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Disturbance observer-based fuzzy adaptive optimal finite-time control for nonlinear systems

Disturbance observer-based fuzzy adaptive optimal finite-time control for nonlinear systems APPLIED MATHEMATICS IN SCIENCE AND ENGINEERING 2023, VOL. 31, NO. 1, 2199211 https://doi.org/10.1080/27690911.2023.2199211 Disturbance observer-based fuzzy adaptive optimal finite-time control for nonlinear systems Zidong Sun, Li Liang and Wei Gao School of Information Science and Technology, Yunnan Normal University, Kunming, People’s Republic of China ABSTRACT ARTICLE HISTORY Received 29 December 2022 This paper investigates the issue of disturbance observer-based Accepted 30 March 2023 fuzzy adaptive optimal finite-time control in light of the backstep- ping approach for strict-feedback nonlinear systems with bias fault KEYWORDS term and full state constraints. Considering that external disturbance Adaptive dynamic and bias fault signal can affect the stability of control and control programming; fuzzy quality, a disturbance observer is constructed to track the external adaptive optimal finite-time disturbances and bias fault online. A disturbance observer-based control; external disturbance observer; state observer finite-time control strategy is proposed to achieve optimized con- trol by utilizing the fuzzy logic system approximation-based adaptive MATHEMATICS SUBJECT dynamic programming method under the critic-actor framework. CLASSIFICATIONS The purpose of the critic is to evaluate control performance and the 34H05; 93C10; 93C42 role of the actor is to execute control behaviour. In addition, it is proved that the proposed fuzzy adaptive optimal finite-time scheme not only realizes all signals in closed-loop system are bounded, but also ensures that system states are restricted within specific sets. Finally, simulation results are shown to demonstrate the effective- ness of the proposed control strategy. 1. Introduction In the past few decades, the research on controller designing for nonlinear systems speedy developed neural networks and fuzzy approximation characteristics [1–4]. The incipient control strategies facilitate the control objectives for nonlinear systems are realized in a inn fi ite time, which means that the stability of systems fails to be guaranteed in a n fi ite time. However, in practical problems, it is imperative to achieve superior control performance inafinite time. Dissimilar to asymptotic stability, n fi ite-time stability converges fast without the requirement of long-term transient response, so it is widely concerned and applied to practical systems such as aircraft flight systems [ 5], cart-pole systems [6], teleoperation systems [7] and so on. Bhat and Bernstein [8] explained n fi ite-time stability based on Lyapunov theory and constructed feedback controller for nonlinear systems to achieve n fi ite-time stability. Qiu et al. [ 9] established a fuzzy finite-time controller for nonlinear systems and realized the tracking error is limited to a bounded set. Li et al. [10]studied CONTACT Wei Gao gaowei@ynnu.edu.cn School of Information Science and Technology, Yunnan Normal University, Kunming 650500, People’s Republic of China © 2023 The Author(s). Published by Informa UK Limited, trading as Taylor & Francis Group This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons. org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The terms on which this article has been published allow the posting of the Accepted Manuscript in a repository by the author(s) or with their consent. 2 Z. SUN ET AL. the problem of finite-time containment control involving unmeasurable states for non- linear multiagent systems with input delay. Zhang et al. [11] designed an adaptive fuzzy control strategy based on finite-time stability criterion for nonlinear systems with unavail- able states, in which all states are conn fi ed to restricted sets. Meng et al. [ 12]developed a n fi ite-time quantized controller for dealing with the nonlinear systems contingent on unknown control directions, which constructed state and disturbance observers simul- taneously. Sui et al. [13] constructed an event-trigger adaptive ni fi te control scheme by combining backstepping and varying threshold condition for stochastic nonlinear systems with unmodelled dynamics. Saravanakumar et al. [14] rst fi investigated the n fi ite-time sta- bility problem. The authors in [15–17] studied the issue of n fi ite-time control for nonlinear systems in accordance with full-state constraints. Nguyen et al. [18] introduced a fuzzy con- trol strategy for parallel manipulators to guarantee that the error of tracking converges fast in a ni fi te time. Wang et al. [ 19] constructed a n fi ite-time control algorithm for stochas- tic nonlinear systems with actuator faults. Nevertheless, the above literature take the ni fi te time into account and the designed control strategies ensure that the control objective can be achieved within the n fi ite time. However, these literature do not involve the optimal controltodealwiththe problemofresourceconsumption. In recent years, optimal control has been a striking topic, and it is concerned with estab- lishing control strategies to achieve control objectives with the least resources in regard to optimal policy. In other words, the goal of optimal control is to consume the least amount of resources to achieve the control objective. For nonlinear systems, it is noteworthy that the resolve of Hamilton–Jacobi–Bellman (HJB) equation is imperative in the control pro- cess, but it is problematic to get analytic solutions because of the dynamic uncertainty and strong nonlinearity in the actual control. To overcome this problem, the dynamic program- ming (DP) was developed by Bellman [20]. NotedthatalthoughDPisaneeff ctivetoolfor obtaining optimal solutions, it is prone to the curse of dimensionality, that is, increasing dimensions bring computational disasters. Adaptive dynamic programming (ADP)-based algorithms were demonstrated to efficiently conquer the problem. Werbos [ 21]developed ADP for discrete systems and Abu-Khalaf and Lewis [22] proposed ADP control scheme for continue-time nonlinear systems where the neural networks approximation structure was established to estimate the value function of the HJB equation. Vamvoudakis et al. [23] developed an online actor-critic scheme combined with neural networks for continue-time systems. Wen et al. [24] designed an optimized formation control strategy in reference with identifier-actor-critic structure and approximation characteristic of fuzzy logic systems. Li and Li [25] introduced a fuzzy adaptive optimal fault-tolerant control scheme for stochastic multiagent systems to incorporate Butterworth low-pass filter, which leverages to compen- sate for negative influence caused by nonlinear fault on the system. Wen et al. [ 26]reviewed the fuzzy adaptive optimized control issue subject to unmeasured states with nonlinear systems, in which the state observer satisefi s the Hurwitz condition that sidestep constant designing. Lan et al. [27] introduced an adaptive optimal formation control technology for multiagent systems with unmeasurable states. Li et al. [28] designed a neural network adaptive optimized control algorithm exposing to immeasurable states and constrained states with nonlinear systems. The authors in [29–31] discussed the problem of adaptive optimal control for nonlinear systems. Wen et al. [32] built an adaptive optimal control scheme by virtue of identifier–critic–actor architecture. To achieve high precision control APPLIED MATHEMATICS IN SCIENCE AND ENGINEERING 3 requirements, it is necessary to consider the influence of external disturbance on the sys- tem, which makes sense to design an optimal finite-time control strategy for the controlled system with external disturbance. However, the proposed approaches in [20–32]ignored the external disturbance in the controller system. It is a prominent prospect to eliminate the influence of external disturbances on the sys- tems [33–38]. Ji et al. [39] designed an adaptive fault-tolerant optimized formation control for multiagent systems with disturbances, in which a disturbance observer was constructed to alleviate the negative eeff cts of external disturbance. Liu et al. [ 40]developed an adap- tive control scheme for Markovian jump systems that suffer from external disturbances and actuator faults, in which external disturbances contain matched and mismatched part. Xu et al. [41] designed a neural network disturbance observer for strict-feedback systems to achieve good control performance in spite of unknown dynamics and time-varying distur- bance. Song and Lewis [42] introduced a robust optimal control scheme with a disturbance observer to estimate disturbances for nonlinear systems. Zerari and Chemachema [43] studied the continuously stirred tank reactor systems containing external disturbances and introduced the compensator to confront the influence of external interference in the designed control strategy. Ran et al. [44] proposed a disturbance rejection optimal control for nonlinear systems, in which the perturbances and other uncertainties are estimated by the designed disturbance observer. Chen and Ge [45] presented an adaptive neural control strategy for nonlinear systems with unmeasured states, hysteresis and disturbances. How- ever, the foregoing advancements rarely studied the optimized finite-time control problem based on fuzzy systems for nonlinear systems with full-state constraints and external disturbances. Motivated by the aforementioned researches, this paper considers the fuzzy adap- tive optimal n fi ite-time control issue for nonlinear systems with full-state constraints, external disturbance and bias fault. Combining backstepping and ADP tricks, a fuzzy adap- tive optimal n fi ite-time control strategy is designed. The unknown system functions are approximated by the fuzzy logic system and disturbance observers are designed to solve the influence of external disturbances and bias fault signal. Virtual and actual controllers are introduced by being incorporated with actor-critic architecture and backstepping framework. The main contributions are summarized as follows. (1) In comparison with [25,27,28], the designed scheme via actor-critic framework and n fi ite-time stability theory can ensure the controller systems not only has a faster con- vergence rate and limits the tracking error derives within a temperate area of the origin inafinite time. (2) An disturbance observer-based fuzzy finite-time strategy based on actor-critic struc- ture is proposed. Discriminating the control schemes in [26,32,46], it not only consid- ers the full-state constraint for strict-feedback nonlinear systems but also addresses the eeff cts of bias fault and external disturbance on the system by designing the disturbance observer. The remainder of this article is organized as follows. In Section 2,the system descrip- tion and preliminary knowledge are presented. Section 3 proposes an observer-based fuzzy optimal control scheme. Subsequently, stability analysis is given in Section 4.Asimulation example is shown in Section 5. Finally, the conclusion is presented in Section 6. 4 Z. SUN ET AL. 2. Preliminaries and problem statement 2.1. System description The strict-feedback nonlinear system consisting of unknown dynamics is formulated by x ˙ = x + φ (x ) + D (t), i i+1 i i 1 x ˙ = u + φ (x ) + D (t), (1) n n n n y(t) = x,1 ≤ i ≤ n, T i where x = [x , x , ... , x ] ∈ R (i = 1, ... , n) represent the states vector of the system, i 1 2 i y ∈ R is the control output. φ (x ) ∈ R are the unknown functions and D (t) denote the i i i external disturbances where i = 1, ... , n. All system states are restricted to a compact set, that is, x < |k | where |k | are positive constants with i = 1, 2, ... , n. i ic ic In actual engineering applications, the actuator bias faults often occur during the operation of the actuator. Thus we consider the actuator bias fault signal as u = u + u,(2) where u denotes the control input and u represents the actuator bias fault signal. Suppose u is boundedand thereisaconstant F that u ≤ F. f f Assumption 2.1 ([39]): The external disturbances D (t) are unknown and bounded, ∗ ∗ ¯ ˙ ¯ i.e. there exist real numbers D and D satisfying |D (t)|≤ D and D (t) ≤ D where i i i i i i i = 1, 2, ... , n. Assumption 2.2 ([25]): The reference trajectory y is known and bounded, and the derivatives of the y , y ˙ ··· y are bounded. r r 2.2. Finite-time theory The following lemmas are beneficial to facilitate the design of the controller. Lemma 2.1 ([15]): For any positive numbers C > 0, C > 0, 0 <δ < 1,0 < l < 1,there 1 2 exist a C function V(x) satisfying V(x) ≤−C V − C V + M,(3) 1 2 1−δ C V (x )+lC 1 2 0 1 then the system is n fi ite-time stable, where the setting time T ≤ ln . 