Elsevier

ISA Transactions

Volume 121, February 2022, Pages 95-104
ISA Transactions

Research article
Adaptive robust dynamic surface asymptotic tracking for uncertain strict-feedback nonlinear systems with unknown control direction

https://doi.org/10.1016/j.isatra.2021.04.009Get rights and content

Highlights

  • The asymptotic tracking control problem is considered for a class of uncertain strict-feedback nonlinear systems with unknown control direction.

  • The proposed design procedure is able to simultaneously overcome the ’differential explosion’ and the unknown control direction problems.

  • The norms of the ideal weight vectors of fuzzy logic systems are taken as the part of estimation parameters, which significantly reduces the calculation amount.

  • Only the desired trajectory itself is required to be known without knowledge of its higher derivatives, which is different from traditional dynamic surface control results.

Abstract

For strict-feedback nonlinear systems (SFNSs) with unknown control direction, this paper synthesizes an asymptotic tracking controller by a combination of the dynamic surface control (DSC) technique, the Nussbaum gain technique (NGT) and fuzzy logic systems (FLSs). The SFNSs under study feature unknown nonlinear uncertainties and external disturbances. By utilizing the DSC technique with nonlinear filters, the issue of ‘differential explosion’ is obviated, in which the adaptive laws are constructed to conquer the effect of unknown functions. The FLSs are exploited to cope with uncertainties without any prior conditions of the ideal weight vectors and the approximation errors. In addition, by introducing the NGT, the unknown control direction problem is solved. Compared with the existing results, the proposed design procedure is able to simultaneously overcome the ‘differential explosion’ and the unknown control direction problems, and asymptotic tracking is accomplished. At the end, a second-order numerical system and a more realistic Norrbin nonlinear mathematical model are applied to confirm the feasibility of the design procedure.

Introduction

As is well-known, nonlinearities and uncertainties exist inherently in many real physical systems [1], [2], [3], [4], [5], [6]. Therefore, in last decades, robust or adaptive control techniques for systems with significant uncertainties have attracted much attention and many robust or adaptive controllers have been proposed [7], [8], [9]. In [10], [11], two adaptive robust controllers are presented for multiple time delays systems, where the former achieves asymptotic tracking and the latter ensures that the steady state error is bounded and its bound can be arbitrarily adjusted. In [12], [13], two sliding mode control schemes are designed to implement the tracking control of a second-order vector system. A common assumption about uncertainties in [10], [11], [12], [13] is that they must be obeying the matching condition. An effective method for relaxing such a requirement is the backstepping technique which was firstly presented in [14]. Through over twenty years’ development, it has become one of the most powerful tools for control systems [15], [16], [17].

Although backstepping technique has been widely studied, it has the defect of ‘differential explosion’, that is, the actual control law will contain the (high order) derivatives of virtual control laws. As a result, it is inevitable to bring about the growth of the complexity of the actual control law when the order of the system increases. Additionally, it has stringent requirements on the smoothness of plant functions. To avoid these thorny problems, there have been several control methods such as the homogeneous domination method [18], [19], the dynamic gain method [20], the low complexity method [21], [22], [23], the dynamic surface control (DSC) method [24], and so on. The most widely used one is the DSC approach which adopts first-order low-pass filters. In [25], the DSC approach is extended to parametric time-delay nonlinear systems. In [26], the adaptive stabilization problem of uncertain SFNSs is addressed. In [27], an adaptive DSC method is presented by virtue of neural networks for constrained SFNSs with unmodeled dynamics. In [28], for semi-strict feedback systems, the output tracking control is considered. In [29], an adaptive dynamic surface controller is constructed, where the persistent excitation condition for parameter convergence is relaxed through a composite learning manner. In [30], by employing nonlinear adaptive filters and novel flat zone introduced Lyapunov functions, an improved adaptive DSC approach is developed to achieve global tracking. More importantly, the DSC technique has been implemented in various practical systems in the world, such as homing missiles [31], permanent magnet synchronous motors [32], multiple marine surface vehicles [33], flexible manipulators [34] and so on. A remarkable character of the above DSC results [24], [25], [26], [27], [28], [29], [30], [31], [32], [33], [34] is that the tracking error or the system state converges to a residual set, and asymptotic convergence is not achieved. As we all known, asymptotic convergence has significant potential both in theory and practical applications. Consequently, it is of practical significance and challenge to investigate the asymptotic control for dynamic systems. Recently, being different from those results [24], [25], [26], [27], [28], [29], [30], [31], [32], [33], [34], the asymptotic convergence could be guaranteed in [35], [36], [37]. By virtue of the adaptive DSC technique and the nonlinear filters, the tracking error can asymptotically approach zero in [35], [36]. In [37], under the neural network approximation framework, the asymptotic control for SFNSs is also implemented. However, the existing approaches only solve the asymptotic tracking issue when the control direction is known. To our current knowledge, in the case where the directions of SFNSs are unknown, the investigation of adaptive dynamic surface asymptotic control methods has not been discussed, which is still open.

