Accepted Manuscript Improving the Performance of Networked Control Systems with Time delay and Data Dropouts Based on Fuzzy Model Predictive Control Ahmad Sakr , Ahmad M. El-Nagar , Mohammad El-Bardini , Mohammed Sharaf PII: DOI: Reference: S0016-0032(18)30488-5 https://doi.org/10.1016/j.jfranklin.2018.07.012 FI 3557 To appear in: Journal of the Franklin Institute Received date: Revised date: Accepted date: 15 February 2018 22 July 2018 31 July 2018 Please cite this article as: Ahmad Sakr , Ahmad M. El-Nagar , Mohammad El-Bardini , Mohammed Sharaf , Improving the Performance of Networked Control Systems with Time delay and Data Dropouts Based on Fuzzy Model Predictive Control, Journal of the Franklin Institute (2018), doi: https://doi.org/10.1016/j.jfranklin.2018.07.012 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. ACCEPTED MANUSCRIPT Improving the Performance of Networked Control Systems with Time delay and Data Dropouts Based on Fuzzy Model Predictive Control Ahmad Sakr a, Ahmad M. El-Nagar b, Mohammad El-Bardini c and Mohammed Sharaf d Department of Control Engineering, High Institute of Eng., Belbies, Egypt. b,c,d b,c,d CR IP T a Department of Industrial Electronics and Control Engineering Faculty of Electronic Engineering, Menoufia University, Menouf, 32852, Egypt. a b eng.ahmed.sakr.1988@gmail.com ahmed.elnagar@el-eng.menofia.edu.eg c dralbardini@el-eng.menofia.edu.eg AN US d msharaf597@gmail.com Abstract M This paper proposes a fuzzy model predictive control (FMPC) combined with the modified Smith predictor for networked control systems (NCSs). The network delays and data dropouts are problems, which greatly reduce the controller performance. For the proposed controller, the ED model of the controlled system is identified on-line using the Takagi – Sugeno (T-S) fuzzy models based on the Lyapunov function. There are two internal loops in the proposed structure. PT The first is the loop around the FMPC, which predicts the future outputs. The other is the loop around the plant to give the error between the system model and the actual plant. The proposed CE controller is designed for controlling a DC servo system through a wireless network to improve the system response. The practical results based on MATLAB/SIMULINK are established. The AC practical results are indicated that the proposed controller is able to respond the networked time delay and data dropouts compared to other controllers. Keywords Networked control systems; Model predictive controller; Smith predictor; DC servo system; T-S fuzzy model 1 ACCEPTED MANUSCRIPT 1. Introduction Networked control system (NCS) is a kind of control system in which its components like, sensor, controller, and actuator are connected through communication channels or a control loop is closed via a form of communication network [1, 2]. After the fast development of CR IP T communication technology and entering it in many industrial applications as communication channels, it is easy to monitor the most process at any time and from anywhere around the world [3]. The advantages of the NCSs over traditional control, which have point to point, are summarized as: 1) it is easy to reform the controlled system. 2) it reduces the cost and the number of cables. 3) the installation becomes simple where it reduces the power requirements AN US and the time spends for adding new sensors to the controlled system [4, 5]. These advantages of the NCS made it more common in many applications such as, DC motor [6, 7], mobile robots [8], manipulators [9, 10], mechanical systems [11], and vehicle engines [12]. The NCS is classified into two parts; the first is the control over the network and the other is the control of the network. Our research focused on the control over the network, which deals M with control design and control strategy to reduce the problems of the NCSs like the network delays and data dropouts [13, 14]. The block diagram of the NCS is shown in Fig. 1, where all ED data transmission between the plant and the controller via communication network may be wired or wireless. Therefore, the delays in the NCS resultant from two communication channels; the PT first is the communication channel between the controller and actuator, which is established to send the control signal from the controller to the actuator. This induced delay from this channel CE is known as the forward delay ( Tca ). The other communication channel is between the sensor and the controller, which is established to send the measured outputs to the controller. The induced AC delay in this channel is known as the feedback delay ( Tsc ). When the network time delays increase due to the limited capacity and network channels, the performance of the fast plants decreases and the controlled system may become unstable [4]. A part of the feedback signal from the sensor and the control signal from the controller may be lost when the data is transmitted via communication network. This problem is called the data losses or data dropout. There are many types of research, which are proposed to overcome the problems of the NCS [15 - 19]. Networked predictive control is proposed to overcome the problems of random network delays 2 ACCEPTED MANUSCRIPT for NCSs [17]. A robust PID controller based on iterative suboptimal algorithms is proposed for the multivariable NCS [18]. The problem of data losses for the NCSs is handled using networked predictive control [19]. Model predictive control (MPC) is a form of control, which gives high-quality control performance compared to traditional controller under changing conditions for many industrial CR IP T problems [20]. The MPC strategy minimizes a predetermined objective function to obtain its control input at each sampling period. After calculating a sequence of predicted control actions, the first one is implemented, and the control input is updated after each sampling time with the new measurements [20, 21]. The MPC has some features that distinguish it from another control strategy, which makes it spread extensively in industrial systems. In addition, it has the ability to AN US handle constraints and the ability to control the MIMO processes [20, 21]. The MPC is proposed for handling the problems of the NCSs [22 – 30]. The MPC is proposed to overcome the random delays in the NCSs [22 - 25]. The strategy of predictive control, which known as a dynamic matrix control (DMC) is proposed for handling the networked