J. Cent. South Univ. (2019) 26: 881-892

DOI: https://doi.org/10.1007/s11771-019-4057-z

Robust sliding mode control for uncertain networked control system with two-channel packet dropouts

ZHANG Yu(张驭)¹, REN Li-tong(任立通)², XIE Shou-sheng(谢寿生)¹,ZHANG Le-di(张乐迪)¹, ZHOU Bin(周彬)¹

1. Aeronautical Engineering Institute, Air Force Engineering University, Xi’an 710038, China;

2. Unit 94314 of Chinese PLA, Zhengzhou 450003, China

Central South University Press and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Abstract: A robust sliding mode control algorithm is developed for a class of networked control system with packet dropouts in both sensor-controller channel and controller-actuator channel, and at the same time mismatched parametric uncertainty and external disturbance are also taken into consideration. A two-level Bernoulli process has been used to describe the packet dropouts existing in both channels. A novel integral sliding surface is proposed, based on which the H_∞ performance of system sliding mode motion is analyzed. Then the sufficient condition for system stability and robustness is derived in the form of linear matrix inequality (LMI). A sliding mode controller is designed which can guarantee a relatively ideal system dynamic performance and has certain robustness against unknown parameter perturbations and external disturbances. The results from numerical simulations are presented to corroborate the validity of the proposed controller.

Key words: networked control system; sliding mode control; packet dropout; uncertainty

Cite this article as: ZHANG Yu, REN Li-tong, XIE Shou-sheng, ZHANG Le-di, ZHOU Bin. Robust sliding mode control for uncertain networked control system with two-channel packet dropouts [J]. Journal of Central South University, 2019, 26(4): 881–892. DOI: https://doi.org/10.1007/s11771-019-4057-z.

1 Introduction

With the emergence of networked control systems, where sensors, controllers, and actuators can be viewed as self-contained entities exchanging information over a common communication network, various benefits can be witnessed. However, before the benefits of networked control can be realized in practice, some obstacles must be overcome, and packet dropout remains one of the most crucial problems that cannot be bypassed. Packet dropouts may occur frequently as a result of imperfect communication mechanism or limited network bandwidth [1, 2]. Therefore, to achieve a further progress towards the adoption of networked control in practical systems, the effect of packet dropouts on system stability and performance must be minimized.

It has been extensively reported over the past few years of researches on networked control systems with packet dropouts [3–6], with two main aspects serving as the focus of most researches, which are system modeling [7] and controller design [8, 9]. In the aspect of system modeling, one of the most commonly used methods is to describe packet dropouts as a random Bernoulli or Markov process [10]. In the aspect of controller design, methods such as H_∞ control [11], predictive control [12], optimal control [13] and sliding mode control [14] are relatively popular as reported in recent literatures. Among the above-mentioned methods, sliding mode control (SMC) stands out for its strong robustness against uncertainties because during the sliding phase (i.e., when system state reaches a specially designed sliding surface), the motion of system state will be totally independent from the influence of uncertainties [15]. Therefore, the sliding mode controller is designed to drive the system trajectory onto the sliding surface or into its neighborhood in finite time and remains there for all subsequent time. The advantages of SMC make it a fairly suitable choice when designing controllers for uncertain systems. XIONG [16] proposed a sliding mode predictive controller, where the predictive approach was applied to compensating the impact of time delay and packet dropout in the network while SMC was employed to overcome the network induced noise. However, the limitation of their work is also obvious that packet dropouts were equivalently transformed as time delay in their work to simplify the analysis process, which makes it impossible to describe explicitly the relation between system performance and packet dropout parameters such as packet dropout rate and the largest permitted number of packet dropouts. JIA [17] proposed a sliding mode control algorithm for a class of random packet dropout system where a packet dropout compensation strategy was adopted to overcome the impact of state loss and a packet dropout rate dependent integral sliding surface was proposed together with a novel sliding mode controller. A similar controller was applied to tackling the packet dropout problem in a networked control system with Markovian jumping parameters and the sliding mode reaching condition depending on packet dropout rate was given [18]. However, the above mentioned literatures only consider the packet dropouts in sensor-controller channel while the effect of controller-actuator packet dropouts have not been discussed. Actually, packet dropouts in controller-actuator channel are also inevitable and harmful. When packet dropouts occur in controller-actuator channel, which means the latest control signal is lost, zero-input or hold-input strategy [19] will be taken by the actuator as an alternative but it is obvious that such input strategies are not able to provide the most suitable control signals for the system. Therefore, controller-actuator channel and sensor-controller channel packet dropouts can both result in system performance deterioration and instability. But to the authors’ best knowledge, the research still remains open on sliding mode controller design for systems with two-channel packet dropouts. When designing sliding mode controller for networked control systems, it is of great importance to figure out the explicit effect of two-channel packet dropouts on the performance and stability of the closed-loop system and this motives the present study.

