Design of decentralized adaptive control approach for large-scale nonlinear systems subjected to input delays under prescribed performance

For the first time, the issue of input delay and prescribed performance control is investigated in the same framework for large-scale nonlinear systems in this study, and an original adaptive decentralized control method is proposed by taking advantage of multi-dimensional Taylor network (MTN) method. Firstly, the problem of input delays is solved by introducing new variables, and a new form of coordinate transformation is introduced before controller design, which simplified the control system. Secondly, the problem of prescribed performance control is coped with by integrating the idea of prescribed performance into the Lyapunov functions of first step of backstepping of each subsystem. Thirdly, MTNs are employed to evaluate the combination of unknown functions, and then, a decentralized MTN-based adaptive control scheme is developed by way of backstepping technology. The theoretical analysis indicates that the proposed control scheme can implement the expected tracking goals under the condition of meeting the prescribed performance control. Finally, two examples are given to show the validity and rationality of the proposed control method.


Introduction
With the rapid development and progress of modem technology, more and more practical systems, such as chemical process systems and power systems, have the characteristics of high dimension and complex structure. Therefore, more and more concerns have been transferred to controller design for large-scale nonlinear systems [1][2][3]. Among them, decentralized control has become an important control method for large-scale nonlinear systems as a result of this control method only depends on the local information of subsystems [4]. Besides, in the light of adaptive backstepping technique provides a simple and systematic recursive method for Lyapunov function design of complex nonlinear systems, this technique plays a vital part in the field of nonlinear systems control, and a lot of significant results have been gotten for nonlinear systems [5][6][7][8] and large-scale nonlinear systems [9][10][11]. However, when the nonlinear systems have uncertain parameters rather than structure uncertainties, adaptive backstepping technique will no longer be applicable.
For the sake of solving the above problem, many nonlinear intelligent control strategies, such as neural network (NN) control, fuzzy control, and multidimensional Taylor network (MTN) control [12][13][14], have been proposed. In particular, by combining the above intelligent control methods with adaptive backstepping technique, the various types of advanced control algorithms have been developed. For example, combining neural networks with adaptive control and backstepping technology, dramatic results have been obtained for perturbed strict-feedback nonlinear systems [15], nonsmooth nonlinear systems [16], and distributed nonlinear multiagent systems [17]. Combining fuzzy logic systems (FLSs) with adaptive backstepping control scheme, remarkable success has been achieved for uncertain fractional-order nonlinear systems [18] and switched nonlinear systems [19]. At the same time, with the help of online function approximation technique of NNs or FLSs or MTN, active efforts have been devoted to control of large-scale nonlinear systems, see, for instance, [20,21] for common large-scale nonlinear systems, [22][23][24] for large-scale stochastic nonlinear systems, [25][26][27][28] for switched nonlinear largescale systems. It is worth pointing out that MTN, as a novel type of neural network, is feasible and has broad use for treating all sorts of nonlinear control problems, such as nonlinear systems [29,30], stochastic nonlinear systems [12][13][14]. Recently, authors in [23,24] proposed adaptive MTN decentralized control strategies and dealt with the tracking problems for large-scale nonlinear systems effectively. However, there are very few research results dealing in control problems of the large-scale nonlinear systems using MTN approach. Meanwhile, when the input or output of systems is limited, the traditional adaptive control framework may not be appropriate. Therefore, system analysis process which considered the input delay is very necessary.
In fact, the input delay is unavoidable in practical engineering, which is one of the factors causing system control performance. Therefore, system analysis process which considered the input delay is very necessary. For example, the input delay issue was discussed for discrete time systems [31], high-order nonlinear systems [32], nonlinear strict-feedback systems [33], and nonlinear stochastic systems [34]. More recently, for a class of large-scale nonlinear systems subject to input delay, the authors in [35] proposed a new output-based decentralized tracking control approach. For a class of switched large-scale nonlinear systems subject to input delay, the authors in [36] developed a commandfiltered backstepping-based adaptive fuzzy decentralized event-triggered control scheme. On the other hand, prescribed performance control, by virtue of it supe-riority in improving performance, has gradually been infiltrated into the area of control, which has become a popular issue, and several adaptive prescribed performance control schemes were developed for nonlinear systems with hysteretic actuator [37], nonlinear systems with nontriangular structure [38], stochastic nonlinear systems [39], uncertain nonlinear MIMO System [40]. However, there is no relevant result on largescale nonlinear systems subjected to both input delays and prescribed performance. Consequently, for largescale nonlinear systems, investigate the input delay and prescribed performance control problems in a unified framework, which inspires the research of this paper.
Based on above analyses, this research tries to study the problem of tracking control for a class of large-scale nonlinear systems subject to input delays under prescribed performance. Integrating the idea of prescribed performance control into the controller design process and employing the MTNs to approximate the combination of nonlinear functions. Based on this solution, an effective adaptive MTN decentralized control algorithm is developed. Compared with the previous results, three main innovations of this research are listed as follows: 1. For the first time, the input delay and the prescribed performance control are simultaneously considered in this study for large-scale nonlinear systems, and a fresh decentralized adaptive control algorithm is developed. Although many adaptive MTN-based control strategies have been proposed for nonlinear systems with input delay [13,41] and largescale nonlinear systems [23,24] and prescribed performance control of nonlinear systems [42], the above approaches cannot be directly applied to solve the systems considered in this paper. Thus, the application scope of MTN-based control method is expanded. 2. The method of dealing with input delay in the work of [43,44] is extended to large-scale nonlinear systems. Specifically, a new variable is introduced for the sake of solving the problem of input delay at the end of step of each subsystem. In addition, although the problem of input delay was investigated in [45][46][47][48], they were only focus on single input single output (SISO) nonlinear systems. Authors in [9][10][11] focused completely on general large-scale systems, without considering the existence of inputoutput constraints. In addition, despite the problem of input delay was investigated in [35,49,50] for large-scale nonlinear systems, they failed to consider the prescribed performance control. In other words, the problems studied in this paper are more general. 3. The computation complexity of the proposed control algorithm is dropped considerably through the following three aspects: Firstly, the problem of input delay is solved well by introducing new variables, and then the controller design process is greatly simplified. Secondly, the property of MTN is utilized to deal with the unknown nonlinear functions in the systems, which has the advantages of simple structure and good approximation performance. Lastly, a new programmatic design process of control strategy is developed via backstepping, so the construct of Lyapunov functions and the design of controller are systematized and structured, which simplify the process of controller design.

