Fuzzy-based adaptive asymptotic tracking control design for uncertain MIMO nonlinear systems with state constraints

This article concentrates on designing an adaptive fuzzy asymptotic tracking control strategy for uncertain multi-input and multi-output (MIMO) nonlinear systems with state constraints and unknown virtual control coeﬃcients (UVCCs). Such design apply the fuzzy logic systems to approximate unknown dynamics, and an asymptotic controller is recursively constructed by employing a bound estimation method and some smooth functions. The predeﬁned state constrains are guaranteed by virtue of barrier Lyapunov function. Diﬀerent from the reporting achievements, the restrictive assumption about the prior knowledge of UVCCs is removed in our context. By fusing the lower bounds of UVCCs into iterative Lyapunov functions, the stability of the controlled system is guaranteed. Finally, simulation example is conducted to exhibit the validity of theoretical ﬁndings.


Introduction
Tracking control nonlinear systems has become a hot and rapidly growing research topic. Over the past decade, many approaches and concepts have been presented in order to achieve precise control for nonlinear systems [1][2][3][4][5]. The backstepping control of nonlinear systems has gained tremendous attention due to its recursively character, with many representative achievements being reported [6][7][8]. Note that these previous work are restricted to the nonlinear systems with known nonlinear dynamics.
Since the preeminent approximation ability of fuzzy logic systems (FLSs) or neural networks (NNs), numerous intelligent fuzzy or NN control designs have been explored for uncertain nonlinear systems (UNS) with unknown dynamics [9][10][11][12][13][14][15][16][17]. In [9], the adaptive fuzzy control (AFC) design was presented for UNS. In [10], the fuzzy-based backstepping design scheme was fabricated for UNS by associating with high-gain filters. With the help of small-gain theorem, an AFC strategy was investigated for UNS with unmodeled dynamics [11]. In [12], the fuzzy-based prescribed performance thought was explored for UNS. Based on the inverse optimal algorithm, an adaptive fuzzy optimal thought was discovered for UNS with dynamic uncertainties [14]. In [15], the approximated-based intelligent neural thought was described for UNS. By applying the reinforcement learning algorithm [16], an adaptive NN optimized backstepping design thought was dexterously presented for UNS. In [17], the observer-based AFC scheme was firstly proposed for UNS. The above approximation-based intelligent control schemes are mainly focused on the single-input and single-output (SISO) nonlinear systems. Meanwhile, for multi-input and multi-output (MIMO) nonlinear systems, numerous fuzzy or NN control designs have been explored by virtue of backstepping technique [18][19][20]. In [18], the NN-based tracking design approach was presented for UNS with unknown dynamics. In [19], the fuzzy decoupling analysis strategy was fabricated for the MIMO chemical systems. By constructing auxiliary systems to address the influence of saturation character, the NN-based tracking design approach was discovered for MIMO nonlinear systems [20]. Meanwhile, some output feedback intelligent control designs were presented for MIMO systems with immeasurable states [21][22][23][24]. It is worth mentioning that the foregoing works are mainly concentrated on the nonlinear systems without state constraints.
Due to the safe specification and performance requirement, various constraints are widespread in the practical physical systems. Therefore, it is essential to explore the effective control strategy to address this problem. Recently, the barrier Lyapunov function (BLF) has been exploited to deal with the state constraints problem. For nonlinear systems with output constraint, the BLF-based tracking design approach was discovered [25]. Based on BLF design thought, the full state constraints were subsequently investigated for UNS with parametric uncertainties [26]. Meanwhile, some intelligent fuzzy or NN control methods were proposed by fusing the BLF into backstepping design framework [27][28][29][30][31][32][33][34]. In [28], the intelligent tracking design scheme was fabricated for state-constrained UNS with faults. In [30], the BLF-based AFC method was presented for MIMO nonlinear systems with uncertainties. Associating with BLF design thought, the approximation-NN tracking control strategy was fabricated for state-constrained MIMO nonlinear systems [31]. By applying filter backstepping technique [33], a NN tracking control design was described for state-constrained MIMO nonlinear systems. Considering the MIMO chemical process with state constraints [34], the AFC algorithm was devised with the help of backstepping technique. Now that these intelligent design methods only achieve the bounded-error control effect, that is to say, the output tracking error converges into a small region, whose size can be adjusted by selecting design parameters.
Obviously, the asymptotic tracking control has better convergence feature than the bounded-error result. Since the remarkable superiority of asymptotic control, much endeavors had been made by various scientific communities [35][36][37]. By introducing positive integrable function to compensate the dead-zone nonlinearity [35], an adaptive asymptotic design scheme was presented for UNS. In [37], the modified backstepping asymptotic tracking method was devised for UNS with input quantization feature. The above asymptotic design thoughts suppose that the unknown virtual control coefficients (UVCCs) of nonlinear systems are constants or known functions. To surmount this restriction, by integrating the lower bounds of UVCCs into Lyapunov function construction, a fuzzy-based asymptotic controller was delicately devised for UNS with UVCCs and unknown dynamics [38]. Moreover, the design thought in [38] was extended into UNS with full state constraints and UVCCs [39]. As far as we know, how to construct an asymptotic tracking strategy for MIMO nonlinear systems with state constraints and UVCCs has not been adequately investigated, which stimulate us to study this problem.
This article will attempt to construct an adaptive fuzzy asymptotic tracking control (AFATC) scheme for uncertain MIMO nonlinear systems with state constraints and UVCCs. More specifically, the BLF is utilized to maintain the prescribed state constraints. By applying the bound estimation method and some positive integral functions, the adverse influences caused by UVCCs are surmounted. The main contributions of the AFATC design are enumerated as follows: (1) In contrast to the published intelligent approximation methods of MIMO nonlinear systems [18][19][20], where only achieve bound-error control effect. By applying backstepping technique and some smooth functions, the asymptotic controller is recursively constructed.
(2) Different with the asymptotic control design [38], which is mainly focused on the SISO nonlinear systems with UVCCs. In our context, the intelligent asymptotic tracking control design is further extended to MIMO nonlinear systems. In addition, the adverse effects of UVCCs are successfully surmounted by adopting the bound estimation method.
(3) By fusing the lower bounds of UVCCs into the BLF construction in every recursively design step, the asymptotic convergence character is guaranteed and the state constraints are not violated.