0 1−δ C (1−δ) 2 M C ( ) +lC 2 1 (1−l)C Lemma 2.2 ([29]): For any τ ∈ R,there existaninteger constant ξ and a positive constant 0 < q ≤ 1,one has q q n n n q 1−q |τ | ≤ |τ | ≤ n |τ |.(4) i=1 i=1 i=1 APPLIED MATHEMATICS IN SCIENCE AND ENGINEERING 5 Lemma 2.3 ([29]): There exists real variables p and p satisfying the following inequality: 1 2 −r r r 1 2 r r +r r r 1 2 2 r +r 1 2 1 2 |p | |p | ≤ r |p + r |p | .(5) 1 2 3 2 1 3 r + r r 1 2 1 where r ,r and r are positive constants. 1 2 3 2.3. Fuzzy logic system A host of real-world systems exit unknown dynamics which aeff ct the control performance, and fuzzy logic system approximation approach facilitates to solve the negative eeff ct prob- lem of unknown dynamics. The knowledge base constitutes IF–THEN rules which are stated in the following forms: k k k k R :If x is G , x is G , ... , x is G , 1 2 n 1 2 l then y is Q , k = 1, 2, ... , N, T l k where x = (x , x , ... , x ) ∈ R and y are the input and output of fuzzy logic system. G 1 2 and Q represent fuzzy sets, and N denotes the number of rules. Then the fuzzy logic system can be described by N l k k y ¯ μ (x ) k=1 i=1 y(x) = ,(6) N l μ (x ) k=1 i=1 G k k k where μ and μ are fuzzy membership functions, and y ¯ = max μ (y). y∈R G Q The basis functions are den fi ed as μ (x ) i=1 G S (x) = .(7) N l μ (x ) k=1 i=1 T T Denote ℵ= (ℵ , ℵ , ... , ℵ ) ,and S(x) = (S (x), S (x), ... , S (x)) ,and (6)can be 1 2 N 1 2 k stated as follows: y(x) =ℵ S(x).(8) Lemma 2.4 ([24]): Let f (x) be acontinuousfunctiondenfi edoncompact set ℘.There exists a positive constant ϕ that satisfies the following inequality: sup |φ(x) −ℵ S(x)|≤ ϕ.(9) x∈℘ By means of (9), the following fuzzy logic systems are served to approximate the functions φ (i = 1, 2, ... , n): ˆ ˆ φ (x |ℵ ) = ℵ S (x ), (10) i i i i i ˆ ˆ where x represents the approximation of x and ℵ denotes the estimations of ℵ .The ideal i i i i weight vectors ℵ can be described as ∗ T ℵ = arg min[sup |φ (x) −ℵ S(x )|], (11) i i i i ℵ ∈ x ∈℘ i i i i where  isabounded set. i 6 Z. SUN ET AL. The control objective of this article is to design a disturbance observer-based fuzzy adap- tive optimal n fi ite-time control scheme for nonlinear systems (1), so that (1) all the signals in systems are finite time stable; (2) all the system states are in constrained sets; (3) the output of systems admits to track the reference trajectory. 3. Optimal controller design In what follows, a fuzzy adaptive optimal n fi ite-time strategy is designed to achieve the control objective by virtue of backstepping process and actor-critic framework, in which the barrier-type function is accustomed to cost functions. Consider the bias function u in (2) and external disturbance D as wholedisturbance.The nonlinearsystems (1)can be expressed as ⎪ x ˙ = x + φ (x ) + d (t), i i+1 i i i (12) x ˙ = u + φ (x ) + d (t), n n n n y(t) = x,1 ≤ i ≤ n, where d = D (i = 1, 2, ... , n − 1) and d = u + D . i i n f n Remark 3.1: Extensive practical systems contain perturbation items, which have a nega- tive effect for control quality. Different from the strategies proposed in [ 40,47], this paper employs disturbance observer to track external disturbances online to mitigate the negative effects on the systems and improve the control performance of the systems. The term u is bounded and u ≤ F, the external disturbance D ≤ D , and we deduce d = D + u ≤ n n n f f ∗ ∗ D + F = d .Thusthe totaldisturbance d is bounded. n n Then the following coordinate transformation is introduced as z = x − y , 1 1 r z = x −ˆ α ,2 ≤ i ≤ n, (13) i i i−1 where α ˆ is the optimal virtual control and y is the desire tracking trajectory. i−1 Step 1: The time derivative of z can be yielded from (1) and (13) as z ˙ = x ˙ − y ˙ 1 1 r = x + φ + d − y ˙ . (14) 2 1 1 r Choose the inn fi ite integral barrier-type performance index function that satisefi s (1) as J (z (0)) = h (z (s), α (z )) ds, (15) 1 1 1 1 1 1 1b 2 where h (z (s), α (z )) = ψ log + α is the cost function where ψ > 0, α is the 1 1 1 1 1 2 2 1 1 k −z 1b 1 virtual controller and a compact set  ={z : |z | < |k |}.Let α be the optimal virtual 1 1 1 1b control. The optimal performance index function is constructed by the following to achieve the minimum control performance index in (15), ∗ ∗ J (z (t)) = h (z (s), α (z )) ds 1 1 1 1 1 1 0 APPLIED MATHEMATICS IN SCIENCE AND ENGINEERING 7 = min h (z (s), α (z )) ds . (16) 1 1 1 1 Taking thetimederivativeof(16),weacquire ∗ ∗ ∂J ∂J 1 1 = (x + φ + d − y ˙ ). (17) 2 1 1 r ∂t ∂z Den fi e HJB equation associating with (17) as ∂J k 1b ∗ 1 ∗2 H z , α , = ψ log + α 1 1 1 1 1 2 2 ∂z k − z 1b ∂J + (α + φ + d − y ˙ ) = 0. (18) 1 1 r ∂z By solving the equation ∂H /∂α = 0, the optimal virtual controller is obtained as 1 ∂J ∗ 1 α =− . (19) 2 ∂z ∂J With the aim of realizing ni fi te-time optimal control, is decomposed as ∂z 2δ−1 ∂J z 9 z 1 1 = 2σ + 2σ ¯ z + 2d + 1 1 1 1 2 2 2 2 δ−1 ∂z (k − z ) 2 k − z 1 1 1b 1b + 2φ + J (z ), (20) 1 1 ∗ o where σ and σ ¯ are designed positive parameters, and ℵ is the optimal weight. J (z ) = 1 1 1 f 1 2δ−1 z ∂J 9 z 1 1 1 −2σ − 2σ ¯ z − 2d − − 2φ + . Merging (19) and (20), we verify 1 2 2 1 1 1 2 2 1 δ−1 2 ∂z (k −z ) k −z 1 1b 1 1b 1 2δ−1 z 9 z ∗ 1 α =−σ −¯ σ z − d − 1 1 1 1 2 2 2 2 δ−1 (k − z ) 4 k − z 1 1 1b 1b − φ − J (z ). (21) 1 1 In (21), J and φ are unknown functions that can be approximated as o ∗T J =ℵ S + ϕ , (22) J1 J1 1 J1 ∗T φ =ℵ S + ϕ , (23) 1 f 1 f 1 f 1 ∗T where ℵ is the ideal weight and S is the basis vector. Combining (20), (21), (22) J1 J1 and (23), we get 2δ−1 z 1 ∗ 1 ∗T α =−σ −¯ σ z − d − ℵ S 1 1 1 1 J1 1 J1 2 2 δ−1 (k − z ) 2 1b 9 z 1 ∗T − −ℵ S − ϕ , (24) f 1 1 f 1 2 2 4 k − z 2 ib 8 Z. SUN ET AL. 2δ−1 ∂J z 9 z 1 1 = 2σ + 2σ ¯ z + 2d + 1 1 1 1 2 2 2 2 δ−1 ∂z (k + 2z ) 2 k − z 1 1 1b 1b ∗T ∗T +ℵ S + ϕ + 2ℵ S , (25) J1 1 f 1 J1 f 1 ∗ ∗T ∗T where ϕ = 2ϕ + ϕ .Itisnoteworthythat α is inaccessible directly since ℵ and ℵ 1 f 1 J1 1 J1 f 1 are unknown ideal weights. The estimate of the unknown function φ can be represented ˆ ˆ ˆ as φ = ℵ S .The adaptive law ℵ is constructed as 1 f 1 f 1 f 1 ˙ 1 ˆ ˆ ℵ = S − m ℵ , (26) f 1 f 1 1 f 1 2 2 k − z 1 1 where m > 0. To obtain available α , the critic-actor structure with the critic and actor adaptive laws is introduced as follows: 2δ−1 ∂J z 9 z 1 1 = 2σ + 2σ ¯ z + 2d + 1 1 1 1 2 2 2 2 δ−1 ∂z 2 (k + 2z ) k − z 1 1 1b 1b T T ˆ ˆ + ℵ S + 2ℵ S , (27) J1 f 1 c1 f 1 ∗ ∗ ∂J ∂J 1 1 where is the estimation of . Design the critic updated law as ∂z ∂z 1 1 ˆ ˆ ℵ =−γ S S ℵ , (28) c1 c1 J1 c1 J1 where γ > 0. The virtual controller consists of actor adaptive law c1 2δ−1 z 9 z ∗ 1 α ˆ =−σ −¯ σ z − d − 1 1 1 1 2 2 2 2 δ−1 (k − z ) k − z 1b 1 1b 1 T T ˆ ˆ − ℵ S − ℵ S . (29) J1 f 1 a1 f 1 Correspondingly, the actor updated law is designed by ˙ T ˆ ˆ ˆ ˆ ℵ =−S S [γ (ℵ − ℵ ) + γ ℵ ], (30) a1 J1 a1 a1 c1 c1 c1 J1 where γ > 0. By substituting (29) and (27) into (18), we obtain a1 ∂J ∗ 1 H z ˆ , α ˆ , 1 1 ∂z 2δ−1 k z 1b = ψ log + −σ −¯ σ z 1 1 1 1 2 2 2 2 δ−1 k − z (k − z ) 1b 1 1b 1 9 z 1 T T ˆ ˆ ˆ −d − − ℵ S − ℵ S 1 J1 f 1 a1 f 1 2 2 4 k − z 2 ib 2δ−1 z 9 z + 2σ + 2σ ¯ z + 2d + 1 1 1 1 2 2 2 2 δ−1 (k − z ) k − z 1b 1 ib 1 APPLIED MATHEMATICS IN SCIENCE AND ENGINEERING 9 2δ−1 T T 1 ˆ ˆ + ℵ S + 2ℵ S −σ −¯ σ z J1 f 1 1 1 1 c1 f 1 2 2 δ−1 (k − z ) 1b 1 1 9 z T T ˆ ˆ −d − ℵ S − ℵ S − + φ 1 J1 1 f 1 a1 2 2 2 2 4 k − z ib 1 + d − y˙ = 0. 1 r The Bellman residual error e is expressed as ∗ ∗ ∂J ∂J ∗ 1 ∗ 1 e = H z , a ˆ , − H z , a ˆ , 1 1 1 1 1 1 1 ∂z ∂z 1 1 ∂J ∗ 1 = H z , a ˆ , . (31) 1 1 ∂z It is emphasized that the optimal virtual controller α ˆ is constructed to guarantee ∗ ∗ ∂J ∂J ∗ 1 ∗ 1 H (z , α ˆ , ) → 0. If H (z , α ˆ , ) = 0has auniquesolution, then oneobtains 1 1 1 1 1 ∂z 1 ∂z 1 1 ∂J ∗ 1 ∂H z , α ˆ , 1 1 1 ∂z ˆ ˆ = S S (ℵ − ℵ ) = 0. (32) J1 a1 c1 J1 ˆ 2 ∂ℵ a1 Construct a positive function ˆ ˆ ˆ ˆ E = (ℵ − ℵ ) (ℵ − ℵ ) = 0. (33) 1 a1 c1 a1 c1 It is obvious that E = 0 which is equal to (32). The actor and critic adaptive laws can be designed in view of the following relation: ∂E ∂E 1 1 ˆ ˆ =− = 2(ℵ − ℵ ). (34) a1 c1 ˆ ˆ ∂ℵ ∂ℵ a1 1 Thus we have ∂E ∂E 1 ˙ 1 ˙ ˆ ˆ E = ℵ + ℵ 1 c1 a1 ˆ ˆ ∂ℵ ∂ℵ c1 a1 ∂E ∂E 1 1 T T =−γ S S ℵ − S S c1 J1 c1 J1 J1 J1 ˆ ˆ ∂ℵ ∂ℵ c1 a1 ˆ ˆ ˆ × [γ (ℵ − ℵ ) + γ ℵ ] a1 a1 c1 c1 c1 γ ∂E a1 1 =− S S ≤ 0. (35) J1 J1 2 ˆ ∂ℵ a1 10 Z. SUN ET AL. Therefore, (28) and (30) enable (32) to be finally realized. The disturbance observer is designed as d = f (x − κ ), 1 1 1 1 ˆ ˆ κ ˙ = x + d + ℵ S , (36) 1 2 1 f 1 f 1 where f > 0. Define the Lyapunov function as follows: 1 1 1 1 1 1b 2 2 2 2 ˜ ˜ ˜ ˜ V = log + ℵ + ℵ + ℵ + d , (37) c1 a1 f 1 1 2 2 2 k − z 2 2 2 2 1b ˜ ˆ where ℵ =ℵ − ℵ . In view of (13), (28) and (30), the time derivative of V is c1 c1 1 c1 ˙ ˙ ˙ ˙ ˙ ˜ ˆ ˜ ˆ ˜ ˆ ˜ ˜ V = z ˙ − ℵ ℵ − ℵ ℵ − ℵ ℵ + d d 1 1 c1 c1 a1 a1 f 1 f 1 1 1 2 2 k − z 1b 2δ−1 z z = z − σ −¯ σ z − d 2 1 1 1 1 2 2 2 2 δ−1 k − z (k − z ) 1b 1 1b 1 1 9 z T T ˆ ˆ − ℵ S − ℵ S − − y ˙ + φ + d J1 r 1 1 f 1 a1 f 1 2 2 2 4 k − z 1b ˜ ˆ ˆ ˆ + ℵ S S (γ (ℵ − ℵ ) + γ ℵ ) a1 J1 a1 a1 c1 c1 c1 J1 ˜ ˆ ˜ ˆ + ℵ S − m ℵ + ℵ γ S S ℵ f 1 f 1 1 f 1 c1 c1 J1 c1 J1 2 2 k − z 1 1 ˜ ˙ ˜ ˜ + d (d − f (d − ℵ S − ϕ )). (38) 1 1 1 1 f 1 f 1 f 1 The following correlations hold by utilizing the Young’s inequality z 1 z 1 1 2 z ≤ + z , (39) 2 2 2 2 k − z 2 (k − z ) 2 1 1 1b 1b z 1 z 1 1 ∗2 ϕ ≤ + ϕ , (40) f 1 f 1 2 2 2 2 2 k − z 2 (k − z ) 2 1b 1 1b 1 z 1 z 1 1 2 − y ˙ ≤ + y ˙ , (41) 2 2 2 2 2 2 k − z (k − z ) 1b 1 1b 1 z 1 z 1 1 2 ˜ ˜ d ≤ + d , (42) 2 2 2 2 2 2 k − z (k − z ) 1b 1 1b 1 1 z 1 z 1 T T T ˆ ˆ ˆ − ℵ S ≤ + ℵ S S ℵ . (43) J1 J1 a1 a1 a1 J1 2 2 2 2 2 4 4 k − z (k − z ) 1b 1 1b 1 From (39)–(43), we yield 2δ 2 z z 1 1 1 1 2 2 ˙ ˜ V ≤−σ −¯ σ + d + z 1 1 1 1 2 2 2 2 2 (k − z ) k − z 2 2 1 1 1b 1b APPLIED MATHEMATICS IN SCIENCE AND ENGINEERING 11 ˜ ˆ ˆ ˆ + ℵ S S (γ (ℵ − ℵ ) + γ ℵ ) a1 J1 a1 a1 c1 c1 c1 J1 ˜ ˆ ˜ ˆ + ℵ S − m ℵ + ℵ γ S S ℵ f 1 f 1 1 f 1 c1 c1 J1 c1 J1 2 2 k − z 1 1 T T ˜ ˙ ˜ ˜ ˆ ˆ + d (d − f (d − ℵ S − ϕ )) + ℵ S S ℵ . (44) 1 1 1 1 f 1 f 1 f 1 J1 a1 a1 J1 ∗ ∗ ∗ ˜ ˆ ˜ ˆ ˜ ˆ In light of ℵ =ℵ − ℵ , ℵ =ℵ − ℵ and ℵ =ℵ − ℵ ,weget c1 c1 a1 a1 f 1 f 1 c1 a1 f 1 T T ∗T T ∗ T T T T ˜ ˆ ˜ ˜ ˆ ˆ γ ℵ S S ℵ = γ [ℵ S S ℵ − ℵ S S ℵ − ℵ S S ℵ ], (45) c1 J1 J1 c1 J1 J1 c1 J1 c1 J1 J1 c1 J1 c1 c1 J1 c1 J1 T T ∗T T ∗ T T T T ˜ ˆ ˜ ˜ ˆ ˆ γ ℵ S S ℵ = γ [ℵ S S ℵ − ℵ S S ℵ − ℵ S S ℵ ], (46) a1 J1 a1 a1 J1 J1 a1 J1 a1 a1 J1 J1 J1 J1 a1 J1 a1 J1 γ − γ c1 a1 T T T T T T ˜ ˆ ˜ ˜ ˆ ˆ (γ − γ )ℵ S S ℵ ≤ [ℵ S S ℵ − ℵ S S ℵ ]. (47) c1 a1 J1 c1 J1 a1 J1 c1 a1 J1 a1 J1 c1 J1 m m 1 1 ∗2 2 ˆ ˜ ˜ −m ℵ ℵ ≤ ℵ − ℵ . (48) 1 f 1 f 1 f 1 f 1 2 2 Substituting (45)–(48) into (44), we confirm 2δ 2 z z γ c1 1 1 T T ˙ ˜ ˜ V ≤−σ −¯ σ − ℵ S S ℵ 1 1 1 J1 c1 c1 J1 2 2 2 2 (k − z ) k − z 1b 1 1b 1 2γ − γ m a1 c1 1 T T 2 ˜ ˜ ˜ − ℵ S S ℵ − ℵ + M J1 a1 1,1 a1 J1 f 1 2 2 1 1 1 T T T T ˆ ˆ ˆ ˆ − γ ℵ S S ℵ − γ − ℵ S S ℵ c1 J1 c1 a1 J1 a1 c1 J1 a1 J1 2 2 4 γ − γ c1 a1 T T ˆ ˆ − ℵ S S ℵ J1 c1 c1 J1 ˜ ˙ ˜ ˜ + d (d − f (d − ℵ − ϕ S )), (49) 1 1 1 1 f 1 f 1 f 1 1 (γ +γ ) 1 m ∗ 1 c1 a1 ∗T T ∗ 2 1 ∗2 where γ >γ /2, γ > and M = (ℵ S S ℵ ) + y ˙ + + ϕ + a1 c1 a1 1,1 J1 2 2 c1 J1 c1 2 r ℵ 2 f 1 f 1 J1 f 1 1 2 1 2 1 2 T k by reason of z < k .Let λ and λ be the minimal eigenvalue of S S and J1 2 min min J1 2 2b 2 2 2b S S , respectively. The following inequalities hold: f 1 f 1 1 1 2 ∗2 ˜ ˙ ˜ d d ≤ d + d , (50) 1 1 1 1 2 2 1 1 2 ∗2 ˜ ˜ −f d ϕ ≤ f d + f ϕ , (51) 1 1 f 1 1 1 1 f 1 2 2 1 1 f 1 2 2 ˜ ˜ ˜ ˜ −f d ℵ S ≤ f d + f λ ℵ , (52) 1 1 f 1 f 1 1 1 min f 1 2 2 γ γ c1 c1 S J1 T T T ˜ ˜ ˜ ˜ − ℵ S S ℵ ≤− λ ℵ ℵ , (53) J1 c1 c1 c1 J1 min c1 2 2 2γ − γ 2γ − γ a1 c1 a1 c1 T T J1 T ˜ ˜ ˜ ˜ − ℵ S S ℵ ≤− λ ℵ ℵ . (54) J1 a1 a1 a1 J1 a1 min 2 2 12 Z. SUN ET AL. Substituting (50)–(54) into (49), we deduce 2δ 2 z z 1 1 V ≤−σ −¯ σ + M 1 1 1 1 2 2 2 2 (k − z ) k − z 1b 1 1b 1 2γ − γ γ S S a1 c1 c1 J1 2 J1 2 ˜ ˜ − λ ℵ − λ ℵ a1 c1 min min 2 2 1 S 1 3 f 1 2 2 ˜ ˜ − (m + f λ )ℵ − f − d , (55) 1 1 1 min f 1 1 2 2 2 3 1 1 ∗2 ∗2 where f > and M = M + d + f ϕ . 