Novel Adaptive Event-Triggered Fuzzy Command Filter Control for Slowly Switched Nonlinear Systems With Constraints

This article addresses the adaptive fuzzy control problem for switched nonlinear systems with state constraints. The unified barrier function (UBF) is introduced to solve the time-varying state constraints, which removes the feasibility conditions. By integrating command filter into backstepping control to avoid the “explosion of complexity.” In addition, a novel event-triggered strategy is designed to deal with the asynchronous switching between subsystems and controllers without limiting the maximum asynchronous time, and mitigate the communication burden. Also, a new threshold function is introduced to overcome the difficulty of discontinuous triggering error at the switching instants. Then, by combining the improved admissible edge-dependent average dwell-time (AED-ADT) method with Lyapunov stability analysis, it is proved that all system signals are bounded and do not violate the predefined constraints under given switching rule. Finally, the numerical simulation results verify the superiority of the proposed algorithm, and the algorithm is applied to a ship maneuvering system.

have been proposed based on the adaptive backstepping technology, such as [1]- [5]. However, the above research results are not feasible when the prior knowledge of the uncertainties of the systems cannot be obtained. Fortunately, the fuzzylogic system (FLS) [6]- [13] or neural network (NN) [14]- [19] has been widely used to solve the uncertainties of nonlinear systems because of their sufficient preponderance in approximating nonlinear functions. It is notable that the above references have a common feature: the repeated differentiations of virtual control inputs inevitably increase serious computing burden. This phenomenon is called "explosion of complexity." To solve this problem, some excellent results of dynamic surface control have been reported (see [20]- [23] and references therein). However, these papers did not consider the compensation errors caused by the first-order filters. Furthermore, in [24], an improved technique called command filter backstepping was introduced. Based on this technique, some outstanding achievements have been popping up in recent works (see [25] and [26]). However, the above results only consider nonswitched nonlinear systems.
Since switched systems are frequently serviceable in modeling engineering systems, the control problem of switched systems has been significant research area and numerous classic results have been reported, such as [27]- [35] and their references. In [27], the average dwell-time (ADT) approach was used to study the domain of attraction estimation, and the exponential stability criteria were proposed. For switched positive linear systems in [28], a modified ADT approach and Lyapunov stability analysis was employed to ensure stability of the concerned systems. For uncertain switched nonlinear systems in [29]- [31], some adaptive control schemes were presented via using the common Lyapunov function (CLF). However, the CLF of subsystems is usually unknown or nonexistent. In this case, it is necessary to use the multiple Lyapunov function to study the stability of the switched systems. Especially, after the concept of modedependent ADT (MDADT) was brought forward in [32], massive fruitful results followed. In [33], the MDADT approach was applied to the slowly switched linear systems, and the scheme was designed to obtain the sufficient condition of stabilization for the studied switched systems. Reference [34]  to improve the stability conditions of linear slowly switched systems, the notion of admissible edge-dependent ADT (AED-ADT) was proposed in [35]. But it should be pointed out that it is an interesting and challenging issue to study the AED-ADT-based stability of adaptive control scheme for nonlinear switched systems. As far as we know, there has been no report on this problem.
It should be emphasized that there are various constraints in real systems. The barrier Lyapunov function (BLF) was first introduced in [36] to cope the output constraint for nonlinear systems. Since then, the BLF has been broadly applied to solve the constrained problem for various nonlinear systems, such as strict-feedback nonlinear systems [37], nonstrict-feedback nonlinear systems [38], pure-feedback nonlinear systems [39], [40], etc. However, all of the above results require that the virtual controllers need to meet the feasibility conditions. Specifically, during the design process of backstepping, the virtual controllers are also constrained within predefined boundaries. Great efforts have been devoted in [41] and [42] to handle this problem. However, all the above papers are devoted to the design of continuous time controller for the studied systems.
