SIP-Based Single Neuron Stochastic Predictive Control for Non ... - MDPI

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SIP-Based Single Neuron Stochastic Predictive Control for Non-Gaussian Networked Control Systems with Uncertain Metrology Delays Xinying Xu, Yalan Zhao, Mifeng Ren *, Lan Cheng and Mingyue Gong College of Information Engineering, Taiyuan University of Technology, Taiyuan 030024, China; [email protected] (X.X.); [email protected] (Y.Z.); [email protected] (L.C.); [email protected] (M.G.) * Correspondence: [email protected]; Tel.: +86-0351-6010051 Received: 21 May 2018; Accepted: 22 June 2018; Published: 26 June 2018







Abstract: In this paper, a novel data-driven single neuron predictive control strategy is proposed for non-Gaussian networked control systems with metrology delays in the information theory framework. Firstly, survival information potential (SIP), instead of minimum entropy, is used to formulate the performance index to characterize the randomness of the considered systems, which is calculated by oversampling method. Then the minimum values can be computed by optimizing the SIP-based performance index. Finally, the proposed strategy, minimum entropy method and mean square error (MSE) are applied to a networked motor control system, and results demonstrated the effectiveness of the proposed strategy. Keywords: networked control systems; data-driven predictive strategy; non-Gaussian random delays; non-Gaussian disturbances; survival information potential

1. Introduction In the last decade, network technology has dramatically been improved. More and more control systems are combined with network technologies [1–3]. Networked control systems (NCSs) are loop-locked control systems in which a closed loop is constituted by communication channels [4,5]. In fact, control system with communications is not a new concept in automation. From tele-operation to distributed control, control theory over a communication network has already been developed for more than 40 years. However, there are many factors that distinguish the current NCSs and previous control systems with communications. Two of them are the most significant: (1) in the previous control with communications, the network was specialized and dedicated for the stability of process operation and timeliness of information exchange, while in the current NCSs the network is general-purpose and public for various concurrent applications, and thus stable operation and real-time communication are no longer ensured; (2) their functionality has been diversified tremendously, and a variety of control and management or administrative functions have been developed, which is different from previous single control [6]. As a new area of control systems, NCSs have advantages of resource sharing, achieved remote monitored and adjusted, low cost, simple installation and high reliability [7,8], which has been applied to many fields such as tele-surgery, tele-manufacturing, museum guidance, space exploration, traffic control, health care and disaster rescue [9], but demands on complexity, diversity, and real-time performance for networked operations have brought new technological challenges to NCSs. Time delays exist inevitably in the communication net [10]. When sampling periods are longer than these delays, the influence of the delays is need not to be considered in general control systems. However, the network control system has high requirements on real-time, so delays induced

