Robust Adaptive Actuator Failure Compensation for a Class of Uncertain Nonlinear Systems

Mahnaz Hashemi Javad Askari Jafar Ghaisari Marzieh Kamali

Mahnaz Hashemi, Javad Askari, Jafar Ghaisari and Marzieh Kamali. Robust Adaptive Actuator Failure Compensation for a Class of Uncertain Nonlinear Systems. International Journal of Automation and Computing, vol. 14, no. 6, pp. 719-728, 2017 doi:  10.1007/s11633-016-1016-0
Citation: Mahnaz Hashemi, Javad Askari, Jafar Ghaisari and Marzieh Kamali. Robust Adaptive Actuator Failure Compensation for a Class of Uncertain Nonlinear Systems. International Journal of Automation and Computing, vol. 14, no. 6, pp. 719-728, 2017 doi:  10.1007/s11633-016-1016-0

doi: 10.1007/s11633-016-1016-0
基金项目: 

Esfahan Regional Electric Company EREC

Robust Adaptive Actuator Failure Compensation for a Class of Uncertain Nonlinear Systems

Funds: 

Esfahan Regional Electric Company EREC

More Information
    Author Bio:

    Mahnaz Hashemi was born in Isfahan, Iran. She received the B. Sc. and M. Sc. degrees in electrical engineering from Department of Electrical and Computer Engineering, Isfahan University of Technology, Iran in 2007 and 2009. She is a Ph. D. degree candidate in Isfahan University of Technology, Iran. She has published about 10 journal and conference papers.
         Her research interests include adaptive control, nonlinear systems and fault tolerant control.
         E-mail:m.hashemi@ec.iut.ac.ir
        ORCID ID:0000-0002-7454-6423

    Javad Askari received the B. Sc. degree in electrical engineering from Isfahan University of Technology, Iran in 1987, and received the M. Sc. degree in electrical engineering from University of Tehran, Iran in 1993. He received the Ph. D. degree in electrical engineering from University of Tehran, Iran in 2001. From 1988 to 1990, he worked at Isfahan Petrochemical Company in Isfahan. From 1999 to 2001, he received a grant from the German Academic Exchange Service (DAAD) and joined Control Engineering Department at Technical University HamburgHarburg in Germany, where he received the Ph. D. degree with professor Lunzes research group. He is currently an associate professor at Control Engineering Department of Isfahan University of Technology, Iran. He has published about 50 journal and conference papers.
         His research interest is control theory, particularly in the field of Hybrid dynamical control systems and fault-tolerant control, identification, discrete-event systems, graph theory and electrical engineering curriculum.
         E-mail:j-askari@cc.iut.ac.ir

    Jafar Ghaisari received the B. Sc., M. Sc., and Ph. D. degrees with honors in electrical engineering from Isfahan University of Technology, Iran in 1996, 1999 and 2006, respectively. During the Ph. D. program, he joined the Department of Electrical and Computer Engineering, Queen's University, Canada, as a visiting researcher from 2004 to 2005, and received research scholarships. Since December 2006, he has been with the Department of Electrical and Computer Engineering, Isfahan University of Technology, Iran. He has published about 50 journal and conference papers.
         His research interests include instrumentation, nonlinear control systems, and control of FACTS devices and power electronics.
         E-mail:ghaisari@cc.iut.ac.ir

    Marzieh Kamali received the B. Sc. degrees in biomedical engineering and electrical engineering from Amirkabir University of Technology, Iran in 2005, and received the M. Sc. degree in biomedical engineering and electrical engineering from Isfahan University of Technology, Iran in 2007. She received the Ph. D. degree in electrical engineering from Isfahan University of Technology, Iran in 2012. Since 2012, she has been with the Department of Electrical and Computer Engineering, Isfahan University of Technology, Iran. She has published about 10 journal and conference papers.
         Her research interests include adaptive control and actuator failure compensation.
         E-mail:m.kamali@cc.iut.ac.ir

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出版历程
  • 收稿日期:  2014-04-19
  • 录用日期:  2015-09-22
  • 网络出版日期:  2016-12-05
  • 刊出日期:  2017-12-01

Robust Adaptive Actuator Failure Compensation for a Class of Uncertain Nonlinear Systems

doi: 10.1007/s11633-016-1016-0
    基金项目:

    Esfahan Regional Electric Company EREC

    作者简介:

    Mahnaz Hashemi was born in Isfahan, Iran. She received the B. Sc. and M. Sc. degrees in electrical engineering from Department of Electrical and Computer Engineering, Isfahan University of Technology, Iran in 2007 and 2009. She is a Ph. D. degree candidate in Isfahan University of Technology, Iran. She has published about 10 journal and conference papers.
         Her research interests include adaptive control, nonlinear systems and fault tolerant control.
         E-mail:m.hashemi@ec.iut.ac.ir
        ORCID ID:0000-0002-7454-6423

    Javad Askari received the B. Sc. degree in electrical engineering from Isfahan University of Technology, Iran in 1987, and received the M. Sc. degree in electrical engineering from University of Tehran, Iran in 1993. He received the Ph. D. degree in electrical engineering from University of Tehran, Iran in 2001. From 1988 to 1990, he worked at Isfahan Petrochemical Company in Isfahan. From 1999 to 2001, he received a grant from the German Academic Exchange Service (DAAD) and joined Control Engineering Department at Technical University HamburgHarburg in Germany, where he received the Ph. D. degree with professor Lunzes research group. He is currently an associate professor at Control Engineering Department of Isfahan University of Technology, Iran. He has published about 50 journal and conference papers.
         His research interest is control theory, particularly in the field of Hybrid dynamical control systems and fault-tolerant control, identification, discrete-event systems, graph theory and electrical engineering curriculum.
         E-mail:j-askari@cc.iut.ac.ir

    Jafar Ghaisari received the B. Sc., M. Sc., and Ph. D. degrees with honors in electrical engineering from Isfahan University of Technology, Iran in 1996, 1999 and 2006, respectively. During the Ph. D. program, he joined the Department of Electrical and Computer Engineering, Queen's University, Canada, as a visiting researcher from 2004 to 2005, and received research scholarships. Since December 2006, he has been with the Department of Electrical and Computer Engineering, Isfahan University of Technology, Iran. He has published about 50 journal and conference papers.
         His research interests include instrumentation, nonlinear control systems, and control of FACTS devices and power electronics.
         E-mail:ghaisari@cc.iut.ac.ir

    Marzieh Kamali received the B. Sc. degrees in biomedical engineering and electrical engineering from Amirkabir University of Technology, Iran in 2005, and received the M. Sc. degree in biomedical engineering and electrical engineering from Isfahan University of Technology, Iran in 2007. She received the Ph. D. degree in electrical engineering from Isfahan University of Technology, Iran in 2012. Since 2012, she has been with the Department of Electrical and Computer Engineering, Isfahan University of Technology, Iran. She has published about 10 journal and conference papers.
         Her research interests include adaptive control and actuator failure compensation.
         E-mail:m.kamali@cc.iut.ac.ir