Research objects and objectives
Consider a class of large-scale nonlinear systems with input delays consists of N subsystems as follows in systems (1), ς i, j indicates the state of the ith system withς i, j = ς i,1 , . . . , ς i, j T ∈ R j and ς i = ς i,1 , . . . , ς i,n i T ∈ R n i , y i indicates the output of the ith system withȳ = [y 1 , . . . , y N ] T ∈ R N , for all i = 1, . . . , N and j = 1, . . . , n i . In addition, h i, j (·) : R j → R indicate unknown smooth nonlinear functions, and satisfy the initial conditions as h i, j (0) = 0. φ i, j (ȳ) denote the interconnections between subsystems. u i ∈ R indicate the system input and τ i are the input delay. d i, j (t) indicate the external bounded disturbance. For the given reference signals y i,r , i = 1, . . . , N , the main task of this study is to design a control strategy for system (1) to achieve the following goals at the same time: (i) A satisfactory control effect is achieved in parallel with all the variables of closed-loop system are bounded. (ii) All the tracking errors e i = y i − y i,r , i = 1, . . . , N are constrained by the prescribed performance function all the time, respectively.

Preparatory knowledge
Firstly, new variables are introduced to overcome the influence of input delay, for this reason, the following Lemma is given.
and ς i,n i +1 satisfieṡ where κ i = 2 τ i . Proof For more information on the proof of Lemma 1, please refer to the work of Li et al. [43].
Remark 2 Obviously, the control objective (ii) can be realized by way of |e i | < ψ i , for i = 1, . . . , N . Therefore, define the error transformation function as , then, the derivative of i can be obtained aṡ Finally, the following Assumptions are given to facilitate controller design.