The control direction describes the movement direction of control systems in response to the control input. In fact, the control directions in many practical systems may be unavailable, for example, the autopilot design of time-varying ships [38]. When the control directions of nonlinear systems are not known in advance, the control complexity will grow considerably. In [39], the NGT was firstly proposed to tackle the unknown signs of control coefficients. Based on this, many meaningful results have been obtained [40], [41], [42], [43], [44], [45], [46], [47], [48], [49], [50], [51]. In [42], a robust adaptive controller is designed for SFNSs that involve unknown control coefficients as well as unknown disturbances. In [43], the idea of [42] is extended to full state-constrained systems. In [44], this idea is further generalized to full state-constrained time-delay systems. The results of [40], [41], [42], [43], [44] only guarantee the boundedness of the state or the tracking error, there are also some asymptotic convergence results [45], [46], [47], [48], [49], [50]. In [47], by employing a special Nussbaum function, the asymptotic stabilization control problem of SFNSs is addressed. In [48], the results of [47] are generalized to full-state constrained systems. In [49], the asymptotic tracking for MIMO systems is considered. In [50], the global stabilization of nonlinear systems is realized, where the growth rate of uncertainty is unknown. Although asymptotic control is achieved for control systems in [45], [46], [47], [48], [49], [50], they still suffer from the ‘differential explosion’ problem.

Inspired by the aforementioned works, this paper proposes an adaptive robust dynamic surface asymptotic tracking control scheme for uncertain SFNSs. According to the universal approximation theorem in [51], the nonlinear functions are approximated by employing FLSs, which have gained considerable research interests [52], [53], [54], [55]. The contributions of this paper can be outlined below: (i) Being different from [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28], [29], [30], [31], [32], [33], [34], [35], [36], [37], [38], [39], [40], [41], [42], [43], [44], [45], [46], [47], [48], [49], [50], by combining the DSC technique and the NGT, the design procedure of this paper is capable of overcoming simultaneously the ‘differential explosion’ and the unknown control direction problems, and asymptotic tracking is achieved. To obviate the issue of ‘differential explosion’, the DSC technique is employed with nonlinear filters. In addition, the NGT is utilized to handle the unknown control direction problem. (ii) Only the desired trajectory itself is needed to be known without knowledge of its high order derivatives, whereas in the DSC results [24], [27], [28], [30], [34], [35], [36], [37], either the upper bounds of the desired trajectory and its high order derivatives (including second-order derivative and first-order derivative) are necessary to be known, or the desired trajectory and its high order derivatives are known, or the desired trajectory needs to be nth order differentiable and all its derivatives up to nth order are bounded. (iii) The unknown nonlinear uncertainties are approximated by constructing FLSs without any prior conditions of the ideal weight vectors and the approximation errors of the FLSs. Compared to [52], [53], [54], [55], the norms of the ideal weight vectors of the FLSs in this paper are set as part of estimation parameters, instead of using adaptive laws to estimate each element of the ideal weight vectors, which significantly reduces the calculation amount.

Notations: Given positive integers i and j, define λi:j=[λi,λi+1,,λj], where λ can represent the symbols z, y, ψ̃ and M̃. Further, we define I[i:j]={i,i+1,,j} and x̃i=[x1,x2,,xi].

Section snippets

SFNSs

In this paper, we focus on the control of the uncertain SFNSs below: ẋ1=f1(x̃1)+x2+d1(t),ẋ2=f2(x̃2)+x3+d2(t),ẋn=fn(x̃n)+gu+dn(t),y=x1, in which yR denotes the output, uR refers to the control input, xiR,iI[1:n] represent the system states, fi()R,iI[1:n] stand for differentiable nonlinear uncertainties, di()R,iI[1:n] denote external disturbances, the unknown nonzero constant g has unknown sign. We will design a control law u(t) to ensure that the output y(t) can accurately track

Main results

For SFNSs (1), we present the control design procedure in this section. From the previous analysis, iI[1:n], we obtain, Wi, such that fi(x̃i)=δi(x̃i)+Wiρi(x̃i), δi(x̃i)δ̄i. Furthermore, for iI[1:n], define σi(t)νieλit, in which νi and λi are greater than 0. Obviously, σi(t) satisfy limt0tσi(τ)dτσ̄i<, where σ̄i are positive constants. Subsequently, without incurring any confusion, we omit variable t to complete algorithm derivation, for instance, σi stand for σi(t).

Step 1:

Simulation results

Two simulation examples are taken into consideration in this section.

Example A

The following second-order system is considered firstly. ẋ1=x2+x1exp(0.5x1)+0.01cos(2t),ẋ2=gu+x1x2+2x12+0.5x2+0.01cos(2t),y=x1, where g=2.

Set the fuzzy membership functions as: μF1(x1)=[exp(5x1+3)+1]1μFl(x1)=exp[(x1+0.80.2l)2],l=2,,6,μF7(x1)=11+exp[5(x10.6)],μF1(x1,x2)=i=1211+exp[5(xi+0.6)],μFl(x1,x2)=i=12exp[(xi+0.80.2l)2],l=2,,6,μF7(x1,x2)=i=1211+exp[5(xi0.6)].

Then, for i=1,2, we have ρil(x̃i)=μFl(x̃i)j=17

Conclusion

To achieve asymptotic tracking for SFNSs, we combine the adaptive DSC technique, the NGT and the FLSs to design the controller. The DSC technique with nonlinear filters is exploited to obviate the issue of ‘differential explosion’, where the adaptive laws with integral functions are designed to eliminate the influence of unknown functions caused by nonlinear filters. By introducing the NGT, the unknown control direction problem is solved. Furthermore, the effect of uncertainties are eliminated

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

This work was supported by the National Natural Science Foundation of China under Grant Number 62073096 and 61773387, and the Major Program of National Natural Science Foundation of China under Grant Numbers 61690210, 61690211 and 61690212, and the Program of Heilongjiang Touyan Team, China .

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