Performance of Networked Control Systems

Abstract

Data packet dropout is a special kind of time delay problem. In this paper, predictive controllers for networked control systems (NCSs) with dual-network are designed by model predictive control method. The contributions are as follows. (1) The predictive control problem of the dual-network is considered. (2) The predictive performance of the dual-network is evaluated. (3) Compared to the popular networked control systems, the optimal controller of the new NCSs with data packets dropout is designed, which can minimize infinite performance index at each sampling time and guarantee the closed-loop system stability. Finally, the simulation results show the feasibility and effectiveness of the controllers designed.

1. Introduction

With the rapid development of computer networks technology, the control system based on NCSs has become one of the hot research tasks in the current international control field. As stability analysis of NCSs subjected to packet dropping has received much attention in [1–3], various approaches for the delay issue in NCSs have been presented in [4–8], and so on. Compared to the traditional control systems, the main advantages of NCSs are lower cost, simpler installation, and higher reliability [9–11]. Because of these attractive advantages, typical application of these systems ranges from a wide field, such as automotive [12], mobile [13], and advanced aircraft [14]. However, the introduction of communication networks in the control loops makes the analysis and design of NCSs complex. For example, network-induced delays and data packet dropout problem may be inevitable during transmitting communication.

Data packet dropout is a special kind of time delay problems. Data packet dropout which is a kind of uncertainty that may happen due to node failures or network congestion is a common problem in networked systems. This loss will deteriorate the performance and may even cause the system to be unstable. Recently, the effect of data packet dropout on the stability and performance of NCSs has received great attention. In [15], the stability of a linear networked control system in the presence of dropped packets was studied. A stability analysis of model-based NCSs can be found in [2, 16–19], where an additional model was used for estimating the plant state between transmission times and generating a control signal. In [20], though turning the model of NCSs into an asynchronous dynamic system, hybrid system technology has been used to handle the system with time delay and data dropouts. A stability condition was obtained for the Try-Once-Discard networked protocol in [3]. In [21], Hadjicostis and Touri analyzed the performance when lost data were replaced by zeros. In [22, 23], Ling and Lemmon posed the problem of optimal compensator design for the case when data loss was independent and identically distributed. Reference [24] addresses the random time delays and packet losses issues of NCSs within the framework of jump linear systems with mode-dependent time delays. Jump linear systems with Markov chains [25, 26] also were used to analyze the effect of dropouts on system stability and performance. In [27], a delay-dependent stability condition was presented for discrete-time jump time delay system where the time delay was dependent on the system mode. In [28], the problem of stability analysis and controller design has been proposed based on a new model with packet dropouts.

Model predictive control (MPC) can now be found in a wide variety of application areas including chemicals, food processing, and so on. MPC is also an effective method to incorporate the input and output constraints into online optimization, which increases the possibility of its application in the synthesis and analysis of NCSs [29]. In [30], the MPC strategy for multivariable plants was presented. Wu et al. [31] introduced MPC into NCSs with time delay and designed an optimal control rule. In [4], the networked predictive control with modified MPC was proposed.

In this paper, the contributions are as follows. (1) The predictive control problem of the dual-network is considered. (2) The predictive performance of the dual-network is evaluated. (3) Compared to the popular networked control systems, the optimal controller of the new NCSs with data packets dropout is designed, which can minimize infinite performance index at each sampling time and guarantee the closed-loop system stability.

The remainder of this paper is organized as follows. The problem descriptions and model of NCSs with packet dropouts are given in Section 2. The optimization method of the new closed-loop systems is proposed in Section 3. The stability analysis is given in Section 4. A numerical example and an industrial example show the effectiveness of the proposed method in Section 5, and some conclusions are given in Section 6.

2. Problem Descriptions and Modeling of NCSs with Packet Dropouts

Consider that NCSs model which is shown in Figure 1, sensors, controllers, and actuators are connected by networks, and we suppose that the communication link between primary sensor and controller (𝑆₁/𝐶₁) is ideal. Based on such a structure, the problem of packet dropouts in network transmission mainly exists in the actuator and secondary sensor and controller (𝑆₂/𝐶₂ and 𝐶₂/𝐴).

Figure 1 

Networked control system model.

In Figure 1, 𝑆_𝑖(𝑖=1,2), 𝐶_𝑖(𝑖=1,2), and 𝑃_𝑖(𝑖=1,2) denote the sensor, controller, and plant of primary and secondary, respectively, and 𝐴 is actuator. 𝑢₁(𝑘) and 𝑢₂(𝑘) are outputs of primary and secondary controllers at sampling time 𝑘. 𝑢₂(𝑘) is input of actuator at sampling time 𝑘, 𝑑^𝐶2𝐴_𝑘 is the quantity of packet dropouts between sampling current time 𝑘 and the latest communicate time successfully (𝑘−𝑑^𝐶2𝐴_𝑘) on 𝐶₂/𝐴 side, and 𝑑^𝑆2𝐶2_𝑘 is the quantity of packet dropouts at current time 𝑘 and the latest communicate successfully (𝑘−𝑑^𝑆2𝐶2_𝑘) on 𝑆₂/𝐶₂ side. The quantity of packet dropouts on 𝑆₂/𝐶₂ and 𝐶₂/𝐴 sides is assumed to satisfy 𝑑_𝑚≤𝑑^𝑆2𝐶2_𝑘≤𝑑_𝑀,𝜏_𝑚≤𝑑^𝐶2𝐴_𝑘≤𝜏_𝑀, (1) where 𝑑_𝑚, 𝑑_𝑀, 𝜏_𝑚, and 𝜏_𝑀 are constant positive scalars representing the minimum and maximum quantities of packet dropouts on 𝑆₂/𝐶₂ and 𝐶₂/𝐴 sides, respectively, where 𝑑^𝑆2𝐶2_𝑘 and 𝑑^𝐶2𝐴_𝑘 are two independent Markov chains. Without loss of generality, define 0≤𝑑^𝑆2𝐶2_𝑘≤𝑑_𝑀,0≤𝑑^𝐶2𝐴_𝑘≤𝜏_𝑀. (2)

