# A Random Access Protocol Approach to Stochastic Jumping Systems with Singular Perturbations

Authors: Saijiang Kai
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

This paper focuses on the stochastic jumping systems with singular perturbation subject to a random access protocol. The key challenge with controller design issue of stochastic jumping systems is how to assess the coordination of communication orders. In this study, a joint Markov process is established, and a novel control law is proposed. In contrast with the existing methods, the developed controller is more general. Finally, a practical example is exhibited to show the effectiveness of the achieved theories.


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## Body

## 1. Introduction

In practical, random faults, random communication failures, and so on are commonly encountered. In these systems, the effect of unpredictable random abrupt events can be modeled as a Markov chain model. Markov jumping systems (MJSs) consist of a series of submodels, in which the switching is determined by the Markov process. In recent years, MJSs have been successfully applied in various areas such as biological systems, networked control systems, hybrid systems, and economic systems [1–3]. Due to the great advantage of MJSs, up to now, many valuable results on the issues of MJSs have been published [4–8], including state estimation [5] and robust control [7, 8].On the contrary, there exists a set of practical systems, whose dynamic behavior is characterized by the multiple time scales (MTSs) property. Especially, in the MTS-based systems, the small parameters lead to difficulty in system performance analysis. Therefore, it is unsolvable by means of the conventional single time-scale technique. It is well known that MTS-based systems can be described by singularly perturbed systems (SPSs). In light of its unique characteristic, a singularly perturbed parameter (SPP) is employed, where the MTS-based systems can be divided into fast and slow dynamic states [9–11]. Recently, quantities of excellent results on SPSs have been exhibited [12]. However, when it comes to Markov jumping SPSs (MJSPSs), the investigation of relevant problems is far away from maturity. How to tackle the sophisticated systems with both SPP and transition probabilities partly is the motive of this study.Note that the information exchange in the networked control systems may result in the heavy communication burden. These limited bandwidths may result in network-induced phenomena, such as data collisions, packet losses, and network-induced delay [13]. Recently, the protocol-based networked systems have attracted increasing interest, for instance, try-once-discard protocol [14] and round-robin protocol (RRP) [15]. Note that the RRP is a static transmission mechanism, which limits the practical application. Compared with the RRP, the random access protocol (RAP) is a dynamic case, in which the sensor node will get access to transmit data in a randomly way [16, 17]. However, to the best of our knowledge, the RAP has not been applied in MJSPSs yet, which partly motives us to fill this gap.Inspired by the above discussion, it is imperative to study the problem of MJSPSs with RAP. To avoid data collisions, the RAP is introduced to regulate the data transmission order. With respect to the dynamic behavior of the MJSPS and the RAP, a joint Markov process is proposed. Therefore, a novel control design technique is presented, which covers the conventional case. Finally, a practical simulation is given to show the effectiveness of the obtained theories.Notations. The notations are fairly standard. ℝm refers to the m-dimensional Euclidean space. λminL denotes the minimum eigenvalue of L. diag… represents a block diagonal matrix. mody1,y2 stands for the nonnegative remainder on division of y1 by y2.


