# Positivel1 State-Bounding Observer Design for Positive Interval Markovian Jump Systems

Authors: Di Zhang; Qingling Zhang; Borong Lyu
Journal: Mathematical Problems in Engineering
 (2016)

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## Abstract

This paper studies the problem of positivel1 state-bounding observer design for a class of positive Markovian jump systems with interval parameter uncertainties by a linear programming approach. For the first, necessary and sufficient conditions are obtained for stochastic stability and l1 performance of positive Markovian jump systems by an “equivalent” deterministic positive linear system. Furthermore, based on the results obtained in this paper, sufficient conditions for the existence of the positive l1 state-bounding observer are derived. The conditions can be solved in terms of linear programming. Finally, a numerical example is used to illustrate the effectiveness of the results obtained.


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## Body

## 1. Introduction

Positive systems whose state and output are nonnegative for any given nonnegative initial state and input have developed a new branch and play an important role in system theory. Positive systems are frequently used in communication, queue processes, and traffic modeling [1]. Recently, many contributions, such as realization, controllability, reachability and stability [2, 3], and positive filtering [4], have been highlighted by many researchers.As we know, if systems have their parameters or structures changed abruptly, it is necessary and natural to describe them as Markovian jump systems. Markovian jump systems have two mechanisms simultaneously. The first one is the time-evolving mechanism and related to the state vector. The second one called system mode is event-driven mechanism and driven by a Markov process taking values in a finite set. Some achievements on Markovian jump systems are given; for instance, see [5–10]. The conditions of stochastic stability on this kind of system are reported in [5–8]. When this system is positive, stochastic stability is investigated in [9, 10]. Also, there are many other results proposed, such as stabilisation [11], l1-gain performance analysis and positive filter design [12], H∞ filtering [13, 14], and l1 control [15]. Sometimes, it is not easy to obtain all the state variables in practical systems. It is necessary to design observer to estimate state variables. The observer design problems for positive systems have been considered in [16–20]. To the best of our knowledge, the observer design for positive Markovian jump systems has not been fully investigated, especially systems with interval parameter uncertainties. We know that the conventional observers estimate the state of the system in an asymptotic way. If we want to obtain the information of the transient state of positive interval Markovian jump systems, we need to design new observers, which motivate the current research.In this paper, we investigate the positivel1 state-bounding observer design problem for positive interval Markovian jump systems. The main contributions of this paper include the following. (1) By an “equivalent” deterministic positive linear system, necessary and sufficient conditions are obtained for stochastic stability and l1 performance of positive Markovian jump systems. (2) For positive interval Markovian jump systems, we design a new observer which is different from the traditional observer. Based on the proposed results, sufficient conditions for the existence of the positive l1 state-bounding observer are derived.The rest of the paper is organized as follows. Preliminaries are presented in Section2. Stochastic stability and l1 performance analysis problem are discussed in Section 3. In Section 4, observer problem of positive interval Markovian jump systems is studied. A numerical example is provided in Section 5. Conclusions are presented in Section 6.Notations. R is the set of real number. RnR+n is the n-dimensional real (nonnegative) vector space; Rp×m is the set of p×m real matrix. Z+ is the set of positive integer. Ω,F,P is a probability space where Ω is sample space, F is the σ-algebra of subsets of the sample space, and P is the probability measure. A⪰0 (A≻0, A⪯0, A≺0) means that all entries of matrix A are nonnegative (positive, nonpositive, and negative). A≻B (A⪰B) means A-B≻0 (A-B⪰0). E· means the mathematical expectation of ·. Ai∈A_i,Ai¯ means A_i⪯Ai⪯A¯i. 1-norm of vector xt∈Rn is denoted by xt1=∑k=1n‍xkt, where xkt is the kth component of xt∈Rn. The l1-norm of a Lebesgue integrable function xt is defined as xtl1=∫0∞‍xt1dt. The space of all vector-valued functions defined on R+n with finite l1-norm is denoted by l1R+n. Ir is the r-dimensional identity matrix. The transpose of a matrix or a vector is expressed as the superscript “T.” A block diagonal matrix with diagonal block A1,A2,…,Ar will be denoted by diag⁡A1,A2,…,Ar. The symbol 1n is the n-dimensional vector whose all entries are equal to 1. ⊗ denotes the Kronecker product.


