 1 0 0 0 0 0 1 2 + 0.015 sin(t) 0 0.03 sin(t) x˙ (t) = −1 −2 0 0.15 sin(t) 0 0.3 sin(t) ([ ] [ ]) 0 0 0.1 + + 0.003 sin(t) u(t), [ 1 ] 0.03 sin(t) 1 0 0 x(t), y(t) =

]

[

]) x(t)

which has matched uncertainties. The eigenvalues for the known parts of the state matrix are {−0.433, 0.7164 ± 2.0266j}; the open-loop system is therefore unstable. Assume that the reference model is 0 0.5 , [ −0.5 ] 0 ym (t) = 1 0 xm (t).

[

0.0011 0.0061 0.0064 0.0 0.0

0.0001 0.0003 0.0004 0.0 0.0

0.0 0.0 0.0 0.0 0.0

⎤

0.0012 ⎢ 0.0050 ⎢ ∆Φ2′ = ⎢ 0.0032 ⎣ 0.0 0.0 ⎡ 0.0017 ⎢ 0.0061 ⎢ ∆Φ3′ = ⎢ 0.0025 ⎣ 0.0 0.0 ⎡ 0.0024 ⎢ 0.0072 ⎢ ∆Φ4′ = ⎢ 0.0014 ⎣ 0.0 0.0

−0.0005 −0.0030 −0.0047 0.0 0.0

0.0018 0.0081 0.0059 0.0 0.0

0.0001 0.0005 0.0004 0.0 0.0

0.0 0.0 0.0 0.0 0.0

⎤

−0.0008 −0.0046 −0.0060 0.0 0.0

0.0028 0.0102 0.0045 0.0 0.0

0.0001 0.0006 0.0004 0.0 0.0

0.0 0.0 0.0 0.0 0.0

⎤

−0.0014 −0.0067 −0.0071 0.0 0.0

0.0039 0.0120 0.0022 0.0 0.0

0.0002 0.0007 0.0003 0.0 0.0

0 .0 0 .0 0 .0 0.0 0.0

⎤

⎢ ⎢ ∆Φ1′ = ⎢ ⎣

⎥ ⎥ ⎥, ⎦

⎥ ⎥ ⎥, ⎦

(67)

⎥ ⎥ ⎥, ⎦

⎥ ⎥ ⎥. ⎦

In this way, it is assumed that EΦi = ∆Φi′ and G′d = I5×5 . LMI i (29) is solved to calculate the following controller gains,

(65)

x˙ m (t) =

−0.0002 −0.0018 −0.0034 0.0 0.0

⎡

4. Simulation studies

([

0.0007 0.0038 0.0036 0.0 0.0

⎡

]

(66)

A sampling period of Ts = 0.01 s was considered by Pai [31]. This model was stabilized by assuming τ = 0.1 s delay in the control loop. A communication network with a random delay shown in Fig. 3 is considered here. Following the off-line procedure described by Remark 7, the parameters of SMVSC are selected as τ v = [0.5, 0.6, 0.7, 0.8] s, S = [1 1 1 10000], and α = 0.1.

K1 K2 K3 K4 K1′

= [1.0636, 2.5050, 0.0658, = [1.0539, 2.8528, 0.5824, = [0.9673, 3.0781, 1.1764, = [0.8142, 3.1748, 1.8050, = K2′ = K3′ = K4′ = −0.0001.

−0.4432], −0.4032], −0.2868], −0.1029],

(68)

In the online procedure, Eqs. (28) and (27) are used to estimate the unknown term dˆ k and then the control law u(k), respectively. The initial conditions are set to be x0 = [0, 0, 0]T and xm0 = [1, 0]T . It must also be noted that the triggering signal is transmitted from the event detector module to sensors, through a local network with a delay upper-bound of τ ′ = 0.01 sec. The implemented control input is shown in Fig. 4. The SMVSC is robust for such a relatively large Internet-induced time delay, as depicted in Fig. 5. Example 2. Let us consider a benchmark cart-mounted inverted pendulum problem which its linearized state-space model

Please cite this article as: B. Rahmani, Sliding mode based-variable selective control for robust tracking of uncertain Internet-based systems. ISA Transactions (2019), https://doi.org/10.1016/j.isatra.2019.08.030.

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Fig. 5. Models output (ymk ) and systems output (yk ) for Example 1. Pai [31] stabilized this uncertain system with τ = 0.1 s.

11

Fig. 6. Internet-induced time delay (s) vs. time (s) for Example 2.

is represented as follows: 0 ⎢ 0 x˙ = ⎣ 0 0

⎡

1 0 0 0

0 0 0 ⎥ ⎢ M1 ⎥ ⎦ x + ⎣ 0 ⎦ u, 1 −1 0 Ml

0

⎤

−mg M

0 g l

⎡

⎤

(69)

]T

where, x = [x1 , x2 , x3 , x4 ]T = y, y˙ , θ, θ˙ , y, and θ , denote the cart position, and the pendulum bob angle, respectively. Here, η˙ represents time rate of the variable η. The parameters of the system are defined as m = 1 kg, M = 10 kg, and l = 3 m. The eigenvalues of the state matrix are {0, 0, ±1.8083}, and the open-loop system is therefore unstable.