In this paper, a robust sliding mode controller is designed for a class of networked control system with two-channel packet dropouts, unknown parameter perturbations and external disturbances. The networked control system model is presented with packet dropouts described as two-level Bernoulli process in both sensor-controller channel and controller-actuator channel, based on which the main contributions of this work are presented 1) a new sliding surface is proposed which can reflect the influence of packet dropouts on system stability; 2) the sliding surface parameter is deduced, which guarantees system asymptotical stability and H∞ robustness during the sliding phase; 3) a sliding mode controller is designed that can drive the system trajectory onto the designed sliding surface and maintain the trajectory within its neighborhood for all subsequent time and the effect of parametric uncertainty and external disturbances is also overcome.

The remainder of this article is organized as follows. Section 2 introduces the control system model. Design of the novel sliding surface is presented in Section 3, followed by the design of robust sliding mode controller in Section 4. Numerical simulation results are shown in Section 5. Section 6 presents some conclusions.

2 Problem formulations

The structure of a class of networked control system with two-channel packet dropouts is shown in Figure 1, where time delay is assumed to be small enough to be neglected and the breakers of the two channels represent the condition of packet dropouts. The sensor and the actuator are both time driven, while the controller is event driven. Data are transmitted in single packet and packet disorder will not happen. Then the discrete system model can be described as follows:

        (1)

where k≥0 is the sampling instant,is system state variable;  is control input; ΔA is mismatched system parameter perturbation; is external disturbance whose upper bound is known; A, B and D are coefficient matrices with appropriate dimensions. In addition, matrix pair (A, B) is assumed to be controllable and B is a matrix of full column rank, which satisfies rank(B)=m≤n. Moreover, all system state variables are measurable.

Figure 1 Structure of a class of networked control system

In this paper, system parameter perturbation ΔA is mismatched but satisfies:

                            (2)

where  are known constant matrices with appropriate dimensions; matrix F(k) is unknown but satisfies F^T(k)F(k)≤I. To simplify the expression, F(k) is expressed as F in the following discussions.

The packet dropout distribution of the two channels is assumed to obey the two-level Bernoulli random process. Denote  as the sensor signal that reaches the controller and  as the control signal that reaches the actuator at time instant k. The hold strategy is adopted here as:

              (3)

where  are parameters used to describe the packet dropout state of the two channels, that is, when ρ_k=0, it is believed that no packet dropout occurs in the sensor-controller channel; when ρ_k=1, it represents that the data packet is lost when being transmitted from sensor to controller. Parameter θ_k is the counterpart of ρ_k in controller-actuator channel. The probability distribution of ρ_k, θ_k is

             (4)

where  are defined as the packet dropout rates of the two channels, respectively. In this paper, the packet dropout rate is assumed to be a constant value.

According to Eq. (3), the system model considering two-channel packet dropouts is described as follows:

        (5)

Then, a sliding mode controller will be designed for system Eq. (5), which is able to guarantee the asymptotical stability of the closed- loop system despite the existence of packet dropouts in both sensor-controller channel and controller-actuator channel and is robust to uncertainties.

