Decentralized Tracking of Large-Scale Stochastic Nonlinear Systems via Output-Feedback

This paper investigates the decentralized output-tracking problem for a class of high-order stochastic nonlinear systems (SNSs). Compared with the existing results, we consider more practical and more general systems, i.e., the system is large-scale and linearization parts may have unstable modes. An output-feedback tracking controller is designed based on a decentralized high-gain homogeneous domination technique. By using advanced stochastic analysis methods, we show that the output tracking errors can be made arbitrarily small while all the states of the closed-loop system remain to be bounded in probability. Finally, the effectiveness of the output-feedback tracking controller is demonstrated by a simulation example.


I. INTRODUCTION
In practical engineering, with the wide application of stochastic control [1], [2], the study of stochastic systems has attracted more and more researchers' attention [3]- [8]. The design includes state-feedback control and output-feedback control. For state-feedback, [9] focuses on the cooperative control problem of multiple SNSs perturbed by second-order moment processes in a directed topology. Reference [10] studies a class of high-order with stochastic inverse dynamics. Subsequently, [11] discusses output-constrained stochastic systems with low-order and high-order nonlinear and high-order nonlinear and stochastic inverse dynamics. Reference [12] considers the case where the diffusion term and the drift term are unknown parameters for stochastic systems with strict feedback. However, due to the internal variable attribute of the state variable, the state cannot be directly measured, which limits the physical composition of the state-feedback. The basic way to solve this problem is to introduce state reconstruction or state estimation, and use the reconstructed state or estimated state as feedback variables to form state feedback. Furthermore, scholars use The associate editor coordinating the review of this manuscript and approving it for publication was Feiqi Deng . an observer to investigate the output-feedback, e.g., [13] is the first to study the output-feedback global stabilization problem of SNSs. [14] studies output-feedback stabilization of SNSs with unknown covariance. Reference [15] considers strict-feedback systems with sensor uncertainties. In addition, for the decentralized output-feedback stabilization problem, [16] investigates SNSs with three types of uncertainties, including nonlinear uncertain interactions, parametric uncertainties, and stochastic inverse dynamics. Reference [17] discusses high-order SNSs.
Besides the stabilization results above, as demonstrated by [18], output tracking is widely used in the military, navigation, and other fields. Recently, [19], [20] discuss the output tracking problem of high-order SNSs with benchmark mechanical systems and stationary Markovian switching respectively. It should be pointed out that [19], [20] only consider the state-feedback output tracking problem, which requires all states of the system to be available. However, as we have previously analyzed, some state information is not always available, so some scholars study the output-feedback tracking control of SNSs [21]- [24]. Specifically, [21] studies adaptive output-feedback tracking control with dynamic uncertainties and unmeasured states. In [23], the output-feedback tracking problem with unstable linearization is studied. Unfortunately, there are few research results on the design of decentralized output-feedback tracking controller for high-order SNSs.
Based on these discussions, we aim to resolve the decentralized output-feedback tracking for high-order SNSs with unstable linearization. The main contributions and characteristics of this paper are two-fold: (1) The system model we take into account is more applicable than the existing results [18]- [24]. Different from the previous results [18]- [20], we consider unmeasurable states. Unlike previous studies in [21]- [24], we investigate the decentralized control system. Decentralized systems with unmeasurable states make the controller design process more complicated and difficult. More advanced stochastic analysis techniques are needed.
(2) In order to deal with the problem of output-feedback tracking control for large-scale high-order SNSs, a new stochastic high gain homogeneous control design method is proposed. Different from centralized systems [13]- [15], [21]- [25], the stability analysis of decentralized systems is more challenging, which is another contribution of this paper.
The rest of this paper is listed as follows. The problem is formulated in Section II. In Section III, an output-tracking controller is designed. Section IV is the stability analysis. A simulation is given in Section V. The conclusions are collected in Section VI.
Notations: R n denotes the n-dimensional space and the set of nonnegative real numbers is represented by R + . R + odd = {q ∈ R: q > 0 and q is a ratio of odd integers }, and X denotes the matrix or vector, its transpose is represented by X T . |X | denotes the Euclidean norm of a vector X . When X is square, Tr{X } denotes its trace. For A ∈ R n×m , |A| denotes the Frobenius norm |A| = ( n i=1 m j=1 A 2 ij ) 1/2 and |A| ∞ = max 1≤i≤n { m j=1 |A ij |}. The set of all functions with continuous ith partial derivatives is represented by C i . Let C 2,1 (R 2 × R + × S; R + ) represent all nonnegative functions V on R 2 × R + × S which are C 2 in x and C 1 in t.