[

For stabilizing such a system, several researches have been performed. Yue et al. [32] and Hu et al. [33] proposed methods for stabilization of this system with τk ≤ 0.23 s and τk ≤ 0.22 s, respectively. Methods which proposed by Hao and Zhao [34], Tian et al. [35] and Li et al. [21], guaranteed the stability of the closedloop system when τk ≤ 0.1 s, τk ≤ 0.2 s and τk ≤ 0.15 s, respectively. For demonstrating the effectiveness of the proposed SMVSC, a communication network with a consequent data packet loss of N = 1 and a randomly varying time delay is assumed. This leads to a delay pattern shown in Fig. 6 with an upper-bound of τβ = 0.8 s. The parameters of SMVSC are selected as τ v = [0.5, 0.6, 0.7, 0.8] s, S = [1 1 1 1 0.1], and α = 0.1. Since the stabilization of this system is considered here, the reference model is considered to be 0 0 .5 0 0 [ ] ym (t) = 1 0 x(t).

[

]

x˙ m (t) =

(70)

LMI (29) is then solved to calculate the following state feedback gains, K1 K2 K3 K4 K1′

= [−0.7798 0.3352 329.4852 182.1228 = [−0.7797 0.3576 395.0377 218.3801 = [−0.7795 0.3803 473.5808 261.8214 = [−0.7796 0.4020 567.5920 313.8163 = K2′ = −9.0005, K3′ = K4′ = −9.0006.

− 2.0469], − 2.7028], − 3.4910], − 4.4368],

Fig. 7. Systems state (xk ) for Example 2.

The initial conditions are assumed to be x0 = [1, 0, 0.05, 0]T and xm0 = [0, 0]T . It is noted that the triggering signal is transmitted from the event detector module to sensors through the local network with a delay upper-bound of τ ′ = 0.01 s. The SMVSC is robust for such a relatively large Internet-induced time delay, as depicted in Fig. 7. The associated control input is shown in Fig. 8. Example 3. The continuous-time system with the following state-space matrices, borrowed from [36], are now considered,

[ (71)

A=

0 −0.5 1 −0.5 0 0 0 0.5

−1

]

[ , B=

0 0 1

] , C=

[

1 0 0 .

]

(72)

Please cite this article as: B. Rahmani, Sliding mode based-variable selective control for robust tracking of uncertain Internet-based systems. ISA Transactions (2019), https://doi.org/10.1016/j.isatra.2019.08.030.

12

B. Rahmani / ISA Transactions xxx (xxxx) xxx

Fig. 8. Control input designed by SMVSC for Example 2.

Fig. 9. Internet-induced time delay (s) vs. time (s), for Example 3.

The eigenvalues of A are {−1, −0.5, 0.5}, and the open-loop system is therefore unstable. A communication network with consequent data packet dropout of N = 5 and Ts = τk = 0.5 sec are assumed by Xiong and Lam [36]. Here, a network with N = 1 and a random delay is assumed which causes an overall Internet-induced delay pattern which is shown in Fig. 9. In order to make the SMVSC feasible, τ v = [1.8, 2.4, 3.0, 3.6] s, S = [1 1 1 0.1], α = 0.2, and the reference model of Eq. (70) are considered. LMI (29) is then solved to design statefeedback gains. x0 = [−0.5, 0, 0.5]T and xm0 = [1, 0]T are assumed as initial condition for the controlled plant and the reference model, respectively. In this example, the effectiveness of the proposed SMVSC method is compared with respect to the

Fig. 10. Output response of Example 3 with τ ′ = 0.

Fig. 11. Control input designed by SMVSC and VSC for Example 3 with τ ′ = 0.

VSC [1]. For this purpose, three simulation studies are done, as follows:

• By setting τ ′ to be zero, the responses of the SMVSC and the VSC are evaluated. Here, τ ′ represents the delay between the sensors and the event detector module. By implementation of the control inputs shown in Fig. 11, output trajectories of Fig. 10 are achieved. • τ ′ = 0.02 s is assumed, and the responses of the SMVSC and the VSC are discussed. The resulting output trajectories of two methods are shown in Fig. 12. Since the VSC cannot tolerate the delay between the event detector module and sensors, the SMVSC can regulate the unstable plant well.

Please cite this article as: B. Rahmani, Sliding mode based-variable selective control for robust tracking of uncertain Internet-based systems. ISA Transactions (2019), https://doi.org/10.1016/j.isatra.2019.08.030.

B. Rahmani / ISA Transactions xxx (xxxx) xxx

13

Fig. 12. Output response of Example 3 with τ ′ ≤ 0.02 s.