3 Design of robust sliding surface

To suppress the effect of packet dropouts on system stability, a novel integral like sliding surface with packet dropout compensation function is constructed as follows:

  (6)

where G is the sliding surface parameter matrix to be designed. It is required as a precondition that GB is nonsingular. This condition can be satisfied by defining G=B^TP,  because matrix B is assumed to be column full rank. It can be obtained from Eqs.(6) and (1) that,

 (7)

When system state trajectory is driven onto the sliding surface, there is s(k+1)=s(k)=0, and then the equivalent control law for ideal sliding mode motion can be obtained as

         (8)

where  Substituting Eq. (8) into Eq. (1) yields the sliding mode motion equation:

              (9)

Since packet dropouts are considered, an augmented system state variable is defined as  According to Eqs. (3), (8) and (9), the augmented sliding mode motion equation is obtained as

               (10)

where    

It can be seen from Eq. (10) that the augmented sliding mode motion will be affected by the packet dropout parameter  and  parameter perturbation ΔA as well as external disturbance d(k), which means that the system is not invariant upon uncertainties. To solve this problem, the to-be- designed sliding surface parameter G is expected to achieve the following target: when d(k)=0, the augmented sliding mode motion (10) can be mean square asymptotically stable despite the existence

of two-channel packet dropouts; when  and zero initial condition is considered, system (10) will have the H_∞ performance defined as where γ is the prescribed H_∞ performance index.

Lemma 1 [20]. For any real vectors a, b and matrix X>0 of appropriate dimensions, the following inequality holds

                 (11)

Lemma 2 [21]. Let H, G and F be real matrices of appropriate dimensions with F satisfying F^TF≤I. Then for arbitrary scalar ε>0, we have

HFG+G^TF^TH^T≤εHH^T+ε^–1G^TG             (12)

Theorem 1. For the networked control system (1) with packet dropout model (5), sliding surface (6) is adopted. Then for the prescribed positive constant γ, the augmented sliding mode motion (10) is asymptotically stable with a noise attenuation level γ upon the external disturbance d(k), if there exist symmetric positive definite matrix P and scalar ε>0 satisfying the following linear matrix inequality (LMI):

 (13)

where 

Thus, the sliding surface parameter matrix can be obtained as G=B^TP.

Proof: Define the Lyapunov function as

                    (14)

Along the trajectory of system (10), we have,

        (15)

Choose   α, β>0, and then it follows from the definition of Θ and  that:

      (16)

It follows from Eqs. (15) and (16) that:

            (17)

Define  then we have

           (18)

According to Eq. (10) and Lemma 1, there is

 (19)

where,,,.

In the same way, it is obtained that:

 (20)

where   

Substituting Eq. (19) and Eq. (20) into Eq. (18), yields the following inequality according to Schur’s complement,

                (21)

where

It can be obtained from Eq. (21) that when Ψ<0, there is

 (22)

It is deduced from Lemma 2 that Eq. (21) holds if and only if there exists a scalar ε>0 satisfying

                (23)

where

According to Schur’s complement, Eq. (23) is equivalent to Eq. (13), which makes Eq. (22) hold. Then when  it can be obtained by summing both sides of Eq. (22) from 0 to ∞ that

    (24)

that is,

              (25)

Thus it is concluded that the sliding mode motion Eq. (10) is asymptotically mean square stable, while the prescribed noise attenuation level γ is achieved.

Remark 1. Here it can be seen that the effect of two-channel packet dropouts on networked control system can be reflected by the packet dropout parameter   and their coupling parameters, such as α, β, λ₁, λ₂, etc. Additionally, the introduction of packet dropout compensation terms  and  enables the designed sliding surface parameter G to be robust against the two-channel packet dropouts.

Corollary. It is noticed that Theorem 1 is presented as a feasible solution for the two-channel packet dropout problem considered in this paper. In order to illustrate the system performance under different packet dropout condition, the H_∞ index γ can be assumed to be an unknown value, and then the feasibility problem Eq. (13) will be converted into the following minimization problem:

                      (26)

Solving problem (26) yields the distribution of largest tolerated packet dropout rates in the two channels.

4 Design of reaching law based sliding mode controller

Ideally, system state is expected to be driven precisely onto the sliding surface, and then maintain s(k+1)=s(k)=0, so that the system state can “slide” smoothly on the sliding surface until it reaches the equilibrium point. However, on one hand, for discrete systems, it is scarcely possible to get a positive integer to make exactly s(x(k))=0. So system state can only be driven into a neighborhood of the sliding surface. On the other hand, the existence of packet dropouts and uncertainties further degrades system dynamic performance, which makes it fairly important to design a controller that can guarantee a relatively ideal system dynamic performance despite the existence of packet dropouts and uncertainties.

