# Interval State Estimation of Linear Multicellular Systems

Authors: Zhaoxia Huang; Jun Liu; Fucai Qian
Journal: Complexity
 (2019)

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## Abstract

Linear multicellular system is a type of differential inclusion system, which can be deemed as an extension of linear control system with set-valued mapping. As an important issue in existing control systems, interval state estimation has been widely applied in engineering practices. Over the years, the objects of the studies on interval state estimation have been extended from the initial linear time-invariant systems to linear time-varying systems, chaotic systems, feedback linearization systems, and nonlinear Lipschitz systems. However, there is no report on the design of interval observer for linear multicellular system. To make up for this gap, this chapter attempts to explore the design of an interval observer for linear multicellular systems.


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## Body

## 1. Introduction

With the rapid development of science and technology, the traditional state estimation theory can no longer accurately estimate the system state, due to the heavy presence of uncertainties in actual systems. The interval observer came into being under this background. In recent years, scholars have paid much attention to the interval observer, especially its applicability.The earliest interval observers were designed for the state monitoring of such biochemical systems as biochemical reactions, wastewater treatment systems, and vehicle control systems. It is impossible to make an accurate model for any of these systems, as they receive uncertain inputs and suffer from external disturbances with unknown statistical laws. In this case, the observation error rarely converges to zero if the observer is constructed according to the traditional method. By contrast, the interval observer can measure the real-time upper and lower bounds of the state change at any time, thus fulfilling the control requirements.The design of interval observer, a core issue in control system design, has been widely applied in engineering practice and intensively discussed in the academia [1–10]. It is generally agreed that the main obstacle in the observer design lies in the wrong assumption that the error observation system is cooperative. In fact, most systems are not cooperative, except for a few monotonous systems. Important progress was made in the study of interval observers around 2010. It is found that system coordination is related to the selected coordinates; under certain conditions, noncooperative systems can be transformed into cooperative ones. The existing designs of interval observers are based on one of the following two theories: the monotonous theory and the positive system theory. The research objects have been extended from the initial linear time-invariant systems to linear time-varying systems, chaotic systems, feedback linearization systems, and nonlinear Lipschitz systems.Linear multicellular system is a type of differential inclusion system, which can be deemed as an extension of linear control system with set-valued mapping. However, there is no report on the design of interval observer for linear multicellular system. Thus, it is very meaningful to make up for this research gap.Considering the uncertain parameters of multicellular system, this chapter gives a rational definition to the interval observer of multicellular system, converts the system into the linear parameter varying (LPV) form, and designs an interval observer for the LPV system. The proposed design algorithm can effectively estimate the system state through the control of the deviation and the adjustment of control gain.


## 2. Description of the Problem and Related Concepts

Vector Comparison  (1) x t ≥ z t x t > z twhere xt,zt∈Rn is the comparison of all corresponding elements in vectors xt and zt. Through the comparison, we have(2)xit≥zitxit>zit,i=1,2,3,…,nMatrix Comparison  (3) A ≥ B A > B ,  A , B ∈ R n × nComparing all corresponding elements in matrices A and B, we have(4)aij≥bijaij>bij,i=1,2,…,n,j=1,2,…,n.Let Ai∈Rn×n,i=1,2,…,s be s matrices. Then, the convex hull formed by Ai can be expressed as(5)coAi,i=1,2⋯s=A=∑i=1sθiAi;θ1,θ2⋯θsT∈ℵwhere(6)ℵ=θ;θ=θ1,θ2⋯θs;0≤θi≤1,i=1,2⋯s,∑i=1sθi=1.If s is a finite integer, coAi can be called a multicellular body composed of Ai,i=1,2,…,s. Thus, a multicellular body can be considered as the convex set of a finite number of linear matrices.Consider the following linear multicellular system:(7)x˙t∈coAixt+Biut,i=1,2⋯s(8)yt=Cxtwhere xt∈Rn and ut∈Rm are the state and input of the system, respectively; Ai∈Rn×n,Bi∈Rn×l,i=1,2⋯,s; yt∈Rm is the output of the system; C∈Rm×n; co∙ is the convex hull of a set.This chapter aims to solve the following problem: based on the measurable informationyt of the original system, two new systems should be established to estimate the upper and lower bounds x+t and x-t of the state of the original system. In other words, for any t>0, there is(9)x-t≤xt≤x+tThe two new systems are called the upper and lower bound observers of system (7). Together, they constitute the interval observer of this system.Definition 1.
MatrixA is a Hurwitz matrix if all eigenvalues of the matrix have a negative real part, a Schur matrix if all eigenvalue norms of the matrix are smaller than one, a non-negative matrix if all elements of the matrix are non-negative, and a Metzler matrix if all non-diagonal elements of the matrix are non-negative.