Fig. 14. Output response of Example 3 with mismatched uncertainty.

Fig. 13. Output response of Example 3 with an small mismatched uncertainty and τ ′ = 0.0.

Fig. 15. Output response of Example 4 with mismatched uncertainty.

• Small mismatched uncertainties of the form, [ ] 0 0 0.0220 ∆A = 0 0.0495 0 sin(t) and 0 0 0.0220 [ ] ∆B =

0 0 0.035

point-to-point wiring communication technologies. On the other hand, the proposed method can tolerate this modeling uncertainty. As the last try, mismatched uncertainties of the form, (73)

sin(t),

are considered for state and input matrices of Eq. (72). The resulting output trajectories of two methods are shown in Fig. 13. Therefore, the need for a close distance between sensors and actuators with negligible delay is not required for the SMVSC. In this way, sensors and actuators can be connected through

[ ∆A =

0 0 0.09 0 0 .2 0 0 0 0.09

]

[ sin(t) and ∆B =

0 0 0.15

] sin(t) , (74)

are added to matrices of Eq. (72) and output tracking of the reference model described by Eq. (66) under a network with τk ≤ 1.8 s is studied. In this regard, τ v = [0.9, 1.2, 1.5, 1.8] s is considered. As shown in Fig. 14, the SMVSC method causes a relatively good model following properties for such an uncertain system.

Please cite this article as: B. Rahmani, Sliding mode based-variable selective control for robust tracking of uncertain Internet-based systems. ISA Transactions (2019), https://doi.org/10.1016/j.isatra.2019.08.030.

14

B. Rahmani / ISA Transactions xxx (xxxx) xxx

Example 4. Consider a synchronous generator with the following matrices for the state-space form of the linearized dynamical model [37],

⎡ A= ⎣

⎢

− 0 0.115 1 3 0.59

β

0

−0.41 β

0 1

⎤

0.1 0 ⎥ ⎣ 0 β1 ⎦ , , B = ⎦

⎡

⎤

(75)

0 0

0

] −0.6013 −0.4656 0.0701 −0.0537 0.0203 = , −0.2855 −0.8443 −0.1726 0.0026 −0.0971 [ ] −0.5321 −0.4097 0.0531 −0.0786 0.0009 = , −0.2792 −0.9875 −0.1684 −0.0125 −0.1425 [ ] −0.4698 −0.3621 0.0370 −0.1003 −0.0149 = , −0.2753 −1.1153 −0.1610 −0.0280 −0.1998 [ ] −0.4147 −0.3225 0.0207 −0.1192 −0.0279 = , −0.2693 −1.2139 −0.1515 −0.0419 −0.2576 [

K2 K3 K4

(76)

K1′ = K2′ = K3′ = K4′ =

[

0.0780 −1.0139

−1.0313 1.0076

]

[

0.0772 −1.0129

−1.0319 1.0081

]

[

0.0766 −1.0128

−1.0321 1.0079

]

[

0.0772 −1.0127

−1.0317 1.0081

]

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. References

where, β ∈ [3.5, 10] is an uncertain parameter varying randomly during the simulation. In this way, Braga et al. [37] considered τk = 0.05 s and used a linear quadratic regulator to stabilize the Internet-based system. A communication network with a random time delay of τk ∈ [0.07, 1.53] s is considered here. To make the SMVSC feasible, τ v = [0.12, 0.6, 1.08, 1.56] s, [ ] 1, 1, 1, 1, 1 S= , and α = 0.1 are chosen. 1, 1, 1, 1, 0.1 LMI (29) is solved to calculate the following control gains, K1

Declaration of competing interest

, , (77)

, .

Fig. 15 shows the effectiveness of the proposed SMVSC to control the uncertain Internet-based control system with such level of time delay and mismatched uncertainty. 5. Conclusion In this research, a control approach was proposed for a class of linear, uncertain, Internet-based systems. It ensured the stability, robustness, an output tracking and model following against the mismatched uncertainties and Internet negative effects. Corresponding to a range of expected time delays, some control signals were designed using a discrete-time sliding mode control methodology. For this purpose, a new sliding surface was firstly designed by solving a set of LMIs. The stability and robustness of such an uncertain closed-loop system were also evaluated using Lyapunov’s second method for stability. The dynamical behavior of four benchmark problems were simulated to show the effectiveness of the proposed strategy with respect to some recent works. The contributions of the present work are summarized as follows: (a) A new framework for model following and robust tracking of linear systems with time delay, jitter, packet loss, packet disordering and parameter uncertainty was established; (b) A new sliding surface was designed for uncertain linear IBCSs by solving a set of LMIs. Accordingly, using SMVSC, the closed-loop control system is robust against unknown matched and mismatched parameter uncertainties; (c) Using SMVSC, sensors, controllers and actuators were connected through local area networks with considerable time delay.

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