## 2. Preliminaries

In the complete probability spaceΩ,F,P, we will consider a class of continuous-time Markovian jump systems described as follows:(1)x˙t=Artxt+Brtwtyt=Crtxt+Drtwt,where xt∈Rn is the system state vector, wt∈Rm is the input, and yt∈Rp is the output. For simplicity, when rt=i, the system matrices Art, Brt, Crt, and Drt are expressed as Ai, Bi, Ci, and Di. Ai∈Rn×n, Bi∈Rn×m, Ci∈Rp×n, and Di∈Rp×m belong to the following interval uncertainty domain:(2)Ai∈A_i,Ai¯,Bi∈B_i,Bi¯,Ci∈C_i,Ci¯,Di∈D_i,Di¯.The jump process rt,t≥0 is a homogeneous Markov process taking values in a finite set S=1,2,3,…,N. System (1) has the following mode transition probabilities:(3)Pr⁡rt+Δt=j∣rt=i=λijΔt+oΔti≠j1+λijΔt+oΔti=j,where Δt>0, limΔt→0oΔt/Δt=0, and λij≥0i,j∈S,i≠j denotes the transition rates from mode i at time t to mode j at time t+Δt, and ∑j=1,j≠iN‍λij=-λii. Furthermore, the transition rate matrix of the Markov process can be expressed as(4)Π=λ11λ12⋯λ1Nλ21λ22⋯λ2N⋮⋮⋱⋮λN1λN2⋯λNN.Definition 1 (see [1]).
System (1) is said to be positive if and only if xt∈R+n, yt∈R+p, for any initial condition x0∈R+n and wt∈R+m, ∀t≥0, ∀r0∈S.Definition 2 (see [10]).
System (1) is stochastically stable if the solution to system (1) for wt=0 satisfies E∫0∞‍xt1dt∣x0,r0<∞, where x0 is the initial condition and r0∈S.Lemma 3 (see [1]).
System (1) is positive if and only if Ai is Metzler matrix, Bi⪰0, Ci⪰0, and Di⪰0, i∈S.Remark 4.
A i, Bi, Ci, and Di belong to the following interval uncertainty domain: Ai∈A_i,Ai¯, Bi∈B_i,Bi¯, Ci∈C_i,Ci¯, and Di∈D_i,Di¯. Ai, Bi, Ci, and Di are uncertain, but A_i, B_i, C_i, and D_i are known. Due to A_i⪯Ai, B_i⪯Bi, C_i⪯Ci, and D_i⪯Di, if A_i is Metzler matrix, B_i⪰0, C_i⪰0, and D_i⪰0 for any i∈S, it is natural that system (1) is positive.Lemma 5 (see [21]).
Consider the following positive system:(5)x˙t=Axt+Bwtyt=Cxt+Dwt.The following statements are equivalent for wt=0:(i)
This system is asymptotically stable.(ii)
A is a Hurwitz.(iii)
There exists a vectorp≻0 such that pTA≺0.Definition 6 (see [19]).
Suppose that positive system (5) is stable; its l1-induced norm is defined as(6)Il1-l1≜supw≠0,w∈l1R+n⁡yl1wl1,where I:l1→l1 denotes the convolution operator; that is, yt=I⁎wt. System (5) has l1-induced performance at the level γ if, under zero initial conditions, Il1-l1<γ, where γ is a given scalar.Lemma 7 (see [4]).
Positive system (5) is asymptotically stable and satisfies yl1≤γwl1 if and only if there exists a vector p≻0 satisfying(7)1TC+pTA≺0pTB+1TD-γ1T≺0.Definition 8.
Suppose that positive system (1) is stable; its l1-induced norm is defined as(8)IMl1-l1≜supw≠0,w∈l1R+n,r0∈S⁡yl1wl1,where I:l1→l1 denotes the convolution operator; that is, yt=I⁎wt. System (1) has l1-induced performance at the level γ if, under zero initial conditions, IMl1-l1<γ, where γ is a given scalar.