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via network communication are becoming more and more important to be analyzed and designed in the NCSs [11,12]. What’s more, the occurrence of disturbances can also degrade their control performance [13]. Therefore, the control of stochastic NCSs drew in large studies. The random network delay only in the forward channel in NCSs was considered in [14]. However, delays usually exist both in the forward and feedback channels, which make the control design and stability analysis much more challenging. In order to solve this problem, [15] proposed a control scheme for NCSs with random network delays in both channels, in which the model of random network-induced delays is utilized by Markov chains. In addition, a model-based predictive control algorithm was developed by Liu et al. [16,17] with the purpose of better control. However, these methods were based on the model. Moreover, the shortcomings of the accurately model building were overcome by data-driven predictive control method [18]. The above presented control methods only considered the random delays in NCSs, it is known that external disturbances are also inevitable in NCS, which may lead to the deterioration of control performance. One- and two-parameter control schemes were researched to track the performance of continuous systems with white Gaussian disturbances in [19,20]. The result was further generalized to other noisy channels of NCSs in [21], where achievable minimal tracking error is used to measure optimal tracking performance. However, these methods are mostly based on the assumption that the system variables follow a Gaussian distribution. Under such an assumption, all possible statistical information of a variable can be extracted from its mean and variance. In fact, most network systems are difficult to meet these pre-conditions because of the mixture of nonlinear factors or different disturbances, which results in non-Gaussian randomness even if the disturbances are of Gaussian types. Stochastic distribution control theory, developed by Wang [22], had been applied to the control of stochastic systems with non-Gaussian disturbances [23,24]. Based on this theory, Zhang et al. applied minimized zero mean entropy method to NCSs, and the corresponding controller was designed [25]. In [26], fault detection of networked control systems via minimum entropy observer was presented and in [27], a minimum entropy Luenberger observer was designed for linear time-invariant (LTI) system and Van der Pol Oscillator. However, system models in the studies mentioned above were all linear. As we all know, in practical engineering, nonlinear systems generally exist, and most of them meet the Lipschitz condition. Ren et al. [28] proposed an entropy-based control algorithm for nonlinear NCSs with non-Gaussian random disturbances and delays. Actually, some drawbacks exist in entropy: (1) it is shift-invariant (i.e., its value remains unchanged even if the location of distribution varies); (2) the value may be negative; (3) When PDF is non-existent, the definition will be ill-suited. In order to overcome these drawbacks, a single neuron control strategy for non-Gaussian stochastic systems based on the survival information potential (SIP) criterion was proposed by Chen et al. [29], in which control input was conservatively considered as a deterministic variable [28,30]. In fact, the randomness of control input exists in practical conditions, so paper [31] proposed a single neuron stochastic predictive PID control algorithm using SIP criterion, in which the performance index not only contains the SIP of tracking error, but also includes the SIP of control input. And it is for nonlinear systems with exact model. But it is usually difficult to obtain accurate model of NCSs. What’s more, the random delays are inevitable in NCSs. Therefore, a data-driven single neuron predictive control structure is constructed by SIP for NCSs with non-Gaussian disturbances and random delays in this paper. This paper is organized as follows: in Section 2, the control problem of the NCSs with non-Gaussian disturbances a time delays is formulated. Specifically, the performance index of SIP is formulated. Based on the proposed performance index, the single neuron stochastic predictive control method is mentioned in Section 3. The efficiency of the proposed control strategy is illustrated through applying method to networked motor control system in Section 4. The last section is a summary of this paper.

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2. Problem Formulation Entropy 2018, 20, 494

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NCSs are dynamic systems and each control loop is closed via a network communication channel, which induces possible random delays. It is known that random disturbances are inevitable inProblem NCSs. Both factors may affect NCSs’ control performance and stability. What’s more, there are 2. Formulation multiple unknown parameters and uncertainties in the NCSs, the system has strong nonlinear NCSs are dynamic systems and each control loop is closed via a network communication channel, characteristic, and its model is difficult to be established. Using the equivalent model and ignoring a which induces possible random delays. It is known that random disturbances are inevitable in NCSs. number of uncertainties are common solutions, which would greatly reduce the accuracy of model Both factors may affect NCSs’ control performance and stability. What’s more, there are multiple and seriously affect the control effect. Data-driven method just need the process data before the unknown parameters and uncertainties in the NCSs, the system has strong nonlinear characteristic, algorithm started, rather than the modeling of the process. The data-driven control model for the and its model is difficult to be established. Using the equivalent model and ignoring a number NCSs can be expressed as: of uncertainties are common solutions, which would greatly reduce the accuracy of model and yk =method f (uk , ωk , djust ,τ kneed ) (1) seriously affect the control effect. Data-driven the process data before the algorithm k started, rather than the modeling of the process. The data-driven control model for the NCSs can be where y and uk are system output and control input. ωk is a kind of random non-Gaussian expressedk as: disturbance. The independent signals, ydk k = and , are random and can be described by the beta f (uk ,τω (1) k k , d k , τk ) distribution respectively, their values are between 2 and 8. where yk and uk are system output and control input. ωk is a kind of random non-Gaussian disturbance. Based on the process described of NCSs above, the tracking error is: The independent signals, dk and τk , are random and can be described by the beta distribution respectively, their values are between 2 and ek =8.f (rk , uk , ωk , dk ,τ k ) (2) Based on the process described of NCSs above, the tracking error is: where rk is the set point. ek can be viewed as a function of dk , τ k and ωk . ek = f (rk , uk , ωk , dk , τk ) (2) f ( ⋅ ) g ( ⋅ ) Remark 1. and are unknown functions. The model is needless in the proposed method, where rk is the set point. ek can be viewed as a function of dk , τk and ωk . Equations (1) and (2) are utilized to depict the relations between and among error, disturbance and delays. Remark 1. f (·) and g(·) are unknown functions. The model is needless in the proposed method, Equations (1) In NCSs, the plant, controller, sensor, actuator and reference command are connected through a and (2) are utilized to depict the relations between and among error, disturbance and delays. network [32]. The scheme of NCSs is shown in Figure 1. Bounded random delays τ k and dk exist

