ADT-based adaptive back-stepping control for the switched non-affine nonlinear system with uncertain parameters

The adaptive back-stepping control issue for the switched non-affine nonlinear system with the uncertain parameter is addressed in this paper. Inspired by the “extended state” conception, the integral item is first employed in the original switched nonlinear system to generate an augmented switched non-affine nonlinear system. Then, the non-affine structures in the system dynamics are adopted to devise the virtual controls directly, and the dynamic surface control is integrated to eliminate the “explosion of complexity” caused by the back-stepping design. Furthermore, the estimation of switched unknown parameter is achieved by the continuous variable, and the individual Lyapunov functions including the estimation errors are constructed for each subsystem. Combined with the multiple Lyapunov functions methods, the uniformly ultimately bounded of the closed-loop switched system could be ensured under any switching law, which satisfies the average dwell time condition. Furthermore, system tracking errors would converge to the neighborhood of the zero. Finally, numerical simulations demonstrate the validity of proposed control approaches.


Introduction
As an efficient recursive methodology, the backstepping control method has been extensively utilized for the nonlinear strict-feedback systems over the past few decades [1,2]. It is worth mentioning that control inputs appearing in these systems are affine. Nevertheless, the plants in many engineering applications are non-affine systems, such as hypersonic vehicles [3,4], rigid and flexible joint manipulators [5,6], and ultrasonic motors [7], and so on. Because the system state cannot be regarded as the virtual control, the non-affine property makes it difficult to apply the back-stepping method straightforwardly. Namely, the control input u is difficult to separate and use directly to compensate the negative effects, caused by unknown parameters or external disturbances. Therefore, the final actual control input cannot be obtained following the way of dealing with nonlinear strict-feedback system [8]. Consequently, it is still an enormous challenge for the nonlinear strict-feedback system with non-affine properties in the control field.
Recently, the control designs based on the backstepping scheme for the nonlinear systems subject to non-affine properties have attracted considerable attention by the worldwide researchers. In [9], the virtual and practical control laws are developed based on implicit function methods, which are constructed by the mean value theorem and neural networks. The idea in [9] is then extended to various classes of non-affine systems including normal form with zero dynamics in [10], perturbed pure-feedback form with dead zero in [11], purefeedback form with uncertainties in [12] and applied to helicopters in vertical flight in [13]. Furthermore, since the inputs of neural network approximation generally contain the control variable, the "circular design problem" widely exists in these works. To address this issue, several approaches are developed from different aspects. The dynamic surface control (DSC) techniques are introduced to eliminate approximations of control inputs in [14], where the repeated differentiations of virtual controls are not required. Similarly, combined with continuous functions and DSC technique, the circular construction problem during the controller design process is removed in [15]. To facilitate DSC design, an affine state variable is constructed at each step in [16] with a filtered version of control signal. The authors in [17] and [18] adopt coordinate transformation to obtain an augmented term in the normal form, and the goal is then converted into designing an output feedback controller. Considering the parametric uncertainties, an adaptive back-stepping control method is proposed in [19]. In particular, non-affine structures are exploited in [20] and [21] to represent the virtual controls as a whole and reduce the conservativeness of designed controller, while the usefulness of these structures is neglected by the aforementioned methods.
On the other hand, switched system, constituted by a series of continuous/discrete subsystems with specific rules orchestrating the switching among these subsystems [22,23], provides a powerful tool of modeling and controller design for many physical control systems including near space vehicles [24], networked control systems [25] and electro-hydraulic servo systems [26] and have attracted great attention in the recent few decades. In particular, switched nonlinear systems with affine appearance of states and control inputs have been extensively addressed by the back-stepping methodology, see [27][28][29][30]. The existing approaches on the stability analysis or stabilization for the switched nonlinear system could be usually classified into two main categories, i.e., common Lyapunov function and multiple Lyapunov functions. The stabil-ity of switched systems can be ensured by the common Lyapunov function method for any arbitrary switching signals [31,32]. However, it would be extremely difficult or even inconceivable to share a common Lyapunov function for all subsystems. Therefore, the multiple Lyapunov functions method is proposed, which can reduce the conservativeness of controller with restrictions on switching signals just like dwell time (AD), average dwell time (ADT), mode-dependent average dwell time (MDADT), and so on [33,34]. However, a restrictive inequation condition V p ≤ μV q with a positive constant μ is required to satisfy for any two subsystems in these studies, which is so difficult to be ensured for the nonlinear controller design. Fortunately, the recent research in [35] extends the multiple Lyapunov functions method to a less restrictive case with a bounded constant , i.e., V p ≤ μV q + . However, the switched systems in [35] belong to the strict-feedback form; there is little related research on switched nonlinear systems subject to non-affine dynamics.
Motivated by previous discussions, the ADT-based adaptive back-stepping control issue for the switched non-affine nonlinear system with uncertain parameters would be addressed in this paper. The original switched non-affine nonlinear systems are first augmented by an integrator to facilitate the controller design. Then, the non-affine structures are employed as a whole to devise the virtual controls, where DSC methods are integrated to obviate "explosion of complexity" problems of backstepping methods. Moreover, the switched unknown parameters are estimated by a continuous variable, and the parameter estimation errors constitute the main difference of the Lyapunov function for each subsystem. For the switching signal with certain ADT, the proposed multiple Lyapunov functions control method could ensure the uniform ultimate boundedness of the switched non-affine nonlinear system, and the convergence of tracking errors is also guaranteed even existing switching uncertain parameters. Finally, simulation results illustrate the correctness of proposed methods.
The remainders of this paper would be arranged as follows. The adaptive back-stepping control issue is formulated in Section II. Section III presents the backstepping control design process and related stability analysis. The simulation results are given in Section IV, and the final conclusion is drawn in Section V.