Multi-dimensional Taylor network
In consideration of nonlinear functions cannot be directly used for controller design, a new type of network approximation method, namely MTN, is introduced to approximate the unknown functions. MTN is a three-layer neural network, and the structure of MTN with n inputs and m highest power is presented in Fig. 1, where z 1 , z 1 , . . . , z n are the inputs of MTN. The basic theories and knowledge of MTN have been presented in [23,24,51]. In the article, only a useful lemma is given.
Lemma 2 [23,24]: MTN can approximate any continuous function f (z) with any precision ε > 0 on a bounded closed set ⊂ R n , namely,

denote input vector and weight
vector of MTN, respectively.

Main results
Before the controller design, the following coordinate transformation is introduced:

Backstepping-based adaptive decentralized control scheme
Step i, 1: Choose the first alternative Lyapunov function ϒ i,1 as follows: According to Assumption 3 and Young's inequality, it is easy to get the following inequalities where ξ i,1 > 0 and ζ i,1 > 0.

Remark 4
In inequality (16), μ i (·) is auxiliary function and it is designed for the stability analysis of the system rather than controller design. Therefore, its value and structure are not required.
In order to facilitate the design of the controller, unknown functionh i,1 should be approximated by a continuous function. Therefore, in the light of Lemma 2, for any given ε i,1 > 0, there must be a MTN as Substituting (17) into (16), and taking ς i,2 = e i,2 + β i,1 into account, one haṡ In addition, the following inequality holds Then, substituting (19) into (18), one haṡ In view of (20), design the intermediate control signal β i,1 as follows where constant r i,1 satisfies r i,1 > 0.
Remark 5 According to the definition of ψ i (t), i and P m i,1 , the intermediate control signal β i,1 is still valid for any value of e i .
Combining (22) and (23), one haṡ Step i, 2: Choose the second alternative Lyapunov function ϒ i,2 as follows: whereθ i,2 = θ i,2 −θ i,2 represents the parameter error. Then, calculating the derivative of function ϒ i,2 with respect to time, one haṡ According to Assumption 3 and Young's inequality, it is easy to get the following inequalities where ξ i,2 > 0 and ζ i,2 > 0.
Substituting (27) and (28) into (26), the following inequality holdṡ In order to facilitate the design of the controller, unknown functionh i,2 should be approximated by a continuous function. Therefore, in the light of Lemma 2, for any given ε i,2 > 0, there must be a MTN as θ T i,2 P m i,2 e i,2 with a bounded error δ i,2 e i,2 , satisfyinḡ where e i,2 = e i,1 , e i,2 T represents the input vector of MTN, and δ i,2 e i,2 ≤ ε i,2 . Substituting (24) and (30) into (29), and taking ς i,3 = e i,3 + β i,2 into account, one haṡ In addition, the following inequalities holds Substituting (32), (33) into (31), one haṡ In view of (34), design the intermediate control signal β i,2 as follows where constant r i,2 satisfies r i,2 > 0. Then, substituting (35) into (34), one haṡ In view of (36), design the adaptation lawθ i,2 as followṡ where constant γ i,2 satisfies γ i,2 > 0. Combining (36) and (37), one haṡ Step i, l: Choose the ith alternative Lyapunov function ϒ i,l as follows: whereθ i,l = θ i,l −θ i,l represents the parameter error. Then, calculating the derivative of function ϒ i,l with respect to time, one haṡ According to Assumption 3 and Young's inequality, it is easy to get the following inequalities where ξ i,l > 0 and ζ i,l > 0.
Substituting (41) and (42) into (40), the following inequality holdṡ whereh i,l = h i,l −β i,l−1 + 1 2 ξ 2 i,l e i,l + 1 2 ζ 2 i,l e i,l + 3 2 e i,l . In order to facilitate the design of the controller, unknown functionh i,l should be approximated by a continuous function. Therefore, in the light of Lemma 2, for any given ε i,l > 0, there must be a MTN as θ T i,l P m i,l e i,l with a bounded error δ i,l e i,l , satisfyinḡ where e i,l = e i,1 , . . . , e i,l T represents the input vector of MTN, and δ i,l e i,l ≤ ε i,l . Substituting (44) into (43), and taking ς i,l+1 = e i,l+1 + β i,l into account, one haṡ In addition, the following inequalities holds Substituting (38), (46) and (47) into (45), one haṡ In view of (48), design the intermediate control signal β i,l as follows where constant r i,l satisfies r i,l > 0. Then, substituting (49) into (48), one haṡ In view of (50), design the adaptation lawθ i,l as followṡ where constant γ i,l satisfies γ i,l > 0. Combining (50) and (51), one haṡ Step i, n i : Choose the n i -th alternative Lyapunov function ϒ i,n i as follows: whereθ i,n i = θ i,n i −θ i,n i represents the parameter error. Then, calculating the derivative of function ϒ i,n i with respect to time, one haṡ According to Assumption 3 and Young's inequality, it is easy to get the following inequalities where ξ i,n i > 0 and ζ i,n i > 0.
Substituting (55) and (56) into (54), the following inequality holdṡ In order to facilitate the design of the controller, unknown functionh i,n i should be approximated by a continuous function. Therefore, in the light of Lemma 2, for any given ε i,n i > 0, there must be a MTN as θ T i,n i P m i,n i e i,n i with a bounded error δ i,n i e i,n i , satisfyinḡ where e i,n i = e i,1 , . . . , e i,n i T represents the input vector of MTN, and δ i,n i e i,n i ≤ ε i,n i . Substituting (58) into (57), one haṡ In addition, the following inequalities holds Then, substituting (60) into (59), one haṡ In view of (61), design the control signal u i as follows where constant r i,n i satisfies r i,n i > 0. Then, substituting (62) into (61), one haṡ In view of (63), design the adaptation lawθ i,n i as followṡ where constant γ i,n i satisfies γ i,n i > 0. Combining (63) and (64), one haṡ To sum up, the detailed of the proposed control method is exhibited by the block diagram in Fig. 2. 3.2 Stability theorem of closed-loop system Proof According to the above backstepping, choose the Lyapunov function ϒ as follows: For the closedloop system, the Lyapunov function as follows: Based on (65), the result of derivative of ϒ is given aṡ In view of γ i, jθ Moreover, the following inequality holds through choosing appropriate auxiliary functions μ i (·), namely Then, employing (68) and (69), one haṡ Next, denote a = min {a 1 , . . . , a N } with a i = min 4r i,1 , 2r i, j , γ i, j j = 1, 2, . . . , n i and b = 1 Then, inequality (70) can be written aṡ Furthermore, inequality (71) implies that Recalling the definition of ϒ in (66), based on (72), we can draw the conclusion that all the signals in the closed-loop system are bounded.
On the other hand, according to (72), the following inequality can be easily obtained Then, for ∀i = 1, . . . , N , the following equation can be obtained from Moreover, the following inequality holds through choosing appropriate design parameters and initial conditions From (74) and (75), it follows that Based on (76), we easily obtain Thus, it is concluded from (77) that all the tracking errors e i = y i − y d,i , i = 1, . . . , N are constrained by the prescribed performance function all the time, respectively.