in both channels to controller and controller to actuator respectively. The control input In NCSs, thefrom plant,sensor controller, sensor, actuator and reference command are connected through aunetwork [32]. The scheme of NCSs is shown in Figure 1. Bounded random delays τ and d d from single neuron predictive controller could be transferred to actuator after time. And kk k exist k in both channels from sensor controller to actuator respectively. control input uk ,toucontroller yk can then, the executive action =u k + dk , is and be utilized to control the process and then The output k could be transferred to actuator after dk time. And then, uek from single neuron predictive controller obtained. The purpose of this paper is to design a single neuron predictive controller to make the the executive action uk , uk = uek+dk , is utilized to control the process and then output yk can be obtained. ek , paper ek =rk −isy kto, approach y k the to zero as closely as possible, is tracking the measured tracking error The purpose of this design a single neuron predictive controllerwhere to make error  = y . e e e eoutput, , e = r − y , approach to zero as closely as possible, where y is the measured output, y = y k k k k k+τ k k k k +τk .

Figure 1. The schematic of NCSs with single neuron predictive controller. Figure 1. The schematic of NCSs with single neuron predictive controller.

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The single neuron controller is in use here. It is a nonlinear processing unit with multiple inputs and single output, which can be shown as Equation (3): 3

uk = uk+1 + K ∑ ωlk xlk /kΣk k

(3)

l =1

3

where ∑k = ∑ ωlk , K > 0 is the proportional coefficient of the neuron. ωlk (l = 1, 2, 3) is the weight l =1

corresponding to each input xlk (l = 1, 2, 3):      x 3k

x1k = ek x2k = ek − ek−1 = ek − 2ek−1 + ek−2

(4)

In Figure 1, delays dk , τk and disturbance ωk follow non-Gaussian distributions. According to the statement above, Gaussian randomness of system could be sufficiently characterized by mean and variance (covariance), which are unavailable for systems with non-Gaussian randomness. Therefore, a criterion should be used to deal with non-Gaussian randomness. 3. SIP-Based Performance Index 3.1. Overview of Entropy Criterion In the previous works, MSE can control well under the assumption that system variables are of Gaussian type, but it is not suitable for non-Gaussian randomness. Entropy is usually chosen for the performance index to reject non-Gaussian randomness of systems. There are many kinds of the entropy [33], but Shannon entropy certainly plays an important role. For a continuous random variable X with PDF γX ( x ), the definition of Shannon entropy is described: HS ( X ) =

Z +∞ −∞

γX ( x ) log γX ( x )dx

(5)

Then, a well-known generalization of Shannon entropy is Renyi entropy defined by: Hα ( X ) =

1 log 1−α

Z +∞ −∞

α γX ( x )dx

(6)

where α is the entropy order and α > 0, α 6= 1. It can be shown that Renyi entropy will reduce to Shannon entropy when α → 1 . The argument of the log in the Renyi entropy (6) is called the information potential1 (IP): Vα ( X ) =

Z +∞ −∞

α γX ( x )dx

(7)

Since Renyi’s entropy Hα ( X ) is a monotonic decreasing function of the IP Vα ( X ) when α > 1, and Vα ( X ) is simpler for computing. From Equation (7), it can be seen that some drawbacks exist in IP: (1) it is shift-invariant (i.e., its value remains unchanged even if the location of distribution varies); (2) the value may be negative; (3) When PDF is non-existent, the definition will be ill-suited. 3.2. SIP Criterion In order to overcome drawbacks of entropy, a more general measure, survival information potential (SIP) is chosen for the performance index. According to [29], SIP can be defined as follows:

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Definition 1. For a random variable X ∈ Rm , SIP of order α(α > 0) is defined by Sα ( X ) =

