Stabilization for Networked Control Systems with Random Sampling Periods

Abstract

This paper investigates the stabilization of networked control systems (NCSs) with random delays and random sampling periods. Sampling periods can randomly switch between three cases according to the high, low, and medium types of network load. The sensor-to-controller (S-C) random delays and random sampling periods are modeled as Markov chains. The transition probabilities of Markov chains do not need to be completely known. A state feedback controller is designed via the iterative linear matrix inequality (LMI) approach. It is shown that the designed controller is two-mode dependent and depends on not only the current S-C delay but also the most recent available sampling period at the controller node. The resulting closed-loop systems are special discrete-time jump linear systems with two modes. The sufficient conditions for the stochastic stability are established. An example of the cart and inverted pendulum is given to illustrate the effectiveness of the theoretical result.

1. Introduction

Networked control systems (NCSs) are feedback control systems whose feedback paths are implemented by a real-time network. Recently, much attention has been paid to the study of NCSs, due to their low cost, reduced weight and power requirements, simple installation and maintenance, high reliability, and so on [1, 2].

A basic problem in NCSs is the stability of the systems. In real-time control systems, three main issues are bandwidth and packet size constraints, time delays, and packet losses, which will degrade the performance of control systems and even make systems unstable; so it is significant to overcome the adverse influences of time delays, packet losses, and dynamic bandwidth [3–7].

On the other hand, the Markov chain, a discrete-time stochastic process with the Markov property, can be effectively used to model NCSs with random time delays and random sampling periods, which are modeled as the Markovian jump linear systems (MJLSs). Recently, there have been considerable research efforts into NCSs [8–12]. For example, Xiao et al. present an V-K iteration algorithm to design stabilizing controllers for specially structured discrete-time jump linear systems with random but bounded delays in the feedback loop [8]. Zhang et al. propose a promising two-mode-dependent-state feedback scheme to stabilize section with the current S-C delay (𝜏_𝑘) and the previous controller-to-actuator (C-A) delay (𝑑_𝑘−1) modeled as two Markov chains [9]. Shi and Yu investigate the output feedback stabilization and robust mixed 𝐻₂/𝐻_∞ control of NCSs, in which it is assumed that at each sampling instant, the current S-C delay (𝜏_𝑘) and the most recent available C-A delay (𝑑_{𝑘−1−𝜏𝑘}) can be obtained [10, 11]. Huang and Nguang discuss the stabilization problem for a class of linear uncertain continuous NCSs, in which it is assumed that random communication S-C delays (𝜏(𝑡)) and random communication C-A delays (𝜌(𝑡)) are modeled two continuous time-discrete-state Markov processes [12].

In the above references, the transition probabilities are assumed completely accessible and considered as the available knowledge for analyzing and designing the NCSs. In practice, this kind of information including the variation of time delays and packet losses is hard to obtain. The problems of partly unknown transition probabilities were investigated [13–16]. Zhang et al. investigate the stability of Markovian jump linear systems with partly unknown transition probabilities [13, 14]. Wang et al. study the partially mode-dependent 𝐻_∞ filtering problem for discrete-time Markovian jump systems with partly unknown transition probabilities [15]. Sun and Qin investigate the stability and stabilization problems of a class of NCSs with bounded packet dropout, in which the transition probabilities are partly unknown due to the complexity of network [16]. However, the developed controllers of these references [13–16] only are either mode independent or one-mode dependent, and the design problem can thus be readily converted into a standard MJLS problem. To the best of the authors’ knowledge, if the transition probabilities are assumed partly accessible, designing the two-mode-dependent controller that simultaneously depends on both the current S-C delay (𝜏_𝑘^𝑠𝑐), and the most recent available sampling period (ℎ_{𝑘−𝜏𝑘}^𝑠𝑐) has not been fully investigated, which is the focus of this work. When considering both 𝜏_𝑘^𝑠𝑐and ℎ_{𝑘−𝜏𝑘}^𝑠𝑐, the resulting closed-loop system can be transformed to a special MJLS, and thus the well-developed results on MJLSs with partly unknown transition probabilities cannot be directly applied [13–16].

In this paper, the stochastic stability of NCSs with random time delays and random sampling periods is studied, in which time delays and sampling periods are driven by two finite-state Markov chains. This paper is organized as follows. In Section 2, the NCSs model with random S-C time delays and random sampling periods is made, which is equivalent to a class of special discrete-time jump linear systems with two modes. Sufficient and necessary conditions of stochastic stability with completely known transition probabilities for the foregoing model are considered in Section 3. Sufficient conditions of stochastic stability with partly unknown transition probabilities for the foregoing model are considered in Section 4. Section 5 is an illustrative example, and our work in this paper is summarized in Section 6.

Notation. In this paper, 𝐑 is the set of all real numbers, 𝐑^𝑛 denotes the 𝑛-dimensional Euclidean space. (Ω,𝐹,𝑃) denotes the probability space. 𝐀^𝑇 and 𝐀⁻¹ denote the transpose and the inverse of a matrix 𝐀, respectively. 𝐀>𝟎(𝐀<𝟎) means that 𝐀 is positive definite (negative definite). 𝟎 and 𝐈 are the zero and identity matrices with appropriate dimensions, respectively. In symmetric block matrices, we use an asterisk (∗) to represent a term that is induced by symmetry.

2. Problem Formulation

The structure of the considered NCSs is shown in Figure 1, where the plant is described by the following linear system model: 𝑥̇(𝑡)=𝐴𝑥(𝑡)+𝐵𝑢(𝑡), (1) where 𝑥(𝑡)∈𝐑^𝑛,𝑢(𝑡)∈𝐑^𝑚 are the system state vector and control input vector, respectively; 𝐴,𝐵 are known constant matrices of appropriate dimensions. Suppose the sensor is clock-driven, the controller and the actuator are event-driven.

Figure 1 

Diagram of NCSs.

Suppose bounded random delays only exist in the link from sensor to controller, as shown in Figure 1. Here, 0≤𝜏_𝑘^𝑠𝑐≤𝜏^𝑠𝑐 represents the S-C delay. The state feedback controller is to be designed.

For NCSs, the shorter the sampling period, the better system performance; however, the short sampling period will increase the possibility of network congestion. If the constant sampling period is adopted, the sampling period should be large enough to avoid network congestion, so network bandwidth cannot be sufficiently used when the network is idle. In [17–19], the variable sampling method is used, and the sampling periods are assumed to switch in a finite discrete set. But when the systems switch too fast, it is apt to cause oscillation and instability of the system. In the actual network, the size of sampling period is closely related to the network load. However, the network load usually is random [20]. As a result of the above, network load can be high load, low load, and medium load. Correspondingly, sampling periods can randomly switch between three cases of maximum, medium, and minimum in this paper. In the following, we consider that sample periods randomly switch between three cases and make NCSs model with random time delays and random sample periods. Thus, not only the network bandwidth can be fully used, but also the conservativeness of the stabilization conditions of NCSs can be reduced.

Suppose ℎ_𝑘 is the length of the kth sampling period, if the network is idle, define the sampling period as ℎ_max, and if the network is occupied by the most users, define the sampling period as ℎ_min, otherwise define the sampling period as (1/2)(ℎ_max+ℎ_min). Then, the sampling period ℎ_𝑘∈{ℎ_min,(1/2)(ℎ_max+ℎ_min),ℎ_max}.

The discrete-time expression of the system (1) is as follows: 𝑥(𝑘+1)=Φℎ_𝑘𝑥(𝑘)+Γℎ_𝑘𝑢(𝑘), (2) where Φ(ℎ_𝑘)=𝑒^𝐴ℎ𝑘,andΓ(ℎ_𝑘)=∫^ℎ𝑘₀𝑒^𝐴𝑡𝑑𝑡𝐵.

In this paper, 𝜏_𝑘^𝑠𝑐 and ℎ_𝑘 are modeled as two homogeneous Markov chains defined in (Ω,𝐹,𝑃) that take values in 𝜑={0,1,…,𝜏^𝑠𝑐} and 𝜙={1,2,3}, and their transition probability matrices are Λ=[𝜆_𝑖𝑗] and Π=[𝜋_𝑟𝑠], respectively, and 𝜆_𝑖𝑗=Pr𝜏^𝑠𝑐_𝑘+1=𝑗∣𝜏_𝑘^𝑠𝑐,𝜋=𝑖_𝑟𝑠=Prℎ_𝑘+1=𝑠∣ℎ_𝑘=𝑟, (3) where 𝜆_𝑖𝑗,𝜋_𝑟𝑠≥0, and ∑^𝜏𝑠𝑐_𝑗=0𝜆_𝑖𝑗=1,∑³_𝑠=1𝜋_𝑟𝑠=1, for all 𝑖,𝑗∈𝜑 and 𝑟,𝑠∈𝜙.

Remark 1. In this paper, we assume that the controller will always use the most recent data [8]. Thus, if we have 𝑥(𝑘−𝜏_𝑘^𝑠𝑐) at sampling instant 𝑡_𝑘, then at sampling time 𝑡_𝑘+1, even if there are delays longer than 1 or package loss, we still have 𝑥(𝑘−𝜏_𝑘^𝑠𝑐) to use. So in our model of the system in Figure 1, the delay 𝜏_𝑘^𝑠𝑐 can increase at most 1 each step, Pr{𝜏^𝑠𝑐_𝑘+1>𝜏_𝑘^𝑠𝑐+1}=0.

In NCSs, to reduce the conservativeness of the stabilization conditions, it is desirable to consider not only the time delay but also the sample period information in the controller design. For the controller node, at time instant 𝑡_𝑘, 𝜏_𝑘^𝑠𝑐 can be obtained by comparing the current time and the time-stamp of the sensor information received. However, this sample period information cannot be received by the controller immediately because it needs to be transmitted through the network from sensor to controller. So if the time delay 𝜏_𝑘^𝑠𝑐 exists, the information ℎ_{𝑘−𝜏𝑘}^𝑠𝑐 of time instant 𝑡_𝑘 would be known at the controller node. Consequently, it is desirable to design the state feedback controller that simultaneously depends on both 𝜏_𝑘^𝑠𝑐 and ℎ_{𝑘−𝜏𝑘}^𝑠𝑐. The mode-dependent state feedback control law is 𝑢(𝑘)=𝐾𝜏_𝑘^𝑠𝑐,ℎ_{𝑘−𝜏𝑘}^𝑠𝑐𝑥(𝑘)=𝐾𝜏_𝑘^𝑠𝑐,ℎ_{𝑘−𝜏𝑘}^𝑠𝑐𝑥𝑘−𝜏_𝑘^𝑠𝑐, (4) where 𝑥(𝑘) is the input vector of the state feedback controller, and 𝐾 is the controller gain to be designed.