Suppose that under the condition 𝑑^𝑆2𝐶2_𝑘=𝑖, the probability that the state of packet dropouts at time 𝑘+1 is 𝑗 (𝑑^𝑆2𝐶2_𝑘+1=𝑗) is Pr𝑑^𝑆2𝐶2_𝑘+1=𝑗∣𝑑^𝑆2𝐶2_𝑘=𝑖=𝜋_𝑖𝑗∀𝑖,𝑗∈𝝌_𝟏, (3) where 𝝌_𝟏={0,1,2,…,𝑑_𝑀}. That is also to say that the transition probability of 𝑑^𝑆2𝐶2_𝑘 jumping from mode 𝑖 to 𝑗 is (3), and the transition probability of 𝑑^𝐶2𝐴_𝑘 jumping from mode 𝑟 to 𝑠 is Pr𝑑^𝐶2𝐴_𝑘+1=𝑠∣𝑑^𝐶2𝐴_𝑘=𝑟=𝜆_𝑟𝑠∀𝑟,𝑠∈𝝌_𝟐, (4) where 𝝌_𝟐={0,1,2,…,𝜏_𝑀}, 𝜋_𝑖𝑗≥0, ∑^𝑑𝑀_𝑗=0𝜋_𝑖𝑗=1, ∑^𝑑𝑀_{𝑗=0,𝑗≠𝑖}𝜋_𝑖𝑗=1−𝜋_𝑖𝑖, and 𝜆_𝑟𝑠≥0, ∑^𝜏𝑀_𝑠=0𝜆_𝑟𝑠=1.

For analysis convenience and without loss of generality, we suppose that Markov chains jump no more than one step. Thus, Markov chain transferring probability matrix is satisfied [28]. Consider 𝜋_𝑖𝑗𝜆=0,if𝑗≠𝑖+1,𝑗≠0,_𝑟𝑠=0,if𝑠≠𝑟+1,𝑠≠0. (5)

And the two equations are used to produce the definition of data packet dropout. 𝑖 denotes the quantity of the packet dropouts at time 𝑘−1. 𝑗 denotes the quantity of packet dropouts at time 𝑘. The system’s conditional probability of data packets dropout is equal to zero when the difference of the quantity of packet dropouts between time 𝑘 and 𝑘−1 is not equal to 1 or there is no dropout at time 𝑘.

Some assumptions of NCSs are introduced as follows. (1)The sensors, controllers, and actuators are clock-driven.(2)The buffers are big enough to hold all the data arrived, and rule of the buffer is last-in-first-out.(3)Transmission link of primary control loop is ideal and data packet dropouts only happen in secondary control loop.(4)The transition of NCSs is single-packet transmission and no timing sequence disordered.

The linear time-invariant discrete-time models of primary and secondary plants in Figure 1 are described as follows: 𝑃₁∶𝑥₁(𝑘+1)=Φ₁𝑥₁(𝑘)+Γ₁𝑦₂(𝑘),𝑦₁(𝑘)=𝐶₁𝑥₁(𝑘),𝑃₂∶𝑥₂(𝑘+1)=Φ₂𝑥₂(𝑘)+Γ₂𝑢₂(𝑘),𝑦₂(𝑘)=𝐶₂𝑥₂(𝑘), (6) where 𝑥₁(𝑘) and 𝑥₂(𝑘) are the states of primary and secondary plants, respectively, at sampling time 𝑘, 𝑢₂(𝑘) is the input of secondary plant 𝑃₂ at sampling time 𝑘, and Φ₁, Γ₁, 𝐶₁, Φ₂, Γ₂, 𝐶₂ are known constant matrixes with appropriate dimensions.

Due to the existence of data packet dropouts, transmission link in network cannot normally communicate. The controller and actuator can use the buffer rule, named first-in-last-out, to pick up signal [28]. Take the controller, for example; when a secondary sensor data 𝑥₂(𝑘) is false to transmit, the secondary controller gets the latest data 𝑥₂(𝑘−1) from buffer and uses it as 𝑥₂(𝑘) to calculate new control input. Otherwise, the new sensor data 𝑥₂(𝑘) will be saved to buffer and used by the secondary controller as 𝑥₂(𝑘). Thus, 𝑥₂(𝑘)=𝑥₂(𝑘),𝑑^𝑆2𝐶2_𝑘=0,𝑥₂(𝑘−1),𝑑^𝑆2𝐶2_𝑘>0. (7)

Similarly 𝑢₂(𝑘)=𝑢₂(𝑘),𝑑^𝐶2𝐴_𝑘𝑢=0,₂(𝑘−1),𝑑^𝐶2𝐴_𝑘>0. (8)

It can easily derive 𝑥₂(𝑘)=𝑥₂𝑘−𝑑^𝑆2𝐶2_𝑘. (9)

From the model of NCSs, as shown in Figure 1, 𝑢₁(𝑘) is the output of primary controller at sampling time 𝑘 as 𝑢₁(𝑘)=𝐹₁𝑥₁(𝑘), (10) where 𝐹₁ is to be designed by MPC method.

𝑢₂(𝑘) is the output of secondary controller as 𝑢₂(𝑘)=𝑢₁(𝑘)+𝐹₂𝑑^𝑆2𝐶2_𝑘𝑥₂(𝑘), (11) where 𝐹₂(𝑑^𝑆2𝐶2_𝑘) is to be designed by MPC method.

Substituting (9) and (10) into (11), we have 𝑢₂(𝑘)=𝐹₁𝑥₁(𝑘)+𝐹₂𝑑^𝑆2𝐶2_𝑘𝑥₂𝑘−𝑑^𝑆2𝐶2_𝑘. (12)

Therefore, (8) can be rewritten as 𝑢₂(𝑘)=𝐹₁𝑥₁(𝑘)+𝐹₂𝑑^𝑆2𝐶2_𝑘𝑥₂𝑘−𝑑^𝑆2𝐶2_𝑘,𝑑^𝐶2𝐴_𝑘𝑢=0,₂(𝑘−1),𝑑^𝐶2𝐴_𝑘>0. (13)