time delays [26]. Networked predictive control (NPC) is proposed for solving the problems of NCSs [27, 28]. The stability M and optimality for the NCSs based on the MPC are introduced in [29]. The MPC is proposed for CE AC Controller Delay Tca Network PT ED handling the problem of data dropout in NCS [30]. Delay Actuator Plant Sensor Tsc Fig. 1: Block diagram of a typical NCS. One of the constraints, which may affect the performance of the MPC as a controller for nonlinear system is the internal model, which is used with MPC to predict the future output over 3 ACCEPTED MANUSCRIPT the prediction horizon. If the model is designed offline without considering the possible presence of disturbance in the process, may lead to inaccurate prediction output and also the decreases the controller performance. In recent years, the T-S fuzzy model concept has been proposed for modeling the complex nonlinear systems [31]. In the T-S fuzzy models, the consequents of the rules are a mathematical expression, which can be any linear functions of any variables [31]. The CR IP T T-S fuzzy model has many successful applications in the industrial process, especially in the modeling and the control of plant with incomplete knowledge [32 - 34]. In [35], the FMPC is designed for controlling a laboratory tank system where the antecedent parameters of the T-S fuzzy model are obtained based on the fuzzy clustering method. However, the consequent parameters are obtained based on the least square method. In [36], the FMPC AN US based on the state-space model for the T-S fuzzy model is introduced. The parameters of the T-S fuzzy model have been estimated using the least square algorithm. In [37], the FMPC is applied for controlling the distillation column system where the parameters of the T-S fuzzy model are estimated based on the least square algorithm. The T-S fuzzy model based on the generalized predictive control (GPC) is proposed for controlling the vehicle system [38]. The parameters of M the T-S fuzzy models are updated based on the gradient algorithm. In [39], the T-S fuzzy model predictive control is proposed for controlling the speed of the electrical vehicle where the T-S ED fuzzy model is represented by continuous time state-space model. The parameters of the T-S fuzzy model are obtained using the local linearization method where the fuzzy system is only an PT approximation of the original nonlinear system. A disadvantage of this method is that there are no longer guidelines on how to choose the linearization points, or how many linearization points should be chosen. In [40], the FMPC is proposed for nonlinear system where the initial values of CE the antecedent parameters of the T-S fuzzy model are obtained using the particle swarm optimization. Then, the gradient algorithm is used for tuning these parameters. However, the AC consequent parameters are obtained based on the sliding-window kernel recursive least square algorithm. All the previous approaches [35-40] have been used off-line identification based on various algorithms to identify the nonlinear systems. The stability analysis for the FMPC have been studied based on various approach such as the linear matrix inequalities [36] and nonquadratic Lyapunov function [39]. In this paper, the FMPC combined with the modified Smith predictor for NCSs is proposed. The internal model of the MPC is identified on-line based on a T-S fuzzy model. The proposed 4 ACCEPTED MANUSCRIPT controller is able to respond the problems of the NCS due to the structure of the modified Smith predictor, which compensates the network delays and improves the system performance with increasing the networked time delays and data dropouts. The FMPC based on the modified Smith predictor is designed for controlling the speed of a DC servo system through a wireless network. The practical results are indicated that the proposed FMPC with the modified Smith predictor is CR IP T able to respond the networked time delays and data dropouts better compared to other controllers. The main novelties and contributions of this paper are summarized as: 1) Proposing the FMPC where the internal model is identified on-line based on the T-S fuzzy model. A new learning algorithm based on the Lyapunov stability theorem is developed to update the AN US antecedent and consequent paramters of the T-S fuzzy model. The optimal value of the learning rate for the adaption algorithm is obtained based on the Lyapunov function. 2) Studying the stability analysis of the proposed FMPC based on the Lyapunov stability criterion. 3) The FMPC is comined with the modified Smith predictor where the model that used in the internal loops around the plant and the controller is identified on-line based on the T-S fuzzy model. 4) M The proposed controller is suitable for on-line networked control systems where it implemented for controlling a DC servo system via a network to overcome the problems of time delays and ED data dropouts. The performance of the proposed controller has achieved better performance indices than other controllers, as shown in the practical results section. PT The rest of this paper is structured as: the nonlinear system modeling using T-S fuzzy models is described in section 2. The model predictive control is presented in section 3. The CE modified Smith predictor is presented in section 4. Section 5, presents the FMPC based on the modified Smith predictor. Section 6, presents the practical results followed by the conclusions AC and relevant references. 