Problem formulation
Consider the following switched non-affine nonlinear system with uncertain parameters, (1) x i ] T ∈ R i and x =x n ∈ R n denote the system states, u ∈ R and y ∈ R are control input and system output, respectively. ϕ i (·) : R i → R l and f i (·) : R i+1 → R represent known smooth nonlinear functions. θ σ (t) ∈ R l is the unknown parameter, where the switching signal σ (t) → Ω = {1, 2, · · · , m} is the piecewise right continuous function, and m denotes the total number of subsystems.
The main objective of this paper is to structure an adaptive back-stepping control scheme for the switched non-affine nonlinear system (1), which could guarantee that the system states are stable and the system output y could track the reference trajectory y r with high precision even existing switching uncertainty parameters or switching occurring.

Assumption 1
The y r ,ẏ r andÿ r are known and bounded.

Assumption 2
For the nonlinear functions f i , the following condition can be satisfied, where x n+1 = u and f 0 > 0 Remark 1 Assumption 1 is commonly employed in extant literature on back-stepping control; the identical assumption can be found in [20,21,35]. Assumption 2 is on the controllability of system (1), which is a basic condition [20,21,32].
Definition 1 [22]. For t ∈ [t, T ), define N σ (T, t) as the total number of switches, if there exist two constants N 0 > 0 and τ a > 0 such that where the τ a is called ADT of the switching law σ (t).
Remark 2 For the switched system, the switching signal is a key issue, which would influence the controller design and stability analysis. Generally speaking, the switching signal can be divided into arbitrary switching signal or designed one [31,32]. The common Lyapunov function approach is needed for the arbitrary signal to stable the switched system, but it is so difficult to find a common Lyapunov function for the switched nonlinear system with so many sub-systems. In order to relax the restrain of common Lyapunov function method, the multiple Lyapunov functions are developed for the switched system with restriction switching signals, just like dwell time (AD), average dwell time (ADT), mode-dependent average dwell time (MDADT), and so on [31,32,38]. In this paper, the switching signal is designed based on the average dwell time (ADT) condition, not arbitrary and the ADT-based adaptive back-stepping control for the switched non-affine nonlinear system with uncertain parameters is investigated in this note.

Main results
The adaptive back-stepping controller would be first designed for the nonlinear non-affine switched system, and stability analysis is also accomplished in this section. To facilitate the controller design by backstepping method, define a "extended state" x n+1 = u and introduce an integrator in system (1) to obtain the augmented switched non-affine nonlinear systems as The controller design for (1) is then transformed to develop an auxiliary control law for (4), where v is regarded as the new control input.

Controller design
The controller design follows the back-stepping design process, which can be described as follows.
Step 1. Define z 1 = x 1 − y r as the tracking error variable, and taking the time derivative of z 1 yieldṡ Regard f 1 as a virtual control input, and corresponding virtual control laws can be designed as follows where theθ denotes the estimation of the switching uncertain parameter θ σ (t) and k 1 > 0. The following modified first-order filter is applied to solve the "explosion of complexity" problem [19,21], where τ 1 > 0 andᾱ 1 is a new intermediate state variable.
Step 2. Denote the second error variable by z 2 = f 1 (x 1 , x 2 ) −ᾱ 1 , and we can attain thaṫ Then, the virtual control law of (8) is proposed as follows.
where k 2 > 0. Similarly,ᾱ 2 can be acquired by, with τ 2 > 0. Step and it can be obtained thaṫ The related virtual control law is expressed as where k i > 0, and then the intermediate variableᾱ i can be generated as follows: where the time constant τ i > 0.

