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Adaptive dynamic surface control of parametric uncertain and disturbed strict-feedback nonlinear systems
- Chenhui Wang^{1}Email authorView ORCID ID profile
https://doi.org/10.1186/s13662-019-1971-1
© The Author(s) 2019
- Received: 31 August 2018
- Accepted: 15 January 2019
- Published: 25 January 2019
Abstract
The construction of backstepping control input needs the derivative of the virtual controller to be available. However, this requirement usually makes the implementation of the controller very difficult and complicated. To overcome this problem, in this paper, an adaptive dynamic surface control (ADSC) is proposed for a class of strict-feedback nonlinear systems with parametric uncertainty and external disturbance. In each step of the backstepping control design, the virtual control input is estimated by an auxiliary signal which is generated by a proposed dynamic surface. This signal’s derivative is easy to obtain, so it is not necessary to achieve the derivative of the virtual control input. By using the Lyapunov stability theorem, an ADSC has been established to guarantee the boundedness of all signals and the convergence of the tracking errors. Finally, a simulation example is given to indicate the effectiveness of our control approach.
Keywords
- Strict-feedback nonlinear system
- Adaptive backstepping control
- Adaptive dynamic surface control
- Mismatched uncertainty
1 Introduction
It is well known that tremendous success has been obtained in controlling nonlinear systems based on the development of adaptive backstepping control (ABC) and feedback linearization (FBL) methods [1, 2]. The main idea of FBL consists in transforming a strict-feedback nonlinear system (SFNS) that satisfies some matching conditions into a linear one. That is to say, this method cannot deal with the nonlinear term directly. To overcome this limitation, the ABC method establishes a systematic framework for controlling SFNSs, whose main idea is using some intermediate variables recursively as pseudo-control signals. If the SFNSs have minimum phase and have known relative degrees, the stability of the closed-loop system can be guaranteed by using the backstepping design method. For ABC of an SFNS without parametric uncertainty or external disturbance, some control methods have been presented, for example, in [2–5]. Based on the main idea of the ABC approach, more complicated conditions are considered in designing ABC for SFNS in [2, 6–8]. However, the above literature did not consider parametric uncertainties and disturbance in the ABC design. Thus, the ABC design needs to be studied further to achieve better robust performance.
As is well known, most physical systems usually suffer from system uncertainty, for example, parametric uncertainty, modeling error, or external disturbance, which will decrease the control performance or even leads to instability of the controlled system [9–21]. Thus, the ABC method usually requires cancelation of the nonlinearities. Sometimes exact knowledge of the system nonlinearities is not available, or these terms change along with time. Therefore, we need to model the nonlinear uncertainties with parametric uncertainties. Some robust control methods have been given to handle these kinds of system uncertainties [19, 22–39]. ABC methods have been studied recently in [40–42]. In [2], an ABC method combined with adaptive fuzzy control was investigated. A fuzzy ABC for SFNSs in the presence of both sampled and delayed measurements was studied in [40]. A command filtered ABC method for unknown SFNSs was proposed in [41], where a first-order filter was introduced to handle the virtual control input. However, in the above literature, external disturbance and/or parametric uncertainty was not considered.
The ABC method suffers from the “explosion of complexity” which is produced by repeatedly differentiating the virtual control input. Then, many control methods have been given to solve this problem. Among these, the adaptive dynamic surface control (ADSC) aims to enhance the drawback of ABC by driving the control input passing through a first-order filter [23, 43–46]. This method not only solve the problem of “explosion of complexity”, but also reduce the requirement of the system model as well as the referenced signal. In [23], by using the ADSC method, an ABC method was proposed for SFNSs with parametric uncertainties. Reference [47] provides a command filtered backstepping control for SFNSs, where the adaptive control method was not considered. Then, based on the work of [47], Ref. [22] provided an adaptive command filtered backstepping control method, and a compensated tracking error was also considered. The above method is based on procedures like nonlinear damping and variable structure as well as their variations, which commonly need prior knowledge of the system uncertainty, for example, utilizing some constant or nonlinear function known in advance as the bounds of the estimated nonlinearity. Consequently, the applications of these kinds of methods may be limited if there is no such prior knowledge.