## 3. Stochastic Stability andl1 Performance Analysis

In this section, we consider the stochastic stability and analyze thel1-induced performance for positive Markovian jump systems.Theorem 9.
The following statements are equivalent:(i)
Positive system (1) is stochastically stable.(ii)
There exist vectorsvi≻0, i∈S, such that (9)AiTvi+∑j=1Nλijvj≺0.(iii)
A~=ΠT⊗In+diag⁡A1,A2,…,AN is Hurwitz.Proof.
(i)⇔(iii) Define the indicator function as (2.2) in [22](10)1rt=iw=1rtw=i,i∈S0otherwise.Let qt=Ext, qit=Ext1rt=i. Then(11)qt=Ext=∑i=1NExt1rt=i=∑i=1Nqit.By [22], we have(12)q˙it=Aiqit+∑j=1Nλjiqjt.Note (13)q^t=q1Ttq2Tt⋯qNTtT,A~=ΠT⊗In+diag⁡A1,A2,…,AN.Then we obtain q^t that satisfies the following system:(14)q^˙t=A~q^tq^0=q1T0q2T0⋯qNT0T⪰0.Also we can conclude the following equation as (16) in [12]:(15)q^t1=1NnTq^t=1nT∑i=1Nqit=1nTExt=Ext1.Then we have(16)E∫0∞xt1dt∣x0,r0=∫0∞Ext1∣x0,r0dt=∫0∞q^t1dt<∞which implies limt→∞⁡q^t1=0; that is, for every q^0⪰0, limt→∞⁡q^t=0. Therefore, the stochastic stability of system (1) is equivalent to asymptotic stability of system (14). By Lemma 5, system (14) being asymptotically stable is equivalent to the matrix A~=ΠT⊗In+diag⁡A1,A2,…,AN being Hurwitz.
(ii)⇔(iii) By Lemma 5, the matrix A~=ΠT⊗In+diag⁡A1,A2,…,AN being Hurwitz is equivalent to the fact that there exists a vector p≻0 satisfying pTA~≺0. Let p=p1Tp2T⋯pnTT, where pi∈Rn; then(17)p1Tp2T⋯pnTA1+λ11Inλ21In⋯λN1Inλ12InA2+λ22In⋯λN2In⋮⋮⋱⋮λ1NInλ2NIn⋯AN+λNNIn≺0which implies piTAi+∑j=1N‍λijpjT≺0; it is equivalent to AiTpi+∑j=1N‍λijpj≺0. The conclusion holds. The proof is completed.Theorem 10.
For positive system (1) and a given γ>0, system (1) is stochastically stable and satisfies yl1≤γwl1 if and only if there exist vectors pi≻0, i∈S, such that(18)1TCi+piTAi+∑j=1NλijpjT≺0(19)piTBi+1TDi-γ1T≺0.Proof.
Define the indicator function as Theorem9. Let (20)z^t=z1Ttz2Tt⋯zNTtTw^t=w1Ttw2Tt⋯wNTtTzit=Eyt1rt=iwit=Ewt1rt=iA~=ΠT⊗In+diag⁡A1,A2,…,ANB~=diag⁡B1,B2,…,BNC~=diag⁡C1,C2,…,CND~=diag⁡D1,D2,…,DN.By [22], it follows that(21)dqit=Edxt1rt=i+xtd1rt=i=EAixt+Biwt1rt=idt+Extd1rt=i=AiExt1rt=idt+BiEwt1rt=idt+∑j=1Nλjiqjtdt=Aiqitdt+Biwitdt+∑j=1Nλjiqjtdtzit=Eyt1rt=i=ECixt+Diwt1rt=i=CiExt1rt=i+DiEwt1rt=i=Ciqit+Diwit.From (21), we obtain the following system:(22)q^˙t=A~q^t+B~w^tz^t=C~q^t+D~w^t.Since the proof of Theorem 9, system (22) is stable if and only if system (1) is stochastically stable. Next, we want to show the relationship of l1-induced performance between system (1) and system (22). Applying the similar way of (15), there are z^t1=Eyt1 and w^t1=Ewt1; then yl1≤γwl1 is equivalent to z^l1≤γw^l1. Therefore, system (1) is stochastically stable and satisfies yl1≤γwl1 if and only if system (22) is stable and satisfies z^l1≤γw^l1. By Lemma 7, there exists a vector p=p1Tp2T⋯pnTT⪰0 satisfying(23)1TC~+piTA~≺0piTB~+1TD~-γ1T≺0.Substitute A~, B~, C~, and D~ to (23); the conclusion is proved. The proof is completed.