1 1 1,1 1 2 2 1 2 f 1 Step i(2 ≤ i ≤ n − 1): The time derivative z can be derived from (12) and (13) as z ˙ = x + φ + d − α ˆ . (56) i i+1 i i i−1 Introduce the performance index function as J (z (0)) = h (z (s), α (z )) ds, (57) i i i i i i ib 2 where h (z (s), α (z )) = ψ log + α represents the cost function and α denotes the i i i i i i 2 2 k −z ib virtual controller. Let α be the optimal virtual controller. Similar to (16), to achieve the minimum control performance index in (57), the optimal performance index function is established as follows: ∗ ∗ J (z (t)) = h (z (s), α (z )) ds i i i i i i = min h (z (s), α (z )) ds . (58) i i i i Taking thetimederivativeof(58),wededuce ∗ ∗ ∂J ∂J i i ∗ = (x + φ + d − α ˆ ) i+1 i i i−1 ∂t ∂z ∂J i ∗ ∗ = (z + α + φ + d − α ˆ ). (59) i+1 i i i i−1 ∂z Define HJB equation associating with (59) as follows: ∂J k ib ∗ i ∗2 H z , α , = ψ log + α i i 1 i i 2 2 ∂z k − z ib i ∂J i ∗ ∗ + (α + φ + d − α ˆ ) = 0. (60) i i i i−1 ∂z Solve the equation ∂H /∂α = 0, the optimal virtual controller is determined by 1 ∂J ∗ i α =− . (61) 2 ∂z For the purpose of the optimal control, ∂J /∂z can be decomposed as 2δ−1 ∂J z 9 z i i = 2σ + 2σ ¯ z + 2d + i i i i 2 2 2 2 δ−1 ∂z (k − z ) 2 k − z i i ib ib APPLIED MATHEMATICS IN SCIENCE AND ENGINEERING 13 + 2φ + J (z ), (62) i 1 2δ−1 z ∂J 9 z o i i i where σ , σ ¯ > 0and J (z ) =−2σ − 2σ ¯ z − 2d − − 2φ + . i i i i i i i i 2 2 δ−1 2 2 i 2 ∂z (k −z ) k −z i i ib ib Thus (61) can be represented as 2δ−1 9 z ∗ i α =−σ −¯ σ z − d − i i i i 2 2 2 2 δ−1 (k − z ) k − z i i ib ib − φ − J (z ). (63) i i In (63), J and φ areapproximatedbyfuzzy logicsystems as o ∗T J =ℵ S + ϕ , (64) Ji Ji i Ji ∗T φ =ℵ S + ϕ . (65) i fi fi fi The following equations are established in light of combining (64) and (65) with (62) and (63) respectively, 2δ−1 z 1 ∗ ∗T α =−σ −¯ σ z − d − ℵ S i i i i J1 i J1 2 2 δ−1 (k − z ) ib 9 z 1 ∗T − −ℵ S − ϕ , (66) f 1 i f 1 2 2 4 2 k − z ib i ∗ 2δ−1 ∂J z 9 z i i = 2σ + 2σ ¯ z + 2d + i i i i 2 2 2 2 δ−1 ∂z 2 (k − z ) k − z ib i ib i ∗T ∗T +ℵ S + ϕ + 2ℵ S , (67) J1 i f 1 J1 f 1 ∗ ∗T ∗T where ϕ = 2ϕ + ϕ .Itisnoticeablethat α is unavailable because ℵ and ℵ are the i fi Ji i Ji fi unknown ideal weights. Resembling (27) and (29), the actor-critic structure is developed as 2δ−1 ∂J z 9 z i i = 2σ +¯ σ z + 2d + i i i i 2 2 2 2 δ−1 ∂z (k + 2z ) 2 k − z i i ib ib T T ˆ ˆ + ℵ S + 2ℵ S . (68) Ji fi ci fi Design the critic updated law, optimal virtual control law and actor updated law as ˆ ˆ ℵ =−γ S S ℵ , (69) ci ci Ji ci Ji 2δ−1 z 9 z ∗ i α ˆ =−σ −¯ σ z − d − i i i i 2 2 δ−1 2 2 (k − z ) 4 k − z ib i ib i T T ˆ ˆ − ℵ S − ℵ S , (70) J1 f 1 a1 f 1 ˆ ˆ ˆ ˆ ℵ =−S S [γ (ℵ − ℵ ) + γ ℵ ], (71) ai Ji ai ai ci ci ci Ji 14 Z. SUN ET AL. where γ and γ are positive numbers. The fuzzy updated law is constructed as ci ai ˆ ˆ ℵ = S − m ℵ , (72) fi fi i fi 2 2 k − z ib where m is a positive constant. By lumping (68), (70) and (72) into (60), we acquire ∂J ∗ i H z , α ˆ , i i ∂z 2δ−1 k z ib = ψ log + −σ −¯ σ z i i i i 2 2 2 2 δ−1 k − z (k − z ) ib i ib i 9 z 1 z i i T T ˆ ˆ − d − − ℵ S − ℵ S i Ji fi ai fi 2 2 2 2 4 2 k − z k − z i i ib ib 2δ−1 z 9 z + 2σ + 2σ ¯ z + 2d + i i i i 2 2 δ−1 2 2 (k + 2z ) k − z ib i ib i 2δ−1 T T i ˆ ˆ + ℵ S + 2ℵ S −σ −¯ σ z Ji fi i i i ci fi 2 2 δ−1 (k − z ) ib 1 1 T T ∗ ˆ ˆ ˆ −d − ℵ S − ℵ S + φ + d − α ˆ = 0. (73) i Ji fi i i ai i−1 2 2 The disturbance observer is devised as d = f (x − κ ), i i i i ˆ ˆ κ ˙ = x + d + ℵ S . (74) i i+1 i fi fi Construct the Lyapunov function as 1 k 1 1 ib 2 2 ˜ ˜ V = V + log + ℵ + ℵ i i−1 ci ai 2 2 2 2 2 k − z ib i 1 1 2 2 ˜ ˜ + ℵ + d , (75) fi 2 2 ∗ ∗ ∗ ˜ ˆ ˜ ˆ ˜ ˆ where ℵ =ℵ − ℵ , ℵ =ℵ − ℵ and ℵ =ℵ − ℵ . By combining (13), (69) ai ai ci ci fi fi ai ci fi and (71), the time derivative of (75) yields ˙ ˙ ˙ ˙ ˙ ˙ ˜ ˆ ˜ ˆ ˜ ˆ ˜ ˜ V = V + z ˙ − ℵ ℵ − ℵ ℵ − ℵ ℵ + d d i i−1 i ci ci ai ai fi fi i i 2 2 k − z ib 2δ−1 z z = z − σ −¯ σ z − d i+1 i i i i 2 2 2 2 δ−1 k − z (k − z ) i i ib ib 1 9 z T T ∗ ˆ ˆ − ℵ S − ℵ S − − α ˆ + φ + d Ji fi i i ai fi i−1 2 2 2 4 k − z ib i APPLIED MATHEMATICS IN SCIENCE AND ENGINEERING 15 ˜ ˆ ˆ ˆ + ℵ S S (γ (ℵ − ℵ ) + γ ℵ ) ai Ji ai ai ci ci ci Ji ˜ ˆ ˜ ˆ + ℵ S − m ℵ + ℵ γ S S ℵ fi fi i fi ci ci Ji ci Ji 2 2 k − z i i ˜ ˙ ˜ ˜ + d (d − f (d − ℵ S − ϕ )). (76) i i i i fi fi fi The following relationship can be deduced by utilizing the Young’s inequality, z 1 z 1 i 2 z ≤ + z , (77) i+1 i+1 2 2 2 2 k − z 2 (k − z ) 2 i i ib ib z 1 z 1 i ∗2 ϕ ≤ + ϕ , (78) fi fi 2 2 2 2 k − z 2 (k − z ) 2 i i ib ib z 1 z 1 ∗ i ∗2 ˙ ˙ − α ˆ ≤ + α ˆ , (79) i−1 i−1 2 2 2 2 2 k − z 2 (k − z ) 2 ib i ib i z 1 z 1 i 2 ˜ ˜ d ≤ + d , (80) 2 2 2 2 2 2 k − z (k − z ) ib i ib i 1 z 1 z 1 T T T ˆ ˆ ˆ − ℵ S ≤ + ℵ S S ℵ . (81) Ji Ji ai ai ai Ji 2 2 2 2 2 4 4 k − z (k − z ) i i ib ib Associate (77)–(81) with (76), we can derive 2δ 2 z z 1 1 i i 2 2 ˙ ˙ V ≤ V − σ −¯ σ + d + z i i−1 1 i 1 2 2 2 2 2 2 2 (k − z ) k − z i i ib ib ˜ ˆ ˆ ˆ + ℵ S S (γ (ℵ − ℵ ) + γ ℵ ) ai Ji ai ai ci ci ci Ji ˜ ˆ ˜ ˆ + ℵ S − m ℵ + ℵ γ S S ℵ i ci ci Ji ci fi fi fi Ji 2 2 k − z i i T T ˜ ˙ ˜ ˜ ˆ ˆ + d (d − f (d − ℵ S − ϕ )) + ℵ S S ℵ . (82) i i 1 i f 1 fi fi Ji ai ai Ji ∗ ∗ ∗ ˜ ˆ ˜ ˆ ˜ ˆ Due to ℵ =ℵ − ℵ , ℵ =ℵ − ℵ and ℵ =ℵ − ℵ the following correlations are ci ci ai ai fi fi ci ai fi inferred: T T ∗T T ∗ T T T T ˜ ˆ ˜ ˜ ˆ ˆ γ ℵ S S ℵ = γ (ℵ S S ℵ − ℵ S S ℵ − ℵ S S ℵ ), (83) ci Ji ci ci Ji Ji ci Ji ci ci Ji Ji Ji Ji ci Ji ci Ji T T ∗T T ∗ T T T T ˜ ˆ ˜ ˜ ˆ ˆ γ ℵ S S ℵ = γ (ℵ S S ℵ − ℵ S S ℵ − ℵ S S ℵ ), (84) ai Ji ai ai Ji Ji ai Ji ai ai Ji Ji Ji Ji ai Ji ai Ji T T ˜ ˆ (γ − γ )ℵ S S ℵ (85) ci ai Ji ci ai Ji γ − γ ci ai T T T T ˜ ˜ ˆ ˆ ≤ (ℵ S S ℵ − ℵ S S ℵ ), (86) Ji ai Ji ci ai Ji ci Ji m m i i ∗2 2 ˆ ˜ ˜ − m ℵ ℵ ≤ ℵ − ℵ . (87) i fi fi fi fi 2 2 16 Z. SUN ET AL. By lumping (83)–(87) into (76), we get 2δ 2 z z ci i i T T ˙ ˙ ˜ ˜ V ≤ V − σ −¯ σ − ℵ S S ℵ i i−1 i i Ji ci ci Ji 2 2 2 2 (k − z ) k − z i i ib ib 2γ − γ m ai ci i T T 2 ˜ ˜ ˜ − ℵ S S ℵ − ℵ + M Ji ai i,1 ai Ji fi 2 2 1 i 1 T T T T ˆ ˆ ˆ ˆ − γ ℵ S S ℵ − γ − ℵ S S ℵ ci Ji ci ai Ji ai ci Ji ai Ji 2 2 4 γ − γ ci ai T T ˆ ˆ − ℵ S S ℵ Ji ci ci Ji ˜ ˙ ˜ ˜ + d (d − f (d − ℵ − ϕ S )), (88) i i i i fi fi fi (γ +γ ) ci ai ∗T T ∗ 1 2 m ∗ 1 ∗2 1 2 ˙ 1 where M = (ℵ S S ℵ ) + α ˆ + ℵ + ϕ + k by virtue of i,1 Ji Ji Ji Ji i−1 2 2 2 fi 2 fi 2 (i+1)b Ji 1 2 1 2 T z < k . Assuming that λ represents the minimal eigenvalue of S S ,the Ji i+1 min Ji 2 2 (i+1)b following inequalities yields: 1 1 2 ∗2 ˜ ˙ ˜ d d ≤ d + d , (89) i i i i 2 2 1 1 2 ∗2 ˜ ˜ −f d ϕ ≤ f d + f ϕ , (90) i i fi i i fi 2 2 1 1 S fi 2 2 ˜ ˜ ˜ ˜ −f d ℵ S ≤ f d + f λ ℵ , (91) i i fi fi i i i min fi 2 2 γ γ ci ci S T T Ji T ˜ ˜ ˜ ˜ − ℵ S S ℵ ≤− λ ℵ ℵ , (92) Ji ci ci ci Ji ci min 2 2 2γ − γ 2γ − γ ai ci ai ai S Ji T T T ˜ ˜ ˜ ˜ − ℵ S S ℵ ≤− λ ℵ ℵ . (93) Ji ai ai ai Ji min ai 2 2 Substituting (92) and (93) into (88), we obtain i 2δ i 2 z  z j j V ≤ −σ − σ ¯ + M i j j i 2 2 2 2 (k − z ) k − z jb j jb j j=1 j=1 i i 2γ − γ γ S S aj cj cj Jj Jj 2 2 ˜ ˜ − λ ℵ − λ ℵ min aj min cj 2 2 j=1 j=1 1 S 1 3 fj 2 2 ˜ ˜ − (m + f λ )ℵ − f − d , (94) j j j min fj 2 2 2 j=1 1 ∗2 1 ∗2 where M = M + d + f ϕ . i i,1 i 2 2 fi Step n: z can be showcased from (12) and (13) as z = x −ˆ α . (95) n n n−1 APPLIED MATHEMATICS IN SCIENCE AND ENGINEERING 17 Thetimederivativeof(95) is stated by z ˙ = x ˙ − α ˆ n n n−1 = u + φ + d − α ˆ . (96) n n n−1 Define the integral performance index function as J(z ) = h (z , u(z )) ds, (97) n n n n nb 2 ∗ where h = ψ log + α .Let u be the optimal actual controller, then the optimal n i 2 2 n k −z nb performance index function is represented by ∗ ∗ J(z ) = h (z , u (z )) ds n n n n = min h (z , u(z )) ds . (98) n n n Akin to (20), we have ∗ ∗ ∂J ∂J n n ∗ ∗ = (u + φ + d − α ˆ ). (99) n n n−1 ∂t ∂z By associating with (95), the HJB equation is formalized by ∂J k nb ∗ n ∗2 H z , α , = ψ log + u n n n 2 2 ∂z k − z nb ∂J n ∗ ∗ + (u + φ + d − α ˆ ) = 0. (100) n n n−1 ∂z Similar to (61), dealing with the (∂H /u ) = 0, we yield 1 ∂J ∗ n u =− . (101) 2 ∂z ∗ 2δ−1 ∂J z n n 7 n o Let = 2σ + 2σ ¯ z + 2d + + 2φ + J (z ),where σ > 0. The n n n n n n n 2 2 2 δ−1 2 n ∂z 2 n (k −z ) k −z n n nb nb optimal actual controller can be stated by 2δ−1 z 7 z ∗ n u =−σ −¯ σ z − d − n n n n 2 2 δ−1 2 2 (k − z ) 4 k − z n n nb nb − φ − J (z ). (102) n n J and φ canbeapproximated by thefuzzy logicsystemas o ∗T J =ℵ S + ϕ , (103) Jn Jn n Jn ∗T φ =ℵ S + ϕ . (104) n fn fn fn 18 Z. SUN ET AL. Merging (103), (104) and (102), we yield 2δ−1 z 1 ∗ ∗T u =−σ −¯ σ z − d − ℵ S n n n n Jn Jn 2 δ−1 (k − z ) nb 7 z 1 ∗T − −ℵ S − ϕ , (105) fn n fn 4 2 k − z nb n ∂J where ϕ = 2ϕ + ϕ Combining with (103), can be formulated by n fn Jn ∂z ∗ 2δ−1 ∂J z 7 z n n = 2σ + 2σ ¯ z + 2d + n n n n 2 2 2 δ−1 2 ∂z (k − z ) 2 k − z n n nb nb ∗T ∗T +ℵ S + 2ℵ S + ϕ . (106) Jn fn n Jn fn The critic for evaluating (106) and critic updated law is conceived as ∗ 2δ−1 ∂J z 7 z n n = 2σ + 2σ ¯ z + 2d + n n n n 2 2 2 δ−1 2 ∂z (k + 2z ) 2 k − z n n nb nb T T ˆ ˆ + ℵ S + 2ℵ S , (107) Jn fn Jn fn ˆ ˆ ℵ =−γ S S ℵ . (108) cn cn Jn cn Jn Thelaw of theactor andthe actual controller areconstructed as 2δ−1 z z u ˆ =−σ −¯ σ z − d − 2 n n n n 2 2 2 δ−1 2 (k − z ) k − z n n nb nb T T ˆ ˆ − ℵ S − ℵ S , (109) Jn fn an fn ˆ ˆ ˆ ˆ ℵ =−S S [γ (ℵ − ℵ ) + γ ℵ ]. (110) an Jn an an cn cn cn Jn The adaptive law ℵ is updated as fn ˙ n ˆ ˆ ℵ = S − m ℵ , (111) fn fn n fn k − z nb where m is a positive constant. The HJB equation is derived as ∂J ∗ n H z , α ˆ , n n ∂z 2δ−1 k z nb = ψ log + −σ −¯ σ z n n n n 2 2 2 2 δ−1 k − z (k − z ) nb n nb n 7 z 1 T T ˆ ˆ ˆ −d − − ℵ S − ℵ S n Jn fn an fn 4 k − z 2 nb 2δ−1 z 7 z + 2σ + 2σ ¯ z + 2d + n n n n 2 2 2 δ−1 2 (k + 2z ) k − z nb n nb n APPLIED MATHEMATICS IN SCIENCE AND ENGINEERING 19 2δ−1 T T n ˆ ˆ + ℵ S + 2ℵ S −σ −¯ σ z Jn fn n n n cn fn 2 2 δ−1 (k − z ) nb 1 1 T T ∗ ˆ ˙ ˆ ˆ − d − ℵ S − ℵ S + φ + d − α ˆ = 0. (112) n Jn n n fn an fn n−1 2 2 The disturbance observer is built as d = f (u − κ ), n n n ˆ ˆ κ ˙ = u + d + ℵ S . (113) n n fn fn The Lyapunov function is selected as follows: 1 k 1 1 nb 2 2 ˜ ˜ V = V + log + ℵ + ℵ n n−1 cn an 2 2 2 k − z nb 1 1 2 2 ˜ ˜ + ℵ + d . (114) fn 2 2 Merging with (108), (110), (113) and (95), the time derivative of (114) is determined by ˙ ˙ ˙ ˙ ˙ ˜ ˆ ˜ ˆ ˜ ˆ V = V + z ˙ − ℵ ℵ − ℵ ℵ − ℵ ℵ n n−1 n cn cn an an fn fn k − z nb ˜ ˜ + d d n n 2δ−1 z z = −σ −¯ σ z − d n n n n 2 2 2 2 δ−1 k − z (k − z ) n n nb nb 1 7 z T T ˆ ˆ − ℵ S − ℵ S − − α ˆ + φ + d Jn fn n−1 n n an fn 2 4 k − z nb n ˜ ˆ ˆ ˆ + ℵ S S (γ (ℵ − ℵ ) + γ ℵ ) an Jn an an cn cn cn Jn ˜ ˆ ˜ ˆ + ℵ S − m ℵ + ℵ γ S S ℵ fn fn n fi cn cn Jn cn Jn 2 2 k − z n n ˜ ˙ ˜ ˜ + d (d − f (d − ℵ S − ϕ )). (115) n n n n fn fn fn By Young’s inequality, we acquire z 1 z 1 n ∗2 ϕ ≤ + ϕ , (116) f 1 fn 2 2 2 2 2 k − z 2 (k − z ) 2 n n nb nb z 1 z 1 ∗ n ∗2 − α ˙ ≤ + α ˙ , (117) n−1 i−1 2 2 2 2 2 k − z 2 (k − z ) 2 n n nb nb z 1 z 1 n 2 ˜ ˜ d ≤ + d , (118) 2 2 2 2 2 k − z 2 (k − z ) 2 n n nb nb 1 z 1 z 1 T n T T ˆ ˆ ˆ − ℵ S ≤ + ℵ S S ℵ . (119) Jn Jn an an an Jn 2 2 2 2 2 2 k − z 4 (k − z ) 4 n n nb nb 20 Z. SUN ET AL. Integrate (116)–(119) into (115), we have 2δ 2 z z 1 n n ˙ ˙ V ≤ V − σ −¯ σ + d n n−1 n n 2 2 2 δ 2 (k − z ) k − z n n nb nb ˜ ˆ ˆ ˆ + ℵ S S (γ (ℵ − ℵ ) + γ ℵ ) an Jn an an cn cn cn Jn ˜ ˆ ˜ ˆ + ℵ S − m ℵ + ℵ γ S S ℵ n cn cn Jn cn fn fn fn Jn 2 2 k − z n n ˜ ˙ ˜ ˜ + d (d − f (d − ℵ S − ϕ )) n n n n fn fn fn T T ˆ ˆ + ℵ S S ℵ . (120) Jn an an Jn ∗ ∗ ∗ ˜ ˆ ˜ ˆ ˜ ˆ From ℵ =ℵ − ℵ , ℵ =ℵ − ℵ and ℵ =ℵ − ℵ , we get the following rela- cn cn an an fn fn cn an fn tions: T T ∗T T ∗ T T T T ˜ ˆ ˜ ˜ ˆ ˆ γ ℵ S S ℵ = γ [ℵ S S ℵ − ℵ S S ℵ − ℵ S S ℵ ], (121) cn Jn cn cn Jn Jn cn Jn cn cn Jn Jn Jn Jn cn Jn cn Jn T T ∗T T ∗ T T T T ˜ ˆ ˜ ˜ ˆ ˆ γ ℵ S S ℵ = γ [ℵ S S ℵ − ℵ S S ℵ − ℵ S S ℵ ], (122) an Jn an an Jn Jn an Jn an an Jn Jn Jn Jn an Jn an Jn γ − γ cn an T T T T T T ˜ ˆ ˜ ˜ ˆ ˆ (γ − γ )ℵ S S ℵ ≤ [ℵ S S ℵ − ℵ S S ℵ ], (123) cn an Jn cn Jn an Jn cn an Jn an Jn cn Jn m m n n ∗2 2 ˆ ˜ ˜ −m ℵ ℵ ≤ ℵ − ℵ . (124) fn fn fn fn 2 2 Invoking (121)–(124) and (94) for (115), we infer 2δ 2 z z n n ˙ ˙ V ≤ V − σ −¯ σ n n−1 n n 2 2 2 δ 2 (k − z ) k − z nb n nb n 2γ − γ m an cn n T T 2 ˜ ˜ ˜ − ℵ S S ℵ − ℵ + M Jn an n,1 an Jn fn 2 2 1 n 1 T T T T ˆ ˆ ˆ ˆ − γ ℵ S S ℵ − γ − ℵ S S ℵ cn Jn cn an Jn an cn Jn an Jn 2 2 4 γ − γ γ cn an cn T T T T ˆ ˆ ˜ ˜ − ℵ S S ℵ − ℵ S S ℵ Jn cn Jn cn cn Jn cn Jn 2 2 ˜ ˙ ˜ ˜ + d (d − f (d − ℵ − ϕ S )), (125) n n n n fn fn fn (γ +γ ) m Jn cn an ∗T T ∗ 1 ∗2 1 ∗ 1 ∗2 where M = (ℵ S S ℵ ) + α ˆ + ℵ + ϕ .Suppose that λ rep- n,1 Jn cn Jn cn n−1 min 2 2 2 fn 2 fn resents the minimal eigenvalue of S S , the following inequalities yield: Jn Jn 1 1 2 ∗2 ˜ ˙ ˜ d d ≤ d + d , (126) n n n n 2 2 1 1 2 ∗2 ˜ ˜ −f d ϕ ≤ f d + f ϕ , (127) n n fn n n n fn 2 2 1 1 S fn 2 2 ˜ ˜ ˜ ˜ −f d ℵ S ≤ f d + f λ ℵ , (128) n n fn fn n n min fn 2 2 APPLIED MATHEMATICS IN SCIENCE AND ENGINEERING 21 γ γ cn cn S Jn T T T ˜ ˜ ˜ ˜ − ℵ S S ℵ ≤− λ ℵ ℵ , (129) Jn cn cn cn Jn min cn 2 2 2γ − γ 2γ − γ an cn an cn T T Jn T ˜ ˜ ˜ ˜ − ℵ S S ℵ ≤− λ ℵ ℵ . (130) Jn an an an Jn an min 2 2 Substituting (129), (130) and similar to (94), we obtain the following inequality: n 2δ n 2 z  z j j V ≤ −σ − σ ¯ + M n j j n 2 2 2 2 (k − z ) k − z jb j jb j j=1 j=1 n n 2γ − γ γ aj cj S cj S Jj Jj 2 2 ˜ ˜ − λ ℵ − λ ℵ aj cj min min 2 2 j=1 j=1 n n 1 1 3 fj 2 2 ˜ ˜ − (m + f λ )ℵ − f − d , (131) j j j min fj 2 2 2 j=1 j=1 1 1 ∗2 ∗2 where M = M + d + f ϕ . n n,1 n 2 2 fi 4. Stability analysis In this section, the stability of the system is demonstrated. Theorem 4.1: Consider the nonlinear system (1) with actuator bias fault signal and external disturbances. Suppose that Assumptions 2.1 and 2.2 hold. Taking into account the designed critic adaptive laws as (28), (69) and (108),actor updatedlawsas (30), (71) and (110), fuzzy adaptive laws (26), (72) and (111), and disturbance observers (36), (74) and (113).The pro- posed fuzzy adaptive optimal n fi ite-time control scheme ensures that (1) all signals within the closed-loop system are bounded; (2) all states are in their specicfi intervals. Proof: Let V = V , on the basis of (131), we have n 2δ n 2 z z j j V ≤− σ − σ ¯ + M j j n 2 2 δ 2 2 (k − z ) k − z jb j jb j j=1 j=1 n n 2γ − γ γ aj cj S cj S Jj Jj 2 2 ˜ ˜ − λ ℵ − λ ℵ aj cj min min 2 2 j=1 j=1 n n 1 1 3 fj 2 2 ˜ ˜ − (m + f )λ ℵ − f − d . (132) j j j min fj 2 2 2 j=1 j=1 Let C = min{2σ ,2σ , ... ,2σ }, C = min{2σ ¯ ,2σ ¯ , ... ,2σ ¯ }, C = min{(2γ − γ ) σ 1 2 n σ ¯ 1 2 n a a1 c1 S S S S S S J1 J2 Jn J1 J2 Jn λ , (2γ − γ )λ , ... , (2γ − γ )λ }, C = min{γ λ , γ λ , ... , γ λ }, a2 c2 an cn c c1 c2 cn min min min min min min S S S f 1 f 2 fn C = min{(m + f )λ , (m + f )λ ··· , (m + f )λ } and C = min{f − , f − f 1 1 2 2 n n d 1 2 min min min 3 3 , ... , f − }. From (132), we acquire 2 2 n 2δ n 2 z  z 1 1 j j V ≤−C − C + M σ σ ¯ n 2 2 2 2 2 2 (k − z ) k − z j j jb jb j=1 j=1 22 Z. SUN ET AL. n n 1 1 2 2 ˜ ˜ − C ℵ − C ℵ a c aj cj 2 2 j=1 j=1 n n 1 1 2 2 − C ℵ − C d . (133) f d fj j 2 2 j=1 j=1 1 T ˜ ˜ In light of (4), den fi e p = 1, p = ℵ ℵ , r = 1 − δ, r = δ, r = 1, one has 1 2 k 1 2 3 k=1 2 k ⎛ ⎞ n n 1 1 2 2 ⎝ ˜ ⎠ ˜ ℵ ≤ 1 − δ + δ ℵ . (134) fj fj 2 2 j=1 j=1 In view of the same fashion as (134), it leads to ⎛ ⎞ 1 1 2 2 ⎝ ˜ ⎠ ˜ d ≤ 1 − δ + δ d , (135) j j 2 2 j=1 ⎛ ⎞ n n 1 1 2 2 ⎝ ˜ ⎠ ˜ ℵ ≤ 1 − δ + δ ℵ , (136) aj aj 2 2 j=1 j=1 ⎛ ⎞ n n 1 1 2 2 ⎝ ˜ ⎠ ˜ ℵ ≤ 1 − δ + ℵ . (137) cj cj 2 2 j=1 j=1 It follows from (5) that n 2δ n 2 z z 1 1 j j δ−1 − C ≤−2 C . (138) σ σ 2 2 2 2 2 2 (k − z ) k − z j j jb jb k=1 k=1 According to (135)–(138), (133) can be rewritten as n 2 n 2 z z 1 1 j j δ−1 V ≤−2 C − C σ σ ¯ 2 2 2 2 2 k − z 2 k − z j j jb jb k=1 j=1 ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ δ δ δ n n n 1 1 1 2 2 2 ⎝ ˜ ⎠ ⎝ ˜ ⎠ ⎝ ˜ ⎠ − C ℵ − C ℵ − C ℵ f a c fj aj cj 2 2 2 j=1 j=1 j=1 ⎛ ⎞ n n 1 1 2 2 ˜ ˜ ⎝ ⎠ − C ℵ − C (1 − δ) ℵ d f cj fj 2 2 j=1 j=1 n n 1 1 2 2 ˜ ˜ − C (1 − δ) ℵ − C (1 − δ) ℵ c a cj aj 2 2 j=1 j=1 + (1 − δ)(C + C + C + C ) + M . (139) c d f a n APPLIED MATHEMATICS IN SCIENCE AND ENGINEERING 23 δ−1 Den fi e C = min{2 C , C , C , C , C , } and C = min{C , C (1 − δ), C (1 − δ),, 1 σ f a c d 2 σ ¯ f c C (1 − δ)}. It is worth noting that z (i = 1, 2, ... , n) in constraint sets, we can get a i 2 2 k z n 1 jb n 1 j log < , rewrite (139) as 2 2 2 2 j=1 2 j=1 2 k −z k −z jb j jb j V ≤−C V − C V + M, (140) 1 2 where M = M + (1 − δ)(C + C + C + C ). In light of (140), we know that V ≤ n c d f a −C V + M, therefore, it is easy to obtain V is bounded. The similarity follows that ||ℵ ||, 2 ai ˜ ˜ ˆ ˆ ˆ ||ℵ || and ||ℵ || are bounded. Therefore, ||ℵ ||, ||ℵ || and ||ℵ || are bounded. From pre- ci fi a1 c1 f 1 vious analysis, we get |z |≤ k ,itcan be furtherdeductedbyAssumption 2.2 that |x |= i ib 1 ∗ ∗ ∗ |z |+|y |≤ k + y = k ,where y is the upper bounded of y . α ˆ is also bounded and 1 r 1b 1c r r r ∗ ∗ α ˆ ≤¯ α ,and we canderivethat |x |=|z |+ αˆ ≤ k +¯ α = k .Inthe same way, it is 1 2 2 2b 1 2c 1 1 true that x ≤ k ,where i = 3, ... , n. Thus all system states are conn fi ed within constraint i ic sets. 1−δ 1 C V (x )+lC 2 0 1 Setting T = ln ,where 0 < l < 1. We can deduce that V(x) ≤ 1−δ C (1−δ) C ( ) +lC 2 1 (1−l)C 1−δ ( ) ,thus (1−l)C 1−δ 2 2 1 k 1 z M 1b 1 log ≤ ≤ V ≤ . (141) 2 2 2 2 2 2 (1 − l)C k − z k − z 1b 1 1b 1 It follows from (141) that 1−δ −2 (1−l)C |z |=|y − y |≤ k 1 − e , (142) 1 r 1b which means that the tracking error remains within the origin of the area after the setting time. Remark 4.1: The adaptive optimal control schemes described in [26,28,32]ensurethatall signals are semi-globally uniformly ultimately bounded with V ≤−C V + M,the con- vergence time may be infinite. The strategy proposed in this paper, the time derivative of V satisefi s V ≤−C V − C V + M, which means that it can achieve the faster response. 1 2 Remark 4.2: It is worth noting that the tracking error is remained within the origin of the area after the setting time by choosing appropriate parameters. Reduce the area of origin by adjusting the value of δ or increasing C . Therefore, the parameters should be chosen carefully. 5. Simulation example This section aims to verify the effectiveness of the proposed fuzzy control method with a simulation instance. 24 Z. SUN ET AL. Example 5.1: A model with external disturbances and bias fault signal is presented below ⎪ x ˙ = x + φ (x ) + D (t), 1 2 1 1 1 x ˙ = u + u + φ (x ) + D (t), (143) 2 f 2 2 2 y(t) = x,1 ≤ i ≤ n, where x = [x , x ] , φ = sin(x ) cos(x ), φ = sin(x + x ), D = 2sin(0.5t), D = sin 2 1 2 1 1 1 2 1 2 1 2 (2t), the output reference trajectory is den fi ed as y = 5sin(0.5t),and 0, t < T, u = (144) 6sin(0.9t − 1), t ≥ T, where T = 10 s. The chosen membership functions are referred to (x − 3) μ (x ) = exp − , (x − 1) μ (x ) = exp − , (x ) μ (x ) = exp − , (x + 1) μ (x ) = exp − , (x + 3) μ (x ) = exp − . (145) Thus the basis function vector S is denoted as f 1 1 5 μ (x ) μ (x ) 1 1 G G S = , ... , . (146) f 1 5 5 i i μ (x ) μ (x ) 1 1 i=1 G i=1 G Analogously, the basis function vectors S , S and S are formulated as f 2 J1 J2 ⎡ ⎤ 2 2 1 1 μ (x ) μ (x ) 2 2 j=1 j=1 G G ⎣ ⎦ S = , ... , , f 2 j j 5 2 5 2 μ (x ) μ (x ) j j i=1 j=1 G i=1 j=1 G 1 5 μ (z ) μ (z ) 1 1 G G S = , ... , , J1 5 5 i i μ (z ) μ (z ) 1 1 i=1 G i=1 G 1 5 μ (z ) μ (z ) 2 2 G G S = , ... , . J2 5 5 i i μ (z ) μ (z ) 2 2 i=1 i=1 G G ˆ ˆ ˆ Theinitial values arecongfi ured as ℵ (0) = ℵ (0) = [0.8, 0.8, 0.8, 0.8, 0.8] , ℵ (0) = f 1 f 2 c1 T T T ˆ ˆ [0.3, 0.3, 0.3, 0.3, 0.3] , ℵ (0) = [0.8, 0.8, 0.8, 0.8, 0.8] , ℵ (0) = [0.1, 0.1, 0.1, 0.1, 0.1] , c2 a1 ℵ (0) = [0.5, 0.5, 0.5, 0.5, 0.5] ,and x (0) = x (0) = 0. a2 1 2 APPLIED MATHEMATICS IN SCIENCE AND ENGINEERING 25 Figure 1. The trajectories of y, y and k . r 1c Figure 2. The trajectories of z and k . 1b The parameters in adaptive laws, optimal virtual controller and optimal actual controller are designed as m = m = 2, γ = γ = 15, γ = γ = 13, k = k = 1 2 c1 c2 a1 a2 1c 2c 5.5, k = 1, k = 2, σ = σ = 30, σ ¯ =¯ σ = 10 and δ = . 1b 2b 1 2 1 2 Figure 1 manifests the trajectories of y, y and constraint bound k ,which showsthe r 1c control performance and the system state staying within the restricted interval. Figure 2 represents the trajectories of tracking error z and k , showing that the tracking error is 1 1b 26 Z. SUN ET AL. Figure 3. The trajectories of x and k . 2 2c ˆ ˆ Figure 4. The curves of ℵ and ℵ . c1 c2 maintained in a small neighbourhood of about 0. Figure 3 shows the trajectories of x and k and the state x remains within the blue dashed line. Figures 4–6 display the curves 2c 2 ˆ ˆ ˆ of ||ℵ ||, ||ℵ || and ||ℵ || where i = 1, 2, and the curves from these gur fi es are decreas- fi ci fi ing. Figure 7 showcases the trajectory of optimal actual controller u.Fromthe simulation results, the proposed scheme in this paper can achieve the desired control objective. APPLIED MATHEMATICS IN SCIENCE AND ENGINEERING 27 ˆ ˆ Figure 5. The curves of ℵ and ℵ . a1 a2 ˆ ˆ Figure 6. The curves of ℵ and ℵ . f1 f2 Example 5.2: Similar to [9]and [48], this paper considers the robotic manipulator system as follows: ¨ ˙ JS + AS + MGr sin(S) = u(t), (147) where S and S are the angle and angular velocity of the link, respectively, M is the total mass of the link, J is the rotational inertia of the motor, G is the gravitational acceleration, A is the damping coefficient. Assuming the eeff ct of external disturbance and bias fault on the 28 Z. SUN ET AL. Figure 7. The trajectory of u. Figure 8. The trajectories of y, y and k . r 1c system is taken into account, (147) is rewritten as ¨ ˙ JS + AS + MGr sin(S) = u (t) + D(t). (148) The parameter selection in (148) is similar to that in [48], that is, J = 1, A = 2, M = 1, G = ˙ ¨ 10, r = 1. Let x = S, x = S thus the system (147) can be rewritten as 1 2 APPLIED MATHEMATICS IN SCIENCE AND ENGINEERING 29 Figure 9. The trajectories of z and k . 1 1b Figure 10. The trajectories of x and k . 2 2c x ˙ = x , 1 1 (149) x ˙ = u − 10 sin(x ) − 2x + D(t), 2 1 2 where f (x ¯ ) =−10 sin(x ) − 2x , u = u(t) + u , the reference signal y = 0.2 sin (t). 2 2 1 2 f r The fuzzy logic system we used and bias fault signal are the same as in Example 5.1. The parameters used in the control strategy are designed as m = m = 2, γ = 6, γ = 1 2 c1 a1 30 Z. SUN ET AL. ˆ ˆ Figure 11. The curves of ℵ and ℵ . c1 c2 ˆ ˆ Figure 12. The curves of ℵ and ℵ . a1 a2 4, γ = 15, γ = 13, k = k = 5.5, k = 1, k = 2, σ = σ = 30, σ ¯ =¯ σ = 10 and c2 a2 1c 2c 1b 2b 1 2 1 2 δ = . The reference trajectory y and external disturbance D(t) are den fi ed as y = r r ˆ ˆ 0.2 sin(t) and D(t) = 0.3 sin(2t). The initial values are configured as ℵ (0) = ℵ (0) = f 1 f 2 T T T ˆ ˆ [0.8, 0.8, 0.8, 0.8, 0.8] , ℵ (0) = [0.2, 0.1, 0.1, 0, 0.1] , ℵ (0) = [0.1, 0, 0, 0.2, 0.1] , c1 c2 T T ˆ ˆ ℵ (0) = [0.1, 0.2, 0.1, 0, 0] , ℵ (0) = [0.1, 0, 0, 0.2, 0.1] ,and x (0) = x (0) = 0. a1 a2 1 2 Figures 8–14 are the simulation results of the robotic manipulator system. Figure 8 demon- strates the trajectories of y, y and k , showing satisfactory control performance. As shown r 1c APPLIED MATHEMATICS IN SCIENCE AND ENGINEERING 31 ˆ ˆ Figure 13. The curves of ℵ and ℵ . f1 f2 Figure 14. The trajectory of u. in Figure 9,the tracking error z represented by the red solid line does not exceed the blue dashed line k and remains within a small neighbourhood relating to the origin. The 1b trajectory of x ,which does notcross thebluedashedline k ,isshown in Figure 10.Fig- 2 2c ˆ ˆ ˆ ures 11–13 display the curves of ||ℵ ||, ||ℵ || and ||ℵ || where i = 1, 2. Figure 14 shows fi ci fi the trajectory of optimal actual controller u. 32 Z. SUN ET AL. 6. Conclusion In this article, the issue of adaptive fuzzy optimal n fi ite-time control for uncertain nonlin- ear systems with bias fault and external disturbances is studied. Consider bias fault term and external disturbance as total disturbance, the disturbance observer is designed to track the total disturbance online, where the total disturbance consists of bias fault term and external disturbance. By combining with backstepping and ADP technologies, an adaptive fuzzy optimal ni fi te-time control approach is proposed. It proves that all signals of closed- loop are n fi ite-time stable, and the all system states in constrained sets. One future research direction is to extend the proposed method to more general systems such as stochastic systems [49], switched nonlinear systems [50,51] and uncertain under-actuated switched nonlinear systems [52]. 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Disturbance observer-based fuzzy adaptive optimal finite-time control for nonlinear systems

