Observer-Based Adaptive Decentralized Control for Interconnected Nonlinear Systems with Input Delay

Authors: qixia  shen
yushan  cen
liang  cao
Venue: IEEE_org_ICIST_2024_Conference
Published: 2024-08-12 04:37:16

Abstract: For nonlinear interconnected systems against input delay, an observer-based fuzzy adaptive decentralized control strategy is designed. First, different from traditional observers, a novel state observer based on tracking error is proposed to estimate unknown states for the interconnected systems. Secondly, a compensation signal is added to handle the time-varying input delay. Then, the dynamic surface control techniques are utilized in the backstepping approach to overcome the “explosion of complex” problem. Meanwhile, through designed schemes, all variables in the closed-loop systems are promised to be bounded. The errors converge into bounded compact set around the origin. The effectiveness of the designed scheme is validated via a simulation example.

Main paper: 
 i. introduction: 
 In the past decade, interconnected systems are typically used to depict complex systems composed of multiple subsystems that have physical couplings or network interactions. Due to their large scale, the decentralized control based on backstepping is widely applied in nonlinear interconnected systems. In [1] , the authors put forward an adaptive fuzzy decentralized control strategy for interconnected systems. In the derivation process of the backstepping method, it is necessary to differentiate the virtual controllers, which leads to the computational complexity problem. To overcome "explosion of complexity" problem, a dynamic surface control (DSC) method was put forward by performing first-order filters at each step. Xu et al. [2] designed a predefined time filter to address "explosion of complex" issue. The decentralized DSC scheme was designed to avoid the repetitive derivative of virtual controllers for largescale power systems [3] . In practical engineering, the stability of the systems are also affected by many factors, such as input delay and unmeasurable system state.Due to limited equipment, nonstandard operations, environmental disturbances and signal transmission, input delay often occurs in practical systems, which may be the source of unstable factors that lead to unexpected degradation of the system. Therefore, how to solve the input delay has become a key issue. In [4] , the input delay was solved by Pade approximation. Xing et al. [5] constructed an auxiliary system to offset the unfavorable effects of input delay. The authors in [6] overcame the influence of the input delay by proposing a compensation signal. In practical application, state variables of interconnected systems are generally unmeasurable. Specially, observers are often adopted to solve this problem [7] . Xu et al. [8] proposed a state observer based on fuzzy logic systems (FLSs) for interconnected systems. A finitedimensional observer was constructed on the basis of system output to asymptotically estimate state information [9] . In [10] , the authors constructed a distributed state observer based on synchronization error for multiagent systems to estimate unmeasured states. Many achievements about observers of interconnected systems based on output information. Inspired by distributed observers, it is significant to design an observer based on tracking errors to estimate unknown states for largescale nonlinear systems against input delay.Motivated by the aforementioned discussions, this paper puts forward an observer-based adaptive decentralized control method for nonlinear interconnected systems against input delay. The main contributions are described below.1) An observer based on tracking errors is investigated for interconnected systems with input delay to estimate unmeasured states. On the basis of traditional state observers, not only the output variables but also the information of the reference signal are considered. 