## 3. Interval Observer Design of Linear Multicellular System

According to the convex analysis theory [11], system (7) is equivalent to(10)x˙t=A0xt+B0ut+∑i=1sθiAixt+Biut=Aθxt+Bθutwhere θi,i=1,2,…,s is an uncertain parameter that satisfies(11)θi0,1,∑i=1sθi=1and(12)Aθ=A0+∑i=1sθiAi,Bθ=B0+∑i=1sθiBiThus, systems (7) and (8) can be transformed as(13)x˙t=Aθxt+Bθut(14)yt=Cxtwhere θ=θ1,θ2⋯θsT a time-varying parameter variable included in the multicellular body ℵ. The vertex of the variable can be expressed as(15)θj,j=1,…,hHypothesis 2.
A θ , C is observable for any θ∈ℵ.Hypothesis 3.
The inputut and output yt in systems (13) and (14) are bounded.
Here, it is assumed that(16)Aθ=A0+θ1A1+⋯θsAs,Bθ=B0+θ1B1+⋯θsBswhere(17)Ai=Ai,0+Ai,1+⋯Ai,s,Bi=Bi,0+Bi,1+⋯Bi,sNext, w+x+,θe+,u and w-x-,θe-,u are defined as(18)w+x+,θe+,u=maxθeA-θe-Aθ0x++B-θe-Bθ0u(19)w-x-,θe-,u=minθeA-θe-Aθ0x-+B-θe-Bθ0uwhere θe∈Rs·n×1 is defined as(20)θe=θ⊗1n=θ1⋯θ1,⋯,θs⋯θswhere all elements of 1n∈Rn×1 are composed of 1s; ⊗ is the Kronecker product. Let ℵe be a set of multicellular bodies with hn as the vertex. Then, A-θe and B-θe in (18) and (19) can be expressed as(21)A-θe=A0+∑i=1s∑j=1nθei-1·n+jAijB-θe=B0+∑i=1s∑j=1nθei-1·n+jBijwhere Ai,j, Bi,j can be expressed as(22)Ai,j=DjAi,Bi,j=DjBiwhere Dj is a matrix in which all elements are zero except for the diagonal elements (whose value is one).
For system (13), an interval observer can be designed as(23)x˙+=Aθ0x++Bθ0u+w+x+,θe+,u+Ly-Cx+(24)x˙-=Aθ0x-+Bθ0u+w-x-,θe-,u+Ly-Cx-where θ0 is the nominal value of the uncertain parameter θ (θ∈ℵ).Lemma 4.
Ifx+, x-, u are bounded, then w+x+,θe+,u and w-x-,θe-,u are bounded, and the optimal decision variables θe+ and θe- are at the vertex of the set of multicellular bodies ℵe.Proof.
From the basic properties of the linear programming problem, it is learned that the optimal decision variablesθe+ and θe- in (18) and (19) are located at a vertex of the set of multicellular bodies ℵe. Since all vertices are finite and the coefficients of θe+ and θe- are the affine functions of bounded variables x+, x-, and u, then w+x+,θe+,u and w-x-,θe-,u must be bounded.Lemma 5.
For a given matrixT∈Rm×n, if T+=max0,T and T-=T+-T, and if v∈Rn is a variable vector satisfying v-≤v≤v+ where v-,v+∈Rn, then(25)T+v--T-v+≤T+v+-T-v-In (18) and (19),(26)A-θe-Aθ0x++B-θe-Bθ0u=A-θex++B-θeu-Aθ0x++Bθ0uwhere θe is an uncertain parameter; A-θex++B-θeu can be written as follows through linear transform:(27)A-θex++B-θeu=A0x++B0u+A11x++B11u⋯Asnx++Bsnu︸T1θe︸vSimilarly, A-θe-Aθ0x-+B-θe-Bθ0u can be expressed as(28)A-θe-Aθ0x++B-θe-Bθ0u=A0x-+B0u-Aθ0x-+Bθ0u+A11x-+B11u⋯Asnx-+Bsnu︸T2θe︸vAccording to Lemma 5, w+x+,θe+,u and w-x-,θe-,u can be expressed as(29)w+x+,θe+,u=A0x-+B0u-Aθ0x++Bθ0u+T1+v+-T1-v-w-x-,θe-,u=A0x-+B0u-Aθ0x-+Bθ0u+T2+v+-T2-v-Among the matrices T1, T2 and v defined in (27) and (28) above, T1+, T1, T2+, and T2 are all time-varying matrices depending on the state estimates of x+ and x-. The adjustable inputs u, v+and v- are, respectively, the upper and lower bounds defined on the vector parameter θe.Lemma 6.
IfA-θe-LC is a Hurwitz matrix for any θe∈ℵ, then Aθ-LC is also a Hurwitz matrix for any θ∈ℵ.Proof.
According to the definition ofA-θe in (21), Ai=∑j=1nAij is equal for any θi,i=1,2⋯s in (17). Thus, we have(30)θi=θei-1n+j,j=1,…,nThen, Aθ can be expressed as(31)Aθ=A0+∑i=1sAi=A0+∑i=1s∑j=nAij=A-θeIt is obvious that Aθ-LC is a special case of A-θe-LC.Lemma 7.
Suppose matrixP is a diagonal positive definite matrix; then matrix R is a Metzler matrix if and only if PR is a Metzler matrix.Proof.
Considering that both matricesP and P-1 are diagonal positive definite matrices and that each element in matrix PR is identical to each element in matrix R, both matrices PR and R must be Metzler matrices.Theorem 8.
In system (13), both Aθ and Bθ are matrix functions relative to θ, if there exist a matrixP and a matrix Q0,Q1⋯Qr that satisfy the following three conditions:
(1)P∈Rn×n is a diagonal positive definite matrix;
(2) For anyθej∈ℵe, there is(32)A-TθejP+PA-θej-CTQT+QC≺0where Q∈Rn×n, θej is thej-th vertex in ℵe.
(3) For anyθ∈ℵ, PAθ-QC is a Metzler matrix, then systems (23) and (24) are the interval observer of system (13) with the gain of L=P-1Q.Proof.
In (18) and (19), θe+ stands for the maximum operator of the optimal decision variable. In (23) and (24), the state equation observer x+ can be expressed as(33)x˙+=A-θe+-LCx++B-θe+u+LyAccording to Lemma 4, θe+ is located at the vertex hn of multicellular body ℵe. This means the state equation observer x+ in (30) is a transformed LPV system. In the theorem, (1) and (2) ensure that there exists a quadratic Lyapunov function in the observer error state matrix A-θe+-LC, ∀θej∈ℵe. Since u,y and state vector x+ are all bounded, the state function of observer error e=x+-x can be expressed as(34)x˙+-x˙=Aθ-LCx+-x+w+x+,θe+,u-Aθ-Aθ0x+-Bθ-Bθ0uThe observer error in (34) e=x+-x is a time-varying linear system with positive bounded inputs and bounded outputs, due to the following reasons:
(a) It can be inferred from Lemma4 that Aθ-LC is a Hurwitz matrix.
(b) According to Lemma5, PAθ-QC is a Metzler matrix for any θ∈ℵ.
(c) In (18) and (19), it is defined that(35)w+x+,θe+,u-Aθ-Aθ0x+-Bθ-Bθ0uis a positive bounded input and that x+ is bounded.
Thus, it is proved thatx+ is the finite upper bound of the true state x. Similarly, we can prove the positive definiteness of x-x-: if the initial state satisfies x-0≤x0≤x+0, then x+t and x-t constitute the interval estimate of xt.(36)x-t≤xt≤x+tThus, (23) and (24) are the interval observer of system (13).