Remark 6
In Theorem 1, it needs to be emphasized that the parameters in control structure, including r i, j in control signals β i, j and γ i, j in adaptation laws inθ i, j with i = 1, . . . , N ; j = 1, . . . , n i , satisfy the conditions of nonnegative in theory. However, those parameters needs to select appropriately for the purpose of obtaining higher control accuracy and satisfactory control effect in practical application.
Remark 8 Examples 1 and 2 show that the proposed control method is validity and rationality for both structured system and practical systems.

Conclusion
For the first time, a new adaptive decentralized control scheme is proposed for large-scale nonlinear systems to address the problem of input delays and prescribed performance control in the same framework. A new variable is introduced for the sake of solving the problem of input delay at the end of step of each subsystem. It is particularly important to note that the computation complexity of the proposed control algorithm is Fig. 11 The responses of e 1 and ψ 1 (t) of system (79) Fig. 12 The responses of e 2 and ψ 2 (t) of system (79) dropped considerably through three aspects, including copy with the input delay by introducing a new variable, low calculation with the benefit of the MTNs' simple structure and the construct of Lyapunov functions and the design of controller are highly stylized.
More recently, finite-time control has been a rapid increase interest because of its advantage of finite-time convergence. Therefore, study on extending the pro- posed control algorithm to the finite-time control of large-scale systems will be our future research direction.
Data availability Data sharing is not applicable to this article as no datasets were generated or analysed during the current study.

Conflict of interest
The author declares that he has no conflict of interest.