2. Nonlinear systems modeling using T-S fuzzy models This section explains the method of estimating the nonlinear model of the process, since many processes can be represented by the following model [41]: y(k ) f y(k 1),, y(k ny ), u(k d 1),, u(k d nu ) n (k ) 5 (1) ACCEPTED MANUSCRIPT Assume, the unknown plant with a single input u k and a single output yk , where n y , nu are the order of the output and the input, respectively, d is the time delay of the plant and n k is a sequence of zero mean Gaussian white noise. In order to find a fuzzy system to model it, it must be excited by an input while data uk , yk , where k 1, 2, 3, ... is taken from the plant. form [37]: CR IP T As the plant is unknown, we can attempt to estimate it as a T-S fuzzy model with rules in the Ri : IF y(k ) is A1i , y(k 1) is A2i and and y(k n y 1) is Ani then y i (k 1) a i ( z 1 ) y(k ) bi ( z 1 ) u (k d ) (k ), Ri represents i th the a i ( z 1 ) a1i a2i z 1 ani y z ( n y 1) fuzzy rule, N is the number of (2) rules, and bi ( z 1 ) b1i b2i z 1 bniu z ( nu 1) . The membership AN US where i 1, 2, 3,, N function Aij (i th rule, j th universe) is given by: 1 y (k j 1) ci 2 j ( y (k j 1)) exp i 2 j (3) M i j ED where c ij and ij are the mean and standard deviation for the Gaussian membership function, respectively. Assume, the time delay of the system d is zero and we neglect the white noise k . PT The fuzzy system is described as: yˆ (k 1) [a11 y (k ) a1ny y (k n y 1) b11 u (k ) bn1u u (k nu 1)]1 (k ) CE [a12 y (k ) an2y y (k n y 1) b12 u (k ) bn2u u (k nu 1)] 2 (k ) (4) AC [a1N y (k ) anNy y (k n y 1) b1N u (k ) bnNu u (k nu 1)] N (k ) where i y (k ), y (k 1),, y (k ny 1) i y (k ),, y (k ny 1) y(k ),, y(k n N i 1 and 6 i y 1) (5) ACCEPTED MANUSCRIPT 1 y (k j 1) ci 2 j i y (k ),, y (k ny 1) exp i 2 j 1 j n (6) 2.1 Learning algorithm CR IP T The antecedent parameters c ij and ij ; the consequent parameters a ij and b ij for the T-S fuzzy model are learned on-line using the Lyapunov function. Let, the Lyapunov function is chosen as: V (k ) 1 em (k )2 (k )2 2 AN US em k yˆ (k 1) y(k 1) (7) (8) where V (k ) is a positive definite function, em (k ) is the model error signal as defined in Eq. (8), yˆ (k 1) is the T-S fuzzy model output and y(k 1) is the desired output. (k ) represents the M parameters vector which include all the T-S fuzzy model parameters. Theorem 1: For the Lyapunov function, V (k ) 1 em (k )2 (k )2 0 , the condition 2 (k ) em (k ) (k ) (k ) em (k ) e (k ) 1 m (k ) CE PT ED V (k ) 0 is satisfied if and only if the parameter adjustment rule is obtained as: 2 (9) Proof 1: Let the change in the Lyapunov function is defined as: AC V (k ) V (k 1) V (k ) 1 em (k 1) em (k ) em (k 1) em (k ) (k 1) (k ) (k 1) (k ) 2 Let em (k ) em (k 1) em (k ) and (k ) (k 1) (k ) , Eq. (10) can be rewritten as: 7 (10) ACCEPTED MANUSCRIPT e (k ) 2 e (k ) 1 2 2 (k ) (k ) em (k ) m V (k ) (k ) 1 m (k ) 2 ( k ) (11) For a very small change, Eq. (11) can be rewritten as: e (k ) 2 e (k ) 1 (k ) 2 1 m 2 (k ) (k ) em (k ) m (k ) 2 ( k ) CR IP T V (k ) (12) To guarantee the stability, the second condition for the stability is V (k ) 0 . Then: AN US e (k ) 2 e (k ) 1 2 2 (k ) (k ) em (k ) m 0 V (k ) (k ) 1 m (k ) 2 (k ) Eq. (13) can be rewritten as: e (k ) 2 e (k ) 1 1 2 2 (k ) (k ) em (k ) m z V (k ) (k ) 1 m 2 (k ) 2 (k ) (13) (14) positive value. So, Eq. (14) becomes: M In order to satisfy the condition in Eq. (13), the value of z in Eq. (14) must have a zero or PT ED e (k ) 1 2 V (k ) (k ) 1 m (k ) 2 2 2 (k ) (k ) e (k ) em (k ) z 0 m (k ) (15) CE Consider a general quadratic equation: a x2 b x c 0 (16) AC The roots of Eq. (16) are given as: b b 2 4a c b b 2 4a c r1 and r2 2a 2a (17) By comparing Eq. (15) with Eq. (16), it is clear that (k ) acts as x in Eq. (16) and the values of a, b and c in Eq. (16) are given as: 8 ACCEPTED MANUSCRIPT 1 e (k ) a 1 m 2 (k ) 2 e (k ) 1 , b (k ) em (k ) m and c z 2 (k ) In order to have a single solution for Eq. (16), the term b 2 4a c must to be equal zero. After b 2 4a c and squaring both sides, we have CR IP T putting the values of a, b and c in (18) 2 2 em (k ) em (k ) 1 z 0 (k ) em (k ) (k ) (k ) So, z is given as: 2 AN US em (k ) (k ) em (k ) (k ) z 2 e (k ) 1 m (k ) Since z 0 , which means (19) (20) 2 (21) ED M em (k ) (k ) em (k ) (k ) 0 2 em (k ) 1 (k ) (k ) b 2a AC CE PT Then, the unique root of Eq. (15) will be given as: em (k ) (k ) (k ) em (k ) e (k ) 1 m (k ) 2 This completes the proof. Then, the updating parameters for the T-S fuzzy model are given as: 9 (22) ACCEPTED MANUSCRIPT (k 1) (k ) (k ) (k ) e (k ) 1 m (k ) (23) 2 CR IP T where is the learning rate constant. em (k ) (k ) (k ) em (k ) Based on the above theorem, the updating equations for the T-S fuzzy model parameters are defined as follow: AN US e (k ) c ij (k ) em (k ) mi c (k ) j c ij (k 1) c ij (k ) c 2 e (k ) 1 mi c (k ) j where (24) ED M i y (k j 1) c i (k ) em (k ) yˆ (k 1) yˆ (k 1) j (25) N . i y (k ),, y (k n 1) 2 i i c j (k ) ( k ) j i y (k ),, y (k n 1) i 1 where CE PT ij (k 1) ij (k ) em (k ) i ( k ) j 2 e (k ) 1 mi (k ) j ij (k ) em (k ) (26) AC i y (k j 1) ci (k ) 2 em (k ) yˆ (k 1) yˆ (k 1) j (27) N . i y (k ),, y (k n 1) 3 i i j (k ) ( k ) j i y (k ),, y (k n 1) i 1 10 ACCEPTED MANUSCRIPT e (k ) a ij (k ) em (k ) mi a (k ) j i i a a j (k 1) a j (k ) 2 e (k ) 1 mi a (k ) j (28) CR IP T where em (k ) i y (k ),, y (k n 1) y (k j 1) a ij (k ) N y ( k ), , y ( k n 1 ) i i 1 (29) AN US e (k ) b ij (k ) em (k ) mi b (k ) j b ij (k 1) b ij (k ) b 2 e (k ) 1 mi b (k ) j M where (30) PT ED em (k ) i y (k ),, y (k n 1) u (k j 1) bij (k ) N i y (k ),, y (k n 1) i 1 (31) CE 2.2 Convergence of the T-S fuzzy model In this section, some convergence theorems are developed to derive appropriate learning AC rates adaptively for the T-S fuzzy model. Assume a Lyapunov function can be defined as: 1 2 v(k ) em (k ) 2 (32) The change in the Lyapunov function is obtained as: v(k ) v(k 1) v(k ) 11 1 2 2 em (k 1) em (k ) 2 (33) ACCEPTED MANUSCRIPT The error difference of T-S fuzzy model can be represented as: e (k ) em (k 1) em (k ) em (k ) em (k ) m T a ij bij ] and the parameter vector change ( ) is obtained from the update rules of Eq. (23) as: ( k ) em ( k ) CR IP T where [cij ij (34) em (k ) ( k ) (k ) em (k ) 2 (35) AN US e ( k ) 1 m ( k ) y ( k ) (k ) em (k ) ( k ) em ( k ) M Let S (k ) is defined as: y ( k ) 1 ( k ) PT y (k ) (k ) (k ) em (k ) ED S (k ) 2 y (k ) 1 (k ) 2 (36) CE Eq. (35) can be rewritten as: AC Theorem 2 : Let c (k ) em (k ) S (k ) (37) a b be the learning rates of the T-S fuzzy model. Then, the asymptotic convergence of the T-S fuzzy model is guaranteed if are chosen to satisfy: 0 2 y (k ) S (k ) T in which is the T-S fuzzy model parameters c ij , ij , a ij and b ij . 12 (38) ACCEPTED MANUSCRIPT Proof 2: From Eqs. (33) and (34), the change in the Lyapunov function is: 1 v(k ) em (k ) em (k ) em (k ) 2 (39) CR IP T T T 1 e (k ) e (k ) v(k ) m em (k ) S (k ).em (k ) m em (k ) S (k ) 2 (40) Since for the T-S fuzzy model em (k ) y(k ) , we obtain: T T 1 y (k ) y ( k ) v(k ) em (k ) S ( k ) e ( k ) e ( k ) S ( k ) m m 2 1 2 v(k ) em (k ) 2 AN US T y(k ) S (k ) 2 where 1 y(k ) T T y(k ) 2 S (k ) em (k ) y(k ) T (42) (43) ED M S (k ) 2 S (k ) 2 (41) If 0 , then v(k ) 0 is satisfied. Thus, the asymptotic convergence of the T-S fuzzy model PT is achieved, and Eq. (38) is obtained. This completes the proof. 