## 2. Problem Formulations

Consider the following MSSPS described as follows:(1)α1k+1=Aμk11α1k+εAμk12α2k+Bμk1uk+Cμk1υk,α2k+1=Aμk21α1k+εAμk22α2k+Bμk2uk+Cμk2υk,zk=Dμk1α1k+εDμk2α2k+Fμkuk+Hμkυk,where α1k∈ℝns means the fast vector and α2k∈ℝnf stands for the slow vector, respectively. uk∈ℝnu symbolizes the control input, zk∈ℝnz signifies the output state, ε>0 symbolizes a singular perturbation parameter, and υk∈l20,∞ renders the disturbance signal. The stochastic variable μk,k≥0 indicates a discrete-time Markov chain (DTMC) obeying a finite set N=1,2,…,N. Clearly, the transition probability matrix (TPM) of μk is defined by Π=φpqN×N, where(2)φpq=Prμk+1=q|μk=p,where φpq≥0 and ∑q∈Nφpq=1,∀p,q∈N. Therefore, it is easy to derive that ∀p∈N, and Aμk11, Aμk12, Aμk21, Aμk22, Bμk1, Bμk2, Cμk1, Cμk2, Dμk1, Dμk2, Fμk, and Hμk can be represented by Ap11, Ap12, Ap21, Ap22, Bp1, Bp2, Cp1, Cp2, Dp1, Dp2, Mp, and Gp, respectively.Letαk=α1⊤kα2⊤k⊤, and the MSSPS (1) can be rewritten as(3)αk+1=ApEεαk+Bpuk+Cpυk,zk=DpEεαk+Mpuk+Gpυk,where(4)Eε=diagIns,εInf,Ap=Ap11Ap12Ap21Ap22,Bp=Bp1Bp2,Cp=Cp1Cp2,Dp=Dp1Dp2.The signal transmission between the plant and the controller via a shared communication networks, which may result in many network-induced phenomena, for example, data collisions. To regulate the data transmission, the RAP is applied between the sensors and the actuators. When the RAP is triggered, only one node can access network at each time instant. Note that the RAP is a dynamic scheduling agreement. Employ a stochastic variableνk∈1,2,…,M to schedule network resources. Here, νk is known as a homogeneous DTMC obeying a set Nc=1,2,…,M, and the TPM Φ=ψfgM×M is inferred as(5)ψfg=Prνk+1=g|νk=f,where ∀f,g∈ℳ, ψfg∈0,1, and ∑g∈ℳψfg=1.Denotingvk=v1⊤kv2⊤k…vnu⊤k and uk=u1⊤ku2⊤k…unu⊤k, where vmk signifies the fth control input state and un indicates the nth actuator. By considering the zero-order hold technique, the fth actuator ufk can be updated as(6)ufk=vfk,ifνk=f,ufk−1,otherwise.In what follows, a Kronecker sign function is given by(7)δF−ℶ=1,ifF=ℶ,0,otherwise.Therefore, by the updated rule (6), for ∀f∈ℳ, the actuator uk is formulated as(8)uk=Φνkvk+I−Φνkuk−1,where Φf≜diagδf1,δf2,…,δfnuf=1,2,…,nu and δϜℶ=δF−ℶ. Clearly, it can be devised from (8) that uk=u1k−1u2k−1…uf−1kvfkuf+1k…unuk⊤.It can be observed from (3) and (8) that two Markov processes μk and νk are involved in this paper. To reveal the relationship between the Markov processes (MPs) μk and νk, the merging strategy is absorbed in Lemma 1.Lemma 1 (see [4]).
The MPsμk and νk can be mapped to a novel MP rk∈1,2,…,S, which can be described as follows:(9)rk=Rμk,νk=μk+νk−1N,where S=NM.
In virtue ofξk, the values of μk and νk could be acquired by ζ1rk and ζ2rk:(10)μk=ζ1rk=modrk−1,N+1,νk=ζ2rk=rk−1N+1.
Obviously, the values ofrk can be determined by the pair μk,νk. The joint TPM of the MP rk is expressed as(11)θhl=Prrk+1=l|rk=h=Prφk+1=ζ1l|φk=ζ1h×Prψk+1=ζ2l|ψk=ζ2h=πζ1hζ1lτζ2hζ2l,where πζ1hζ1l and τζ2hζ2l are in (2) and (4).
By resorting to the joint MP, the control lawvk is designed as(12)vk=KrkEεxk,where Krk are matrices being solved.Remark 1.
The matrixKrk in (12) is assumed to be dependent on the joint Markov process rk, which implies that Krk is characterized by both the corresponding mode μk and the RAP case νk. Otherwise, the controller gain is reduced to Kμk,νk, which will rise the difficulty in the MSSPS analysis.
According to (3), (8), and (12), the augmented MSPS (13) is established:(13)xk+1=AhEεxk+BiI−Ψmuk−1+Ciυk,zk=DhEεxk+FiI−Ψmuk−1+Hiυk,where(14)Ah=Aim=Ai+BiΨmKh,Dh=Dim=Di+FiΨmKh.Definition 1.
(see [18]). The MSSPS (13) with υk=0 is said to be stochastic stable (SS), if for any δ0,ϑ0, such that(15)E∑k=0∞δk2|δ0,ϑ0<∞.Definition 2.
(see [18]). The MSSPS (13) with is called SS with a ℋ∞ performance level γ, if the MSSPS (11) is SS, and under zero initial condition, one gets(16)∑k=0∞Eδk2<γ2∑k=0∞Eϱk2.Lemma 2 (see [11]).
For symmetric matricesZtt=1,2,3,4 and matrix Z5 satisfies(17)Z1>0,Z1+ϵ¯Z3ϵ¯Z5⊤∗ϵ¯Z2>0,Z1+ϵ¯Z3ϵ¯Z5⊤∗ϵ¯Z2+ϵ¯2Z4>0,such that EϵZϵ=Zϵ⊤Eϵ>0 for all ϵ∈0,ϵ¯, where(18)Zϵ=Z1+ϵZ3ϵZ5⊤∗ϵZ2+ϵ2Z4.Lemma 3 (see [11]).
For given a scalarϵ¯>0, W1, W2, and W3 are with suitable dimensions. For all ϵ∈0,ϵ¯, if W1+ϵW1+ϵ2W1>0, one gets(19)W1>0,W1+ϵ¯W1>0,W1+ϵ¯W2+ϵ¯2W3>0.