Recently, to mitigate communication burden and save resources, many scholars have focused on event-triggered control (ETC). Unlike time-triggered control that the control signals are continuously transmitted to the actuator regardless of whether the systems need or not, the ETC fully considers the behavior of concerned systems. Therefore, ETC [43]- [47] has received considerable recent attention. It should be point out that for ETC of switched systems, the asynchronous switching problem between subsystems and controllers caused by switching and triggering should not be ignored. Unfortunately, most of the existing results directly make strict assumptions about the information structure to avoid asynchronous switching. For example, literature [43] only allows switching when triggered. Recently, literature [45] solved the asynchronous switching problem by assuming that the prior knowledge of the maximum triggering interval and the maximum asynchronous interval length are known. Also, the above results are only applicable to switched linear systems, and the methods cannot be directly used to solve the asynchronous switching problem in ETC of switched nonlinear systems. In addition, since the triggering error is discontinuous at the switching instants, which may lead to additional continuous triggering. Therefore, a problem naturally arises: how to settle the asynchronous switching problem in the ETC of the switched nonlinear systems and avoid Zeno behavior? Motivated by the above mentioned, this article will concentrate on the issue of command filter-based adaptive eventtriggered fuzzy control for nonlinear switched systems with state constraints in the frame of AED-ADT. The main contributions of this dissertation are as follows.
1) The adaptive control for switched nonlinear systems under AED-ADT is considered for the first time.
Compared with the case studied in [32]- [34], the AED-ADT method is the evolutionary form of the MDADT method, which could relax the limitations of the MDADT technique.
2) Different from the results in [37] and [40], which handle the constraints of the systems by using traditional BLF. The unified barrier function (UBF) is introduced in this article, and the backstepping method is combined to deal with the asymmetric state constraints, so as to avoid the feasibility conditions. 3) By integrating the information of switching and triggering into ETC, the asynchronous problem between subsystems and controllers is solved without the tedious calculation on asynchronous intervals as [27] and [45]. At the same time, the number of switches within two consecutive triggering instants is not limited. 4) Unlike the results in the literature [46], [47], which add a dwell time in ETC. On the basis of removing the dwell-time assumption of the above results, this article considers the jump characteristic of triggering error at the switching instants, and a cumulative switching threshold condition is elegantly designed to avoid Zeno behavior.

II. PROBLEM FORMULATION AND PRELIMINARIES
Consider the following system: where y ∈ R,x i = [x 1 , . . . , x i ] T denote the system output and states, respectively; g where exist constantsk a i and k b i satisfying k a i (t) <k a i and k b i (t) > k b i . In the following, the argument in functions or variable is sometimes deleted if confusion is not possible.
Remark 1: The constraints of the system studied in this article are the dynamic asymmetric state constraints. If only x 1 is constrained, the state constraints become the output constraint in [36]. If k a i (t) and k b i (t) are constants, the corresponding constraints reduce to static, which become the case studied in [37]. However, the above control schemes always require feasibility conditions, that is, the virtual controllers The process of finding suitable design parameters meeting feasibility conditions is both complex but time consuming.
To construct the event-triggered tracking controller, the following lemmas and assumptions are introduced.
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Assumption 3: For t > 0, y d and its time derivatives up to the n-order are bounded and continuous.
Definition 1 [35]: For ∀(p, q) ∈ W × W, let N σ p q (T, t) denote the switching numbers from qth subsystem to pth subsystem on [t, T] and T p q (T, t) denote the total running time of the pth subsystem whenever the switching from the qth subsystem to the pth subsystem on [t, T]. σ (t) has AED-ADT τ ap q , and N 0p q > 0 (N 0p q called the admissible edge-dependent chatter bounds) such that Remark 2: It is worth noting that we can obtain then we can obtain the inequality of the same form as MDADT. From the above mentioned, we can see that the AED-ADT switching is more versatile than the MDADT switching. At the same time, the switching signal used in this article has the characteristic of "slow switching," and the designed algorithm could guarantee the phenomenon of infinitely fast switching to be avoided.