Z

α

Rm +

F | X | ( x )dx

(8)

where F | X | ( x ) = P(| X | > x ) = E[ I (| X | > x )] is the survival function (or equivalently, the distribution function) of the random variable | X |, and R+ = { x ∈ R : x ≥ 0}. I (·) is the indicator function. There are some advantages that can be concluded from the definition of SIP: (1) it is not shift-invariant (i.e., its value remains unchanged even if the distribution location varies); (2) it has good robustness because of more regular distribution function; (3) it is easy to compute values from the sample data to avoid computing the kernel and choosing the kernel width. The control input is also non-Gaussian according to the single neuron controller Equations (3) and (4), so it is improper to consider conservatively the control input as a deterministic variable. What’s more, predictive control strategy is used to obtain the optimal control input here, which has achieved well performance, as above. Both tracking error and control input are considered, the following predictive performance index J is utilized to optimize the weights of single neuron: P

M

i =1

j =1

J=

∑ S α ( e k + i −1 ) + λ ∑ S α ( u k + j −1 )

(9)

where ek+i is the i-step ahead prediction of the tracking error relating to disturbance and delays. uk+i is the i-step ahead prediction of the control input. P and M are the prediction horizon and the control horizon, respectively, and M ≤ P. It’s worth noting that the control input would not change after M steps, i.e., uk+ j−1 = uk+ M−1 ( j > M ). It is not easy to describe the distribution of the error ek with integrated first-principle model in actual industry. Therefore, the data-driven empirical SIP, instead of the theoretical SIP, is used here. Suppose there are error samples (e1k , e2k , · · · , e Nk ) at instant k, and without the loss of generality, that |e1k | ≤ |e2k | ≤ · · · ≤ |e Nk |, the empirical SIP of error as follows [29]: Sˆα (ek ) =

N

∑

µm |emk |

(10)

m =1

where µm =



N − m +1 N

α

−



N −m N

α

.

According to Equation (10), the empirical SIP Sˆα (uk+m−1 ) of control input can similarly also be formulated. The performance index Jˆ with empirical SIP can be expressed as: Jˆ =

P

N

∑∑

i =1 m =1

M

µm |emk+i−1 | + λ ∑

N

∑

j =1 m =1

µm umk+ j−1

(11)

 α  α −m . where µm = N −Nm+1 − N N The calculation of empirical SIP needs samples. Taking into account the realization of actual operation, we handle samples with an oversampling method [34]. Assuming the over-sampling rate is N, it is a positive integer. Figure 2 displays the structure diagram of NCSs with an over-sampling technique. Like in general closed-loop system, NCSs contain process model G p and controller of which the control period is T. A zero-order holder is behind controller, which is used to generate piecewise control input. It is noting that the output sampling time equals ∆ = T/N, and then the signal y∆ (m)(m = k, k + ∆, k + 2∆, · · · , k + ( N − 1)∆) can be generated,

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while a conventional output signal is sampled at a period of T. Specially, under the effect of zero-order Entropy 2018, x FORsignal PEER REVIEW 6 of 13 holding, the20, input u∆ (m) is actually generated as: u∆ (m)u =( mu)(=k )u, ((km),(= · ·, k, k++ −Δ1))∆) m k, = kk ,+ k +∆,Δk, k++2∆, 2 Δ ,· ( N( N − 1) Δ

(12) (12)

Figure 2. Over-sampling technique in NCSs. Figure 2. Over-sampling technique in NCSs.

The over-sampling method is presented above for an NCS. Particularly, the value of T is one The over-sampling is presented above foratan NCS. Particularly, the value T is until one will beofheld second. In simple terms,method the control input is sampled a period of T and then second. In simple terms, the control input is sampled at a period of T and then will be held until next next sampling due to zero-order holding, but the output is sampled at a period of Δ = T N . sampling due to zero-order holding, but the output is sampled at a period of ∆ = T/N. Therefore, Therefore, N samples of output can be sampled after T time, and then, N samples of error N samples of output can be sampled after T time, and then, N samples of error between N samples between N samples of output and corresponding reference trajectory can be obtained. Similarly, of output and corresponding reference trajectory can be obtained. Similarly, substituting N error substituting N error samples into Equations (3) and (4), N control input samples also can be samples into Equations (3) and (4), N control input samples also can be obtained. After substituting N obtained. After substituting N samples of error and control input into Equation (11), the samples of error and control input into Equation (11), the performance index can be calculated through performance index can be calculated through empirical SIP. empirical SIP. OptimalControl ControlAlgorithm Algorithm 4.4.Optimal Toachieve achieve optimal weights of single the single neuron, considering the structure of theneuron single To thethe optimal weights of the neuron, considering the structure of the single neuron adaptive from Equation (3), the performance index from Equation (9) will be: adaptive controllercontroller from Equation (3), the performance index from Equation (9) will be: w * = arg min J w∗k k= argminJ w k ∈ 3 M

(13) (13)

wk ∈R3M

T

where w k =  w1, k , w2, k , w3, k , w1, k +1 , w2, k +1 , w3, k +1 , , w1, k + M −1 , w2, k + M −1 , w3, k + M −1  ∈  3 M . T where wk = [w1,k , w2,k , w3,k , w1,k+1 , w2,k+1 , w3,k+1 , · · · , w1,k+ M−1 , w2,k+ M−1 , w3,k+ M−1 ] ∈ R3M . There are many optimal algorithms solving the optimal of Equation (13), among There are many optimal algorithms for for solving the optimal valuevalue of Equation (13), among which which the gradient descent method is the earliest and most commonly used optimization method the gradient descent method is the earliest and most commonly used optimization method [35]. [35]. The gradient descent method is simple. When the objective function is convex, the gradient The gradient descent method is simple. When the objective function is convex, the gradient descent descentismethod is the global solution. idea is that the gradient negative direction gradient direction current method the global solution. Its idea isIts that make themake negative of currentofposition position search direction, which is the fastestdirection. decline direction. it also is as “the as search as direction, which is the fastest decline Therefore,Therefore, it also is known asknown “the steepest steepest descent method”. Based on the gradient descent method and Equation (11), the optimal descent method”. Based on the gradient descent method and Equation (11), the optimal weight can be weight can updated by: be updated by: ∂ Jˆ (14) wk+1 = wk − η ∂Jˆkk w k +1 = w k − η ∂wk (14) ∂w k where η > 0 denotes adaptation gain. where > 0 denotes Theηoptimal inputadaptation uk of NCSsgain. can be computed according to Equation (3). The following steps describe process of proposed control in detail: uk of NCSs can algorithm be computed according to Equation (3). The following steps Thethe optimal input describe the process of proposed control algorithm in detail: Step 1: Giving initial value of weight vector w 0 and operation time kmax . Setting parameters of step-size η and α . Step 2: Calculate the performance index at present instant from Equation (11).

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Step 1: Giving initial value of weight vector w0 and operation time kmax . Setting parameters of Step η3:and Obtain the update weight vector w k +1 through the gradient descent method by step-size α. Step 2: Calculate the performance index at present instant from Equation (11). Equation (14). Step 3: Obtain the update weight vectorinto wk+Equation method 1 through Step 4: Substitute the optimal weight vector (3),the andgradient then the descent next control inputby is Equation (14). obtained. Step 5: 4: Substitute the optimal vectoraccording into Equation (3), and then the next input, controlwhich input Step Process outputs could weight be collected to corresponding control is obtained. could be used to update the next performance index. Then implement repeatedly processes from 5: Process outputs couldstep, be collected control input, which could Step Step 2 to Step 5 for the next time , until k =tokcorresponding . k = k + 1according max be used to update the next performance index. Then implement repeatedly processes from Step 2 to Step 5 for the next time step, k = k + 1, until k = kmax . 5. Simulation Results 5. Simulation Results A networked DC motor control test rig is utilized to demonstrate the performance of the proposed method,DC which consists oftest tworig NetControllers and a DC motor with its driver, as shown in A networked motor control is utilized to demonstrate the performance of the proposed Figure 3. Both the networked controller board (NCB) and networked implementation board (NIB) method, which consists of two NetControllers and a DC motor with its driver, as shown in Figure 3. have thenetworked same hardware structure. NIB isand onnetworked the DC motor, which is located in the Huazhong Both the controller board (NCB) implementation board (NIB) have the same University of Science and Technology (Wuhan, China), while the NCB is located remotely at the hardware structure. NIB is on the DC motor, which is located in the Huazhong University of Science North China Electric Power University (Beijing, China). The two parts are connected by the internet and Technology (Wuhan, China), while the NCB is located remotely at the North China Electric Power which contains non-Gaussian delays, by and communication protocol between University (Beijing, China). The disturbaces two parts areand connected thethe internet which contains non-Gaussian them is the UDP. The test rig adopts a 57BL (3)-10-30 DC servo motor, which is made by disturbaces and delays, and the communication protocol between them is the UDP. The test rigZhuhai adopts Motion Control Motor Co. Ltd. (city, country). The model parameters and mathematical model of a 57BL (3)-10-30 DC servo motor, which is made by Zhuhai Motion Control Motor Co. Ltd. The model the considered motor can be obtained Ren et al. parameters andDC mathematical model of thefrom considered DC[28]. motor can be obtained from Ren et al. [28].