3. Model predictive control CE 3.1 Predictive control law The MPC depends primarily on the accuracy of the process model selection in one of the AC different ways, such as system identification or fuzzy models [42]. So that it can predict the future results of the output properly and give a much better system response. Therefore, it is able to solve many of the problems that appear in the industry, clearly the most important one that this research cares, network delay and data dropouts. The T-S fuzzy model that of the form in Eq. (2) can be rewritten as follows: A( z 1 ) y(k ) B( z 1 ) u(k 1) n(k ) 13 (44) ACCEPTED MANUSCRIPT where A( z 1 ) 1 A1 z 1 An y z n y , B( z 1 ) 1 B1 z 1 Bnu z nu and n(k ) is the disturbances model that described by: n(k ) C ( z 1 ) e( k ) (45) CR IP T The plant parameters are obtained every instance with the T-S fuzzy model. Therefore, the model of predictive control is not estimated off-line, but on-line using the T-S fuzzy model, which makes the model is more accurate. Assume the polynomial C ( z 1 ) 1 and 1 z 1 . Eq. (44) can be rewritten as: e(k ) 1 z 1 AN US A( z 1 ) y(k ) B( z 1 ) u (k 1) (46) where e(k ) is the disturbance or noise with zero mean. To obtain the j step prediction of y(k j ) , the following identity must be considered: M 1 E j ( z 1 ) A( z 1 ) z 1Fj ( z 1 ), deg E j j 1, deg Fj ny , Gj E j B (47) ED where E j and F j are polynomials uniquely defined and the prediction interval j is given. Multiplying Eq. (46) by E j z j and after some mathematical operations, we obtain the multi- PT step ahead predictor of y(k ) as: AC where CE yˆ (k j | k ) G j ( z 1 ) u(k j 1) Fj ( z 1 ) y(k ) G j ( z 1 ) E j ( z 1 ) B( z 1 ) B( z 1 )[1 z j Fj ( z 1 )] A( z 1 ) (48) (49) The main target of an objective function is to make future outputs ( y(k j ) ) on considered horizon follows the reference signal ( wr (k ) ). Therefore, there are different cost functions for obtaining the control law, which depend on various MPC algorithms. The GPC algorithm depends on minimizing the objective function J over receding prediction horizon: 14 ACCEPTED MANUSCRIPT J ( N1 , N 2 , N u ) N2 Nu l N1 l 1 (l )[ yˆ (k l ) wr (k l )]2 (l )[u(k l 1)]2 (50) where N1 , N 2 and N u are the minimum, maximum prediction horizons, and control horizon, respectively, (l ) and (l ) are the weighting sequence, where the choice of these parameters is described in details in [43 - 45]. Longer horizons tend to produce more aggressive control action, CR IP T more overshoot, and faster response. As presented in [46], when (l ) =1 is the case where there are no constraints. Then, the j th steps ahead predictor y(k j ) can be expressed in the vector form as: AN US y(k j) G u(k ) F y(k ) H u(k 1) (51) and the corresponding control law is given by [43, 45]: u(k ) d T (wr y0 ) (52) where wr wr (k 1) wr (k N2 ) , y0 y0 (k 1 k ) y0 (k N2 k ), g0 g 1 G g N 1 2 PT ED d T 1,0,0(GT G I )1 GT and M T g N 2 2 g N 2 Nu 0 0 g0 CE Here, the step response coefficient is g j for the model and if the future control increments are AC all zeros then y 0 (k j k ) is the free response [30]. 3.2 Stability analysis The T-S fuzzy model has been shown stable where the updating parameters are derived via the Lyapunov stability theorem as shown in Theorem 1 and 2 (Section 2). The obtained control low (Eq. (52)) has zero steady-state error due to 1 z 1 which introduced into the Eq. (46) in order to provide an integral action and so, the steady-state error is eliminated. The following theorem state that the closed loop system based on the FMPC is stable. 15 ACCEPTED MANUSCRIPT Theorem 3: Let that the T-S fuzzy model (Eq. (4)) is controlled by Eq. (52), the estimated parameters c ij , ij , a ij and b ij are bounded, all set-points are constant ( wr (k ) wr ) and the steady-state error has zero ( lim j E{wr y(k j )} 0) . Then the closed loop is stable. Proof 3: Assume a Lyapunov function is chosen as: n (k ) lim n E{wr y(k j 1)} CR IP T 2 j 1 The change in the Lyapunov function is defined as: (k ) (k 1) (k ) n n lim n E{wr y (k j )} lim n E{wr y (k j 1)} 2 (54) j 1 AN US j 1 2 (53) Eq. (54) can be rewritten as: n (k ) lim n E{wr y (k j 1)} E{wr y (k )} lim j E{wr y (k j )} 2 j 1 n 2 lim n E{wr y (k j 1)} j 1 wr E{ yˆ (k ) (k )} ( wr yˆ (k )) 2 (55) ED 2 M 2 2 PT From Eq. (55), we find that (k ) 0 is satisfied. Thus, the closed loop system based on CE the FMPC is stable. 4. Smith predictor In this section, we introduce the structure of the Smith predictor. The simplified block AC diagram of the NCS is shown in Fig. 2. X (s) is the input signal and C (s) is the output signal. Tca is the networked time delay between the controller and the actuator, Tsc is the networked time delay between the controller and the sensor, Gc (s) is the controller transfer function (TF) and the GP (s) is the real system TF without delay time. The overall TF for the closed loop NCS is obtained as: (where Tc Tca Tsc ) 16 ACCEPTED MANUSCRIPT E ( s) X ( s) C ( s) e Tsc s (56) T s Gc ( s) G p ( s) e ca C ( s) X ( s) 1 Gc ( s) G p ( s) e (Tca Tsc ) s (57) T s (58) CR IP T Gc ( s) G p ( s) e ca C ( s) X ( s) 1 Gc ( s) G p ( s) e Tc s It is clear from Eq. (58), there is a time delay in the denominator of the TF, which affects on the performance of the system. E (s) Gc (s) - e Tca s G p (s) C (s) AN US X (s) + Tsc s M e Fig. 2: Simplified block diagram of NCS. ED 4.1 Traditional Smith predictor The traditional Smith predictor for the NCSs is shown in Fig. 3, which contains an internal PT loop around the TF of the controller. This internal loop is the main part of the Smith predictor, which makes compensation of time delays [47, 48]. G pm (s) is the nominal model of the system CE without the delay time. The TF of the overall system ( Gs (s) ) when the Smith predictor is AC involved, is obtained as: E(s) X (s) C2 (s) C1 (s) C (s) e Tsc s (59) If G pm (s) G p (s) , Tca Tsc and Tcom Tc , then: E ( s) X ( s) C ( s) 17 (60) ACCEPTED MANUSCRIPT Gc ( s) 1 (1 e ) Gc ( s) G pm ( s) (61) G p ( s) Gc ( s) Tca C ( s) e X ( s ) 1 G p ( s ) Gc ( s ) (62) Gs ( s) Tcom CR IP T From Eqs. (58) and (62), the Smith predictor is able to remove the time delay. In this traditional Smith predictor, there are some constraints like Tca Tsc and Tc Tcom which must be achieved to improve the controller stability and compensate the delays. These constraints make the controller must know the delays in the network and need an extra part to compensate