Stochastic stabilization of Markovian jump cloud control systems based on max-plus algebra

WANG Jin ,YANG Hongjiu,* ,XIA Yuanqing ,and YAN Ce

1.School of Electrical and Information Engineering,Tianjin University,Tianjin 300072,China;2.School of Automation,Beijing Institute of Technology,Beijing 100081,China

Abstract: In this paper,stochastic stabilization is investigated by max-plus algebra for a Markovian jump cloud control system with a reference signal.For the Markovian jump cloud control system,there exists framework adjustment whose evolution is satisfied with a Markov chain.Using max-plus algebra,a maxplus stochastic system is used to describe the Markovian jump cloud control system.A causal feedback matrix is obtained by exponential stability analysis for a causal feedback controller of the Markovian jump cloud control system.A sufficient condition is given to ensure existence on the causal feedback matrix of the causal feedback controller.Based on the causal feedback controller,stochastic stabilization in probability is analyzed for the Markovian jump cloud control system with a reference signal.Simulation results are given to show effectiveness of the causal feedback controller for the Markovian jump cloud control system.

Keywords: Markovian jump cloud control system,causal feedback controller,max-plus algebra,max-product algebra,stochastic stabilization.

1.Introduction

Since its birth,cloud computing has attracted much attention for significate advantages in data processing,computing power and communication security [1]. With expansion of application scenarios,cloud computing is applied into many complex applications which usually have time-varying characteristics and switching structures [2,3].A time-varying data cloud computing system is used to reduce communication burden of individual agents in a low-power nonlinear multi-agent system [2].In [3],a multi-order Markov chain framework is designed for anomaly detection uncertainties which are caused by structural change of a cloud server system.Nowadays,new-type systems on cloud computing are investigated such as cloud control systems [4],mobile cloud computing systems [5] and mCloud systems [6].For a cloud control system,there exist numerous delays which are generated by information transmission and computing processes [7,8].Based on cloud predictive control schemes,networked delays from information transmission are handled to achieve consensus control for a networked multiagent system with cloud computing [7].In [8],time delays of computing processes are analyzed on predictive cloud control for a networked multiagent system with quantized signals and deniel of service (DoS) attacks.During a cloud control process,a processing framework of a cloud control system is usually changeable for the reason of that the processing framework is adjusted according to control expectation in real time [9].Moreover,framework adjustment of a cloud control system is often dependent on states in the last sampling time solely such that it is reasonable to take the framework adjustment as a Markovian jump process [10].Therefore,it is an interesting work to study stochastic control for a cloud control system with a Markovian jump process and time delays.

It is well known that discrete-event systems are suitable to be taken as cloud control systems which are used to model many engineered systems [11].Nonlinearity can be solved by linear equations of max-plus algebra in a discrete-event system such that there exists a good idea to apply max-plus algebra in cloud control systems [12,13].Based on max-plus algebra,a very large scale integration array processor is modeled as a max-plus linear system for dimension reduction and feedback stabilization [12].In [13],time schedule of a multi-legged robot is proposed as max-plus linear equation sets which transform discrete states into continuous states.Moreover,there are some stochastic control strategies on max-plus algebra for discrete-event switching systems [14].A model predictive control strategy is designed for a switching max-pluslinear system which consists of a switching process within different operation modes [15].It is known from[16] that model predictive control with a nonlinear nonconvex constraint is given to achieve linear programming for a switching max-plus-linear system.Note that operability is dependent on stability of Markovian jump systems in which stochastic stabilization should be considered for feedback controller design [17,18].With a memoryless state feedback controller,sufficient conditions for exponential stabilization are proposed for a stochastic Cohen-Grossberg neural network with Markovian jump parameters [19].In [20],remote stabilization was investigated for an erasure channel using multi-step Lyapunov bounds.To the best of our knowledge,there are few results on max-plus based stochastic control for Markovian jump cloud control systems,which motivates our current research work.