In this paper, we address the ADSC method for SFNSs with mismatched parametric uncertainties and disturbances. The proposed method described herein needs only the signals \(x_{d}(t)\) and the state variables of the controlled system to be available. The dynamic surface idea is used to obtain a practical generalization of the conventional backstepping technique. A main motivation of this work is to simplify the process of determining the command derivatives required in the backstepping procedure. Our controller ensures the boundedness of all signals and the convergence of the tracking errors. The main contributions of this work can be presented as follows. (1) In this paper, an ADSC is proposed for SFNSs with parametric uncertainties and disturbances. Note that it is very difficult to obtain exact analytical expressions for the dynamic surface, the proposed ADSC method works well even in the presence of parametric uncertainty as well as of an external disturbance. An effective first-order dynamic surface, which is easy to use, has been proposed. (2) Based on the proposed dynamic surface, the conventional “explosion of complexity” will not occur in our work. The virtual control input’s derivative is replaced by an auxiliary variable which can track the virtual control input in arbitrary degree of accuracy. Compared with the method proposed in [48, 49], our method combined with adaptation laws has a more concise construction and is easier to implement.
2 Problem formulation
Throughout this paper, \(\mathcal{R}\), \(\mathcal{R}^{i}\) represent the spaces of real numbers and real i-vectors, respectively. \(\Vert x(t) \Vert \) is the standard 2-norm of \(x(t)\), \(\operatorname{sgn}(\cdot )\) denotes the signum function, \(\mathcal{L}_{\infty }\) represents the space of the bounded variable, \(\varOmega _{c}\) is the ball of radius c, and \(\mathcal{C}^{i}\) is the space of functions for which all ith-order derivatives exist and are continuous.
To facilitate the controller design, we need the following assumptions.
Assumption 1
The nonlinear functions \(f_{i}(\bar{\pmb{x}}(t))\), \(\pmb{\varphi } _{n}(\bar{\pmb{x}}(t))\) are of \(\mathcal{C}^{1}\).
Assumption 2
The desired signal \(x_{d}(t)\) and its derivative are continuous functions of \(\mathcal{L}_{\infty }\).
Assumption 3
The external disturbance \(d(t)\) is bounded, i.e., there exists a positive constant \(d^{*}\) such that \(\vert d(t) \vert \leq d^{*}\).
Assumption 4
Assume that \(\varOmega _{c}\) is an open set which includes the referenced signal, the origin and the initial conditions of the system 1. Suppose that, for \(i=1,\ldots,n\), \(\frac{\partial ^{j} f _{i}(\bar{\pmb{x}}_{i}(t))}{\partial t^{j}}\), \(\frac{\partial ^{j} x _{d}(t)}{\partial t^{j}}\), where \(j=1,\ldots,n\), are all bounded on \(\bar{\varOmega }_{c}\).
Remark 1
It is worth mentioning that above four assumptions are in line with the practical situation. Firstly, the Assumption 1 is satisfied in most real-world systems. Secondly, in most literature, the referenced signal is a smooth function, that is to say, Assumption 2 is reasonable. Thirdly, in this paper, it is assumed that the unknown external disturbance \(d(t)\) is bounded. In fact, lots of common disturbance functions are bounded. In addition, the upper bound \(d^{*}\) is assumed to be unknown. Finally, Assumption 4 is needed in the stability analysis, and this assumption is common in much related literature, for example, [22, 23, 47].
3 Controller design and stability analysis
3.1 The ADSC design
For convenience, this section will design the controller based on the backstepping procedure which has n steps.
Step 1.
Step 2.
Step i, \(3\leq i\leq n-1\).
Step n.
3.2 Stability analysis
The proposed filters, i.e., (6), (9), (13) can guarantee that \(\tilde{\nu }_{i}(t)\) is sufficiently small eventually. To prove this result, we give the following lemma first.
Lemma 1
Proof
Based on above result, we can easily obtain the following theorem.
Theorem 1
Under Assumptions 3 and 4, we see that, for any \(\mu _{i}>0\), there exists a constant \(T>0\), for all \(t>T\), \(\vert \tilde{\nu }_{i}(t) \vert \leq \mu _{i}\) if sufficiently large parameters \(\sigma _{i}\) are chosen.
Proof
Now, let us give the following main results.