In order to simplify the expression of the closed-loop control systems, a function 𝛼=^{0,𝑑𝐶2𝐴𝑘=0}_{1,𝑑𝐶2𝐴𝑘>0} is introduced, which is dependent on whether packet dropped or not, instead of the quantity of packet dropouts. Combining (6) and (13), it can obtain that 𝑢₂(𝑘)=𝛼𝑢₂(𝑘−1)+[1−𝛼]𝐹₁𝑥₁(𝑘)+[1−𝛼]𝐹₂𝑑^𝑆2𝐶2_𝑘𝑥₂𝑘−𝑑^𝑆2𝐶2_𝑘,𝑥₁(𝑘+1)=Φ₁𝑥₁(𝑘)+Γ₁𝐶₂𝑥₂(𝑘),𝑥₂(𝑘+1)=Φ₂𝑥₂(𝑘)+Γ₂𝛼𝑢₂(𝑘−1)+Γ₂[1−𝛼]𝐹₁𝑥₁(𝑘)+Γ₂[1−𝛼]×𝐹₂𝑑^𝑆2𝐶2_𝑘𝑥₂𝑘−𝑑^𝑆2𝐶2_𝑘. (14)(15)(16)

Combining (10), (14), (15), and (16) and augmenting the state vectors, the new resulting closed-loop control systems and the augmenting vector (18) formed by predictive controller (10) and (14) designed by MPC method are as follows: 𝑥(𝑘+1)=𝐴𝑑^𝐶2𝐴_𝑘𝑥(𝑘)+𝐵𝑑^𝑆2𝐶2_𝑘,𝑑^𝐶2𝐴_𝑘𝑥𝑘−𝑑^𝑆2𝐶2_𝑘,𝑢(𝑘)=𝑢₁(𝑘)𝑢₂(𝑘)=𝐶𝑑^𝐶2𝐴_𝑘𝑥(𝑘)+𝐷𝑑^𝑆2𝐶2_𝑘,𝑑^𝐶2𝐴_𝑘𝑥𝑘−𝑑^𝑆2𝐶2_𝑘, (17)(18) where 𝑥(𝑘)=𝑥^Τ₁(𝑘)𝑥^Τ₂(𝑘)𝑢^Τ₂(𝑘−1)^Τ, 𝑥𝑘−𝑑^𝑆2𝐶2_𝑘=𝑥^Τ₁𝑘−𝑑^𝑆2𝐶2_𝑘𝑥^Τ₂𝑘−𝑑^𝑆2𝐶2_𝑘𝑢^Τ₂𝑘−𝑑^𝑆2𝐶2_𝑘−1^T,𝐴𝑑^𝐶2𝐴_𝑘=Φ₁Γ₁𝐶₂0Γ₂[1−𝛼]𝐹₁Φ₂Γ₂𝛼[1−𝛼]𝐹₁,𝐵𝑑0𝛼_{𝑆2𝐶2𝑘},𝑑^𝐶2𝐴_𝑘=0000Γ₂[1−𝛼]𝐹₂𝑑^𝑆2𝐶2_𝑘00[1−𝛼]𝐹₂𝑑^𝑆2𝐶2_𝑘0,𝐶𝑑^𝐶2𝐴_𝑘=𝐹₁[]𝐹001−𝛼₁,𝐷𝑑0𝛼_{𝑆2𝐶2𝑘},𝑑^𝐶2𝐴_𝑘=0[]𝐹0001−𝛼₂𝑑^𝑆2𝐶2_𝑘0. (19)

Remark 1. The new closed-loop control systems (17) and the augmenting vector (18) formed by predictive controller (10) and (14) are linear jumping systems, where their communications are described by Markov chain which is the description of the quantity of packet dropouts at sampling time 𝑘 on 𝐶₂/𝐴 and 𝑆₂/𝐶₂ sides. The value of 𝛼 depends on whether the designed control signal is successfully transmitted or not. Therefore we can use the results of linear jumping system with delay to analyze this class of NCSs with packets dropped.

3. Optimum Analysis Based on MPC

Assume that predictive horizon 𝑝=∞, control horizon 𝑞=∞, and the 𝑚 steps control sequences 𝑢(𝑘+𝑚∣𝑘), 𝑚=0,1,2,…,∞, are computed by minimizing the following performance function: 𝐽_∞=^∞_𝑚=0‖𝑥(𝑘+𝑚∣𝑘)‖²_𝑄+‖𝑢(𝑘+𝑚∣𝑘)‖²_𝑅. (20)

The norm terms in the performance function are defined as ‖𝑥‖²_𝑄=𝑥^Τ𝑄𝑥. (21)

The exact measurement state of NCSs at each sampling time 𝑘 is 𝑥(𝑘∣𝑘)=𝑥(𝑘). (22)

By using the control input given in (18), the first control move is 𝑢(𝑘)()𝑑=𝑢𝑘∣𝑘=𝐶^𝐶2𝐴_𝑘∣𝑘𝑥()𝑑𝑘∣𝑘+𝐷^𝑆2𝐶2_𝑘∣𝑘,𝑑^𝐶2𝐴_𝑘∣𝑘𝑥𝑘−𝑑^𝑆2𝐶2_𝑘∣𝑘∣𝑘. (23)

According to (17) and (22), the predicted state at time 𝑘+𝑚 is obtained as (24) which is predicted based on the exact measurement state 𝑥(𝑘∣𝑘). Consider 𝑥()𝑑𝑘+𝑚+1∣𝑘=𝐴^𝐶2𝐴_{𝑘+𝑚∣𝑘}𝑥(𝑘+𝑚∣𝑘)+𝐵𝑑^𝑆2𝐶2_{𝑘+𝑚∣𝑘},𝑑^𝐶2𝐴_{𝑘+𝑚∣𝑘}×𝑥𝑘+𝑚−𝑑^𝑆2𝐶2_{𝑘+𝑚∣𝑘}∣𝑘, (24) where 𝑑^𝑆2𝐶2_{𝑘+𝑚∣𝑘} is model-dependent time invariable delay on 𝑆₂/𝐶₂ side, which is 𝑚-step ahead prediction based on the measurement time 𝑘. 𝑑^𝐶2𝐴_{𝑘+𝑚∣𝑘} is the predicted quantity of packet dropouts at time 𝑘+𝑚.

And the key of predictive controller rule is to compute the matrixes 𝐹₁ and 𝐹₂(𝑑^𝑆2𝐶2_𝑘) in (14). The predicted controller at time 𝑘+𝑚 based on the first control move 𝑢(𝑘∣𝑘) is 𝑢(𝑘+𝑚∣𝑘)=𝐶𝑑^𝐶2𝐴_{𝑘+𝑚∣𝑘}𝑥()𝑑𝑘+𝑚∣𝑘+𝐷^𝑆2𝐶2_{𝑘+𝑚∣𝑘},𝑑^𝐶2𝐴_{𝑘+𝑚∣𝑘}𝑥𝑘+𝑚−𝑑^𝑆2𝐶2_{|𝑘𝑘+𝑚}∣𝑘, (25) where 𝐶(𝑑^𝐶2𝐴_𝑘) and 𝐷(𝑑^𝑆2𝐶2_𝑘,𝑑^𝐶2𝐴_𝑘) are matrixes which contain the control gain matrixes 𝐹₁ and 𝐹₂(𝑑^𝑆2𝐶2_𝑘), respectively.