English Abstract

Mahnaz Hashemi, Javad Askari, Jafar Ghaisari and Marzieh Kamali. Robust Adaptive Actuator Failure Compensation for a Class of Uncertain Nonlinear Systems. International Journal of Automation and Computing, vol. 14, no. 6, pp. 719-728, 2017 doi:  10.1007/s11633-016-1016-0
Citation: Mahnaz Hashemi, Javad Askari, Jafar Ghaisari and Marzieh Kamali. Robust Adaptive Actuator Failure Compensation for a Class of Uncertain Nonlinear Systems. International Journal of Automation and Computing, vol. 14, no. 6, pp. 719-728, 2017 doi:  10.1007/s11633-016-1016-0
    • Unexpected faults such as loss of effectiveness fault and stuck failure are frequently encountered in actuators because of the harsh working environment. These problems not only degrade the control performance of the system, but also sometimes lead to instability or even catastrophic accidents. Therefore, the problem of controller design for systems to maintain acceptable performance despite the presence of unknown actuator faults is of both practical and theoretical importance, especially for critical systems such as flight control systems. A specific control design technique in the presence of actuator faults is a fault tolerant control (FTC)approach which is divided into passive and active approaches. The active FTC approach requires a fault detection unit to detect the occurred faults. In this method, when a fault is detected by the fault detection unit, the controller is redesigned to achieve the desired control performance. The active fault tolerant control approach has some problems such as false detection, complexity and a time delay between the detection of the faults and the reconfiguration or accommodation of the controller[1]. Besides, the passive fault tolerant controller utilizes the same control scheme before and after fault, without specific accommodation parameters, typically by introducing a conservative control law. If a fault occurs out of the prescribed faults, the passive fault tolerant controller cannot tolerate it. By considering the mentioned drawbacks of the passive and active fault tolerant controllers, in this paper, the adaptive actuator failure compensation approach is proposed that does not require any explicit fault detection. The proposed controller compensates all the considered actuator faults by redesigning adaptively. It means that, the unknown parameters of the proposed controller are updated so that the system achieves the desired control objective even though some actuators fail. At present, numerous research results are available for fault tolerant control of linear systems, such as linear matrix inequalities (LMI)scheme[2], sliding mode control [3-5] and adaptive control [6-12]. For nonlinear systems, passive fault tolerant control schemes were proposed in [13-15] and active fault tolerant controllers were proposed in [15-18] for some classes of nonlinear systems. In [19, 20], adaptive backstepping was investigated for nonlinear systems. It was concluded that the advantages of backstepping approach lie in its flexibility, due to its recursive use of Lyapunov functions and its robustness against unmodeled dynamics of the systems. Due to the drawbacks of passive and active fault tolerant control approaches, some valuable research and practical results have been achieved in adaptive failure compensation based on the backstepping design method and without any need for fault detection. For example in [21-29], adaptive actuator failure compensation schemes were proposed for some classes of uncertain nonlinear systems based on the backstepping design method. The considered actuator failures in [21-26] were modeled as stuck at some unknown values and the considered failures in[27-29] were modeled to cover both loss of effectiveness and stuck at some unknown constant values. In [30], a neural network based failure compensator was proposed based on the backstepping design method for a class of nonlinear systems. In [31], an adaptive fuzzy controller was proposed for a class of nonlinear systems with unknown parameters and actuator failures in the presence of disturbances with known bounds. The proposed methods in[30, 31] proved the semi-global boundedness of the closed loop signals. In [32], an adaptive fuzzy failure compensation scheme was proposed for a class of uncertain stochastic nonlinear systems in strict feedback form with known control gains.The considered faults in [30-32] were modeled to cover both loss of effectiveness and constant stuck failures. The proposed fuzzy adaptive actuator failure compensator in [32] promised the semi-global boundedness of the closed loop signals. However, the tracking problem was not considered.

      In this paper, a robust adaptive control is proposed for a class of parametric-strict-feedback nonlinear systems with unknown parameters and in the presence of time varying actuator failures with unknown bounds. The proposed adaptive method can compensate a large class of actuator failures without any need for explicit fault detection. The considered faults are modeled to cover loss of effectiveness faults as well as stuck at some unknown time varying values. The values, times and patterns of the considered failures are unknown, i.e., during system operation it is completely unknown that at which times, how many of the actuators and which of them fail. Compared with the existing results, the main contributions of this paper are as follows: 1) The control problem is investigated for a class of nonlinear systems with unknown parameters subjected to unknown time varying actuator failures and disturbances with unknown bounds. 2) The considered actuator failure is more general than the previous literature [2-18, 21-32]. It covers both loss of effectiveness and unknown time varying stuck failures. 3) Appropriate Lyapunov-Krasovskii functionals are introduced to design new adaptive laws to compensate unknown time varying actuator failures as well as uncertainties from unknown parameters. 4) The proposed systematic backstepping design method proves that, with no need for explicit failure detection, not only are all the signals in the closed loop system globally bounded, but also the tracking error converges to a small neighborhood of the origin.

      The paper is organized as follows. In Section 2, the system description is given along with the necessary assumptions. In Section 3, the design and analysis of an adaptive actuator failure compensation scheme are explained. In Section 4, simulation examples are studied to illustrate the effectiveness of the proposed compensation approach. Finally, the paper is concluded in Section 5.

    • Consider a class of strict-feedback nonlinear multi input single output systems in the form of (1).

      \begin{align}\begin{split}\begin{array}{lll}\dot{x_i}(t)=x_{i+1}(t)+\theta_{f_i}^{\rm T}F_i(\bar{x}_i(t))+\delta_{f_i}(\bar{x}_i(t)), \\\qquad \qquad i=1, \cdots, n-1\\\dot{x_n}(t)=\varphi_0(x(t))+b^{\rm T}u(t)+\theta_{f_n}^{\rm T}F_n(x(t))+ \delta_{f_n}(x(t))\\y = x_1(t)\end{array}\end{split}\end{align}

      (1)

      where $ \bar{x}_i=[x_1, x_2, \cdots, x_i]^{\rm T} $, $x=[x_1, x_2, \cdots, x_n] $, $ u~\in~{{\bf R}^m} $, $ y \in {\bf R} $, are the state variables, system input and output, respectively. $F_i(\cdot) $ is a known smooth function vector, $ b $ and $\theta_{f_i}, i=1, \cdots, n $, are unknown constant parameter vectors. $ \delta_{f_i}(\cdot) $ is an unknown function that expresses the external disturbances or unknown nonlinearities and satisfies

      \begin{align} \vert \delta_{f_i}(\bar{x}_i(t))\leq c_{f_i}\phi_i(\bar{x}_i(t))\vert, i=1, \cdots, n\end{align}

      (2)

      where $ c_{f_i} $ is an unknown positive constant parameter and $\phi_i(\bar{x}_i(t)) $ is a known nonnegative smooth function. The control objective is to design an adaptive actuator failure compensator for plant (1) in order to assure that all the closed loop signals are globally bounded and the plant output $ y(t) $ tracks a desired signal $ y_d(t) $ despite the presence of unknown plant parameters and unknown time varying actuator failures. The block diagram of the proposed compensator is shown in Fig. 1.

      Figure 1.  The proposed failure compensation approach without explicit failure detection

      Assumptions 1-4 are considered.

      Assumption 1. The signs of nonzero elements $ b_j, j=1, 2, \cdots, m$ are known.

      Assumption 2. The desired signal $ y_d(t) $ and its first $n$-th order derivatives $ y_d^{(i)}(i=1, \cdots, n) $ are known bounded and piecewise continuous.

      Assumption 3. All the states are measurable. The stuck type actuator failure being considered is modeled as

      \begin{align}&u_j(t)=\bar{u}_j(t), \hspace{1.2 cm} t\geq t_j \notag \\&j=j_1, j_2, \cdots, j_p, \hspace{0.5 cm} 1\leq p\leq m-1 \notag \\&\bar{u}_j(t)=\bar{\bar{u}}_j+\bar{d}_j(t)+\Delta_j(t)\end{align}

      (3)

      where the constant value $ \bar{\bar{u}}_j $, the failure time instant $ t_j $ and the failure index $ j $ are unknown and $\bar{d}_j(t) $ satisfies

      \begin{align} \bar{d}_j(t)=\sum\limits_{l=1}^{h} \bar{d}_{jl}g_{jl}(t), j=j_1, j_2, \cdots, j_p, 1\leq p\leq m-1\end{align}

      (4)

      where the scalar constants $ \bar{d}_{jl} $ are unknown and the scalar bounded signals $ g_{jl}(t)$, $j=j_1, j_2, \cdots, j_p$, $l=1, \cdots, h, h\geq 1$ are known. In addition, $ \Delta_j(t) $ satisfies

      \begin{align}\begin{array}{lll}\vert \Delta_j(t)\vert \leq c_{\Delta_j} \phi_{j}(x(t)), j=j_1, j_2, \cdots, j_p, 1\leq p\leq m-1 \end{array}\end{align}

