Stabilization of Discrete-Time Markovian Jump Systems via Controllers with Partially Mode-Dependent Characterization

Abstract

A kind of stabilizing controller in terms of being partially mode-dependent is developed for discrete-time Markovian jump systems (MJSs). The property referred to be partially mode-dependent is described by the Bernoulli variable. Based on the established model, the stabilization for MJSs over unreliable networks is considered, where both network-induced delay and packet dropout take place in system modes and states. Such effects of network are taken into account in controller design. All the conditions are derived in terms of linear matrix inequalities (LMIs). Finally, illustrative examples are presented to show the effectiveness and applicability of the proposed method.

1. Introduction

Markovian jump systems (MJSs) [1–9] are regarded as a special class of hybrid systems with finite operation modes. Their structures are subject to random abrupt changes, which may result from abrupt phenomena such as random component failures or repairs, changes in subsystem interconnections, and sudden environmental changes and modification of the operating point of a linearized model of a nonlinear system. Over the past decade, many important issues have been investigated for discrete-time Markovian jump systems (DMJSs), such as stability and stabilization [10, 11] and 𝐻₂ and 𝐻_∞ control [12–15]. By investigating such references, it is known that system mode is assumed to be available online and plays important roles in system analysis and synthesis. This will have the application scope of mode-dependent results limited. In order to remove such ideal assumption, a mode-independent stabilization problem for DMJSs was considered in [16]. The operation mode is not necessary in the work of the designed controller and is totally ignored.

However, the feedback control systems are usually closed via real time networks. Instead of point-to-point wiring, the use of shared communication networks is mainly due to its great advantages, such as low cost, reduced weight and power requirements, simple installation and maintenance, and high reliability. Despite the advantages and potentials, the introduction of communication networks in feedback control loops complicates the networked control systems (NCSs) analysis and synthesis [17], which also proposes new interesting and challenging problems. Network-induced delays and packet dropouts are usually considered in NCSs. Because of such effects, the data transmitted through networks is usually accessible with some probability. Based on this fact, it is said that the mode-independent control method is an overdesign method, which sometimes will lead to no solution to the desired control law. Since mode-independent and mode-dependent controller designs are worst-case and best-case design methods, respectively, a question whether there is a control method to bridge the aforementioned two extreme cases is proposed naturally.

Via modeling two random delays as two independent Markov chains, sufficient and necessary condition on the existence of stabilizing controller is established in [18]. On the other hand, [19, 20] considered the stability and controller design of NCSs with packet dropouts modeled as Markov chains. It is seen that the plant of the underlying NCSs in most literature is a deterministic system. That means that there is only one operation mode in the original system. When the controlled plant is an MJS, few results are available. Then, it is natural to consider how the insertion of networks affects the stability and performance of the closed-loop MJSs. Recently, for DMJSs with time-delays in both system state and mode signal, the authors in [21] first showed that the closed-loop system is a time-varying delayed MJS with extended state space. A general case that the time delays in signal mode and system state are the same and time-varying is considered in [22]. To the best of the authors’ knowledge, the stabilization problem of MJSs via networks has not been investigated fully and still remains challenging. All the observations motivate the current research.

In this paper, a partially mode-dependent controller design method is developed to solve the stabilization problem of DMJS. Instead of a common Lyapunov function selected for all system modes, a stochastic Lyapunov function is exploited. Moreover, we utilize the proposed technique to stabilize NCSs, whose controlled plant is an MJS. Both time-induced delays and packet dropouts are taken into account in controller design. All the conditions for the solvability of the stabilizing controllers are established in terms of LMIs.

Notation.  ℝ^𝑛 denotes the 𝑛 dimensional Euclidean space and ℝ^𝑚×𝑛 is the set of all 𝑚×𝑛 real matrices. |⋅| denotes the Euclidean norm. ℰ{⋅} is the expectation operator with respect to some probability measure. In symmetric block matrices, we use “∗” as an ellipsis for the terms induced by symmetry, diag{⋯} for a block-diagonal matrix, and (𝑀)^⋆≜𝑀+𝑀^𝑇.

2. Preliminaries

Fix the probability space (Ω,𝔽,ℙ) and consider the following DMJS 𝑥(𝑘+1)=𝐴(𝜃(𝑘))𝑥(𝑘)+𝐵(𝜃(𝑘))𝑢(𝑘), (1) where 𝑥(𝑘)∈ℝ^𝑛 is the state vector, 𝑢(𝑘)∈ℝ^𝑚 is the control input, and {𝜃(𝑘),𝑘≥0} is a discrete-time homogeneous Markov chain taking values in a finite set 𝕊={1,2,…,𝑁} with generator Λ=(𝜆_𝑖𝑗); that is 𝜆_𝑖𝑗≜Pr𝜃(𝑘+1)=𝑗∣𝜃(𝑘)=𝑖, (2) where 𝜆_𝑖𝑗≥0∀𝑖, 𝑗∈𝕊, and ∑^𝑁_𝑗=1𝜆_𝑖𝑗=1. For notation simplification, a matrix 𝑀(𝜃(𝑘)) will be denoted by 𝑀_𝑖.