2) With the framework of backstepping control, a compensation signal is added to deal with input delay, and the system stability is assured. In order to decrease the computational burden, DSC and first-order filters are adopted to handle complexity explosion problems.The interconnected systems against time-varying input delay are modeled as follows:EQUATIONwhere xi,m = [x i,1 , . |∆ i,m (ȳ)| 2 ≤ N ∑ p=1 ρ 2 i,m,p (y p )Lemma 1: Let f (x) on the compact set Ω be the continuous function. There is an FLS y(x) = Θ T φ(x) for any constant δ > 0 satisfyingsup x∈Ω |f (x) -Θ T φ(x)| ≤ δIn the light of Lemma 1, the unknown smooth function in (1) can be approximated by the following FLSs:fi,m ( xi,m |Θ i,m ) = Θ T i,m φ i,m ( xi,m )Define the optimal parameter vectors Θ * i,m asΘ * i,m = arg min Θ * i,m ∈Ωi,m [ sup xi,m∈ Ûi,m | fi,m ( xi,m |Θ i,m ) -f i,m ( xi,m )]in which the compact regions of Θ * i,m and xi,m are showed as Ω i,m and Ûi,m , respectively.The following fuzzy minimum approximation errors δ i,m and approximation errors ω i,m are defined:δ i,m = f i,m ( xi,m ) -fi,m ( xi,m |Θ * i,m ) ω i,m = f i,m ( xi,m ) -fi,m ( xi,m |Θ i,m ) Assumption 4: Two known constants δ * i,m > 0 and ω * i,m > 0 fulfill |δ i,m | ≤ δ * i,m and |ω i,m | ≤ ω * i,m . Lemma 2: There exist positive real numbers k i,1 , . . . , k i,ni , Hurwitz matrix A i , P i = P T i > 0 and ς i = ς T i > 0 fulfilling A T i P i + P i A i ≤ -ς iwhereA i =      -k i,1 . . . I (ni-1)×(ni-1) -k i,ni-1 -k i,ni • • • 0      .Due to the unavailability of the states in (1), an observer based on the tracking error is designedEQUATIONwhere e i,1 expresses the tracking error, which will be provided later.Define the following observation error:EQUATIONFrom ( 2) and ( 3), one hasEQUATIONwhereL i = [1, 0, . . . , 0] and B i,m = [0, . . . , 1 m , . . . , 0] T .The following Lyapunov function is considered:V i,0 = ε T i P i ε i (5)In view of the above equation, one getsEQUATIONWith Young's inequality, Assumptions 2-4, one hasEQUATIONEQUATIONEQUATIONEQUATIONwhereω * i = [ω * i,1 , . . . , ω * i,ni ] T . ρp,m,i (y i ) means an unknown smooth function.By substituting ( 7)-( 10) into (6), one obtains Vi,0 ≤ -EQUATIONTo decrease the computational burden, the first-order filter is added, and the coordinate transformation is given (12) where α i,j-1 and α i,ni stand for the virtual controllers. ᾱi,j-1 and ᾱi,ni indicate the first-order filters, and e αi,j-1 and e αi,n i show the output errors of filters. The auxiliary signal η i will be provided later.               e i,1 = y i -y i,d e i,j = xi,j -ᾱi,j-1 , j = 2, . . . , n i -1 e i,ni = xi,ni -ᾱi,ni-1 + η i e αi,j-1 = ᾱi,j-1 -α i,j-1 e αi,n i = ᾱi,ni -α i,niFor fear of repeated differentiation of virtual controllers, this paper introduces a first-order filters ᾱi,j-1 and ᾱi,ni , and one hasEQUATIONwhere σ i,j-1 and σ i,ni are positive constants. Based on ( 12) and ( 13), it yields thatEQUATIONwhere O i,j-1 (•) = -αi,j-1 and O i,ni (•) = -αi,ni denote the continuous functions.Step i, 1: It follows from ( 1), ( 2) and ( 12) that ėi,1 = e i,2 + e αi,1 + f i,1 (x i,1 ) + α i,1EQUATIONConstruct the Lyapunov function asV i,1 = V i,0 + 1 2 e 2 i,1 + 1 2 e 2 αi,1 + 1 2Υ i,1 Θ2 i,1 + 1 2ζ i,1 W 2 i (16)where 1 and W i denote the estimations of Θ * i,1 and W * i , respectively. W * i will be explained later. Υ i,1 and ζ i,1 are both designed constants.Θi,1 = Θ * i,1 -Θ i,1 and Wi = W * i -W i . Θ i,In view of (15) and ( 16), the derivative of V i,1 isVi,1 = Vi,0 + e i,1( ε i,2 + e i,2 + e αi,1 + f i,1 (x i,1 ) -ẏi,d +α i,1 + ∆ i,1 ) + e αi,1 ( - e αi,1 σ i,1 + O i,1)EQUATIONAccording to the Young's inequality, Lemma 1 and Assumption 4, one has e i,1 (e αi,1 + e i,2EQUATIONEQUATIONin which the constant q i,1 is positive. This paper applies the FLS bi (y 11) and (21).i |W i ) = W T i ϕ i (y i ) to approximate the interconnection terms y i ∑ N p=1 ∑ np m=1 ρ2 p,m,i (y i ) and y i ∑ N p=1 ρ2 p,1,i (y i ) in (Define the optimal vector W * i asEQUATIONwhere Π i and Ψ i are compact regions for W * i and y i , respectively. Define the minimum approximation error ξ i asξ i = y i ∑ N p=1 ∑ np m=1 ρ2 p,m,i (y i ) + y i ∑ N p=1 ρ2 p,1,i (y i ) -bi (y i |W * i ).There is a known positive constant ξ * i such that |ξ i | ≤ ξ * i . Based on the above equation, one getsEQUATIONFurther, we obtainEQUATIONΘi,EQUATIONin which γ i,1 > 0 and l i > 0 are both designed parameters. Based on the above analysis, Vi,1 is transformed intoEQUATION)EQUATIONin whichµ i = λ min (ς i ) -2λ max (P i K i L i ) -3λ 2 max (P i )ι 2 i,1 -λ max (P i K i K T i P i ) - ∑ ni m=1 ι 2 i,m -ι 2 i,1 -1 2 .Step i, j (2 ≤ j ≤ n -1): The expression of the Lyapunov function isEQUATIONwhere Υ i,j expresses a normal number. Θ i,j denotes the estimation of Θ * i,j , and the error between them is Θi,j = Θ * i,j -Θ i,j .In the light of ( 12), the differentiation of V i,j is calculated as Vi,j = e i,j ( k i,j e i,1 + e i,j+1 + ΘT i,j φ i,j ( xi,j ) + α i,j + e αi,j (30)EQUATIONEQUATIONin which the constant q i,j is positive.The following α i,j and Θi,j are established asEQUATION2) e i,j -Θ T i,j φ i,ni ( xi,j )+ αi,j -k i,j e i,1 (33) Θi,j =Υ i,j (e i,j φ i,j ( xi,j ) -γ i,j Θ i,j )where the designed parameter γ i,j is positive. From (30)-( 34), ( 29) is changed asEQUATION)EQUATIONStep i, n i : Similar to [6] , the following compensation signal is added:ηi = -λ i η i + u i (t) -u i (t -τi ) (36)where λ i is design parameter. From (12) , the error dynamic yieldsEQUATIONThe following Lyapunov function is selected:V i,ni = V i,ni-1 + 1 2 e 2 i,ni + 1 2Υ i,ni Θ2 i,ni + M i + Q i + N i + G i + 1 2 e 2 αi,n i (38) in which M i = 1 2(1-i) ∫ t t-τi(t) |u i (p)| 2 dp, Q i = 1 1-i ∫ t t-τi(t) ( ∫ t ϑ |u i (p)| 2 dp)dϑ, N i = 1 2 ∫ t t-τi |u i (p)| 2 dp and G i = ∫ t t-τi ( ∫ t ϑ |u i (p)| 2 dp)dϑ, respectively. Υ i,ni > 0 is a constant.Based on the Young's inequality, one obtainsEQUATIONThe following controller u i (t) and adaptive law Θi,ni are established:EQUATIONin which γ i,ni > 0 stands for the designed parameter.According to (39)-(44) and the definitions of Θi,j and Wi , one hasEQUATION)EQUATIONThrough the definition of Q i and G i , we can easily obtainQ i ≤ τi 1 -i ∫ t t-τi(t) |u i (p)| 2 dp (46) G i ≤ τi ∫ t t-τi |u i (p)| 2 dp (47)From (43), we can see that the design controller u i is composed of bounded signals. Hence, it fulfills |u i (t)| ≤ ūi , which ūi > 0 indicates a constant.Combining ( 46) and (47), the inequality (45) is rewritten asEQUATIONFor the compensation signal η i , we chose V ηi = 1 2 η 2 i , and its derivative isEQUATION)Based on the above-mentioned analysis and the decentralized controller design, there exist the following results.Theorem 1: All signals can be kept to be bounded for largescale nonlinear systems (1) with time-varying input delay, and the tracking error of each subsystem can be made arbitrarily small by utilizing the designed observer (2), adaptive laws (25)-( 26), ( 34) and (44), virtual controllers (24) and (33) and controller (43).Proof: For the whole closed-loop system, the Lyapunov function is described asEQUATIONThe differential of the above equation isEQUATION)EQUATION)whereC = min {C 1 , C 2 , . . . , C N } and ℵ = ∑ N i=1 { 1 2 (q 2 i,1 +δ * 2 i,1 ) + 1 2 ∑ ni j=2 (q 2 i,j + δ * 2 i,j + ω * 2 i,j ) + ∑ ni j=1 γi,j 2 Θ * 2 i,j + y 2 i,d +( 1 2 + li 2 )W * 2 i + ∥ω * i ∥ 2 + ( 4-3 i 2(1-i) + 2-i 1-i τi ) ū2 i + ξ * 2 i + 1 2 (q 2 i,1 + δ * 2 i,1 ) } . One defines C i = min {µ i /λ min (P i ), 2c i,j , 2( 1 σi,j -1 2q 2 i,j O 2 i,j -1 2 ), γ i,j Υ i,j , l i ζ i,1 , (1 -i ), 1-i 2τi , 1, -1 2τi , 2(λ i -1) } .IV. SIMULATION RESULTS The presented control strategy is applied in the two inverted pendulum systems which are connected through a spring with θ 1 = x 1,1 , θ 2 = x 2,1 , θ1 = x 1,2 and θ2 = x 2,2 , and the systems are modeled by the following form [11] :EQUATION) sin(x 1,1 )EQUATIONEQUATIONEQUATIONwhere the pendulum end masses are defined as m 1 and m 2 . k and l denote spring constant and the natural length of the spring, respectively. J 1 and J 2 represent the moments of inertia. The distance between the pendulum hinges, gravitational acceleration and pendulum height are indicated as b, g and r, respectively. The reference signal is given as y i,d = sin(t). The time-varying input delay is expressed as τ i (t) = 0.005 sin(t) + 0.01. The initial values and parameters of the systems are described in Table I . Through the designed adaptive laws (25), ( 26) and (44), real controller (43) and virtual controller (24) to control two inverted pendulum systems, Figs. 1 2 3 4 display the results of the simulation. Fig. 1 and Fig. 2 depict the good tracking performance of the control outputs of each subsystem on the ideal reference trajectory. Fig. 3 and Fig. 4 plot the trajectory of the control input u i with τ i (t) = 0.005 sin(t) + 0.01. In summary, the simulation results verify the availability of the proposed strategy.