To solve the full-state constraints and remove the feasibility conditions, the UBF [41] is given under The equality (4) for χ i can be rewritten as So it can be derived from (5) that where . . , n. By differentiating (5), one haṡ where Then, the system (1) is converted to the following form: n + η 2n . Remark 3: Based on UBF (4), system (1) is transformed into the system (8), where χ i are the state variables. The constraint problem of x i can be reduced to the problem for the boundedness of χ i . It is obvious that the control input of the system (8) is the same as that of system (1), that is, both system are equivalent.
To approximate the unknown nonlinear function, the following FLS is employed.
By the singleton function, central mean fuzzy decomposition, and product inference [8], the FLS can be given as Then, the FLS (9) can be reformulated as where Lemma 2 [10]: Letf (x) be a continuous function defined on a compact set * . Then, for any ε > 0, there is an FLS (10) that satisfies III. ADAPTIVE EVENT-TRIGGERED CONTROLLER DESIGN AND STABILITY ANALYSIS To develop the backstepping design process, the following coordinate transformations are defined as: where ς i denotes the error signals, , and α i−1,f refers to the output of the first-order filter, the first-order filter is defined as where α i is the input of the first-order filter, ε i is a positive constant, and α i, The following compensating signals are accepted as follows: (15) where c i , i = 1, . . . , n are design parameters. Then, the compensated tracking error signals are defined as Step 1: The derivative of e 1 iṡ Choose the Lyapunov function candidate as Authorized licensed use limited to the terms of the applicable license agreement with IEEE. Restrictions apply.
whereθ 1 = θ 1 −θ 1 ,θ 1 is the estimation of θ 1 , and l 1 is positive design parameter. Its derivative along (18) yieldṡ In view of the FLS approximation property, one has Using Young's inequality, we obtain where The virtual controller α 1 and the adaptive lawθ 1 are designed as Replacing (23) and (24) whereΥ Step j j j (2 ≤ j ≤ n − 1): From (13) and (16), we havė Choose the Lyapunov function candidate as whereθ j = θ j −θ j , l j is positive design parameter. Its derivative iṡ ,θ 1 , . . . ,θ j−1 , y d , . . . , y (j) d ] T . As is the same case of (21), the following inequality can be obtained: where Now, construct the virtual controller α j as and the adaptive law aṡ Based on (32) Step n: In general, the asynchronous switching caused by switching within the triggering interval will affect the system performance. We will analyze the pth subsystem within the triggering interval [t r , t r+1 ). Let r 0 = t r , r k+1 = t r+1 , on the time interval [t r , t r+1 ), r 1 , r 2 , . . . , r k are the switching times. The switching event-triggered communication mechanism (SECM) is designed as follows: u p (t) = −(1 + η(t)) α n tanh e n g p n η 1n α n ρ p Authorized licensed use limited to the terms of the applicable license agreement with IEEE. Restrictions apply.