these X (s) + - + + C2 ( s) AN US delays like an extra network. E (s) Controller Gc (s) e Tcom C1 ( s) e Tca s G p (s) C (s) Delay G pm (s) e Tsc s Delay ED M Delay U (s) Fig. 3: Traditional Smith predictor for NCSs. PT 4.2 Modified Smith Predictor CE Fig. 4, shows the block diagram of the modified Smith predictor. It contains two internal loops; the first is around the controller and the other with delay Tpm around the real plant. This AC structure improves the function of the Smith predictor where there is no need to know the network delay. Therefore, there are no constraints for forward delay Tca and feedback delay Tsc . The closed loop TF for the plant with delay T p is given as follows: G p ( s) Gc ( s) Tca s T p s C ( s) e e , G p ( s) G pm ( s) and Tp Tpm X ( s) 1 G p ( s)Gc ( s) 18 (63) ACCEPTED MANUSCRIPT where the prediction value of T p is represented by Tpm . Eq. (63) shows that the modified Smith predictor is able to respond the networked time delay. As shown in Fig. 4, the controller is dependent on the feedback signal, which obtained from the system model G pm (s) at the internal loop around the controller. So, when there is a data dropout in the measured output C (s) , the CR IP T controller will respond to this data dropout due to the internal loop around the controller. The modified Smith predictor is dependent on the model that obtained for the controlled system. So, in this study the system model around the plant and around the controller is identified on-line based on the T-S fuzzy model that described in section 2. - + C 2 ( s) E (s) Controller Gc (s) C1 ( s) U (s) e Tca s G p ( s )e AN US X (s) + T p s C (s) Delay G pm (s) G pm ( s)e M e T pm s - + Tsc s Delay ED Fig. 4: Modified Smith predictor for NCSs. PT 5. FMPC based on modified Smith predictor Fig. 5, shows the overall structure of the proposed controller, which combines the FMPC, CE which described in sections 2 and 3 with the modified Smith predictor, which described in section 4. In Fig. 5, the reference signal is wr (k ) , u (k ) is the manipulated variable, y(k ) is the AC measured output from the controlled system, yˆ1 (k ) and yˆ 2 (k ) are predicted outputs of the controlled system model and cm (k ) is the result of Smith predictor where: cm (t ) y(k ) yˆ 2 (k ) yˆ1 (k ) (64) The FMPC block has two inputs; wr (k ) and cm (k ) and one output u (k ) . There is an internal loop with the model of the system around the FMPC. There is another loop around the system to give the error between the model of the system and the actual system. As shown in Fig. 19 ACCEPTED MANUSCRIPT 5, the system model at the internal loops is identified on-line based on the T-S fuzzy model. If there are no disturbances, or noise and the model of the system are obtained accurately, the error between the model and actual system will equal to zero ( y(k ) yˆ 2 (k ) 0 ). It is clear that the input signal cm (k ) for the proposed controller, which combines the modified Smith predictor with the FMPC is not affected with network delay and data dropout. In addition, it depends on CR IP T the plant model, which is designed using the T-S fuzzy model. This structure makes the controller like double predictor one with Smith predictor and the other in the FMPC block. FMPC wr (k ) u (k ) MPC cm (k ) + Adaptive T-S fuzzy model R1 + yˆ 2 (k ) - ED cˆ(k ) T-S Fuzzy Model Tsc PT yˆ1 (k ) y(k ) M Rn - System T-S Fuzzy Model On-line Adaption + Tca AN US + CE Fig. 5: Overall structure of the proposed controller (FMPC with modified Smith predictor). The computations at each iteration k can be summarized as the following steps: AC Step 1: Set wr (k ) , n y , nu , d , N , N1 , N 2 and N u as described in section 3. Step 2: Measure the current system output y(k ) , obtain the output of the T-S fuzzy model and the difference between them as described in section 3 and 4. Step 3: Update c ij , ij , a ij and b ij of the FMPC using Eqs. (24), (26), (28) and (30) as described in section 2. 20 ACCEPTED MANUSCRIPT Step 4: Define the learning rate based on Eq. (38) as described in section 2. Step 5: Determine the change of control signal u (k ) from Eq. (52). Then send the control signal u (k ) to the controlled system. Step 6: Repeat steps 2-5. CR IP T 6. Practical results In this section, the proposed FMPC with modified Smith predictor is designed for controlling a DC servo system through a wireless network. 6.1 Model validation AN US In this section, we show the model validation using the T-S fuzzy model based on the Lyapunov function, which described in Section 2 to clarify the robustness of the proposed T-S fuzzy model to identify the DC servo system. Fig. 6, shows the input test signal, which applied practically to the DC servo system and also applied to the T-S fuzzy model. Fig. 7, shows the M measured output and the T-S fuzzy model output. It is clear that the T-S fuzzy model based on AC CE PT ED the Lyapunov function is able to identify the DC servo system. Fig. 6: Input signal. 21 CR IP T ACCEPTED MANUSCRIPT Fig. 7: Measured and T-S fuzzy outputs. AN US 6.2 Experimental setup Fig. 8, shows the experimental setup used in this study. The network consists of two PCs. The PC1 is used to implement the proposed FMPC with the modified Smith predictor that shown in Fig. 5. The implementation for the proposed controller is done based on the MATLAB/SIMULINK as shown in Fig. 9. As shown earlier in Fig. 5, the internal loop around M the DC servo system is implemented using the PC2. The implementation of the internal loop is done based on the MATLAB/SIMULINK as shown in Fig. 10. The PC2 sends and receives the ED signal to and from the controller (PC1) through the network and deliver the signal to the servo system. The Arduino Mega Kit is connected to PC2 through USB port and works as interface PT module between the DC servo system and the PC2. The PC2 receives the tacho signal through analog pin (A0) of the Arduino Kit and sends the feedback signal to the PC1 through the CE network. In addition, it receives the control signal from PC1 and delivers it to the servo system through PWM pin (A3) of the Arduino Kit. The communication network between the two PCs is AC executed through Wi-Fi. 22 PT ED M AN US Fig. 8: Experimental setup. CR IP T ACCEPTED MANUSCRIPT AC CE Fig. 9: Implementation of the FMPC with modified Smith predictor (PC1). 23 AN US CR IP T ACCEPTED MANUSCRIPT Fig. 10: Implementation of internal loop around the DC servo system (PC2). 