A New Adaptive Controller for Nonlinear Systems with Uncertain Virtual Control Gains

This paper addresses the adaptive asymptotic tracking control problem for nonlinear systems whose virtual control gains are unknown nonlinear functions of system states. Only in the ﬁrst step, the Nussbaum gain technique is utilized to handle the uncertain virtual control gain. In the remaining steps, virtual control gains are dealt with by constructing novel control laws without the approximation of the uncertain nonlinear functions and external disturbances by neural networks or fuzzy logic. New adaptive laws are deﬁned to compensate for unknown virtual control gains, uncertain parameters, and external disturbances. Finally, an adaptive tracking controller is designed and applied to the control of a 3-order robot system, which guarantees the boundedness of all the signals in the closed-loop system and asymptotic stability of the tracking error.


Introduction
e tracking control has received considerable attention for the purpose of ensuring the output of the system is tracking a desired trajectory. To deal with the uncertainties in nonlinear systems, one of the most popular methods, adaptive control, has been introduced for controller design. ere have been many related results in this area [1][2][3][4][5][6][7][8][9][10][11]. An adaptive data-driven controller was designed for nonlinear systems using goal representation heuristic and dynamic programming [4]. In view of unknown nonlinear fractionalorder systems, an adaptive control scheme was proposed [5]. For nonlinear systems with dead-zone and actuator failure, two novel finite-time adaptive tracking controllers were developed [7][8][9]. By introducing a new performance function, [10,11] they constructed two adaptive tracking controllers via prescribed performance control and funnel control, respectively. It needs to be emphasized that the VCGs (virtual control gains) of systems in the above papers are assumed to be known.
However, the VCGs are uncertain for many actual systems. In response to this challenge, the Nussbaum gain technique was first proposed [12]. ere have been many achievements for nonlinear systems, whose VCGs were unknown constants [13][14][15][16]. Two adaptive control strategies were given for unknown nonlinear SISO (single input, single output) systems and stochastic nonlinear systems based on state observers [13,14]. In consideration of unmodeled dynamics and an unknown dead-zone, an improved control strategy was addressed [15]. In view of nonlinear systems with actuator faults and state/input constraints, a controller was designed based on dynamic surface and Nussbaum gain [16]. Taking into account uncertain time-varying VCGs, an improved adaptive control method was given [17], which was further improved [18] such that the Nussbaum gain technique was applied to nonlinear systems whose virtual control gains were unknown nonlinear functions of system states. To handle time-varying uncertain control gains, new Nussbaum functions were defined [19]. Two adaptive robust control schemes were proposed [20] for nonlinear systems with certain and uncertain signs of VCGs. Aiming at nonlinear systems with uncertain dead-zone output, an adaptive fuzzy control scheme was designed using a state observer [21]. To achieve full state constraints, adaptive tracking control was studied by the Barrier Lyapunov function [22]. Considering MIMO (multiple input, multiple output) nonlinear systems with input saturations and uncertain control gains, two adaptive NN (neural network) controllers were constructed [23,24]. A dynamic surface control strategy was given for nonlinear systems with uncertain VCGs using fuzzy logic [25].
Fuzzy logic and neural networks were also usually applied to handle the unknown VCGs. e VCGs were supposed to have known upper and lower bounds and were approximated by fuzzy logic, and then the adaptive fuzzy tracking controllers were constructed for strict feedback [26] and switched nonlinear systems [27]. Two adaptive fuzzy control schemes were proposed for nonaffine [28] and nonstrict-feedback [29] nonlinear systems based on funnel control and dynamic surface control, respectively. Several adaptive NN tracking controllers were constructed for different-type uncertain nonlinear systems [30][31][32]. Besides, by utilizing |z| − z 2 / ������ z 2 + δ 2 < δ with δ > 0 and z being any real number, two controllers were designed by invoking the lower bounds of unknown VCGs [33,34]. By describing UVCCs in terms of their known and unknown parts, a new adaptive tracking controller was constructed and applied to the attitude control of quadrotors [35]. More research results on nonlinear systems with unknown VCGs can be seen in [36,37].
In summary, only the boundedness of tracking error can be obtained in the aforementioned achievements. e better tracking performance of asymptotic stability was not researched for nonlinear systems with uncertain VCGs being functions of system states. Recently, an adaptive asymptotic tracking control method was presented in [38]. ere were also some others that dealt with unknown nonlinear functions and external disturbances by neural networks. However, according to the literature review, there has been no work reported on adaptive asymptotic tracking controllers for nonlinear systems with VCGs being uncertain state functions without using fuzzy logic or neural networks.
Inspired by the mentioned research achievements, the paper is concerned with adaptive tracking control for nonlinear systems with external disturbances, whose VCGs are uncertain functions of system states. An adaptive tracking controller is designed in a stepwise strategy and applied to the control of a robot system, which ensures both the boundedness of all the signals in the closed-loop system and the asymptotic stability of the tracking error. e effectiveness and practicability of the developed control strategy are validated by both theoretical analysis and simulations. e paper possesses the following features: (1) Different from the control methods based on the Nussbaum gain technique [13][14][15][16][17][18][19][20][21][22][23][24][25], a Nussbaum function is only employed in the first step and an improved adaptive law for the Nussbaum variable is defined such that the asymptotic stability of the tracking error is achieved. is is the main improvement on [13][14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30][31][32], which can only guarantee that the tracking error is bounded. Without using fuzzy logic or neural networks [26][27][28][29][30][31][32], and the lemma of |z| − z 2 / ������ z 2 + δ 2 < δ [33,34], novel control laws are constructed in the remaining steps without the bounds of the unknown VCGs.
(2) To compensate for unknown VCGs, unknown external disturbances, and parameter uncertainties, new adaptive laws are adopted so that the assumption of unknown VCGs can be relaxed as in Assumption 2, where unknown VCGs only have unknown lower and upper bounds [22,25,26,29,31,34,38]. Unknown VCGs are assumed to be bound by with known upper and lower bounds [14,18], known lower bounds [27,30], and known upper bounds [13,16]. In [32], unknown VCGs are assumed to be strictly either positive or negative. All the above assumptions are more restrictive. erefore, the proposed controller can be suitable for more nonlinear systems.
(3) e reference signal is only required to be differentiable and the assumption is written as Assumption 1, which is less restrictive than the related assumption that the reference signal and its time derivatives up to the n-th order are continuous and bounded [13, 17, 18, 20-22, 26, 28-31, 33, 38]. e outline of this paper is presented below. In Section 2, the preliminaries and problem formulation are described. e design of an adaptive control algorithm and stability analysis are given in Section 3. Sections 4 and 5 provide the simulation examples of a second-order nonlinear system and a robotic system, respectively, and the summary of the paper.

