Adaptive Control for Complex Systems with Dynamics and Time-Varying Powers

Tis work focuses on presenting a control algorithm to investigate nonlinear systems, which contain time-varying powers, inverse dynamics, and uncertainties. First, some appropriate transformations are introduced to obtain a new system. Ten, a Lyapunov function, which covers quadratic and high-order components, is recursively constructed for control design. Subsequently, by introducing the neural networks, the uncertain functions encountered during the design are approximated. Based on the inequality techniques, the nonlinear terms are skillfully estimated. By defning the bounds of some unknown parameters and using the adaptive technique, some virtual controllers are selected in each step to dominate the nonlinear functions and guarantee that the derivative of the Lyapunov function satisfes the required form. Finally, a new adaptive controller is constructed and semiglobal practical fnite time stability (SGPFS) is guaranteed. Te proposed approach is verifed with a numerical example.


Introduction
Control of complex nonlinear systems is viewed as a challenging issue in practical engineering. Since different systems are modeled with diferential equations that have disparate nonlinear characters, the related control schemes constantly vary from each other. Recently, plenty of outstanding approaches have been raised, as shown in the sliding mode control [1], intelligent control [2,3], event-based control [4], and so on. It should be mentioned that complex systems often have diversifed uncertainties, including unknown parameters, uncertain dynamics, and unpredictable disturbances. Te uncertainties actually add more obstacles for the system analysis and control synthesis. Besides, practical systems often sufer from time-varying powers [5,6], which lead to the inapplicability of conventional methods of systems with constant powers. So far, when systems have timevarying powers, lots of control challenges have not been solved. It is interesting to study such systems and raise a feasible control strategy.
For uncertain systems, the adaptive control strategy has been viewed as one of the efective tools for control design, see [7][8][9]. During the control design steps of nonlinear systems described by diferential equations, this strategy utilizes the idea of certain equivalence and always results in a dynamic controller. Particularly, by presenting an adaptive backstepping method [10], the global asymptotic stability was guaranteed for systems that sufered from unknown parameters. By proposing a dynamic event-triggered control-based output-feedback control method, Cao et al. [11] studied the adaptive issue of systems with immeasurable states and input delay. By proposing an adaptive NN fxedtime control method, Cao et al. [12] further designed the controller for systems that included dynamic uncertainties and communication resources. By raising an adaptive fuzzy command fltered method, Li et al. [13] considered systems with uncertain dynamics and ensured the system to be semiglobally stable. Recently, high-order nonlinear system has been a hot topic and many excellent results have been reported, see [14][15][16]. Specially, by introducing a Lyapunov-Krasovskii functional and employing the adaptive neural network control technique, Duan et al. [14] skillfully designed the adaptive controller for delayed systems. Utilizing one power integrator strategy, Niu et al. [15] studied the adaptive stabilization issue of stochastic systems with arbitrary switching. Based on the idea of homogeneous domination and dynamic control approach, Shen and Zhai [16] designed the controller for uncertain high-order systems. In fact, the system powers in the above works are assumed to be constants. When the system powers are timevarying functions, there are some results discussing the control issues. For instance, Chen et al. [17] presented a feedback control approach for systems that involved timevarying powers. Yoo [18] discussed uncertain systems with time-varying functions and investigated the fault accommodation control issue. However, fnite time control issues are still challenging, especially when the system powers vary with a large range.
Finite time control has received lots of concerns in last years. Such kind of control has better system responses such as faster convergent rate, higher tracking precision, and better robustness. Tere are many splendid results for this control issue. Specially, for low-order systems, Yu et al. [19] studied the tracking control problem by presenting a fnite time command fltered backstepping method. Polyakov et al. [20] considered the stability problems and provided a robust control method. As for high-order systems, Gao et al. [21] discussed the output feedback control approach, Chen et al. [22] further considered the output constraint and raised the output feedback method to solve fnite stabilization problem. For system with uncertainties, Li et al. [23] investigated the adaptive fnite time regulation for lower-order systems. Sun et al. [24] raised one fast fnite time control approach and applied it to systems with constant powers. However, the adaptive fnite time control issue is still open for systems that have dynamics and time-varying powers. Naturally, we raise the following problem: Can we regulate the uncertain system with dynamics and time-varying powers via a fnite time control approach?
We will consider the above problem. Te study has two advantages as follows: (i) Te considered system has a more general form. In practical engineering, some dynamic models have time-varying powers, see the model of a boilerturbine unit in [17] and the underactuated mechanical system in reference [25]. However, few results considered control issue of systems with time-varying powers. Also, due to the inaccuracy of measurement or the modeling errors, some inverse dynamics cannot be directly neglected. Besides, the practical systems are often in nontriangular structure and sufer from multiple uncertainties. Tese factors motivate us to study the more general system, which contains time-varying powers, inverse dynamics, nontriangular structure, and uncertainties. Te system is hence more general.
(ii) A new adaptive control approach is presented. Noting that the existing methods of systems are mainly for constant powers, they cannot be applied to the system of this work. Besides, since the fnite time control of the system has inverse dynamics, nontriangular structure, and uncertainties are still challenging, we are inspired to raise a new adaptive control approach. In this work, a Lyapunov function, which includes quadratic and high-order components, is constructed. By estimating the complex nonlinear terms with the neural network and employing the adaptive control design, the adaptive controller is skillfully constructed for the considered system. Te merits of the method lie in two aspects. One is that, it provides an efective solution for adaptive fnite control of complicated nonlinear system. As can be seen, for the complicated system (1), the fnite time control problem is still challenging. Te presented method overcomes a series of obstacles and fnally provides a feasible solution to solve the above problem. Te second is that the proposed method adopts few parameter estimations and does not introduce complicated basis functions of neural network in the control input. Hence, the designed adaptive controller has a simple form.