Now, consider a partially mode-dependent, state feedback control law 𝑢(𝑘)=[𝛼(𝑘)𝐾(𝜃(𝑘))+(1−𝛼(𝑘))𝐾]𝑥(𝑘), (3) where 𝐾(𝜃(𝑘)) and 𝐾 are control gains to be determined and 𝛼(𝑘) is a Bernoulli distributed sequence with Pr{𝛼(𝑘)=1}=ℰ{𝛼(𝑘)}=𝛼,Pr{𝛼(𝑘)=0}=1−𝛼, (4) where 0≤𝛼≤1 is a constant. Furthermore, from (4), we have ℰ{𝛼(𝑘)−𝛼}=0,𝛽²≜ℰ(𝛼(𝑘)−𝛼)²=𝛼(1−𝛼). (5)

Remark 1. It is said that the stochastic variable 𝛼(𝑘) is used to reflect the observation probability of system mode. If 𝛼=1, controller (3) will become a mode-dependent controller, and, when 𝛼=0, we will have a mode-independent controller. It is worth mentioning that controller (3) is more advantageous than a totally mode-dependent or mode-independent controller. Compared with mode-independent controller in [16], one advantage of (3) avoids the overdesign problem which may lead to no solution to the desired control law. To the contrary, compared with mode-dependent control methods [11–14], another advantage is that it can reduce the burden of data transmission. That is, we can measure mode signal with some probability instead of measuring them exactly online. Both the two advantages are shown via numerical examples.

Remark 2. It should be remarked that the introduction of (3) applied to (1) does not destroy the Markov process in (1) with respect to 𝜃(𝑘). Because the Bernoulli process is a special case of Markov process, it is seen as another Markov process. In this paper, two stochastic variables 𝜃(𝑘) and 𝛼(𝑘) are assumed to be independent from each other.
Applying the controller in (3) to the system in (1) results in the following closed-loop system: 𝑥(𝑘+1)=𝐴_𝑖+𝐵_𝑖𝛼𝐾_𝑖+()𝐾𝑥(𝑘)+(𝛼(𝑘))𝐵1−𝛼−𝛼_𝑖𝐾_𝑖−𝐾𝑥(𝑘), (6) where for all𝜃(𝑘)=𝑖∈𝕊.

Definition 3. The closed-loop system (6) is stochastically stable if ℰ^∞_𝑘=0‖𝑥(𝑘)‖²∣𝑥₀,𝜃₀<∞, (7) for every every initial condition 𝑥₀=𝑥(0)∈ℝ^𝑛 and 𝜃₀=𝜃(0)∈𝕊.

3. Main Results

Firstly, we give the stabilization condition for DMJS (1) via controller (3).

Theorem 4. Considering DMJS (1), there exists a controller (3) such that closed-loop system (6) is stochastically stable if there exist matrices 𝑃_𝑖>0, 𝑌_𝑖, 𝑌, and 𝐺, satisfying the following LMIs, for all 𝑖∈𝕊: −𝑃_𝑖𝐺^𝑇𝐴^𝑇_𝑖+𝛼𝑌^𝑇_𝑖+(1−𝛼)𝑌^𝑇𝐵^𝑇_𝑖𝛽𝑌^𝑇_𝑖−𝑌^𝑇𝐵^𝑇_𝑖∗(−𝐺)^⋆+∑^𝑁_𝑗=1𝜆_𝑖𝑗𝑃_𝑗0∗∗(−𝐺)^⋆+∑^𝑁_𝑗=1𝜆_𝑖𝑗𝑃_𝑗<0. (8) Moreover, the corresponding control laws are given by 𝐾_𝑖=𝑌_𝑖𝐺⁻¹,𝐾=𝑌𝐺⁻¹. (9)