      (5)

      where $ c_{\Delta_j} $ is an unknown positive constant parameter and $ \phi_{j}(x(t)) $ is a known nonnegative smooth function. The loss of effectiveness model of actuator failure to be considered is modeled as

      \begin{align}u_i(t)=\rho_iv_i(t), \rho_i \in [\underline{\rho}_i, \bar{\rho}_i], 0<\underline{\rho}_i\leq1, \bar{\rho}_i=1\nonumber\\ i=\{1, \cdots, m\}\cap\overline{\{j_1, j_2, \cdots, j_p\}}\end{align}

      (6)

      where $ \rho_i $ and $ \underline{\rho}_i $ are unknown constant parameters. For plant (1) with actuator failures (3) -(6), the input vector can be expressed as

      \begin{align}&u(t)=\rho v(t)+\delta(\bar{u}-\rho v(t)) \notag \\&\bar{u}(t)=[\bar{u}_1(t), \bar{u}_2(t), \cdots, \bar{u}_m(t)]^{\rm T} \notag \\&v(t)=[v_1(t), v_2(t), \cdots, v_m(t)]^{\rm T} \notag \\&\delta={\rm{diag}}\{\delta_1, \delta_2, \cdots, \delta_m\}, \delta_i=\begin{cases} 1, \hspace{0.1 cm}u_i(t)=\bar{u}_i \notag \\ 0, \hspace{0.1 cm} u_i(t)\neq\bar{u}_i\end{cases}\\&\rho={\rm{diag}}\{\rho_1, \rho_2, \cdots, \rho_m\}\end{align}

      (7)

      where $ v(t) $ is the applied control input to be designed later.With this description, a general type of actuator failures including loss of effectiveness and stuck failures are considered. Loss of effectiveness faults can occur due to loss of a part of a control surface, engine malfunction or icing. Variant stuck failures can occur due to hydraulic failures that can produce unintended movements in the control surfaces of an aircraft.

      Remark 1. The failures that may occur in a practical industrial system may be time varying and their precise bound values are very difficult to obtain due to the complex environments. By considering this fact, this paper considered the unknown time varying actuator failures with unknown bounds. However, the previous literature considered unknown constant failures or time varying ones with known bounds [2-18, 21-32].

      Table 1 describes different failure situations of the considered failure model. For systems in which actuators may fail during the operation of the system, one common approach is to use actuator redundancy. In this way, when one actuator fails, some others could compensate the effect[22].

      Table 1.  Failure model

      Assumption 4 [21-32]. For plant (1) with known plant parameters and failure parameters, if any up to $ m-1 $ actuators stuck as (3-5), the others may lose effectiveness as (6), the closed loop system can still be driven to achieve a desired control objective. To develop a design procedure, Lemma 1 is introduced.

      Lemma 1 [32]. The following inequality holds for any $ \varepsilon>0$ and $\xi \in \bf{R} $

      $ 0\leq\vert\xi\vert -\xi{\rm{tanh}}(\dfrac{\xi}{\varepsilon}) \leq L\varepsilon$

      where $L$ is a constant that satisfies $L={\rm{e}}^{-(L+1) }$, i.e., $L=0.2785.$

    • In this section, an adaptive actuator failure compensation approach is developed for the multi input single output nonlinear system (1) by employing the backstepping design method and without explicit failure detection. The proposed design method contains stages. At each stage, an intermediate control function and updating laws are designed so that the considered system achieves the desired control objective even though some actuators fail. To design both the control laws and updating laws, the following state transformation is considered.

      $ z_1=x_1-y_d, \hspace{0.4 cm} z_i=x_i-\alpha_{i-1}, \hspace{0.4 cm} i=2, \cdots, n.$

      The transformed system in the new coordinates is obtained as

      \begin{align}&\dot{z}_1(t)=z_2(t)+\alpha_1(t)+\theta_{f_1}^{\rm T}F_1(x_1(t))+\delta_{f_1}(x_1(t))-\dot{y}_d(t)\nonumber\\&\dot{z}_i(t)=z_i+1(t)+\alpha_i(t)+\theta_{f_i}^{\rm T}F_i(\bar{x}_i(t))-\dot{\alpha}_{i-1}(t)+\nonumber\\&\qquad \quad ~~~~\delta_{f_1}(\bar{x}_i(t)), \; i=2, \cdots, n-1\nonumber\\&\dot{z}_n(t)=\varphi_0(x(t))-\dot{\alpha}_{n-1}+b^{\rm T}\delta\bar{u}+b^{\rm T}(I-\delta)\rho v+\nonumber\\&\qquad \quad ~~~\theta_{f_n}^{\rm T}F_n(x(t))+\delta_{f_n}(x(t)).\end{align}

      (8)

      As mentioned earlier, the backstepping design method for system(8) contains stages that will be presented in the following three steps.

      Step 1. In this step, the $ z_1 $ subsystem is considered and the controller will be designed for this subsystem. First, the Lyapunov functions are selected as

      \begin{align}&V_{z_i}=\frac{1}{2}z_i^2(t)\end{align}

      (9)

      \begin{align}&V_{\theta_i}=\frac{1}{2}\tilde{\theta}_i^{\rm T}\Gamma_i^{-1}\tilde{\theta_i}\end{align}

      (10)

      \begin{align}&V_{\theta^\prime_i}=\frac{1}{2}\gamma_i^{-1}\tilde{\theta}^{\prime^2}_i\end{align}

      (11)

      where $ \gamma_i $ is a positive constant, matrix $\Gamma_i=\Gamma_i^{\rm T}>0, \tilde{\theta}_i=\hat{\theta}_i-\theta_i$ and $ \tilde{\theta}^{\prime}_i=\hat{\theta}^{\prime}_i-\theta^\prime_i$, in which $ \hat{\theta}_i, \hat{\theta}^\prime_i $ are the estimates of $ \theta_i $ and $ \theta^{\prime}_i $ to be defined later. Therefore, the considered Lyapunov function for the $z_1 $ subsystem becomes

      \begin{align} V_1=V_{z_1}+V_{\theta_1}+V_{\theta^\prime_1}.\end{align}

      (12)

      Along system (8), the time derivative of $ V_{z_1} $ satisfies

      \begin{align}\begin{split}\begin{array}{lll}\dot{V}_{z_1}=&z_1(t)z_2(t)+z_1(t)\alpha_1(t)-z_1(t)\dot{y}_d(t)+\\&z_1(t)\theta_{f_1}^{\rm T}F_1(x_1(t))+z_1(t)\delta_{f_1}(x_1(t)).\\\end{array}\end{split}\end{align}

      (13)

      By using (2) and (10) -(12), the time derivative of $ V_1(t) $ becomes

      \begin{align}\dot{V}_1=&\dot{V}_{z_1}+\dot{V}_{\theta_1}+\dot{V}_{\theta^{\prime}_1}\leq z_1(t)z_2(t)+z_1(t)\alpha_1(t)+\nonumber\\&\vert z_1(t)\vert c_{f_1}\phi_1(x_1(t))-z_1(t)\dot{y_d}(t)+\widetilde{\theta}_1^{\rm T}\Gamma_1^{-1}\hat{\theta}_1+\nonumber\\&z_1(t)\theta_1^{\rm T}\varphi_1+\gamma_1^{-1}\tilde{\theta}^{\prime}_1\hat{\theta}^{\prime}_1\end{align}

      (14)

      where

      \begin{align} \theta_1=\theta_{f_1}, \varphi_1=F_1(x_1(t)).\end{align}

      (15)

      By applying Lemma 1, the time derivative of $ V_1(t) $ becomes

      \begin{align}\dot{V}_1&\leq z_1(t)z_2(t)+z_1(t)\alpha_1(t)+ z_1(t)\theta_1^{\rm T}\varphi_1+z_1(t)\theta^{\prime}_1\varphi^{\prime}_1+\nonumber\\&\qquad L\varepsilon_1\theta^{\prime}_1-z_1(t)\dot{y}_d(t)+\tilde{\theta}^{\rm T}_1\Gamma_1^{-1}\dot{\hat{\theta}}_1+\gamma_1^{-1}\tilde{\theta}^\prime_1\dot{\hat{\theta}}'_1\end{align}