It can be seen that there are unknown uncertainty terms  and  in Eq. (7) that must be compensated during the design of controller to achieve a relatively ideal reaching phase performance. A sliding mode controller is thus proposed here based on the uncertainty upper bound.

Defineand then it is obtained according to Eq.(2) and the bound of external disturbance d(k) that the following inequality holds,

                      (27)

with  Δ_dmax= and  being the upper bound vector of the external disturbance.Furthermore, we define with  and being the corresponding elements of vectors  and  respectively. Then the sliding mode controller is proposed based on the exponential reaching law [22]:

                   (28)

where T is the system sampling period;  and  are coefficient matrices of the reaching law. It must be satisfied that e_i>0, q_i>0 and 1–q_iT>0 (i=1, 2, …, m) when designing the controller. In addition, is a specially designed term to compensate the effect of parameter perturbation and external disturbance so that the performance of the closed-loop system can be guaranteed.

Theorem 2. For the networked control system Eq.(1) with packet dropout model Eq. (5), the sliding mode control law Eq. (28) is adopted. Then the system trajectory can be driven onto the designed sliding surface Eq. (6) in finite time and stay in a reasonable neighborhood of it for all subsequent time.

Proof: It follows from Eqs. (1), (3) and (7) that

 (29)

Then it can be obtained from formula (27) that

  (30)

Substituting formula (30) into Eq. (29), yields:

 (31)

Therefore, the reaching condition is satisfied, which means that system (1) will be driven onto the sliding surface and stay there for all subsequent time.

5 Simulation results and analysis

Consider the following uncertain system in the form of Eq.(1):

.

Subsequently, the parameters for packet dropouts, parameter perturbation and external disturbance are illustrated as

The specific quantitative settings of the three above-mentioned uncertainties are given as follows:

1) Two-channel random packet dropouts

The packet dropout rates of the sensor- controller channel and controller-actuator channel are  and  respectively.

2) System parameter perturbation

According to Eq. (2), define F(k)=sin(k·T), with T being the system sampling period and here it is given that T=0.02 s. It is obvious that F^T(k)F(k)≤I.

3) External disturbance

Define

                       (32)

whereis the Gaussian white noise;  is the upper bound of noise intensity and here we define N_i=0.1, i=1, 2, 3.

It is given that γ=0.68, then according to Theorem 1, the sliding surface parameter matrix can be obtained by solving LMI (13), that is , ε=0.4888.

Then it follows from formula (26) that  and

Furthermore, it is obtained that

In the same way, we have  

The reaching law parameters are defined as e_i=0.2 and q_i=10 (i=1, 2).

To verify the robustness of the proposed sliding surface, a sliding surface designed based on pole assignment method [23] is introduced as a comparison. Then, for the reduced order normal form of system (9), the pole assignment method based coefficient matrix is calculated as

with expected pole as λ_g=–1. Leave the other parameters remaining the same as the proposed controller and substitute G_p into Eq. (28), the first contrastive controller can be obtained, which is denoted as P-RSMC (pole-assignment based reaching law sliding mode controller). Meanwhile, another contrastive controller is introduced to reflect the advantage of the proposed controller in dynamic performance. Compared with the proposed controller given by Eq. (28), the control law of the second contrastive controller does not have the uncertainty compensation term  and we denote the second contrastive controller as RSMC (reaching law sliding mode controller). Then, for the easiness of presentation, the proposed controller given by   Eq. (28) is denoted as UR-RSMC (uncertainty bound based robust reaching law sliding mode controller).

