- Research
- Open Access
Adaptive tracking control for a class of uncertain nonlinear systems with infinite number of actuator failures using neural networks
- Wenshun Lv^{1} and
- Fang Wang^{1}Email author
https://doi.org/10.1186/s13662-017-1426-5
© The Author(s) 2017
- Received: 30 July 2017
- Accepted: 13 November 2017
- Published: 1 December 2017
Abstract
We consider adaptive compensation for infinite number of actuator failures in the tracking control of uncertain nonlinear systems. We construct an adaptive controller by combining the common Lyapunov function approach and the structural characteristic of neural networks. The proposed control strategy is feasible under the presupposition that the systems have a nonstrict-feedback structure. We prove that the states of the closed-loop system are bounded and the tracking error converges to a small neighborhood of the origin under the designed controllers, even though there are an infinite number of actuator failures. At last, the validity of the proposed control scheme is demonstrated by two examples.
Keywords
- nonlinear systems
- actuator failures
- adaptive control
- backstepping
- neural networks
- nonstrict feedback
1 Introduction
In recent years, many approximation-based adaptive fuzzy or neural backstepping controllers have been developed for uncertain nonlinear systems; see [1–19]. Among them, to eliminate the problem of ‘explosion of complexity’ inherent in the existing method, in [13] a control design strategy was developed for a class of nonlinear systems in strict-feedback form with arbitrary uncertainty. To deal with the state unmeasured problem, a novel control scheme was introduced in [18]. To address the control problem of nonsmooth hysteresis nonlinearity in the actuator, adaptive neural controllers were constructed for nonlinear strict-feedback systems with unknown hysteresis in [19]. It should be noted that the control schemes mentioned are under the presupposition that the systems have a nonstrict-feedback structure.
In the nonlinear systems without strict-feedback structure, the unknown nonlinear functions involve all the state variables, so they cannot be approximated with current states. To deal with such a structural restriction, in [20] a variable separation method was proposed. The control scheme in [20] assured that the tracking performance is achieved as time variable goes to infinity. Besides the proposed control scheme, many efforts have been made in relaxing such a restriction of system structure; see [21–25]. In practical application, the actuator component is usually employed to execute control actions on the plant. However, the actuation mechanism may suffer from failures, which results in the actuator losses of partial or total effectiveness. To prevent the emergence of performance deterioration and instability of the closed-loop system caused by actuator faults, accommodating actuator failures should be taken into account in the control design.
In recent years, many control schemes have been proposed to accommodate actuator failures; see, for example, [26–32]. By applying backstepping technique for the linear systems, a systematic actuator failure compensation control was presented in [26]. Then, in [27] the proposed control method was extended to nonlinear systems with actuator failures; in [28] the problem of accommodating actuator failures was investigated for a lass of uncertain nonlinear systems with hysteresis input as a follow-up extension. In practice, the failure pattern in an actuator may change repeatedly, which makes failure parameters suffer from an infinite number of jumps. Consequently, the considered Lyapunov function would experience infinite number of jumps. In [29], this problem was addressed by applying a new tuning function under the frame of adaptive control. However, the proposed control strategy can only apply to the strict-feedback systems.
- (1)
The control scheme in this paper relaxes the restriction of system structure so that a better approach is proposed to deal with the problem of compensation for an infinite number of actuator failures, which is more meaningful in practical application in comparison with [29].
- (2)
In this paper, combining neural networks and a new piecewise Lyapunov function analysis, we establish an adaptive control scheme for a class of uncertain nonlinear systems with a nonstrict-feedback structure.
The remainder of the paper is organized as follows. In the next section, the problem description and preliminaries are presented. Section 3 shows the major result. In Section 4, the simulation result expounds the validity of the proposed control scheme. Finally, we give a simple summary.
2 Preliminaries and problem description
2.1 A. System description
Remark 1
The unknown time-varying value \(u_{fj,k}(t)\) can be linearly parameterized as \(u_{fj,k}(t) = u_{j0,k}+\Sigma_{i=1}^{q_{j}}\zeta _{ji,k}\phi_{ji}(t)\) with known functions \(\phi_{ji}(t)\) and unknown constants \(u_{j0,k}\) and \(\zeta_{ji,k}\).
