Dynamic Output Feedback Control for Nonlinear Networked Control Systems with Random Packet Dropout and Random Delay

Abstract

This paper investigates the dynamic output feedback control for nonlinear networked control systems with both random packet dropout and random delay. Random packet dropout and random delay are modeled as two independent random variables. An observer-based dynamic output feedback controller is designed based upon the Lyapunov theory. The quantitative relationship of the dropout rate, transition probability matrix, and nonlinear level is derived by solving a set of linear matrix inequalities. Finally, an example is presented to illustrate the effectiveness of the proposed method.

1. Introduction

Networked control systems (NCSs) are feedback control loops closed through a real time network. Compared with conventional point-to-point control systems, NCSs have many attractive advantages of low cost, simple installation, reduced wiring, easy maintenance, and so on. On the other hand, the introduction of network also presents some challenging problems such as quantization [1–4], packet dropout [5–8], and time delays [9–11], which bring difficulties for analysis and design of NCSs. Hence, NCSs have attracted considerable research interests in recent years [12–14].

Generally, time delays and packet dropout are two essential issues that need careful consideration in NCSs design. They can be handled separately or simultaneously. Some results have been available. Markov chain and independent Bernoulli random process were commonly used to model the packet loss processes. The results applying the Bernoulli process assume that the packet losses are independent and identically distributed [5, 7], while those using the Markov chain assume that the packet losses are burst and occur according to a Markov chain [6, 8]. Based on a similar Bernoulli random delay model, the controller design problem for NCSs with both sensor-to-controller and controller-to-actuator delays has been studied in [9]. A delay-dependent approach for NCSs controller design has been provided in [10], where the feedback gain of a memoryless controller was determined by solving a set of linear matrix inequalities. In [1, 2, 11], the network-induced random delays were modeled as Markov chains such that the closed-loop system was a jump linear system. In [15], a nonlinear stochastic system model was proposed to describe the NCSs with both random packet dropout and network-induced time-varying delay. Based on this model, a sufficient condition was derived to the exponential stability of the nonlinear NCSs. However, [15] only studied the static output feedback controller and time-varying delay. In this paper, we extend the results in [15] to general random delay and dynamic output feedback controller.

This paper proposes a new nonlinear networked control systems model with both random packet dropout and random delay. In this model, two random variables are included and they are independent of each other. The random delay is modeled as a Markov chain with partly known transition probability matrix, which covers the completely known and completely unknown transition probabilities as its two special cases. The random packet dropout is modeled as Bernoulli distributed white sequences. A sufficient condition is presented for the stabilization of the closed systems, and a dynamic output feedback controller is designed based upon the Lyapunov theory. An example is presented to illustrate the effectiveness of the proposed method.

Notation 1. Throughout the paper, 𝑅^𝑛 denotes the 𝑛-dimensional Euclidean space. The superscripts “𝑇” and “–1” stand for the transpose and inverse of a matrix, respectively. diag{⋯} stands for a block-diagonal matrix. The symbol * denotes the symmetric part in a symmetric matrix. The notation 𝑃>0 means that 𝑃 is real symmetric and positive definite matrix. In addition, 𝐸{𝑥} and 𝐸{𝑥∣𝑦} denote expectation of 𝑥 and 𝑥 conditional on 𝑦, respectively. Matrices, if their dimensions are not explicitly stated, are assumed to be compatible for algebraic operations.

2. Problem Statement and Preliminaries

The structure of the NCSs to be discussed in this paper is depicted in Figure 1.

Figure 1 

Block diagram of the considered NCSs.

The plant model is described by the following nonlinear discrete-time system: 𝑥()(𝑘)(𝑘)(𝑥(𝑘𝑦(𝑘)(𝑘),𝑘+1=𝐴𝑥+𝐵𝑢+𝑓))=𝐶𝑥 (1) where 𝑥(𝑘)∈𝑅^𝑛, 𝑢(𝑘)∈𝑅^𝑚, and 𝑦(𝑘)∈𝑅^𝑝 are the state, control input, and measured output, respectively, and 𝐴, 𝐵, 𝐶 are known real matrices with appropriate dimensions. The function 𝑓(𝑥(𝑘)) satisfies the following inequality: 𝑓^𝑇(𝑥(𝑘))𝑓(𝑥(𝑘))≤𝛽²𝑥^𝑇(𝑘)𝐺^𝑇𝐺𝑥(𝑘), (2) where 𝛽>0 is a scalar parameter and 𝐺 is a constant matrix.

Throughout the paper, the following assumptions are needed.

Assumption 1. The state 𝑥(𝑘) is not measurable.

Assumption 2. For convenience of analysis, it is assumed that the network induced packet dropout and delay occur only in the sensor-to-controller side.

In what follows, we model signal transmission delay and random packet dropout mathematically.