Problem Formulation and Preliminaries
e uncertain nonlinear system is considered as follows: y ∈ R are the state vector, input, and output of the system, respectively, i � 1, . . . , n. g i (x i ) ∈ R denotes the VCG, which is an unknown smooth non-zero nonlinear function. θ ∈ R s 1 is an unknown parameter vector, and φ i (x i ) ∈ R s 1 . denotes a nonlinear function, which is known and smooth, s 1 is a positive integer; d i (t) ∈ R is an unknown bounded external disturbance.
Assumption 1 (see [24]). e ideal output vector y 1 . T is absolutely continuous and bounded.

Complexity
Assumption 2 (see [22]). Let us assume that the sign of g i (x i ). does not change for all x i and g i (x i ). satisfies with g im and g iM being unknown positive constants. It is reasonable that g i (x i ). is bigger than a positive constant g im due to the system controllable condition of g i (x i ) being away from 0, which could eliminate the controller singularity problem and has been given in many control strategies [22,25,26,29,31,34,38] and the references therein. It is worth noting that the VCGs in the above references possess the following synthesis: the lower and upper bounds of the VCGs are only used in the procedures of the controller design and are not required in the controller, which means that g im and g iM can be unknown. Without loss of generality, we assume that g i (x i ) > 0.
Lemma 1 (see [18]). Let g(x) ∈ [g l , g u ] with g l , g u being non-zero real numbers and 0 ∉ [g l , g u ], N(ς) is an even Nussbaum function, z(t) is an absolutely continuous function, and x(t) is another function. If there exist l > 0 and a real constant c such that a function V(t) should be subject to According to Proposition 2 in [39], t f can be extended to +∞, when the solution of the closed-loop system is bounded.

