Zhiyu Duan·Xianfu Zhang·Shuai Liu·Airong Wei

Abstract The H∞ output feedback control problem for a class of large-scale nonlinear systems with time delay in both state and input is considered in this paper.It is assumed that the interconnected nonlinearities are limited by constant multiplied by unmeasured states,delayed states and external disturbances.Different from existing methods to study the H∞ control of large-scale nonlinear systems,the static gain control technique is utilized to obtain an observer-based output feedback control strategy,which makes the closed-loop system globally asymptotically stable and attenuates the effect of external disturbances.An example is finally carried out to show the feasibility of the proposed control strategy.

Keywords Large-scale nonlinear systems·H∞control·Output feedback·Time delay

1 Introduction

Large-scale systems can be applied to many practical systems,such as multi-robot systems,power systems,aerospace systems,and so on.In the past few decades,many control design methods have been used to study large-scale nonlinear systems [1–4].For large-scale nonlinear systems with lower-triangular structure,making use of the backstepping design method,a novel neural network-based nonlinear decentralized adaptive controller was constructed in [1],and a decentralized state feedback controller was designed in [2].For large-scale nonlinear system with upper-triangular structure,using the forwarding and saturation design method,the decentralized robust stabilization problem was considered in [3],and the pseudo-decentralized adaptive stabilization problem was addressed in [4].

As we all know,time delay may cause performance degradation or even instability of the control systems.Thus,many scholars have studied the large-scale nonlinear timedelay systems [5–11].In [8],for large-scale stochastic highorder upper-triangular nonlinear systems with state delay,a series of state feedback controllers was constructed by adding one power integrator method.Subsequently,inspired by the homogeneous domination idea,the output feedback control problem of large-scale stochastic high-order uppertriangular nonlinear systems with multi-state delays was addressed in [9].To our best knowledge,the results on largescale upper-triangular nonlinear systems with both state and input delays are limited in the existing literatures [10,11];and however,they did not consider the H∞output feedback control problem.

In practice,it is inevitable that control systems may be affected by external disturbances,and their impact on the output should be reduced as much as possible.Therefore,the H∞control problem has aroused widespread attention.Great achievements have been made in single-input–single-output(SISO) nonlinear systems [12–16].Then,it has been extended to the large-scale nonlinear systems with multiple inputs[17–21].The disturbance attenuation problem of large-scale nonlinear systems was addressed by the backstepping design method in [17] and [18].In [19],for large-scale nonlinear systems with lower-triangular structure,the backstepping-based adaptive neural H∞control problem was investigated.Using the linear matrix inequality (LMI) method,the decentralized H∞state feedback control problem of large-scale nonlinear systems with time-varying delay was studied in [20],and the decentralized H∞output feedback control problem of largescale T-S fuzzy systems with time-varying delay was considered in [21].However,none of the above studies involve the large-scale upper-triangular nonlinear systems with both state and input delays,which promotes the research of this paper.

Besides the design methods mentioned above,the static gain control technique is also an effective approach to study the large-scale nonlinear systems [22].In this paper,using the static gain control technique,the H∞output feedback control problem of large-scale upper-triangular systems with both state and input delays is addressed.Compared with the existing results,the contributions of this paper are as follows:

(1) In this paper,the SISO upper-triangular nonlinear system is extended to the large-scale upper-triangular nonlinear time-delay systems,which makes our work more challenging.

(2) The static gain control technique used in this paper is different from the existing control methods for largescale nonlinear systems,which can simplify the design process and the controller structure.

(3) The proposed output feedback control strategy is proved to have the ability to relax the requirement of system states.That is to say,one need not ensure that all the states of the system are measurable,which makes our result more practical than the state feedback control strategy.

The rest of this paper is arranged as below.Section 2 presents the model of system,a necessary assumption and several lemmas.Section 3 shows the solving process of H∞output feedback control problem.Section 4 proves the feasibility of the output feedback control strategy by an example.Section 5 summarizes the paper.

NotationIn this paper,one denotesX(t) byX,X(t?τ) by,whereτ≥0 is a known delay.‖?‖ refers to the Euclidean norm.The real field is represented by ?,and then-dimensional vector space is represented by ?n.Ishows the identity matrix with appropriate dimensions,0shows the zero matrix with appropriate dimensions.LetX∶(0,T)→?kis in

2 Problem statement

A class of large-scale nonlinear systems with time delay consisting ofNsubsystems is denoted as

Remark 1From system (1),one knows that the connection between subsystems is realized by the nonlinear functionsfi,j,j=1,…,ni,i=1,…,N.When the external disturbances are not considered,a similar system can be found in [11].Different from the global state regulation problem considered in [11],one considers the H∞output feedback control problem.

The control objective is to address the H∞control problem,that is,to design the continuous output feedback controller for each subsystem in system (1) such that system (1)satisfies the following features:

(i) Whenw(t)=0,the closed-loop system is globally asymptotically stable.

