Disturbance rejection of interconnected positive systems

This paper investigates the issue of stability of interconnected systems, where the systems are composed of perturbed positive subsystems and a interconnection matrix. To address the challenge posed by disturbances, a disturbance observer is formulated for disturbance estimation. The estimated disturbance is incorporated into the input of a controller, which is crafted to produce control inputs reliant on both the system state and the estimated disturbance. Using linear copositive functions, a controller design methodology based on linear programming is established. Ultimately, the efficacy of the proposed approach is substantiated through the illustration of numerical example.


Introduction
In recent times, the increasing complexity and scale of systems in engineering, biology, economics, and other domains have drawn considerable focus to control challenges, particularly concerning large-scale systems like scalable control [1].Positive systems are characterized by states that remain within the nonnegative orthant, which aligns with the inherent constraints of many physical quantities in practical scenarios such as population, prices, and production.Consequently, positive systems have extensive applications in fields like biology, thermodynamics, power systems, statistics (see e.g., [2][3][4][5]), necessitating control strategies that fully account for positivity constraints.
Disturbance control is a vital aspect of control engineering due to the frequent impact of disturbances on real-world systems.A comprehensive review of perturbation/uncertainty estimation and attenuation techniques for general systems is given in [6].Recently, there has been increased focus on studying disturbance resistance in positive systems.The 1 L norm is commonly used to assess disturbance attenuation capabilities, while other performance indexes such as 2 H performance [7] and performance [8] have been adopted for positive systems.Disturbance rejection is more effective than disturbance attenuation in eliminating the influence of disturbance information [9].In [10], a disturbance observer-based control (DOBC) was developed to estimate generalized modeling disturbances and the stabilize system.For linear Metzlerian systems, an uncertain input observer was studied in [11].Additionally, in [12], a positive unknown input observer was designed to enable the concurrent estimation of both states and unknown inputs.A procedure for estimating the unknown input also provided in [12].While applying disturbance compensation to individual subsystems can restore stability, the interconnected nature of these subsystems introduces the potential for instability.Therefore, developing disturbance-resistant approaches to guarantee stability in interconnected systems becomes necessary.Y. Ebihara et al. had extensively explored the properties of large-scale systems with non-negative interconnection matrices in numerous papers [13][14][15].However, there still exists research gaps concerning interconnected systems affected by disturbances.
This paper focuses on tackling the stabilization problem of interconnected positive systems under the influence of exogenous disturbances.To address this, a disturbance observer (DO) is designed, utilizing both system output and disturbance information.The estimation error of the disturbance is incorporated as an augmented system state.By leveraging linear copositive Lyapunov theory, this article derives a necessary and full condition for the presence of a stable state feedback controller in interconnected positive systems affected by disturbances, enabling the formulation of an appropriate gain matrix.
The remaining section will be structured as follows.A clear explanation of the relevant definitions in relation to positive systems can be found in Section 2. Hence, Section 3 provides the main results.Simulation studies appear in Section 4. Section 5 provides a summary of the paper's findings.
Notations: denotes the real space, is the set of -dimension real vectors, and is the set of real matrices of size .Given , represents its inverse matrix, and is its transpose matrix.Given , represents its Moore-Penrose pseudoinverse matrix.represents the -dimensional all-one vector.Hence, define and .Likewise, introduce and with obvious modifications.A matrix ( ) is called strictly positive (positive), denoted by .Furthermore, diag represents a block diagonal matrix composed of blocks .denotes the set of the Metzler matrix with dimensions .The strictly positive diagonal matrix is represented as , and is the -dimension Hurwitz stable matrix.

Preliminaries
Some basic definitions and lemmas are listed in this section.
Definition 1: [16] A matrix is called Metzler if all of its non-diagonal elements are nonnegative, which means that .Lemma 1: [16] For a given matrix , the following statements are equal: a) there exists such that ; b) there exists such that ; c) matrix is a Hurwitz stable matrix, i.e., .In view of the linear system : (1) Where , , are the state, internal input and output of system, respectively, and are positive constants., , , are system matrices.Then, a definition of positive systems along with the relevant lemmas are provided.
Definition 2: [16] the linear system (1) is positive if for any initial state and input , state and output are nonnegative.
In the consider problem, we assume that there exists positive subsystems described by: 3 Where , , , are system matrices, and , , are positive constants.
Defining , the state space realization of is expressed as: ( Additionally, the interconnection matrix is defined as and the specific form of will be discussed in detail later.This paper will focus on analysing the stability of the interconnected system described by equation ( 3) and .For better understanding, the concept of "Admissibility" is introduced at this point.Admissibility guarantees the inheritance of subsystems positivity in interconnected system.Further details can be found in [13] and its cited references.It is crucial to guarantee that is a Metzler matrix and Hurwitz stable in order to ensure the admissibility of the interconnected system .Under the presupposition of admissibility, the state-space model of can be expressed as: (5) : [13] The interconnected system is considered admissible and stable if and only if the Metzler matrix is Hurwitz stable.(6) In practical engineering, most systems are influenced by disturbances.In this context, our focus is on the positive system (2) influenced by generalized external disturbances, which is represented by: (7) Where , are the control input, external disturbance of system (7), respectively., are known coefficient matrices, and , .Similar to [5], the external disturbance can be formulated as (8) In which is the state of this external system, , are known coefficient matrices.Assumption 1 holds throughout this paper.
Assumption 1: the coefficient matrices and are column full rank.
Remark 1: equation ( 8) represents a general dynamic expression for common disturbances.Assumption 1 is established to guarantee the feasibility of maintaining a positive closed-loop system.This assumption is explained in detail in [10].