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APPLIED MATHEMATICS IN SCIENCE AND ENGINEERING 2023, VOL. 31, NO. 1, 2199211 https://doi.org/10.1080/27690911.2023.2199211 Disturbance observer-based fuzzy adaptive optimal finite-time control for nonlinear systems Zidong Sun, Li Liang and Wei Gao School of Information Science and Technology, Yunnan Normal University, Kunming, People’s Republic of China ABSTRACT ARTICLE HISTORY Received 29 December 2022 This paper investigates the issue of disturbance observer-based Accepted 30 March 2023 fuzzy adaptive optimal finite-time control in light of the backstep- ping approach for strict-feedback nonlinear systems with bias fault KEYWORDS term and full state constraints. Considering that external disturbance Adaptive dynamic and bias fault signal can affect the stability of control and control programming; fuzzy quality, a disturbance observer is constructed to track the external adaptive optimal finite-time disturbances and bias fault online. A disturbance observer-based control; external disturbance observer; state observer finite-time control strategy is proposed to achieve optimized con- trol by utilizing the fuzzy logic system approximation-based adaptive MATHEMATICS SUBJECT dynamic programming method under the critic-actor framework. CLASSIFICATIONS The purpose of the critic is to evaluate control performance and the 34H05; 93C10; 93C42 role of the actor is to execute control behaviour. In addition, it is proved that the proposed fuzzy adaptive optimal finite-time scheme not only realizes all signals in closed-loop system are bounded, but also ensures that system states are restricted within specific sets. Finally, simulation results are shown to demonstrate the effective- ness of the proposed control strategy. 1. Introduction In the past few decades, the research on controller designing for nonlinear systems speedy developed neural networks and fuzzy approximation characteristics [1–4]. The incipient control strategies facilitate the control objectives for nonlinear systems are realized in a inn fi ite time, which means that the stability of systems fails to be guaranteed in a n fi ite time. However, in practical problems, it is imperative to achieve superior control performance inafinite time. Dissimilar to asymptotic stability, n fi ite-time stability converges fast without the requirement of long-term transient response, so it is widely concerned and applied to practical systems such as aircraft flight systems [ 5], cart-pole systems [6], teleoperation systems [7] and so on. Bhat and Bernstein [8] explained n fi ite-time stability based on Lyapunov theory and constructed feedback controller for nonlinear systems to achieve n fi ite-time stability. Qiu et al. [ 9] established a fuzzy finite-time controller for nonlinear systems and realized the tracking error is limited to a bounded set. Li et al. [10]studied CONTACT Wei Gao gaowei@ynnu.edu.cn School of Information Science and Technology, Yunnan Normal University, Kunming 650500, People’s Republic of China © 2023 The Author(s). Published by Informa UK Limited, trading as Taylor & Francis Group This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons. org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The terms on which this article has been published allow the posting of the Accepted Manuscript in a repository by the author(s) or with their consent. 2 Z. SUN ET AL. the problem of finite-time containment control involving unmeasurable states for non- linear multiagent systems with input delay. Zhang et al. [11] designed an adaptive fuzzy control strategy based on finite-time stability criterion for nonlinear systems with unavail- able states, in which all states are conn fi ed to restricted sets. Meng et al. [ 12]developed a n fi ite-time quantized controller for dealing with the nonlinear systems contingent on unknown control directions, which constructed state and disturbance observers simul- taneously. Sui et al. [13] constructed an event-trigger adaptive ni fi te control scheme by combining backstepping and varying threshold condition for stochastic nonlinear systems with unmodelled dynamics. Saravanakumar et al. [14] rst fi investigated the n fi ite-time sta- bility problem. The authors in [15–17] studied the issue of n fi ite-time control for nonlinear systems in accordance with full-state constraints. Nguyen et al. [18] introduced a fuzzy con- trol strategy for parallel manipulators to guarantee that the error of tracking converges fast in a ni fi te time. Wang et al. [ 19] constructed a n fi ite-time control algorithm for stochas- tic nonlinear systems with actuator faults. Nevertheless, the above literature take the ni fi te time into account and the designed control strategies ensure that the control objective can be achieved within the n fi ite time. However, these literature do not involve the optimal controltodealwiththe problemofresourceconsumption. In recent years, optimal control has been a striking topic, and it is concerned with estab- lishing control strategies to achieve control objectives with the least resources in regard to optimal policy. In other words, the goal of optimal control is to consume the least amount of resources to achieve the control objective. For nonlinear systems, it is noteworthy that the resolve of Hamilton–Jacobi–Bellman (HJB) equation is imperative in the control pro- cess, but it is problematic to get analytic solutions because of the dynamic uncertainty and strong nonlinearity in the actual control. To overcome this problem, the dynamic program- ming (DP) was developed by Bellman [20]. NotedthatalthoughDPisaneeff ctivetoolfor obtaining optimal solutions, it is prone to the curse of dimensionality, that is, increasing dimensions bring computational disasters. Adaptive dynamic programming (ADP)-based algorithms were demonstrated to efficiently conquer the problem. Werbos [ 21]developed ADP for discrete systems and Abu-Khalaf and Lewis [22] proposed ADP control scheme for continue-time nonlinear systems where the neural networks approximation structure was established to estimate the value function of the HJB equation. Vamvoudakis et al. [23] developed an online actor-critic scheme combined with neural networks for continue-time systems. Wen et al. [24] designed an optimized formation control strategy in reference with identifier-actor-critic structure and approximation characteristic of fuzzy logic systems. Li and Li [25] introduced a fuzzy adaptive optimal fault-tolerant control scheme for stochastic multiagent systems to incorporate Butterworth low-pass filter, which leverages to compen- sate for negative influence caused by nonlinear fault on the system. Wen et al. [ 26]reviewed the fuzzy adaptive optimized control issue subject to unmeasured states with nonlinear systems, in which the state observer satisefi s the Hurwitz condition that sidestep constant designing. Lan et al. [27] introduced an adaptive optimal formation control technology for multiagent systems with unmeasurable states. Li et al. [28] designed a neural network adaptive optimized control algorithm exposing to immeasurable states and constrained states with nonlinear systems. The authors in [29–31] discussed the problem of adaptive optimal control for nonlinear systems. Wen et al. [32] built an adaptive optimal control scheme by virtue of identifier–critic–actor architecture. To achieve high precision control APPLIED MATHEMATICS IN SCIENCE AND ENGINEERING 3 requirements, it is necessary to consider the influence of external disturbance on the sys- tem, which makes sense to design an optimal finite-time control strategy for the controlled system with external disturbance. However, the proposed approaches in [20–32]ignored the external disturbance in the controller system. It is a prominent prospect to eliminate the influence of external disturbances on the sys- tems [33–38]. Ji et al. [39] designed an adaptive fault-tolerant optimized formation control for multiagent systems with disturbances, in which a disturbance observer was constructed to alleviate the negative eeff cts of external disturbance. Liu et al. [ 40]developed an adap- tive control scheme for Markovian jump systems that suffer from external disturbances and actuator faults, in which external disturbances contain matched and mismatched part. Xu et al. [41] designed a neural network disturbance observer for strict-feedback systems to achieve good control performance in spite of unknown dynamics and time-varying distur- bance. Song and Lewis [42] introduced a robust optimal control scheme with a disturbance observer to estimate disturbances for nonlinear systems. Zerari and Chemachema [43] studied the continuously stirred tank reactor systems containing external disturbances and introduced the compensator to confront the influence of external interference in the designed control strategy. Ran et al. [44] proposed a disturbance rejection optimal control for nonlinear systems, in which the perturbances and other uncertainties are estimated by the designed disturbance observer. Chen and Ge [45] presented an adaptive neural control strategy for nonlinear systems with unmeasured states, hysteresis and disturbances. How- ever, the foregoing advancements rarely studied the optimized finite-time control problem based on fuzzy systems for nonlinear systems with full-state constraints and external disturbances. Motivated by the aforementioned researches, this paper considers the fuzzy adap- tive optimal n fi ite-time control issue for nonlinear systems with full-state constraints, external disturbance and bias fault. Combining backstepping and ADP tricks, a fuzzy adap- tive optimal n fi ite-time control strategy is designed. The unknown system functions are approximated by the fuzzy logic system and disturbance observers are designed to solve the influence of external disturbances and bias fault signal. Virtual and actual controllers are introduced by being incorporated with actor-critic architecture and backstepping framework. The main contributions are summarized as follows. (1) In comparison with [25,27,28], the designed scheme via actor-critic framework and n fi ite-time stability theory can ensure the controller systems not only has a faster con- vergence rate and limits the tracking error derives within a temperate area of the origin inafinite time. (2) An disturbance observer-based fuzzy finite-time strategy based on actor-critic struc- ture is proposed. Discriminating the control schemes in [26,32,46], it not only consid- ers the full-state constraint for strict-feedback nonlinear systems but also addresses the eeff cts of bias fault and external disturbance on the system by designing the disturbance observer. The remainder of this article is organized as follows. In Section 2,the system descrip- tion and preliminary knowledge are presented. Section 3 proposes an observer-based fuzzy optimal control scheme. Subsequently, stability analysis is given in Section 4.Asimulation example is shown in Section 5. Finally, the conclusion is presented in Section 6. 4 Z. SUN ET AL. 2. Preliminaries and problem statement 2.1. System description The strict-feedback nonlinear system consisting of unknown dynamics is formulated by x ˙ = x + φ (x ) + D (t), i i+1 i i 1 x ˙ = u + φ (x ) + D (t), (1) n n n n y(t) = x,1 ≤ i ≤ n, T i where x = [x , x , ... , x ] ∈ R (i = 1, ... , n) represent the states vector of the system, i 1 2 i y ∈ R is the control output. φ (x ) ∈ R are the unknown functions and D (t) denote the i i i external disturbances where i = 1, ... , n. All system states are restricted to a compact set, that is, x < |k | where |k | are positive constants with i = 1, 2, ... , n. i ic ic In actual engineering applications, the actuator bias faults often occur during the operation of the actuator. Thus we consider the actuator bias fault signal as u = u + u,(2) where u denotes the control input and u represents the actuator bias fault signal. Suppose u is boundedand thereisaconstant F that u ≤ F. f f Assumption 2.1 ([39]): The external disturbances D (t) are unknown and bounded, ∗ ∗ ¯ ˙ ¯ i.e. there exist real numbers D and D satisfying |D (t)|≤ D and D (t) ≤ D where i i i i i i i = 1, 2, ... , n. Assumption 2.2 ([25]): The reference trajectory y is known and bounded, and the derivatives of the y , y ˙ ··· y are bounded. r r 2.2. Finite-time theory The following lemmas are beneficial to facilitate the design of the controller. Lemma 2.1 ([15]): For any positive numbers C > 0, C > 0, 0 <δ < 1,0 < l < 1,there 1 2 exist a C function V(x) satisfying V(x) ≤−C V − C V + M,(3) 1 2 1−δ C V (x )+lC 1 2 0 1 then the system is n fi ite-time stable, where the setting time T ≤ ln . 