In the following, we make the closed-loop system model. Substituting the formula (4) into (2), we can obtain the following closed-loop NCSs: 𝑥(𝑘+1)=Φℎ_𝑘𝑥(𝑘)+Γℎ_𝑘𝐾𝜏_𝑘^𝑠𝑐,ℎ_{𝑘−𝜏𝑘}^𝑠𝑐𝑥𝑘−𝜏_𝑘^𝑠𝑐. (5)

Augment the plant’s state variable as 𝑋(𝑘)=𝑥^𝑇(𝑘)𝑥^𝑇(𝑘−1)…𝑥^𝑇𝑘−𝜏^𝑠𝑐𝑇, (6) where 𝑋(𝑘)∈𝐑^{(𝜏𝑠𝑐+1)𝑛}, then we have 𝑋(𝑘+1)=𝐴ℎ_𝑘+𝐵ℎ_𝑘𝐾𝜏_𝑘^𝑠𝑐,ℎ_{𝑘−𝜏𝑘}^𝑠𝑐𝐶𝜏_𝑘^𝑠𝑐𝑋(𝑘)=𝐴𝜏_𝑘^𝑠𝑐,ℎ_{𝑘−𝜏𝑘}^𝑠𝑐,ℎ_𝑘𝑋(𝑘), (7) where 𝐴ℎ_𝑘=Φℎ_𝑘,0⋯00𝐼0⋯000𝐼⋯00⋮⋮⋱⋮⋮00⋯𝐼0𝐵ℎ_𝑘=Γℎ_𝑘00⋮0,𝐶𝜏_𝑘^𝑠𝑐=[],𝐴𝜏0⋯⋯0𝐼0⋯0_𝑘𝑠𝑐,ℎ_{𝑘−𝜏𝑘}^𝑠𝑐,ℎ_𝑘=𝐴ℎ_𝑘+𝐵ℎ_𝑘𝐾𝜏_𝑘^𝑠𝑐,ℎ_{𝑘−𝜏𝑘}^𝑠𝑐𝐶𝜏_𝑘^𝑠𝑐, (8) and 𝐶(𝜏_𝑘^𝑠𝑐) has all elements being zero except for the 𝜏_𝑘^𝑠𝑐+1th block being an identity matrix.

Remark 2. Because the controller in (4) is two-mode dependent, the resulting closed-loop system in (7) depends on 𝜏_𝑘^𝑠𝑐, ℎ_𝑘, ℎ_{𝑘−𝜏𝑘}^𝑠𝑐 and can be transformed to a special MJLS. In addition, ℎ_{𝑘−𝜏𝑘}^𝑠𝑐 is related to both 𝜏_𝑘^𝑠𝑐and ℎ_𝑘.

It can be seen that the closed-loop system in (7) is a jump linear system with two modes modeled by different homogeneous Markov chains. The objective of this paper is to design the state feedback controller (4) to guarantee the stochastic stability of the NCSs in (7). For the stochastic stability, we adopt the following definition [9].

Definition 3. The system in (7) is stochastically stable if for every finite matrix 𝑋₀=𝑋(0), initial mode 𝜏₀^𝑠𝑐=𝜏^𝑠𝑐(0)∈𝜑, and ℎ_−𝜏0^𝑠𝑐=ℎ(−𝜏₀^𝑠𝑐)∈𝜙, there exists a finite matrix 𝑊>0 such that the following holds: 𝜀^∞_𝑘=0‖𝑋(𝑘)‖²∣𝑋₀,𝜏₀^𝑠𝑐,ℎ_−𝜏0^𝑠𝑐<𝑋^𝑇₀𝑊𝑋₀. (9)

3. The Case of Completely Known Transition Probabilities

In this section, we first give the sufficient and necessary conditions for the state feedback stabilization of NCSs with completely known transition probabilities in (7) and then derive the equivalent conditions of LMIs with nonconvex constraints.

Because ℎ_{𝑘−𝜏𝑘}^𝑠𝑐 is to be considered into the controller design, the multistep jump problem of Markov chains is involved in the system design. According to the Chapman-Kolmogorov (C-K) equation of stochastic process [21], we give the multistep transition probability as follows.

Lemma 4. If the transition probability matrix from ℎ_𝑘=𝑠₁ to ℎ_𝑘+1=𝑠 is Π=(𝜋_𝑠1𝑠),𝑠,𝑠₁∈𝜙, then the transition probability matrix from ℎ_𝑘−𝑖=𝑟 to ℎ_𝑘=𝑠₁ is Π^(𝑖)=(𝜋^(𝑖)_𝑟𝑠1),𝑟,𝑠₁∈𝜙, which is a multistep transition probability matrix of the Markov chain. Particularly, when 𝑖=0, the transition probability matrix is Π⁽⁰⁾=𝐼.

Proof. The transition probability from ℎ_𝑘−𝑖=𝑟 to ℎ_𝑘=𝑠₁ is 𝜋^(𝑖)_𝑟𝑠1, then 𝜋^(𝑖)_𝑟𝑠1=𝑃ℎ_𝑘=𝑠₁∣ℎ_𝑘−𝑖ℎ=𝑟=𝑃_{(𝑘−𝑖)+1+(𝑖−1)}=𝑠₁∣ℎ_𝑘−𝑖=𝑟=_𝑙1∈𝜙𝑃ℎ_{(𝑘−𝑖)+1}=𝑙₁∣ℎ_𝑘−𝑖ℎ=𝑟×𝑃_{(𝑘−𝑖)+1+(𝑖−1)}=𝑠₁∣ℎ_𝑘−𝑖=𝑟,ℎ_{(𝑘−𝑖)+1}=𝑙₁=_𝑙1∈𝜙𝑃ℎ_{(𝑘−𝑖)+1}=𝑙₁∣ℎ_𝑘−𝑖ℎ=𝑟×𝑃_{(𝑘−𝑖)+1+(𝑖−1)}=𝑠₁∣ℎ_{(𝑘−𝑖)+1}=𝑙₁=_𝑙1∈𝜙𝜋_𝑟𝑙1𝜋_𝑙^(𝑖−1)_1𝑠1=_𝑙1∈𝜙𝜋_{𝑟𝑙1𝑙2∈𝜙}𝑃ℎ_{(𝑘−𝑖)+2}=𝑙₂∣ℎ_{(𝑘−𝑖)+1}=𝑙₁×𝑃ℎ_{(𝑘−𝑖)+2+(𝑖−2)}=𝑠₁∣ℎ_{(𝑘−𝑖)+2}=𝑙₂=_𝑙1∈𝜙𝜋_{𝑟𝑙1𝑙2∈𝜙}𝜋_𝑙1𝑙2𝜋_𝑙^(𝑖−2)_2𝑠1=⋯=_{𝑙1∈𝜙𝑙2∈𝜙}⋯_{𝑙𝑖−1∈𝜙}𝜋_𝑟𝑙1𝜋_𝑙1𝑙2⋯𝜋_{𝑙𝑖−1𝑠1}. (10) From matrix multiplication, the transition probability matrix from ℎ_𝑘−𝑖 to ℎ_𝑘 is Π^(𝑖).

Remark 5. It is noted that (3) to (4) are based on the ineffectiveness theory of Markov chain. With Definition 3, the necessary and sufficient conditions on the stochastic stability of closed-loop system in (7) can be obtained.

Theorem 6. The closed-loop system with completely known transition probabilities in (7) is stochastically stable if and only if there exist the matrices 𝑃(𝑖,𝑟)>0 such that the following matrix inequalities: 𝐿(𝑖,𝑟)=^{𝜏𝑠𝑐3}_𝑗=0𝑠2³_=1𝑠1=1𝜆_𝑖𝑗𝜋^{(𝑖+1−𝑗)}_𝑟𝑠2𝜋^(𝑖)_𝑟𝑠1×𝐴𝑠₁+𝐵𝑠₁𝐾(𝑖,𝑟)𝐶(𝑖)^𝑇𝑃𝑗,𝑠₂×𝐴𝑠₁+𝐵𝑠₁𝐾(𝑖,𝑟)𝐶(𝑖)−𝑃(𝑖,𝑟)<0 (11) hold for all 𝑖∈𝜑 and 𝑟∈𝜙.

