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.
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 Comparisonwhere is the comparison of all corresponding elements in vectors and . Through the comparison, we have
Matrix ComparisonComparing all corresponding elements in matrices and , we haveLet be matrices. Then, the convex hull formed by can be expressed aswhereIf is a finite integer, can be called a multicellular body composed of . Thus, a multicellular body can be considered as the convex set of a finite number of linear matrices.
Consider the following linear multicellular system:where and are the state and input of the system, respectively; ; is the output of the system; ; is the convex hull of a set.
This chapter aims to solve the following problem: based on the measurable information of the original system, two new systems should be established to estimate the upper and lower bounds and of the state of the original system. In other words, for any , there isThe 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. Matrix 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 towhere is an uncertain parameter that satisfiesandThus, systems (7) and (8) can be transformed aswhere a time-varying parameter variable included in the multicellular body . The vertex of the variable can be expressed as
Hypothesis 2. is observable for any .
Hypothesis 3. The input and output in systems (13) and (14) are bounded.
Here, it is assumed thatwhereNext, and are defined aswhere is defined aswhere all elements of are composed of ; is the Kronecker product. Let be a set of multicellular bodies with as the vertex. Then, and in (18) and (19) can be expressed aswhere , can be expressed aswhere 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 aswhere is the nominal value of the uncertain parameter ().
Lemma 4. If , , are bounded, then and are bounded, and the optimal decision variables and are at the vertex of the set of multicellular bodies .
Proof. From the basic properties of the linear programming problem, it is learned that the optimal decision variables and in (18) and (19) are located at a vertex of the set of multicellular bodies . Since all vertices are finite and the coefficients of and are the affine functions of bounded variables , , and , then and must be bounded.
Lemma 5. For a given matrix , if and , and if is a variable vector satisfying where , then
In (18) and (19),where is an uncertain parameter; can be written as follows through linear transform:Similarly, can be expressed asAccording to Lemma 5, and can be expressed asAmong the matrices , and defined in (27) and (28) above, , , , and are all time-varying matrices depending on the state estimates of and . The adjustable inputs , and are, respectively, the upper and lower bounds defined on the vector parameter .
Lemma 6. If is a Hurwitz matrix for any , then is also a Hurwitz matrix for any .
Proof. According to the definition of in (21), is equal for any in (17). Thus, we haveThen, can be expressed asIt is obvious that is a special case of .
Lemma 7. Suppose matrix is a diagonal positive definite matrix; then matrix is a Metzler matrix if and only if is a Metzler matrix.
Proof. Considering that both matrices and are diagonal positive definite matrices and that each element in matrix is identical to each element in matrix , both matrices and must be Metzler matrices.
Theorem 8. In system (13), both and are matrix functions relative to , if there exist a matrix P and a matrix that satisfy the following three conditions:
(1) is a diagonal positive definite matrix;
(2) For any , there iswhere , is the j-th vertex in .
(3) For any , is a Metzler matrix, then systems (23) and (24) are the interval observer of system (13) with the gain of .
Proof. In (18) and (19), stands for the maximum operator of the optimal decision variable. In (23) and (24), the state equation observer can be expressed asAccording to Lemma 4, is located at the vertex of multicellular body . This means the state equation observer in (30) is a transformed LPV system. In the theorem, and ensure that there exists a quadratic Lyapunov function in the observer error state matrix , . Since and state vector are all bounded, the state function of observer error can be expressed asThe observer error in (34) is a time-varying linear system with positive bounded inputs and bounded outputs, due to the following reasons:
(a) It can be inferred from Lemma 4 that is a Hurwitz matrix.
(b) According to Lemma 5, is a Metzler matrix for any .
(c) In (18) and (19), it is defined thatis a positive bounded input and that is bounded.
Thus, it is proved that is the finite upper bound of the true state . Similarly, we can prove the positive definiteness of : if the initial state satisfies , then and constitute the interval estimate of .Thus, (23) and (24) are the interval observer of system (13).
4. Simulation
In system (13), stands for system state and stands for the known input of the system. The following definitions are given for the simulation:wherewhere is not a Hurwitz matrix. Using the linear matrix inequality toolbox of Matlab, we have the observer gain:Figure 1 shows the interval state estimation of the system. It can be seen that , 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.
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.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This work was supported by National Natural Science Foundation of China (61773016, 61473222, and U1534208) and Science and Technology Project of Shaanxi Province (2016GY-108).