0 1−δ C (1−δ) 2 M C ( ) +lC 2 1 (1−l)C Lemma 2.2 ([29]): For any τ ∈ R,there existaninteger constant ξ and a positive constant 0 < q ≤ 1,one has q q n n n q 1−q |τ | ≤ |τ | ≤ n |τ |.(4) i=1 i=1 i=1 APPLIED MATHEMATICS IN SCIENCE AND ENGINEERING 5 Lemma 2.3 ([29]): There exists real variables p and p satisfying the following inequality: 1 2 −r r r 1 2 r r +r r r 1 2 2 r +r 1 2 1 2 |p | |p | ≤ r |p + r |p | .(5) 1 2 3 2 1 3 r + r r 1 2 1 where r ,r and r are positive constants. 1 2 3 2.3. Fuzzy logic system A host of real-world systems exit unknown dynamics which aeff ct the control performance, and fuzzy logic system approximation approach facilitates to solve the negative eeff ct prob- lem of unknown dynamics. The knowledge base constitutes IF–THEN rules which are stated in the following forms: k k k k R :If x is G , x is G , ... , x is G , 1 2 n 1 2 l then y is Q , k = 1, 2, ... , N, T l k where x = (x , x , ... , x ) ∈ R and y are the input and output of fuzzy logic system. G 1 2 and Q represent fuzzy sets, and N denotes the number of rules. Then the fuzzy logic system can be described by N l k k y ¯ μ (x ) k=1 i=1 y(x) = ,(6) N l μ (x ) k=1 i=1 G k k k where μ and μ are fuzzy membership functions, and y ¯ = max μ (y). y∈R G Q The basis functions are den fi ed as μ (x ) i=1 G S (x) = .(7) N l μ (x ) k=1 i=1 T T Denote ℵ= (ℵ , ℵ , ... , ℵ ) ,and S(x) = (S (x), S (x), ... , S (x)) ,and (6)can be 1 2 N 1 2 k stated as follows: y(x) =ℵ S(x).(8) Lemma 2.4 ([24]): Let f (x) be acontinuousfunctiondenfi edoncompact set ℘.There exists a positive constant ϕ that satisfies the following inequality: sup |φ(x) −ℵ S(x)|≤ ϕ.(9) x∈℘ By means of (9), the following fuzzy logic systems are served to approximate the functions φ (i = 1, 2, ... , n): ˆ ˆ φ (x |ℵ ) = ℵ S (x ), (10) i i i i i ˆ ˆ where x represents the approximation of x and ℵ denotes the estimations of ℵ .The ideal i i i i weight vectors ℵ can be described as ∗ T ℵ = arg min[sup |φ (x) −ℵ S(x )|], (11) i i i i ℵ ∈ x ∈℘ i i i i where  isabounded set. i 6 Z. SUN ET AL. The control objective of this article is to design a disturbance observer-based fuzzy adap- tive optimal n fi ite-time control scheme for nonlinear systems (1), so that (1) all the signals in systems are finite time stable; (2) all the system states are in constrained sets; (3) the output of systems admits to track the reference trajectory. 3. Optimal controller design In what follows, a fuzzy adaptive optimal n fi ite-time strategy is designed to achieve the control objective by virtue of backstepping process and actor-critic framework, in which the barrier-type function is accustomed to cost functions. Consider the bias function u in (2) and external disturbance D as wholedisturbance.The nonlinearsystems (1)can be expressed as ⎪ x ˙ = x + φ (x ) + d (t), i i+1 i i i (12) x ˙ = u + φ (x ) + d (t), n n n n y(t) = x,1 ≤ i ≤ n, where d = D (i = 1, 2, ... , n − 1) and d = u + D . i i n f n Remark 3.1: Extensive practical systems contain perturbation items, which have a nega- tive effect for control quality. Different from the strategies proposed in [ 40,47], this paper employs disturbance observer to track external disturbances online to mitigate the negative effects on the systems and improve the control performance of the systems. The term u is bounded and u ≤ F, the external disturbance D ≤ D , and we deduce d = D + u ≤ n n n f f ∗ ∗ D + F = d .Thusthe totaldisturbance d is bounded. n n Then the following coordinate transformation is introduced as z = x − y , 1 1 r z = x −ˆ α ,2 ≤ i ≤ n, (13) i i i−1 where α ˆ is the optimal virtual control and y is the desire tracking trajectory. i−1 Step 1: The time derivative of z can be yielded from (1) and (13) as z ˙ = x ˙ − y ˙ 1 1 r = x + φ + d − y ˙ . (14) 2 1 1 r Choose the inn fi ite integral barrier-type performance index function that satisefi s (1) as J (z (0)) = h (z (s), α (z )) ds, (15) 1 1 1 1 1 1 1b 2 where h (z (s), α (z )) = ψ log + α is the cost function where ψ > 0, α is the 1 1 1 1 1 2 2 1 1 k −z 1b 1 virtual controller and a compact set  ={z : |z | < |k |}.Let α be the optimal virtual 1 1 1 1b control. The optimal performance index function is constructed by the following to achieve the minimum control performance index in (15), ∗ ∗ J (z (t)) = h (z (s), α (z )) ds 1 1 1 1 1 1 0 APPLIED MATHEMATICS IN SCIENCE AND ENGINEERING 7 = min h (z (s), α (z )) ds . (16) 1 1 1 1 Taking thetimederivativeof(16),weacquire ∗ ∗ ∂J ∂J 1 1 = (x + φ + d − y ˙ ). (17) 2 1 1 r ∂t ∂z Den fi e HJB equation associating with (17) as ∂J k 1b ∗ 1 ∗2 H z , α , = ψ log + α 1 1 1 1 1 2 2 ∂z k − z 1b ∂J + (α + φ + d − y ˙ ) = 0. (18) 1 1 r ∂z By solving the equation ∂H /∂α = 0, the optimal virtual controller is obtained as 1 ∂J ∗ 1 α =− . (19) 2 ∂z ∂J With the aim of realizing ni fi te-time optimal control, is decomposed as ∂z 2δ−1 ∂J z 9 z 1 1 = 2σ + 2σ ¯ z + 2d + 1 1 1 1 2 2 2 2 δ−1 ∂z (k − z ) 2 k − z 1 1 1b 1b + 2φ + J (z ), (20) 1 1 ∗ o where σ and σ ¯ are designed positive parameters, and ℵ is the optimal weight. J (z ) = 1 1 1 f 1 2δ−1 z ∂J 9 z 1 1 1 −2σ − 2σ ¯ z − 2d − − 2φ + . Merging (19) and (20), we verify 1 2 2 1 1 1 2 2 1 δ−1 2 ∂z (k −z ) k −z 1 1b 1 1b 1 2δ−1 z 9 z ∗ 1 α =−σ −¯ σ z − d − 1 1 1 1 2 2 2 2 δ−1 (k − z ) 4 k − z 1 1 1b 1b − φ − J (z ). (21) 1 1 In (21), J and φ are unknown functions that can be approximated as o ∗T J =ℵ S + ϕ , (22) J1 J1 1 J1 ∗T φ =ℵ S + ϕ , (23) 1 f 1 f 1 f 1 ∗T where ℵ is the ideal weight and S is the basis vector. Combining (20), (21), (22) J1 J1 and (23), we get 2δ−1 z 1 ∗ 1 ∗T α =−σ −¯ σ z − d − ℵ S 1 1 1 1 J1 1 J1 2 2 δ−1 (k − z ) 2 1b 9 z 1 ∗T − −ℵ S − ϕ , (24) f 1 1 f 1 2 2 4 k − z 2 ib 8 Z. SUN ET AL. 2δ−1 ∂J z 9 z 1 1 = 2σ + 2σ ¯ z + 2d + 1 1 1 1 2 2 2 2 δ−1 ∂z (k + 2z ) 2 k − z 1 1 1b 1b ∗T ∗T +ℵ S + ϕ + 2ℵ S , (25) J1 1 f 1 J1 f 1 ∗ ∗T ∗T where ϕ = 2ϕ + ϕ .Itisnoteworthythat α is inaccessible directly since ℵ and ℵ 1 f 1 J1 1 J1 f 1 are unknown ideal weights. The estimate of the unknown function φ can be represented ˆ ˆ ˆ as φ = ℵ S .The adaptive law ℵ is constructed as 1 f 1 f 1 f 1 ˙ 1 ˆ ˆ ℵ = S − m ℵ , (26) f 1 f 1 1 f 1 2 2 k − z 1 1 where m > 0. To obtain available α , the critic-actor structure with the critic and actor adaptive laws is introduced as follows: 2δ−1 ∂J z 9 z 1 1 = 2σ + 2σ ¯ z + 2d + 1 1 1 1 2 2 2 2 δ−1 ∂z 2 (k + 2z ) k − z 1 1 1b 1b T T ˆ ˆ + ℵ S + 2ℵ S , (27) J1 f 1 c1 f 1 ∗ ∗ ∂J ∂J 1 1 where is the estimation of . Design the critic updated law as ∂z ∂z 1 1 ˆ ˆ ℵ =−γ S S ℵ , (28) c1 c1 J1 c1 J1 where γ > 0. The virtual controller consists of actor adaptive law c1 2δ−1 z 9 z ∗ 1 α ˆ =−σ −¯ σ z − d − 1 1 1 1 2 2 2 2 δ−1 (k − z ) k − z 1b 1 1b 1 T T ˆ ˆ − ℵ S − ℵ S . (29) J1 f 1 a1 f 1 Correspondingly, the actor updated law is designed by ˙ T ˆ ˆ ˆ ˆ ℵ =−S S [γ (ℵ − ℵ ) + γ ℵ ], (30) a1 J1 a1 a1 c1 c1 c1 J1 where γ > 0. By substituting (29) and (27) into (18), we obtain a1 ∂J ∗ 1 H z ˆ , α ˆ , 1 1 ∂z 2δ−1 k z 1b = ψ log + −σ −¯ σ z 1 1 1 1 2 2 2 2 δ−1 k − z (k − z ) 1b 1 1b 1 9 z 1 T T ˆ ˆ ˆ −d − − ℵ S − ℵ S 1 J1 f 1 a1 f 1 2 2 4 k − z 2 ib 2δ−1 z 9 z + 2σ + 2σ ¯ z + 2d + 1 1 1 1 2 2 2 2 δ−1 (k − z ) k − z 1b 1 ib 1 APPLIED MATHEMATICS IN SCIENCE AND ENGINEERING 9 2δ−1 T T 1 ˆ ˆ + ℵ S + 2ℵ S −σ −¯ σ z J1 f 1 1 1 1 c1 f 1 2 2 δ−1 (k − z ) 1b 1 1 9 z T T ˆ ˆ −d − ℵ S − ℵ S − + φ 1 J1 1 f 1 a1 2 2 2 2 4 k − z ib 1 + d − y˙ = 0. 1 r The Bellman residual error e is expressed as ∗ ∗ ∂J ∂J ∗ 1 ∗ 1 e = H z , a ˆ , − H z , a ˆ , 1 1 1 1 1 1 1 ∂z ∂z 1 1 ∂J ∗ 1 = H z , a ˆ , . (31) 1 1 ∂z It is emphasized that the optimal virtual controller α ˆ is constructed to guarantee ∗ ∗ ∂J ∂J ∗ 1 ∗ 1 H (z , α ˆ , ) → 0. If H (z , α ˆ , ) = 0has auniquesolution, then oneobtains 1 1 1 1 1 ∂z 1 ∂z 1 1 ∂J ∗ 1 ∂H z , α ˆ , 1 1 1 ∂z ˆ ˆ = S S (ℵ − ℵ ) = 0. (32) J1 a1 c1 J1 ˆ 2 ∂ℵ a1 Construct a positive function ˆ ˆ ˆ ˆ E = (ℵ − ℵ ) (ℵ − ℵ ) = 0. (33) 1 a1 c1 a1 c1 It is obvious that E = 0 which is equal to (32). The actor and critic adaptive laws can be designed in view of the following relation: ∂E ∂E 1 1 ˆ ˆ =− = 2(ℵ − ℵ ). (34) a1 c1 ˆ ˆ ∂ℵ ∂ℵ a1 1 Thus we have ∂E ∂E 1 ˙ 1 ˙ ˆ ˆ E = ℵ + ℵ 1 c1 a1 ˆ ˆ ∂ℵ ∂ℵ c1 a1 ∂E ∂E 1 1 T T =−γ S S ℵ − S S c1 J1 c1 J1 J1 J1 ˆ ˆ ∂ℵ ∂ℵ c1 a1 ˆ ˆ ˆ × [γ (ℵ − ℵ ) + γ ℵ ] a1 a1 c1 c1 c1 γ ∂E a1 1 =− S S ≤ 0. (35) J1 J1 2 ˆ ∂ℵ a1 10 Z. SUN ET AL. Therefore, (28) and (30) enable (32) to be finally realized. The disturbance observer is designed as d = f (x − κ ), 1 1 1 1 ˆ ˆ κ ˙ = x + d + ℵ S , (36) 1 2 1 f 1 f 1 where f > 0. Define the Lyapunov function as follows: 1 1 1 1 1 1b 2 2 2 2 ˜ ˜ ˜ ˜ V = log + ℵ + ℵ + ℵ + d , (37) c1 a1 f 1 1 2 2 2 k − z 2 2 2 2 1b ˜ ˆ where ℵ =ℵ − ℵ . In view of (13), (28) and (30), the time derivative of V is c1 c1 1 c1 ˙ ˙ ˙ ˙ ˙ ˜ ˆ ˜ ˆ ˜ ˆ ˜ ˜ V = z ˙ − ℵ ℵ − ℵ ℵ − ℵ ℵ + d d 1 1 c1 c1 a1 a1 f 1 f 1 1 1 2 2 k − z 1b 2δ−1 z z = z − σ −¯ σ z − d 2 1 1 1 1 2 2 2 2 δ−1 k − z (k − z ) 1b 1 1b 1 1 9 z T T ˆ ˆ − ℵ S − ℵ S − − y ˙ + φ + d J1 r 1 1 f 1 a1 f 1 2 2 2 4 k − z 1b ˜ ˆ ˆ ˆ + ℵ S S (γ (ℵ − ℵ ) + γ ℵ ) a1 J1 a1 a1 c1 c1 c1 J1 ˜ ˆ ˜ ˆ + ℵ S − m ℵ + ℵ γ S S ℵ f 1 f 1 1 f 1 c1 c1 J1 c1 J1 2 2 k − z 1 1 ˜ ˙ ˜ ˜ + d (d − f (d − ℵ S − ϕ )). (38) 1 1 1 1 f 1 f 1 f 1 The following correlations hold by utilizing the Young’s inequality z 1 z 1 1 2 z ≤ + z , (39) 2 2 2 2 k − z 2 (k − z ) 2 1 1 1b 1b z 1 z 1 1 ∗2 ϕ ≤ + ϕ , (40) f 1 f 1 2 2 2 2 2 k − z 2 (k − z ) 2 1b 1 1b 1 z 1 z 1 1 2 − y ˙ ≤ + y ˙ , (41) 2 2 2 2 2 2 k − z (k − z ) 1b 1 1b 1 z 1 z 1 1 2 ˜ ˜ d ≤ + d , (42) 2 2 2 2 2 2 k − z (k − z ) 1b 1 1b 1 1 z 1 z 1 T T T ˆ ˆ ˆ − ℵ S ≤ + ℵ S S ℵ . (43) J1 J1 a1 a1 a1 J1 2 2 2 2 2 4 4 k − z (k − z ) 1b 1 1b 1 From (39)–(43), we yield 2δ 2 z z 1 1 1 1 2 2 ˙ ˜ V ≤−σ −¯ σ + d + z 1 1 1 1 2 2 2 2 2 (k − z ) k − z 2 2 1 1 1b 1b APPLIED MATHEMATICS IN SCIENCE AND ENGINEERING 11 ˜ ˆ ˆ ˆ + ℵ S S (γ (ℵ − ℵ ) + γ ℵ ) a1 J1 a1 a1 c1 c1 c1 J1 ˜ ˆ ˜ ˆ + ℵ S − m ℵ + ℵ γ S S ℵ f 1 f 1 1 f 1 c1 c1 J1 c1 J1 2 2 k − z 1 1 T T ˜ ˙ ˜ ˜ ˆ ˆ + d (d − f (d − ℵ S − ϕ )) + ℵ S S ℵ . (44) 1 1 1 1 f 1 f 1 f 1 J1 a1 a1 J1 ∗ ∗ ∗ ˜ ˆ ˜ ˆ ˜ ˆ In light of ℵ =ℵ − ℵ , ℵ =ℵ − ℵ and ℵ =ℵ − ℵ ,weget c1 c1 a1 a1 f 1 f 1 c1 a1 f 1 T T ∗T T ∗ T T T T ˜ ˆ ˜ ˜ ˆ ˆ γ ℵ S S ℵ = γ [ℵ S S ℵ − ℵ S S ℵ − ℵ S S ℵ ], (45) c1 J1 J1 c1 J1 J1 c1 J1 c1 J1 J1 c1 J1 c1 c1 J1 c1 J1 T T ∗T T ∗ T T T T ˜ ˆ ˜ ˜ ˆ ˆ γ ℵ S S ℵ = γ [ℵ S S ℵ − ℵ S S ℵ − ℵ S S ℵ ], (46) a1 J1 a1 a1 J1 J1 a1 J1 a1 a1 J1 J1 J1 J1 a1 J1 a1 J1 γ − γ c1 a1 T T T T T T ˜ ˆ ˜ ˜ ˆ ˆ (γ − γ )ℵ S S ℵ ≤ [ℵ S S ℵ − ℵ S S ℵ ]. (47) c1 a1 J1 c1 J1 a1 J1 c1 a1 J1 a1 J1 c1 J1 m m 1 1 ∗2 2 ˆ ˜ ˜ −m ℵ ℵ ≤ ℵ − ℵ . (48) 1 f 1 f 1 f 1 f 1 2 2 Substituting (45)–(48) into (44), we confirm 2δ 2 z z γ c1 1 1 T T ˙ ˜ ˜ V ≤−σ −¯ σ − ℵ S S ℵ 1 1 1 J1 c1 c1 J1 2 2 2 2 (k − z ) k − z 1b 1 1b 1 2γ − γ m a1 c1 1 T T 2 ˜ ˜ ˜ − ℵ S S ℵ − ℵ + M J1 a1 1,1 a1 J1 f 1 2 2 1 1 1 T T T T ˆ ˆ ˆ ˆ − γ ℵ S S ℵ − γ − ℵ S S ℵ c1 J1 c1 a1 J1 a1 c1 J1 a1 J1 2 2 4 γ − γ c1 a1 T T ˆ ˆ − ℵ S S ℵ J1 c1 c1 J1 ˜ ˙ ˜ ˜ + d (d − f (d − ℵ − ϕ S )), (49) 1 1 1 1 f 1 f 1 f 1 1 (γ +γ ) 1 m ∗ 1 c1 a1 ∗T T ∗ 2 1 ∗2 where γ >γ /2, γ > and M = (ℵ S S ℵ ) + y ˙ + + ϕ + a1 c1 a1 1,1 J1 2 2 c1 J1 c1 2 r ℵ 2 f 1 f 1 J1 f 1 1 2 1 2 1 2 T k by reason of z < k .Let λ and λ be the minimal eigenvalue of S S and J1 2 min min J1 2 2b 2 2 2b S S , respectively. The following inequalities hold: f 1 f 1 1 1 2 ∗2 ˜ ˙ ˜ d d ≤ d + d , (50) 1 1 1 1 2 2 1 1 2 ∗2 ˜ ˜ −f d ϕ ≤ f d + f ϕ , (51) 1 1 f 1 1 1 1 f 1 2 2 1 1 f 1 2 2 ˜ ˜ ˜ ˜ −f d ℵ S ≤ f d + f λ ℵ , (52) 1 1 f 1 f 1 1 1 min f 1 2 2 γ γ c1 c1 S J1 T T T ˜ ˜ ˜ ˜ − ℵ S S ℵ ≤− λ ℵ ℵ , (53) J1 c1 c1 c1 J1 min c1 2 2 2γ − γ 2γ − γ a1 c1 a1 c1 T T J1 T ˜ ˜ ˜ ˜ − ℵ S S ℵ ≤− λ ℵ ℵ . (54) J1 a1 a1 a1 J1 a1 min 2 2 12 Z. SUN ET AL. Substituting (50)–(54) into (49), we deduce 2δ 2 z z 1 1 V ≤−σ −¯ σ + M 1 1 1 1 2 2 2 2 (k − z ) k − z 1b 1 1b 1 2γ − γ γ S S a1 c1 c1 J1 2 J1 2 ˜ ˜ − λ ℵ − λ ℵ a1 c1 min min 2 2 1 S 1 3 f 1 2 2 ˜ ˜ − (m + f λ )ℵ − f − d , (55) 1 1 1 min f 1 1 2 2 2 3 1 1 ∗2 ∗2 where f > and M = M + d + f ϕ . 