(ii) Whenw(t)∈L2(0,T),for given real numberγ>0 and anyT≥0,there holds

An assumption and several lemmas are presented to better realize the control objective.

Remark 2Obviously,Assumption 1 is a linear growth condition,which is typical and general enough in nonlinear systems [16,23].WhenN=1,the inputuand the time delayτare not included in Assumption 1,the disturbance attenuation problem of upper-triangular nonlinear systems was studied in [16].WhenN=1,τ=τ(t),the inputuand the disturbanceware not included in Assumption 1,the state feedback stabilization problem of stochastic upper-triangular nonlinear systems was addressed in [23].Therefore,Assumption 1 is meaningful.

Lemma 1[24]For any integrable vector function g(z) ,the following condition holds:

3 Output feedback controller design

In this section,an output feedback control strategy will be constructed to solve the H∞control problem of system (1).Inspired by [10],the observer and controller for theith subsystem of system (1) are designed as

wherei=1,…,N,αi,jandbi,jare constants defined in Lemma 2,φi,jis the estimation ofzi,j,andr≥1 is a constant to be determined later.

Remark 3The traditional observer design means that for any controller,the states that cannot be measured in a given system can be estimated by observer [26].However,we mainly focus on controller design,the observer (2) is just a formal observer,not a traditional observer.It is only effective for the certain designed controller.This kind of observer has been widely used in the study of nonlinear systems [10,15,16,25].

Forj=1,…,ni,i=1,…,N,define the following transformations:

Theorem 1Under Assumption1,theH∞control problem of system(1)is solvable by the output feedback control strategy in the form of(3).

ProofFirst,under (3),(6),and (7),one derives the following closed-loop system,

To deal with the time-delay terms in (12),two Lyapunov functional candidates are introduced

wherer=max{1,Δ+1},andΔ=2D1τ+D2+2D3.

From (14),whenw=0,one knows that the closed-loop system consisting of (1)–(3) is globally asymptotically stable;moreover,whenw∈L2(0,T),one has

Remark 4In this paper,the static gain control technique is adopted to investigate the H∞control problem of large-scale nonlinear time-delay systems with upper-triangular structure.Compared with the existing methods,our method has lower complexity.

Remark 5The observer-based output feedback control strategy (3) is designed in this paper.Compared with the state feedback control strategy [17,19,20],our strategy does not need all the system states to be measurable,which makes our research more practical.

Remark 6When the constant delayτin system (1) is considered as time-varying delayτ(t),one needs to makeτ(t)be an unknown but bounded time-varying delay satisfyingwhereβ1andβ2are known positive constants,to guarantee that Theorem 1 still holds.And to deal with the problems caused by the timevarying delay,two new L-K functional candidates are designed as

4 An example

In this section,a numerical example is given to show the feasibility of the proposed control strategy.Consider the large-scale nonlinear time-delay systems described by

Obviously,the nonlinear terms in system (15) satisfy Assumption 1 withci,1=ci,2=0.1.

Applying Theorem 1,one can construct the observer and the controller as below,

In the simulation,fors∈[?0.1,0],the following initial conditions are given

Figure 1 gives the curves of the control signalsu1andu2.Figures 2 and 3 show the state trajectories of the first subsystem and the second subsystem respectively.They illustrate that the closed-loop system consisting of (15) with (16) is globally asymptotically stable whenw1=0 andw2=0 .And each subsystem has a better stabilization performance no matter whether the external disturbance exists,that is,the effect of external disturbances is attenuated by the proposed control strategy.

Remark 7Fromr=max{1,Δ+ρ} withΔ=2D1τ+D2+2D3,it can be seen that the selection ofrdepends on the time delayτ.In theory,one can always find a controller for anyτ>0 .However,when the time delay is larger,the control gain will be smaller and the time to stabilize the system will be longer.Therefore,to obtain better simulation results,the time delay should be as small as possible.

5 Conclusions

In this paper,the static gain control technique has been adopted to address the H∞output feedback control problem of large-scale nonlinear time-delay systems with upper-triangular structure.The connection between every subsystem was realized by the nonlinear functions.Two L-K functional candidates were designed to deal with the problems caused by state delay and input delay,respectively.The output feedback control strategy made the closed-loop system globally asymptotically stable and attenuated the effect of external disturbances.In the future,the H∞tracking control problem of nonlinear systems with triangular structure may be studied.

Fig.1 Input trajectories of the closed-loop system consisting of (15)with (16)

Fig.2 State trajectories of the first subsystem with w1=0 and w1=

Fig.3 State trajectories of the second subsystem with w2=0 and w2=

AcknowledgementsThe work was supported by the National Natural Science Foundation of China (Nos.61973189,62073190,61873334),the Research Fund for the Taishan Scholar Project of Shandong Province of China (No.ts20190905),and the Foundation for Innovative Research Groups of National Natural Science Foundation of China(No.61821004).