Problem Formulation
We study the system as follows: x n � a n (t, x)⌈u⌉ s n (t) + f n (ϱ, ξ, x), where ξ ∈ R m and x � [x 1 , . . . , x n ] ⊤ ∈ R n are state vectors of the system, ϱ ∈ R l denotes a parameter vector, g ∈ R m represents a vector of continuous functions, and u ∈ R denotes the input. For 1 ≤ i ≤ n, a i > 0 denotes the control coefcients, f i (·) are unknown continuous functions, s i (t) > 0 are unknown system powers, and ⌈·⌉ s i (t) � sign(·)| · | s i (t) . System (1) describes a class of complex nonlinear systems. It covers the inverse dynamics, which generates because of the inaccuracy of mathematical modeling or measuring errors and always leads to poor system responses. Also, it contains time-varying powers, which are involved in many models in practical engineering. Besides, the system has unknown coefcients and involves unknown functions that are in nontriangular form. In practical life, many dynamic models can be transformed into system (1). Tus, the research of such kind of systems is of practical signifcance. However, the corresponding control issue is still challenging. In this work, we will provide a new adaptive control strategy for it.
Assumption 2. Tere exists s m ≤ s i (t) ≤ s h , where 0 < s m ≤ 1 and s h ≥ s m are constants that belong to R odd � q 1 / q 2 | q 1 > 0 and q 2 > 0 are odd integers}.
where d 0 is a positive constant and ϕ 0 (·) ≥ 0 is a continuous function.
Lemma 6 (see [6]). Let 0 ≤ l 1 ≤ · · · ≤ l n , m 1 > 0, . . ., and m n > 0. For y ∈ R, there exists the following: Remark 7. We emphasize two points as follows: (i) In this paper, the system powers refer to s i (t), 1 ≤ i ≤ n in (1) and are not the order n. Tey are time-varying functions, which are more general than the constants in [14,15,22,24]. From Assumption 1, we see that they can belong to a larger interval [s m , s h ], which cover the powers smaller than one or bigger than one. (ii) In engineering, there are practical models, which have the form of system (1), see the system [17,26]. System (1) actually provides a general form to describe similar dynamic models.
Remark 8. Assumptions 1-3 of system (1) are reasonable and much weaker than those of the existing research. Assumption 1 implies that the bounds of control coefcients can be unknown. It is more general than the known cases of [2,21]. Assumption 2 shows that the powers can be changing in a larger scope. Assumption 3 implies that the inverse dynamics of system (1) satisfes a weaker stability condition. It should be emphasized that the control issue under Assumptions 1-3 is still difcult. Next, we will study the fnite control issue and raise an adaptive control strategy.

Adaptive Control Design
Te adaptive control method of this paper adapts to the control law according to the parameter variation. As can be seen, there are many uncertainties in the nonlinear functions. To design the controller, we adopt the neural networks to approximate these unknown functions and subsequently introduce some unknown ideal constant weight vectors. During the control design steps, the obtained nonlinear bounds contain unknown parameters, see (7), (9), (24), and (26). By defning the maximum value θ i of those unknown parameters in each step, we can design adaptive laws (18) and (28) to approximate parameters θ i , 1 ≤ i ≤ n. Ten, we construct the virtual controllers with a recursive method by using θ i and a new Lyapunov function. In the last step, the adaptive controller is successfully designed. For the details, see Figure 1.