Next we will introduce a method to solve the optimal control sequence which can minimize the following performance index at each sampling time: min_{𝑢()𝑘+𝑚∣𝑘,𝑚=0,1,2,…,∞}𝐽_∞=^∞_𝑚=0𝑥^Τ(𝑘+𝑚∣𝑘)𝑄𝑥(𝑘+𝑚∣𝑘)+𝑢^Τ(𝑘+𝑚∣𝑘)𝑅𝑢(𝑘+𝑚∣𝑘). (26)

𝑄 and 𝑅 are symmetric positive definite weight matrixes.

In order to simplify, let 𝑑^𝑆2𝐶2_{𝑘+𝑚∣𝑘}=𝜐, 𝑑^𝐶2𝐴_{𝑘+𝑚|𝑘}=𝜍, 𝑑^𝑆2𝐶2_{𝑘+𝑚+1∣𝑘}=𝜐₁.

First, consider a Lyapunov function candidate 𝑉(𝑋(𝑘+𝑚∣𝑘),𝜐) with 𝑋(𝑘+𝑚∣𝑘)=[𝑥^Τ(𝑘+𝑚∣𝑘),𝑥^Τ(𝑘+𝑚−1∣𝑘),…,𝑥^Τ(𝑘+𝑚−𝜐∣𝑘)]^Τ as follows: 𝑉(𝑋(𝑘+𝑚∣𝑘),𝜐)=𝑉₁+𝑉₂+𝑉₃, (27) where 𝑉₁(𝑋(𝑘+𝑚∣𝑘),𝜐)=𝑥^Τ()𝑃(𝜐)𝑥(),𝑉𝑘+𝑚∣𝑘𝑘+𝑚∣𝑘₂(𝑋(𝑘+𝑚∣𝑘),𝜐)=^{𝑘+𝑚−1}_{𝑙=𝑘+𝑚−𝜐}𝑥^Τ()(),𝑉𝑙∣𝑘𝑆𝑥𝑙∣𝑘₃(𝑋())=𝑘+𝑚∣𝑘,𝜐1−𝜋_𝑚⁰_{𝜃=−𝑑𝑀+1}^{𝑘+𝑚−1}_{𝑙=𝑘+𝑚+𝜃}𝑥^Τ(𝑙∣𝑘)𝑆𝑥(𝑙∣𝑘). (28)

Matrixes 𝑃(𝜐) and 𝑆 are positive definite with appropriate dimensions.

Second, suppose that at sampling time 𝑘, 𝑉(𝑋(𝑘+𝑚∣𝑘),𝜐) satisfies (29) for any 𝑥(𝑘+𝑚∣𝑘) and 𝑢(𝑘+𝑚∣𝑘). Consider 𝑉𝑋(𝑘+𝑚+1∣𝑘),𝜐₁(𝑋())𝑥−𝑉𝑘+𝑚∣𝑘,𝜐≤−^Τ(𝑘+𝑚∣𝑘)𝑄𝑥(𝑘+𝑚∣𝑘)+𝑢^Τ(𝑘+𝑚∣𝑘)𝑅𝑢(𝑘+𝑚∣𝑘). (29)

For the infinite performance index 𝐽_∞(𝑘) to be finite, we must set 𝑥(∞∣𝑘)=0; hence, 𝑉(𝑥(∞∣𝑘),𝑑^𝑆2𝐶2_∞∣𝑘)=0. Summing (29) from 𝑚=0 to 𝑚=∞, it can be obtained that 𝐽_∞(𝑘)≤𝑉𝑋(𝑘∣𝑘),𝑑^𝑆2𝐶2_𝑘∣𝑘. (30)

Therefore, the infinite optimization problem at time 𝑘 has transformed into optimizing upper function 𝑉(𝑋(𝑘∣𝑘),𝑑^𝑆2𝐶2_𝑘∣𝑘) at each sampling time 𝑘. As a standard in MPC, only the first control move 𝑢(𝑘|𝑘) is implemented, and at the next sampling time 𝑘+1, the state 𝑥(𝑘+1) is measured and the optimization is repeated to compute matrixes 𝐹₁ and 𝐹₂(𝑑^𝑆2𝐶2_𝑘).

Third, a sufficient condition for existing of a set of control sequence is 𝐽_∞<Υ, (31) where Υ is a suitable nonnegative coefficient to be minimized.

Before proceeding, the following lemma needs to be introduced.

Lemma 2 (Schur Complement Lemma). Given constant matrixes 𝑍₁, 𝑍₂, 𝑍₃, where 𝑍₁=𝑍^Τ₁, 𝑍₂=𝑍^Τ₂, then 𝑍₁+𝑍^Τ₃𝑍₂⁻¹𝑍₃<0 holds if ^𝑍1𝑍Τ3_{𝑍3−𝑍2}<0 or ^{−𝑍2𝑍3}_𝑍Τ3𝑍1<0.

Assumption 3. The values of 𝑑^𝑆2𝐶2_𝑘 and 𝛼 are known to the controller.