## 3. Main Results

Theorem 1.
The augmented MSSPS (13) is said to be SS with a given ℋ∞ performance, if there exists symmetric matrices P¯h>0, Q¯h>0, Z1, Z2, Z3, Z4, and Z5, K¯ϵf, one has(20)Z1>0,(21)Z1+ϵ¯Z3ϵ¯Z5⊤ϵ¯Z5ϵ¯Z2>0,(22)Z1+ϵ¯Z3ϵ¯Z5⊤ϵ¯Z5ϵ¯Z2+ϵ¯2Z4>0,(23)Θh1tΘh2t⊤ThΘh3t⊤ThΘh4t⊤∗Θh500∗∗Θh60∗∗∗−I<0,t=ℓ,ı,j.
Meanwhile, the controller gains can be achieved:(24)Kh=K¯εhY1⊤+ϵY1⊤−1,h∈S,where(25)Θh1ℓ=diagP¯h−V1+V1⊤,Q¯h−Yh+Yh⊤,−γ2I,Θh1ı=diagP¯h−Zɛ¯+Zɛ¯⊤,Q¯h−Yh+Yh⊤,−γ2I,Θh1j=diagP¯h−Zɛ¯+Zɛ¯⊤,Q¯h−Yh+Yh⊤,−γ2I,Θh2ℓ=AiY1+BiΨhK¯εfE0BiI−ΨhYhCi,Θh2ı=AiY2+BiΨhK¯εfEɛ¯BiI−ΨhYhCi,Θh2j=AiY3+BiΨhK¯εfEɛ¯BiI−ΨhYhCi,Θh3ℓ=ΨhK¯ɛfE0I−ΨhYh0,Θh3ı=Θh3j=ΨhK¯ɛfEɛ¯I−ΨhYh0,Θh4ℓ=DiY1+FiΨhK¯ɛfE0FiI−ΨhYhHi,Θh4ı=DiY2+FiΨhK¯ɛfEɛ¯FiI−ΨhYhHi,Θh4j=DiY3+FiΨhK¯ɛfEɛ¯FiI−ΨhZɛ¯Hi,Θh5=diag−P¯1,−P¯2,…,−P¯S,Θh6=diag−Q¯1,−Q¯2,…,−Q¯S,Th=θf1Iθf2I…θfNrI,Zε¯=V1+ɛ¯V2,Y1=W1,Y2=W1+ɛ¯W2,Y3=W1+ɛ¯W2+ɛ¯2W3,V1=Z10Z5Z2,V2=Z3Z5⊤0Z4,W1=Z1000,W2=Z3Z5⊤Z5Z2,W3=000Z4,E0=Ins000,Eɛ¯=Ins00ɛ¯Inh.Proof.
Recalling the LMIs (20)–(22) and by means of Lemma 2, for all ϵ∈0,ϵ¯, we have(26)Θ˜h1Θ˜h2⊤ThΘ˜h3⊤ThΘ˜h4⊤∗Θh500∗∗Θh60∗∗∗−I<0,where(27)Θ˜h1=diagP¯h−Zϵ+Zϵ⊤,Q¯h−Zϵ+Zϵ⊤,−γ2I,Θ˜h2=AiEϵZϵ+BiΨhKhZϵ⊤EϵBiI−ΨhZϵCi,Θ˜h3=ΨhKhZϵ⊤EϵI−ΨhZϵ0,Θ˜h4=DiEϵZϵ+FiΨhKhZϵ⊤EϵFiI−ΨhZϵHi.
By virtue of Lemma3 and the LMIs (13)–(21), for all ϵ∈0,ϵ¯, one derives the fact that EϵZϵ=Zϵ⊤Eϵ>0. Furthermore, it is well recognized that the following inequalities Zϵ⊤−P¯hPhZϵ−P¯h≥0 and Zϵ⊤−Q¯hQhZϵ−Q¯h≥0 hold. Denote P¯h=Ph−1 and Q¯h=Qh−1, and we obtain(28)Θ^h1Θ^h2⊤ThΘ^h3⊤ThΘ^h4⊤∗Θh500∗∗Θh60∗∗∗−I<0,where(29)Θ^h1=diag−Zϵ⊤PhZϵ,−Zϵ⊤QhZϵ,−γ2I,Θ^h2=AhEϵZϵBiI−ΨhZϵCi,Θ^h3=ΨhKhEϵZϵI−ΨhZϵ0,Θ^h4=DhEϵZϵFiI−ΨhZϵHi,Θ^h5=diag−P1−1,−P2−1,…,−PS−1,Θ^h6=diag−Q1−1,−Q2−1,…,−QS−1.