## 4. Simulation

In system (13), xt∈R2 stands for system state and ut stands for the known input of the system. The following definitions are given for the simulation:(37)ut=1,Bθ=B0=0.1;0.1∈R2×1,C=0,1∈R1×2Aθ=-0.632-0.16sin⁡t0.1cos⁡3t-0.14cos⁡2t0.06sin⁡t=A0+∑i=13Aiθiwhere(38)θ=θ1;θ2;θ3=sint;cos⁡3t;cos⁡2tA0=-0.632,0;0,0,A1=-0.16,0;0,0.06A2=0,0.1;0,0,A3=0,0;-0.14,0where A0 is not a Hurwitz matrix. Using the linear matrix inequality toolbox of Matlab, we have the observer gain:(39)L=0;4.368Figure 1 shows the interval state estimation of the system. It can be seen that xi-t≤xit≤xi+t,i=1,2,3, is valid at any time. Thus, it is proved that the proposed interval observers (23) and (24) can realize the interval estimation of the system state.Figure 1
Interval state estimation of linear multicelluar systems.The simulation results show that the state of the linear multicellular system (13) always fell between the results of the interval observers (23) and (24), achieving the expected estimation effect. Thus, the proposed design algorithm is proved valid.


## 5. Conclusions

This chapter explores the interval state estimation problem of linear multicellular systems and develops an interval observer capable of estimating the real-time upper and lower bounds of the state at any time. The proposed design method can solve many practical control problems. Specifically, the linear multicellular system was converted into an LPV system through convex analysis; then, the author proposed a design method for interval state observer of linear multicellular systems according to the positive system theory. Through strict reasoning, the proposed design method was proved correct and effective.



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