Figure 3. 3. Networked Networked DC DC motor motor control control system. system. Figure

The goal goalofofthe thecontrol control design is not to reach the target value of rotational of the design is not onlyonly to reach the target value of rotational speed ofspeed the armature but also to reduce the variation of the corresponding variable and armature r = 3000 rad / min r = 3000 rad/min but also to reduce the variation of the corresponding variable and random delays. random delays. Considering of the convenience of analysis, round trip timeFigures delay is measured. Considering the convenience analysis, the round trip timethe delay is measured. 4 and 5 show Figures andthe 5 distribution show samples andtime the delay, distribution of theThe timeoutput delay,isrespectively. output is samples 4and of the respectively. affected by a The non-Gaussian affected by a non-Gaussian measurement (ω ) . The PDF of ω is given by: measurement disturbance (ω ). The PDF ofdisturbance ω is given by: (  − 1 −1 μ +11 λ λ   0.01 λ+λµ+ + + 1, − x− )μ x, x)µ∈, (0,0.01) 0.01 ββ((λλ + 1,μµ++1)1) x x(0.01 (0.01 x ∈ (0, 0.01)  γω ( x ) γ=ω ( x)=  0, otherwise 0,otherwise 

(15) (15)

R1 1 µ where ββ(λ 1) =x λ0(1x−λ x(1)μ− dx, where (λ++ 1, 1, µ μ+ + 1)= dxx, )λ = 4,λμ = = 34,. µ = 3. 0 Three statistical information indexes, mean value, standard deviation and the SIP value of three Three statistical information indexes, mean value, standard deviation and the SIP value of control methods are listed in Table 1, which illustrates that the proposed SIP-based control algorithm is three control methods are listed in Table 1, which illustrates that the proposed SIP-based control better than the entropy-based method. The control performances of the networked DC motor control algorithm is better than the entropy-based method. The control performances of the networked DC systems are shown in Figures 6–12. motor control systems are shown in Figures 6–12. The performances of the proposed optimal control algorithm based on SIP, MSE and the minimum entropy control (MEE) in controlling a networked DC motor control system are compared. The conventional MSE method could only control well based on the assumption that the