Remark 2
In this paper, the switched non-affine nonlinear systems with unknown parameters are investigated, i.e.,ẋ n = θ T σ (t) ϕ n (x n ) + f n (x n , u). Because of the non-affine properties, the control input u is difficult to separate and use directly to compensate the negative effects, caused by unknown parameters or switching instant. Combined with the idea of "extended state", a new virtual extended state x n+1 is designed, and define as x n+1 = u , andẋ n+1 =u = v. Then, the new augmented switched non-affine nonlinear system can be obtained and the controller design in Step n+1, and the similar way is adopted in [36].
For i = 1, it can be verified from (6) and (7) thaṫα where B 1 (·) represents the continuous function subject to α 1 andα 1 . Moreover, for i = 2, · · · , n, one obtains with B i (·) being continuous functions with respect to the time derivatives of α i , i = 2, · · · , n. For any p ∈ M, the corresponding Lyapunov function is selected as whereθ p = θ p −θ denotes the parameter estimation error for the p-th subsystem. With (16) and (18)- (23), the derivative of V p can be obtained as follows, It can be seen that Substituting (26) and (27) into (25) giveṡ To show the boundedness of B i (·) , we need to ensure that the arguments of B i (·) are bounded. According to assumption 1, for a given constant C 0 > 0, the set Π 0 = y 2 r +ẏ 2 r +ÿ 2 r C 0 } is compact in R 3 . On the other hand, the sets Π i p = n+1 j=1 z 2 j + 1 κθ T pθ p + i j=1α 2 j C i p are also compact in R (n+l+i+1) for given positive constants C i p , (i = 1, · · · , n) . Sinceθ = θ p −θ p and θ p is a bounded constant, θ is also bounded on Π i p . Hence, z 1 , · · · , z n+1 , α 1 , · · · ,α i ,θ,θ p , y r ,ẏ r ,ÿ r are bounded on Π 0 × Π i p . We now turn to the boundedness of x 1 , · · · , x n+1 .
Notice that x 1 = z 1 + y r , then x 1 is bounded. Moreover, since f 1 (x 1 , x 2 ) = z 2 +ᾱ 1 = z 2 +α 1 +α 1 and α 1 is a continuous function of z 1 ,θ , x 1 andẏ r , f 1 (x 1 , x 2 ) takes a bounded value. Applying mean value theorem to f 1 (x 1 , x 2 ) yields where ξ ∈ (min (0, x 2 ) , max (0, x 2 )) and by Assumption 2, we have A combination of (29) and (30) leads to the boundedness of x 2 . By the same token, we can deduce that x 3 , · · · , x n+1 are bounded. As a result, there exist positive constants D i p such that |B i (·)| D i p on Π 0 ×Π i p . By the Young's inequality, we obtain Using (31) and (32) in (28) results iṅ where a = min 2k 1 , · · · , 2k n+1 , k 0 , 2 For ∀ p, q ∈ M, it is noticed that the only difference between the corresponding Lyapunov functions V p and V q lies in 1 2κθ I pθ p and 1 2κθ I qθ q . Define μ = 2, and μY = 1 κ Θ 2 , where Θ = max p,q∈M θ p − θ q , then one has Note thatθ p =θ q + θ p − θ q , it can then be verified that 1 2κθ A combination of (34)-(36) implies With the help of above analysis and back-stepping scheme design, the following theorem can be obtained. (1) subject to uncertain parameters, if the assumptions 1 and 2 are fulfilled, control laws and adaptive laws are proposed as (15) and (16), then it can be obtained that all signals are uniformly ultimately bounded for the switching signal τ a , which conforms to ADT conditions, i.e., τ a > log μ a . Besides, the tracking error satisfies the following inequality,
Proof Define the Lyapunov function as W (t) = e at V σ (t) (t), and then one haṡ By integration on the time interval [t , t +1 ), one obtains Choosing an arbitrary T > t 0 = 0 and iterating bode sides of (41) over [0, T ), we see that t j be at dt Note that τ a > logμ/a, and for ∀ε ∈ (0, a − (log μ /τ a )), we can obtain that τ a > logμ/(a − ε). Besides, it follows that N σ (T, 0) − j N σ T, t j+1 + 1 by the definition of N σ (T, t j+1 . Hence, Moreover, the last term of the right-hand side of (42) satisfies Substituting (43) and (44) into the right-hand side of (42) and in view of the definition of W (t), one has which shows that V σ (t) (t) bounded, and z 1 , · · · , z n+1 , α 1 , · · · ,α i ,θ,θ σ (t) are all bounded as a result. Similar to the foregoing procedure that uses (29) and (30), the boundedness of x 1 , · · · , x n+1 can then be deduced. Therefore, all signals of the closed-loop system are bounded for the switching law σ (t) corresponding to ADT τ a > logμ/a. Moreover, notice that z 2 1 T − 2V σ (T − ) T − and taking T → ∞ yield (38).