      (16)

      where

      \begin{align} &\theta^\prime_1=c_{f_1} \nonumber \\&\varphi^{\prime}_1=\phi_1(x_1(t)){\rm{tanh}}\dfrac{z_1(t)\phi_1(x_1(t))}{\varepsilon_1}.\end{align}

      (17)

      Accordingly, the intermediate control input is selected as

      \begin{align}\alpha_1(x_1(t))=-\hat{\theta}_1^\prime\varphi^\prime_1-z_1-\frac{1}{2}\gamma_1z_1+\dot{y}_d\end{align}

      (18)

      where $ \gamma_1 $ is a positive constant. Therefore, the time derivative of $ V_1(t) $ becomes

      \begin{align}\dot{V}_1\leq &, \frac{1}{2} z_2^2-\frac{1}{2}\gamma_1z_1^2-z_1(t)\tilde{\theta}_1^2\varphi_1-z_1\tilde{\theta}_1^\prime\varphi_1^\prime+\nonumber\\&L\varepsilon_1\theta_1^\prime+\tilde{\theta}_1^{\rm T}\Gamma_1^{-1}\dot{\hat{\theta}}_1+\gamma_1^{-1}\tilde{\theta}_1^\prime\dot{\hat{\theta}}_1^\prime.\end{align}

      (19)

      Accordingly the updating laws for parameters $ \theta_1 $ and $\theta^\prime_1 $ are selected as

      \begin{align}&\hat{\theta}_1=\Gamma_1(z_1\varphi_1-\sigma_1\hat{\theta}_1) \nonumber\\&\hat{\theta}^\prime_1=\gamma_1(z_1\varphi^{\prime}_1-\sigma^\prime_1\hat{\theta}^\prime_1) \end{align}

      (20)

      where $ \sigma_1, \sigma_1'>0 $ are small positive constants. By using the inequalities $-\sigma_1\tilde{\theta}_1^{\rm T}\hat{\theta}_1\leq\frac{-1}{2}\sigma_1\parallel\tilde{\theta}_1\parallel^2+\frac{1}{2}\sigma_1\parallel\theta_1\parallel^2$, the time derivative of $ V_1(t) $ becomes

      \begin{align}&\dot{V}_1\leq-C_1V_1+\frac{1}{2}z_2^2+\mu_1 \nonumber\\&C_1={\rm{min}}\big (\gamma_1, \dfrac{\sigma_1}{λ_{\rm max}(\Gamma_1^{-1})}, \sigma_1^\prime\gamma_1 \big)\nonumber \\&\mu_1=\frac{1}{2}\sigma_1\parallel\theta_1\parallel^2+\dfrac{1}{2}\sigma_1^\prime\theta^{\prime^2}_1+L\varepsilon_1\theta^\prime_1.\end{align}

      (21)

      As can be seen from the above inequality, the time derivative of $V_1(t)$ is dependent on the boundedness of the signal $z_2$ to be regulated in the following.

      Step 2. Similar procedures are taken for each step when $i=2, \cdots, n-1$ as in step 1. The $z_i$ subsystem is considered as

      \begin{align}&\dot{z_i}(t)=z_{i+1}(t)+\alpha_i(t)+\theta_{f_i}^{\rm T}F_i(\bar{x}_i(t))+\nonumber\\&\qquad \quad \delta_{f_i}(\bar{x}_i(t))-\dot{\alpha}_{i-1}(t).\end{align}

      (22)

      For the $z_i$, $i=2, \cdots, n-1$, subsystems, the following Lyapunov function is considered.

      \begin{align} V_i=V_{z_i}+V_{\theta_i}+V_{\theta'_i}\end{align}

      (23)

      where $V_{z_i}$, $V_{\theta_i}$ and $V_{\theta'_i}$, are defined in (9) -(11).

      The intermediate controllers $\alpha_i(t)$, $i=1, \cdots, n$, are functions of $\bar{x}_i(t), {\theta}_1, \cdots, {\theta}_i, {\theta'}_1, \cdots, {\theta'}_i, y_d, {y_d}^{(1) }, {y_d}^{(i)}$, hence $\dot{\alpha}_{i-1}(t)$ becomes

      \begin{align}&\dot{\alpha}_{i-1}(t)\!=\!\displaystyle\sum\limits_{k=1}^{i-1} \frac{\partial \alpha_{i-1}}{\partial x_k} \{x_{k+1}(t)\!+\!\theta_{f_k}^{\rm T}F_k(\bar{x}_k(t))\!+\!\delta_{f_k}(\bar{x}_k(t))\}\!+\!\nonumber\\& \quad \displaystyle\sum\limits_{k=1}^{i-1} \Big [\frac{\partial\alpha_{i-1}}{\partial {\hat{\theta}}_k}\dot{\hat{\theta}}_k+ \frac{\partial \alpha_{i-1}}{\partial {\hat{\theta}}'_k}\dot{\hat{\theta}}'_k \Big]+\displaystyle\sum\limits_{k=1}^{i} \frac{\partial \alpha_{i-1}}{\partial y_d^{(k-1) }} y_d^{(k)}.\end{align}

      (24)

      By using (2), (23) and (24), the time derivative of $V_i(t)$ becomes

      \begin{align}\begin{split}\begin{array}{ll }\dot{V}_i=&\dot{V}_{z_i}+\dot{V}_{\theta_i}+\dot{V}_{\theta'_i}\leq \\& z_i(t)z_{i+1}(t)+z_i(t)\alpha_{i}(t)+z_i(t){\theta_i}^{\rm T} \varphi_i+\\& \vert z_i(t) \vert c_{f_i} \phi_i(\bar{x}_i(t))-\\&z_i(t) \displaystyle\sum\limits_{k=1}^{i-1} \bigg[\frac{\partial \alpha_{i-1}}{\partial x_k} x_{k+1}(t) + \frac{\partial \alpha_{i-1}}{\partial {\hat{\theta}}_k}\dot{\hat{\theta}}_k + \frac{\partial \alpha_{i-1}}{\partial {\hat{\theta}}'_k}\dot{\hat{\theta}}'_k \bigg]+\\& \vert z_i(t) \vert \displaystyle\sum\limits_{k=1}^{i-1} \big \vert \frac{\partial \alpha_{i-1}}{\partial x_k} \big \vert c_{f_k}\phi_k(\bar{x}_k(t)) -\\&z_i(t) \displaystyle\sum\limits_{k=1}^{i} \frac{\partial \alpha_{i-1}}{\partial y_d^{(k-1) }} y_d^{(k)}+\tilde{\theta}^{\rm T}_i\Gamma_i^{-1}\dot{\hat{\theta}}_i+\gamma_i^{-1}\tilde{\theta}'_i\dot{\hat{\theta}}'_i\\\end{array}\end{split}\end{align}

      (25)

      where $\theta_i$ and $\varphi_i$ are defined as

      \begin{align}&\theta_i=[\theta_{f_i}^{\rm T} \theta_{f_{i-1}}^{\rm T}, \cdots, \theta_{f_1}^{\rm T}]^{\rm T}\end{align}

      (26)

      \begin{align}&\varphi_i=\bigg[F_i^{\rm T}, -\frac{\partial \alpha_{i-1}}{\partial x_{i-1}}F_{i-1}^{\rm T}, \cdots, -\frac{\partial\alpha_{i-1}}{\partial x_{1}}F_{1}^{\rm T}\bigg]^{\rm T}.\end{align}

      (27)

      By applying Lemma 1, the time derivative of $V_i(t)$ becomes

      \begin{align}\begin{split}\begin{array}{ll }\dot{V}_i=&\dot{V}_{z_i}+\dot{V}_{\theta_i}+\dot{V}_{\theta'_i}\leq z_i(t)z_{i+1}(t)+\\&z_i(t)\alpha_{i}(t)+z_i(t){\theta_i}^{\rm T} \varphi_i+ z_i(t) \theta'_i \varphi'_i +\displaystyle\sum\limits_{k=1}^{i} L \varepsilon_k \theta'_i -\\&z_i(t) \displaystyle\sum\limits_{k=1}^{i-1} [\frac{\partial\alpha_{i-1}}{\partial x_k} x_{k+1}(t)+ \frac{\partial \alpha_{i-1}}{\partial {\hat{\theta}}_k}\dot{\hat{\theta}}_k + \frac{\partial \alpha_{i-1}}{\partial {\hat{\theta}}'_k}\dot{\hat{\theta}}'_k]-\\&z_i(t) \displaystyle\sum\limits_{k=1}^{i} \frac{\partial \alpha_{i-1}}{\partial y_d^{(k-1) }} y_d^{(k)}+\gamma_i^{-1}\tilde{\theta}'_i\dot{\hat{\theta}}'_i+{\tilde{\theta}_i}^{\rm T}\Gamma_i^{-1}\dot{\hat{\theta}}_i\\\end{array}\end{split}\end{align}