2.1. Signal Transmission Delay

Let 𝑑_𝑘 represent the random delay from sensor to controller. Assume that 𝑑_𝑘 take the positive integers between 𝑑_𝑚 and 𝑑_𝑀. In real communication systems, current time delay is usually affected by previous time delays. It is reasonable to model the time delay 𝑑_𝑘 as a Markov chain and its transition probability matrix is Π=[𝜆_𝑟𝑠]; that is, 𝑑_𝑘 jump from mode 𝑟 to 𝑠 with probability 𝜆_𝑟𝑠, which is defined by 𝜆_𝑟𝑠=𝑃_𝑟𝑑_𝑘+1=𝑠∣𝑑_𝑘=𝑟, (3) where 𝜆_𝑟𝑠>0, ∑^𝑑𝑀_{𝑠=𝑑𝑚}𝜆_𝑟𝑠=1, for all 𝑟, 𝑠∈Λ, where Λ={𝑑_𝑚,𝑑_𝑚+1,…,𝑑_𝑀}. In reality, the transition probabilities of the Markov chain in this paper are usually partially available; namely, some elements in matrix Π are unknown. For instance, 𝑑_𝑘 with four modes will have the transition probability matrix Π as 𝜆Π=¹¹?𝜆₁₃????𝜆₂₄𝜆₃₁??𝜆₃₄??𝜆₄₃𝜆₄₄, (4) where “?” represents the unavailable elements. For notational clarity, for all 𝑟∈Λ, we denote 𝐼^𝑟_𝜅={𝑠∶𝜆_𝑟𝑠isknown}, 𝐼^𝑟_𝑢𝜅={𝑠∶𝜆_𝑟𝑠isunknown}. Moreover, if 𝐼^𝑟_𝜅≠∅ and have 𝑚 elements, 𝐼^𝑟_𝜅 and 𝐼^𝑟_𝑢𝜅 can be further described as 𝐼^𝑟_𝜅={𝜅^𝑟₁,…,𝜅^𝑟_𝑚}, 𝐼^𝑟_𝑢𝜅={𝜅^𝑟₁,…,𝜅^𝑟_𝑀−𝑚}. Where 𝜅^𝑟_𝑚∈𝑍⁺ represents the 𝑚th known element with the index 𝜅^𝑟_𝑚 in the 𝑟th row of matrix Π and 𝜅^𝑟_𝑀−𝑚∈𝑍⁺ represents the (𝑀−𝑚)th unknown element with the index 𝜅^𝑟_𝑀−𝑚 in the 𝑟th row of matrix Π.

2.2. Random Packet Dropout

The Bernoulli distributed white sequence 𝛼_𝑘 is introduced to reflect the random packet dropout from the sensor to the controller, which takes value in {0,1} and satisfies Prob𝛼_𝑘=1=𝐸𝛼_𝑘=𝛼,0≤𝛼𝛼≤1,Prob_𝑘=0=1−𝐸𝛼_𝑘=1−𝛼𝛼,var_𝑘=𝐸𝛼_𝑘−𝛼²=(1−𝛼)𝛼=𝛼₁. (5)

Assumption 3. The random variables 𝛼_𝑘 and 𝑑_𝑘 are independent of each other.
In this paper, the measurement with packet dropout and random delay is described by 𝑦(𝑘)=1−𝛼_𝑘𝑦𝑘−𝑑_𝑘+𝛼_𝑘𝑦(𝑘−1). (6)

By 𝛼_𝑘=0, we mean that the delayed packet is correctly delivered; that is, 𝑦(𝑘)=𝑦(𝑘−𝑑_𝑘). While by 𝛼_𝑘=1, we mean that the packet is lost; that is 𝑦(𝑘)=𝑦(𝑘−1).

Remark 4. If the random delay is transformed into the time-varying delay, the model (6) is the same as the one used in [15]. If the transmission delay 𝑑_𝑘 is not taken into account, then (6) reduces to the one used in [6, 7]. This, in turn, implies that the model (6) is more general than those in [6, 7, 15].
The output feedback controller to be designed is of the following form: 𝑥_𝑐(𝑘+1)=𝐴𝑥_𝑐(𝑘)+𝐵𝑢(𝑘)+𝐿𝑦(𝑘)−𝐶𝑥_𝑐(𝑘)𝑢(𝑘)=𝐾𝑥_𝑐(𝑘), (7) where the gain matrices 𝐿 and 𝐾 will be designed later.
Define that 𝑒(𝑘)=𝑥(𝑘)−𝑥_𝑐(𝑘). (8)
Substituting (6), (7) into (8), we have 𝑒()=)𝑒(𝑘)(𝑘)−(𝑘+1(𝐴−𝐿𝐶+𝐿𝐶𝑥1−𝛼)𝐿𝐶𝑥𝑘−𝑑_𝑘−𝛼𝐿𝑦(+𝛼𝑘−1)_𝑘−𝛼𝐿𝐶𝑥𝑘−𝑑_𝑘−𝐿𝑦((𝑥(𝑘.𝑘−1)+𝑓)) (9)
Seting 𝜉(𝑘)=[𝑥^𝑇(𝑘)𝑒^𝑇(𝑘)𝑦^𝑇(𝑘−1)]^𝑇, we obtain the augmented NCSs with random packet dropout and random delay: 𝜉(𝑘+1)=𝐴𝜉(𝑘)+𝐵𝜉𝑘−𝑑_𝑘+𝐶𝑓(𝑥(𝑘)), (10) where 𝐴=𝐴₁+𝛼_𝑘−𝛼𝐴₂,𝐵=(1−𝛼)+𝛼−𝛼_𝑘𝐵₁,𝐴₁=𝐴+𝐵𝐾−𝐵𝐾0𝐿𝐶𝐴−𝐿𝐶−𝛼𝐿00𝛼,𝐴₂=,𝐵00000−𝐿00−𝐼₁=,000−𝐿𝐶00𝐶00𝐼𝐼0𝐶=. (11)

3. Main Results

In this section, we will present our main results. To this end, the following definition and lemmas will be introduced.