Control objective: design the control input \(\check{u}_{j}\) to compensate for the actuator failures modeled as (2)-(3) such that all the closed-loop signals are bounded and the system output y tracks the given reference signal \(y_{d}\) with a tracking error converging to a residual around zero. The following assumptions are general in the literature on the adaptive actuator failure compensation control.
Assumption 1
The number of failed actuators undergoing TLOE faults simultaneously is allowed to be at most \({m-1}\), and achieving control objective with the remaining actuators is still possible.
Assumption 2
The reference signal \(y_{d}(t)\) and its first nth-order time derivatives \(y_{d}(t)\) (\(i = 1, \ldots, n\)) are known, smooth, and bounded.
Assumption 3
\(\operatorname{sign}(b_{j})\) are known for \(j = 1, 2,\ldots,m\).
Assumption 4
The conditions \({\sigma}_{j}(\cdot)\neq0\), \(0 < b_{0} \leq \vert b_{j}\vert \leq b_{M}\), and \(\vert u_{fj,k}(t)\vert \leq u_{fM}\) are satisfied. In addition, for the PLOE faults, \(0 < \kappa_{0} \leq \kappa_{j,k}(t) < 1\). Note that \(b_{0}\), \(b_{M}\), \(u_{fM}\), and \(\kappa_{0}\) are known constants.
Assumption 5
The failure parameters \(\kappa_{j,k}(t)\) and \(u_{fj,k}(t)\) are smooth and continuous over \([t_{k}, t_{k+1})\). Moreover, their change rates are bounded, that is, \(\sup_{t\in[t_{k},t_{k+1})} \vert \dot{\kappa}_{j,k}(t)\vert \leq d_{1}\) and \(\sup_{t\in[t_{k},t_{k+1})} \vert \dot{u}_{fj,k}(t)\vert \leq d_{2}\), where \(d_{1}\) and \(d_{2}\) denote two unknown positive constants.
2.2 B. RBF neural networks
Lemma 1
([34])
Lemma 2
([35])
3 Adaptive tracking control design and stability analysis
In this section, based on a backstepping technique and neural networks, we design an adaptive actuator failure compensation control scheme. The control goal of this manuscript is to establish an adaptive controller such that the system output y follows a desired reference signal \(y_{d}\).
3.1 A. Adaptive tracking control design
To ensure the boundedness of the jumping size of the Lyapunov function at failure instants, we design the adaptation laws for updating parameter estimators with projection operation.
With the developed projection-based tuning function approach, we further propose new piecewise Lyapunov function analysis to establish the closed-loop system stability.
3.2 B. Stability analysis
Theorem 1
- (1)
All the signals of the closed-loop system are bounded;
- (2)
The tracking error converges to a small neighborhood of zero.
Proof
Remark 2
Inequality (28) makes a vital contribution to the backstepping design because it builds the relation between \(x_{i}\) and \(z_{i}\), which makes a backstepping-based design procedure viable.
Remark 3
The adaptive parameters \(\hat{W}_{i}\), ε̂, τ̂ are utilized to estimate \(W_{i}\), ε, τ, respectively. \(\tilde{W}_{i} = W _{i}-\hat{W}_{i}\), \(\tilde{\varepsilon}=\epsilon-\hat{\varepsilon}\), and \(\tilde{\tau}=\tau-\hat{\tau}\) denote the estimated errors. Note that the failure parameter \(\kappa_{j,k}\) is allowed to be time varying during each time interval \([t_{k}, t_{k+1})\) for \(k = 0,1,\ldots\) and \(b_{j}\) (\(j=1,\ldots,m\)) are unknown control coefficients. The instability cannot be ensured when \([t_{k}, t_{k+1})\) and \(b_{j}\) are not contained in the Lyapunov function.