Theorem 4. Suppose the states 𝑥(𝑘∣𝑘),𝑥(𝑘−1∣𝑘),…,𝑥(𝑘−𝑑_𝑀∣𝑘) of system (17) are measured; then, the control rule in (18) makeing (29) and 𝐽_∞<Υ hold, if existing matrixes 𝑃(𝜐)>0, 𝑆>0, 𝐶(𝜍), 𝑋(𝜐)>0, 𝐷(𝜐,𝜍), 𝐹, and scalar Υ>0 make the following optimization feasible: minΥ (32)
subject to 𝑥^Τ_{𝑘+𝑚∣𝑘}𝑃(𝜐)𝑥_{𝑘+𝑚∣𝑘}+^{𝑘+𝑚−1}_{𝑙=𝑘+𝑚−𝑑𝑘𝑆𝐶}𝑥^Τ_𝑙∣𝑘𝑆𝑥_𝑙∣𝑘+1−𝜋_𝑚⁰_{𝜃=−𝑑𝑀+1}^{𝑘+𝑚−1}_{𝑙=𝑘+𝑚+𝜃}𝑥^Τ_𝑙∣𝑘𝑆𝑥_𝑙∣𝑘≤Υ,_{(𝜐)+−𝑃1+𝜇𝑆+𝑄0𝐴Τ(𝜍)𝐶Τ(𝜍)0−𝑆𝐵Τ()𝐷𝜐,𝜍Τ()𝐴(𝜍)𝐵()−𝑃𝜐,𝜍𝜐,𝜍−1(𝜐)0𝐶(𝜍)𝐷()𝜐,𝜍0−𝑅−1}<0, (33)(34)
or (𝜐)+𝐴(𝜍)𝐵(𝜍)𝐼̃(𝜍)+𝐽̃(𝜍)𝐹𝐻−𝑃1+𝜇𝑆+𝑄∗∗∗0−𝑆∗∗−Λ∗₁𝐾(𝜍)𝐹𝐻0−𝑅⁻¹<0,𝑃𝐼𝐼𝑋≥0,𝑃𝑋=𝐼, (35) where 𝜋_𝑚=min{𝜋_𝑖𝑖,𝑖∈𝝌_𝟏}, 𝑃(𝜐)=∑^𝑑𝑀_{𝑗=𝑑𝑚,𝑗≠𝑖}𝜋_𝑖𝑗𝑃(𝑗), 𝑖 denotes the quantity of packet dropouts, 𝑖∈𝝌_𝟏, 𝝌_𝟏={0,1,2,…,𝑑_𝑀}, 𝜇=𝑑_𝑀(1−𝜋_𝑚), 𝐴(𝜍)=[√𝜋_𝑖0𝐴(𝜍),√𝜋_𝑖1𝐴(𝜍),…,𝜋_𝑖𝑑𝑀𝐴(𝜍)]^Τ, 𝐵(𝜍)=[√𝜋_𝑖0𝐵(𝜍),√𝜋_𝑖1𝐵(𝜍),…,𝜋_𝑖𝑑𝑀𝐵(𝜍)]^Τ, Λ=diag(𝑋₀,𝑋₁,…,𝑋_𝑖), 𝑖∈(0,1,2,…,𝑑_𝑀),𝑋(𝜐)=𝑃⁻¹(𝜐), Θ=diag[−[1+(1−𝜋_𝑚)(𝑑_𝑀−1)𝑆]⁻¹,−[1+(1−𝜋_𝑚)(𝑑_𝑀−2)𝑆]⁻¹,…,−[(1−𝜋_𝑚)𝑆]⁻¹],𝑥_𝑝=[𝑥(𝑘−1∣𝑘),𝑥(𝑘−2∣𝑘),…,𝑥(𝑘−𝑑_𝑀+1∣𝑘)]^T, 𝐹=[𝐹₁,𝐹₂].

In order to express conveniently, let 𝑥_{𝑘+𝑚∣𝑘}=𝑥(𝑘+𝑚∣𝑘), 𝑥_{𝑘+𝑚−𝜐∣𝑘}=𝑥(𝑘+𝑚−𝜐∣𝑘), 𝑢_{𝑘+𝑚∣𝑘}=𝑢(𝑘+𝑚∣𝑘), 𝑥_𝑙∣𝑘=𝑥(𝑙∣𝑘). Thus, (24) becomes 𝑥_{𝑘+𝑚+1∣𝑘}=𝐴(𝜍)𝑥_{𝑘+𝑚∣𝑘}+𝐵(𝜐,𝜍)𝑥_{𝑘+𝑚−𝜐∣𝑘}.

The proof of the previous theorem is divided into the following two steps: Step  1, we design a Lyapunov function 𝑉(𝑋(𝑘+𝑚∣𝑘),𝜐) for the systems; Step  2, an optimal controller is designed such that the closed-loop systems are stable and the optimal characteristics are satisfied.