## 4. Design of Observer

We know that the conventional observers estimate the state of the system in an asymptotic way. If we want to obtain the information of the transient state of positive interval Markovian jump systems, we need to design new observers. Therefore, we design a pair of positivel1 state-bounding observers that can bound the state xt all the time.For system (1), observers are considered as follows:(24)x⌢˙t=F¯ix⌢t+G¯iyt+K¯iwt(25)x˘˙t=F_ix˘t+G_iyt+K_iwt,where i∈S, x⌢t∈Rn, and x˘t∈Rn are the upper-bounding and lower-bounding estimated state of state xt; F¯i∈Rn×n, G¯i∈Rn×p, K¯i∈Rn×m, F_i∈Rn×m, G_i∈Rn×m, and K_i∈Rn×m are observer parameters to be determined.Definee⌢t=x⌢t-xt. By systems (1) and (24), we have(26)e⌢˙t=F¯i+G¯iCi-Aixt+F¯ie⌢t+G¯iDi+K¯i-Biwt.We let z⌢ot=Lie⌢t, where z⌢ot is the output of error state; Li⪰0 (i∈S) are known.Define(27)x⌢ot=xte⌢t,A¯oi=Ai0F¯i+G¯iCi-AiF¯iB¯oi=BiG¯iDi+K¯i-Bi,C¯oi=0Li.Then by (26) and (27), we have the system as follows:(28)x⌢˙ot=A¯oix⌢ot+B¯oiwtz⌢ot=C¯oix⌢ot.Observer (24) is designed for positive system (1) to approximate xt by x⌢t. Therefore, the estimate x⌢t is required to be positive; that is, the observer (24) is positive. By Lemma 3, we know it needs that F¯i is Metzler, G¯i⪰0 and K¯i⪰0.Therefore, the upper-bounding observer problem can be stated as follows: design a positive observer in the form of (24) such that system (28) is positive and stochastically stable and satisfies the performance z⌢ol1≤γwl1 under zero initial conditions.Similarly, definee˘t=xt-x˘t. By systems (1) and (25), we have(29)e˘˙t=Ai-G_iCi-F_ixt+F_ie˘t+Bi-G_iDi-K_iwt.We let z˘ot=Lie˘t, where zot is the output of error state; Li⪰0 (i∈S) are known. Define(30)x˘ot=xte˘t,A_oi=Ai0Ai-G_iCi-F_iF_iB_oi=BiBi-G_iDi-K_i,C_oi=0Li.Then by (29) and (30), we have the system as follows:(31)x˘˙ot=A_oixot+B_oiwtz˘ot=C_oixot.Therefore, the lower-bounding observer problem can be stated as follows: design a positive observer in the form of (25) such that system (31) is positive and stochastically stable and satisfies the performance z˘ol1≤γwl1 under zero initial conditions.Next, we give the existence condition of the upper-bounding and lower-bounding observer. Before giving the condition, we denote system matrix as follows:(32)A¯i=A¯i1A¯i2⋯A¯inTA_i=A_i1A_i2⋯A_inTB¯i=B¯i1B¯i2⋯B¯inTB_i=B_i1B_i2⋯B_inTC¯i=C¯i1C¯i2⋯C¯inC_i=C_i1C_i2⋯C_in,where A¯it∈Rn, B¯it∈Rm, C¯it∈Rp, A_it∈Rn, B_it∈Rm, C_it∈Rp, i∈S, t=1,2,…,n, s=1,2,…,m.Theorem 11.