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The performances of the proposed optimal control algorithm based on SIP, MSE and the minimum entropy control (MEE) in controlling a networked DC motor control system are compared. Entropy 2018, 20, x FOR PEER REVIEW 8 of 13 The conventional MSE method could only control well based on the assumption that the system variables follow a Gaussian the disturbances anddisturbances delays are of non-Gaussian types system variables follow adistribution, Gaussian but distribution, but the and delays are of in Figure 3. Therefore, the MEE strategy, instead of MSE, is utilized to deal with non-Gaussian non-Gaussian types in Figure 3. Therefore, the MEE strategy, instead of MSE, is utilized to deal with randomness. the MEE strategy somestrategy shortcomings as mentioned above, SIP was non-GaussianHowever, randomness. However, thehas MEE has some shortcomings as so mentioned proposed to overcome the entropy drawbacks. Figure 6 shows the response of the DC motor above, so SIP was proposed to overcome the entropy drawbacks. Figure 6 shows the responsespeed, of the the line is thethe response of is thethe DC motor speed the proposed method. It’s obvious that DCsolid motor speed, solid line response of theusing DC motor speed using the proposed method. the SIP-based control method could control well and stabilize the speed around the set point, which It’s obvious that the SIP-based control method could control well and stabilize the speed around the has and its speed fluctuation small.fluctuation Accordingistosmall. the SIP value from 1, set rapid point, convergence, which has rapid convergence, and itsisspeed According to Table the SIP the randomness is reduced and approaches zero with proposed method. Although the fluctuation value from Table 1, the randomness is reduced andthe approaches zero with the proposed method. of speed with the entropy method and SIP value is also small, it converges to the target value at Although the fluctuation of speed with the entropy method and SIP value is also small, it converges atovery slow speed. thespeed. MSE However, control strategy could notstrategy control could well, which has high the target value atHowever, a very slow the MSE control not control well, standard deviation. It is clear that the DC motor speed control system with SIP criterion has superior which has high standard deviation. It is clear that the DC motor speed control system with SIP performance that based on entropy controller. Figure shows the weights of the single criterion hasthan superior performance thanand thatMSE based on entropy and7 MSE controller. Figure 7 shows neuron. Obviously, the required control force under the proposed control is smaller than that with the weights of the single neuron. Obviously, the required control force law under the proposed control entropy and MSE seen in Figure 8. Figure 9 demonstrates the SIP error value is decreasing overall law is smaller than that with entropy and MSE seen in Figure 8. Figure 9 demonstrates the SIP error with progress of overall time, which represents that control is getting better.performance In order to valuethe is decreasing with the progress of the time, whichperformance represents that the control demonstrate the SIP value to is demonstrate close to zero during stationary SIP performance is getting better. In order the SIP the value is close phase, to zerothe during the stationaryindexes phase, after two moments are showed in the amplification region. the SIP performance indexes after two moments are showed in the amplification region. Table Table1.1.Control Controlperformance performanceindexes. indexes.

Disturbances Method Method Disturbances MSE MSE Non-Gaussian Entropy Entropy Non-Gaussian SIP SIP

The The Mean Mean Value Value 49.9153 49.9153 49.6720 49.6720 49.8978 49.8978

The SIPSIP Value TheStandard StandardDeviation Deviation The The Value 22.6583 3.0000 22.6583 3.0000 10.9471 0.4997 10.9471 0.4997 8.1570 0.1054 8.1570 0.1054

80

70

Delay(ms)

60

50

40

30

20

10

0

50

100

150

200 250 300 Samples number

350

400

Figure4. 4. Data Data graph graph of of time time delays. delays. Figure

450

500

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0.6 0.6 0.6 0.5 0.5 0.5 Probability Probability Probability

0.4 0.4 0.4 0.3 0.3 0.3 0.2 0.2 0.2 0.1 0.1 0.1 0 0 0 00 0

10 10 10

20 20 20

30 30 30

40 40 40 Delay(ms) Delay(ms) Delay(ms)

50 50 50

60 60 60

70 70 70

80 80 80

Figure5.5. 5.The Thedistribution distributionof oftime timedelays. delays. Figure Figure The distribution of time delays. Figure 5. The distribution of time delays.

set point point set set point MSE based control control MSE based MSE based control Entropy based control Entropy based control Entropy based control SIP based control SIP based control SIP based control

120 120 120 100 100 100

Speed(r/s) Speed(r/s) Speed(r/s)

80 80 80 60 60 60 40 40 40 20 20 20 0 0 0 -20 -20 -20

0.5 0.5 0.5

1 1 1

1.5 1.5 1.5

2 2 2

2.5 2.5 2.5 time(s) time(s) time(s)

3 3 3

3.5 3.5 3.5

4 4 4

4.5 4.5 4.5

5 5 5

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Entropy Entropy2018, 2018,20, 20,xxFOR FORPEER PEERREVIEW REVIEW Entropy Entropy2018, 2018,20, 20,494 x FOR PEER REVIEW MSE based control MSE based control Entropy based control Entropy based control MSE based control SIP based control SIP based control Entropy based control SIP based control

Voltage(V) Voltage(V) Voltage(V)