Definition 5. The closed-loop system (10) is said to be stochastically stable if for any initial condition 𝑥₀∈𝑅^𝑛 and 𝑑₀∈Λ, the following inequality 𝐸^∞_𝑘=0‖𝑥(𝑘)‖²𝑥₀,𝑑₀<∞holds. (12)

Lemma 6 (see [16]). For given scalar 𝜆_𝑖≥0 and matrix 𝑃_𝑖≥0, 𝑖=1,2,…,𝑁, one has ^𝑁_𝑖=1𝜆_𝑖𝑃_𝑖≤^𝑁_𝑖=1𝜆_𝑖^𝑁_𝑖=1𝑃_𝑖. (13)

Lemma 7 (see [17]). For any positive definite matrix 𝑊∈𝑅^𝑛×𝑛, two positive integers 𝑟 and 𝑟₀ satisfying 𝑟≥𝑟₀≥1, and vector function 𝑥(𝑖)∈𝑅^𝑛, one has ^𝑟_𝑖=𝑟0𝑥(𝑖)^𝑇𝑊^𝑟_𝑖=𝑟0𝑥(𝑖)≤𝑟̃^𝑟_𝑖=𝑟0𝑥^𝑇(𝑖)𝑊𝑥(𝑖), (14) where 𝑟̃=𝑟−𝑟₀+1.

In the following theorem, we will derive a sufficient condition such that the closed-loop system (10) is stochastically stable.

Theorem 8. For the given scalars 0≤𝛼≤1, 𝛽>0, and matrices 𝐾, 𝐿, the closed-loop system (10) is stochastically stable for each mode 𝑟∈Λ, if there exist matrices 𝑃₁(𝑟)>0, 𝑃₂(𝑟)>0, 𝑃₃(𝑟)>0, 𝑍₁>0, 𝑍₂>0, 𝑄₁>0, 𝑄₂>0, 𝑄₃>0 and scalar Γ>0 satisfying the following linear matrix inequality: Φ₁₁Φ₁₂Φ₁₃Φ₁₄𝑍₂0Φ₁₇∗Φ₂₂Φ₂₃Φ₂₄00Φ₂₇∗∗Φ₃₃000Φ₃₇∗∗∗Φ₄₄Φ₄₅Φ₄₆Φ₄₇∗∗∗∗Φ₅₅00∗∗∗∗∗Φ₆₆0∗∗∗∗∗∗Φ₇₇<0, (15) where Φ₁₁=(𝐴+𝐵𝐾)^𝑇𝑃₁(𝑠)(𝐴+𝐵𝐾)−𝑃₁(𝑟)+(𝐿𝐶)^𝑇𝑃₂(𝑠)(𝐿𝐶)−𝑍₂+𝑑_𝑀−𝑑_𝑚+1𝑄₁+𝑄₂+𝑄₃+𝑑_𝑀−𝑑_𝑚²(𝐴+𝐵𝐾−𝐼)^𝑇𝑍₁(𝐴+𝐵𝐾−𝐼)+𝑑²_𝑚(𝐴+𝐵𝐾−𝐼)^𝑇𝑍₂(𝐴+𝐵𝐾−𝐼)+Γ𝛽²𝐺^𝑇Φ𝐺,₁₂=−(𝐴+𝐵𝐾)^𝑇𝑃₁(𝑠)+())(𝐵𝐾𝐿𝐶^𝑇𝑃₂(𝑠)(𝐴−𝐿𝐶)−𝑑²_𝑚(𝐴+𝐵𝐾−𝐼)^𝑇𝑍₂()−𝑑𝐵𝐾_𝑀−𝑑_𝑚²(𝐴+𝐵𝐾−𝐼)^𝑇𝑍₁(),Φ𝐵𝐾₁₃=−𝛼(𝐿𝐶)^𝑇𝑃₂(𝑠)Φ𝐿,₁₄=−(1−𝛼)(𝐿𝐶)^𝑇𝑃₂(𝑠),Φ)(𝐿𝐶₁₇=(𝐴+𝐵𝐾)^𝑇𝑃₁(𝑠)+(𝐿𝐶)^𝑇𝑃₂(𝑠)+𝑑_𝑀−𝑑_𝑚²(𝐴+𝐵𝐾−𝐼)^𝑇𝑍₁+𝑑²_𝑚(𝐴+𝐵𝐾−𝐼)^𝑇𝑍₂,Φ₂₂=(𝐵𝐾)^𝑇𝑃₁(𝑠)+())(𝐵𝐾𝐴−𝐿𝐶^𝑇𝑃₂(𝑠)+𝑑)(𝐴−𝐿𝐶_𝑀−𝑑_𝑚²(𝐵𝐾)^𝑇𝑍₁(𝐵𝐾)+𝑑²_𝑚(𝐵𝐾)^𝑇𝑍₂(𝐵𝐾)−𝑃₂(𝑟),Φ₂₃=−𝛼(𝐴−𝐿𝐶)^𝑇𝑃₂(𝑠)Φ𝐿,₂₄=−(1−𝛼)(𝐴−𝐿𝐶)^𝑇𝑃₂(𝑠),Φ)(𝐿𝐶₂₇=−(𝐵𝐾)^𝑇𝑃₁(𝑠)+(𝐴−𝐿𝐶)^𝑇𝑃₂(𝑠)−𝑑_𝑀−𝑑_𝑚²(𝐵𝐾)^𝑇𝑍₁−𝑑²_𝑚(𝐵𝐾)^𝑇𝑍₂,Φ₃₃=𝛼²𝐿^𝑇𝑃₂(𝑠)𝐿+𝛼₁𝐿^𝑇𝑃₂(𝑠)𝐿+𝛼²𝑃₃(𝑠)+𝛼₁𝑃₃(𝑠)−𝑃₃(𝑟),Φ₃₇=−𝛼𝐿^𝑇𝑃₂(𝑠),Φ₄₄=(1−𝛼)²(𝐿𝐶)^𝑇𝑃₂(𝑠)(𝐿𝐶)+𝛼₁(𝐿𝐶)^𝑇𝑃₂(𝑠)(𝐿𝐶)+(1−𝛼)²𝐶^𝑇𝑃₃(𝑠)𝐶+𝛼₁𝐶^𝑇𝑃₃(𝑠)𝐶−𝑄₁−2𝑍₁,Φ₄₅=𝑍₁,Φ₄₆=𝑍₁,Φ₄₇=−(1−𝛼)(𝐿𝐶)^𝑇𝑃₂(𝑠),Φ₅₅=−𝑄₂−𝑍₁−𝑍₂,Φ₆₆=−𝑄₃−𝑍₁,Φ₇₇=𝑃₁(𝑠)+𝑃₂(𝑠)+𝑑_𝑀−𝑑_𝑚²𝑍₁+𝑑²_𝑚𝑍₂−Γ𝐼. (16)