Proof. Choose a stochastic Lyapunov function for the system in (6) as 𝑉(𝑥(𝑘),𝜃(𝑘))=𝑥^𝑇(𝑘)𝑃(𝜃(𝑘))𝑥(𝑘). (10)
By condition (5), one has ℰ{𝑉(𝑥(𝑘),𝜃(𝑘))}=ℰ𝑥^𝑇()𝑃(𝜃(()𝑘+1𝑘+1))×𝑥𝑘+1∣𝜃_𝑘=𝑖−𝑥^𝑇(𝑘)𝑃_𝑖𝑥(𝑘)=ℰ𝑥^𝑇(𝑘)𝐴_𝑖+(𝛼(𝑘)−𝛼)𝐵_𝑖^𝑇(𝜃(××𝑃𝑘+1))𝐴_𝑖+(𝛼(𝑘)−𝛼)𝐵_𝑖𝑥(𝑘)−𝑥^𝑇(𝑘)𝑃_𝑖𝑥(𝑘)=𝑥^𝑇(𝑘)Φ_𝑖𝑥(𝑘), (11) where Φ_𝑖=𝐴^𝑇_𝑖𝑃_𝑖𝐴_𝑖+𝛽²𝐵^𝑇_𝑖𝑃_𝑖𝐵_𝑖−𝑃_𝑖,𝑃_𝑖=^𝑁_𝑗=1𝜆_𝑖𝑗𝑃_𝑗,𝐴_𝑖=𝐴_𝑖+𝐵_𝑖𝛼𝐾_𝑖+(1−𝛼)𝐾,𝐵_𝑖=𝐵_𝑖𝐾_𝑖−𝐾. (12) From (8), it is concluded that 𝐺 is nonsingular. Pre- and postmultiplying (8) with diag{𝐺^−𝑇,𝐼,𝐼} and its transpose, respectively, and letting 𝑃_𝑗=𝐺^−𝑇𝑃_𝑗𝐺⁻¹, we have −𝑃_𝑖𝐴^𝑇_𝑖𝛽𝐵^𝑇_𝑖∗(−𝐺)^⋆+𝐺^𝑇𝑃_𝑖()𝐺0∗∗−𝐺^⋆+𝐺^𝑇𝑃_𝑖𝐺<0. (13) Since ∑^𝑁_𝑗=1𝜆_𝑖𝑗𝐺^𝑇𝑃_𝑗𝐺=𝐺^𝑇𝑃_𝑖𝐺>0, we have −𝑃_𝑖⁻¹=−𝐺𝐺⁻¹𝑃_𝑖⁻¹𝐺^−𝑇𝐺^𝑇≤(−𝐺)^⋆+𝐺^𝑇𝑃_𝑖𝐺. (14) By the Schur complement and (14), it is known that (13) implies Φ_𝑖<0. Therefore, we have ℰ{𝑉(𝑥()((𝑥(𝑘)(𝑘‖𝑥(𝑘𝑘+1,𝜃𝑘+1))}−𝑉,𝜃))≤−𝜆)‖²<0, (15) where 𝜆=inf_𝑖∈𝕊(−Φ_𝑖)>0. From Dynkin’s formula and summing up (15) form 0 to ∞ with respect to 𝑘 yields, we obtain ℰ^∞_𝑘=0‖𝑥(𝑘)‖²≤1𝜆ℰ𝑉₀<∞. (16) By Definition 3, system (6) is stochastically stable. This completes the proof.

Remark 5. It is worth mentioning that controller with partially mode-dependent property (3) is established in terms of LMIs via a stochastic Lyapunov function instead of a common Lyapunov function, where both totally mode-dependent and mode-independent controllers are contained. It is said that controller (3) bridges such two extreme cases. It is also seen that 𝛼 in partially mode-dependent method plays an important role, which should be known. When it is unknown or unavailable, how to design an effective controller is necessary to be considered. Two possible ways may deal with such problem. The first one is to use a robust method to handle (6). On the other hand, this general case can be done by use an adaptive control. The unknown probability 𝛼 will be estimated by an adaption law.
Next, we will apply key idea of (3) to NCSs. The controlled plant is an MJS as shown in Figure 1.

Figure 1 

Structure of NCS with network-induced delays.

As demonstrated in Figure 1, system state 𝑥(𝑘) and mode signal 𝜃(𝑘) are transmitted through networks with communication delays 𝑑(𝑘) and 𝜏(𝑘), respectively. Considering the effect of random delays 𝑑(𝑘) and 𝜏(𝑘), the control law is described as 𝑢(𝑘)=𝐾(𝜃(𝑘−𝜏(𝑘)))𝑥(𝑘−𝑑(𝑘)), (17) where 𝜏_𝑚≤𝜏(𝑘)≤𝜏_𝑀 and 𝑑_𝑚≤𝑑(𝑘)≤𝑑_𝑀.

Remark 6. If delay 𝜏(𝑘) of system mode is constant, controller (17) is similar to that in [21] where the authors first showed that the resulting closed-loop system is a new time-varying delayed MJS with extended state space. For the case that delay 𝜃(𝑘) is similar to delay 𝑑(𝑘), the resulting closed-loop system is modeled as an MJS with two modes via augmentation technique [22]. Both of the aforesaid references constructed a mode-dependent stabilizing controller with an assumption that there is no packet missing in channel 1. However, it is a more realistic situation that not only network-induced delays but also packet dropouts exist in channel 1, when system mode is transmitted. Since both network-induced delays and packet dropouts affect system mode 𝜃(𝑘), it is natural to consider both of them in the controller design. Unfortunately, controller (17) will not be applicable to this case.
Based on the key idea of (3), we know that the effect of dropouts on system mode 𝜃(𝑘) does not destroy the Markov process with respect to 𝜃(𝑘). So we construct the following controller: 𝑢(𝑘)=𝛼(𝑘)𝐾𝜃𝑘−𝜏_𝑚+(1−𝛼(𝑘))𝐾𝑥(𝑘), (18) where 𝛼(𝑘) is function as 𝛼(𝑘)=1,𝜏(𝑘)=𝜏_𝑚andtransmittedsuccessfully,0,otherwise. (19)
Thus, 𝛼(𝑘) is a Bernoulli distributed sequence satisfying (4) and (5). From [11], we know that constant delay in system mode 𝜃(𝑘) results in the original system being no longer traditional MJSs. However, the closed system is a time-varying delayed MJS with extended state space. In addition to Remark 2, we know that the closed-loop system (1) via controller (18) is also an MJS with respect to 𝜃(𝑘)≜[𝜃(𝑘)𝜃(𝑘−1)⋯𝜃(𝑘−𝜏_𝑚)]^𝑇.
Define 𝜑_𝜏𝑚+1≜[1,2,…,𝑁^𝜏𝑚+1], and applying controller (18) to system (1) results in the following closed-loop system: 𝑥()𝜃(𝑘𝑥(𝑘)+𝐴𝑘+1=𝐴)_𝑑𝜃(𝑘)+(𝛼(𝑘)−𝛼)𝐴_𝑑𝜃(𝑘)×𝑥(𝑘−𝑑(𝑘)), (20) where 𝐴𝜃(𝑘(𝜃(𝑘,𝐴)=𝐴))_𝑑𝜃(𝑘)=𝐵𝜃_𝑘𝜃(𝑘)+()𝐾,𝐵𝜃(𝑘)(𝜃(𝑘,𝐾𝜃(𝑘)𝜃𝛼𝐾1−𝛼=𝐵))=𝐾𝑘−𝜏_𝑚,𝐴_𝑑𝜃(𝑘)=𝐵𝜃(𝑘)𝐾𝜃(𝑘)−𝐾. (21)

Definition 7. Let 𝑥(𝑘,𝜙(⋅),𝜅(⋅)) be the solution to closed-loop system (20). The closed-loop system is stochastically stable if ℰ^∞_𝑘=0𝑥𝑘,𝜙(⋅),𝜅(⋅)²∣𝜙(⋅),𝜅(⋅)<∞, (22) for every initial conditions 𝜙(⋅)∈ℝ^𝑛, 𝑘=−𝑑_𝑀,−𝑑_𝑀+1,…,0, and 𝜅(⋅)∈𝕊, 𝑘=−𝜏_𝑚,−𝜏_𝑚+1,…,0.
Now, we will consider the stabilization condition of MJS (20).

Theorem 8. Considering DMJS (1), there exists a controller (18) such that closed-loop system (20) is stochastically stable if there exist matrices 𝑃_𝜐>0,  𝑄_𝑞>0,  𝑍_𝑞>0,  𝑌_{𝑖−𝜏𝑚}, 𝑌, and 𝐺, satisfying the following LMIs, for all 𝑖∈𝕊, 𝜐∈𝜑_𝜏𝑚+1 and 𝑞=1,2,3: Θ_𝜐0𝑍₃𝑍₁𝐺^𝑇𝐴^𝑇_𝑖𝐺^𝑇𝐴_𝑖∗Ω−𝐼00₁𝑍₂𝑍₂Ψ^𝑇_𝑖Ψ^𝑇_𝑖𝛽Ψ^𝑇_𝑖𝛽Ψ^𝑇_𝑖∗∗Ω₂Ω00000∗∗∗₃0000∗∗∗∗Π𝑖,𝜇000∗∗∗∗∗Ω₄000∗∗∗∗∗∗Π𝑖,𝜇∗∗∗∗∗∗∗Ω₄<0, (23) where Θ_𝜐=−𝑃_𝜐−𝑍₁−𝑍₃+(𝑑+1)𝑄₁+𝑄₂+𝑄₃,Ω₁=−𝑄₁−2𝑍₂,Ω₂=−𝑄₃−𝑍₂−𝑍₃,Ψ_𝑖=𝐵_𝑖𝛼𝑌_{𝑖−𝜏𝑚}+(1−𝛼)𝑌,Ψ_𝑖=𝐵_𝑖𝑌_{𝑖−𝜏𝑚},Ω−𝑌₃=−𝑄₂−𝑍₁−𝑍₂,Π𝑖,𝜇=(−𝐺)^⋆+𝑃,𝑃=𝑖,𝜇𝑖,𝜇^𝑁_𝑗=1𝜆_𝑖𝑗𝑃_𝜇+𝑗,Ω₄=(−𝐺)^⋆+𝑍,𝑍=𝑑²_𝑀𝑍₁+𝑑²𝑍₂+𝑑²_𝑚𝑍₃,𝑑=𝑑_𝑀−𝑑_𝑚,𝜇=(𝑖−1)𝑁+𝑖₋₁−1𝑁²+𝑖+⋯_{−𝜏𝑚+2}−1𝑁^𝜏𝑚−1+𝑖_{−𝜏𝑚+1}−1𝑁^𝜏𝑚. (24)
Moreover, the corresponding control laws are given by 𝐾_{𝑖−𝜏𝑚}=𝑌_{𝑖−𝜏𝑚}𝐺⁻¹,𝐾=𝑌𝐺⁻¹. (25)