Proof (sufficiency). For the closed-loop system in (7), consider the Lyapunov function 𝑉(𝑋(𝑘),𝑘)=𝑋^𝑇(𝑘)𝑃𝜏_𝑘^𝑠𝑐,ℎ_{𝑘−𝜏𝑘}^𝑠𝑐𝑋(𝑘). (12) Then 𝜀{(𝑋(𝑘)𝑉(𝑋())(𝑋(𝑘))(𝑘)Δ𝑉,𝑘)}=𝜀𝑘+1,𝑘+1−𝑉,𝑘∣𝑋,𝜏_𝑘^𝑠𝑐=𝑖,ℎ_{𝑘−𝜏𝑘}^𝑠𝑐𝑋=𝑟=𝜀^𝑇(𝑘+1)𝑃𝜏^𝑠𝑐_𝑘+1,ℎ_{𝑘+1−𝜏}^𝑠𝑐_𝑘+1×𝑋(𝑘+1)∣𝑋(𝑘),𝜏_𝑘^𝑠𝑐=𝑖,ℎ_{𝑘−𝜏𝑘}^𝑠𝑐=𝑟−𝑋^𝑇(𝑘)𝑃(𝑖,𝑟)𝑋(𝑘). (13)
Define 𝜏^𝑠𝑐_𝑘+1=𝑗, ℎ_𝑘=𝑠₁, and ℎ_{𝑘+1−𝜏𝑠𝑐𝑘+1}=𝑠₂. To evaluate the first term in (13), we need to apply the three probability transition matrices: 𝜏_𝑘^𝑠𝑐→𝜏^𝑠𝑐_𝑘+1ℎ∶Λ,_{𝑘−𝜏𝑘𝑠𝑐}→ℎ_𝑘∶Π^(𝑖),ℎ_{𝑘−𝜏𝑘}^𝑠𝑐→ℎ_{𝑘+1−𝜏}^𝑠𝑐_𝑘+1∶Π^{(𝑖+1−𝑗)}. (14) Then, (13) can be written as 𝜀Δ𝑉(𝑋(𝑘),𝑘)∣𝑋(𝑘),𝜏_𝑘^𝑠𝑐=𝑖,ℎ_{𝑘−𝜏𝑘}^𝑠𝑐=𝑟=𝑋^𝑇(𝑘)^{𝜏𝑠𝑐3}_𝑗=0𝑠2³_=1𝑠1=1𝜆_𝑖𝑗𝜋^{(𝑖+1−𝑗)}_𝑟𝑠2𝜋^(𝑖)_𝑟𝑠1×𝐴𝑠₁+𝐵𝑠₁𝐾(𝑖,𝑟)𝐶(𝑖)^𝑇×𝑃𝑗,𝑠₂×𝐴𝑠₁+𝐵𝑠₁𝐾(𝑖,𝑟)𝐶(𝑖()𝑋(𝑘))−𝑃𝑖,𝑟=𝑋^𝑇(𝑘)𝐿(𝑖,𝑟)𝑋(𝑘)<0. (15) Thus, if 𝐿(𝑖,𝑟)<0, then 𝜀Δ𝑉(𝑋(𝑘),𝑘)∣𝑋(𝑘),𝜏_𝑘^𝑠𝑐=𝑖,ℎ_{𝑘−𝜏𝑘}^𝑠𝑐𝑉(𝑋())(𝑘)=𝑟=𝜀𝑘+1,𝑘+1∣𝑋,𝜏_𝑘^𝑠𝑐=𝑖,ℎ_{𝑘−𝜏𝑘}^𝑠𝑐(𝑋(𝑘))=𝑟−𝑉,𝑘≤−𝜆_min(−𝐿(𝑖,𝑟))𝑋(𝑘)^𝑇𝑋(𝑘)≤−𝛽𝑋^𝑇(𝑘)𝑋(𝑘)=−𝛽‖𝑋(𝑘)‖², (16) where 𝛽=inf{𝜆_min(−𝐿(𝑖,𝑟)),𝑖∈𝜑,𝑟∈𝜙}>0.From inequality (16), we can see that for any 𝑇≥1, 𝜀{𝑉(𝑋(𝑇+1),𝑇+1)}−𝜀𝑉𝑋₀,0≤−𝛽𝜀^𝑇_𝑡=0‖𝑋(𝑡)‖²∣𝑋₀,𝜏₀^𝑠𝑐,ℎ_−𝜏0^𝑠𝑐. (17) Furthermore 𝜀^𝑇_𝑡=0‖𝑋(𝑡)‖²∣𝑋₀,𝜏₀^𝑠𝑐,ℎ_−𝜏0^𝑠𝑐≤1𝛽𝜀𝑉𝑋₀{𝑉(𝑋()≤1,0−𝜀𝑇+1,𝑇+1)}𝛽𝜀𝑉𝑋₀,0. (18) Let 𝑇→∞, then 𝜀^∞_𝑡=0‖𝑋(𝑡)‖²∣𝑋₀,𝜏₀^𝑠𝑐,ℎ_−𝜏0^𝑠𝑐≤1𝛽𝜀𝑉𝑋₀=1,0𝛽𝑋^𝑇(0)𝑃𝜏₀^𝑠𝑐,ℎ_−𝜏0^𝑠𝑐𝑋(0). (19)
From Definition 3, the closed-loop system in (7) is stochastically stable.
Necessity. Assume that the closed-loop system in (7) is stochastically stable. Then, we have 𝜀^∞_𝑘=0‖𝑋(𝑘)‖²∣𝑋₀,𝜏₀^𝑠𝑐,ℎ_−𝜏0^𝑠𝑐<𝑋₀^𝑇𝑊𝑋₀. (20)
Define 𝑋^𝑇(𝑡)𝑃𝑇−𝑡,𝜏_𝑡^𝑠𝑐,ℎ_{𝑡−𝜏𝑡}^𝑠𝑐𝑋(𝑡)≜𝜀^𝑇_𝑘=𝑡𝑋^𝑇(𝑘)𝑄𝜏_𝑘^𝑠𝑐,ℎ_{𝑘−𝜏𝑘}^𝑠𝑐𝑋(𝑘)∣𝑋_𝑡,𝜏_𝑡^𝑠𝑐,ℎ_{𝑡−𝜏𝑡}^𝑠𝑐, (21) with 𝑄(𝜏_𝑘^𝑠𝑐,ℎ_{𝑘−𝜏𝑘}^𝑠𝑐)>0. Assume that 𝑋(𝑘)≠0, since 𝑄(𝜏_𝑘^𝑠𝑐,ℎ_{𝑘−𝜏𝑘}^𝑠𝑐)>0, as 𝑇 increases, 𝑋^𝑇(𝑡)𝑃(𝑇−𝑡,𝜏_𝑡^𝑠𝑐,ℎ_{𝑡−𝜏𝑡}^𝑠𝑐)𝑋(𝑡) is monotonically increasing, for 𝑘≥𝑡. From (20), 𝑋^𝑇(𝑡)𝑃(𝑇−𝑡,𝜏_𝑡^𝑠𝑐,ℎ_{𝑡−𝜏𝑡}^𝑠𝑐)𝑋(𝑡) is upper bounded. Furthermore, its limit exists and can be represented as 𝑋^𝑇(𝑡)𝑃(𝑖,𝑟)𝑋(𝑡)≜lim_𝑇→∞𝑋^𝑇(𝑡)𝑃𝑇−𝑡,𝜏_𝑡^𝑠𝑐=𝑖,ℎ_{𝑡−𝜏𝑡}^𝑠𝑐=𝑟𝑋(𝑡)=lim_𝑇→∞𝜀^𝑇_𝑘=𝑡𝑋^𝑇(𝑘)𝑄𝜏_𝑘^𝑠𝑐,ℎ_{𝑘−𝜏𝑘}^𝑠𝑐×𝑋(𝑘)∣𝑋_𝑡,𝜏_𝑡^𝑠𝑐=𝑖,ℎ_{𝑡−𝜏𝑡}^𝑠𝑐=𝑟. (22) Since this is valid for any 𝑋(𝑡), we have 𝑃(𝑖,𝑟)=lim_𝑇→∞𝑃𝑇−𝑡,𝜏_𝑡^𝑠𝑐=𝑖,ℎ_{𝑡−𝜏𝑡}^𝑠𝑐=𝑟. (23) From (22), we have 𝑃(𝑖,𝑟)>0, since 𝑄(𝜏_𝑘^𝑠𝑐,ℎ_{𝑘−𝜏𝑘}^𝑠𝑐)>0. Consider 𝜀𝑋^𝑇(𝑡)𝑃𝑇−𝑡,𝜏_𝑡^𝑠𝑐,ℎ_{𝑡−𝜏𝑡}^𝑠𝑐𝑋(𝑡)−𝑋^𝑇()×𝑃𝑡+1𝑇−𝑡−1,𝜏^𝑠𝑐_𝑡+1,ℎ_{𝑡+1−𝜏}^𝑠𝑐_𝑡+1×𝑋(𝑡+1)∣𝑋_𝑡,𝜏_𝑡^𝑠𝑐=𝑖,ℎ_{𝑡−𝜏𝑡}^𝑠𝑐=𝑟=𝑋^𝑇(𝑘)𝑄(𝑖,𝑟)𝑋(𝑘). (24) By using Lemma 4, the second term in (24) can be written as 𝜀𝑋^𝑇(𝑡+1)𝑃𝑇−𝑡−1,𝜏^𝑠𝑐_𝑡+1,ℎ_{𝑡+1−𝜏}^𝑠𝑐_𝑡+1𝑋(𝑡+1)∣𝑋_𝑡,𝜏_𝑡^𝑠𝑐=𝑖,ℎ_{𝑡−𝜏𝑡}^𝑠𝑐=𝑟=𝑋^𝑇(𝑡)^{𝜏𝑠𝑐3}_𝑗=0𝑠2³_=1𝑠1=1𝜆_𝑖𝑗𝜋^{(𝑖+1−𝑗)}_𝑟𝑠2𝜋^(𝑖)_𝑟𝑠1×𝐴𝑠₁+𝐵𝑠₁𝐾(𝑖,𝑟)𝐶(𝑖)^𝑇×𝑃𝑇−𝑡−1,𝑗,𝑠₂×𝐴𝑠₁+𝐵𝑠₁𝐾(𝑖,𝑟)𝐶(𝑖)𝑋(𝑡). (25) Substituting (25) into (24), we can get 𝑋^𝑇(𝑡)𝑃𝑇−𝑡,𝜏_𝑡^𝑠𝑐,ℎ_{𝑡−𝜏𝑡}^𝑠𝑐−^𝜏3_𝑗=0𝑠2³_=1𝑠1=1𝜆_𝑖𝑗𝜋^{(𝑖+1−𝑗)}_𝑟𝑠2𝜋^(𝑖)_𝑟𝑠1×𝐴𝑠₁+𝐵𝑠₁𝐾(𝑖,𝑟)𝐶(𝑖)^𝑇×𝑃𝑇−𝑡−1,𝑗,𝑠₂×𝐴𝑠₁+𝐵𝑠₁𝐾(𝑖,𝑟)𝐶(𝑖)𝑋(𝑡)=𝑋^𝑇(𝑡)𝑄(𝑖,𝑟)𝑋(𝑡). (26) Letting 𝑇→∞ and noticing (22) and 𝑄(𝑖,𝑟)>0, we have 𝑃(𝑖,𝑟)−^𝜏3_𝑗=0𝑠2³_=1𝑠1=1𝜆_𝑖𝑗𝜋^{(𝑖+1−𝑗)}_𝑟𝑠2𝜋^(𝑖)_𝑟𝑠1×𝐴𝑠₁+𝐵𝑠₁𝐾(𝑖,𝑟)𝐶(𝑖)^𝑇×𝑃𝑗,𝑠₂𝐴𝑠₁+𝐵𝑠₁𝐾(𝑖,𝑟)𝐶(𝑖)>0. (27)
This completes the proof.

Remark 7. Although the proof of Theorem 6 is similar to the literature [10], the considered problem is not the same. The S-C random delays and random sampling periods are considered in this paper; however, the S-C and C-A random delays are considered in [10]. In addition, the continuous systems are considered in this paper and the discrete systems are considered in [10].

Theorem 6 gives the sufficient and necessary conditions on the existence of the state feedback controller, when transition probabilities are completely accessible. However, the conditions in (11) are nonlinear in the controller matrices. In the following theorem, the equivalent LMI conditions to (11) are given.

Theorem 8. There exists a controller (4) such that the closed-loop system in (7) is stochastically stable if and only if there exist matrices 𝐾(𝑖,𝑟) and 𝑀(𝑗,𝑠₂)>0, 𝑃(𝑖,𝑟)>0 such that the following matrix inequalities −𝑃(𝑖,𝑟)𝑉^𝑇()()𝑃𝑖,𝑟∗−𝑌𝑖,𝑟<0,𝑗,𝑠₂𝑀𝑗,𝑠₂=𝐼 (28) with V(𝑖,𝑟)=𝑉^𝑇₀(𝑖,𝑟)𝑉^𝑇₁(𝑖,𝑟)⋯𝑉^𝑇_𝜏𝑠𝑐(𝑖,𝑟)^𝑇,𝑉_𝑗(𝑖,𝑟)=𝑉^𝑇_𝑗1(𝑖,𝑟)𝑉^𝑇_𝑗2(𝑖,𝑟)𝑉^𝑇_𝑗3(𝑖,𝑟)^𝑇,𝑉_𝑗𝑠2()=𝜆𝑖,𝑟_𝑖𝑗𝜋^{(𝑖+1−𝑗)}_𝑟𝑠2𝜋^(𝑖)_𝑟1^1/2𝐴(1)+𝐵(1)𝐾(𝑖,𝑟)𝐶(𝑖𝜆_𝑖𝑗)𝜋^{(𝑖+1−𝑗)}_𝑟𝑠2𝜋^(𝑖)_𝑟2^1/2𝐴(2)+𝐵(2)𝐾(𝑖,𝑟)𝐶(𝑖𝜆_𝑖𝑗)𝜋^{(𝑖+1−𝑗)}_𝑟𝑠2𝜋^(𝑖)_𝑟3^1/2𝐴(3)+𝐵(3)𝐾(𝑖,𝑟)𝐶(𝑖,𝑌()𝑌)𝑖,𝑟=diag₀(𝑖,𝑟)𝑌₁(𝑖,𝑟)⋯𝑌_𝜏𝑠𝑐(),𝑌𝑖,𝑟_𝑗(𝑖,𝑟)=diag𝑌_𝑗1(𝑖,𝑟)𝑌_𝑗2(𝑖,𝑟)𝑌_𝑗3(),𝑌𝑖,𝑟_𝑗𝑠2(𝑖,𝑟)=diag𝑀𝑗,𝑠₂𝑀𝑗,𝑠₂𝑀𝑗,𝑠₂ (29) hold for all 𝑖,𝑗∈𝜑 and 𝑟,𝑠₁,and𝑠₂∈𝜙.