Recalling (28), with the help of term, diagZϵ−⊤,Zϵ−⊤,I,…,I and its transpose, for any Zϵ=Y1+ϵY2, the following inequality holds:(30)Θ¯h1Θ¯h2⊤ThΘ¯h3⊤ThΘ¯h4⊤∗Θh500∗∗Θh60∗∗∗−I<0,where(31)Θ¯h1=diag−Ph,−Qh,−γ2I,Θ¯h2=AhEϵBiI−ΨhCi,Θ¯h3=ΨhKhEϵI−Ψh0,Θ¯h4=DhEϵFiI−ΨhHi.
Next, choose a Lyapunov functional candidate as(32)Vk,xk,uk,rk=x⊤kPrkxk+u⊤k−1Qrkuk−1.
Calculating the time derivative of (32), one gets Vk,xk,uk,rk, and one has(33)EVk+1,xk+1,uk+1,rk+1=g|k,xk,uk,f,−Vk,xk,uk,rk,x⊤k+1Phxk+1−x⊤kPhxk+u⊤kQhuk,−u⊤k−1Qhuk−1,x⊤kPh+EɛAh⊤PhAhEɛxk+x⊤kEɛAh⊤PhBiI−Ψhuk−1,+u⊤k−1I−ΨhBi⊤PhAhEɛxk,+x⊤kEɛAh⊤PhCiυk+υ⊤kCi⊤PhAhEɛxk,+u⊤k−1I−Ψh⊤Bi⊤PhBiI−Ψhuk−1,+u⊤k−1I−ΨhBi⊤PhCiυk,+υ⊤kCi⊤PhBiI−Ψhuk−1+υ⊤kCi⊤PhCiυk,+ΨhKhEɛxk+I−Ψhuk−1⊤Qh,×ΨhKhEɛxk+I−Ψhuk−1,+x⊤kEɛKh⊤Ψh⊤QhΨhKhEɛxk,+x⊤kEɛKh⊤Ψh⊤QhI−Ψhuk−1,+u⊤k−1I−ΨhQhΨhKhEɛxk,+u⊤k−1I−ΨhQhI−Ψhuk−1.where Ph=∑g∈NrθhlPg,Qh=∑g∈NrθhlQg.
Note that inequality (33) can be rewritten as(34)EΔVk=ϑ⊤kΘ⟶hϑk,where(35)ϑk=x⊤ku⊤k−1υ⊤k,Θ⟶h1=diag−Ph,−Qh,0,Θ⟶h=Θ⟶h1+Θ¯h2⊤PhΘ¯h2+Θ¯h3⊤QhΘ¯h3.
In case ofυk=0, by resorting to (34), it is clear that(36)EΔVk≤ϑ¯⊤kΘ¯hϑ¯k≤−χExk2,where ϑ¯k=x⊤ku⊤k−1, Θ¯h=Θ¯h1+Θ¯h2⊤PhΘ¯h2+Θ¯h3⊤QhΘ¯h3, Θ¯h1=diag−Ph,−Qh, Θ¯h2=AhEϵBiI−Ψh, Θ¯h3=DhEϵFiI−Ψh, and ϖ=minf∈S,p,q∈1,2,…,rλminΘ¯h. Apparently, condition (30) implies ϖ>0. Therefore, we have(37)E∑k=0∞xk2<−1ϖE∑k=0∞ΔVk,≤1ϖEV0,x0,u0,r0<∞.
By Definition1, in case of υk=0, one concludes that the MSSPS (13) is SS.