      (28)

      where $\theta'_i$ and $\varphi'_i$ are defined as

      \begin{align} \theta'_i={\rm{max}}\{ c_{f_{i}}, c_{f_{i-1}}, \cdots, c_{f_{1}} \}\qquad\qquad\qquad\qquad\;\;\end{align}

      (29)

      \begin{align}\begin{split}\begin{array}{ll}\varphi'_i=\phi_i {\rm{tanh}}\left(\dfrac{z_i(t) \phi_i(\bar{x}_i(t))}{\varepsilon_i}\right)+\\\qquad \displaystyle\sum\limits_{k=1}^{i-1} \dfrac{\partial\alpha_{i-1}}{\partial x_k}\phi_k(\bar{x}_k(t)){\rm{tanh}}\bigg(\dfrac{z_i(t) \dfrac{\partial \alpha_{i-1}}{\partial x_k}\phi_k(\bar{x}_k(t))}{\varepsilon_k} \bigg).\end{array}\end{split}\end{align}

      (30)

      Therefore, the updating laws and the intermediate controller are selected as

      \begin{align} \dot{\hat{\theta}}_i=\Gamma_i (z_i \varphi_i -\sigma_i\hat{\theta}_i), \hspace{1 cm} 1<i<n\qquad\qquad\quad\;\end{align}

      (31)

      \begin{align} \dot{\hat{\theta}}'_i=\gamma_i (z_i \varphi'_i-\sigma'_i \hat{\theta'}_i), \hspace{1 cm}1<i<n\qquad\qquad\quad\end{align}

      (32)

      \begin{align}\alpha_i(t)=&-\hat{\theta}^{\rm T}_i \varphi_i -\hat{\theta'}_i \varphi'_i -z_i-\frac{1}{2} \gamma_i z_i +\nonumber\\&\displaystyle\sum\limits_{k=1}^{i-1} \Bigg[\frac{\partial\alpha_{i-1}}{\partial x_k} x_{k+1}(t)+ \frac{\partial \alpha_{i-1}}{\partial {\hat{\theta}}_k}\dot{\hat{\theta}}_k + \frac{\partial \alpha_{i-1}}{\partial {\hat{\theta}}'_k}\dot{\hat{\theta}}'_k\Bigg]+\nonumber\\&\displaystyle\sum\limits_{k=1}^{i} \frac{\partial \alpha_{i-1}}{\partial y_d^{(k-1) }} y_d^{(k)}\end{align}

      (33)

      where $\gamma_i >0$ and $\sigma_i$, $\sigma'_i$ are small positive constants.

      By using $-\sigma_i \tilde{\theta}_i^{\rm T}\hat{\theta}_i \leq \frac{-1}{2}\sigma_i \Vert \tilde{\theta}_i\Vert^2 + \frac{1}{2}\sigma_i \Vert \theta_i \Vert^2$ and $-\sigma'_i \tilde{\theta}'_i \hat{\theta}'_i \leq\frac{-1}{2}\sigma'_i \tilde{\theta_i}'^2+ \frac{1}{2} \sigma'_i{\theta_i}'^2$, the time derivative of $V_i(t)$ becomes

      \begin{align}&\dot{V}_i \leq-C_i V_i(t)+\dfrac{1}{2}z_{i+1}^2+\mu_i \nonumber\\&C_i={\rm{min}}(\gamma_i, \frac{\sigma_i}{λ_{\rm max} (\Gamma_{i}^{-1})}, {\sigma_i}' \gamma_i) \nonumber\\&\mu_i=\dfrac{1}{2}\sigma_i \Vert \theta_i \Vert^2+\dfrac{1}{2}\sigma'_i {\theta_i}'^2 +\displaystyle\sum\limits_{k=1}^{i} L\varepsilon_k \theta'_i.\end{align}

      (34)

      It can be seen from the above inequality that the time derivative of $V_i(t)$ is dependent on the boundedness of signal $z_{i+1}$.

      Step 3. In the final step, the $z_n$ subsystem is considered as

      \begin{align}&\dot{z}_n(t)= \varphi_0(x(t))- \dot{\alpha}_{n-1} + b^{\rm T} \delta \bar{u} +b^{\rm T} (1-\delta)\rho v+ \nonumber\\&\qquad \quad \theta_{f_n}^{\rm T} F_n(x(t))+ \delta_{f_n}(x(t)).\end{align}

      (35)

      By assuming that the unknown actuator failures and unknown parameter $b$ are completely known, the structure of the ideal controller becomes as

      $ v_j =k_{1, j}(t) v_0 + k'_{1, j}(t), , j=1, 2, \cdots, m$

      where $v_0$ is the nominal control to be designed later, $k_{1, j}\in{\bf R}$ and $k'_{1, j}\in {\bf R}$ are some constant parameters which satisfy

      \begin{align}&[k_{1, 1}, k_{1, 2}, \cdots, k_{1, m}](1-\delta)\rho b=1\end{align}

      (36)

      \begin{align}& [k'_{1, 1}, k'_{1, 2}, \cdots, k'_{1, m}](1-\delta)\rho b=-b^{\rm T} \delta \bar{\bar{u}}\end{align}

      (37)

      with the knowledge of $b$ and actuator failures, $k_{1, j}$ and $k'_{1, j}$ can be achieved from the above equations. By considering Assumption 4, the above equations always have a solution. For unknown parameter $b$ and unknown actuator failures, the adaptive control input becomes as

      \begin{align} v_j =\hat{k}_{1, j}(t) v_0 + \hat{k'}_{1, j}(t), , j=1, 2, \cdots, m\end{align}

      (38)

      where $\hat{k}_{1, j}$ and $\hat{k'}_{1, j}$ are the estimates of $k_{1, j}$ and $k'_{1, j}$. For stability analysis of $z_n$ subsystem, the following Lyapunov functions are considered.

      \begin{align}&V_{\theta"} =\tilde{\theta"}^{\rm T} \Gamma"^{-1}\tilde{\theta"}\end{align}

      (39)

      \begin{align}&V_{\bar{u}}=\displaystyle\sum\limits_{j=1}^{m}(1-\delta_j) \frac{\vert b_j \vert}{2} \rho_j \tilde{k}_j^{\rm T}(t) \bar{\Gamma}_j^{-1}\tilde{k}_j\end{align}

      (40)

      \begin{align}&V_{n}=V_{z_n}+V_{\theta_n}+V_{\theta'_n}+V_{\theta"}+V_{\bar{u}}\end{align}

      (41)

      where $V_{z_n}$, $V_{\theta_n}$ and $V_{\theta'_n}$ are defined in(9) -(11), $\tilde{k}_j=\hat{k}_j-k_j$, $\tilde{\theta"}=\hat{\theta"}-\theta"$ in which $\hat{k}_j$ and $\hat{\theta"}$ are the estimates of $k_j=[k_{1, j}, k'_{1, j}]^{\rm T}$ and $\theta"$, matrices $\Gamma" = \Gamma"^{\rm T} >0$ and $\bar{\Gamma}_j= \bar{\Gamma}_j^{\rm T}> 0$.