Proof. Choose the Lyapunov function candidate as 𝑉𝑥(𝑘),𝑑_𝑘,𝛼_𝑘=⁵_𝑖=1𝑉_𝑖𝑥(𝑘),𝑑_𝑘,𝛼_𝑘,𝑉₁𝑥(𝑘),𝑑_𝑘,𝛼_𝑘=𝑥^𝑇(𝑘)𝑃₁𝑑_𝑘𝑥(𝑘)+𝑒^𝑇(𝑘)𝑃₂𝑑_𝑘𝑒(𝑘)+𝑦^𝑇(𝑘−1)𝑃₃𝑑_𝑘𝑦(),𝑉𝑘−1₂𝑥(𝑘),𝑑_𝑘,𝛼_𝑘=^𝑘−1_{𝑖=𝑘−𝑑𝑘}𝑥^𝑇(𝑖)𝑄₁𝑥(𝑖)𝑉₃𝑥(𝑘),𝑑_𝑘,𝛼_𝑘=^𝑘−1_{𝑖=𝑘−𝑑𝑚}𝑥^𝑇(𝑖)𝑄₂𝑥(𝑖)+^𝑘−1_{𝑖=𝑘−𝑑𝑀}𝑥^𝑇(𝑖)𝑄₃𝑥(𝑖),𝑉₄𝑥(𝑘),𝑑_𝑘,𝛼_𝑘=^−𝑑𝑚_{𝑗=−𝑑𝑀+1}^𝑘−1_{𝑖=𝑘+𝑗}𝑥^𝑇(𝑖)𝑄₁𝑥(𝑖)𝑉₅𝑥(𝑘),𝑑_𝑘,𝛼_𝑘=^−𝑑𝑚_{𝑗=−𝑑𝑀+1}^𝑘−1_{𝑖=𝑘+𝑗−1}𝑑_𝑀−𝑑_𝑚𝜂^𝑇(𝑖)𝑍₁𝜂(𝑖)+⁻¹_{𝑗=−𝑑𝑚}^𝑘−1_{𝑖=𝑘+𝑗}𝑑_𝑚𝜂^𝑇(𝑖)𝑍₂𝜂(𝑖), (3) where 𝜂(𝑘)=𝑥(𝑘+1)−𝑥(𝑘)=(𝐴+𝐵𝐾−𝐼)𝑥(𝑘)−𝐵𝐾𝑒(𝑘)+𝑓(𝑥(𝑘)).
Then, for 𝑑_𝑘+1=𝑠, 𝑑_𝑘=𝑟, 𝑃(𝑠)=∑^𝑑𝑀_{𝑠=𝑑𝑚}𝜆_𝑟𝑠𝑃(𝑠), we have 𝐸Δ𝑉₁=(𝐴+𝐵𝐾)𝑥(𝑘)−𝐵𝐾𝑒(𝑘)+𝑓(𝑥(𝑘))^𝑇×𝑃₁(𝑠)(𝐴+𝐵𝐾)𝑥(𝑘)−𝐵𝐾𝑒(𝑘)+𝑓(𝑥(𝑘))−𝑥^𝑇(𝑘)𝑃₁(𝑟)𝑥(𝑘)+)𝑒(𝑘)(𝑘)−((𝐴−𝐿𝐶+𝐿𝐶𝑥1−𝛼)𝐿𝐶𝑥(𝑘−𝑟)−𝛼𝐿𝑦(𝑘−1)^𝑇×𝑃₂(𝑠)𝑒(𝑘)(𝑘)−()(𝐴−𝐿𝐶+𝐿𝐶𝑥1−𝛼)𝐿𝐶𝑥(𝑘−𝑟)−𝛼𝐿𝑦()𝑒(𝑘)(𝑘)𝑘−1)+2(𝐴−𝐿𝐶+𝐿𝐶𝑥−(1−𝛼)𝐿𝐶𝑥(𝑘−𝑟)−𝛼𝐿𝑦(𝑘−1)^𝑇×𝑃₂(𝑠)𝑓(𝑥(𝑘))+𝛼₁𝐿𝐶𝑥(𝑘−𝑟)−𝐿𝑦(𝑘−1)^𝑇×𝑃₂(𝑠)𝐿𝐶𝑥(𝑘−𝑟)−𝐿𝑦(𝑘−1)+𝑓^𝑇(𝑥(𝑘))𝑃₂(𝑠)𝑓(𝑥(𝑘))−𝑒^𝑇(𝑘)𝑃₂(𝑟)𝑒(𝑘)−𝑦^𝑇(𝑘−1)𝑃₃(𝑟)𝑦()+𝑘−1(1−𝛼)𝐶𝑥(𝑘−𝑟)+𝛼𝑦(𝑘−1)^𝑇×𝑃₃(𝑠)(1−𝛼)𝐶𝑥(𝑘−𝑟)+𝛼𝑦(𝑘−1)+𝛼₁𝑦(𝑘−1)−𝐶𝑥(𝑘−𝑟)^𝑇×𝑃₃(𝑠)𝑦()(,𝐸𝑘−1−𝐶𝑥𝑘−𝑟)Δ𝑉₂=𝑥^𝑇(𝑘)𝑄₁𝑥(𝑘)−𝑥^𝑇(𝑘−𝑟)𝑄₁𝑥()+𝑘−𝑟_{𝑘−1𝑖=𝑘+1−𝑑𝑘+1}𝑥^𝑇(𝑖)𝑄₁𝑥(𝑖)−^𝑘−1_{𝑖=𝑘+1−𝑑𝑘}𝑥^𝑇(𝑖)𝑄₁𝑥(𝑖). (18)(19)
Note that ^𝑘−1_{𝑖=𝑘+1−𝑑𝑘+1}𝑥^𝑇(𝑖)𝑄₁𝑥(𝑖)=^𝑘−1_{𝑖=𝑘+1−𝑑𝑚}𝑥^𝑇(𝑖)𝑄₁𝑥(𝑖)+^{𝑘−𝑑𝑚}_{𝑖=𝑘+1−𝑑𝑘+1}𝑥^𝑇(𝑖)𝑄₁𝑥(𝑖)≤^𝑘−1_{𝑖=𝑘+1−𝑑𝑘}𝑥^𝑇(𝑖)𝑄₁𝑥(𝑖)+^{𝑘−𝑑𝑚}_{𝑖=𝑘+1−𝑑𝑀}𝑥^𝑇(𝑖)𝑄₁𝑥(𝑖). (20)