In the following, whenυk≠0, we first denoting the H∞ performance as(38)JT=E∑k=0Tz⊤kzk−γ2υ⊤kυk.
Taking (34) and (38) into consideration, JT is shown as follows:(39)JT≤E∑k=0Tz⊤kzk−γ2υ⊤kυk+ΔVk≤ϑ⊤kΘh'ϑk,where(40)Θh'=Θ⟶h+Θh4⊤Θh4,Θh4=DhEϵFiI−ΨhHi.
Clearly, it is easy to derive the following condition from (30) and (39) that(41)JT<0.
LettingT⟶∞, it can be derived from (41) that(42)∑k=0∞Ezk2≤γ2∑k=0∞Eυk2.
Therefore, recalling the Definition2, we can conclude that the MSSPS (13) is SS with ℋ∞ performance. This ends our proof.Remark 2.
Remarkably, in updated actuatoruk, the compensation strategy is applied to schedule the information exchange. By omitting the compensation scheme, ∀m=1,2,…,nu, the received measurement signal (RMS) can be debased into(43)ufk=vfk,ifνk=m,0,otherwise.
Consequently, the RMS is degraded as(44)uk=Ψνkvk.
Accordingly, the closed-loop FMSSPS (13) will be reformulated as(45)xk+1=AiEϵxk+BiΨhKhEϵxk+Ciυk,zk=DiEϵxk+FiΨhKhEϵxk+Hiυk.
To exploit the dynamic behavior of the augmented MSSPS (45), the sufficient condition is forwarded in Corollary 1.Corollary 1.
The augmented MSSPS (45) is called SS with a prescribed ℋ∞ performance level γ, if there exists symmetric matrices P¯h>0, Q¯h>0, Z1, Z2, Z3, Z4, and matrices Z5, K¯ϵf, such that (20)–(22) hold and(46)Θh1tΘ¯h2t⊤ThΘ¯h4t⊤∗Θh50∗∗−I<0,t=ℓ,ı,j.
Meanwhile, theϵ-dependent controller gains are achieved as(47)Kh=K¯εfY1⊤+ɛY1⊤−1,where(48)Θ¯h2ℓ=AiY1+BiΨhK¯εfE0Ci,Θ¯h2ı=AiY2+BiΨhK¯εfEϵ¯Ci,Θ¯h2j=AiY3+BiΨhK¯εfEϵ¯Ci,Θ¯h4ℓ=DiY1+FiΨhK¯ϵfZϵ⊤E0Hi,Θ¯h4ı=DiY2+FiΨhK¯ϵfZϵ⊤Eϵ¯Hi,Θ¯h4j=DiY3+FiΨhK¯ϵfZϵ⊤Eϵ¯Hi.