      By using(3) -(6), the time derivative of $V_{z_n}$ becomes

      \begin{align}\begin{split}\begin{array}{ll }\dot{V}_{z_n}=&z_n(t) \Big[\varphi_0 (x(t))-\dot{\alpha}_{n-1}+b^{\rm T} \delta \bar{\bar{u}}+\\&\theta_{f_n}^{\rm T} F_n(x(t))+\delta_{f_n}(x)+\displaystyle\sum\limits_{j=1}^{m} \{ b_j \delta_j \displaystyle\sum\limits_{l=1}^{h}\bar{d}_{jl}g_{jl}(t)+\\&b_j \delta_j \Delta_j(t) + b_j (1-\delta_j) \rho_j(\hat{k}_{1, j}(t)v_0 +\hat{k'}_{1, j}(t)) \}\Big].\end{array}\end{split}\end{align}

      (42)

      By using (2), (5), (24) and applying lemma 1, the time derivative of $V_n(t)$ becomes

      \begin{align}\dot{V}_{n}=&\dot{V}_{z_n}+\dot{V}_{\theta_n}+\dot{V}_{\theta'_n}+\dot{V}_{\theta"}+\dot{V}_{\bar{u}}\leq z_n \varphi_o(x) + z_n b^{\rm T} \delta \bar{\bar{u}} +\nonumber\\&z_n \displaystyle\sum\limits_{j=1}^{m} [b_j (1-\delta_j) \rho_j(\hat{k}_{1, j}v_0+ \hat{k'}_{1, j})] +z_n {\theta_n}^{\rm T} \varphi_n +\nonumber\\&z_n \theta'^{\rm T} \varphi' +\displaystyle\sum\limits_{k=1}^{n} L\varepsilon_k \theta'_n + \vert z_n\vert \theta"^{\rm T} \varphi"-\nonumber\\&z_n \displaystyle\sum\limits_{k=1}^{n-1} \Bigg[\frac{\partial \alpha_{n-1}}{\partial x_k} x_{k+1}+\frac{\partial \alpha_{n-1}}{\partial {\hat{\theta}}_k}\dot{\hat{\theta}}_k + \frac{\partial \alpha_{n-1}}{\partial {\hat{\theta}}'_k}\dot{\hat{\theta}}'_k \Bigg]-\nonumber\\&z_n \displaystyle\sum\limits_{k=1}^{n} \frac{\partial\alpha_{n-1}}{\partial y_d^{(k-1) }} y_d^{(k)}+\tilde{\theta}_n^{\rm T} \Gamma_n^{-1} \dot{\hat{\theta}}_n+ \gamma_n^{-1} \tilde{\theta'}_n \dot{\hat{\theta'}}_n+\nonumber\\&\tilde{\theta"}^{\rm T}\Gamma"^{-1}\dot{\hat{\theta}}"+\displaystyle\sum\limits_{j=1}^{m}(1-\delta_j)\vert b_j \vert \rho_j \tilde{k}_j^{\rm T} \bar{\Gamma}_j^{-1}\dot{\hat{k}}_j\end{align}

      (43)

      and $\theta_n$ and $\varphi_n$ are defined as

      \begin{align}&\theta_n=[\theta_{fn}^{\rm T}, \theta_{f(n-1) }^{\rm T}, \cdots, \theta_{f1}^{\rm T}, \Theta_{1}^{\rm T} \vert b_1 \vert \delta_1, \Theta_{2}^{\rm T} \vert b_2 \vert \delta_2, \cdots, \nonumber\\& \qquad \Theta_{m}^{\rm T} \vert b_m \vert \delta_m]^{\rm T}\end{align}

      (44)

      \begin{align}&\varphi_n=\Bigg[F_{n}^{\rm T}, -\frac{\partial \alpha_{n-1}}{\partial x_{n-1}} F_{n-1}^{\rm T}, \cdots, -\frac{\partial \alpha_{n-1}}{\partial x_{1}} F_{1}^{\rm T}, {\rm{sgn}}(b_1) G_1^{\rm T}, \nonumber\\&\qquad {\rm{sgn}}(b_2) G_2^{\rm T}, \cdots, {\rm{sgn}}(b_m)G_m^{\rm T} \Bigg]^{\rm T}\end{align}

      (45)

      where $\Theta_j=[\bar{d}_{j1}, \bar{d}_{j2}, \cdots, \bar{d}_{jh}]^{\rm T}$ and $G_j=[g_{j1}, g_{j2}, \cdots, $$g_{jh}]^{\rm T}$, $j=1, \cdots, m$, $h\geq 1$. In addition, $\theta'_n$ and $\varphi'_n$ are defined as

      \begin{align} \theta'_n={\rm{max}}\{c_{f_{n}}, c_{f_{n-1}}, \cdots, c_{f_{1}} \}\qquad\qquad\qquad\qquad\;\end{align}

      (46)

      \begin{align}\begin{split}\begin{array}{l }\varphi'_n=\phi_n {\rm{tanh}}\left(\dfrac{z_n(t) \phi_n(x(t))}{\varepsilon_n}\right)+\\\qquad \displaystyle\sum\limits_{k=1}^{n-1} \dfrac{\partial\alpha_{n-1}}{\partial x_k}\phi_k(\bar{x}_k(t)){\rm{tanh}}\left(\dfrac{z_n(t)\frac{\partial \alpha_{n-1}}{\partial x_k}\phi_k(\bar{x}_k(t))}{\varepsilon_k}\right)\end{array}\end{split}\end{align}

      (47)

      and $\theta"$ and $\varphi"$ are defined as

      \begin{align}& \theta"= [\vert b_1 \vert \delta_ 1 c_{\Delta_1}, \vert b_2\vert \delta_ 2 c_{\Delta_2}, \cdots, \vert b_m \vert \delta_ m c_{\Delta_m}]^{\rm T}\end{align}

      (48)

      \begin{align}& \varphi"=[\phi_1 (x(t)), \phi_2 (x(t)), \cdots, \phi_m(x(t))]^{\rm T}.\end{align}

      (49)

      Therefore,

      \begin{align}&\dot{\hat{\theta}}_n= \Gamma_n (z_n \varphi_n -\sigma_n \hat{\theta}_n) \end{align}

      (50)

      \begin{align}&\dot{\hat{\theta}}'_n= \gamma_n (z_n \varphi'_n - \sigma'_n\hat{\theta}'_n)\end{align}

      (51)

      \begin{align}&\dot{\hat{\theta}}"= \Gamma"(z_n {\rm{sgn}} (z_n)\varphi" -\sigma" \hat{\theta}")\end{align}

      (52)

      \begin{align}&{\hat{k}}_j= \bar{\Gamma}_j (-{\rm{sgn}} (b_j) z_n [v_0 ~~ 1]^{\rm T} -\bar{\sigma}_j \hat{k}_j), j = 1, \cdots, m\end{align}

      (53)

      \begin{align}v_0=&-\varphi_0(x)-\frac{\gamma_n}{2}z_n -z_n - \hat{\theta}_n^{\rm T} \varphi_n -\hat{\theta}'_n \varphi'_n-\nonumber\\&\hat{\theta}"^{\rm T} \varphi" {\rm{sgn}}(z_n)+\displaystyle\sum\limits_{k=1}^{n-1} \Bigg[\frac{\partial \alpha_{n-1}}{\partial x_k} x_{k+1}(t)+\nonumber\\& \frac{\partial \alpha_{n-1}}{\partial{\hat{\theta}}_k}\dot{\hat{\theta}}_k + \frac{\partial\alpha_{n-1}}{\partial {\hat{\theta}}'_k}\dot{\hat{\theta}}'_k\Bigg]+ \displaystyle\sum\limits_{k=1}^{n} \frac{\partial\alpha_{n-1}}{\partial y_d^{(k-1) }} y_d^{(k)}\end{align}