Therefore, we have 𝐸Δ𝑉₂≤𝑥^𝑇(𝑘)𝑄₁𝑥(𝑘)−𝑥^𝑇(𝑘−𝑟)𝑄₁𝑥()−𝑘−𝑟_{𝑘−𝑑𝑚𝑖=𝑘+1−𝑑𝑀}𝑥^𝑇(𝑖)𝑄₁𝑥(𝑖)𝐸Δ𝑉₃=𝑥^𝑇(𝑘)𝑄₂𝑥(𝑘)−𝑥^𝑇𝑘−𝑑_𝑚𝑄₂𝑥𝑘−𝑑_𝑚+𝑥^𝑇(𝑘)𝑄₃𝑥(𝑘)−𝑥^𝑇𝑘−𝑑_𝑀𝑄₃𝑥𝑘−𝑑_𝑀𝐸Δ𝑉₄=𝑑_𝑀−𝑑_𝑚𝑥^𝑇(𝑘)𝑄₁𝑥(𝑘)−^{𝑘−𝑑𝑚}_{𝑖=𝑘+1−𝑑𝑀}𝑥^𝑇(𝑖)𝑄₁𝑥(𝑖)𝐸Δ𝑉₅=𝑑_𝑀−𝑑_𝑚²𝜂^𝑇(𝑘)𝑍₁𝜂(𝑘)−^{𝑘−𝑑𝑚−1}_{𝑖=𝑘−𝑑𝑀}𝑑_𝑀−𝑑_𝑚𝜂^𝑇(𝑖)𝑍₁𝜂(𝑖)+𝑑²_𝑚𝜂^𝑇(𝑘)𝑍₂𝜂(𝑘)−^𝑘−1_{𝑖=𝑘−𝑑𝑚}𝑑_𝑚𝜂^𝑇(𝑖)𝑍₂𝜂(𝑖). (21)(22)(23)(24)
Using Lemma 7, we have the following inequality −^𝑘−1_{𝑖=𝑘−𝑑𝑚}𝑑_𝑚𝜂^𝑇(𝑖)𝑍₂𝜂(𝑖)−^{𝑘−𝑑𝑚−1}_{𝑖=𝑘−𝑑𝑀}𝑑_𝑀−𝑑_𝑚𝜂^𝑇(𝑖)𝑍₁𝜂(𝑖)=−^𝑘−1_{𝑖=𝑘−𝑑𝑚}𝑑_𝑚𝜂^𝑇(𝑖)𝑍₂𝜂(𝑖)−^{𝑘−𝑑𝑚−1}_{𝑖=𝑘−𝑟}𝑑_𝑀−𝑑_𝑚𝜂^𝑇(𝑖)𝑍₁𝜂(𝑖)−^{𝑘−𝑟−1}_{𝑖=𝑘−𝑑𝑀}𝑑_𝑀−𝑑_𝑚𝜂^𝑇(𝑖)𝑍₁𝜂(𝑖)≤−𝑥(𝑘)−𝑥(𝑘−𝑑_𝑚)^𝑇×𝑍₂𝑥(𝑘)−𝑥𝑘−𝑑_𝑚−𝑥(𝑘−𝑑_𝑚)−𝑥(𝑘−𝑟)^𝑇×𝑍₁𝑥𝑘−𝑑_𝑚(−−𝑥𝑘−𝑟)𝑥(𝑘−𝑟)−𝑥(𝑘−𝑑_𝑀)^𝑇×𝑍₁𝑥(𝑘−𝑟)−𝑥𝑘−𝑑_𝑀. (25)
From (2), for any scalar Γ>0, we have Γ𝛽²𝑥^𝑇(𝑘)𝐺^𝑇𝐺𝑥(𝑘)−𝑓^𝑇(𝑥(𝑘))𝑓(𝑥(𝑘))≥0. (26)
Then, from (18), (21)–(26), we have 𝐸Δ𝑉𝑥(𝑘),𝑑_𝑘,𝛼_𝑘≤𝜁^𝑇Φ𝜁, (27) where 𝜁(𝑘)=𝜉^𝑇(𝑘)𝑥^𝑇(𝑘−𝑟)𝑥^𝑇(𝑘−𝑑_𝑚)𝑥^𝑇(𝑘−𝑑_𝑀)𝑓^𝑇(𝑥(𝑘))^𝑇ΦΦ=¹¹Φ₁₂Φ₁₃Φ₁₄𝑍₂0Φ₁₇∗Φ₂₂Φ₂₃Φ₂₄00Φ₂₇∗∗Φ₃₃000Φ₃₇∗∗∗Φ₄₄Φ₄₅Φ₄₆Φ₄₇∗∗∗∗Φ₅₅00∗∗∗∗∗Φ₆₆0∗∗∗∗∗∗Φ₇₇. (28)(29) From (15), we conclude that Φ<0; thus we have 𝐸Δ𝑉𝑥(𝑘),𝑑_𝑘,𝛼_𝑘≤𝜁^𝑇Φ𝜁<0. (30) Following a similar line in the proof of Theorem  1 in [18], the stochastic stability of the closed-loop system (10) can be confirmed.
Now, we apply the result obtained in Theorem 8 to solve the controller design problem.