6.3 Experimental tasks In order to show the improvements of the proposed FMPC with modified Smith predictor, the practical results are compared with the results of the FMPC. The sample period for the DC M servo system is 0.01sec, the control horizon is 2, the prediction horizon is 10, the weight of control signal is 7 and the weight of output is 1. The reference speed is 1000 RPM. The main ED objective of the proposed controller is improving the performance of the system that connected via a wireless network to reduce the effect of the network time delays and the data dropouts. PT The practical results are performed based on the practical network consists of two PCs as shown in Fig. 8. In all experimental tasks, there are data dropouts due to the practical CE communication network, where the packet loss occurs when one or more packets of data traveling across a computer network fail to reach their destination. Five practical tasks are AC considered to show the effect of delays and load to the system. Three performance indices are employed as quantitative measures for comparing the proposed controller and the FMPC; the mean absolute errors (MAE), integral of square error (ISE) and the root mean square of errors (RMSE), which are defined as [49]: MAE 24 N 1 | e(k ) | k 1 N (65) ACCEPTED MANUSCRIPT ISE e 2 (t ) . dt (66) 0 RMSE N 1 (e(k ))2 k 1 N (67) Task 1: Variable delays in NCSs and load effect. CR IP T This task shows the influence of variable time delays and an external load. The time-varying delay is performed using a block in MATLAB/SIMULINK, which have two parameters; minimum and maximum, which added to the real network delay in order to simulate the increasing of real network delay. The external load is presented using the additional part in our system called the magnetic load. This magnetic load has a potentiometer to adjust the value of AN US the load and keep the load level. In this task, there is a variable delay with values between (0.01 and 0.32) sec. and a 25% load effect, which is inserted at 6 sec. The response of the proposed controller and the FMPC for this task is shown in Fig. 11. The performance of the proposed FMPC with modified Smith predictor has a lower overshoot and settling time than that obtained M for the FMPC. Task 2: Normal Network delays in NCSs with a drop in the network and load effect. ED This practical experiment shows the influence of the normal network delays with drop or cut in the network between the two PCs form instant 4 sec to instant 6.5 sec where the external load is inserted at instant 6 sec before the network reconnected between the two PCs. Fig. 12, shows PT the response of the DC servo system for this task. The response of the proposed FMPC with the modified Smith predictor has achieved a good performance without an overshoot after the CE connection backs between the two PCs because there is an internal loop with a system model around the controller. The response for the FMPC has an overshoot after the connection backs AC between the two PCs. 25 AN US CR IP T ACCEPTED MANUSCRIPT AC CE PT ED M Fig. 11: Response of the DC servo system in NCS (Task 1). Fig. 12: Response of the DC servo system in NCS (Task 2). 26 ACCEPTED MANUSCRIPT Task 3: Normal network delays in NCSs with a cut in the network. This practical task shows the effect of the normal network delays with a cut in the network between the two PCs from the starting point to 0.35 sec. Fig. 13, shows the response of the DC servo system for this task. In this case, we display two control signals; the first is the control signal at PC1 (before sending to the system via the network) and the other is the controller signal CR IP T at PC2 (after sending to the system via the network). The delay between two control signals is due to the cut in the network. It is clear that the response of the proposed FMPC with the modified Smith predictor has a settling time and an overshoot lower than that of the FMPC. Therefore, the FMPC with the modified Smith predictor is able to respond the problem of data AC CE PT ED M AN US dropouts in the NCSs better than the FMPC. Fig. 13: Response of the DC servo system (Task 3). 27 ACCEPTED MANUSCRIPT Task 4: Variable delays in NCSs with data dropouts. This practical task shows the effect of variable delays in NCSs with weakening the connection between the two PCs to increase the dropout of the signal. This can be achieved by increasing the distance between the two PCs from 8 sec to 14 sec. The variable delays are between (0.01 and 0.34) sec. Fig. 14, shows the response of the DC servo system for this task. It CR IP T is clear that the performance of the FMPC with the modified Smith predictor has a small overshoot and settling time. However, the response for the FMPC has a large settling time, an overshoot and an oscillation around the set-point after weakening the connection between the two PCs. Therefore, the proposed FMPC with the modified Smith predictor is superior to Fig. 14: Response of the DC servo system (Task 4). AC CE PT ED M AN US respond the effect of the data dropouts rather than the FMPC with increasing network delay. Task 5: Variable delays in NCSs under variable set-point This practical task shows the effect of the step change with a variable delay. For the first step, the amplitude is 1000 RPM and the variable delay is between (0.01 and 0.2) sec. For the second step, the amplitude is 800 RPM. For the final step, the amplitude is 1200 RPM. Fig. 15, shows the response of the DC servo system for this task. The response of the proposed FMPC with the modified Smith predictor is better than the FMPC with variable delay and step change. 28 ACCEPTED MANUSCRIPT Therefore, the FMPC with the modified Smith predictor is superior to respond the set-point changes rather than the FMPC. As shown above in all experimental tasks, the proposed FMPC with the modified Smith predictor is able to respond to the problems of the NCSs due to the following: 1) the structure of the modified Smith predictor, which dependent on the T-S fuzzy model. This structure is able to CR IP T compensate the network time delay and the data dropouts. 