Proof. Let 𝜉(𝑘)=𝑥(𝑘+1)−𝑥(𝑘). Choose a stochastic Lyapunov function for system in (20) as 𝑉𝑥_𝑘,𝜃(𝑘)=𝑉₁𝑥_𝑘,𝜃(𝑘)+𝑉₂𝑥_𝑘,𝜃(𝑘)+𝑉₃𝑥_𝑘,𝜃(𝑘)+𝑉₄𝑥_𝑘,𝜃(𝑘)+𝑉₅𝑥_𝑘,𝜃(𝑘), (26) where 𝑉₁𝑥_𝑘,𝜃(𝑘)=𝑥^𝑇(𝑘)𝑃𝜃(𝑘𝑥(𝑘),𝑉₂)𝑥_𝑘,𝜃(𝑘)=^𝑘−1_{𝑙=𝑘−𝑑𝑘}𝑥^𝑇(𝑙)𝑄₁𝑥(𝑙)+^𝑘−1_{𝑙=𝑘−𝑑𝑀}𝑥^𝑇(𝑙)𝑄₂𝑥(𝑙)+^𝑘−1_{𝑙=𝑘−𝑑𝑚}𝑥^𝑇(𝑙)𝑄₃𝑥(𝑙),𝑉₃𝑥_𝑘,𝜃(𝑘)=^−𝑑𝑚_{𝑞=−𝑑𝑀+1}^𝑘−1_{𝑙=𝑘+𝑞}𝑥^𝑇(𝑙)𝑄₁𝑥(𝑙),𝑉₄𝑥_𝑘,𝜃_𝑘=𝑑_𝑀⁻¹_{𝑞=−𝑑𝑀}^𝑘−1_{𝑙=𝑘+𝑞}𝜉^𝑇(𝑙)𝑍₁𝜉(𝑙),𝑉₅𝑥_𝑘,𝜃(𝑘)=𝑑^{−𝑑𝑚−1}_{𝑞=−𝑑𝑀}^𝑘−1_{𝑙=𝑘+𝑞}𝑥^𝑇(𝑙)𝑍₂𝑥(𝑙),𝑉₆𝑥_𝑘,𝜃_𝑘=𝑑_𝑚⁻¹_{𝑞=−𝑑𝑚}^𝑘−1_{𝑙=𝑘+𝑞}𝜉^𝑇(𝑙)𝑍₃𝜉(𝑙). (27)
From [21], we have 𝜆_𝜐𝜂=Pr𝜃(𝑘+1)=𝜂∣𝜃(𝑘)=𝜐=𝜆_𝑖𝑗𝛿𝑖,𝑗₋₁𝛿𝑖₋₁,𝑗₋₂⋯𝛿𝑖_{−𝜏𝑚+1},𝑗_−𝜏𝑚, (28) where 𝛿(⋅,⋅) is similar to that in [21], and 𝜆_𝜐,𝜂=𝜆_𝑖𝑗, if 𝜂=𝜇+𝑗; otherwise, 𝜆_𝜐𝜂=0. Thus, we have ^{𝑁𝜏𝑚+1}_𝜂=1𝜆_𝜐𝜂𝑃_𝜂=^𝑁_𝑗=1𝜆_𝑖𝑗𝑃_𝜇+𝑗=𝑃𝑖,𝜇. (29)
Then, by computation, we have ℰ𝑉₁(𝑘+1)−𝑉₁(𝑘)=𝐴_𝑖𝑥(𝑘)+𝐴_𝑑𝑖𝑥(𝑘−𝑑(𝑘))^𝑇×𝑃𝑖,𝜇𝐴_𝑖𝑥(𝑘)+𝐴_𝑑𝑖𝑥(𝑘−𝑑(𝑘))+𝛽²𝐴_𝑑𝑖𝑥(𝑘−𝑑(𝑘))^𝑇×𝑃𝑖,𝜇𝐴_𝑑𝑖𝑥(𝑘−𝑑(𝑘))−𝑥^𝑇(𝑘)𝑃_𝜐𝑥(𝑘),ℰ𝑉₂(𝑘+1)−𝑉₂(𝑘)≤𝑥^𝑇(𝑘)𝑄₁+𝑄₂+𝑄₃𝑥(𝑘)+^{𝑘−𝑑𝑚}_{𝑙=𝑘−𝑑𝑀+1}𝑥^𝑇(𝑙)𝑄₁𝑥(𝑙)−𝑥^𝑇(𝑘−𝑑(𝑘))𝑄₁𝑥(𝑘−𝑑(𝑘))−𝑥^𝑇𝑘−𝑑_𝑀𝑄₂𝑥𝑘−𝑑_𝑀−𝑥^𝑇𝑘−𝑑_𝑚𝑄₃𝑥𝑘−𝑑_𝑚,ℰ𝑉₃(𝑘+1)−𝑉₃(𝑘)≤𝑑𝑥^𝑇(𝑘)𝑄₁𝑥(𝑘)−^{𝑘−𝑑𝑚}_{𝑙=𝑘−𝑑𝑀+1}𝑥^𝑇(𝑙)𝑄₁𝑥(𝑙),ℰ𝑉₄(𝑘+1)−𝑉₄(𝑘)=𝑑²_𝑀ℰ𝜉^𝑇(𝑘)𝑍₁𝜉(𝑘)−𝑑_𝑀^𝑘−1_{𝑙=𝑘−𝑑𝑀}𝜉^𝑇(𝑙)𝑍₁𝜉(𝑙)≤𝑑²_𝑀𝐴_𝑖−𝐼𝑥(𝑘)+𝐴_𝑑𝑖𝑥(𝑘−𝑑(𝑘))^𝑇×𝑍₁𝐴_𝑖−𝐼𝑥(𝑘)+𝐴_𝑑𝑖𝑥(𝑘−𝑑(𝑘))