Theorem 9. For the given scalars 0≤𝛼≤1, 𝛽>0, 𝜀_𝑙>0, 𝑙=1,2,3,4,5, the closed-loop system (10) is stochastically stable for each mode 𝑟∈Λ, if there exist matrices 𝑋₁(𝑟)>0, 𝑋₂(𝑟)>0, 𝑋₃(𝑟)>0, 𝑍₁>0, 𝑍₂>0, 𝑄₁>0, 𝑄₂>0, 𝑄₃>0, 𝐾, 𝐿 and scalar Γ>0, satisfying the following linear matrix inequality:Ω₀Ω₁Ω₂Ω₃Ω₃Ω₄Ω₄Ω₅Ω₅Ω₆Ω₆Ω₇Ω₇∗−2𝜀₁𝐼+𝜀²₁𝑍₁00000000000∗∗−2𝜀₂𝐼+𝜀²₂𝑍₂0000000000∗∗∗Ω₈000000000∗∗∗∗Ω₈00000000∗∗∗∗∗Ω₉0000000∗∗∗∗∗∗Ω₉000000∗∗∗∗∗∗∗Ω₉00000∗∗∗∗∗∗∗∗Ω₉0000∗∗∗∗∗∗∗∗∗Ω₁₀000∗∗∗∗∗∗∗∗∗∗Ω₁₀00∗∗∗∗∗∗∗∗∗∗∗Ω₁₀0∗∗∗∗∗∗∗∗∗∗∗∗Ω₁₀<0, (31) where Ω₀=Φ₁₁000𝑍₂00∗−2𝜀₄𝐼+𝜀²₄𝑋₂(𝑟)00000∗∗−2𝜀₅𝐼+𝜀²₅𝑋₃(𝑟)0000∗∗∗−𝑄₁−2𝑍₁Φ₄₅Φ₄₆0∗∗∗∗Φ₅₅00∗∗∗∗∗Φ₆₆0,∗∗∗∗∗∗−Γ𝐼Φ₁₁=−2𝜀₃𝐼+𝜀²₃𝑋₁(𝑟)−𝑍₂+𝑑_𝑀−𝑑_𝑚+1𝑄₁+𝑄₂+𝑄₃+Γ𝛽²𝐺^𝑇𝐺,𝑃⁻¹(𝑟)(𝑟),Θ=𝑋₁=𝜆_𝑟𝜅^𝑟₁⋯𝜆_{𝑟𝜅𝑟𝑚},Θ₂=√𝜐_𝑟⋯√𝜐_𝑟,Ω₁=(𝑑_𝑀−𝑑_𝑚)(𝐴+𝐵𝐾−𝐼)−(𝑑_𝑀−𝑑_𝑚)(𝐵𝐾)0000(𝑑_𝑀−𝑑_𝑚)𝐼^𝑇,Ω₂=𝑑_𝑚(𝐴+𝐵𝐾−𝐼)−𝑑_𝑚(𝐵𝐾)0000𝑑_𝑚𝐼^𝑇,Ω₃=Θ₁(𝐴+𝐵𝐾)−𝐵𝐾0000𝐼^𝑇,Ω₃=Θ₂(𝐴+𝐵𝐾)−𝐵𝐾0000𝐼^𝑇,Ω₄=Θ₁𝐿𝐶𝐴−𝐿𝐶−𝛼𝐿−(1−𝛼)(𝐿𝐶)00𝐼^𝑇,Ω₄=Θ₂𝐿𝐶𝐴−𝐿𝐶−𝛼𝐿−(1−𝛼)(𝐿𝐶)00𝐼^𝑇,Ω₅=Θ₁00√𝛼₁𝐿−√𝛼₁(𝐿𝐶)000^𝑇,Ω₅=Θ₂00√𝛼₁𝐿−√𝛼₁(𝐿𝐶)000^𝑇,Ω₆=Θ₁00𝛼𝐼(1−𝛼)𝐶000^𝑇,Ω₆=Θ₂00𝛼𝐼(1−𝛼)𝐶000^𝑇,Ω₇=Θ₁00√𝛼₁𝐼−√𝛼₁𝐶000^𝑇,Ω₇=Θ₂00√𝛼₁𝐼−√𝛼₁𝐶000^𝑇,Ω₈=−diag𝑋₁𝜅^𝑟₁⋯𝑋₁𝜅^𝑟_𝑚,Ω₈=−diag𝑋₁𝜅^𝑟₁⋯𝑋₁𝜅^𝑟_𝑀−𝑚,Ω₉=−diag𝑋₂𝜅^𝑟₁⋯𝑋₂𝜅^𝑟_𝑚,Ω₉=−diag𝑋₂𝜅^𝑟₁⋯𝑋₂𝜅^𝑟_𝑀−𝑚,Ω₁₀=−diag𝑋₃𝜅^𝑟₁⋯𝑋₃𝜅^𝑟_𝑚,Ω₁₀=−diag𝑋₃𝜅^𝑟₁⋯𝑋₃𝜅^𝑟_𝑀−𝑚. (32)