Proof. By applying the Schur complement and letting 𝑀(𝑗,𝑠₂)=𝑃⁻¹(𝑗,𝑠₂), the proof can be readily completed.

Remark 9. The conditions in Theorem 8 are a set of LMIs with nonconvex constraints. This can be solved by several existing iterative LMI algorithms. Using a cone complementarity approach [22–24], we can solve this nonconvex feasibility problem by formulating it into an optimisation problem subject to LMI constraints.
Now, using a cone complementarity approach, we suggest the following nonlinear minimisation problem involving LMI conditions instead of the original nonconvex feasibility problem formulated in Theorem 8.
Minimise Tr^{𝜏𝑠𝑐3}_{𝑗=0𝑠2=1}𝑃_𝑗𝑠2𝑀_𝑗𝑠2 (30) subject to (28) and 𝑃𝑗,𝑠₂𝐼𝐼𝑀𝑗,𝑠₂≥0. (31)

Algorithm 10. (1) Find a feasible point (𝑃_𝑗𝑠2(0),𝑀_𝑗𝑠2(0)) satisfying (30). If there are none, set 𝑘=0.
(2) Solve the following LMI problem MinTr^{𝜏𝑠𝑐3}_{𝑗=0𝑠2=1}𝑃^𝑘_𝑗𝑠2𝑀_𝑗𝑠2+𝑃_𝑗𝑠2𝑀^𝑘_𝑗𝑠2subjectto(31). (32) Set 𝑃^𝑘+1_𝑗𝑠2=𝑃_𝑗𝑠2,𝑀^𝑘+1_𝑗𝑠2=𝑀_𝑗𝑠2.
(3) If the conditions in (31) are satisfied, then return to (1). If the conditions in (31) are not satisfied within a specified number (𝑘_max) of iterations, then exit. Otherwise, set 𝑘=𝑘+1 and go to (2).

4. The Case of Partly Known Transition Probabilities

In the above section, the transition probabilities are assumed completely accessible and considered as the available knowledge for analyzing the NCSs. In practice, this kind of information including the variation of time delay and actual sampling period is hard to obtain. This variation can degrade the control performance and even make the system unstable.

In this section, the transition probabilities are assumed partially accessible. The transition probabilities in (3) are considered to be partially available; namely, some elements in matrices Π and Λ are time invariant but unknown. For instance, 𝜋Π=¹¹???𝜋₂₂?𝜋₃₁𝜋₃₂𝜋₃₃𝜆,Λ=¹¹?𝜆₁₃??𝜆₂₂?𝜆₂₄??𝜆₃₃𝜆₃₄𝜆₄₁𝜆₄₂𝜆₄₃𝜆₄₄, (33) where ? represents the unavailable elements. For notation clarity, 𝑖∈𝜑,𝑟∈𝜙, we denote 𝜑^𝑖_𝜒=𝑗∶if𝜆_𝑖𝑗,𝜑isknown_𝑖𝜇𝜒=𝑗∶if𝜆_𝑖𝑗,𝜙isunknown_𝑟𝜒=𝑠∶if𝜋_𝑟𝑠,𝜙isknown_𝑟𝜇𝜒=𝑠∶if𝜋_𝑟𝑠isunknown. (34)

Moreover, if 𝜑^𝑖_𝜒≠0,𝜙^𝑟_𝜒≠0, it is further described as 𝜑^𝑖_𝜒=𝜒^𝑖₀,𝜒^𝑖₁,…,𝜒^𝑖_𝑚,0≤𝑚≤𝜏^𝑠𝑐;𝜙^𝑟_𝜒=𝜒^𝑟₁,…,𝜒^𝑟_𝑛𝜑,1≤𝑛≤3,_𝑖𝜇𝜒=𝜒^𝑖₁,𝜒^𝑖₂,…,𝜒^𝑖_{𝜏𝑠𝑐−𝑚},0≤𝑚≤𝜏^𝑠𝑐,𝜙^𝑟_𝜇𝜒=𝜒^𝑟₁,…,𝜒^𝑟_3−𝑛,1≤𝑛≤3, (35) where 𝜒^𝑖_𝑚 represents the 𝑚th known element with the index in the 𝑖th row of matrix Λ, 𝜒^𝑖_𝑗(𝑗=1,2,…,𝜏^𝑠𝑐−𝑚) represents the 𝑗th unknown element with the index in the 𝑖th row of matrix Λ, 𝜒^𝑟_𝑛 represents the 𝑛th known element with the index in the 𝑟th row of matrix Π, and 𝜒^𝑟_𝑗(𝑗=1,2,…,3−𝑛) represents the 𝑗th unknown element with the index in the 𝑟th row of matrix Π.

Remark 11. Note that the expression of (33)–(35) was first introduced for regular state-space Markovian systems [13, 14], and it covers completely known and completely unknown probabilities. If 𝜑^𝑖_𝜒=𝜑,𝜑^𝑖_𝜇𝜒=0 or 𝜙^𝑟_𝜒=𝜙,and𝜙^𝑟_𝜇𝜒=0, the transition probabilities are completely available. If 𝜑^𝑖_𝜒=0,𝜑^𝑖_𝜇𝜒=𝜑 or 𝜙^𝑟_𝜒=0,and𝜙^𝑟_𝜇𝜒=𝜙, the transition probabilities are completely unavailable.
Now, the following theorem presents a sufficient condition for the stochastic stability of the system described by (7) with partially known transition probabilities (33).

Theorem 12. The system described by (7) with partially known transition probabilities (33) is stochastically stable if there exist matrices 𝑃(𝑖,𝑟)>0 such that the following matrix inequalities _{𝑗∈𝜑𝑖𝜒𝑠1∈𝜙𝑟𝜒𝑠2∈𝜙𝑟𝜒}𝜆_𝑖𝑗𝜋^{(𝑖+1−𝑗)}_𝑟𝑠2𝜋^(𝑖)_𝑟𝑠1+_{𝑠2∈𝜙𝑟𝜇𝜒}𝜆_𝑖𝑗𝜋_𝑟^{(𝑖+1−𝑗)}𝜋^(𝑖)_𝑟𝑠1+_{𝑗∈𝜑𝑖𝜒𝑠1∈𝜙𝑟𝜇𝜒𝑠2∈𝜙𝑟𝜒}𝜆_𝑖𝑗𝜋^{(𝑖+1−𝑗)}_𝑟𝑠2𝜋_𝑟^(𝑖)+_{𝑠2∈𝜙𝑟𝜇𝜒}𝜆_𝑖𝑗𝜋_𝑟^{(𝑖+1−𝑗)}𝜋_𝑟^(𝑖)+_{𝑗∈𝜑𝑖𝜇𝜒𝑠1∈𝜙𝑟𝜒𝑠2∈𝜙𝑟𝜒}𝜆_𝑖𝜋^{(𝑖+1−𝑗)}_𝑟𝑠2𝜋^(𝑖)_𝑟𝑠1+_{𝑠2∈𝜙𝑟𝜇𝜒}𝜆_𝑖𝜋_𝑟^{(𝑖+1−𝑗)}𝜋^(𝑖)_𝑟𝑠1+_{𝑗∈𝜑𝑖𝜇𝜒𝑠1∈𝜙𝑟𝜇𝜒𝑠2∈𝜙𝑟𝜒}𝜆_𝑖𝜋^{(𝑖+1−𝑗)}_𝑟𝑠2𝜋_𝑟^(𝑖)+_{𝑠2∈𝜙𝑟𝜇𝜒}𝜆_𝑖𝜋_𝑟^{(𝑖+1−𝑗)}𝜋_𝑟^(𝑖)∗𝐴^𝑇𝑖,𝑟,𝑠₁𝑃𝑗,𝑠₂𝐴𝑖,𝑟,𝑠₁−𝑃(𝑖,𝑟)<0,𝑖=0,1,2,_{𝑗∈𝜑𝜒𝑖−23𝑠1=1𝑠2∈𝜙𝑟𝜒}𝜆_𝑖𝑗+_{𝑠2∈𝜙𝑟𝜇𝜒}𝜆_𝑖𝑗+_{𝑗∈𝜑𝑖−2𝜇𝜒}³_{𝑠1=1𝑠2∈𝜙𝑟𝜒}𝜆_𝑖+_{𝑠2∈𝜙𝑟𝜇𝜒}𝜆_𝑖+_{𝑗∈𝜑𝜒𝑖−2}³_{𝑠1=1𝑠2∈𝜙𝑟𝜒}𝜆_𝑖𝑗𝜋^{(𝑖+1−𝑗)}_𝑟𝑠2+_{𝑠2∈𝜙𝑟𝜇𝜒}𝜆_𝑖𝑗𝜋_𝑟^{(𝑖+1−𝑗)}+_{𝑗∈𝜑𝑖−2𝜇𝜒}³_{𝑠1=1𝑠2∈𝜙𝑟𝜒}𝜆_𝑖𝜋^{(𝑖+1−𝑗)}_𝑟𝑠2+_{𝑠2∈𝜙𝑟𝜇𝜒}𝜆_𝑖𝜋_𝑟^{(𝑖+1−𝑗)}∗𝐴^𝑇𝑖,𝑟,𝑠₁𝑃𝑗,𝑠₂𝐴𝑖,𝑟,𝑠₁−𝑃(𝑖,𝑟)<0,3≤𝑖≤𝜏^𝑠𝑐 (36)(37) hold for all i∈𝜑 and r∈𝜙, where 𝐴𝑖,𝑟,𝑠₁=𝐴𝑠₁+𝐵𝑠₁𝐾(𝑖,𝑟)𝐶(𝑖),𝜆_𝑖=1−_{𝑗∈𝜑𝑖𝜒}𝜆_𝑖𝑗,𝜋_𝑟^{(𝑖+1−𝑗)}=1−_{𝑠2∈𝜙𝑟𝜒}𝜋^{(𝑖+1−𝑗)}_𝑟𝑠2,𝜋_𝑟^(𝑖)=1−_{𝑠1∈𝜙𝑟𝜒}𝜋^(𝑖)_𝑟𝑠1,𝜑_𝜒^𝑖−2=𝑗∶𝑖𝑓𝜆_𝑖𝑗,𝜑𝑖𝑠𝑘𝑛𝑜𝑤𝑛𝑎𝑛𝑑𝑗≤𝑖−2_{𝑖−2𝜇𝜒}=𝑗∶𝑖𝑓𝜆_𝑖𝑗𝑖𝑠𝑢𝑛𝑘𝑛𝑜𝑤𝑛𝑎𝑛𝑑𝑗≤𝑖−2,𝜑_𝜒^𝑖−2=𝑗∶𝑖𝑓𝜆_𝑖𝑗𝑖𝑠𝑘𝑛𝑜𝑤𝑛𝑎𝑛𝑑𝑗>𝑖−2,𝜑^𝑖−2_𝜇𝜒=𝑗∶𝑖𝑓𝜆_𝑖𝑗𝑖𝑠𝑢𝑛𝑘𝑛𝑜𝑤𝑛𝑎𝑛𝑑𝑗>𝑖−2. (38)