The initial system state is chosen as  and the simulation results are shown in Figures 2–4. It can be seen that the system with the proposed controller has better dynamic performance and smaller steady state error than those with the two contrastive controllers. In addition, the system with the proposed controller presents a shorter convergence time with the sliding mode motion being achieved in 0.6 s. In contrast, system with RSMC shows a longer settling time. It takes the system 1.1 s to reach the sliding mode motion state, the reason of which can be explained as follows: the absence of uncertainty compensation term limits RSMC’s ability to suppress the effect of parameter perturbation and external disturbance during the reaching phase, which leads to the falling of convergence speed. However, since the same robust sliding surface has been adapted, RSMC shares a same error level with UR-RSMC. Compared with the first two controllers, P-RSMC shows obvious disadvantage in steady state error both for state variable x(k) and switching function s(k), the reason of which can be explained as follows: the pole assignment method is conducted on the nominal system without parameter perturbation and external disturbance, thus the obtained sliding surface parameter cannot guarantee a small steady state error when uncertainties exist.

Figure 2 Response curves of system state variables:

Figure 3 Convergence curves of switching function s(k):

Figure 4 Convergence curves of control input u(k):

In this paper, it is assumed that data are transmitted in single packet, which means the information of one variable will be transmitted through one single data packet without separation, so in other words, when packet dropout occurs, the sensor or controller information of this time instant is lost entirely. Therefore, when system suffers from a high packet dropout rate, consecutive packet dropouts will occur which will severely worsen the control effectiveness. Considering such condition, the packet dropout rate is redefined as  then the system packet dropout situation is presented in Figure 5, where μ is the packet dropout description value. It can be seen that higher packet dropout rate results in severer consecutive packet dropout. In Figure 6, the system state response is given under the packet dropout condition A certain degree of performance degradation can be witnessed but system’s robustness to uncertainties is still maintained, which verifies that the proposed method is suitable for the situation of consecutive packet dropouts.

In order to obtain the maximum permissive packet dropout rate, LMI Eq. (26) is solved iteratively following the change of  and  and the change curve of optimal H_∞ performance index  is shown in Figure 7. It is seen that the higher packet the dropout rate, the bigger the value of  which leads to a worse robustness. When the packet dropout rate increases to an extremely high level, the optimal solution of LMI (26) does not exist. Therefore, Figure 7 can reflect the distribution of the maximum permissive packet dropout rate.

6 Conclusions

A robust sliding mode control algorithm is developed for a class of uncertain networked control system with packet dropouts in both sensor-controller channel and controller-actuator channel. The following conclusions are obtained:

1) A robust sliding surface is proposed which is proved to have better robustness against two- channel packet dropouts, parameter perturbation, and external disturbances, thus the robustness of the system sliding mode motion can be well guaranteed.

2) A sliding mode controller with uncertainty compensation term is proposed, through which the influence of uncertainties on system reaching phase is effectively restrained, resulting in a favorable system dynamic performance.

Figure 5 Packet dropout distribution under different packet dropout rates:

Figure 6 Response curves of system state variables when

Figure 7 Changing curve of optimal H_∞ performance index

3) When packet dropout rate increases, consecutive packet dropout will occur more frequently, thus the dynamic and steady state performance of the closed-loop system will be affected to a certain degree. However, as long as Theorem 1 is solvable, the robustness of the closed-loop system can still be guaranteed, which illustrates the validity of the proposed method.

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(Edited by HE Yun-bin)

中文导读

存在双通道丢包的不确定网络控制系统鲁棒滑模控制

摘要：针对一类同时存在传感器-控制器以及控制器-执行器丢包的网络控制系统，提出了一种鲁棒滑模控制算法。首先，基于随机Bernoulli建模方法建立了丢包系统模型。在此基础上提出了一种具有丢包补偿项的鲁棒滑模面，以及基于干扰上界的趋近率滑模控制器，并应用Lyapunov稳定性分析方法，分析了系统滑动模态的稳定性以及控制器的到达条件。仿真结果表明，所设计的控制器能够有效抑制双通道数据丢包对系统性能的影响，同时对非匹配不确定性和外部干扰具有良好的鲁棒性。

关键词：网络控制系统；滑模控制；数据丢包；不确定性

Foundation item: Projects(51476187, 51506221, 51606219) supported by the National Natural Science Foundation of China

Received date: 2017-12-13; Accepted date: 2018-09-20

Corresponding author: ZHANG Yu, PhD; Tel: +86-13325382530; E-mail: frank_sharon1314@126.com; ORCID: 0000-0001-6609- 583X