+𝛽²𝑑²_𝑀𝐴_𝑑𝑖𝑥(𝑘−𝑑(𝑘))^𝑇×𝑍₁𝐴_𝑑𝑖𝑥((𝑘−𝑥(𝑘)𝑘−𝑑))−𝑥𝑘−𝑑_𝑀^𝑇×𝑍₁𝑥(𝑘)−𝑥𝑘−𝑑_𝑀,ℰ𝑉₅(𝑘+1)−𝑉₅(𝑘)=𝑑²ℰ𝜉^𝑇(𝑘)𝑍₂𝜉(𝑘)−𝑑^{𝑘−𝑑(𝑘)−1}_{𝑙=𝑘−𝑑𝑀}+^{𝑘−𝑑𝑚−1}_{(𝑘)𝑙=𝑘−𝑑}𝜉^𝑇(𝑙)𝑍₁𝜉(𝑙)≤𝑑²𝐴_𝑖−𝐼𝑥(𝑘)+𝐴_𝑑𝑖𝑥(𝑘−𝑑(𝑘))^𝑇×𝑍₂𝐴_𝑖−𝐼𝑥(𝑘)+𝐴_𝑑𝑖𝑥(𝑘−𝑑(𝑘))+𝛽²𝑑²𝐴_𝑑𝑖𝑥(𝑘−𝑑(𝑘))^𝑇×𝑍₂𝐴_𝑑𝑖𝑥((𝑘−𝑥((𝑘𝑘−𝑑))𝑘−𝑑))−𝑥𝑘−𝑑_𝑀^𝑇×𝑍₂𝑥(𝑘−𝑑(𝑘))−𝑥𝑘−𝑑_𝑀−𝑥𝑘−𝑑_𝑚−𝑥(𝑘−𝑑(𝑘))^𝑇×𝑍₂𝑥𝑘−𝑑_𝑚((𝑘,ℰ𝑉−𝑥𝑘−𝑑))₆(𝑘+1)−𝑉₆(𝑘)=𝑑²_𝑚ℰ𝜉^𝑇(𝑘)𝑍₃𝜉(𝑘)−𝑑_𝑚^𝑘−1_{𝑙=𝑘−𝑑𝑚}𝜉^𝑇(𝑙)𝑍₃𝜉(𝑙)≤𝑑²_𝑚𝐴_𝑖−𝐼𝑥(𝑘)+𝐴_𝑑𝑖𝑥(𝑘−𝑑(𝑘))^𝑇×𝑍₃𝐴_𝑖−𝐼𝑥(𝑘)+𝐴_𝑑𝑖𝑥(𝑘−𝑑(𝑘))+𝛽²𝑑²_𝑚𝐴_𝑑𝑖𝑥(𝑘−𝑑(𝑘))^𝑇×𝑍₃𝐴_𝑑𝑖𝑥((𝑘−𝑥(𝑘)𝑘−𝑑))−𝑥𝑘−𝑑_𝑚^𝑇×𝑍₃𝑥(𝑘)−𝑥𝑘−𝑑_𝑚. (30)
Taking into account (30), we conclude ℰ{𝑉(𝑘+1)}−𝑉(𝑘)≤𝜁^𝑇(𝑘)Γ𝑖,𝜇𝜁(𝑘)<0, (31) where 𝜁^𝑇(𝑘)=𝑥^𝑇(𝑘)𝑥^𝑇(𝑘−𝑑(𝑘))𝑥^𝑇𝑘−𝑑_𝑚𝑥^𝑇𝑘−𝑑_𝑀,Γ=Θ𝑖,𝜇^𝜐0𝑍₃𝑍₁∗Ω₁𝑍₂𝑍₂∗∗Ω₂0∗∗∗Ω₃+𝐴^𝑇_𝑖𝐴^𝑇_𝑑𝑖00𝑃𝐴𝑖,𝜇^𝑇𝑖𝐴^𝑇_𝑑𝑖00^𝑇+𝐴_𝑖−𝐼^𝑇𝐴^𝑇_𝑑𝑖00𝑍𝐴_𝑖−𝐼^𝑇𝐴^𝑇_𝑑𝑖00^𝑇+𝛽²0𝐴^𝑇_𝑑𝑖00𝑃0𝑖,𝜇+𝑍𝐴^𝑇_𝑑𝑖00^𝑇,Θ_𝜐=−𝑃_𝜐−𝑍₁−𝑍₃+(𝑑+1)𝑄₁+𝑄₂+𝑄₃,Ω₁=−𝑄₁−2𝑍₂,Ω₂=−𝑄₃−𝑍₂−𝑍₃,Ω₃=−𝑄₂−𝑍₁−𝑍₂,𝑍=𝑑²_𝑀𝑍₁+𝑑²𝑍₂+𝑑²_𝑚𝑍₃. (32)
From (23), we have that 𝐺 is nonsingular. Defining 𝑃_𝜐=𝐺^𝑇𝑃_𝜐𝐺, 𝑄_𝑞=𝐺^𝑇𝑄_𝑞𝐺, and 𝑍_𝑞=𝐺^𝑇𝑍_𝑞𝐺, we have Θ_𝜐=𝐺^𝑇Θ𝐺, 𝑍=𝐺^𝑇𝑍𝐺, and 𝑃(𝑖,𝜇)=𝐺^𝑇𝑃(𝑖,𝜇)𝐺. Similar to (14), we have −𝑃⁻¹𝑖,𝜇=−𝐺𝑃𝑖,𝜇⁻¹𝐺^𝑇≤Π𝑖,𝜇,−𝑍⁻¹=−𝐺𝑍⁻¹𝐺^𝑇≤Ω₄. (33)
In addition, by the Schur complement, we obtain that Γ(𝑖,𝜇) is equivalent to the following:

Θ_𝜐0𝑍₃𝑍₁𝐴^𝑇_𝑖𝐴_𝑖−𝐼^𝑇00∗Ω₁𝑍₂𝑍₂𝐴^𝑇_𝑑𝑖𝐴^𝑇_𝑑𝑖𝛽𝐴^𝑇_𝑑𝑖𝛽𝐴^𝑇_𝑑𝑖∗∗Ω₂00000∗∗∗Ω₃0000∗∗∗∗−𝑃⁻¹𝑖,𝜇000∗∗∗∗∗−𝑍⁻¹00∗∗∗∗∗∗−𝑃⁻¹𝑖,𝜇0∗∗∗∗∗∗∗−𝑍⁻¹<0. (34)

Pre- and postmultiplying (34) by diag{𝐺^𝑇,𝐺^𝑇,𝐺^𝑇,𝐺^𝑇,𝐼,𝐼,𝐼,𝐼} and its transpose, respectively, and taking into account (23), (25), and (33), we have that (34) holds, which implies (31) holds. Then, we have that the system in (20) is stochastically stable via the similar lines in [13]. This completes the proof.

Remark 9. It should be pointed out that it is not appreciate to model transmission delays or packet losses as Markov process, since both of them occur in channel 1 simultaneously. However, the control technique based on partially mode-pendent controller can be used to tackle this case. The scalar 𝛼 may reflect the jam degree of networks. That is, the lager value of 𝛼 denotes the more expedite of networks. Comparing to mode-independent controller with 𝛼=0, it is less conservative in the sense that the desired controller is also constructed, when there is no solution to a totally mode-independent controller. When 𝛼=1 in terms of being a mode-dependent controller, a partially mode-dependent controller (18) is less dependent on the transmitted system modes, which could reduce the burden of communication networks. Both of the advantages can be shown via a numerical example. In addition, the solvability of controller (18) is established in terms of LMIs, which is solved directly by Matlab LMI toolbox.