Proof. From Theorem 6, we know 𝐿(𝑖,𝑟)=^{𝜏𝑠𝑐3}_𝑗=0𝑠2³_=1𝑠1=1𝜆_𝑖𝑗𝜋^{(𝑖+1−𝑗)}_𝑟𝑠2𝜋^(𝑖)_𝑟𝑠1𝐴^𝑇𝑖,𝑟,𝑠₁𝑃𝑗,𝑠₂×𝐴𝑖,𝑟,𝑠₁−𝑃(𝑖,𝑟)<0. (39)
However, from (30), we know that some elements are inaccessible in Λ and Π. According to the property of Markov chains, we obtain 𝑃𝑗,𝑟=³_𝑠2=1𝜋^{(𝑖+1−𝑗)}_𝑟𝑠2𝑃𝑗,𝑠₂=_{𝑠2∈𝜙𝑟𝜒}𝜋^{(𝑖+1−𝑗)}_𝑟𝑠2𝑃𝑗,𝑠₂+_{𝑠2∈𝜙𝑟𝜇𝜒}𝜋^{(𝑖+1−𝑗)}_𝑟𝑠2𝑃𝑗,𝑠₂≤_{𝑠2∈𝜙𝑟𝜒}𝜋^{(𝑖+1−𝑗)}_𝑟𝑠2𝑃𝑗,𝑠₂+_{𝑠2∈𝜙𝑟𝜇𝜒}𝜋_𝑟^{(𝑖+1−𝑗)}𝑃𝑗,𝑠₂,𝑃𝑖,𝑗,𝑟=³_𝑠1=1𝜋^(𝑖)_𝑟𝑠1𝐴^𝑇𝑖,𝑟,𝑠₁𝑃𝑗,𝑟𝐴𝑖,𝑟,𝑠₁=_{𝑠1∈𝜙𝑟𝜒}𝜋^(𝑖)_𝑟𝑠1𝐴^𝑇𝑖,𝑟,𝑠₁𝑃𝑗,𝑟𝐴𝑖,𝑟,𝑠₁+_{𝑠1∈𝜙𝑟𝜇𝜒}𝜋^(𝑖)_𝑟𝑠1𝐴^𝑇𝑖,𝑟,𝑠₁𝑃𝑗,𝑟𝐴𝑖,𝑟,𝑠₁≤_{𝑠1∈𝜙𝑟𝜒}𝜋^(𝑖)_𝑟𝑠1𝐴^𝑇𝑖,𝑟,𝑠₁𝑃𝑗,𝑟𝐴𝑖,𝑟,𝑠₁+_{𝑠1∈𝜙𝑟𝜇𝜒}𝜋_𝑟^(𝑖)𝐴^𝑇𝑖,𝑟,𝑠₁𝑃𝑗,𝑟𝐴𝑖,𝑟,𝑠₁,_{𝜏𝑠𝑐𝑗=0}𝜆_𝑖𝑗𝑃𝑖,𝑗,𝑟=_{𝑗∈𝜑𝑖𝜒}𝜆_𝑖𝑗𝑃𝑖,𝑟,𝑗+_{𝑗∈𝜑𝑖𝜇𝜒}𝜆_𝑖𝑗𝑃𝑖,𝑟,𝑗≤_{𝑗∈𝜑𝑖𝜒}𝜆_𝑖𝑗𝑃𝑖,𝑟,𝑗+_{𝑗∈𝜑𝑖𝜇𝜒}𝜆_𝑖𝑃𝑖,𝑟,𝑗, (40)(41)(42) where 𝐴(𝑖,𝑟,𝑠₁), 𝜆_𝑖, 𝜋_𝑟^{(𝑖+1−𝑗)},𝜋_𝑟^(𝑖) are defined in (38).
According to Lemma 4, 𝜋⁽³⁾_𝑟𝑠=∑_𝑙2∑_𝑙1𝜋_𝑟𝑙1𝜋_𝑙1𝑙2𝜋_𝑙2𝑠. We obtain 𝜋⁽³⁾_𝑟𝑠(forall𝑟,𝑠∈𝜙), that is inaccessible in Π⁽³⁾, even if there is an inaccessible element in Π. So we obtain 𝜋^(𝑖)_𝑟𝑠(forall𝑟,𝑠∈𝜙,𝑖≥3), that is inaccessible in Π^(𝑖), and _{𝑠∈𝜙𝑟𝜇𝜒}𝜋^(𝑖)_𝑟𝑠=_𝑠∈𝜙𝜋^(𝑖)_𝑟𝑠=1. (43)
Substituting the formulae (40)–(43) into (39), we can obtain (36) and (37). This completes the proof.

Theorem 12 gives the sufficient conditions on the existence of the state feedback controller, when transition probabilities are partly accessible. However, the conditions in (36) and (37) are nonlinear in the controller matrices. In the following theorem, the equivalent LMI conditions to (36) and (37) are given.