Proof Step  1. Consider a Lyapunov function candidate 𝑉(𝑋(𝑘+𝑚∣𝑘),𝜐)=𝑉₁+𝑉₂+𝑉₃ with 𝐱≠0, where 𝑉₁(𝑋(𝑘+𝑚∣𝑘),𝜐)=𝑥^Τ_{𝑘+𝑚∣𝑘}𝑃(𝜐)𝑥_{𝑘+𝑚∣𝑘},𝑉₂(𝑋(𝑘+𝑚∣𝑘),𝜐)=^{𝑘+𝑚−1}_{𝑙=𝑘+𝑚−𝜐}𝑥^Τ_𝑙∣𝑘𝑆𝑥_𝑙∣𝑘,𝑉₃(𝑋(𝑘+𝑚∣𝑘),𝜐)=1−𝜋_𝑚⁰_{𝜃=−𝑑𝑀+1}^{𝑘+𝑚−1}_{𝑙=𝑘+𝑚+𝜃}𝑥^Τ_𝑙∣𝑘𝑆𝑥_𝑙∣𝑘,Δ𝑉₁=Ε𝑉₁𝑋_{𝑘+𝑚+1∣𝑘},𝜐−𝑉₁𝑋_{𝑘+𝑚∣𝑘}𝑥,𝜐=Ε^Τ_{𝑘+𝑚+1∣𝑘}𝑃(𝜐)𝑥_{𝑘+𝑚+1∣𝑘}−𝑥^Τ_{𝑘+𝑚∣𝑘}𝑃(𝜐)𝑥_{𝑘+𝑚∣𝑘}=𝑥^Τ_{𝑘+𝑚+1∣𝑘}𝑃(𝜐)𝑥_{𝑘+𝑚+1∣𝑘}−𝑥^Τ_{𝑘+𝑚∣𝑘}𝑃(𝜐)𝑥_{𝑘+𝑚∣𝑘}=𝜂^Τ_{𝑘+𝑚∣𝑘}𝐴^Τ(𝜍)𝑃(𝜐)(𝜍)(𝜐)𝐴×𝐴−𝑃^Τ(𝜍)𝑃(𝜐)𝐵(𝜐,𝜍)∗𝐵^Τ()𝑃(𝜐)𝐵()𝜂𝜐,𝜍𝜐,𝜍^{𝑘+𝑚∣𝑘}, (36)(37) where 𝜂_{𝑘+𝑚∣𝑘}=𝑥_{𝑘+𝑚∣𝑘}𝑥_{𝑘+𝑚−𝜐}^Τ, and E is the mathematical expectation. Consider Δ𝑉₂=Ε𝑉₂𝑋_{𝑘+𝑚+1∣𝑘},𝜐₁−𝑉₂𝑋_{𝑘+𝑚∣𝑘},𝜐=Ε^𝑘+𝑚_{𝑙=𝑘+𝑚+1−𝑑𝑘𝑆𝐶}𝑥^Τ_𝑙|𝑘𝑆𝑥_𝑙|𝑘−^{𝑘+𝑚−1}_{𝑙=𝑘+𝑚−𝑑𝑘𝑆𝐶}𝑥^Τ_𝑙∣𝑘𝑆𝑥_𝑙∣𝑘=𝑥^Τ_{𝑘+𝑚∣𝑘}𝑆𝑥_{𝑘+𝑚∣𝑘}−𝑥^Τ_{𝑘+𝑚−𝑑𝑘𝑆𝐶∣𝑘}𝑆𝑥_{𝑘+𝑚−𝑑𝑘}^𝑆𝐶_∣𝑘+^𝑑𝑀_{𝑗=0,𝑖≠𝑗}𝜋_𝑖𝑗^{𝑘+𝑚−1}_{𝑙=𝑘+𝑚+1−𝑑𝑆𝐶𝑘+1}−^{𝑘+𝑚−1}_{𝑙=𝑘+𝑚−𝑑𝑘𝑆𝐶}𝑥^Τ_𝑙∣𝑘𝑆𝑥_𝑙∣𝑘, (38)
and ∑^𝑑𝑀_{𝑗=0,𝑗≠𝑖}𝜋_𝑖𝑗=1−𝜋_𝑖𝑖≤1−𝜋_𝑚, 0≤𝜐≤𝑑_𝑀; thus, Δ𝑉₂=Ε𝑉₂𝑋_{𝑘+𝑚+1∣𝑘},𝜐₁−𝑉₂𝑋_{𝑘+𝑚∣𝑘},𝜐≤𝑥^Τ_{𝑘+𝑚∣𝑘}𝑆𝑥_{𝑘+𝑚∣𝑘}−𝑥^Τ_{𝑘+𝑚−𝜐∣𝑘}𝑆𝑥_{𝑘+𝑚−𝜐∣𝑘}+1−𝜋_𝑚^𝑘+𝑚_{𝑙=𝑘+𝑚−𝑑𝑀+1}𝑥^Τ_𝑙∣𝑘𝑆𝑥_𝑙∣𝑘. (39)
Similarly, Δ𝑉₃=Ε𝑉₃𝑋_{𝑘+𝑚+1∣𝑘},𝜐₁−𝑉₃𝑋_{𝑘+𝑚∣𝑘},𝜐≤𝑑_𝑀1−𝜋_𝑚𝑥^Τ_𝑙∣𝑘𝑆𝑥_𝑙∣𝑘−1−𝜋_𝑚^𝑘+𝑚_{𝑙=𝑘+𝑚−𝑑𝑀+1}𝑥^Τ_𝑙∣𝑘𝑆𝑥_𝑙∣𝑘. (40)
Therefore, Δ𝑉=Δ𝑉₁+Δ𝑉₂+Δ𝑉₃ and ≤𝜂^Τ_{𝑘+𝑚∣𝑘}^{𝐴Τ(𝜍)𝑃(𝜐)𝐴(𝜍)}_(𝜐)+𝑆^𝐴_{−𝑃1+𝜇}^{Τ(𝜍)𝑃(𝜐)𝐵()𝜐,𝜍}_{∗𝐵Τ()𝑃(𝜐)𝐵()𝜐,𝜍𝜐,𝜍−𝑆}𝜂_{𝑘+𝑚∣𝑘}, (A1) where ∗ denotes the block determined by symmetry.
Substituting (25) and (A1) into (29), we have 𝑉𝑋_{𝑘+𝑚+1∣𝑘},𝜐₁−𝑉𝑋_{𝑘+𝑚∣𝑘},𝜐+𝑥^Τ_{𝑘+𝑚∣𝑘}𝑄𝑥_{𝑘+𝑚∣𝑘}+𝑢^Τ_{𝑘+𝑚∣𝑘}𝑅𝑢_{𝑘+𝑚∣𝑘}≤𝜂^Τ_{𝑘+𝑚∣𝑘}Ξ𝜂_{𝑘+𝑚∣𝑘},Ξ=Φ𝐴^Τ(𝜍)𝑃(𝜐)𝐵(𝜐,𝜍)+𝐶^Τ(𝜍)𝑅𝐷(𝜐,𝜍)∗𝐵^Τ(𝜐,𝜍)𝑃(𝜐)𝐵(𝜐,𝜍)−𝑆+𝐷^Τ()(),𝜐,𝜍𝑅𝐷𝜐,𝜍 (41) where Φ=𝐴^Τ(𝜍)𝑃(𝜐)𝐴(𝜍)−𝑃(𝜐)+(1+𝜇)𝑆+𝑄+𝐶^Τ(𝜍)𝑅𝐶(𝜍).
According to Lemma 2, we obtain (𝜐)+Ξ=−𝑃1+𝜇𝑆+𝑄0𝐴^Τ(𝜍)𝐶^Τ(𝜍)0−𝑆𝐵^Τ(𝜐,𝜍)𝐷^Τ()𝐴(𝜍)𝐵()−𝑃𝜐,𝜍𝜐,𝜍⁻¹(𝜐)0𝐶(𝜍)𝐷(𝜐,𝜍)0−𝑅⁻¹. (42)
According to (34), then 𝜂^Τ_{|𝑘𝑘+𝑚}Ξ𝜂_{𝑘+𝑚|𝑘}<0. (43)
Hence, (29) holds.
And 𝑥^Τ_{𝑘+𝑚∣𝑘}𝑃(𝜐)𝑥_{𝑘+𝑚∣𝑘}+^{𝑘+𝑚−1}_{𝑙=𝑘+𝑚−𝜐}𝑥^Τ_𝑙∣𝑘𝑆𝑥_𝑙∣𝑘+1−𝜋_𝑚⁰_{𝜃=−𝑑𝑀+1}^{𝑘+𝑚−1}_{𝑙=𝑘+𝑚+𝜃}𝑥^Τ_𝑙∣𝑘𝑆𝑥_𝑙∣𝑘≤Υ. (44)
That is, 𝑉𝑋_{𝑘+𝑚∣𝑘},𝜐≤Υ. (45)
From (30), we have 𝐽_∞(𝑘)<Υ. (46)
Step 2. A congruence transformation to (17) and (18) leads to 𝐴(𝜍)=𝐷(𝜍)+𝐸(𝜍)𝐹𝐻₁()=𝐸(𝜍)𝐹𝐶(𝜍)=𝐼̃(𝜍)+𝐽̃(𝜍)𝐹𝐻,𝐵𝜐,𝜍𝐻,₁,𝐷(𝜐,𝜍)=𝐾(𝜍)𝐹𝐻, (A2) where 𝐷(𝜍)=Φ₁Γ₁𝐶₂00Φ₂Γ₂𝛼,𝐸(𝜍)=0Γ00𝛼₂[][]1−𝛼𝐹1−𝛼,𝐹=₁,𝐹₂,,𝐼̃(𝜍)=,𝐽̃(𝜍)=Ι[],𝐻𝐻=0000Ι000000𝛼1−𝛼¹=,𝐾(𝜍)=0[].Ι000001−𝛼 (47)
Substituting (A2) into (34), (35) can be obtained.
The proof of Theorem 4 is completed.