## 4. Numerical Examples

Example 1.
(a dc motor via gear train model (DMGTM)).
To explain the validity of the attained scheme, a practical DMGTM is borrowed from [19]. The dynamic equation is governed by(49)θ.pt=x2t,θ..pt=glsinθpt+NKhml2Iat,LaI.at=−KbNθ.−RiIat+u1t+u2t+ωt,where the physical meaning of (49) is expressed in Table 1. The parameters are elicited as g=9.8h/s2, Kh=0.1Nm/s, Kb=0.1Vs/rad, Nr=10, l=1h, and La=50mH. For any i=1,2, letting x1t=θpt, x2t=θ.pt, and x3t=Iat, xt=x1⊤tx2⊤tx3⊤t⊤ and ϵ=La, and one gets(50)Eϵx˙t=Aixt+Biut+Ciωt,where(51)A1=010gl0NKhml20−KbNr−Ri,Bi=000011,Ci=001⊤,i=1,2,Eϵ=diag1,1,ϵ.
By means of a zero-order and sampling periodT=0.05s, (49) can be discretized into the following model:(52)xk+1=AiEɛxk+Biuk+Ciυk,where(53)A1=1.01220.04990.01840.48880.99390.6302−0.1800−0.63027.0938,A2=1.01220.04990.01420.48940.99810.4316−0.1389−0.43162.5584,B1=0.00030.00030.01840.01840.62690.6269,B2=0.00030.00030.01420.01420.42900.4290,C1=0.00030.01840.6269,C2=0.00030.01840.6269,D1=111,D2=111,F1=0.20.3,F2=0.30.7,H1=0.5,H2=0.4.
The TPM of the corresponding FMSSPS (1) is selected as Π=0.250.750.650.35. For another MP Φ in the RAP (6), the TPM is chosen as Φ=0.40.60.50.5. Consequently, by virtue of Lemma 1, the joint TPM can be inferred as(54)Θ=0.10.30.150.450.260.140.390.210.1250.3750.1250.3750.3250.1750.3250.175.
Lettingϵ=0.05 and γ=5. In view of LMIs of Theorem 1, the controller gains Kh can be listed as follows:(55)K1=−143.230930.83979.2890000,K2=−149.081427.72777.4494000,K3=000−151.469135.29649.9320,K4=000−154.408628.51137.4207.
In this section, according to the TPMs of corresponding plant and RAP, Figure1 depicts the possible mode switching of Markov process rk. The evolution of states x1k, x2k, and x3k over 50 realizations is depicted in Figures 2–4, respectively. With the designed control law, the control input over 50 realizations is shown in Figure 5. Meanwhile, the evolution of the output zk over 50 realizations is shown in Figure 6. From the above results, it is obvious that the RAP is effective in regulating the data transmission.Table 1
The meaning of parameters.
θptAngle of the pendulumIatCurrent of the motorKmMotor torque constantlLength of shaftRii=1,2Resistanceθ.ptAngular velocitygGravitational constantKbBack emf constantNrGear ratioLaInductanceFigure 1
The possible mode switching of MPrk.Figure 2
The evolution of statesx1k.Figure 3
The evolution of statesx2k over 50 realizations.Figure 4
The evolution of statesx3k over 50 realizations.Figure 5
The control inputuk over 50 realizations.Figure 6
The outputzk over 50 realizations.


## 5. Conclusions

This paper has studied a type of MJSPS with RAP. A dynamic RAP was adopted to regulate the data transmission, in which only one sensor node has gained the access to transmit data each time. Furthermore, a joint Markov process is established, and sufficient conditions are achieved. Finally, a practical example has been exhibited to show the effectiveness of the achieved theories. Besides, to extend the achieved results to sliding mode-based filter/controller is an issue [20, 21], and we will devote ourselves to tacking in near future.



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