2) the internal model of the FMPC, which is identified on-line based on the T-S fuzzy model using the Lyapunov function. To show the ability of the proposed controller to respond the problems of the NCSs, the comparisons between the proposed controller and other controllers based on other structures of the Smith predictor are done. Tables 1 - 3 list the MAE, RMSE, and ISE values for the FMPC, PID AN US controller with the Smith predictor [50, 51], the fuzzy PI (FPI) controller with the Smith predictor [51] and the proposed FMPC with the modified Smith predictor for all the above experimental tasks. The values of the MAE, RMSE, and ISE for the proposed FMPC with the modified Smith predictor are lower than that obtained for other controllers. Therefore, the proposed FMPC with the modified Smith is preferable than other controllers where it improves M the performance of the system and it is able to overcome the problems of networked time delays AC CE PT ED and data dropouts. Fig. 15: Response of the DC servo system (Task 5). 29 ACCEPTED MANUSCRIPT Table 1: MAE values for all practical tasks. Task 2 Task 3 Task 4 Task 5 FMPC 0.1194 0.1265 0.3076 0.1122 0.085 PID with Smith [47] 0.1061 0.1166 0.2313 0.0812 0.0611 FPI with Smith [51] 0.1033 0.1121 0.2231 0.0787 0.0581 MPC with Smith 0.1011 0.0998 0.2077 0.0678 0.0551 Proposed controller 0.0925 0.0957 0.2064 0.0611 0.0530 Task 4 Task 5 0.5340 0.5781 CR IP T Task 1 Task 1 Task 2 Task 3 FMPC 0.4953 0.4518 0.8377 PID with Smith [47] 0.4711 FPI with Smith [51] 0.4621 MPC with Smith 0.4431 Proposed controller 0.4200 AN US Table 2: RMSE values for all practical tasks. 0.7822 0.4612 0.5314 0.4233 0.7512 0.4551 0.5193 0.4112 0.7231 0.4038 0.5032 0.3878 0.6887 0.3480 0.5029 M 0.4411 Table 3: ISE values for all practical tasks. Task 2 Task 3 Task 4 Task 5 2.4557 2.0430 3.5160 4.5660 5.0171 PID with Smith [47] 2.2301 1.8123 2.8320 2.4155 4.2113 FPI with Smith [51] 2.0311 1.7909 2.6540 2.1909 4.0480 MPC with Smith 1.8541 1.6055 2.6048 2.0145 3.8210 1.7658 1.5052 2.3763 1.9387 3.7962 CE PT FMPC ED Task 1 AC Proposed controller 7. Conclusions This paper proposed the FMPC with the modified Smith predictor for NCSs. The proposed controller is able to respond and overcome the problems of the NCSs; the time delays and data dropouts. The structure of the modified Smith predictor compensates the network time delays. The internal model for the FMPC is identified on-line based on the T-S fuzzy models using the 30 ACCEPTED MANUSCRIPT Lyapunov function. In addition, the convergence analysis of the T-S fuzzy model is studied to find the appropriate range for the learning rate. The stability of the FMPC is studied based on the Lyapunov theorem to achieve the stability of the closed loop system. The structure of the modified Smith predictor has two internal loops; the first is around the FMPC and the other is around the system. The proposed FMPC with the modified Smith predictor is applied practically CR IP T for controlling a DC servo system through a wireless network. The practical results are established to show the robustness of the proposed controller. The proposed controller is evaluated using five practical tasks including variable time delay with load effect, normal network delays with a drop in the network and the load effect, normal network delays with a cut in the network at the starting point, variable time delays with data dropouts and variable time AN US delays with step-point change. To show the robustness of the proposed controller, the experimental tasks are compared with the results of the PID controller with Smith predictor, a fuzzy PI controller with Smith predictor, MPC with Smith predictor and the FMPC. The MAE, ISE, and RMSE are measured for all the controllers to show the controller performance. The obtained values for the proposed FMPC with modified Smith predictor are lower than that M obtained for other controllers, which indicate that the proposed controller is able to overcome the problems of the networked time delays and data dropouts. This method proved with the practical ED experimental superiority and the ability to ingress in many industrial applications that need networks. PT References [1] J. Qiu, H. Gao, S. X. Ding, Recent advances on fuzzy-model-based nonlinear networked CE control systems: a survey. IEEE Transactions on Industrial Electronics. 63(2016) 1207-1217. [2] T. Wang, H. Gao, J. Qiu, A combined fault-tolerant and predictive control for network-based AC industrial processes. IEEE Transactions on Industrial Electronics. 63(2016) 2529-2536. [3] Y. Xia, W. Xie, B. Liu, X. Wang, Data-driven predictive control for networked control systems. Information Sciences. 235(2013) 45-54. [4] J. Qiu, H. Gao, S. X. Ding, Recent advances on fuzzy-model-based nonlinear networked control systems: a survey. IEEE Transactions on Industrial Electronics. 63(2016) 1207-1217. 31 ACCEPTED MANUSCRIPT [5] B. Xuhui, H. Zhongsheng, J. Shangtai, C. Ronghu, An iterative learning control design approach for networked control systems with data dropouts. International Journal of Robust and Nonlinear Control. 26(2016) 91-109. [6] H. N. Tran, V. Q. Nguyen, N. V. P. Tran, K. M. Le, J. W. Jeon, Implementation of adaptive fuzzy dual rate PID controller for networked control systems. 2017-43rd Annual Conference CR IP T of the IEEE In Industrial Electronics Society, IECON, pp. 3390-3395, 2017. [7] J. Vilela, L. F. C. Figueredo, J. Y. Ishihara, H∞ control for networked control systems with dynamic controllers in the feedback loop. 2017 IEEE Conference on Control Technology and Applications (CCTA), pp. 1390-1395, 2017. [8] A. Borri, G. Pola, M. D. Benedetto, Design of Symbolic Controllers for Networked Control AN US Systems. IEEE Transactions on Automatic Control, 64(2019), 1-13. [9] A. Kheirkhah, D. Aschenbrenner, M. Fritscher, F. Sittner, K. Schilling, Networked Control Systems with Application in the Industrial Tele-Robotics. IFAC-PapersOnLine, 48(2015), 147-152. [10] N. Jiang , J. Xu , S. Zhang, Neural network control of networked redundant manipulator M system with weight initialization method, Neurocomputing, 2018, in press. [11] L. Liu, X. Liu, C. Man, C. Xu, Delayed observer-based H∞ control for networked control ED systems. Neurocomputing, 179(2016), 101-109. [12] H. Zhang, Y. Shi, J. Wang, H. Chen, A New Delay-Compensation Scheme for Networked Control Systems in Controller Area Networks. IEEE Transactions on Industrial PT Electronics, 65(2018), 7239-7247. [13] R. A. Gupta, M. Y. Chow, Networked control system: overview and research trends. IEEE CE Transactions on Industrial Electronics. 