Remark 5. According to the proposed method, the optimal input is unique. Υ is a scalar and Υ>0. Unique resolution is given finally by using Toolbox of Matlab. Its convergence can be found in Figure 3.

4. Stability Analysis of NCSs with Packet Dropouts

Theorem 6. Suppose that the optimization problem in Theorem 4 is feasible at time 𝑘; then, the new resulting closed-loop systems in (17) are asymptotically stable by the optimal control law in (18) which is obtained from Theorem 4.

Proof. First, the Lyapunov function, 𝑉(𝑋(𝑘+𝑚∣𝑘),𝜐)=𝑉₁+𝑉₂+𝑉₃ is discussed and 𝑉(𝑋(𝑘+𝑚∣𝑘),𝑑^𝑆2𝐶2_{𝑘+1∣𝑘})−𝑉(𝑋(𝑘∣𝑘),𝑑^𝑆2𝐶2_𝑘∣𝑘)<0 holds only when 𝑥≠0.
Next, in the following we will consider the situation at time 𝑘 (i.e., 𝑚=0): 𝑉𝑋(𝑘∣𝑘),𝑑^𝑆𝐶_𝑘∣𝑘=𝑥^Τ_𝑘∣𝑘𝑃_𝑘𝑥_𝑘∣𝑘+^𝑘−1_{𝑙=𝑘−𝑑𝑆𝐶𝑘∣𝑘}𝑥^Τ_𝑙∣𝑘𝑆_𝑘𝑥_𝑙∣𝑘+1−𝜋_𝑚⁰_{𝜃=−𝑑𝑀+1}^𝑘−1_{𝑙=𝑘+𝜃}𝑥^Τ_𝑙∣𝑘𝑆_𝑘𝑥_𝑙∣𝑘, (48) where 𝑃_𝑘>0, 𝑆_𝑘>0 are obtained from the optimal solution at time 𝑘. Suppose that the optimization problem in Theorem 4 is feasible at time 𝑘, and according to the introduction in [8], these optimization problems are also feasible for all 𝑘+1. Let us note values of 𝑃_𝑘>0, 𝑆_𝑘>0, 𝐹_1𝑘, 𝐹_2𝑘, Υ_𝑘, and 𝑃_𝑘+1>0, 𝑆_𝑘+1>0, 𝐹_1𝑘+1, 𝐹_2𝑘+1 obtained from the optimal solution at time 𝑘 and 𝑘+1, respectively. Therefore, we have 𝑥^Τ_{𝑘+1∣𝑘+1}𝑃_𝑘+1𝑥_{𝑘+1∣𝑘+1}+^𝑘_{𝑙=𝑘+1−𝑑𝑆𝐶𝑘+1∣𝑘+1}𝑥^Τ_𝑙|𝑘+1𝑆_𝑘+1𝑥_𝑙|𝑘+1+1−𝜋_𝑚⁰_{𝜃=−𝑑𝑀}^𝑘_{+1𝑙=𝑘+1+𝜃}𝑥^Τ_𝑙|𝑘+1𝑆_𝑘+1𝑥_𝑙|𝑘+1≤𝑥^Τ_{𝑘+1∣𝑘+1}𝑃_𝑘𝑥_{𝑘+1∣𝑘+1}+^𝑘_{𝑙=𝑘+1−𝑑𝑆𝐶𝑘∣𝑘}𝑥^Τ_{𝑙∣𝑘+1}𝑆_𝑘𝑥_{𝑙∣𝑘+1}+1−𝜋_𝑚⁰_{𝜃=−𝑑𝑀}^𝑘_{+1𝑙=𝑘+1+𝜃}𝑥^Τ_{𝑙∣𝑘+1}𝑆_𝑘𝑥_{𝑙∣𝑘+1}. (49)
That is because 𝑃_𝑘+1>0, 𝑆_𝑘+1>0 are optimal, whereas 𝑃_𝑘>0, 𝑆_𝑘>0 are only feasible at time 𝑘.
From (29), it can be obtained that 𝑉𝑋(𝑘+1∣𝑘),𝑑^𝑆2𝐶2_{𝑘+1∣𝑘}−𝑉𝑋(𝑘∣𝑘),𝑑^𝑆2𝐶2_𝑘∣𝑘≤−𝑥^Τ(𝑘∣𝑘)𝑄𝑥(𝑘|𝑘)+𝑢^Τ(𝑘∣𝑘)𝑅𝑢(𝑘∣𝑘)≤−𝑥^Τ(𝑘∣𝑘)𝑄𝑥(𝑘∣𝑘)≤−𝜆_min(𝑄)‖𝑥(𝑘∣𝑘)‖². (50)
Because 𝑄 is symmetric positive definite, we have 𝜆_min(𝑄)>0 (the minimal eigenvalue of matrix 𝑄), and then 𝑉𝑋(𝑘+1∣𝑘),𝑑^𝑆2𝐶2_{𝑘+1∣𝑘}−𝑉𝑋(𝑘∣𝑘),𝑑^𝑆2𝐶2_𝑘∣𝑘<0. (51)
When 𝑚=0, (24) is becoming 𝑥(𝑘+1∣𝑘)=𝐴𝑑^𝐶2𝐴_𝑘∣𝑘𝑥()𝑑𝑘∣𝑘+𝐵^𝑆2𝐶2_𝑘∣𝑘,𝑑^𝐶2𝐴_𝑘∣𝑘𝑥𝑘−𝑑^𝑆2𝐶2_𝑘∣𝑘∣𝑘. (52)
Because of the measured state, 𝑥()()𝑑𝑘+1∣𝑘+1=𝑥𝑘+1=𝐴^𝐶2𝐴_𝑘𝑥(𝑘)+𝐵𝑑^𝑆2𝐶2_𝑘,𝑑^𝐶2𝐴_𝑘𝑥𝑘−𝑑^𝑆2𝐶2_𝑘. (53)
So we have 𝑥(𝑘+1∣𝑘+1)=𝑥(𝑘+1∣𝑘). (54)
Comparing with (A1), it can be obtained that 𝑉𝑋(𝑘+1∣𝑘+1),𝑑^𝑆2𝐶2_{𝑘+1∣𝑘+1}−𝑉𝑋(𝑘∣𝑘),𝑑^𝑆2𝐶2_𝑘∣𝑘<0. (55)
Therefore, the Lyapunov function𝑉(𝑋(𝑘∣𝑘),𝑑^𝑆2𝐶2_𝑘∣𝑘) is decreasing for the new closed loop, and lim_𝑘→∞𝑥(𝑘∣𝑘)=0 is concluded. The stochastic stability is obtained.