Consider positive system (1). For a given γ>0, there exists positive upper-bounding observer (24) such that system (28) is positive and stochastically stable and satisfies z⌢ol1≤γwl1 if there exist Metzler matrix η¯i=η¯i1η¯i2⋯η¯inT, ξ¯i=ξ¯i1ξ¯i2⋯ξ¯inT⪰0, θ¯i=θ¯i1θ¯i2⋯θ¯inT⪰0, αi≻0, and βi=βi1βi2⋯βinT≻0 with αi∈Rn, βi∈Rn, η¯it∈Rn, ξ¯it∈Rp, θ¯it∈Rm, i∈S, t=1,2,…,n, such that(33)η¯itT+ξ¯itTC_i-βitA¯itT⪰0(34)ξ¯itTD_i+θ¯itT-βitB¯itT⪰0(35)αiTA¯i+∑j=1nη¯ijT+∑j=1nξ¯ijTC¯i-βiTA_i+∑j=1NλijαjT≺0(36)1TLi+∑j=1nη¯ijT+∑j=1NλijβjT≺0(37)αiTB¯i+∑j=1nξ¯ijTD¯i+∑j=1nθ¯ijT-βiTB_i-γ1T≺0.Then, the parameters of the observer are given by(38)G¯i=G¯i1G¯i2⋯G¯inT=βi1-1ξ¯i1βi2-1ξ¯i2⋯βin-1ξ¯inTF¯i=F¯i1F¯i2⋯F¯inT=βi1-1η¯i1βi2-1η¯i2⋯βin-1η¯inTK¯i=K¯i1K¯i2⋯K¯inT=βi1-1θ¯i1βi2-1θ¯i2⋯βin-1θ¯inT.Proof.
Sinceβi≻0, ξ¯i⪰0, and θ¯i⪰0 and η¯i is Metzler, it follows that F¯i is Metzler, G¯i⪰0 and K¯i⪰0 from (38). Thus, observer (24) is positive.
From (33), (34), and βi≻0, we obtain(39)βit-1η¯itT+βit-1ξ¯itTC_i-A¯itT⪰0βit-1ξ¯itTD_i+βit-1θ¯itT-B¯itT⪰0.By (38), we have(40)F¯itT+G¯itTC_i-A¯itT⪰0G¯itTD_i+K¯itT-B¯itT⪰0.Further, we obtain(41)F¯i+G¯iC_i-A¯i⪰0G¯iD_i+K¯i-B¯i⪰0.From (41) and G¯i⪰0, it follows that, for any Ai∈A_i,Ai¯, Bi∈B_i,Bi¯, Ci∈C_i,Ci¯, and Di∈D_i,Di¯,(42)F¯i+G¯iCi-Ai⪰F¯i+G¯iC_i-A¯i⪰0G¯iDi+K¯i-Bi⪰G¯iD_i+K¯i-B¯i⪰0which imply A¯oi is Metzler and B¯oi⪰0 in (27); therefore, system (28) is positive.
From (38), we have(43)∑j=1nξ¯ijT=∑j=1nβijG¯ijT=βiTG¯i∑j=1nη¯ijT=∑j=1nβijF¯ijT=βiTF¯i∑j=1nθ¯ijT=∑j=1nβijK¯ijT=βiTK¯i.According to (43), (35)–(37) become(44)αiTA¯i+βiTF¯i+βiTG¯iC¯i-βiTA_i+∑j=1NλijαjT≺01TLi+βiTF¯i+∑j=1NλijβjT≺0αiTB¯i+βiTG¯iD¯i+βiTK¯i-βiTB_i-γ1T≺0which imply that(45)1T0Li+piTA¯i0F¯i+G¯iC¯i-A_iF¯i+∑j=1NλijpjT≺0piTB¯iG¯iD¯i+K¯i-B_i-γ1T≺0,where piT=αiTβiT. For any Ai∈A_i,Ai¯, Bi∈B_i,Bi¯, Ci∈C_i,Ci¯, and Di∈D_i,Di¯, we obtain(46)F¯i+G¯iCi-Ai⪯F¯i+G¯iC¯i-A_iG¯iDi+K¯i-Bi⪯G¯iD¯i+K¯i-B_i.Further, we have(47)1T0Li+piTAi0F¯i+G¯iCi-AiF¯i+∑j=1NλijpjT≺0piTBiG¯iDi+K¯i-Bi-γ1T≺0.By Theorem 10, system (28) is stochastically stable and satisfies yl1≤γwl1. The proof is completed.Similarly, we give the existence condition of the lower-bounding observer.Theorem 12.
Consider positive system (1). For a given γ>0, there exists positive lower-bounding observer (25) such that system (31) is positive and stochastically stable and satisfies z˘ol1≤γwl1 if there exist Metzler matrix η_i=η_i1η_i2⋯η_inT, ξ_i=ξ_i1ξ_i2⋯ξ_inT⪰0, θ_i=θ_i1θ_i2⋯θ_inT⪰0, and αi≻0, βi=βi1βi2⋯βinT≻0 with αi∈Rn, βi∈Rn, η_it∈Rn, ξ_it∈Rp, θ_it∈Rm, i∈S, t=1,2,…,n, such that(48)βitA_itT-ξ_itTC¯i-η_itT⪰0βitB_itT-ξ_itTD¯i-θ_itT⪰0αiTA¯i+βiTA¯i-∑j=1nξ_ijTC_i-∑j=1nη_ijT+∑j=1NλijαjT≺0∑j=1nη_ijT+∑j=1NλijβjT+1TLi≺0αiTB¯i+βiTB¯i-∑j=1nξ_ijTD_i-∑j=1nθ_ijT-γ1T≺0.Then, the parameters of the observer are given by(49)G_i=G_i1G_i2⋯G_inT=βi1-1ξ_i1βi2-1ξ_i2⋯βin-1ξ_inTF_i=F_i1F_i2⋯F_inT=βi1-1η_i1βi2-1η_i2⋯βin-1η_inTK_i=K_i1K_i2⋯K_inT=βi1-1θ_i1βi2-1θ_i2⋯βin-1θ_inT.