Proof. Note that (15) is equivalent to Φ₁₁000𝑍₂00∗−𝑃₂(𝑟)00000∗∗−𝑃₃(𝑟)0000∗∗∗−𝑄₁−2𝑍₁Φ₄₅Φ₄₆0∗∗∗∗Φ₅₅00∗∗∗∗∗Φ₆₆0+∗∗∗∗∗∗−Γ𝐼𝐴+𝐵𝐾−𝐵𝐾0000𝐼^𝑇𝑃₁(𝑠)×+𝐴+𝐵𝐾−𝐵𝐾0000𝐼𝐿𝐶𝐴−𝐿𝐶−𝛼𝐿−(1−𝛼)(𝐿𝐶)00𝐼^𝑇𝑃₂(𝑠)×𝐿𝐶𝐴−𝐿𝐶−𝛼𝐿−(1−𝛼)+√)(𝐿𝐶00𝐼00𝛼₁𝐿−√𝛼₁(𝐿𝐶)000^𝑇𝑃₂(𝑠)×00√𝛼₁𝐿−√𝛼₁()+𝐿𝐶00000𝛼𝐼(1−𝛼)𝐶000^𝑇𝑃₃(𝑠)×00𝛼𝐼(1−𝛼)+√𝐶00000𝛼₁𝐼−√𝛼₁𝐶000^𝑇𝑃₃(𝑠)×00√𝛼₁𝐼−√𝛼₁+𝑑𝐶000_𝑚(𝐴+𝐵𝐾−𝐼)−𝑑_𝑚(𝐵𝐾)0000𝑑_𝑚𝐼^𝑇𝑍₂×𝑑_𝑚(𝐴+𝐵𝐾−𝐼)−𝑑_𝑚(𝐵𝐾)0000𝑑_𝑚𝐼+^{𝑑𝑀−𝑑𝑚)(𝐴+𝐵𝐾−𝐼𝑇−𝑑𝑀−𝑑𝑚()𝐵𝐾𝑇0}_{000𝑑𝑀−𝑑𝑚𝐼}𝑍₁^{𝑑𝑀−𝑑𝑚)(𝐴+𝐵𝐾−𝐼𝑇−𝑑𝑀−𝑑𝑚()𝐵𝐾𝑇0}_{000𝑑𝑀−𝑑𝑚𝐼}^𝑇<0, (33) where Φ₁₁=−𝑃₁(𝑟)−𝑍₂+(𝑑_𝑀−𝑑_𝑚+1)𝑄₁+𝑄₂+𝑄₃+Γ𝛽²𝐺^𝑇𝐺.
Via Lemma 6, we obtain ^𝑀_𝑠=1𝜆_𝑟𝑠𝑃𝜏_𝑠=_{𝑠∈𝐼𝑟𝜅}𝜆_𝑟𝑠𝑃𝜏_𝑠+_{𝑠∈𝐼𝑟𝑢𝜅}𝜆_𝑟𝑠𝑃𝜏_𝑠≤𝑃𝐼^𝑟_𝜅+𝜐_𝑟𝑃𝐼^𝑟_𝑢𝜅, (34) where 𝑃𝐼^𝑟_𝜅=_{𝑠∈𝐼𝑟𝜅}𝜆_𝑟𝑠𝑃𝜏_𝑠,𝑃𝐼^𝑟_𝑢𝜅=_{𝑠∈𝐼𝑟𝑢𝜅}𝑃𝜏_𝑠,𝜐_𝑟=1−_{𝑠∈𝐼𝑟𝜅}𝜆_𝑟𝑠. (35)
From (34) and Schur complement Lemma, (33) can be transformed into the following inequality:Ω₀Ω₁Ω₂Ω₃Ω₃Ω₄Ω₄Ω₅Ω₅Ω₆Ω₆Ω₇Ω₇∗−𝑍₁⁻¹00000000000∗∗−𝑍₂⁻¹0000000000∗∗∗Ω₈000000000∗∗∗∗Ω₈00000000∗∗∗∗∗Ω₉0000000∗∗∗∗∗∗Ω₉000000∗∗∗∗∗∗∗Ω₉00000∗∗∗∗∗∗∗∗Ω₉0000∗∗∗∗∗∗∗∗∗Ω₁₀000∗∗∗∗∗∗∗∗∗∗Ω₁₀00∗∗∗∗∗∗∗∗∗∗∗Ω₁₀0∗∗∗∗∗∗∗∗∗∗∗∗Ω₁₀<0, (36)where Ω₀=Φ₁₁000𝑍₂00∗−𝑃₂(𝑟)00000∗∗−𝑃₃(𝑟)0000∗∗∗−𝑄₁−2𝑍₁Φ₄₅Φ₄₆0∗∗∗∗Φ₅₅00∗∗∗∗∗Φ₆₆0∗∗∗∗∗∗−Γ𝐼. (37)
For scalars 𝜀_𝑙, 𝑙=1,2,3,4,5, the following inequalities hold: 𝐼−𝜀₁𝑍₁^𝑇𝑍₁⁻¹𝐼−𝜀₁𝑍₁≥0,𝐼−𝜀₂𝑍₂^𝑇𝑍₂⁻¹𝐼−𝜀₂𝑍₂≥0,𝐼−𝜀₃𝑃₁⁻¹(𝑟)^𝑇𝑃₁(𝑟)𝐼−𝜀₃𝑃₁⁻¹(𝑟)≥0,𝐼−𝜀₄𝑃₂⁻¹(𝑟)^𝑇𝑃₂(𝑟)𝐼−𝜀₄𝑃₂⁻¹(𝑟)≥0,𝐼−𝜀₅𝑃₃⁻¹(𝑟)^𝑇𝑃₃(𝑟)𝐼−𝜀₅𝑃₃⁻¹(𝑟)≥0. (38) Then we have −𝑍₁⁻¹≤−2𝜀₁𝐼+𝜀²₁𝑍₁,−𝑍₂⁻¹≤−2𝜀₂𝐼+𝜀²₂𝑍₂,−𝑃₁(𝑟)≤−2𝜀₃𝐼+𝜀²₃𝑃₁⁻¹(𝑟),−𝑃₂(𝑟)≤−2𝜀₄𝐼+𝜀²₄𝑃₂⁻¹(𝑟),−𝑃₃(𝑟)≤−2𝜀₅𝐼+𝜀²₅𝑃₃⁻¹(𝑟). (39)
According to the inequalities (39), it is obtained that if (31) holds, then (36) holds.