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The4-D 4-Dmesh meshplots plotsofofthe thePDFs PDFsof oftracking trackingerror errorare areshown shownin inFigures Figures10 10and and11. 11.Comparing Comparing the PDFs of tracking error are shown in Figures 10 and 11. Comparing The Figures 10 with 11, the shape of the PDFs of tracking error in Figure 10 becomes narrower and Figures1010with with 11, the of the PDFs of of tracking error and Figure Figure 11, shape the shape of the PDFs tracking errorininFigure Figure10 10becomes becomes narrower narrower and sharperalong along with sampling time; indicates that the proposed proposed control system has small sharper along with sampling time; itit indicates that the control system aa small sharper with sampling time; it indicates that the proposed control system has a smallhas uncertainty uncertainty in its closed loop operation. The final PDF of three control laws are presented in Figure uncertainty its operation. closed loopThe operation. The of three laws are in its closed in loop final PDF of final threePDF control laws control are presented in presented Figure 12, in in Figure which 12, in which PDFs of proposed method is sharper than that under MSE law and entropy-based 12, in which PDFs of proposed method is sharper than that under MSE law and entropy-based PDFs of proposed method is sharper than that under MSE law and entropy-based control law, which controllaw, law, which indicates thatthe the proposed controlbetter strategy canobtain obtainbetter betterperformance. performance. control indicates that proposed strategy can indicates thatwhich the proposed control strategy cancontrol obtain performance. Conclusions 6.6.Conclusions Inthis this work, new single single neuron neuronmethodology control methodology methodology for systems networked systems with with In work, a new neuron control for networked with systems non-Gaussian In this work, aa single new control for networked non-Gaussian delay and disturbances is presented. Instead of the entropy criterion, a SIP-based delay and disturbances is presented. Instead of the entropy criterion, a SIP-based performance function non-Gaussian delay and disturbances is presented. Instead of the entropy criterion, a SIP-based performance function is adopted to design the controller, and the tracking control problem is adopted to design theiscontroller, the tracking control problem is converted an optimization performance function adoptedand to design the controller, and the tracking into control problem isis converted into an optimization one. In order to calculate the tracking error SIP, the oversampling one. In order to an calculate the tracking SIP, to thecalculate oversampling methoderror is used simplicity, converted into optimization one. error In order the tracking SIP,here. the For oversampling method usedhere. here.For Foris simplicity, the optimization problem solvedby byusing usingdescent thewell-known well-known the optimization problem solved bythe using the well-known andisis effective gradient method. method isisused simplicity, optimization problem solved the and effective gradient descent method. The proposed control strategy is applied in a networked DC The control strategy applied in a networked DC motor control and proposed effective gradient descentismethod. The proposed control strategy is system. applied Itinisaconfirmed networkedfrom DC motor control system. It is confirmed from the simulation results that the proposed SIP-based the simulation theconfirmed proposed from SIP-based predictive strategy can achieve a betterSIP-based tracking motor control results system.that It is the simulation results that the proposed predictivestrategy strategy canachieve achieve betterTo tracking performance than MSEand and entropy. Tosum sumhas up,the the performance than MSE and entropy. sum up, the proposed optimal control algorithm predictive can aabetter tracking performance than MSE entropy. To up, proposed optimal control algorithm has the following advantages: following proposed advantages: optimal control algorithm has the following advantages: (1) This Thisstrategy strategyisisdata-driven, data-driven,which whichcan canavoid avoidmodeling modelingerror errorand andcomplexity complexityin inbuilding buildingthe the (1) model. model.

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(1) (2) (3)

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This strategy is data-driven, which can avoid modeling error and complexity in building the model. A general measure, SIP, is used to formulate the performance index to design the control algorithm, which could overcome shortcomings of entropy. Oversampling method is used to calculate the SIP of tracking error, which is more practical for online controller design problem.

Extending SIP to Multiple-Input-Multiple-Output (MIMO) NCS is a relevant research problem. In the MIMO system, there is an interaction between the variables. Some preprocessing of output variables should be done to eliminate multi-colinearity. Extending our approach to MIMO systems will be included in our future research topics. Author Contributions: X.X., M.R. and L.C. conceived the project. X.X. searched relevant literatures. X.X. carried out the formal analysis. L.C. and Y.Z. provided the simulation results. Y.Z. finished original draft. X.X., M.R. and M.G. wrote review & editing. Correspondence and requests for materials should be addressed to M.R. All authors have read and approved the final manuscript. Funding: This work was supported by the Natural Science Foundation of China (61503271, 61374052) and Shanxi (201701D221112). Conflicts of Interest: The authors declare no conflict of interest.

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