## 5. Numerical Examples

Consider a three-dimensional continuous-time uncertain Markovian jump system of form (1) with rt∈S=1,2, and its parameters are given by(50)A1=-1.5±0.010.10.50.5-2±0.010.20.10.7±0.02-0.9,A2=-1.4±0.020.20.4±0.010.7-1.50.30.20.5-0.8±0.03,B1=0.20.50.3,B2=0.10.20.3±0.01,C1=0.90.81±0.1,C2=0.8±0.030.90.9,D1=1,D2=1.2±0.02.The transition rate matrix is given as (51)Π=-0.50.50.4-0.4.Here, we choose L1=0.20.30.4, L2=0.30.20.4 and assume that γ=0.1. Solving the LP problem in Theorems 11 and 12, the parameters of the positive upper-bounding observer and lower-bounding observer are given by(52)G_1=0.02420.01730.0080T,G_2=0.02330.01300.0057TF_1=-1.53150.08070.47330.4066-2.02370.18100.07220.6736-0.9088F_2=-1.43930.17910.36910.6892-1.51170.28830.19530.4949-0.8051K_1=0.17570.48260.2920T,K_2=0.07160.18420.2831TG¯1=0.00170.00110.0007T,G¯2=0.12440.00050.0100TF¯1=-1.49140.09870.49840.4990-1.98900.19900.10060.7206-0.9006F¯2=-1.47580.12300.29790.7010-1.49120.29950.22300.7160-0.7790K¯1=0.19840.49890.2993T,K¯2=00.19930.2980T.With input wt=2.3e-tsin⁡4t and the initial conditions x0=000, we have the simulation of system mode shown in Figure 1. The system state, upper-bounding, and lower-bounding estimated states are showed in Figures 2, 3, and 4.Figure 1
The simulation of system modert.Figure 2
System state and estimated system statex1(t).Figure 3
System state and estimated system statex2(t).Figure 4
System state and estimated system statex3(t).


## 6. Conclusions

In this paper, positivel1 state-bounding observer design for a class of positive Markovian jump systems with interval parameter uncertainties is investigated. First, necessary and sufficient conditions are obtained for stochastic stability and l1 performance of positive Markovian jump systems by an “equivalent” deterministic positive linear system. Then based on the proposed results, sufficient conditions for existence of the positive l1 state-bounding observer are derived. The conditions can be solved in terms of linear programming. Finally, a numerical example is used to demonstrate the effectiveness of the proposed results.So far, many results on Markov jump systems have been applied to networked control systems, such as [23–27]. However, the state of networked control systems may need to be nonnegative. Therefore, the results of positive Markov jump systems can also be applied to networked control systems, which have been proposed in a few papers, such as [20]. In the future, we can try our best to extend our results to deal with the problem of networked control systems.



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