5. Simulation Examples

In order to show the effectiveness of the proposed method, we will give some cases and simulations.

5.1. A Simple Example

Considering the coefficient matrixes of 𝑃₁ and 𝑃₂Φ₁=0.95121.051−0.6065−1.1618,Γ₁=,8.553029.4735𝐶₁=,Φ−0.00450.1004₂=−0.3667−0.011110,Γ₂=10,𝐶₂=00.0111. (56)(57)(58)

Markov models of the whole closed control systems are consisted. Suppose that 𝑑^𝑆2𝐶2_𝑘=𝑖∈(0,1) and 𝑑^𝐶2𝐴_𝑘=𝑟∈(0,1), as shown in Figures 2 and 5, which means that the number of data packet dropouts is 𝑖 at sensor to controller of secondary side and, similarly, is 𝑟 at secondary controller to actuator side at time 𝑘. In the method, Υ is a scalar and Υ>0. Its convergence is shown in Figure 3. Unique resolution is given finally by using Toolbox of Matlab. The simulation results under the optimal state feedback controller are shown in Figure 4.

Figure 2 

The number of data packet dropouts of 𝑆 to 𝐶 and 𝐶2 to 𝐴.

Figure 3 

The upper of performance index.

(a)

(b)

(c)

(d)

(e)

(f)

Figure 4 

(a) State response of state 𝑥2 with designed optimal controller, (b) state response of state 𝑥1 with designed optimal controller, (c) state trajectories of state 𝑥2, (d) state trajectories of state 𝑥1, (e) state response of state 𝑥2, and (f) state trajectories of state 𝑥1.

Figure 5 

The number of data packet dropouts of 𝑆 to 𝐶 side.

From the simulation (1) Figure 3 is the upper of performance index. (2) In Figures 4(a) and 4(b), system state trajectories will eventually be more stable. It is shown that NCSs with data packet dropouts are stable with the optimal controller which is designed by Theorem 4. (3) Through comparing to LQR controller, which is showed in Figures 4(c), 4(d), 4(e), and 4(f), the system overshoots are shortened, and state trajectory paths are more superior. Thus, the optimal controller designed in our paper is effective.

5.2. Industrial Systems

In order to verify the method proposed earlier, a networked control system for main furnace temperature in industrial systems is taken for an example. It is assumed that the transfer functions of the inertial and leading sections are 𝐺_𝑝1(𝑠)=(1/[(30𝑠+1)(3𝑠+1)]) and 𝐺_𝑝2(𝑠)=(1/[(𝑠+1)²(10𝑠+1)]), respectively. After discretization, the following state space models are available: 𝑃₁∶𝑥₁(𝑘+1)=𝑥0.6887−0.00930.83560.9951¹(𝑘)+𝑦0.83560.4437²(𝑘),𝑦₁(𝑘)=00.0111𝑥₁(𝑘),𝑃₂∶𝑥₂(𝑘+1)=𝑥−0.0342−0.4364−0.03420.34250.6849−0.02540.25420.87620.9899²(𝑘)+𝑢0.34250.25420.1008²(𝑘),𝑦₂(𝑘)=000.1𝑥₂(𝑘). (59)

Figures 6(a) and 6(b) can be obtained, from which the systems in (17) are stabilized by our designed controller. Through comparing to LQR controller, as shown in Figures 6(c), 6(d), 6(e), and 6(f), and the system overshoots shortened, state trajectories are more superior and performance of the control systems is further improved. Thus, the proposed optimal controller designed is effective and feasible.

(a)

(b)

(c)

(d)

(e)

(f)

Figure 6 

(a) State response of state 𝑥2 with designed optimal controller, (b) state response of state 𝑥1 with designed optimal controller, (c) state trajectories of state 𝑥2, (d) state trajectories of state 𝑥1, (e) state response of state 𝑥2, and (f) state trajectories of state 𝑥1.

6. Conclusions

In this paper, the modeling, optimal, and control problems for a class of NCSs under data packet dropout effect have been studied. The data packet dropouts are described by Markov chains. The Markov chains describe that the quantity of packet dropouts between current time 𝑘 and its latest communicates successfully instead of whether the data packets dropped or not. Though augmenting the state vectors, the resulting closed-loop control system is transformed into a jump system with time delays. Model predictive control method is applied to study optimization and stability problems of the resulting closed-loop NCSs. Sufficient conditions are proposed for the optimization and stability of the resulting closed-loop NCSs. Some simulations are given in the last section and it can be seen that the designed controllers are feasible and effective. In this work, the full state feedback controllers are designed. However, it is difficult to obtain the full state. Output feedback controllers should be considered. Dynamic output feedback can be investigated in the future work.