Remark 10. In Theorem 9, the inequality (31) is a linear matrix inequality, which can easily be solved by using the Matlab LMI Toolbox.

Remark 11. Compared with the results in [9, 10], partially unknown elements in the transition probability matrix do not require the complete knowledge or even the lower and upper bounds of the transition probabilities in the jump process, which means that it is more powerful and desirable.

Remark 12. The results in this paper derived based on the assumption that the network exists only in the sensor-to-controller side, but when the network exists both in the sensor-to-controller and controller-to-actuator sides, this needs further study.

4. Numerical Example

The system data of (1) is described as follows: 10,.𝐴=0.080.130.410.19,𝐵=𝐶=0.200.6−0.1 (40)

The random delay is 𝑑_𝑘∈{1,2,3}, and its transition probability matrix is given by Π₁=0.90.050.050.50.40.10.10.10.8,Π₂=,Π0.9???0.4?0.10.10.8₃=.????????? (41)

The maximum allowable nonlinear level 𝛽 and the maximum packet dropout rate 𝛼 for three different transition probability matrices are given in Tables 1 and 2, respectively, in which “—” means that the results are not applicable to the corresponding cases. Thus, from Tables 1 and 2, it is easily seen that the more transition probability matrix we have, the larger maximum dropout rate 𝛼 for different 𝛽 and the larger maximum allowable nonlinear level 𝛽 for different 𝛼 can be obtained for ensuring stability.

Table 1 

The maximum dropout rate 𝛼 for different 𝛽 and Π.

Table 2 

The maximum allowable nonlinear level 𝛽 for different 𝛼 and Π.

Give that𝑥(−2)=𝑥(−1)=𝑥(0)=1−0.5^𝑇,𝑦(−2)=𝑦(−1)=𝑦(0)=00^𝑇,𝛼=0.5,𝛽=0.3,𝜀₁𝜀=3.5,₂=2.8,𝜀₃=10,𝜀₄𝜀=1.1,₅=0.6,𝐺=1. (42)

Assuming that the nonlinear function is 𝑓(𝑥(𝑘))=0.1sin(𝑥(𝑘)), by Theorem 9, we can obtain the controller gain and observer gain with Π₂ as follows: ,.𝐾=0.0017−0.0137𝐿=0.00280.00200.01480.0361 (43)

Based on the designed controller and observer, the simulation results of the state responses are given in Figure 2. It is obvious that the designed novel dynamic output feedback controller is feasible and ensures the stability of the closed-loop system despite the random packet dropout and random delay.

Figure 2 

State response of the closed-loop system.

5. Conclusion

In this paper, a novel dynamic output feedback controller is designed for nonlinear networked control systems with both random packet dropout and random delay. A sufficient condition is presented for the stabilization of the closed systems based upon the Lyapunov theory. The quantitative relationship of the maximum dropout rate, the transition probability matrix, and the maximum allowable nonlinear level is derived by solving a set of linear matrix inequalities. An example is presented to illustrate the effectiveness of the proposed method.

Acknowledgments

This work is supported by NNSFC (60974139), and by the Fundamental Research Funds for the central universities (72103676 and K50511070008).