Theorem 13. Consider the system described by (7) with partially known transition probabilities (30), the corresponding system is stochastically stable if there exist matrices 𝐾(𝑖,𝑟) and 𝑋(𝜒^𝑖_𝑐,𝜒^𝑟_𝑎)>0,𝑋(𝜒^𝑖_𝑑,𝜒^𝑟_𝑏)>0, 𝑋(𝜒^𝑖_𝑐,𝜒^𝑟_𝑏)>0, 𝑋(𝜒^𝑖_𝑑,𝜒^𝑟_𝑎)>0, 𝑋(𝜒_𝑔^𝑖−2,𝜒^𝑟_𝑎)>0, 𝑋(𝜒_𝑔^𝑖−2,𝜒^𝑟_𝑏)>0, 𝑋(𝜒_𝑒^𝑖−2,𝜒^𝑟_𝑎)>0, 𝑋(𝜒_𝑒^𝑖−2,𝜒^𝑟_𝑏)>0, 𝑋(𝜒_𝑓^𝑖−2,𝜒^𝑟_𝑎)>0, 𝑋(𝜒_𝑓^𝑖−2,𝜒^𝑟_𝑏)>0, 𝑋(𝜒_ℎ^𝑖−2,𝜒^𝑟_𝑎)>0, 𝑋(𝜒_ℎ^𝑖−2,𝜒^𝑟_𝑏)>0, 𝑃(𝜒^𝑖_𝑐,𝜒^𝑟_𝑎)>0, 𝑃(𝜒^𝑖_𝑑,𝜒^𝑟_𝑏)>0, 𝑃(𝜒^𝑖_𝑐,𝜒^𝑟_𝑏)>0, 𝑃(𝜒^𝑖_𝑑,𝜒^𝑟_𝑎)>0, 𝑃(𝜒_𝑔^𝑖−2,𝜒^𝑟_𝑎)>0, 𝑃(𝜒_𝑔^𝑖−2,𝜒^𝑟_𝑏)>0, 𝑃(𝜒_𝑒^𝑖−2,𝜒^𝑟_𝑎)>0, 𝑃(𝜒_𝑒^𝑖−2,𝜒^𝑟_𝑏)>0, 𝑃(𝜒_𝑓^𝑖−2,𝜒^𝑟_𝑎)>0, 𝑃(𝜒_𝑓^𝑖−2,𝜒^𝑟_𝑏)>0, 𝑃(𝜒_ℎ^𝑖−2,𝜒^𝑟_𝑎)>0, 𝑃(𝜒_ℎ^𝑖−2,𝜒^𝑟_𝑏)>0, such that the following matrix inequalities: −𝑃(𝑖,𝑟)𝑉^𝑇(𝑖,𝑟)𝑉^𝑇()()0𝑌()𝑃𝜒𝑖,𝑟∗−𝑌𝑖,𝑟∗∗−𝑖,𝑟<0,_𝑖𝑐,𝜒^𝑟_𝑎𝑋𝜒^𝑖_𝑐,𝜒^𝑟_𝑎𝑃𝜒=𝐼,_𝑖𝑐,𝜒^𝑟_𝑎𝑋𝜒^𝑖_𝑐,𝜒^𝑟_𝑏𝑃=𝐼,𝜒^𝑖_𝑑,𝜒^𝑟_𝑎𝑋𝜒^𝑖_𝑑,𝜒^𝑟_𝑎𝑃=𝐼,𝜒^𝑖_𝑑,𝜒^𝑟_𝑏𝑋𝜒^𝑖_𝑑,𝜒^𝑟_𝑏𝑃𝜒=𝐼,_𝑔𝑖−2,𝜒^𝑟_𝑎𝑋𝜒_𝑔^𝑖−2,𝜒^𝑟_𝑎𝑃𝜒=𝐼,_𝑔𝑖−2,𝜒^𝑟_𝑏𝑋𝜒_𝑔^𝑖−2,𝜒^𝑟_𝑏𝑃𝜒=𝐼,_𝑒𝑖−2,𝜒^𝑟_𝑎𝑋𝜒_𝑒^𝑖−2,𝜒^𝑟_𝑎𝑃𝜒=𝐼,_𝑒𝑖−2,𝜒^𝑟_𝑏𝑋𝜒_𝑒^𝑖−2,𝜒^𝑟_𝑏𝑃=𝐼,𝜒_𝑓^𝑖−2,𝜒^𝑟_𝑎𝑋𝜒_𝑓^𝑖−2,𝜒^𝑟_𝑎𝑃=𝐼,𝜒_𝑓^𝑖−2,𝜒^𝑟_𝑏𝑋𝜒_𝑓^𝑖−2,𝜒^𝑟_𝑏𝑃=𝐼,𝜒_ℎ^𝑖−2,𝜒^𝑟_𝑎𝑋𝜒_ℎ^𝑖−2,𝜒^𝑟_𝑎𝑃=𝐼,𝜒_ℎ^𝑖−2,𝜒^𝑟_𝑏𝑋𝜒_ℎ^𝑖−2,𝜒^𝑟_𝑏=𝐼 (44)(45)(46)(47)(48)(49)(50) with 𝑉𝑖=0,1,2,_𝑇(𝑖,𝑟)=𝑉^𝑇₁(𝑖,𝑟)𝑉^𝑇₂(),𝑉𝑖,𝑟_𝑇()=𝑉𝑖,𝑟^𝑇1(𝑖,𝑟)𝑉^𝑇₂(),𝑉𝑖,𝑟_𝑖(𝑖,𝑟)=𝑉^𝑇_{𝑖𝜒𝑖0}(𝑖,𝑟)𝑉^𝑇_{𝑖𝜒𝑖1}(𝑖,𝑟)⋯𝑉^𝑇_{𝑖𝜒𝑖𝑚}(𝑖,𝑟)^𝑇,𝑉_𝑖(𝑖,𝑟)=𝑉^𝑇_𝑖𝜒^𝑖₁(𝑖,𝑟)𝑉^𝑇_𝑖𝜒^𝑖₂(𝑖,𝑟)⋯𝑉^𝑇_𝑖𝜒^𝑖_{𝜏𝑠𝑐−𝑚}(𝑖,𝑟)^𝑇,𝑉_{1𝜒𝑖𝑐}(𝑖,𝑟)=𝑉^𝑇_{1𝜒𝑖𝑐𝜒𝑟1}(𝑖,𝑟)𝑉^𝑇_{1𝜒𝑖𝑐𝜒𝑟2}(𝑖,𝑟)⋯𝑉^𝑇_{1𝜒𝑖𝑐𝜒𝑟𝑛}(𝑖,𝑟)^𝑇,𝑉_{2𝜒𝑟𝑎}(𝑖,𝑟)=𝑉^𝑇_{2𝜒𝑖𝑐𝜒}^𝑟₁(𝑖,𝑟)𝑉^𝑇_{2𝜒𝑖𝑐𝜒}^𝑟₂(𝑖,𝑟)⋯𝑉^𝑇_{2𝜒𝑖𝑐𝜒}^𝑟_3−𝑛(𝑖,𝑟)^𝑇,𝑉_1𝜒^𝑖_𝑑(𝑖,𝑟)=𝑉^𝑇_1𝜒^𝑖_𝑑𝜒^𝑟₁(𝑖,𝑟)𝑉^𝑇_1𝜒^𝑖_𝑑𝜒^𝑟₂(𝑖,𝑟)⋯𝑉^𝑇_1𝜒^𝑖_{𝑑𝜒𝑟𝑛}(𝑖,𝑟)^𝑇,𝑉_2𝜒^𝑖_𝑑(𝑖,𝑟)=𝑉^𝑇_2𝜒^𝑖_𝑑𝜒^𝑟₁(𝑖,𝑟)𝑉^𝑇_2𝜒^𝑖_𝑑𝜒^𝑟₂(𝑖,𝑟)⋯𝑉^𝑇_2𝜒^𝑖_𝑑𝜒^𝑟_3−𝑛(𝑖,𝑟)^𝑇,𝑉_{1𝜒𝑖𝑐𝜒𝑟𝑎}()=𝜆𝑖,𝑟^{𝑖𝜒𝑖𝑐}𝜋^{(𝑖+1−𝜒𝑖𝑐)}_{𝑟𝜒𝑟0}𝜋^(𝑖)_{𝑟𝜒𝑟𝑎}^1/2𝐴𝑖,𝑟,𝜒^𝑟_𝑎⋮𝜆_{𝑖𝜒𝑖𝑐}𝜋^{(𝑖+1−𝜒𝑖𝑐)}_{𝑟𝜒𝑟𝑚}𝜋^(𝑖)_{𝑟𝜒𝑟𝑎}^1/2𝐴𝑖,𝑟,𝜒^𝑟_𝑎𝜆_{𝑖𝜒𝑖𝑐}𝜋^{(𝑖+1−𝜒𝑖𝑐)}_𝑟𝜋^(𝑖)_{𝑟𝜒𝑟𝑎}^1/2𝐴𝑖,𝑟,𝜒^𝑟_𝑎⋮𝜆_{𝑖𝜒𝑖𝑐}𝜋^{(𝑖+1−𝜒𝑖𝑐)}_𝑟𝜋^(𝑖)_{𝑟𝜒𝑟𝑎}^1/2𝐴𝑖,𝑟,𝜒^𝑟_𝑎,𝑉_{2𝜒𝑖𝑐𝜒}^𝑟_𝑏()=𝜆𝑖,𝑟^{𝑖𝜒𝑖𝑐}𝜋^{(𝑖+1−𝜒𝑖𝑐)}_{𝑟𝜒𝑟0}𝜋_𝑟^(𝑖)1/2𝐴𝑖,𝑟,𝜒^𝑟_𝑏⋮𝜆_{𝑖𝜒𝑖𝑐}𝜋^{(𝑖+1−𝜒𝑖𝑐)}_{𝑟𝜒𝑟𝑚}𝜋_𝑟^(𝑖)1/2𝐴𝑖,𝑟,𝜒^𝑟_𝑏𝜆_{𝑖𝜒𝑖𝑐}𝜋^{(𝑖+1−𝜒𝑖𝑐)}_𝑟𝜋_𝑟^(𝑖)1/2𝐴𝑖,𝑟,𝜒^𝑟_𝑏⋮𝜆_{𝑖𝜒𝑖𝑐}𝜋^{(𝑖+1−𝜒𝑖𝑐)}_𝑟𝜋_𝑟^(𝑖)1/2𝐴𝑖,𝑟,𝜒^𝑟_𝑏,𝑉_1𝜒^𝑖_{𝑑𝜒𝑟𝑎}()=𝜆𝑖,𝑟^𝑖𝜋^{(𝑖+1−𝜒𝑖𝑑)}_{𝑟𝜒𝑟0}𝜋^(𝑖)_{𝑟𝜒𝑟𝑎}^1/2𝐴𝑖,𝑟,𝜒^𝑟_𝑎⋮𝜆_𝑖𝜋^{(𝑖+1−𝜒𝑖𝑑)}_{𝑟𝜒𝑟𝑚}𝜋^(𝑖)_{𝑟𝜒𝑟𝑎}^1/2𝐴𝑖,𝑟,𝜒^𝑟_𝑎𝜆_𝑖𝜋^{(𝑖+1−𝜒𝑖𝑑)}_𝑟𝜋^(𝑖)_{𝑟𝜒𝑟𝑎}^1/2𝐴𝑖,𝑟,𝜒^𝑟_𝑎⋮𝜆_𝑖𝜋^{(𝑖+1−𝜒𝑖𝑑)}_𝑟𝜋^(𝑖)_{𝑟𝜒𝑟𝑎}^1/2𝐴𝑖,𝑟,𝜒^𝑟_𝑎,𝑉_2𝜒^𝑖_𝑑𝜒^𝑟_𝑏()=𝜆𝑖,𝑟^𝑖𝜋^{(𝑖+1−𝜒𝑖𝑑)}_{𝑟𝜒𝑟0}𝜋_𝑟^(𝑖)1/2𝐴𝑖,𝑟,𝜒^𝑟_𝑏⋮𝜆_𝑖𝜋^{(𝑖+1−𝜒𝑖𝑑)}_{𝑟𝜒𝑟𝑚}𝜋_𝑟^(𝑖)1/2𝐴𝑖,𝑟,𝜒^𝑟_𝑏𝜆_𝑖𝜋^{(𝑖+1−𝜒𝑖𝑑)}_𝑟𝜋_𝑟^(𝑖)1/2𝐴𝑖,𝑟,𝜒^𝑟_𝑏⋮𝜆_𝑖𝜋^{(𝑖+1−𝜒𝑖𝑑)}_𝑟𝜋_𝑟^(𝑖)1/2𝐴𝑖,𝑟,𝜒^𝑟_𝑏,𝑌(𝑖,𝑟)=diag𝑌₁(𝑖,𝑟),𝑌₂(,𝑌()𝑌𝑖,𝑟)𝑖,𝑟=diag₁(𝑖,𝑟),𝑌₂(,𝑌𝑖,𝑟)_𝑖(𝑖,𝑟)=diag𝑌_𝑖𝜒^𝑖₀(),𝑌𝑖,𝑟_{𝑖𝜒𝑖1}(𝑖,𝑟),…,𝑌_{𝑖𝜒𝑖𝑚}(,𝑌𝑖,𝑟)_𝑖(𝑖,𝑟)=diag𝑌_𝑖𝜒^𝑖₁(),𝑌𝑖,𝑟_{𝑖𝜒𝑖2}(𝑖,𝑟),…,𝑌_𝑖𝜒^𝑖_{𝜏𝑠𝑐−𝑚}(,𝑌𝑖,𝑟)_{1𝜒𝑖𝑐}(𝑖,𝑟)=diag𝑌_{1𝜒𝑖𝑐𝜒}^𝑟₁(),𝑌𝑖,𝑟_{1𝜒𝑖𝑐𝜒𝑟2}(𝑖,𝑟),…,𝑌_{1𝜒𝑖𝑐𝜒𝑟𝑛}(,𝑌𝑖,𝑟)_{2𝜒𝑟𝑎}(𝑖,𝑟)=diag𝑌_{2𝜒𝑖𝑐𝜒}^𝑟₁(),𝑌𝑖,𝑟_{2𝜒𝑖𝑐𝜒𝑟2}(𝑖,𝑟),…,𝑌_{2𝜒𝑖𝑐𝜒}^𝑟_3−𝑛(,𝑌𝑖,𝑟)_{1𝜒𝑖𝑐𝜒𝑟𝑎}(𝑖,𝑟)=𝑌_{2𝜒𝑟𝑎𝜒}^𝑟_𝑏()𝑋𝜒𝑖,𝑟=diag^𝑖_𝑐,𝜒^𝑟₀,…,𝑋𝜒^𝑖_𝑐,𝜒^𝑟_𝑚,𝑋𝜒^𝑖_𝑐,𝜒^𝑟₁,…,𝑋𝜒^𝑖_𝑐,𝜒^𝑟_{𝜏𝑠𝑐−𝑚},𝑌_1𝜒^𝑖_𝑑(𝑖,𝑟)=diag𝑌_1𝜒^𝑖_{𝑑𝜒𝑟1}(),𝑌𝑖,𝑟_{1𝜒𝑖𝑑𝜒𝑟2}(𝑖,𝑟),…,𝑌_1𝜒^𝑖_{𝑑𝜒𝑟𝑛}(,𝑌𝑖,𝑟)_{2𝜒𝑟𝑏}(𝑖,𝑟)=diag𝑌_2𝜒^𝑖_𝑑𝜒^𝑟₁(),𝑌𝑖,𝑟_{2𝜒𝑖𝑑𝜒𝑟2}(𝑖,𝑟),…,𝑌_2𝜒^𝑖_𝑑𝜒^𝑟_3−𝑛(,𝑌𝑖,𝑟)_{1𝜒𝑖𝑑𝜒𝑟𝑎}(𝑖,𝑟)=𝑌_2𝜒^𝑟_𝑏𝜒^𝑟_𝑏()𝑋𝑖,𝑟=diag𝜒^𝑖_𝑑,𝜒^𝑟₀,…,𝑋𝜒^𝑖_𝑑,𝜒^𝑟_𝑚,𝑋𝜒^𝑖_𝑑,𝜒^𝑟₁,…,𝑋𝜒^𝑖_𝑑,𝜒^𝑟_{𝜏𝑠𝑐−𝑚},3≤𝑖≤𝜏^𝑠𝑐𝑉^𝑇(𝑖,𝑟)=𝑉^𝑇₁(𝑖,𝑟)𝑉^𝑇₂(,𝑉𝑖,𝑟)_𝑇(𝑖,𝑟)=𝑉^𝑇₁(𝑖,𝑟)𝑉^𝑇₂(,𝑉𝑖,𝑟)₁(𝑖,𝑟)=𝑉^𝑇_{1𝜒0𝑖−2}(𝑖,𝑟)⋯𝑉^𝑇_{1𝜒𝑖−2𝑚}(𝑖,𝑟)^𝑇,𝑉₂(𝑖,𝑟)=𝑉^𝑇_{2𝜒𝑖−2𝑚+1}(𝑖,𝑟)⋯𝑉^𝑇_{2𝜒𝑚𝑖−2}(𝑖,𝑟)^𝑇,𝑉₁(𝑖,𝑟)=𝑉^𝑇_1𝜒1^𝑖−2(𝑖,𝑟)⋯𝑉^𝑇_1𝜒𝑚^𝑖−2(𝑖,𝑟)^𝑇,𝑉₂(𝑖,𝑟)=𝑉^𝑇_2𝜒^𝑖−2_𝑚+1(𝑖,𝑟)⋯𝑉^𝑇_2𝜒𝜏^𝑖−2_{𝑠𝑐−𝑚}(𝑖,𝑟)^𝑇,𝑉_{1𝜒𝑔𝑖−2}()=𝑉𝑖,𝑟_{𝑇1𝜒𝑔𝑖−21}(𝑖,𝑟)𝑉^𝑇_{1𝜒𝑔𝑖−22}(𝑖,𝑟)𝑉^𝑇_{1𝜒𝑔𝑖−23}(𝑖,𝑟)^𝑇,𝑉_2𝜒ℎ^𝑖−2()=𝑉𝑖,𝑟_{𝑇2𝜒ℎ𝑖−21}(𝑖,𝑟)𝑉^𝑇_{2𝜒ℎ𝑖−22}(𝑖,𝑟)𝑉^𝑇_{2𝜒ℎ𝑖−23}(𝑖,𝑟)^𝑇,𝑉_{1𝜒𝑒𝑖−2}()=𝑉𝑖,𝑟_{𝑇1𝜒𝑒𝑖−21}(𝑖,𝑟)𝑉^𝑇_{1𝜒𝑒𝑖−22}(𝑖,𝑟)𝑉^𝑇_{1𝜒𝑒𝑖−23}(𝑖,𝑟)^𝑇,𝑉_2𝜒𝑓^𝑖−2()=𝑉𝑖,𝑟_{𝑇2𝜒𝑓𝑖−21}(𝑖,𝑟)𝑉^𝑇_2𝜒𝑓^𝑖−2₂(𝑖,𝑟)𝑉^𝑇_2𝜒𝑓^𝑖−2₃(𝑖,𝑟)^𝑇,𝑉_{1𝜒𝑔𝑖−2𝑠1}()=𝜆𝑖,𝑟^{𝑖𝜒𝑔𝑖−21/2}𝐴𝑖,𝑟,𝑠₁⋮𝜆_{𝑖𝜒𝑔𝑖−2}^1/2𝐴𝑖,𝑟,𝑠₁𝜆_{𝑖𝜒𝑔𝑖−2}^1/2𝐴𝑖,𝑟,𝑠₁⋮𝜆_{𝑖𝜒𝑔𝑖−2}^1/2𝐴𝑖,𝑟,𝑠₁,𝑉_{2𝜒ℎ𝑖−2𝑠1}()=𝜆𝑖,𝑟^𝑖1/2𝐴𝑖,𝑟,𝑠₁⋮𝜆_𝑖^1/2𝐴𝑖,𝑟,𝑠₁𝜆_𝑖^1/2𝐴𝑖,𝑟,𝑠₁⋮𝜆_𝑖^1/2𝐴𝑖,𝑟,𝑠₁,𝑉_{1𝜒𝑒𝑖−2𝑠1}()=𝜆𝑖,𝑟^{𝑖𝜒𝑒𝑖−2}𝜋^{(𝑖+1−𝜒𝑒𝑖−2)}_{𝑟𝜒𝑟1}^1/2𝐴𝑖,𝑟,𝑠₁⋮𝜆_{𝑖𝜒𝑒𝑖−2}𝜋^{(𝑖+1−𝜒𝑒𝑖−2)}_{𝑟𝜒𝑟𝑚}^1/2𝐴𝑖,𝑟,𝑠₁𝜆_{𝑖𝜒𝑒𝑖−2}𝜋^{(𝑖+1−𝜒𝑒𝑖−2)}_𝑟^1/2𝐴𝑖,𝑟,𝑠₁⋮𝜆_{𝑖𝜒𝑒𝑖−2}𝜋^{(𝑖+1−𝜒𝑒𝑖−2)}_𝑟^1/2𝐴𝑖,𝑟,𝑠₁,𝑉_2𝜒𝑓^𝑖−2_𝑠1()=𝜆𝑖,𝑟^𝑖𝜋^{(𝑖+1−𝜒𝑓𝑖−2)}_{𝑟𝜒0𝑠𝑐}^1/2𝐴𝑖,𝑟,𝑠₁⋮𝜆_𝑖𝜋^{(𝑖+1−𝜒𝑓𝑖−2)}_{𝑟𝜒𝑚𝑠𝑐}^1/2𝐴𝑖,𝑟,𝑠₁𝜆_𝑖𝜋^{(𝑖+1−𝜒𝑓𝑖−2)}_𝑟^1/2𝐴𝑖,𝑟,𝑠₁⋮𝜆_𝑖𝜋^{(𝑖+1−𝜒𝑓𝑖−2)}_𝑟^1/2𝐴𝑖,𝑟,𝑠₁,𝑌(𝑖,𝑟)=diag𝑌₁(𝑖,𝑟),𝑌₂(,𝑌𝑖,𝑟)₁(𝑖,𝑟)=diag𝑌₁(𝑖,𝑟),𝑌₂(,𝑌𝑖,𝑟)₁(𝑖,𝑟)=diag𝑌_1𝜒0^𝑖−2(𝑖,𝑟),…,𝑌_1𝜒^𝑖−2_𝑚(,𝑌𝑖,𝑟)₂()𝑌𝑖,𝑟=diag_2𝜒^𝑖−2_𝑚+1(𝑖,𝑟),…,𝑌_{2𝜒𝑚𝑖−2}(,𝑌𝑖,𝑟)₁()𝑌𝑖,𝑟=diag_1𝜒1^𝑖−2(𝑖,𝑟),…,𝑌_1𝜒𝑚^𝑖−2(,𝑌𝑖,𝑟)₂()𝑌𝑖,𝑟=diag_2𝜒^𝑖−2_𝑚+1(𝑖,𝑟),…,𝑌_2𝜒𝜏^𝑖−2_{𝑠𝑐−𝑚}(,𝑌𝑖,𝑟)_{1𝜒𝑔𝑖−2}()𝑌𝑖,𝑟=diag_{1𝜒𝑔𝑖−21}(),𝑌𝑖,𝑟_{1𝜒𝑔𝑖−22}(𝑖,𝑟),𝑌_{1𝜒𝑔𝑖−23}(,𝑌𝑖,𝑟)_{2𝜒ℎ𝑖−2}()𝑌𝑖,𝑟=diag_2𝜒ℎ^𝑖−2₁(),𝑌𝑖,𝑟_{2𝜒ℎ𝑖−22}(𝑖,𝑟),𝑌_2𝜒ℎ^𝑖−2₃(,𝑌𝑖,𝑟)_{1𝜒𝑒𝑖−2}()𝑌𝑖,𝑟=diag_{1𝜒𝑒𝑖−21}(),𝑌𝑖,𝑟_{1𝜒𝑒𝑖−22}(𝑖,𝑟),𝑌_{1𝜒𝑒𝑖−23}(,𝑌𝑖,𝑟)_{2𝜒𝑓𝑖−2}()𝑌𝑖,𝑟=diag_2𝜒𝑓^𝑖−2₁(),𝑌𝑖,𝑟_{2𝜒𝑓𝑖−22}(𝑖,𝑟),𝑌_2𝜒𝑓^𝑖−2₃(,𝑌𝑖,𝑟)_{1𝜒𝑔𝑖−2𝑠1}()𝑋𝜒𝑖,𝑟=diag_𝑔^𝑖−2,𝜒^𝑟₁,…,𝑋𝜒_𝑔^𝑖−2,𝜒^𝑟_𝑛,𝑋𝜒_𝑔^𝑖−2,𝜒^𝑟₁,…,𝑋𝜒_𝑔^𝑖−2,𝜒^𝑟_3−𝑛,𝑌_2𝜒ℎ^𝑖−2_𝑠1()𝑋𝑖,𝑟=diag𝜒_ℎ^𝑖−2,𝜒^𝑟₁,…,𝑋𝜒_ℎ^𝑖−2,𝜒^𝑟_𝑛,𝑋𝜒_ℎ^𝑖−2,𝜒^𝑟₁,…,𝑋𝜒_𝜏^𝑖−2_{𝑠𝑐−𝑚},𝜒^𝑟_3−𝑛,𝑌_{1𝜒𝑒𝑖−2𝑠1}()𝑋𝜒𝑖,𝑟=diag_𝑒^𝑖−2,𝜒^𝑟₁,…,𝑋𝜒_𝑒^𝑖−2,𝜒^𝑟_𝑛,𝑋𝜒_𝑒^𝑖−2,𝜒^𝑟₁,…,𝑋𝜒_𝑒^𝑖−2,𝜒^𝑟_3−𝑛,𝑌_2𝜒𝑓^𝑖−2_𝑠1()𝑋𝑖,𝑟=diag𝜒_𝑓^𝑖−2,𝜒^𝑟₁,…,𝑋𝜒_𝑓^𝑖−2,𝜒^𝑟_𝑛,𝑋𝜒_𝑓^𝑖−2,𝜒^𝑟₁,…,𝑋𝜒_𝑓^𝑖−2,𝜒^𝑟_3−𝑛 (51) hold for all 𝑖,𝑗∈𝜑, 𝑟,𝑠₁,𝑠₂∈𝜙, 𝜒^𝑟_𝑎∈𝜙^𝑟_𝜒,𝜒^𝑟_𝑏∈𝜙^𝑟_𝜇𝜒,𝜒^𝑖_𝑐∈𝜑^𝑖_𝜒,𝜒^𝑖_𝑑∈𝜑^𝑖_𝜇𝜒,𝜒_𝑔^𝑖−2∈𝜑_𝜒^𝑖−2,𝑎𝑛𝑑𝜒_ℎ^𝑖−2∈𝜑^𝑖−2_𝜇𝜒, where 𝜑_𝜒^𝑖−2=𝜒₀^𝑖−2,𝜒₁^𝑖−2,…,𝜒^𝑖−2_𝑚,0≤𝑚≤𝑚,𝜑_𝜒^𝑖−2=𝜒^𝑖−2_𝑚+1,𝜒^𝑖−2_𝑚+2,…,𝜒_𝑚^𝑖−2,0≤𝑚≤𝜏^𝑠𝑐,𝜑^𝑖−2_𝜇𝜒=𝜒₁^𝑖−2,𝜒₂^𝑖−2,…,𝜒_𝑚^𝑖−2,1≤𝑚≤𝜏^𝑠𝑐−𝑚,𝜑^𝑖−2_𝜇𝜒=𝜒^𝑖−2_𝑚+1,𝜒^𝑖−2_𝑚+2,…,𝜒_𝜏^𝑖−2_{𝑠𝑐−𝑚},0≤𝑚≤𝜏^𝑠𝑐,𝜋^{(𝑖+1−𝜒𝑒𝑖−2)}_𝑟=1−_{𝑠2∈𝜙𝑟𝜒}𝜋^{(𝑖+1−𝜒𝑒𝑖−2)}_𝑟𝑠2,𝜋^{(𝑖+1−𝜒𝑓𝑖−2)}_𝑟=1−_{𝑠2∈𝜙𝑟𝜒}𝜋^{(𝑖+1−𝜒𝑓𝑖−2)}_𝑟𝑠2,𝜋^{(𝑖+1−𝜒𝑖𝑐)}_𝑟=1−_{𝑠2∈𝜙𝑟𝜒}𝜋^{(𝑖+1−𝜒𝑖𝑐)}_𝑟𝑠2,𝜋^{(𝑖+1−𝜒𝑖𝑑)}_𝑟=1−_{𝑠2∈𝜙𝑟𝜒}𝜋^{(𝑖+1−𝜒𝑖𝑑)}_𝑟𝑠2,∀𝜒_𝑒^𝑖−2∈𝜑_𝜒^𝑖−2,𝜒_𝑓^𝑖−2∈𝜑^𝑖−2_𝜇𝜒,𝜒^𝑖_𝑐∈𝜑^𝑖_𝜒,𝜒^𝑖_𝑑∈𝜑^𝑖_𝜇𝜒, (52) and 𝐴(𝑖,𝑟,𝑠₁),𝜆_𝑖, and𝜋_𝑟^(𝑖) are defined in (38).