In this paper,a max-plus stochastic system with a Markov chain is taken as the Markovian jump cloud control system.A causal feedback controller is obtained by exponential stability of an autonomous max-product system obtained from the max-plus stochastic system.A sufficient condition is proposed for the causal feedback controller on a causal feedback matrix in the max-plus stochastic system.Main contributions of this paper are summarized as follows:

(i) Max-plus algebra is utilized for model establishment on a Markovian jump cloud control system with a reference signal.

(ii) A causal feedback matrix is introduced to a causal feedback controller for the Markovian jump cloud control system.

(iii) A sufficient condition of stochastic stabilization is used for exponential stability of the Markovian jump cloud control system.

2.Problem statement and preliminaries

2.1 Max-plus algebra

For the real number set R,an algebraic structure (R∪{-∞},⊕,⊗)is defined as max-plus alge bra Rmaxin which⊕and ⊗arethe mainoperations inmax-plus algebra such that

where P={p1,p2,···,pm} represents the location set,Q={q1,q2,···,qn} represents the transition set,Frepresents the set of directed arcs,Wrepresents the weight function of directed arcs,Mrepresents the status marking which is also called the Token,M0represents the initial marking.Furthermore,the operation ⊗ is sometimes omitted for simplification.

2.2 Cloud control system

In a cloud control system,an instruction of a cloud parallel processor is often divided into three subprocesses,such as instruction fetch,instruction analysis and instruction execution [21].Therefore,the cloud control system is obtained as shown in Fig.1.

Fig.1 Cloud control system

There exist multiple latches in the cloud control system to avoid instruction congestion [22].During a cloud control process,the instruction processed by each subprocess is locked up by a latch until the next instruction is delivered to the subprocess.By a Petri net,the cloud control system is established as [23] in Fig.2.

Fig.2 Petri net of the cloud control system

According to Fig.2,it is obtained for thekth instruction that the cloud control system is satisfied with

whereu(k) is the control input of the cloud parallel processor,x1(k),x2(k) andx3(k) are the beginning epochs of instruction fetch,instruction analysis and instruction execution,respectively,represent the processing time units of instruction fetch,instruction analysis and instruction execution,respectively,represents the transforming time unit between instruction fetch and instruction analysis,represents the transforming time unit between instruction analysis and instruction execution.Therefore,the cloud control system is modeled as follows:

According to [24],there exists instruction dependency of the cloud parallel processor in a cloud control process.During instruction dependency,the (k+1)th instruction has to enter the instruction analysis after thekth instruction leaves the instruction execution for the reason of that both thekth and (k+1)th instructions have the same source operand reference.Therefore,the cloud control system is transformed into the following form as Fig.3.

Fig.3 Petri net with instruction dependency

In Fig.3,the cloud control system is modeled as follows:

2.3 Objective statement

The following definitions and lemmas are proposed based on [25] and [26] to show main results in this paper.

Definition 1A matrix is row G-astic if it has at least one nonzero in each row of the matrix.

Definition 2An autonomous max-product system

is exponentially stable if there exista＞1 andL＞0 such that

Definition 3A max-product system

is bounded input bounded output in probability (BIBipO)stable if there exists a positive constant Mzfor a positive constant σ and an initial condition χ (0) such thatP[χi(k)≤Mz]＞σ with σ ∈(0,1) in which χi(k) represents the entry at theit h row of χ(k) . ϱ(k) respresents the control input.