Design and Analysis of the Controller
e procedure of the adaptive controller design and stability analysis are presented.
Firstly, the common coordinate transformation is introduced as with α i− 1 being the intermediate control law\designed later. e system (1) is changed into In order to estimate the unknown VCGs g i (x i ), unknown parameter vector θ, and uncertain external disturbances d i (t), define where ‖ · ‖ is the 2-norm of vectors, Γ i is an unknown bounded vector given later; ϖ i is the approximation error with ϖ i being the estimation of ϖ * i . In what follows, the controller will be constructed in a stepwise strategy via backstepping. For convenience, let Step 1. We choose the first Lyapunov function candidate as where q 1 is a positive constant. Differentiating V 1 and invoking (9) produce Substituting _ z 1 in (7) into (11) and considering (6) yield Based on Young inequality, we have where a 1 , b 1 , and c 1 are positive constants and _ y dM is the maximum of | _ y d | and unknown. By invoking (13), _ V 1 is described as Let us define Substituting (15) and (16) into (14) results in In the following, the virtual control signal α 1 is defined as where N(ς) is a Nussbaum-type even function and k 1 is a positive design parameter. ς is adjusted according to the following law: where λ > 0 is a design parameter.
Remark 1. From (19) and the definition of α 1 , ϖ 1 , it can be known that all the terms of _ ς are nonnegative, which is the key to proving the asymptotic stability of the tracking error later.
By invoking (18) and (20), _ V 1 is shown as e adaptive law _ ϖ 1 is defined as With σ 1 > 0 being a design parameter. Replacing _ ϖ 1 in (21) by (22), _ V 1 can be given as Step 2. A Lyapunov function candidate is given as 4 Complexity where q 2 > 0 is a constant. Based on (23) and _ z 2 in (7), we have From (18), _ α 1 is expressed as Substituting (26) into (25) leads to According to Young inequality, the following inequalities hold where a 2 , b 2 , c 2 , l 2 , and m 2 are positive constants. Substituting (28) into (27) leads to Γ 2 and Υ 2 are written as Invoking (30) and (31) produces Designing the following virtual control signal as and substituting it into (32) give where k 2 > 0 is a design parameter.
_ V 2 is changed to

Complexity
We construct an adaptive law as where σ 2 is a positive constant. Substituting (36) into (35) produces Step j (3 ≤ j ≤ n − 1): For the j th subsystem of (6), a Lyapunov function candidate is iteratively chosen from the previous step.
where q j > 0 is a design parameter.
By an induction argument, _ V j− 1 is deduced as where Substituting (40) into (39) gives where Furthermore, _ V j is expressed as Based on the completion of squares, one has Complexity g j g jm z j z j+1 ≤ 1 2 where a j , b j , c j , l j and m j are positive design parameters and g * jM � max g 2 1M , . . . , g 2 j− 1,M .
Invoking (44), _ V j is rewritten as Similarly, we define By invoking (46) and (47), _ V j is shown as Defining the j th virtual control law as and substituting it into (48) to yield 8 Complexity where k j > 0 is a constant. Similar to (36), _ ϖ j can be specified as with σ j > 0 being a design parameter. en, _ V j is given as Step n: following the similar procedure for (38) in Step j, the Lyapunov function candidate for the last step is designed as and _ V n is presented as where q n > 0 is a constant. It follows from (52) that where a i , b i , c i , l i , m i , σ i , and q i are all positive constants. _ V n is given as where Furthermore, _ V n is reformulated as g n− 1,M g n− 1,m z 2 n + g n g nm z n u + z n d n − z n g nm

Complexity
Repeating the similar procedures in Step j, we have where a n , b n , c n , l n , and m n are positive design parameters and g * nM � max g 2 1M , . . . , g 2 n− 1,M .

Substituting (60) and (61) into (58) results in
Similar to (49) and (51), the real control law and n th adaptive law can be defined, respectively, as follows: and where k n and σ n are positive design parameters. By some direct calculations, _ V n can be described as According to the above controller design, the main conclusion is summarized as eorem 1. For system (1) with Assumptions 1-3, there exists an adaptive controller with the adaptive laws in (21), (35), (51), (65), and real control signal u in (64), which can guarantee the boundedness of all the signals in the closedloop system for t ∈ [0, +∞) and asymptotic convergence of the tracking error.
Furthermore, using the definition of _ ς, it can be seen that Based on the boundedness of ς(t), the following inequality holds.
From the definition of _ z 1 in (6) and the boundedness of x 1 , x 2 , and _ y d , it is easily shown that _ z 1 is bounded. us, by utilizing Barbalat's Lemma, one has lim t⟶∞ z 1 (t) � 0. (74) which means that the tracking error is asymptotically stable.
Control parameters selection. e selection ranges of σ i and k i are wide. e bigger the parameters σ i and k i , the faster the asymptotic convergence and response. Besides, λ is the other key parameter, which affects the asymptotic convergence of the tracking error. According to the simulation tests (see

Simulation Results
To illustrate the feasibility of the developed adaptive control approach, an uncertain nonlinear system and a single-link robot are considered.