Problem formulation and preliminaries
Consider the MIMO nonlinear systems as follows: 3 . . .
wherex i,s = [x i,1 , x i,2 , . . . , x i,s ] T ∈ R s (i = 1, . . . , m, s = 1, . . . , n i ) is the state vector for the ith subsystem; T ; u i ∈ R and y i ∈ R denote the control input and system output; f i,s (·) and g i,s (·) denote unknown functions. All states are demanded to stay in the sets |x i,s | < k ci,s with k ci,s > 0 being constants.
This article aims to construct an AFATC strategy for system (1), such that the system output y i can asymptotically follow the desired signal y di , and the prescribed state constraints are not violated.
Assumption 2 The signs of g i,s (·) are known, i.e., g i,s (·) are either positive or negative. Without losing generality, it is further supposed that g i,s (·) ≥ g i,s > 0 with g i,s being positive constants. (2) Lemma 2 [37] For positive continuous function σ(t) and any x ∈ R, we can get whereσ > 0 is a constant.

Lemma 3 [9]
Define a continuous unknown function F(℘) on the compact set Ω. Therefore, for C > 0, there is a fuzzy logic system, then T is the center vector and κ i is the width, respectively.

Controller design
In this section, an AFATC is presented to achieve control aim. To this end, introduce the following coordinate transformation: where α i,s−1 are the virtual control laws to be devised later.
CalculatingV i,1 along (8) yieldṡ where Since the functions F i,1 are unknown, which can not directly apply to control design process. Associating with Lemma 3, the FLS where Inserting (11) into (10) yieldṡ where the definitions of θ i,1 and β i, .
The time derivative of V i,s iṡ where κ ζi,s = ζi,s In accordance with Lemma 3, the FLS P T i,s Q i,s (℘ i,s ) can be utilized to approximate F i,s as follows: where where θ i,s = Pi,s g i,s and β i,s =C i,s g i,s . Associating with Lemma 2, for σ i,s > 0, we have From (26)- (27), the (25) can be resolved aṡ The α i,s andθ i,s ,β i,s are formulated as where c i,s , γ i,s and λ i,s are positive constants. Substituting (29)-(31) into (28) results iṅ Step i, n i : Since wherė Consider the following Lyapunov function where γ i,ni > 0 and λ i,ni > 0 are constants;θ i,ni = θ i,ni −θ i,ni andβ i,ni = β i,ni −β i,ni are the estimation errors,θ i,ni andβ i,ni are the estimations of θ i,ni and β i,ni , whose definitions will be given in the later.
Differentiating V i,ni along (33) generateṡ From Lemma 3, the F i,ni can be reconstructed by the FLS, that is to say where Taking (37) into (36) yieldṡ where θ i,ni = Pi,n i g i,n i and β i,ni =C i,n i g i,n i . By applying Lemma 2, for σ i,ni > 0, we can obtain Inserting (39)-(40) into (38) giveṡ The actual control law u i and the adaptive lawsθ i,ni ,β i,ni are formulated as where c i,ni , γ i,ni and λ i,ni are positive constants. From (42)-(44), the (41) can be resolved aṡ Based on Young's inequality, there arẽ then the (45) will resolved aṡ where

Stability analysis
Theorem 1 Consider MIMO nonlinear systems (1) in presence of state constraints and UVCCs. The constructed AFATC strategy can make the output tracking error asymptotically converge to zero, and all states never transgress their prescribed regions.
Then, the asymptotic convergence about ζ i,s will be proved. In view of (49), we can get Since all internal system signals are bounded, we can obtain that and time derivative of ζ i,k are also bounded, which shows is uniformly continuous. Applying Barbalat Lemma, we have which implies lim t→∞ ζ i,k = 0, then the asymptotic convergence has been proved.
The fuzzy membership functions are set as

Conclusion
In this article, for MIMO nonlinear systems with state constraints, UVCCS and unknown dynamics, an AFATC strategy has been developed. More specifically, the bounded estimation scheme is explored to counteract the adverse effect deriving from UVCCs. By introducing some positive integral functions, associating with backstepping technique, the asymptotic tracking controller is recursively constructed. Such asymptotic control design not only achieves the asymptotic convergence of tracking errors, but also guarantees the prescribed state constraints. Simulation studies are conducted to elucidate the validity of the presented AFATC scheme.

Data availability statements
Data sharing not applicable to this article as no datasets were generated or analysed during the current study.

Funding
This work was supported in part by the National Natural Science Foundation of China under Grant 61673129.

Conflicts of interest
The authors declare that they have no conflict of interest.