λ 2 = 70 k 1,1 = 1 k 1,2 = 20 k 2,1 = 1 k 2,2 = 5 Υ 1,2 = 0.5 Υ 2,2 = 0.01 γ 1,2 = 2Observer-based fuzzy adaptive decentralized control strategy has been studied for nonlinear interconnected systems against input delay. By introducing the tracking error into the designed of the state observer, the unmeasurable states have been estimated. Moreover, the compensation signal has been prompted to compensate for the impact of input delay. The "explosion of complexity" problem have been overcome by combining DSC and first-order filters. It has displayed that the presented control strategy guarantees that all variables of whole system are semiglobally uniformly ultimately bounded. Finally, the inverted pendulum systems has been applied to clarify that the proposed method is valid. In the future work, we will strive to apply the presented method in this paper to other systems [12] . 
 
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Reviews: review: title: title: This article is very interesting and a good one
main_review: main_review: In this paper, an observer-based fuzzy adaptive decentralized control strategy was proposed for nonlinear interconnected systems against input delay. The obtained result is valuable and can be accepted if the following problems can be clarified. 
(1) In the introduction, the shortages of those relevant studies are suggested to be further summarized. 
(2) The definition of Lemma 1 should be provided in detail, including each variable
 (3) In the end of Section 1, the organization of this study is suggested to be summarized. 
(4) There exist several spelling and grammar errors. Please check carefully and further polish
 (5) In the simulation section, more analysis can be added to better explain the main results of this paper, that's not enough.
title: title: This article is quite fascinating and of high quality.
main_review: main_review: The paper titled "Observer-Based Adaptive Decentralized Control for Interconnected Nonlinear Systems with Input Delay" design the observer-based fuzzy adaptive decentralized control strategy. Firstly, a novel state observer based on tracking error is proposed to estimate unknown states for the interconnected systems. A compensation signal is added to handle the time-varying input delay. Then, the dynamic surface control techniques are utilized in the backstepping approach to overcome the “explosion of complex” problem. Meanwhile, through designed schemes, all variables in the closed-loop systems are promised to be bounded. The errors converge into bounded compact set around the origin. Finally, the effectiveness of the designed scheme is validated via a simulation example. My specific feedback is as follows: 1) The advantages of the author's method compared to other methods are not stated in the contribution. 2) Some formatting issues need to be addressed.
title: title: Accept
main_review: main_review: In this paper, " Observer-Based Adaptive Decentralized Control for Interconnected Nonlinear Systems with Input Delay", an observer-based fuzzy adaptive decentralized control strategy is proposed. Firstly, a tracking error-based state observer is proposed to estimate the unknown state of interconnected systems. Then, a compensating signal is added to handle the time-varying input delay. The article has clear logic and organization, but there are still some problems. My specific feedback is as follows :1) In the introduction section, the interconnection system and input delay are not introduced enough. 2) What are the advantages of tracking error-based observers compared with traditional state observers?

Comments: title: This article is very interesting and a good one
comment: In this paper, an observer-based fuzzy adaptive decentralized control strategy was proposed for nonlinear interconnected systems against input delay. The obtained result is valuable and can be accepted if the following problems can be clarified.
(1)	In the introduction, the shortages of those relevant studies are suggested to be further summarized.
(2)	The definition of Lemma 1 should be provided in detail, including each variable
(3)	In the end of Section 1, the organization of this study is suggested to be summarized.
(4)	There exist several spelling and grammar errors. Please check carefully and further polish
(5)	In the simulation section, more analysis can be added to better explain the main results of this paper, that's not enough.