      (54)

      where $\gamma_n >0$, $\sigma_n$, $\sigma'_n$, $\sigma"$ and $\bar{\sigma}_j$, $j=1, \cdots, m$, are small positive constants. By using $-\sigma_n \tilde{\theta}_n^{\rm T} \hat{\theta}_n \leq-\frac{1}{2} \sigma_n \Vert \tilde{\theta}_n \Vert^2 + \frac{1}{2}\sigma_n \Vert {\theta}_n \Vert^2 $, $-\sigma'_n \tilde{\theta'}_n\hat{\theta'}_n \leq -\frac{1}{2} \sigma'_n \tilde{\theta'}_n^2+\frac{1}{2} \sigma'_n {\theta'}_n^2$ and $-\sigma"\tilde{\theta"}^{\rm T} \hat{\theta"} \leq -\frac{1}{2} \sigma"\Vert \tilde{\theta"} \Vert^2 + \frac{1}{2} \sigma" \Vert{\theta"} \Vert^2$, the time derivative of $V_n(t)$ becomes as

      \begin{align*}&\dot{V}_n \leq -C_n V_n + \mu_n\\&C_n = {\rm{min}}\left(\gamma_n, \frac{{\sigma_n}}{λ_{\rm max}({\Gamma}_n^{-1})}, \sigma'_n \gamma_n, \frac{{\sigma"}}{λ_{\rm max}({\Gamma"}^{-1})}, C'\right)\\&C' = {\rm{min}}\left(\frac{\bar{\sigma}_1}{λ_{\rm max}(\bar{\Gamma}_1^{-1})}, \cdots, \frac{\bar{\sigma}_m}{λ_{\rm max}(\bar{\Gamma}_m^{-1})}\right)\end{align*}

      \begin{align}\begin{split}\begin{array}{ll }\quad \mu_n=&\dfrac{1}{2} \sigma_n \Vert \theta_n \Vert^2 + \dfrac{1}{2} \sigma'_n {\theta'_n}^2 + \dfrac{1}{2} \sigma" \Vert \theta" \Vert^2 + \displaystyle\sum\limits_{k=1}^{n} L \varepsilon_k \theta'_n+\\&\dfrac{1}{2} \displaystyle\sum\limits_{j=1}^{m} (1- \delta_j)\vert b_j\vert \rho_j \bar{\sigma_j} \Vert k_j \Vert^2.\end{array}\nonumber\end{split}\end{align}

      The result shows that $V_n(t)$ is bounded. So far, adaptive compensation controller design has been completed. Now the main result is summarized by Theorem 1.

      Theorem 1. Consider the closed loop system (1). Under Assumptions 1-4, the controller structure (38) with the parameter updating laws (31), (32) and(50) -(53), assure that all the closed loop signals are globally bounded and the signal $z(t)=[z_1, z_2, \cdots, z_n]$ converges to the following compact set:

      \begin{align}\begin{split} A_z=\left\{ z \mid \Vert z \Vert \leq \sqrt{\frac{2\mu}{C}}\right\}\end{split}\end{align}

      (55)

      where $C={\rm{min}}\{C_1, C_2, \cdots, C_n \}$ and $\mu=\displaystyle\sum_{i=1}^{n} \mu_i$.

      Proof. The following Lyapunov function is considered:

      \begin{align}\begin{split}V(t)=\displaystyle\sum\limits_{i=1}^{n} V_i \nonumber\end{split}\end{align}

      where $V_i (t)$ for $i=1, \cdots, n$ are defined in (12), (23) and(41). Therefore, the time derivative of $V(t)$ becomes

      \begin{align}&\dot{V}(t) \leq -CV(t) +\mu \nonumber\\&V(t) \leq \left[V(0)-\frac{\mu}{C}\right]{\rm e}^{-Ct}+ \frac{\mu}{C}\nonumber\\&\Vert z \Vert \leq \sqrt{2 [(V(0)-\frac{\mu}{C}){\rm e}^{-Ct}+\frac{\mu}{C}]}. \nonumber\end{align}

      Thus, it is concluded that $V(t)$ is bounded. Accordingly, all the closed loop signals are bounded and the signal $z(t)=[z_1, z_2, \cdots, z_n]$ converges to the compact set defined in(55).

      To summarize and to clarify more, the proposed design method is shown through Algorithm 1.

      In addition, block diagram of the proposed approach is given in Fig. 2.

      Figure 2.  Block diagram of the proposed robust adaptive compensator

      Algorithm 1. Adaptive failure compensation design

      Begin

      for $i= 1$ do

          Transform the state $ z_i=x_i-y_d $

          Determine the intermediate controller $ \alpha_i $ and the updating laws by (18), (20)

      end for

      for $i= 1$ to $ n-1 $ do

          Transform the state $ z_i=x_i-\alpha_{i-1} $

          Determine the intermediate controller $ \alpha_i $ and the updating laws by (31), (33)

      for $i= n$ do

          Transform the state $ z_i=x_i-\alpha_{i-1} $

          Determine the intermediate controller $ v_o $ and the updating laws by (50), (54)

      end for

      for $j= 1$ to $ m $ do

          Determine the adaptive control input by using (38)

          as $ v_j=\hat{k}_{1, j}(t)v_o+\hat{k}^\prime_{1, j}(t) $

      end for

      End

    • In this section, the obtained results are simulated to verify the effectiveness of the proposed method. For this purpose, two simulation examples are considered. In the first example, the actuator failure compensation scheme is investigated for a two-axis positioning stage system, which is driven by two linear motors. In the second example, an aircraft wing motion is considered in the presence of actuator failures and disturbances.

      Example 1. Two-axis positioning stage

      The dynamics of a two axis positioning stage system in the presence of disturbances is described as [25].

      \begin{align}&\dot{x}_1(t)=x_2(t)+\theta_{f_1}^{\rm T} F_1(t)+ \delta_1 (t, x(t))) \nonumber\\&\dot{x}_2(t)= b^{\rm T}u(t)+ \theta_{f_2}^{\rm T} F_2(t)+ \delta_2(t, x(t)) \nonumber\\&y(t)=x_1(t)\end{align}

      (56)

      where $x_1$ and $x_2$ are the states, $y$ represents the position of the inertia load of the motor, $u(t)$ is the voltage signal to the driving motor, $P$ is the motor magnets pitch and $P=60{\rm mm}$, $\theta_{f_1}\in {\bf R}$ is an unknown constant parameter, $ b \in{\bf R}^3 $ and $ \theta_{f_2}\in {\bf R}^4 $ are unknown constant vectors, $\delta_i (\cdot)$, $i=1, 2$, express the uncertain disturbances, $F_1 (t)=-x_1(t)$ and $F_2(t)=[{\rm{cos}}(\frac{2\pi y}{P}), {\rm{sin}}(\frac{2\pi y}{P}), {\rm{cos}}(\frac{6\pi y}{P}), {\rm{sin}}(\frac{6\pi y}{P})]^{\rm T}$. For simulation purpose, $\theta_{f_1}=2$, $\theta_{f_2}=[-1, 1, -1, 1]^{\rm T}$ and $b=[2,3,2]^{\rm T}$. The control objective is to track the desired signal $y_d (t)=0.1 {\rm{sin}}(0.5t)$ under both loss of effectiveness and stuck failures in actuators.

      The failure model considered in this simulation is

      \begin{align}& u_1(t), = \left\{ \begin{array}{lr} v_1(t),& t< 40\\ 0.04{\rm{cos}}(t) + 2 +\Delta_1(t),&t\geq 40 \end{array}\right. \nonumber \\& u_2(t)= \left\{ \begin{array}{lc} v_2(t),& t< 20\\ 0.04{\rm{sin}}(t) + 1.5+\Delta_2(t),&t\geq 20 \end{array}\right. \nonumber \\& u_3(t)= \left\{ \begin{array}{lc} v_3(t),& t< 60\\ 0.2 v_3(t),&t\geq 60 \end{array} \right.\end{align}

      (57)

      where $ \vert \Delta_1(t) \vert \leq c_{\Delta_1} ({\rm e}^{x_2(t)}+1) $ and $ \vert \Delta_2(t) \vert \leq c_{\Delta_2}(0.5 {\rm e}^{x_1(t)x_2(t)}+0.5) $. The uncertain disturbancesare the following functions:

      \begin{align}&\delta_1(t, x(t))=0.5 {v}_1(t)x_1^2(t) \nonumber\\&\delta_2(t, x(t))={v}_2(t)(0.1x_2^2(t)+1) \end{align}

      (58)

      where $\vert {v}_i(t) \vert \leq \bar{{ v}}_i$, in which $\bar{{v}}_i$, $i=1, 2$, are unknown positive constants.

