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

Xiao, Wei; Yin, Zhixiang; Zhou, Tianyue; Ye, Zi; Yang, Xiaoqi

doi:https://doi.org/10.1155/2022/7408077

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Research Article | Open Access

Volume 2022 | Article ID 7408077 | https://doi.org/10.1155/2022/7408077

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

Wei Xiao,^1,2Zhixiang Yin,³Tianyue Zhou,⁴Zi Ye,⁵and Xiaoqi Yang⁶

Academic Editor: Dan Selişteanu

Received01 Nov 2021

Revised06 Feb 2022

Accepted23 Feb 2022

Published28 Mar 2022

Abstract

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.

1. Introduction

The 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. There have been many related results in this area [1–11]. An adaptive data-driven controller was designed for nonlinear systems using goal representation heuristic and dynamic programming [4]. In view of unknown nonlinear fractional-order 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 [7–9]. 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]. There have been many achievements for nonlinear systems, whose VCGs were unknown constants [13–16]. 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. The 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 [30–32]. Besides, by utilizing with and 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. The 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]. There 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. The effectiveness and practicability of the developed control strategy are validated by both theoretical analysis and simulations.

The paper possesses the following features:(1)Different from the control methods based on the Nussbaum gain technique [13–25], 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. This is the main improvement on [13–32], which can only guarantee that the tracking error is bounded. Without using fuzzy logic or neural networks [26–32], and the lemma of [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. Therefore, the proposed controller can be suitable for more nonlinear systems.(3)The 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 -th order are continuous and bounded [13, 17, 18, 20–22, 26, 28–31, 33, 38].

The outline of this paper is presented below. In Section 2, the preliminaries and problem formulation are described. The 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.

2. Problem Formulation and Preliminaries

The uncertain nonlinear system is considered as follows:where are the state vector, input, and output of the system, respectively, . denotes the VCG, which is an unknown smooth non-zero nonlinear function. is an unknown parameter vector, and denotes a nonlinear function, which is known and smooth, is a positive integer; is an unknown bounded external disturbance.

Assumption 1 (see [24]). The ideal output vector is absolutely continuous and bounded.

Assumption 2 (see [22]). Let us assume that the sign of does not change for all and satisfieswith and being unknown positive constants.
It is reasonable that is bigger than a positive constant due to the system controllable condition of 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 and can be unknown. Without loss of generality, we assume that .

Assumption 3. There exists an unknown positive constant satisfying

Definition 1 (see [12]). A continuous function is named as Nussbaum function when the following conditions hold:There are many Nussbaum-type functions, such as , , , and .

Lemma 1 (see [18]). Let with , being non-zero real numbers and , is an even Nussbaum function, is an absolutely continuous function, and is another function. If there exist and a real constant such that a function should be subject tothen , and are all bounded on .
According to Proposition 2 in [39], can be extended to , when the solution of the closed-loop system is bounded.

3. Design and Analysis of the Controller

The procedure of the adaptive controller design and stability analysis are presented.

Firstly, the common coordinate transformation is introduced aswith being the intermediate control law\designed later.

The system (1) is changed into

In order to estimate the unknown VCGs , unknown parameter vector , and uncertain external disturbances , definewhere is the 2-norm of vectors, is an unknown bounded vector given later; is the approximation error with being the estimation of .

In what follows, the controller will be constructed in a stepwise strategy via backstepping. For convenience, let , , and .

Step 1. We choose the first Lyapunov function candidate aswhere is a positive constant.
Differentiating and invoking (9) produceSubstituting in (7) into (11) and considering (6) yieldBased on Young inequality, we havewhere , , and are positive constants and is the maximum of and unknown.
By invoking (13), is described asLet us defineSubstituting (15) and (16) into (14) results in In the following, the virtual control signal is defined aswhere is a Nussbaum-type even function and is a positive design parameter.
is adjusted according to the following law:where is a design parameter.

Remark 1. From (19) and the definition of , , it can be known that all the terms of are nonnegative, which is the key to proving the asymptotic stability of the tracking error later.
By invoking (18) and (20), is shown asThe adaptive law is defined asWith being a design parameter.
Replacing in (21) by (22), can be given as

Step 2. A Lyapunov function candidate is given aswhere is a constant.
Based on (23) and in (7), we have From (18), is expressed asSubstituting (26) into (25) leads to According to Young inequality, the following inequalities holdwhere , , , , and are positive constants.
Substituting (28) into (27) leads to and are written asInvoking (30) and (31) producesDesigning the following virtual control signal asand substituting it into (32) givewhere is a design parameter.