are design parameters. For any given initial value 0 < η(0) ≤ 0.5, one has η(t) ∈ (0, 0.5] in t > 0. Remark 4: The triggering error in the SECM depends on the switching signal, which is the key to deal with asynchronous switching. However, because the triggering error is discontinuous at the switching instants, the switches may lead to additional triggers, resulting in Zeno behavior. We innovatively design a cumulative threshold function integrating the jump information at the switching instants to avoid the above problem. Remark 5: One of the main features of the SECM in (39) is that the threshold parameter η(t) can be dynamically adjusted without constant fixation when switching is not considered. If choosing η(0) = 0, d = 0 in (39), and ι = 0 in (39), the SECM can be written as t r+1 = inf{t ∈ R||β σ (t) (t)| ≥ d}, which is the classical sample-data control. Moreover, if setting η(0) = 0, d = 0 in (39), and ι = 0 in (39), the developed dynamic event-triggered controller in (39) is simplified to the static event-triggered controller. Thus, the SECM is more flexible than the static event-triggered strategy or classic sampled data control, and it is actually covers both mechanisms.  1]. Then, the actual controller can be denoted as Select the following Lyapunov function candidate: whereθ n = θ n −θ n , l n is positive design parameter. Differentiating The following virtual control law and adaptive law are devised as: From (45)- (47), it follows that: Then, using the similar derivation given in 1 , we obtain e n η 1n g p n u(t) = e n η 1n g p n u p (t r ) ≤ e n η 1n g p n u p (t) 1 + η + e n η 1n g p Same to the procedures from 1 can obtaiṅ  1, 2, . . . , k (k > 0). The SECM (39) can ensure that

Part 2 (Asynchronous Interval): This interval is nonempty
Then, using the similar derivation given in Part 1 situation 2 , we obtain e n η 1n g p n u(t) = e n η 1n g p n u σ (t r ) (t r ) ≤ e n η 1n g p n u p (t) 1 + η + e n η 1n g p The right-hand side of the inequality (50) is only related to the pth subsystem. In other words, the effect of the asynchronous switching is eliminated by the designed SECM. Same to the procedures from Part 1 situation 2 can obtaiṅ Step n+1: Construct the Lyapunov function for the pth subsystem as follows: Then, we can acquirė According to the lemma in [24], |(α j,f − α j )| ≤ j can be obtained. Thus, we havė Remark 6: Significantly, the designed command filterbased backstepping approach has many advantages. First, compared with the traditional backstepping design method in [2], [6] and [17], because the filter (14) is used at the design steps, the virtual controllers do not need the repeated differentiations, which avoids the problem of "explosion of complexity." Second, different from the dynamic surface control in [20]- [23], the error compensation mechanism is considered in the design process to reduce the filtering errors.
In the next, the following positive constants are defined: By selecting the Lyapunov function candidate V p = V p n , we haveV and Theorem 1: For the switched nonlinear system (1) under Assumptions 1-3, the actual controller (37) and the adaptive laws in (24), (34), and (47) are constructed for σ (t) with AED-ADT τ ap q ≥ τ * ap q = (lnμ p,q /λ p ), the following conclusions can be drawn: 1) all the resulting closed-loop system signals are bounded; 2) the out tracking error eventually converges to the vicinity of the origin; 3) all states never transgress their prescribed regions, meanwhile, the feasibility conditions are removed; and 4) the SECM (39) is Zeno-free.
Based on the above discussion, the Zeno phenomenon can be successfully avoided. The proof is completed here.
Remark 7: In [27], the switching signal that satisfies the assumption of slow switching was considered. The AED-ADT switching signal used in this article and the switching signal in the assumption are both typical slow switching signals. Therefore, for consecutive discontinuities r 1 , r+1 1 , there exists a positive number τ * satisfying r+1 1 − r 1 ≥ τ * . Remark 8: Based on the discussion of the above three cases, the designed SECM ensures that the distance between any two triggering instants is strictly positive or that any finitetime period has a finite number of triggers. From the formal definition of the event-triggered condition, we conclude that Zeno can be avoided.

IV. SIMULATION EXAMPLES
In this part, the simulation examples validate the effectiveness of the designed scheme.

V. CONCLUSION
In this article, a novel event-triggered fuzzy tracking controller is proposed for switched nonlinear systems with the AED-ADT switching rule in the presence of state constraints. First, the state constraints are solved by establishing the UBF on the constrained states to eliminate the requirement of feasibility conditions. Then, command filter is exploited to handle with the "explosion of complexity," FLS is utilized to deal with unknown nonlinearities, and the SECM is proposed to save communication resources. Finally, the theoretical results are verified by two simulation examples. In the future, we will study the prescribed performance control design of multiple-input-multiple-output systems, large-scale systems, and multiagent systems.