      The following design parameters are adopted in the simulation:

      $ (0)={{[0.1,-0.1]}^{\text{T}}} \\ {{\gamma }_{1}}={{\gamma }_{2}}=30 \\ {{\Gamma }_{1}}={{{{\Gamma }'}}_{1}}={{\Gamma }_{2}}={\Gamma }''=10I \\ {{{\bar{\Gamma }}}_{1}}={{{{\Gamma }'}}_{2}}=I,{{{\bar{\Gamma }}}_{2}}=8I,{{{\bar{\Gamma }}}_{3}}=3I \\ {{\sigma }_{1}}={{{{\sigma }'}}_{1}}=0.001 \\ {{\sigma }_{2}}={{{{\sigma }'}}_{2}}={\sigma }''={{{\bar{\sigma }}}_{1}}={{{\bar{\sigma }}}_{2}}={{{\bar{\sigma }}}_{3}}=0.005 \\ {{\theta }_{1}}(0)=1,{{{{\theta }'}}_{1}}(0)=0.2,{{\theta }_{2}}(0)={{{{\theta }'}}_{2}}(0)={\theta }''(0)=0 \\ {{k}_{1}}(0)={{[0.5,0.5]}^{\text{T}}},{{k}_{2}}(0)={{[-0.5,-0.5]}^{\text{T}}} \\ {{k}_{3}}(0)={{[0.2,0.3]}^{\text{T}}}. \\ $

      The simulation results are shown in Figs. 3-5. In all of the Figures, "*" denotes the occurrence time of the actuator failure. It can be seen from Fig. 3 that the output tracking is ensured even though there are actuator failures with unknown time instants, values and patterns during an operation.

      Figure 3.  Tracking performance

      Figure 4.  Control inputs

      Figure 5.  Parameter estimates

      At the time instant when one failure occurs, there is a transient response in the tracking error that disappears as time goes on.Figs. 4 and 5 show the boundedness of the control inputs and the estimates of the parameters in the control loop. As can be seen, the unknown parameters of the proposed controller (38) and the considered system (57) are updated so that the system achieves the desired control objective even though some actuators fail.

      Example 2. Aircraft wing

      The motion of the aircraft wing in the presence of disturbances is described as Section 10.1.3 of [22].

      \begin{align}&\dot{x}_1(t)=x_2(t)+\delta_1 (t, x_1(t)) \nonumber\\&\dot{x}_2(t)=x_3(t)+\theta_{f_2}^{\rm T}F_2(t)+\delta_2 (t, \bar{x}_2(t))\nonumber\\&\dot{x}_3(t)=\dfrac{1}{\tau} b^{\rm T} u(t)- \dfrac{1}{\tau} x_3(t)+\delta_3 (t, x(t))\nonumber\\&y(t)=x_1(t)\end{align}

      (59)

      where the states $x_1$, $x_2$ and $x_3$ represent the roll angle, roll rate and aileron deflection angle, respectively, $u(t)\in {\bf R}^2$ is the control input and $\tau \in {\bf R}$ is the aileron time constant which is unknown, $b \in {\bf R}^2$ and $\theta_{f_2}\in {\bf R}^5$ are unknown constant vectors and $ F_2(t)=[1, x_1, x_2, \vert x_1 \vert x_2, \vert x_2 \vert x_2]^{\rm T}$.The functions $\delta_i({\cdot})$, $i=1, 2, 3$, express the uncertain disturbances. The control objective is to track the desired signal $y_d (t)=0$ under disturbances and actuator failures.

      For simulation purpose, $\tau= \frac{1}{15}$, $b=[0.06, 0.08] ^{\rm T}$ and $\theta_{f_2}=[0, -2.667, 0.86485, -2.9225, 0]^{\rm T}$.

      The considered failure model in this simulation is

      \begin{align}& u_1(t)= \left\{ \begin{array}{ll} v_1(t) & t< 50\\ 0.5{\rm{cos}}(t) +1.5+ \Delta_1(t),&t\geq 50 \end{array}\right. \nonumber\\&u_2(t)=v_2(t)\end{align}

      (60)

      where $ \vert \Delta_1(t) \vert \leq c_{\Delta_1} ({\rm e}^{x_1 (t)}+1) $. The uncertain disturbances are the following functions.

      \begin{align}&\delta_1(t, x_1(t))=0.5 { v}_1(t)x_1^2(t) \nonumber\\&\delta_2(t, \bar{x}_2(t))=0.5{ v}_2(t)(0.1x_2^2(t)+0.01) \nonumber\\&\delta_3(t, x_1(t))=0.5 { v}_3(t)x_3^2(t) {\rm e}^{0.01x_3}\end{align}

      (61)

      where $\vert { v}_i(t) \vert \leq \bar{{ v}}_i$, in which $\bar{{v}}_i$ is an unknown positive constant.

      For controller design, $F_2$ should be smooth thus based on Lemma 1, $F_2$ can be approximated by

      $\begin{split}\begin{array}{l}F_2(t)=\left[1, x_1, x_2, x_1x_2{\rm{tanh}}(\dfrac{x_1}{0.1}), x_2^2{\rm{tanh}}(\dfrac{x_2}{0.1})\right]^{\rm T}. \nonumber\end{array}\end{split}$

      The following design parameters are adopted in the simulation.

      $\begin{array}{ll }x(0) =[0.1, -0.1, 0]^{\rm T}\\\gamma_1=25, \gamma_2=2, \gamma_3=10\\\Gamma_1^\prime=\Gamma_2=\Gamma_2^\prime=I, \Gamma_3=\Gamma_3^\prime=\Gamma^{\prime\prime}=0.1I\\\sigma_1^\prime=\sigma_2^\prime=\sigma_2=\sigma_3=\sigma^\prime_3=\sigma^{\prime\prime}=\bar{\sigma}_1=\bar{\sigma}_2=0.005\\\theta^{\prime}_1(0) =1, \theta_2(0) =[0.2, 0.2, 0.2, 0.2, 0.2]\\\theta^\prime_2(0) =0.2\\\theta_3(0) =[0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5]^{\rm T}\\\theta^\prime_3(0) =[0.5], \theta^{\prime\prime}(0) =[1, 1]^{\rm T}\\k_1(0) =[0.5, 0.5]^{\rm T}, k_2(0) =[0.1, 0.1]^{\rm T}\\\Gamma_{k_1}=0.5I, \Gamma_{k_2}=0.1I.\end{array}\nonumber$

      The simulation results are shown in Figs. 6 and 7. It can be seen from Figs. 6(a) that the output tracking is ensured even though, during an operation, there is an actuator failure with unknown occurrence time, pattern and value. At the time instant when a failure occurs, there is a transient response in the output that disappears as time goes on. Figs. 6 and 7 represent the boundedness of the control inputs and the estimates of the parameters in the control loop. As can be seen, the unknown parameters of the proposed controller (38) and the considered system (59) are updated so that the system achieves the desired control objective even though an actuator fails.

      Figure 6.  System performance

      Figure 7.  Parameter estimates

      Because of the existence of the sgn ($ \cdot $) function in the control input (54), the control input may change rapidly which is not practically possible, therefore in the simulations, the ${\rm{sgn}(\cdot)} $ function is replaced by the $ {\rm{sat}(\cdot)}$ function. It can be seen from the results, that the designed controller compensates the unknown time varying actuator failures(3) -(6) without any need for explicit fault detection. The unknown parameters of the proposed controller (38) and system (1) are updated so that the system achieves the desired control objective even though some actuators fail. The above simulation results demonstrate the merits of the proposed adaptive failure compensation design method.

    • In this paper, a robust adaptive actuator failure compensation approach is proposed for a class of parametric-strict-feedback nonlinear systems. The considered actuator faults are modeled to cover both loss of effectiveness and stuck at some unknown time varying values. The offered controller compensates the time varying actuator faults without any need for explicit fault detection. The uncertainties from unknown parameters and unknown time varying actuator failures have been compensated by using appropriate Lyapunov-Krasovskii functionals. The proposed systematic backstepping design method can guarantee global boundedness of all the closed loop signals in addition to the convergence of the system output to a small neighborhood of the desired signal. Simulations have been conducted to verify the effectiveness of the proposed failure compensation method.

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