57(2010) 2527-2535. [14] M. Vallabhan, S. Seshadhri, S. Ashok, S. Ramaswmay, R. Ayyagari, An analytical AC framework for analysis and design of networked control systems with random delays and packet losses. arXiv preprint arXiv: (2015). [15] H. S. Park, Y. H. Kim, D. S. Kim, W. H. Kwon, A scheduling method for network-based control systems. IEEE Transactions on Control Systems Technology. 10(2002) 318-330. [16] P. V. Zhivoglyadov, R. H. Middleton, Networked control design for linear systems. Automatica. 39 (2003) 743-750. 32 ACCEPTED MANUSCRIPT [17] G. P. Liu, Y. Xia, J. Chen, D. Rees, W. Hu, Networked predictive control of systems with random network delays in both forward and feedback channels. IEEE Transactions on Industrial Electronics. 54(2007) 1282-1297. [18] H. Zhang, Y. Shi, A. S. Mehr, Robust H∞ PID control for multivariable networked control systems with disturbance/noise attenuation. International Journal of Robust and Nonlinear CR IP T Control. 22(2012)183-204. [19] S. Li, G.P. Liu, Networked predictive control for nonlinear systems with stochastic disturbances in the presence of data losses. Neurocomputing, 194(2016) 56-64. [20] Camacho, Eduardo F., and Carlos Bordons Alba. Model predictive control. Springer Science & Business Media, 2013. AN US [21] J. M. Maciejowski, Predictive Control with Constraints. Pearson education; 2002. [22] T. Wang, H. Gao, J. Qiu, A combined adaptive neural network and nonlinear model predictive control for multirate networked industrial process control. IEEE Transactions on Neural Networks and Learning Systems, 27(2016) 416-425. [23] Z. Li, D. Sun, Y. Shi, L. Wang, A stabilizing model predictive control for networked control M system with data packet dropout. Journal of Control Theory and Applications. 7(2009) 281284. ED [24] P. M. Marusak, Advantages of an easy to design fuzzy predictive algorithm in control systems of nonlinear chemical reactors. Applied Soft Computing. 9(2009) 1111-1125. [25] D. Srinivasagupta, H. Schättler, J B. oseph, Time-stamped model predictive control: an PT algorithm for control of processes with random delays.Computers and Chemical Engineering. 28(2004) 1337-1346. CE [26] Q. Zhang, C. Lin, P. Chen, Time-stamped dynamic matrix control for networked control systems. In 2005 IEEE International Conference on Industrial Technology. (2005) 1159- AC 1163. [27] G. P. Liu, J. Mu, D. Rees, Networked predictive control of systems with random communication delay. In UKACC Intl. Conf. on Control.(2004). [28] J. Mu, G. P. Liu, D. Rees, Design of robust networked predictive control systems. In Proceedings of the 2005, American Control Conference. (2005) 638-643. 33 ACCEPTED MANUSCRIPT [29] P. Tang, C. W. de Silva, Stability and optimality of constrained model predictive control with future input buffering in networked control systems. American Control Conference. New York: IEEE Press. (2005) 1245 – 1250. [30] P. Varutti, B. Kern, T. Faulwasser, R. Findeisen, Event-based model predictive control for networked control systems. Proceedings of the 48th IEEE Conference on Decision and CR IP T Control, (2009) 567-572. [31] Lilly JH. Fuzzy control and identification. John Wiley & Sons; 2011. [32] Zaidi, Salman, and Andreas Kroll. "NOE TS fuzzy modelling of nonlinear dynamic systems with uncertainties using symbolic interval-valued data." Applied Soft Computing 57 (2017): 353-362. AN US [33] Khooban, Mohammad Hassan, Navid Vafamand, and Taher Niknam. "T–S fuzzy model predictive speed control of electrical vehicles." ISA transactions 64 (2016): 231-240. [34] Chang, Wen-Jer, and Chong-Cheng Shing. "Robust covariance control for discrete system by Takagi-Sugeno fuzzy controllers." ISA transactions 43, no. 3 (2004): 377-387. [35] D. Sáez, E. Kemerer, Fuzzy predictive control strategies and its application to a laboratory M tank. In European Control Conference (ECC), pp. 1063-1068, 2003. [36] S. Blažič, I. Škrjanc, Design and stability analysis of fuzzy model-based predictive control– ED a case study. Journal of Intelligent and Robotic Systems, 49(2007), 279-292. [37] R. Sivakumar, K. S. Manic, V. Nerthiga, R. Akila, K. Balu, Application of fuzzy model predictive control in multivariable control of distillation column. International Journal of PT Chemical Engineering and Applications, 1(2010), 38-42. [38] G. Tang, D. Huang, Z. Deng, TS fuzzy model based generalized predictive control of CE vehicle yaw stability. Kybernetes, 41(2012), 1261-1268. [39] M. H. Khooban, N. Vafamand, T. Niknam, T–S fuzzy model predictive speed control of AC electrical vehicles. ISA transactions, 64 (2016), 231-240. [40] I. Boulkaibet, K. Belarbi, S. Bououden, T. Marwala, M. Chadli, A new TS fuzzy model predictive control for nonlinear processes. Expert Systems with Applications, 88(2017), 132151. [41] C. H. Lu, C. C. Tsai, Generalized predictive control using recurrent fuzzy neural networks for industrial processes. Journal of process control, 17(2007), 83-92. 34 ACCEPTED MANUSCRIPT [42] Clarke, David W., C. Mohtadi, and P. S. Tuffs. "Generalized predictive control—Part I. The basic algorithm." Automatica23, no. 2 (1987): 137-148. [43] E. Zafiriou, M. Morari, Design of robust digital controllers and sampling-time selection for SISO systems. International Journal of Control, 44(1986), 711-735. [44] R. Scattolini, S. Bittanti, On the choice of the horizon in long-range predictive control— CR IP T some simple criteria. Automatica, 26(1990), 915-917. [45] B.W. Hogg, N. M. El-Rabaie, Generalized predictive control of steam pressure in a drum boiler. IEEE Transactions on Energy Conversion. 5(1990) 485-492. [46] M. Ellis, J. Liu, P. D. Christofides. Introduction. In Economic Model Predictive Control (2017) (pp. 1-19). Springer, Cham. AN US [47] Smith O. Closer control of loops with dead time. Chem. Eng. Prog., 1957, vol. 53, no. 5, pp. 217-219. [48] C. E. Garcia, M. Morari: Internal model control - 1. A unifying review and some new results. Ind. Engng. Chem. Process Des. Dev., 1982, vol. 21, pp. 308-323. [49] A. M. El-Nagar, Embedded intelligent adaptive PI controller for an electromechanical M system. ISA Transactions, 64 (2016) 314-327. [50] H. Zhang, Z. Li, Simulation of Networked Control System based on Smith Compensator ED and Single Neuron Incomplete Differential Forward PID. JNW, 6 (2011), 1675-1681. [51] A. Sakr, A. M. El-Nagar, M. El-Bardini, M. Sharaf, Fuzzy smith predictor for networked control systems. 11th International Conference on Computer Engineering & Systems AC CE PT (ICCES), 2016 (pp. 437-443). 35

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