Remark 3
For each subsystem, the Lyapunov function is requested strictly decreasing in the multiple Lyapunov functions method. This constraint is relaxed in [33,37], but the inequation V p ≤ μV q is needed to be satisfied, which is difficult to be ensured for the nonlinear control design. Fortunately, the research in [35] extends the multiple Lyapunov functions method to a less restrictive case with a bounded constant , i.e., V p ≤ μV q + . Under such a weaker condition, the global boundedness of all the closed-loop signals is ensured by the extended multiple Lyapunov functions method in [35]. However, it can be found that the nonaffine peculiarity is not considered in [35], and there is little related research on switched nonlinear systems subject to non-affine peculiarity. Therefore, inspired by the [35], the less restrictive case with a bounded constant is employed in this work to deal with the subject to non-affine dynamics.

Numerical simulation
The following second-order switched non-affine nonlinear systems are considered to illustrate the effectiveness of the proposed adaptive back-stepping control methods, where the switching signal σ (t) ∈ Ω = {1, 2}, i.e., there are two subsystems. The unknown parameters are chosen as θ 1 = 0.4 and θ 2 = 0.1. The system initial states are selected as x 1 (0) = 1, x 2 (0) = u(0) = 0, and the reference trajectory y r = sin(t).
For the proposed control parameters, we take k 1 = 5, k 2 = 5, k 3 = 5, τ 1 = 0.05, τ 2 = 0.05, κ = 1 and k 0 = 2. It can be verified that a = 2, and the closed-loop system is ensured to be uniformly ultimately bounded for every σ (t) with average dwell time τ a > 0.3466 = log2/2. The switching signal is shown in Fig 1, and it could be not difficult to see that the dwell time τ a = 2.5 > 0.3466 with N 0 = 2, which is satisfied with the ADT condition.
Applying the proposed ADT-based adaptive backstepping control scheme, the simulation results are given in Figs. 2, 3, 4, 5 and 6. In addition, the same adaptive back-stepping control problem for the nonlinear system with uncertain parameters is addressed in [19], and the uncertain parameters θ(t) is also consider and handled by the adaptive estimations, which are also employed in our work. Thus, the reference [19] is used as the comparison simulation, and the results are also  Fig. 6 The control input u depicted in Figs. 2, 3, 4, 5 and 6 It is obvious that the tracking missions for the uncertain switched system with non-affine could be successfully completed by the two adaptive back-stepping control schemes, but a better tracking performance can be guaranteed by the proposed control method. Figure 2 shows the responses of system output y, and it can be found that the y can successfully track the reference signal y r even the subsystem switching and in presence of unknown parameters. Noticeably, a higher tracking precision can be guaranteed by the proposed back-stepping control scheme at subsystem switching occurring. The similar conclusion could be obtained in Fig. 3, which denotes the tracking errors y − y r . As displayed in Fig. 3, the tracking errors would converge quickly and errors is not exceeding 0.06, whereas the tracking errors is 0.26 by the back-stepping control method in [19]. Figures 4 and 5 indicate the responses of the system state x 2 andθ , respectively. Furthermore, the control input u is shown in Fig. 6, and the required control energies are limited, which is reasonable for the engineering applications.

Conclusions
The ADT-based adaptive back-stepping control schemes for the switched non-affine nonlinear system with uncertain parameters are proposed in this paper. The application of back-stepping approach is made feasible by augmenting the original systems with an integrator. The virtual controls are developed with direct utilization of the structures in the system dynamics at each step. By employing the DSC technique, the "explosion of complexity" problem is avoided, and a continuous variable is structured to estimate the switched unknown parameters. Combined with multiple Lyapunov functions and ADT, the stability of the switched system is analyzed. The uniformly ultimately bounded of the closed-loop system and tracking errors would be guaranteed by the designed adaptive back-stepping control schemes. The actuator saturation faults are important issues for the trajectory tracking mission, which are not considered in this paper and may be our future work. Data availability All data generated or analyzed during this study are included in this article.