Proof. By applying the Schur complement and letting 𝑋(𝜒^𝑖_𝑐,𝜒^𝑟_𝑎)=𝑃⁻¹(𝜒^𝑖_𝑐,𝜒^𝑟_𝑎),𝑋(𝜒^𝑖_𝑑,𝜒^𝑟_𝑏)=𝑃⁻¹(𝜒^𝑖_𝑑,𝜒^𝑟_𝑏), 𝑋(𝜒^𝑖_𝑐,𝜒^𝑟_𝑎)=𝑃⁻¹(𝜒^𝑖_𝑐,𝜒^𝑟_𝑎), 𝑋(𝜒^𝑖_𝑑,𝜒^𝑟_𝑏)=𝑃⁻¹(𝜒^𝑖_𝑑,𝜒^𝑟_𝑏),𝑋(𝜒^𝑖_𝑐,𝜒^𝑟_𝑏)=𝑃⁻¹(𝜒^𝑖_𝑐,𝜒^𝑟_𝑏), 𝑋(𝜒^𝑖_𝑑,𝜒^𝑟_𝑎)=𝑃⁻¹(𝜒^𝑖_𝑑,𝜒^𝑟_𝑎), 𝑋(𝜒_𝑔^𝑖−2,𝜒^𝑟_𝑎)=𝑃⁻¹(𝜒_𝑔^𝑖−2,𝜒^𝑟_𝑎), 𝑋(𝜒_𝑔^𝑖−2,𝜒^𝑟_𝑏)=𝑃⁻¹(𝜒_𝑔^𝑖−2,𝜒^𝑟_𝑏), 𝑋(𝜒_𝑒^𝑖−2,𝜒^𝑟_𝑎)=𝑃⁻¹(𝜒_𝑒^𝑖−2,𝜒^𝑟_𝑎), 𝑋(𝜒_𝑒^𝑖−2,𝜒^𝑟_𝑏)=𝑃⁻¹(𝜒_𝑒^𝑖−2,𝜒^𝑟_𝑏), 𝑋(𝜒_𝑓^𝑖−2,𝜒^𝑟_𝑎)=𝑃⁻¹(𝜒_𝑓^𝑖−2,𝜒^𝑟_𝑎), 𝑋(𝜒_𝑓^𝑖−2,𝜒^𝑟_𝑏)=𝑃⁻¹(𝜒_𝑓^𝑖−2,𝜒^𝑟_𝑏), 𝑋(𝜒_ℎ^𝑖−2,𝜒^𝑟_𝑏)=𝑃⁻¹(𝜒_ℎ^𝑖−2,𝜒^𝑟_𝑏), and𝑋(𝜒_ℎ^𝑖−2,𝜒^𝑟_𝑎)=𝑃⁻¹(𝜒_ℎ^𝑖−2,𝜒^𝑟_𝑎), the proof can be readily completed.