Theorem 2
Proof
On the other hand, we know that if \(\mu _{i}\), \(\lambda _{12}\) and \(\lambda _{22}\) are small enough, then α will be small enough, too. Thus, the tracking errors \(e_{i}(t)\) will eventually converge to a small region at zero if α is small enough. □
Remark 2
From the results of Theorem 2 we know that to drive the tracking errors \(e_{i}(t)\) small enough, we should choose small enough \(\mu _{i}\), \(\lambda _{12}\) and \(\lambda _{22}\). We know that \(\mu _{i}\) will be arbitrarily small if \(\sigma _{i}\) are chosen large enough. With respect to \(\lambda _{12}\) and \(\lambda _{22}\), in some literature, these parameters are set to be zero, for example, in [50–54]. However, we know in this case that the boundedness of the updated parameters cannot be guaranteed. Thus, in our method, these parameters are introduced for the purpose to guarantee the boundedness of all signals in the closed-loop system. In the simulation, to achieve good control performance, we can set these parameters sufficiently small.
Remark 3
In this paper, the dynamic surfaces (6), (10) and (14) are used to get the estimation of the virtual control inputs and their derivatives. It should be mentioned that the proposed Lemma 1 plays an important role in the stability analysis, which can guarantee the estimation error to be as small as possible. The estimation error can be adjusted by the design parameter \(\sigma _{i}\). In fact, in practical applications, one does not to select too large \(\sigma _{i}\), which is indicated by the following simulation results.
Remark 4
In the conventional ABC method, every middle variable is treated as an input, and by using Lyapunov stability theorems, a virtual control input is designed. In the next step, the derivative of the virtual input is needed. However, as the order increases, it is more and more difficult to get the exact value of the derivative of the virtual input. Thus, the “explosion of complexity” occurs. To overcome this problem, in [2, 40–42], the derivative of the virtual input was estimated by using a fuzzy logic system, however, more control energy is needed and more computational burden will be added to the control system. In this paper, the ADSC method was proposed for SFNS with parametric uncertainty. The ADSC is an extension of the ABC, which is effective for handling SFNS. By using the proposed dynamical surface, i.e. (6), (10) and (14), the estimation of the derivatives of the virtual inputs is easy to obtain. As a result, the “explosion of complexity” problem can be solved effectively.
4 Simulation example
The controller design parameters are chosen as \(k_{1}=k_{2}=k_{3}=1.5\), \(\sigma _{1}=\sigma _{2}=\sigma _{3}=20\), \(\lambda _{11}=\lambda _{21}=10\), \(\lambda _{21}=\lambda _{22}=0.05\). The true value of \(\pmb{\vartheta }\) is \(\pmb{\vartheta }=[-0.5, 0.1, 0.5]^{T}\). The initial conditions for \(\hat{\pmb{\vartheta }}(t) \) and \(\hat{d}^{*}(t)\) are \(\hat{\pmb{\vartheta }}(0)=\pmb{0}\) and \(\hat{d}^{*}(0)=0\), respectively.
5 Conclusions
In this paper, we present a strict stability analysis for a practical generalization of the conventional ABC method. Our main idea consists in designing a backstepping controller by using an auxiliary variable, i.e., out of a dynamic surface, whose derivative is easy to obtain to approximate the virtual controller. The proposed method is feasible even when the controlled system suffers from parametric uncertainty and external disturbances. It has been proven that our method is feasible for a wider range of SFNSs than conventional ABC and the dynamic surface can also be utilized to enhance constraints on the state trajectories. The proposed ADSC guarantees that all signals in the closed-loop system keep bounded and tracking errors converge to a sufficiently small region. The effectiveness of our method has been verified by a simulation results. Our future research direction include: (1) Design an extended controller consider how to deduce the requirement of the system model; (2) Combined our method with some robust control approach, for example, adaptive fuzzy control, adaptive neural network control, sliding mode control.
Declarations
Acknowledgements
Not applicable.
Funding
This work is supported by the National Natural Science Foundation of China (Grant No. 11302184) and the Young and Middle-aged Teacher Education and Science Research Foundation of Fujian Province of China (Grant No. JAT170423).
Authors’ contributions
All authors contributed equally to the writing of this paper. All authors conceived of the study, participated in its design and coordination, read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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