1 1 1,1 1 2 2 1 2 f 1 Step i(2 ≤ i ≤ n − 1): The time derivative z can be derived from (12) and (13) as z ˙ = x + φ + d − α ˆ . (56) i i+1 i i i−1 Introduce the performance index function as J (z (0)) = h (z (s), α (z )) ds, (57) i i i i i i ib 2 where h (z (s), α (z )) = ψ log + α represents the cost function and α denotes the i i i i i i 2 2 k −z ib virtual controller. Let α be the optimal virtual controller. Similar to (16), to achieve the minimum control performance index in (57), the optimal performance index function is established as follows: ∗ ∗ J (z (t)) = h (z (s), α (z )) ds i i i i i i = min h (z (s), α (z )) ds . (58) i i i i Taking thetimederivativeof(58),wededuce ∗ ∗ ∂J ∂J i i ∗ = (x + φ + d − α ˆ ) i+1 i i i−1 ∂t ∂z ∂J i ∗ ∗ = (z + α + φ + d − α ˆ ). (59) i+1 i i i i−1 ∂z Define HJB equation associating with (59) as follows: ∂J k ib ∗ i ∗2 H z , α , = ψ log + α i i 1 i i 2 2 ∂z k − z ib i ∂J i ∗ ∗ + (α + φ + d − α ˆ ) = 0. (60) i i i i−1 ∂z Solve the equation ∂H /∂α = 0, the optimal virtual controller is determined by 1 ∂J ∗ i α =− . (61) 2 ∂z For the purpose of the optimal control, ∂J /∂z can be decomposed as 2δ−1 ∂J z 9 z i i = 2σ + 2σ ¯ z + 2d + i i i i 2 2 2 2 δ−1 ∂z (k − z ) 2 k − z i i ib ib APPLIED MATHEMATICS IN SCIENCE AND ENGINEERING 13 + 2φ + J (z ), (62) i 1 2δ−1 z ∂J 9 z o i i i where σ , σ ¯ > 0and J (z ) =−2σ − 2σ ¯ z − 2d − − 2φ + . i i i i i i i i 2 2 δ−1 2 2 i 2 ∂z (k −z ) k −z i i ib ib Thus (61) can be represented as 2δ−1 9 z ∗ i α =−σ −¯ σ z − d − i i i i 2 2 2 2 δ−1 (k − z ) k − z i i ib ib − φ − J (z ). (63) i i In (63), J and φ areapproximatedbyfuzzy logicsystems as o ∗T J =ℵ S + ϕ , (64) Ji Ji i Ji ∗T φ =ℵ S + ϕ . (65) i fi fi fi The following equations are established in light of combining (64) and (65) with (62) and (63) respectively, 2δ−1 z 1 ∗ ∗T α =−σ −¯ σ z − d − ℵ S i i i i J1 i J1 2 2 δ−1 (k − z ) ib 9 z 1 ∗T − −ℵ S − ϕ , (66) f 1 i f 1 2 2 4 2 k − z ib i ∗ 2δ−1 ∂J z 9 z i i = 2σ + 2σ ¯ z + 2d + i i i i 2 2 2 2 δ−1 ∂z 2 (k − z ) k − z ib i ib i ∗T ∗T +ℵ S + ϕ + 2ℵ S , (67) J1 i f 1 J1 f 1 ∗ ∗T ∗T where ϕ = 2ϕ + ϕ .Itisnoticeablethat α is unavailable because ℵ and ℵ are the i fi Ji i Ji fi unknown ideal weights. Resembling (27) and (29), the actor-critic structure is developed as 2δ−1 ∂J z 9 z i i = 2σ +¯ σ z + 2d + i i i i 2 2 2 2 δ−1 ∂z (k + 2z ) 2 k − z i i ib ib T T ˆ ˆ + ℵ S + 2ℵ S . (68) Ji fi ci fi Design the critic updated law, optimal virtual control law and actor updated law as ˆ ˆ ℵ =−γ S S ℵ , (69) ci ci Ji ci Ji 2δ−1 z 9 z ∗ i α ˆ =−σ −¯ σ z − d − i i i i 2 2 δ−1 2 2 (k − z ) 4 k − z ib i ib i T T ˆ ˆ − ℵ S − ℵ S , (70) J1 f 1 a1 f 1 ˆ ˆ ˆ ˆ ℵ =−S S [γ (ℵ − ℵ ) + γ ℵ ], (71) ai Ji ai ai ci ci ci Ji 14 Z. SUN ET AL. where γ and γ are positive numbers. The fuzzy updated law is constructed as ci ai ˆ ˆ ℵ = S − m ℵ , (72) fi fi i fi 2 2 k − z ib where m is a positive constant. By lumping (68), (70) and (72) into (60), we acquire ∂J ∗ i H z , α ˆ , i i ∂z 2δ−1 k z ib = ψ log + −σ −¯ σ z i i i i 2 2 2 2 δ−1 k − z (k − z ) ib i ib i 9 z 1 z i i T T ˆ ˆ − d − − ℵ S − ℵ S i Ji fi ai fi 2 2 2 2 4 2 k − z k − z i i ib ib 2δ−1 z 9 z + 2σ + 2σ ¯ z + 2d + i i i i 2 2 δ−1 2 2 (k + 2z ) k − z ib i ib i 2δ−1 T T i ˆ ˆ + ℵ S + 2ℵ S −σ −¯ σ z Ji fi i i i ci fi 2 2 δ−1 (k − z ) ib 1 1 T T ∗ ˆ ˆ ˆ −d − ℵ S − ℵ S + φ + d − α ˆ = 0. (73) i Ji fi i i ai i−1 2 2 The disturbance observer is devised as d = f (x − κ ), i i i i ˆ ˆ κ ˙ = x + d + ℵ S . (74) i i+1 i fi fi Construct the Lyapunov function as 1 k 1 1 ib 2 2 ˜ ˜ V = V + log + ℵ + ℵ i i−1 ci ai 2 2 2 2 2 k − z ib i 1 1 2 2 ˜ ˜ + ℵ + d , (75) fi 2 2 ∗ ∗ ∗ ˜ ˆ ˜ ˆ ˜ ˆ where ℵ =ℵ − ℵ , ℵ =ℵ − ℵ and ℵ =ℵ − ℵ . By combining (13), (69) ai ai ci ci fi fi ai ci fi and (71), the time derivative of (75) yields ˙ ˙ ˙ ˙ ˙ ˙ ˜ ˆ ˜ ˆ ˜ ˆ ˜ ˜ V = V + z ˙ − ℵ ℵ − ℵ ℵ − ℵ ℵ + d d i i−1 i ci ci ai ai fi fi i i 2 2 k − z ib 2δ−1 z z = z − σ −¯ σ z − d i+1 i i i i 2 2 2 2 δ−1 k − z (k − z ) i i ib ib 1 9 z T T ∗ ˆ ˆ − ℵ S − ℵ S − − α ˆ + φ + d Ji fi i i ai fi i−1 2 2 2 4 k − z ib i APPLIED MATHEMATICS IN SCIENCE AND ENGINEERING 15 ˜ ˆ ˆ ˆ + ℵ S S (γ (ℵ − ℵ ) + γ ℵ ) ai Ji ai ai ci ci ci Ji ˜ ˆ ˜ ˆ + ℵ S − m ℵ + ℵ γ S S ℵ fi fi i fi ci ci Ji ci Ji 2 2 k − z i i ˜ ˙ ˜ ˜ + d (d − f (d − ℵ S − ϕ )). (76) i i i i fi fi fi The following relationship can be deduced by utilizing the Young’s inequality, z 1 z 1 i 2 z ≤ + z , (77) i+1 i+1 2 2 2 2 k − z 2 (k − z ) 2 i i ib ib z 1 z 1 i ∗2 ϕ ≤ + ϕ , (78) fi fi 2 2 2 2 k − z 2 (k − z ) 2 i i ib ib z 1 z 1 ∗ i ∗2 ˙ ˙ − α ˆ ≤ + α ˆ , (79) i−1 i−1 2 2 2 2 2 k − z 2 (k − z ) 2 ib i ib i z 1 z 1 i 2 ˜ ˜ d ≤ + d , (80) 2 2 2 2 2 2 k − z (k − z ) ib i ib i 1 z 1 z 1 T T T ˆ ˆ ˆ − ℵ S ≤ + ℵ S S ℵ . (81) Ji Ji ai ai ai Ji 2 2 2 2 2 4 4 k − z (k − z ) i i ib ib Associate (77)–(81) with (76), we can derive 2δ 2 z z 1 1 i i 2 2 ˙ ˙ V ≤ V − σ −¯ σ + d + z i i−1 1 i 1 2 2 2 2 2 2 2 (k − z ) k − z i i ib ib ˜ ˆ ˆ ˆ + ℵ S S (γ (ℵ − ℵ ) + γ ℵ ) ai Ji ai ai ci ci ci Ji ˜ ˆ ˜ ˆ + ℵ S − m ℵ + ℵ γ S S ℵ i ci ci Ji ci fi fi fi Ji 2 2 k − z i i T T ˜ ˙ ˜ ˜ ˆ ˆ + d (d − f (d − ℵ S − ϕ )) + ℵ S S ℵ . (82) i i 1 i f 1 fi fi Ji ai ai Ji ∗ ∗ ∗ ˜ ˆ ˜ ˆ ˜ ˆ Due to ℵ =ℵ − ℵ , ℵ =ℵ − ℵ and ℵ =ℵ − ℵ the following correlations are ci ci ai ai fi fi ci ai fi inferred: T T ∗T T ∗ T T T T ˜ ˆ ˜ ˜ ˆ ˆ γ ℵ S S ℵ = γ (ℵ S S ℵ − ℵ S S ℵ − ℵ S S ℵ ), (83) ci Ji ci ci Ji Ji ci Ji ci ci Ji Ji Ji Ji ci Ji ci Ji T T ∗T T ∗ T T T T ˜ ˆ ˜ ˜ ˆ ˆ γ ℵ S S ℵ = γ (ℵ S S ℵ − ℵ S S ℵ − ℵ S S ℵ ), (84) ai Ji ai ai Ji Ji ai Ji ai ai Ji Ji Ji Ji ai Ji ai Ji T T ˜ ˆ (γ − γ )ℵ S S ℵ (85) ci ai Ji ci ai Ji γ − γ ci ai T T T T ˜ ˜ ˆ ˆ ≤ (ℵ S S ℵ − ℵ S S ℵ ), (86) Ji ai Ji ci ai Ji ci Ji m m i i ∗2 2 ˆ ˜ ˜ − m ℵ ℵ ≤ ℵ − ℵ . (87) i fi fi fi fi 2 2 16 Z. SUN ET AL. By lumping (83)–(87) into (76), we get 2δ 2 z z ci i i T T ˙ ˙ ˜ ˜ V ≤ V − σ −¯ σ − ℵ S S ℵ i i−1 i i Ji ci ci Ji 2 2 2 2 (k − z ) k − z i i ib ib 2γ − γ m ai ci i T T 2 ˜ ˜ ˜ − ℵ S S ℵ − ℵ + M Ji ai i,1 ai Ji fi 2 2 1 i 1 T T T T ˆ ˆ ˆ ˆ − γ ℵ S S ℵ − γ − ℵ S S ℵ ci Ji ci ai Ji ai ci Ji ai Ji 2 2 4 γ − γ ci ai T T ˆ ˆ − ℵ S S ℵ Ji ci ci Ji ˜ ˙ ˜ ˜ + d (d − f (d − ℵ − ϕ S )), (88) i i i i fi fi fi (γ +γ ) ci ai ∗T T ∗ 1 2 m ∗ 1 ∗2 1 2 ˙ 1 where M = (ℵ S S ℵ ) + α ˆ + ℵ + ϕ + k by virtue of i,1 Ji Ji Ji Ji i−1 2 2 2 fi 2 fi 2 (i+1)b Ji 1 2 1 2 T z < k . Assuming that λ represents the minimal eigenvalue of S S ,the Ji i+1 min Ji 2 2 (i+1)b following inequalities yields: 1 1 2 ∗2 ˜ ˙ ˜ d d ≤ d + d , (89) i i i i 2 2 1 1 2 ∗2 ˜ ˜ −f d ϕ ≤ f d + f ϕ , (90) i i fi i i fi 2 2 1 1 S fi 2 2 ˜ ˜ ˜ ˜ −f d ℵ S ≤ f d + f λ ℵ , (91) i i fi fi i i i min fi 2 2 γ γ ci ci S T T Ji T ˜ ˜ ˜ ˜ − ℵ S S ℵ ≤− λ ℵ ℵ , (92) Ji ci ci ci Ji ci min 2 2 2γ − γ 2γ − γ ai ci ai ai S Ji T T T ˜ ˜ ˜ ˜ − ℵ S S ℵ ≤− λ ℵ ℵ . (93) Ji ai ai ai Ji min ai 2 2 Substituting (92) and (93) into (88), we obtain i 2δ i 2 z  z j j V ≤ −σ − σ ¯ + M i j j i 2 2 2 2 (k − z ) k − z jb j jb j j=1 j=1 i i 2γ − γ γ S S aj cj cj Jj Jj 2 2 ˜ ˜ − λ ℵ − λ ℵ min aj min cj 2 2 j=1 j=1 1 S 1 3 fj 2 2 ˜ ˜ − (m + f λ )ℵ − f − d , (94) j j j min fj 2 2 2 j=1 1 ∗2 1 ∗2 where M = M + d + f ϕ . i i,1 i 2 2 fi Step n: z can be showcased from (12) and (13) as z = x −ˆ α . (95) n n n−1 APPLIED MATHEMATICS IN SCIENCE AND ENGINEERING 17 Thetimederivativeof(95) is stated by z ˙ = x ˙ − α ˆ n n n−1 = u + φ + d − α ˆ . (96) n n n−1 Define the integral performance index function as J(z ) = h (z , u(z )) ds, (97) n n n n nb 2 ∗ where h = ψ log + α .Let u be the optimal actual controller, then the optimal n i 2 2 n k −z nb performance index function is represented by ∗ ∗ J(z ) = h (z , u (z )) ds n n n n = min h (z , u(z )) ds . (98) n n n Akin to (20), we have ∗ ∗ ∂J ∂J n n ∗ ∗ = (u + φ + d − α ˆ ). (99) n n n−1 ∂t ∂z By associating with (95), the HJB equation is formalized by ∂J k nb ∗ n ∗2 H z , α , = ψ log + u n n n 2 2 ∂z k − z nb ∂J n ∗ ∗ + (u + φ + d − α ˆ ) = 0. (100) n n n−1 ∂z Similar to (61), dealing with the (∂H /u ) = 0, we yield 1 ∂J ∗ n u =− . (101) 2 ∂z ∗ 2δ−1 ∂J z n n 7 n o Let = 2σ + 2σ ¯ z + 2d + + 2φ + J (z ),where σ > 0. The n n n n n n n 2 2 2 δ−1 2 n ∂z 2 n (k −z ) k −z n n nb nb optimal actual controller can be stated by 2δ−1 z 7 z ∗ n u =−σ −¯ σ z − d − n n n n 2 2 δ−1 2 2 (k − z ) 4 k − z n n nb nb − φ − J (z ). (102) n n J and φ canbeapproximated by thefuzzy logicsystemas o ∗T J =ℵ S + ϕ , (103) Jn Jn n Jn ∗T φ =ℵ S + ϕ . (104) n fn fn fn 18 Z. SUN ET AL. Merging (103), (104) and (102), we yield 2δ−1 z 1 ∗ ∗T u =−σ −¯ σ z − d − ℵ S n n n n Jn Jn 2 δ−1 (k − z ) nb 7 z 1 ∗T − −ℵ S − ϕ , (105) fn n fn 4 2 k − z nb n ∂J where ϕ = 2ϕ + ϕ Combining with (103), can be formulated by n fn Jn ∂z ∗ 2δ−1 ∂J z 7 z n n = 2σ + 2σ ¯ z + 2d + n n n n 2 2 2 δ−1 2 ∂z (k − z ) 2 k − z n n nb nb ∗T ∗T +ℵ S + 2ℵ S + ϕ . (106) Jn fn n Jn fn The critic for evaluating (106) and critic updated law is conceived as ∗ 2δ−1 ∂J z 7 z n n = 2σ + 2σ ¯ z + 2d + n n n n 2 2 2 δ−1 2 ∂z (k + 2z ) 2 k − z n n nb nb T T ˆ ˆ + ℵ S + 2ℵ S , (107) Jn fn Jn fn ˆ ˆ ℵ =−γ S S ℵ . (108) cn cn Jn cn Jn Thelaw of theactor andthe actual controller areconstructed as 2δ−1 z z u ˆ =−σ −¯ σ z − d − 2 n n n n 2 2 2 δ−1 2 (k − z ) k − z n n nb nb T T ˆ ˆ − ℵ S − ℵ S , (109) Jn fn an fn ˆ ˆ ˆ ˆ ℵ =−S S [γ (ℵ − ℵ ) + γ ℵ ]. (110) an Jn an an cn cn cn Jn The adaptive law ℵ is updated as fn ˙ n ˆ ˆ ℵ = S − m ℵ , (111) fn fn n fn k − z nb where m is a positive constant. The HJB equation is derived as ∂J ∗ n H z , α ˆ , n n ∂z 2δ−1 k z nb = ψ log + −σ −¯ σ z n n n n 2 2 2 2 δ−1 k − z (k − z ) nb n nb n 7 z 1 T T ˆ ˆ ˆ −d − − ℵ S − ℵ S n Jn fn an fn 4 k − z 2 nb 2δ−1 z 7 z + 2σ + 2σ ¯ z + 2d + n n n n 2 2 2 δ−1 2 (k + 2z ) k − z nb n nb n APPLIED MATHEMATICS IN SCIENCE AND ENGINEERING 19 2δ−1 T T n ˆ ˆ + ℵ S + 2ℵ S −σ −¯ σ z Jn fn n n n cn fn 2 2 δ−1 (k − z ) nb 1 1 T T ∗ ˆ ˙ ˆ ˆ − d − ℵ S − ℵ S + φ + d − α ˆ = 0. (112) n Jn n n fn an fn n−1 2 2 The disturbance observer is built as d = f (u − κ ), n n n ˆ ˆ κ ˙ = u + d + ℵ S . (113) n n fn fn The Lyapunov function is selected as follows: 1 k 1 1 nb 2 2 ˜ ˜ V = V + log + ℵ + ℵ n n−1 cn an 2 2 2 k − z nb 1 1 2 2 ˜ ˜ + ℵ + d . (114) fn 2 2 Merging with (108), (110), (113) and (95), the time derivative of (114) is determined by ˙ ˙ ˙ ˙ ˙ ˜ ˆ ˜ ˆ ˜ ˆ V = V + z ˙ − ℵ ℵ − ℵ ℵ − ℵ ℵ n n−1 n cn cn an an fn fn k − z nb ˜ ˜ + d d n n 2δ−1 z z = −σ −¯ σ z − d n n n n 2 2 2 2 δ−1 k − z (k − z ) n n nb nb 1 7 z T T ˆ ˆ − ℵ S − ℵ S − − α ˆ + φ + d Jn fn n−1 n n an fn 2 4 k − z nb n ˜ ˆ ˆ ˆ + ℵ S S (γ (ℵ − ℵ ) + γ ℵ ) an Jn an an cn cn cn Jn ˜ ˆ ˜ ˆ + ℵ S − m ℵ + ℵ γ S S ℵ fn fn n fi cn cn Jn cn Jn 2 2 k − z n n ˜ ˙ ˜ ˜ + d (d − f (d − ℵ S − ϕ )). (115) n n n n fn fn fn By Young’s inequality, we acquire z 1 z 1 n ∗2 ϕ ≤ + ϕ , (116) f 1 fn 2 2 2 2 2 k − z 2 (k − z ) 2 n n nb nb z 1 z 1 ∗ n ∗2 − α ˙ ≤ + α ˙ , (117) n−1 i−1 2 2 2 2 2 k − z 2 (k − z ) 2 n n nb nb z 1 z 1 n 2 ˜ ˜ d ≤ + d , (118) 2 2 2 2 2 k − z 2 (k − z ) 2 n n nb nb 1 z 1 z 1 T n T T ˆ ˆ ˆ − ℵ S ≤ + ℵ S S ℵ . (119) Jn Jn an an an Jn 2 2 2 2 2 2 k − z 4 (k − z ) 4 n n nb nb 20 Z. SUN ET AL. Integrate (116)–(119) into (115), we have 2δ 2 z z 1 n n ˙ ˙ V ≤ V − σ −¯ σ + d n n−1 n n 2 2 2 δ 2 (k − z ) k − z n n nb nb ˜ ˆ ˆ ˆ + ℵ S S (γ (ℵ − ℵ ) + γ ℵ ) an Jn an an cn cn cn Jn ˜ ˆ ˜ ˆ + ℵ S − m ℵ + ℵ γ S S ℵ n cn cn Jn cn fn fn fn Jn 2 2 k − z n n ˜ ˙ ˜ ˜ + d (d − f (d − ℵ S − ϕ )) n n n n fn fn fn T T ˆ ˆ + ℵ S S ℵ . (120) Jn an an Jn ∗ ∗ ∗ ˜ ˆ ˜ ˆ ˜ ˆ From ℵ =ℵ − ℵ , ℵ =ℵ − ℵ and ℵ =ℵ − ℵ , we get the following rela- cn cn an an fn fn cn an fn tions: T T ∗T T ∗ T T T T ˜ ˆ ˜ ˜ ˆ ˆ γ ℵ S S ℵ = γ [ℵ S S ℵ − ℵ S S ℵ − ℵ S S ℵ ], (121) cn Jn cn cn Jn Jn cn Jn cn cn Jn Jn Jn Jn cn Jn cn Jn T T ∗T T ∗ T T T T ˜ ˆ ˜ ˜ ˆ ˆ γ ℵ S S ℵ = γ [ℵ S S ℵ − ℵ S S ℵ − ℵ S S ℵ ], (122) an Jn an an Jn Jn an Jn an an Jn Jn Jn Jn an Jn an Jn γ − γ cn an T T T T T T ˜ ˆ ˜ ˜ ˆ ˆ (γ − γ )ℵ S S ℵ ≤ [ℵ S S ℵ − ℵ S S ℵ ], (123) cn an Jn cn Jn an Jn cn an Jn an Jn cn Jn m m n n ∗2 2 ˆ ˜ ˜ −m ℵ ℵ ≤ ℵ − ℵ . (124) fn fn fn fn 2 2 Invoking (121)–(124) and (94) for (115), we infer 2δ 2 z z n n ˙ ˙ V ≤ V − σ −¯ σ n n−1 n n 2 2 2 δ 2 (k − z ) k − z nb n nb n 2γ − γ m an cn n T T 2 ˜ ˜ ˜ − ℵ S S ℵ − ℵ + M Jn an n,1 an Jn fn 2 2 1 n 1 T T T T ˆ ˆ ˆ ˆ − γ ℵ S S ℵ − γ − ℵ S S ℵ cn Jn cn an Jn an cn Jn an Jn 2 2 4 γ − γ γ cn an cn T T T T ˆ ˆ ˜ ˜ − ℵ S S ℵ − ℵ S S ℵ Jn cn Jn cn cn Jn cn Jn 2 2 ˜ ˙ ˜ ˜ + d (d − f (d − ℵ − ϕ S )), (125) n n n n fn fn fn (γ +γ ) m Jn cn