II. PROBLEM FORMULATION
Consider the following interconnected large-scale high-order SNSs: where ζ = (ζ T 1 , · · · , ζ T m ) T ∈ R mn , ζ i = (ζ i1 , · · · , ζ in ) T ∈ R n , y i ∈ R and u i ∈ R,i = 1, · · · , m, are the system and subsystem state, output and input, respectively. ζ i2 , · · · , ζ in are unmeasurable. y ir (t) is the reference signal to be tracked. ω is an r-dimensional standard Wiener process defined on a probability space ( , F, P), with being a sample space, F being a filtration, and P being a probability measure. For i = 1, · · · , m, j = 1, · · · , n, p ij ∈ R ≥1 odd = {q ∈ R: q ≥ 1 and q is a ratio of odd integers}, the functions f ij : R mn → R and g ij : R mn → R r , are assumed to be C 1 , vanishing at the origin. The following assumptions are made on system (1). Assumption 1: There are constants b, c > 0, and τ ≥ 0 for i = 1, · · · , m, j = 1, · · · , n, such that and a ij = i0 ij , i = 1, · · · , m, j = 1, · · · , n. Furthermore, for fixed i, one of the following conditions should be satisfied: odd . Remark 1: In Assumption 1, it should be emphasized that compared with [17], the constant term c is added, which guarantees the existence of additive bounded disturbances in the drift and diffusion terms. Compared with [21]- [23], we study large-scale SNSs with cross terms shown in Assumption 1. New design schemes should be developed.
For system (1), our objective is to design an output-feedback controller such that the output tracking errors can be made arbitrarily small while all the states of the closed-loop system remain to be bounded in probability.

III. OUTPUT-TRACKING CONTROLLER DESIGN
The design process is divided into two steps: • Firstly, we analyze the nominal system of (1); • Secondly, by adopting homogeneous domination method, we design an output-tracking controller for system (1).

C. STABILITY ANALYSIS
In this part, for the ith closed-loop subsystem, we first give two crucial lemmas, which is useful to deal with the Hessian terms. Then, we present the main results of the stability analysis. Lemma 2: If Assumption 1 is satisfied, there is a positive constant β i0 such that Proof: By choosing l and σ , we can get that which with ij > 0 yields which yields From (19), we can choose arbitrarily constant β i0 > 0 satisfying so (17) is true, and the proof is thus done.
The following lemma uses homogeneous theory to estimate the ith gradient term and Hessian term.
Since L > 1, we choose a sufficient large L such thať By using (38) and (39), we get From [26, Lemma 2] and 1, there are positive constantsδ 1 and δ 2 such that By using (41), with Young's inequality, we get Substituting (42) into (40), with (41), we have By using (43) and [16,Th.1], we obtain that the whole closed-loop system (16) has an almost surely unique solution on [0, ∞) and all the states of the closed-loop system are bounded in probability. Let When t ≥ t 0 , choose t l = min{σ l , t}. We can obtain that bounded | (·)| on interval [t 0 , t l ] a.s., which means that V ( ) is bounded on [t 0 , t l ] a.s. By using (43), we get that LV is bounded in the interval [t 0 , t l ] a.s. Thus by using [27, Lemma 1.9], we can get Note that lim l→∞ σ l = ∞. Then, letting l → ∞, with (44), we get which together with (43) implies or equivalently, By the definition ofβ and tuning the gain L thatβ can be made any small, which with (47) yields that EV ( (t)) can be tuned to be any small. Therefore, noting i = (ζ i1 − y ir , . . . , ξ in , η i2 , . . . , η in ), we can obtain that for initial value x(t 0 ) and arbitrarily given ε, there exists a sufficient large L and finite-time T (x(t 0 ), ε) and such that The proof is thus done. VOLUME 10, 2022 Remark 2: Different from the analysis results of existing centralized systems [21]- [24] (i.e. m = 1), our analysis method is more general, because it is suitable for decentralized systems and is also effective for centralized systems. Unlike the deterministic systems [28], [29], by using Itô formula, the stochastic system will produce more nonlinear terms, which increases the difficulty of analyzing the stability. To solve this problem, we use homogeneous technology, which can be found in (20) and (21). Different from the decentralized system [30], it is not necessary to estimate the boundary of diffusion and drift term step by step, which simplifies the design process and improves the design efficiency, see (28) and (32).
Remark 3: It should be pointed out that our method can not only deal with the case that the drift term and diffusion term are functions, but also deal with the case that they are constants, as shown in (49). This is one of the advantages of this paper.

V. CONCLUSION
We investigate the decentralized output-feedback tracking problem for a class of large-scale high-order SNSs in this paper. Specifically, by developing a decentralized high-gain homogeneous domination technique, we design an output-feedback tracking controller. By using advanced stochastic analysis methods, we show that the expectation of the tracking error can be made arbitrarily small while all the states of the closed-loop system remain to be bounded in probability.
There are many related problems to be considered, such as how to extend the result to more general systems [31]- [33].