4. Numerical Examples

In this section, two examples are given to demonstrate the effectiveness of the proposed approaches in this paper.

Example 1. Consider the following inverted pendulum system [23]: 𝑥̇(𝑡)=3()𝒢01𝑀+𝑚𝑙()0𝑥(𝑡)+0−34𝑀+𝑚𝑙()𝑢(𝑡)4𝑀+𝑚, (35) where 𝒢=9.8m/s², 𝑙 is the half length of the pendulum, and 𝑀 and 𝑚 are the masses of the cart and the pendulum, respectively. Here, we choose two samples; that is, 𝑀₁=8, 𝑚₁=2, and 𝑙₁=0.5 and 𝑀₂=6, 𝑚₂=3, and𝑙₂=0.7, respectively. Assuming the sampling period 𝑇_𝑠=0.01s, we have two discrete-time models for the system in (35) described as 𝐴₁=1.00090.01000.17301.0009,𝐵₁=,𝐴−0.0000−0.0018₂=1.00070.01000.14001.0007,𝐵₂=.−0.0000−0.0016 (36)
The transition probability matrix is given by Λ=0.30.70.20.8. (37)
Under the initial condition 𝑥(0)=[2−1]^𝑇, the state response of open system is shown in Figure 2. Obviously, it is not stable. By Theorem 4, we have the following controllers with 𝛼=0.2: 𝐾₁=,𝐾359.5431566.5793₂=,.373.6697634.0686𝐾=361.4663597.2739 (38)
Applying such controllers to the above system, the trajectory of the closed loop is illustrated in Figure 3.

Figure 2 

Open-loop system 𝑥1(𝑘)(−) and 𝑥2(𝑘)(−⋅).

Figure 3 

Closed-loop system 𝑥1(𝑘)(−) and 𝑥2(𝑘)(−⋅).

Example 2. Consider the following Markovian jump system described as 𝐴₁=1.3100.4,𝐵₁=01,𝐴₂=0.300.41.1,𝐵₂=1.0.07 (39)
The transition probability matrix is given by Λ=0.10.90.40.6. (40)
Let 𝑑_𝑚=1, 𝑑_𝑀=3, and 𝜏_𝑚=1. The initial conditions are 𝜙(𝑘)=[2−1] and 𝑘=−3,…,0 and 𝜅(𝑘)=1 and𝑘=−1,0, respectively. The state response of the open-loop system (with 𝑢(𝑘)=0) is shown in Figure 4, which is not stable. By Theorem 8, it is known that there is no solution, when 0≤𝛼≤0.203. That means that we cannot construct a totally mode-independent controller. Instead, we have the following controllers with 𝛼=1 and 𝛼=0.4, respectively.

Figure 4 

Open-loop system 𝑥1(𝑘)(−) and 𝑥2(𝑘)(⋅).

Case 𝐾∶𝛼=1𝐾𝐾₁=−0.1094−0.1787,𝐾𝐾₂=−0.0934−0.1526,𝐾𝐾=00. (41)

Case 𝐾̌∶𝛼=0.4𝐾₁=,𝐾−0.1231−0.2018₂=−0.0851−0.1395,𝐾=−0.0991−0.1617. (42)

The state trajectories 𝑥₁(𝑘) of the closed-loop system via controllers 𝐾 and 𝐾̌ are shown in Figure 5, respectively, so is 𝑥₂(𝑘) shown in Figure 6. Though 𝐾̌ is a partially mode-pendent controller, it can also stabilize this system. Because 𝛼=0.4, it means that the system mode of controller 𝐾̌ is not necessarily transmitted through network online. In this sense, it is said that the partially mode-pendent controller is advantageous over a mode-dependent controller in terms of reducing network expedite.

Figure 5 

Closed-loop system 𝑥1(𝑘) under 𝐾(−) and 𝐾̌(∗).

Figure 6 

Closed-loop system 𝑥2(𝑘) under 𝐾(⋅) and 𝐾̌(•).

5. Conclusions

This paper has investigated the stabilization problem for DMJSs via a partially mode-dependent controller. Sufficient condition for such controller is developed within LMI framework. Moreover, the key idea of this control scheme has been utilized to networked control system. In this case, the controlled plant is an MJS with simultaneous consideration of network-induced delays and data packet dropouts in system mode transmission. Numerical simulations are used to illustrate the effectiveness and applicability of the developed results.

Acknowledgments

This work was supported by the National Natural Science Foundation of China under Grants 61104066 and 61374043, the China Postdoctoral Science Foundation funded project under Grant 2012M521086, and the Program for Liaoning Excellent Talents in University under Grant LJQ2013040.