Design of DO
By utilizing the information from the exogenous system, the DO can be constructed as follows to estimate (9) (10) Where represents an auxiliary variable, is an estimate for , is the estimation of , and is the gain of DO to be designed.The DO combines the outputs and the control inputs with all the state information to estimate the unknown disturbances.
The dynamic characteristics of estimation errors and are the following: Remark 2: Obviously, with the appropriate selection of the gain matrix in disturbance observer, ( 11) is an autonomous system, and the estimate error could approach zero.However, unlike typical systems, it is crucial for to exhibit both asymptotically stable and remain within the positive quadrant.This constraint arises from its incorporation into the extended system.

Closed-loop System with DO and Control Law
Through the combination of state feedback control and the disturbance compensation, the composite controller could be developed in the following way: (12) Where is parameter matrices to be designed, and satisfied The parameter and the term in the system model ( 7) are introduced to handle the dimension mismatch between the inputs and disturbances.When the dimensions are consistent, setting and would be sufficient.

Stabilization of Controller Synthesis
This result considers a special structure of an interconnected system where all subsystems are Multiple-Input Multiple-Output (MIMO) systems, each subsystem exchanges information with other systems, and the output of each system serves as the input to other systems.Note that the and is internal input and output, respectively, which is represented as , for , .The mathematical form of this structure is as follows: (14) It is worth noting that each and is not a scalar but can be multidimensional.In this result, our assumption is that the dimension of and are the same.From Lemma 3, the necessary and full condition for ensuring the admissibility, positivity and stability of the can be developed.Theorem 1: The interconnected system, which composed by subsystems (7) (7), we apply the controller (12) and disturbance observer (11), then the closed-loop subsystems can be obtained as : Taking as the augmented state of the subsystem, the corresponding augmented system can be expressed as (23) In which (24) This enables us to conveniently apply (6) in the subsequent analysis.We define , , , From (14), it is evident that in this case, the interconnection matrix is actually a permutation matrix, i. e. , As the matrix is a permutation matrix, it is evident that , which leads to that and .Then, the fact that matrix can be derived.According to Assumption 1, the condition holds.Furthermore, by combining equations ( 16) and ( 17), it can be concluded that belongs to the set , implying that the composite matrix also belongs to .Accordingly, it can be inferred from Lemma 2 that each subsystem is positive.Additionally, it is straightforward to deduce that (25) By Lemma 3, one has that the is admissible and stable if and only if (25) is Hurwitz stable.This is equivalent to that there exist vectors and such that . From (26), we can obtain that By utilizing (15), inequality (27) can be divided into inequalities as: Examining the structure of matrices , and the assembly of , it becomes evident that equation (28) is valid if and only if equations ( 18)-(20) are satisfied.
At this point, we have guaranteed the stability and admissibility of the interconnected system.. Admissibility allows the to inherit the positivity of its individual subsystems, thus, the positivity of the interconnected system is also guaranteed.The proof is now complete.

Simulation
In this Section, a interconnected system comprised of three subsystems is considered to showcase the efficacy of the developed method.
Example 1: Each subsystem is modeled by (7).It is assumed that and for all subsystems.The gain matrices are set as: , , , , , , .The original values of state are , and the start values of disturbances are .Hence, the coefficient matrices of disturbance systems are formulated as: , .As show in figure1.In cases where prior information about the initial disturbance estimation is lacking, a rational approach is to initialize the disturbance estimate to .It is assumed that the coefficient matrices of the three disturbance systems are identical in order to facilitate the expression.In practical, different subsystems may encounter distinct disturbances, which requires the establishment of different disturbance models.The feasible solutions can be computed by utilizing Theorem 1 that Then, by Theorem1, the parameter matrices of controller (12) and DO (11) are designed as: The simulation results driven by the investigated scheme are displayed in 1-4. 1 exhibits the state of the three stabilized subsystems after forming the interconnected system, without considering the presence of disturbance.Fig. 2 portrays the state following the introduction of disturbances, revealing the unstable state of all three subsystems.
After successful implementation of the controller and disturbance observer, the system attains asymptotic stabilization while preserving its positive nature.The outcomes are exemplified in Fig. 3. Furthermore, Fig. 4 serves as compelling evidence that the proposed method efficiently diminishes the estimation error of disturbances in the interconnected positive system over time, gradually converging to zero.These simulation results provide compelling evidence for the effectiveness of Theorem 1.The application of this approach ensures the stability of interconnected subsystems, while the accurate estimation of external disturbances by the DO is evident from the rapid convergence towards zero.

Summary
This article deals with the stability control of interconnected positive systems with exogenous disturbances, interconnected through a non-negative interconnection matrix.A disturbance observer is developed in this paper, and both the necessary and full conditions for ensuring the feasibility of maintaining positivity and stability within the perturbed interconnected system are established.Significantly, it demonstrates that the proposed design methodology can facilitate the development of controllers and observers, even when considering the distinct dynamics of each subsystem and each exogenous disturbance system, as long as the internal inputs and outputs of each subsystem align with the provided interconnection matrix.Our findings highlight that addressing the involved challenges can be effectively addressed through the application of linear programming methods.

Figure 1 .
Figure 1.The state of interconnected system without disturbance.

Figure 2 .
Figure 2. The state of practical interconnected system with disturbance.

Figure 3 .
Figure 3. Resulting from the interconnected system with the controller.

Figure 4 .
Figure 4.The estimate error of disturbance.