Lemma 1[27] Consider the autonomous max-product system (2).If the autonomous max-product system(2) is mean norm exponentially stable,then a nonautonomous max-product system

3.Main results

In this section,a causal feedback controller is investigated for the max-plus stochastic system (1) with a reference signal.According to [16],a reference signal is denoted asr(k)=kT+ξ(k),whereTis the positive constant,ξ(k) is the bounded vector.Moreover,the max-plus stochastic system (1) is rewritten as follows:

Considering the max-plus stochastic system (3),a causal feedback controller is designed as

whereKis the causal feedback matrix to be designed,(k)=Kx(k-1) represents the regulating part of the causal feedback controller,v(k)=kTrepresents the input part of the causal feedback controller.Based on the maxplus stochastic system (3),a stochastic system is calculated as

With the additional controllerv(k)=ε,an autonomous stochastic system is obtained as

In [27],a max-product stochastic system is obtained from the autonomous stochastic system (5) as

with

Lemma 2Consider the max-product stochastic system (6) with 0 ＜α ＜1 andi=1,2,···,M.If there exists a positive vectorh(yk) such that

then the max-product stochastic system (6) is exponentially stable.

ProofSelect a Lyapunov functionBased on the Markov chainyk,expectation on the Lyapunov functionV(yk+1) is obtained as

Therefore,the max-product stochastic system (6) is exponentially stable.This completes the proof. □

Theorem 1If there exists at least an stochastic feedback matrixKwhich is satisfied with

ProofNote that transformation from the autonomous stochastic system (5) to the max-product stochastic system (6) is invertible [27].Therefore,it is reasonable to transform the inequality (7) to the following inequality as

Taking natural logarithms on both sides of the inequality(8),it is obtained that

By shifting items and residuation theory,one has that

and the causal feedback matrixKsuch that the maxproduct stochastic system (6) is mean norm exponentially stable,then the autonomous stochastic system (5) is stable in probability with a positive constantMxsuch that

Remark 2According to [31],state variables have corresponding exponential upper bounds in max-plus stochastic systems with a stochastic distribution.Therefore,it is feasible to further reduce the differencesxi(k)-kTbetween the state variables and reference signals in the max-plus stochastic system (3) with the specific stochastic distribution.Moreover,network attacks sometimes happen such that the time delays fluctuate in the cloud control system [32].That is,there has to be at least one additional possible realization for the Markov chainykof the max-plus stochastic system (3) with the network attacks.In this case,it is interesting to improve the proposed method for the Markovian jump cloud control system (1) under network attacks.

4.Simulation results

In this section,two frameworks of the Markovian jump cloud control system (1) are selected as structures whose Petri nets are shown in Fig.2 and Fig.3.A max-plus stochastic system (3) for numerical simulation is chosen as a Markovian jump cloud control system (1) with

and the initial statex(0)=[0 3 4]T.The Markov chain for the Markovian jump cloud control system (1) is obtained asyk={1,2} in which the transition probability matrixCand the observation probability matrix Φ are satisfied with

One of the possible realizations for the Markov chainykis shown in Fig.4.

Fig.4 Possible realization of Markov chain yk

Based on Theorems 2 and 3,there existK=[8 ε ε]TandT=10 in the causal feedback controlleru(k).Therefore,the differencexi(k)-kTon the Markov chainykis presented in Fig.5.

Fig.5 Differences xi(k)-kT in the sample 1 of the Markov chain yk

It is shown from Fig.5 that the Markovian jump cloud control system (1) is stable in probability with the proposed causal feedback controller (4).Moreover,tracking between the statex(k) and the reference signalr(k) are given in Fig.6.Based on the causal feedback controller(4),the inputu(k) is shown in Fig.7.

Fig.6 Tracking between x(k) and r(k)

Fig.7 Input u(k)

5.Conclusions

In this paper,a max-plus stochastic system with a Markov chain is established as the Markovian jump cloud control system.Based on residuation theory and exponential stability,a causal feedback matrix has been designed for a causal feedback controller to track a reference signal.For the causal feedback controller,a sufficient condition has been proposed for existence of the causal feedback matrix.By BIBipO stability analysis,stochastic stabilization has been shown for the Markovian jump cloud control system with the reference signal.Simulation results have shown effectiveness of the causal feedback controller for the Markovian jump cloud control system.