Remark 14. The nonconvex feasibility problem (45)–(50) can be formulated as an optimisation problem subject to LMI constraints by using Algorithm 10.

5. Number Example

To illustrate the effectiveness of the proposed method, we consider the cart and inverted pendulum problem [8–10] in Figure 2, where 𝑚₁ is the cart mass, 𝑚₂ is the pendulum mass, L is the length from the point of rotation to the center of gravity of the pendulum, x is the cart position, 𝜃 is the pendulum angular position, and u is the input force. This is a fourth-order unstable system. The state variables are 𝑥₁=𝑥,𝑥₂=𝑥̇,𝑥₃=𝜃,and𝑥₄=𝜃̇. The parameters used are 𝑚₁=1kg,𝑚₂=0.5kg,and𝐿=1m and smooth surfaces. The state-space model is 𝑥̇(𝑡)=𝐴𝑥(𝑡)+𝐵𝑢(𝑡), (53) where ,.𝐴=1.00001.0000−0.00050.000501.0000−3.2663000−0.00021.00000019.6007−0.0002𝐵=−0.00030.8448−0.0003−1.3328 (54)

Figure 2 

Cart and inverted pendulum system.

Assume the minimum sampling period is 0.1𝑠, the maximum sampling period is 0.3𝑠, then ℎ_𝑘∈{0.1𝑠,0.2𝑠,0.3𝑠}. Assume the case where random delays only exist from the sensor to the controller, and 𝜏_𝑘^𝑠𝑐∈{0,1}, and their transition probability matrices are given by Λ=0.40.60.50.5,Π=0.50.40.10.20.50.30.10.40.5. (55)

Figure 3 shows part of the simulated run of the S-C delays 𝜏_𝑘^𝑠𝑐and sampling periods ℎ_𝑘 governed by their corresponding transition probability matrices, respectively.

Figure 3 

Modes with 𝜏𝑘 and ℎ𝑘.

In the above assumption, we discretize the system (53) and obtain Φ₁=,Φ1.10520.1105−0.0177−0.000501.1052−0.3546−0.0172001.09960.1033002.02471.0996₂=,Φ1.22140.2443−0.0794−0.004901.2214−0.8165−0.0745001.41820.2272004.45271.4182₃=,Γ1.34990.4050−0.2055−0.018401.3499−1.4816−0.1872002.01940.3963007.76762.0194₁=0.00450.0896−0.0068−0.1377,Γ₂=,Γ0.01960.1934−0.0285−0.3029₃=.0.04820.3187−0.0694−0.5285 (56)

The eigenvalues of 𝐴 are 1, 1, 4.4272, and −4.4275, the eigenvalues of Φ₁ are 1.1052, 1.1052, 0.6423, and 1.5569, the eigenvalues of Φ₂ are 1.2214, 1.2214, 2.4240, and 0.4125, the eigenvalues of Φ₃ are 1.3499, 1.3499, 3.7739, and 0.2649. Hence, the system is unstable. The state trajectory of the open-loop system is shown in Figure 4.

Figure 4 

States of the open-loop system.

By Theorem 8 and Algorithm 10, when transition probabilities are completely known, we can obtain the controllers with the following matrices 𝐾(𝜏_𝑘^𝑠𝑐,ℎ_{𝑘−𝜏𝑘}^𝑠𝑐): ,,,,,.𝐾01=−0.1722−1.448812.09053.2261𝐾02=−0.5648−1.86620.88114.2691𝐾03=−0.5648−1.86620.88114.2690𝐾11=−0.4764−1.63411.12883.6632𝐾12=−0.5657−1.86870.87834.2466𝐾13=−0.5655−1.83970.88064.2296 (57)

By Theorem 13, when transition probabilities are partly known; that is, Λ=0.40.6??,Π=0.5???0.5?0.10.40.5, (58) we can obtain the controllers with the following matrices 𝐾(𝜏_𝑘^𝑠𝑐,ℎ_{𝑘−𝜏𝑘}^𝑠𝑐): ,,,,,.𝐾01=−0.15341.548210.19022.2463𝐾02=0.5648−1.8662−0.581414.2531𝐾03=−0.5648−1.86620.88113.9625𝐾11=−0.3561−1.40261.47232.9558𝐾12=−0.5659−1.84320.87914.2211𝐾13=−0.5646−1.76490.88694.1893 (59)

Assume that the initial condition for simulation is θ(0) = 0.1 rad, and all other initial states are zero. The state trajectories of the closed-loop system caused by the discretized model are shown in Figures 5 and 6, respectively.

Figure 5 

States of the closed-loop system (completely known).

Figure 6 

States of the closed-loop system (partly known).

6. Conclusions

This paper proposes a state feedback controller design method for NCSs with random time delays and random sampling periods. Sampling periods also can randomly switch between three cases according to the network load. The S-C random delays and random sampling periods are modeled as Markov chains. The transition probabilities do not need to be completely known. The closed-loop systems can be expressed as jump linear systems with two modes. Sufficient conditions of stochastic stability are obtained in terms of a set of LMIs with matrix inversion constraints. Further, the state feedback controller is designed via the iterative linear matrix inequality approach and the state feedback gain depends on the two modes.

Acknowledgments

This work was supported in part by the Natural Science Foundation of China (NSFC) under Grants nos. 61273008, 61074029, and 61104093.