Design the actual adaptive controller
and analyze the stability of the closed-loop system. u = -β n z n , β n = 3r n 2 1/sm , r n = θ n ϕ n + ρ,θ
Step k (2 ≤ k ≤ n). In step k − 1, we assume that there exist transformations as follows: a candidate Lyapunov function V k− 1 , and the adaptive laws such that where β j and ϕ j are smooth functions and σ j , ϵ j and d k− 1 are positive constants. In this step, we defne θ k (t) � θ k (t) − θ k , where θ k denotes the unknown constant and θ k (t) is an estimation of θ k , and we choose V k � V k− 1 + 1/λ 1 (z 2 k + z s+1 k ) + 1/(2ϵ k θ 2 k ), where ϵ k > 0 is a constant. We obtain the following: Following the proof in Appendix, we have the following: where θ k1 > 0 is an unknown constant, d k0 is a positive constant, and φ k1 > 0 is a smooth function. It is deduced that

Complexity
Utilizing Lemma 4, there exists a neural network the approximation error, and |η k (t)| ≤ δ. By the proof in Appendix, it follows that where θ k2 and d k1 , . . . , d k4 are positive constants and ϕ k2 > 0 is a smooth function. Defning θ k � max θ k1 , θ k2 and φ k � φ k1 + φ k2 , substituting (24) and (26) into (23), and considering ⌈x k+1 ⌉ s k � ⌈x k+1 ⌉ s k − ⌈π k ⌉ s k + ⌈π k ⌉ s k , we have the following: Now, choosing the adaptive law, we get the following: where σ k ≥ 0 is a constant and selecting the virtual control, we get the following: where r k � θ k ϕ k + 2ρ and the derivative of V k satisfes the following: where d k � d k− 1 + 4 j�0 d kj . Tis completes the recursive design.
Actually, we can extend the method to the system as follows: ⋮ _ x n � a n (t, x)⌈u⌉ s n (t) + f n (ϱ, x), where the symbols are given in (1). If Assumptions 1 and 2 hold, we can also design a fnite-time adaptive controller as (31). For this case, we select V n � 1/λ 1 n j�1 (z 2 j + z s+1 j ) + n j�1 1/(2ϵ j θ 2 j ). Utilizing the same design procedures, we can obtain a similar conclusion as Teorem 9. □ Remark 10. During the design of the controller, we adopt some strategies to handle the constraints of the system. To deal with the unknown coefcients a k , we introduce a term 1/λ 1 in the Lyapunov function. Since a k /λ 1 ≥ 1, we can obtain ) by using the defnition of π k . Besides, by employing the inequality of Lemma 4, the nonlinear terms that contain a k can be estimated skillfully. Ten, we can defne an enough big constant θ k and design the adaptive law θ k to estimate it and utilize θ k for the control design. For the constraint of timevarying power s i (t), we introduce quadratic components and higher-order components in the Lyapunov function, so that nonlinear terms with s i (t) can be estimated with the appropriate bounds, see (15), (24), and (26) for instance. For the constraint of dynamics and nonlinear functions, we employ the inequality skills and introduce the neural network Φ ⊤ i S i (Ξ) to approximate uncertain terms in each design step, see (6) and (26).
Remark 11. Tere are some advantages and limitations of this work. Te advantages are in three aspects. In detail, this work provides a universal method for system powers in the intervals (0, 1) and [1, s h ]. Also, this work considers complicated nonlinear conditions and provides an adaptive controller with a simple form. In addition, the system is rendered semiglobal fnite time stable, which is more general than the semiglobal stable. Tere are also some limitations of this work. For example, despite that the system is complicated, we do not consider the infuence of time delay and random disturbances. Also, this work achieves a semiglobal result. A better result is to obtain fnite time stabilization globally. Besides, we consider the problem via the state feedback and do not consider the output feedback control.

Conclusion
Tis work has studied the regulation of complex systems. Diferent from the reported researches, the considered systems involve time-varying powers, inverse dynamics, and uncertainties. Besides, the system powers can be either smaller than one or bigger than one. By constructing a new Lyapunov function and utilizing the adaptive control algorithm and neural network, a semiglobal fnite time adaptive controller is successfully designed. Indeed, there are also some interesting but challenging problems. For example, when the system contains time delay, can we design the adaptive controller to guarantee the stability of the system? Can we develop the proposed approach to study stochastic systems? When the system output is the only measured signal, can we construct the state observer and design the output feedback controller?