an ∗T T ∗ 1 ∗2 1 ∗ 1 ∗2 where M = (ℵ S S ℵ ) + α ˆ + ℵ + ϕ .Suppose that λ rep- n,1 Jn cn Jn cn n−1 min 2 2 2 fn 2 fn resents the minimal eigenvalue of S S , the following inequalities yield: Jn Jn 1 1 2 ∗2 ˜ ˙ ˜ d d ≤ d + d , (126) n n n n 2 2 1 1 2 ∗2 ˜ ˜ −f d ϕ ≤ f d + f ϕ , (127) n n fn n n n fn 2 2 1 1 S fn 2 2 ˜ ˜ ˜ ˜ −f d ℵ S ≤ f d + f λ ℵ , (128) n n fn fn n n min fn 2 2 APPLIED MATHEMATICS IN SCIENCE AND ENGINEERING 21 γ γ cn cn S Jn T T T ˜ ˜ ˜ ˜ − ℵ S S ℵ ≤− λ ℵ ℵ , (129) Jn cn cn cn Jn min cn 2 2 2γ − γ 2γ − γ an cn an cn T T Jn T ˜ ˜ ˜ ˜ − ℵ S S ℵ ≤− λ ℵ ℵ . (130) Jn an an an Jn an min 2 2 Substituting (129), (130) and similar to (94), we obtain the following inequality: n 2δ n 2 z  z j j V ≤ −σ − σ ¯ + M n j j n 2 2 2 2 (k − z ) k − z jb j jb j j=1 j=1 n n 2γ − γ γ aj cj S cj S Jj Jj 2 2 ˜ ˜ − λ ℵ − λ ℵ aj cj min min 2 2 j=1 j=1 n n 1 1 3 fj 2 2 ˜ ˜ − (m + f λ )ℵ − f − d , (131) j j j min fj 2 2 2 j=1 j=1 1 1 ∗2 ∗2 where M = M + d + f ϕ . n n,1 n 2 2 fi 4. Stability analysis In this section, the stability of the system is demonstrated. Theorem 4.1: Consider the nonlinear system (1) with actuator bias fault signal and external disturbances. Suppose that Assumptions 2.1 and 2.2 hold. Taking into account the designed critic adaptive laws as (28), (69) and (108),actor updatedlawsas (30), (71) and (110), fuzzy adaptive laws (26), (72) and (111), and disturbance observers (36), (74) and (113).The pro- posed fuzzy adaptive optimal n fi ite-time control scheme ensures that (1) all signals within the closed-loop system are bounded; (2) all states are in their specicfi intervals. Proof: Let V = V , on the basis of (131), we have n 2δ n 2 z z j j V ≤− σ − σ ¯ + M j j n 2 2 δ 2 2 (k − z ) k − z jb j jb j j=1 j=1 n n 2γ − γ γ aj cj S cj S Jj Jj 2 2 ˜ ˜ − λ ℵ − λ ℵ aj cj min min 2 2 j=1 j=1 n n 1 1 3 fj 2 2 ˜ ˜ − (m + f )λ ℵ − f − d . (132) j j j min fj 2 2 2 j=1 j=1 Let C = min{2σ ,2σ , ... ,2σ }, C = min{2σ ¯ ,2σ ¯ , ... ,2σ ¯ }, C = min{(2γ − γ ) σ 1 2 n σ ¯ 1 2 n a a1 c1 S S S S S S J1 J2 Jn J1 J2 Jn λ , (2γ − γ )λ , ... , (2γ − γ )λ }, C = min{γ λ , γ λ , ... , γ λ }, a2 c2 an cn c c1 c2 cn min min min min min min S S S f 1 f 2 fn C = min{(m + f )λ , (m + f )λ ··· , (m + f )λ } and C = min{f − , f − f 1 1 2 2 n n d 1 2 min min min 3 3 , ... , f − }. From (132), we acquire 2 2 n 2δ n 2 z  z 1 1 j j V ≤−C − C + M σ σ ¯ n 2 2 2 2 2 2 (k − z ) k − z j j jb jb j=1 j=1 22 Z. SUN ET AL. n n 1 1 2 2 ˜ ˜ − C ℵ − C ℵ a c aj cj 2 2 j=1 j=1 n n 1 1 2 2 − C ℵ − C d . (133) f d fj j 2 2 j=1 j=1 1 T ˜ ˜ In light of (4), den fi e p = 1, p = ℵ ℵ , r = 1 − δ, r = δ, r = 1, one has 1 2 k 1 2 3 k=1 2 k ⎛ ⎞ n n 1 1 2 2 ⎝ ˜ ⎠ ˜ ℵ ≤ 1 − δ + δ ℵ . (134) fj fj 2 2 j=1 j=1 In view of the same fashion as (134), it leads to ⎛ ⎞ 1 1 2 2 ⎝ ˜ ⎠ ˜ d ≤ 1 − δ + δ d , (135) j j 2 2 j=1 ⎛ ⎞ n n 1 1 2 2 ⎝ ˜ ⎠ ˜ ℵ ≤ 1 − δ + δ ℵ , (136) aj aj 2 2 j=1 j=1 ⎛ ⎞ n n 1 1 2 2 ⎝ ˜ ⎠ ˜ ℵ ≤ 1 − δ + ℵ . (137) cj cj 2 2 j=1 j=1 It follows from (5) that n 2δ n 2 z z 1 1 j j δ−1 − C ≤−2 C . (138) σ σ 2 2 2 2 2 2 (k − z ) k − z j j jb jb k=1 k=1 According to (135)–(138), (133) can be rewritten as n 2 n 2 z z 1 1 j j δ−1 V ≤−2 C − C σ σ ¯ 2 2 2 2 2 k − z 2 k − z j j jb jb k=1 j=1 ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ δ δ δ n n n 1 1 1 2 2 2 ⎝ ˜ ⎠ ⎝ ˜ ⎠ ⎝ ˜ ⎠ − C ℵ − C ℵ − C ℵ f a c fj aj cj 2 2 2 j=1 j=1 j=1 ⎛ ⎞ n n 1 1 2 2 ˜ ˜ ⎝ ⎠ − C ℵ − C (1 − δ) ℵ d f cj fj 2 2 j=1 j=1 n n 1 1 2 2 ˜ ˜ − C (1 − δ) ℵ − C (1 − δ) ℵ c a cj aj 2 2 j=1 j=1 + (1 − δ)(C + C + C + C ) + M . (139) c d f a n APPLIED MATHEMATICS IN SCIENCE AND ENGINEERING 23 δ−1 Den fi e C = min{2 C , C , C , C , C , } and C = min{C , C (1 − δ), C (1 − δ),, 1 σ f a c d 2 σ ¯ f c C (1 − δ)}. It is worth noting that z (i = 1, 2, ... , n) in constraint sets, we can get a i 2 2 k z n 1 jb n 1 j log < , rewrite (139) as 2 2 2 2 j=1 2 j=1 2 k −z k −z jb j jb j V ≤−C V − C V + M, (140) 1 2 where M = M + (1 − δ)(C + C + C + C ). In light of (140), we know that V ≤ n c d f a −C V + M, therefore, it is easy to obtain V is bounded. The similarity follows that ||ℵ ||, 2 ai ˜ ˜ ˆ ˆ ˆ ||ℵ || and ||ℵ || are bounded. Therefore, ||ℵ ||, ||ℵ || and ||ℵ || are bounded. From pre- ci fi a1 c1 f 1 vious analysis, we get |z |≤ k ,itcan be furtherdeductedbyAssumption 2.2 that |x |= i ib 1 ∗ ∗ ∗ |z |+|y |≤ k + y = k ,where y is the upper bounded of y . α ˆ is also bounded and 1 r 1b 1c r r r ∗ ∗ α ˆ ≤¯ α ,and we canderivethat |x |=|z |+ αˆ ≤ k +¯ α = k .Inthe same way, it is 1 2 2 2b 1 2c 1 1 true that x ≤ k ,where i = 3, ... , n. Thus all system states are conn fi ed within constraint i ic sets. 1−δ 1 C V (x )+lC 2 0 1 Setting T = ln ,where 0 < l < 1. We can deduce that V(x) ≤ 1−δ C (1−δ) C ( ) +lC 2 1 (1−l)C 1−δ ( ) ,thus (1−l)C 1−δ 2 2 1 k 1 z M 1b 1 log ≤ ≤ V ≤ . (141) 2 2 2 2 2 2 (1 − l)C k − z k − z 1b 1 1b 1 It follows from (141) that 1−δ −2 (1−l)C |z |=|y − y |≤ k 1 − e , (142) 1 r 1b which means that the tracking error remains within the origin of the area after the setting time. Remark 4.1: The adaptive optimal control schemes described in [26,28,32]ensurethatall signals are semi-globally uniformly ultimately bounded with V ≤−C V + M,the con- vergence time may be infinite. The strategy proposed in this paper, the time derivative of V satisefi s V ≤−C V − C V + M, which means that it can achieve the faster response. 1 2 Remark 4.2: It is worth noting that the tracking error is remained within the origin of the area after the setting time by choosing appropriate parameters. Reduce the area of origin by adjusting the value of δ or increasing C . Therefore, the parameters should be chosen carefully. 5. Simulation example This section aims to verify the effectiveness of the proposed fuzzy control method with a simulation instance. 24 Z. SUN ET AL. Example 5.1: A model with external disturbances and bias fault signal is presented below ⎪ x ˙ = x + φ (x ) + D (t), 1 2 1 1 1 x ˙ = u + u + φ (x ) + D (t), (143) 2 f 2 2 2 y(t) = x,1 ≤ i ≤ n, where x = [x , x ] , φ = sin(x ) cos(x ), φ = sin(x + x ), D = 2sin(0.5t), D = sin 2 1 2 1 1 1 2 1 2 1 2 (2t), the output reference trajectory is den fi ed as y = 5sin(0.5t),and 0, t < T, u = (144) 6sin(0.9t − 1), t ≥ T, where T = 10 s. The chosen membership functions are referred to (x − 3) μ (x ) = exp − , (x − 1) μ (x ) = exp − , (x ) μ (x ) = exp − , (x + 1) μ (x ) = exp − , (x + 3) μ (x ) = exp − . (145) Thus the basis function vector S is denoted as f 1 1 5 μ (x ) μ (x ) 1 1 G G S = , ... , . (146) f 1 5 5 i i μ (x ) μ (x ) 1 1 i=1 G i=1 G Analogously, the basis function vectors S , S and S are formulated as f 2 J1 J2 ⎡ ⎤ 2 2 1 1 μ (x ) μ (x ) 2 2 j=1 j=1 G G ⎣ ⎦ S = , ... , , f 2 j j 5 2 5 2 μ (x ) μ (x ) j j i=1 j=1 G i=1 j=1 G 1 5 μ (z ) μ (z ) 1 1 G G S = , ... , , J1 5 5 i i μ (z ) μ (z ) 1 1 i=1 G i=1 G 1 5 μ (z ) μ (z ) 2 2 G G S = , ... , . J2 5 5 i i μ (z ) μ (z ) 2 2 i=1 i=1 G G ˆ ˆ ˆ Theinitial values arecongfi ured as ℵ (0) = ℵ (0) = [0.8, 0.8, 0.8, 0.8, 0.8] , ℵ (0) = f 1 f 2 c1 T T T ˆ ˆ [0.3, 0.3, 0.3, 0.3, 0.3] , ℵ (0) = [0.8, 0.8, 0.8, 0.8, 0.8] , ℵ (0) = [0.1, 0.1, 0.1, 0.1, 0.1] , c2 a1 ℵ (0) = [0.5, 0.5, 0.5, 0.5, 0.5] ,and x (0) = x (0) = 0. a2 1 2 APPLIED MATHEMATICS IN SCIENCE AND ENGINEERING 25 Figure 1. The trajectories of y, y and k . r 1c Figure 2. The trajectories of z and k . 1b The parameters in adaptive laws, optimal virtual controller and optimal actual controller are designed as m = m = 2, γ = γ = 15, γ = γ = 13, k = k = 1 2 c1 c2 a1 a2 1c 2c 5.5, k = 1, k = 2, σ = σ = 30, σ ¯ =¯ σ = 10 and δ = . 1b 2b 1 2 1 2 Figure 1 manifests the trajectories of y, y and constraint bound k ,which showsthe r 1c control performance and the system state staying within the restricted interval. Figure 2 represents the trajectories of tracking error z and k , showing that the tracking error is 1 1b 26 Z. SUN ET AL. Figure 3. The trajectories of x and k . 2 2c ˆ ˆ Figure 4. The curves of ℵ and ℵ . c1 c2 maintained in a small neighbourhood of about 0. Figure 3 shows the trajectories of x and k and the state x remains within the blue dashed line. Figures 4–6 display the curves 2c 2 ˆ ˆ ˆ of ||ℵ ||, ||ℵ || and ||ℵ || where i = 1, 2, and the curves from these gur fi es are decreas- fi ci fi ing. Figure 7 showcases the trajectory of optimal actual controller u.Fromthe simulation results, the proposed scheme in this paper can achieve the desired control objective. APPLIED MATHEMATICS IN SCIENCE AND ENGINEERING 27 ˆ ˆ Figure 5. The curves of ℵ and ℵ . a1 a2 ˆ ˆ Figure 6. The curves of ℵ and ℵ . f1 f2 Example 5.2: Similar to [9]and [48], this paper considers the robotic manipulator system as follows: ¨ ˙ JS + AS + MGr sin(S) = u(t), (147) where S and S are the angle and angular velocity of the link, respectively, M is the total mass of the link, J is the rotational inertia of the motor, G is the gravitational acceleration, A is the damping coefficient. Assuming the eeff ct of external disturbance and bias fault on the 28 Z. SUN ET AL. Figure 7. The trajectory of u. Figure 8. The trajectories of y, y and k . r 1c system is taken into account, (147) is rewritten as ¨ ˙ JS + AS + MGr sin(S) = u (t) + D(t). (148) The parameter selection in (148) is similar to that in [48], that is, J = 1, A = 2, M = 1, G = ˙ ¨ 10, r = 1. Let x = S, x = S thus the system (147) can be rewritten as 1 2 APPLIED MATHEMATICS IN SCIENCE AND ENGINEERING 29 Figure 9. The trajectories of z and k . 1 1b Figure 10. The trajectories of x and k . 2 2c x ˙ = x , 1 1 (149) x ˙ = u − 10 sin(x ) − 2x + D(t), 2 1 2 where f (x ¯ ) =−10 sin(x ) − 2x , u = u(t) + u , the reference signal y = 0.2 sin (t). 2 2 1 2 f r The fuzzy logic system we used and bias fault signal are the same as in Example 5.1. The parameters used in the control strategy are designed as m = m = 2, γ = 6, γ = 1 2 c1 a1 30 Z. SUN ET AL. ˆ ˆ Figure 11. The curves of ℵ and ℵ . c1 c2 ˆ ˆ Figure 12. The curves of ℵ and ℵ . a1 a2 4, γ = 15, γ = 13, k = k = 5.5, k = 1, k = 2, σ = σ = 30, σ ¯ =¯ σ = 10 and c2 a2 1c 2c 1b 2b 1 2 1 2 δ = . The reference trajectory y and external disturbance D(t) are den fi ed as y = r r ˆ ˆ 0.2 sin(t) and D(t) = 0.3 sin(2t). The initial values are configured as ℵ (0) = ℵ (0) = f 1 f 2 T T T ˆ ˆ [0.8, 0.8, 0.8, 0.8, 0.8] , ℵ (0) = [0.2, 0.1, 0.1, 0, 0.1] , ℵ (0) = [0.1, 0, 0, 0.2, 0.1] , c1 c2 T T ˆ ˆ ℵ (0) = [0.1, 0.2, 0.1, 0, 0] , ℵ (0) = [0.1, 0, 0, 0.2, 0.1] ,and x (0) = x (0) = 0. a1 a2 1 2 Figures 8–14 are the simulation results of the robotic manipulator system. Figure 8 demon- strates the trajectories of y, y and k , showing satisfactory control performance. As shown r 1c APPLIED MATHEMATICS IN SCIENCE AND ENGINEERING 31 ˆ ˆ Figure 13. The curves of ℵ and ℵ . f1 f2 Figure 14. The trajectory of u. in Figure 9,the tracking error z represented by the red solid line does not exceed the blue dashed line k and remains within a small neighbourhood relating to the origin. The 1b trajectory of x ,which does notcross thebluedashedline k ,isshown in Figure 10.Fig- 2 2c ˆ ˆ ˆ ures 11–13 display the curves of ||ℵ ||, ||ℵ || and ||ℵ || where i = 1, 2. Figure 14 shows fi ci fi the trajectory of optimal actual controller u. 32 Z. SUN ET AL. 6. Conclusion In this article, the issue of adaptive fuzzy optimal n fi ite-time control for uncertain nonlin- ear systems with bias fault and external disturbances is studied. Consider bias fault term and external disturbance as total disturbance, the disturbance observer is designed to track the total disturbance online, where the total disturbance consists of bias fault term and external disturbance. By combining with backstepping and ADP technologies, an adaptive fuzzy optimal ni fi te-time control approach is proposed. It proves that all signals of closed- loop are n fi ite-time stable, and the all system states in constrained sets. One future research direction is to extend the proposed method to more general systems such as stochastic systems [49], switched nonlinear systems [50,51] and uncertain under-actuated switched nonlinear systems [52]. 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Journal

Applied Mathematics in Science and EngineeringTaylor & Francis

Published: Dec 31, 2023

Keywords: Adaptive dynamic programming; fuzzy adaptive optimal finite-time control; external disturbance observer; state observer; 34H05; 93C10; 93C42

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