Remark 2. . Based on (32), Assumption 2, and the definition of , it is easy to check that all the items of are nonpositive, and , which means that can be replaced by 1 in the controller design.
is changed toWe construct an adaptive law aswhere is a positive constant.
Substituting (36) into (35) producesStep j: For the th subsystem of (6), a Lyapunov function candidate is iteratively chosen from the previous step.where is a design parameter.
Differentiating producesBy an induction argument, is deduced aswhere , , , , , , are all positive constants.
Substituting (40) into (39) giveswhereFurthermore, is expressed asBased on the completion of squares, one haswhere , , , and are positive design parameters and .
Invoking (44), is rewritten asSimilarly, we defineBy invoking (46) and (47), is shown asDefining the th virtual control law asand substituting it into (48) to yieldwhere is a constant.
Similar to (36), can be specified aswith being a design parameter.
Then, is given asStep n: following the similar procedure for (38) in Step j, the Lyapunov function candidate for the last step is designed asand is presented aswhere is a constant.
It follows from (52) thatwhere , , , , , , and are all positive constants.
is given aswhereFurthermore, is reformulated asRepeating the similar procedures in Step j, we have where , , , , and are positive design parameters and .
Substituting (60) and (61) into (58) results in Similar to (49) and (51), the real control law and th adaptive law can be defined, respectively, as follows: andwhere and are positive design parameters.
By some direct calculations, can be described asAccording to the above controller design, the main conclusion is summarized as
Theorem 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 in (64), which can guarantee the boundedness of all the signals in the closed-loop system for and asymptotic convergence of the tracking error.

Proof: . since , it is easy to check thatSubstituting (66) into (65) leads towithMoving to the left-hand side and then multiplying both sides of (67) by produceIntegrating (69) givesBy some direct calculations, (70) is described aswhere is a positive constant.
By virtue of Lemma 1, it is shown that , , are bound for . Therefore, the boundedness of all the signals in the closed-loop system is certified for .
Furthermore, using the definition of , it can be seen thatBased on the boundedness of , the following inequality holds.From the definition of in (6) and the boundedness of , , and , it is easily shown that is bounded.
Thus, by utilizing Barbalat’s Lemma, one haswhich means that the tracking error is asymptotically stable.
Control parameters selection. The selection ranges of and are wide. The bigger the parameters and , the faster the asymptotic convergence and response. Besides, is the other key parameter, which affects the asymptotic convergence of the tracking error. According to the simulation tests (see Figure 1), the bigger the faster the transient process and the smaller the steady-state errors. But it cannot be too big. In addition, the choices of , , , and are flexible.

4. Simulation Results

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

4.1. Simulation of a Second-Order Nonlinear System

Let us consider the strict-feedback nonlinear system aswhere with and , with , and , , , , .

The desired trajectories are selected as and In order to see the controller performance under different initial conditions for the system states, the initial value of is given as 1.5 and 3, respectively; ; and other initial conditions are chosen to be 0.

The intermediate control signal , Nussbaum variable , and actual control law of the developed control strategy are defined as (17), (19), and (64), where , , and , , , 0.5, 1, and 3. It is worth noting that the upper limit of is 3.5. Otherwise, the controller is easy to be ill-defined such that the states in the system are divergent.

and are designed as (21) and (65), with , , and .

The proposed controller (PC) is compared with the adaptive fuzzy (AFC) [28] and adaptive controllers (AC) [34], respectively.

The AFC is presented aswhere and are defined in [28], , , , and are taken the same values as the PC, , 2.

The AC is given aswhere , , , , , , , and are as same as the PC, , 2. , are positive constants, , .

Figures. 1–11 display the simulation results. In Figure 1, the tracking errors are drawn under different . It can be seen that the transient processes of the tracking errors are faster and the steady-state errors decrease as increase. Under different initial values and 3, the comparisons of tracking performance for the desired trajectory 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 , and the similar conclusion can be obtained as Figures 2–4. The inputs are given in Figure 8, which are large at the initial stage and converge rapidly to a range defined by . The states and adaptive parameters , are presented in Figures 9 and 10, which are all bounded. Finally, Figure 11 shows the Nussbaum variables , which tends to positive constants from the initial values 1 as time goes on.

4.2. 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 , , and denote the angle, angular velocity, and motor current of the manipulator, respectively. is the system inertia, represents the coefficient between the current and moment with being the constant torque, and expressing the joint reduction ratio. Furthermore, represents the uncertain disturbance of torque by the external environment, is the armature inductance and unknown, denotes the back electromotive force constant, displays the circuitous resistance, and is the input voltage. The formulae of , , and are introduced as follows:where expresses the coefficient of the electromechanical conversion of armature current to torque and shows the coefficient of viscous friction at the joint. is the gravity coefficient, denotes the rotor inertia, represents the link length, is the radius of the load, and and display the mass of the link and load. Moreover, the parameter values are as

The external disturbance is defined as the Gaussian white noise written as with being the time series.

The initial condition vector is selected as . The reference trajectory is chosen to be a step signal. To satisfy the first order differentiability, it is written as with .

The Nussbaum variable , control laws , , and adaptive laws , , are defined as (19), (17), (32), (64), (21), (35), and (65) with , , , , , , , , , and . The other parameters are equal to 1.

Figures 12–15 show the simulation results of the 3-order single-link robot. The angle tracking error is displayed in Figure 12, which converges to the initial value rapidly and oscillates at the step time . In Figure 13, the output can track the reference trajectory well. This is a step signal. The input voltage is presented in Figure 14. It is easily seen that the oscillation also occurs at the step time. The angular velocity and motor current are given in Figure 15, respectively, both of which are bounded.

5. 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

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

Conflicts of Interest

The authors declare that they have no conflicts of interest.

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Copyright © 2022 Wei Xiao et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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