Remark 4
The failure-related parameters τ contained in the Lyapunov function (15) will undergo a sudden jump at unknown time instant \(t_{k}\), and it follows that decreasing of the Lyapunov function, shown as in (38), is only valid at the time interval \([t_{k}, t_{k+1})\) during which the Lyapunov function \(\bar{V}_{k}(t)\) is differentiable. To establish the closed-loop system stability under the case of actuator failures or faults, we consider the overall Lyapunov function defined in (41).
4 Simulation example
In this section, we use two examples to expound our design scheme and testify the results obtained.
4.1 A. Example 1
4.2 B. Example 2
Parameters of the cascade chemical reactor system
q = 2.8317 m^{3}/h | \(q_{R}= 1.4158 \mbox{m}^{3}/\mbox{h}\) | V = 1.3592 m^{3} |
\(T_{1}^{d}=750\ {}^{\circ}\mbox{C}\) | \(T_{2}^{d}=737.5\ {}^{\circ }\mbox{C}\) | \(C_{A_{2}}^{d}= 10.4178\mbox{ mol}/\mbox{m}^{3}\) |
α = 7.08 × 10^{10} h^{−1} | λ = −3.1644 × 10^{7} h^{−1} | ρ = 800.9189 kg/m^{3} |
\(c_{\rho}= 1395.3\mbox{ J}/\mbox{kg}{}^{\circ}\mbox{C}\) | \(E_{a}= 3.1644\times10^{7}\mbox{ J}/\mbox{mol}\) | \(R = 1679.2\mbox{ J}/\mbox{mol}{}^{\circ}\mbox{C}\) |
\(U = 1.3625\times10^{7}\mbox{ J}/\mbox{hm}^{2}{}^{\circ}\mbox{C}\) | A = 23.2 m^{2} | \(T_{j2}^{d}=727.6\ {}^{\circ}\mbox{C}\) |
\(q_{j2}=1.4130\mbox{ m}^{3}/\mbox{h}\) | \(T_{j20}^{d}=608.2\ {}^{\circ }\mbox{C}\) | \(\rho_{j}=997.9450\mbox{ kg}/\mbox{m}^{3}\) |
\(c_{j}= 1860.3\mbox{ J}/\mbox{kg}{}^{\circ}\mbox{C}\) | V = 0.1090 m^{3} |
4.3 C. Conclusions
In this papedr, we investigate the issue of adaptive finite-time tracking for a class of nonlinearity systems with hysteresis. On the basis of the approximation capability of fuzzy logic systems, we give an adaptive law and an intermediate control function. Moreover, we proved that the aforementioned approach can make the system SGPFS.
In comparison with the tuning function control schemes in [39–44], which also focus on adaptive actuator failure compensation problem, the framework of our control is further simplified by using RBFNN, as seen in (9)-(13). Moreover, with respect to the previous fault-tolerant controllers such as [45, 46], our scheme additionally contains an optimized neural network adaptation mechanism, which renders that there are only two estimates, that is, \(\hat{W_{i}}\) and ε need to be computed online. The control scheme in [47] is under the presupposition that the systems have a nonstrict-feedback structure. Our control scheme relaxes the restriction of system structure so that the method in this note is more meaningful. In this sense, the presented study is more computationally attractive and thus feasible in practical implementation in comparison with other existing results.
Declarations
Acknowledgements
The authors greatly appreciate the reviewers suggestions, the editors’ encouragement, and Yan Li’s contribution to this paper. Yan Li is currently professor at Shandong University of Science and Technology. She made a significant contribution to this paper, including control scheme design, paper writing, and revision. This work is supported partially by the National Natural Science Foundation of China (Grant No. 61503223), in part by the Project of Shandong Province Higher Educational Science and Technology Program (J15LI09), in part by China Postdoctoral Science Foundation-funded project 2016M592140, in part by Shandong innovation postdoctoral program 201603066, and in part by the SDUST Research Fund (2014TDJH102).
Authors’ contributions
All authors contributed equally to writing this paper. All authors 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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