Simulation of a Second-Order Nonlinear
System. Let us consider the strict-feedback nonlinear system as where g 1 � 1 + 0.1 sin x 1 with g 1m � 0.9 and g 1M � 1.1, g 2 � 1 + 0.1 cos x 1 sin x 2 with g 2m � 0.9, and g 2M � 1.1, e desired trajectories are selected as y d1 (t) � 4 sin(0.2πt). and y d2 (t) � 2.5(sin(0.5t) + sin t). In order to see the controller performance under different initial conditions for the system states, the initial value of x 1 is given as 1.5 and 3, respectively; ς(0) � 1; and other initial conditions are chosen to be 0. e intermediate control signal α 1 , Nussbaum variable _ ς, and actual control law u of the developed control strategy are defined as (17), (19), and (64), where n � 2, k 1 � 5, and k 2 � 10, a 1 � a 2 � b 1 � b 2 � c 1 � c 2 � l 2 � m 2 � 10, λ � 0.1, 0.5, 1, and 3. It is worth noting that the upper limit of λ is 3.5. Otherwise, the controller is easy to be ill-defined such that the states in the system are divergent.
Figures. 1-11 display the simulation results. In Figure 1, the tracking errors are drawn under different λ. It can be seen that the transient processes of the tracking errors are faster and the steady-state errors decrease as λ increase. Under different initial values x 1 (0) � 1.5 and 3, the comparisons of tracking performance for the desired trajectory y d1 between the AFC, AC, and PC are displayed in Figures 2-4. It is shown that the tracking performance with the PC is better and the tracking error is much smaller. Furthermore, the AFC can only ensure the boundedness of tracking errors, and the PC guarantees the asymptotic convergence of tracking errors, which is obvious in Figure 2. In addition, the absolute mean and root mean square values of tracking errors with the PC are smaller than the values with AFC and AC. Figures 5-7 draw the comparisons of tracking errors and tracking performance for the ideal trajectory y d2 , and the similar conclusion can be obtained as . e inputs u are given in Figure 8, which are large at the initial stage and converge rapidly to a range defined by ±10. e states x 2 and adaptive parameters ϖ * 1 , ϖ * 2 are presented in Figures 9 and 10, which are all bounded. Finally, Figure 11 shows the Nussbaum variables ς, which tends to positive constants from the initial values 1 as time goes on.

Simulation of a 3-Order Robot.
In order to demonstrate the practicability, the developed controller is applied to the following 3-order robot system [40]. where x 1 , x 2 , and x 3 denote the angle, angular velocity, and motor current of the manipulator, respectively. M is the system inertia, K m � KN c represents the coefficient between the current and moment with K being the constant torque, and N c expressing the joint reduction ratio. Furthermore, τ e represents the uncertain disturbance of torque by the external environment, L is the armature inductance and unknown, K e denotes the back electromotive force constant, R displays the circuitous resistance, and V is the input voltage. e formulae of B, N, and M are introduced as follows: where K τ expresses the coefficient of the electromechanical conversion of armature current to torque and B 0 shows the coefficient of viscous friction at the joint. g is the gravity     e reference trajectory is chosen to be a step signal. To satisfy the first order differentiability, it is written as y d � 0.5(tanh(p(t − 5)) + t1) with p � 2. e Nussbaum variable _ ς, control laws α 1 , α 2 u, and adaptive laws _ ϖ 1 , _ ϖ 2 , _ ϖ 3 are defined as (19), (17), (32), (64), (21), (35), and (65) with k 1 � 10, k 2 � 12, k 3 � 20, q 1 � 0.02, q 2 � 0.01, q 3 � 0.025, σ 1 � 15, σ 2 � 12, σ 3 � 28, and λ � 10.
e other parameters are equal to 1. Figures 12-15 show the simulation results of the 3-order single-link robot. e angle tracking error is displayed in Figure 12, which converges to the initial value 0.1ra d rapidly and oscillates at the step time t � 5s. In Figure 13, the output can track the reference trajectory well. is is a step signal. e input voltage u is presented in Figure 14. It is easily seen that the oscillation also occurs at the step time.
e angular velocity x 2 and motor current x 3 are given in Figure 15, respectively, both of which are bounded.

Conclusion
In the article, an adaptive asymptotic tracking controller has been designed for nonlinear systems with uncertain VCGs. Compared with these existing achievements, the developed control method could not only handle the VCGs in the form of unknown nonlinear functions but also achieve the asymptotic stability of the tracking error, which was carried out without the approximation by fuzzy logic or neural network and repetitive use of Nussbaum-type functions. New adaptive laws were defined to compensate for unknown virtual control gains, uncertain parameters, and external disturbances. Finally, the proposed control scheme was designed and applied to the control of a robot system. Both theoretical analysis and simulation were used to validate the effectiveness and practicability of the developed control strategy. In the future, we plan to apply the proposed controller to MIMO nonlinear systems by combining dead zone, saturation, hysteresis, and so on.

Data Availability
e data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest
e authors declare that they have no conflicts of interest.