Adaptive Fuzzy Sliding Mode Control of Under-actuated Nonlinear Systems

Amir Hossein Davaie Markazi Mohammad Maadani Seyed Hassan Zabihifar Nafiseh Doost-Mohammadi

Amir Hossein Davaie Markazi, Mohammad Maadani, Seyed Hassan Zabihifar and Nafiseh Doost-Mohammadi. Adaptive Fuzzy Sliding Mode Control of Under-actuated Nonlinear Systems. International Journal of Automation and Computing, vol. 15, no. 3, pp. 364-376, 2018 doi:  10.1007/s11633-017-1108-5
Citation: Amir Hossein Davaie Markazi, Mohammad Maadani, Seyed Hassan Zabihifar and Nafiseh Doost-Mohammadi. Adaptive Fuzzy Sliding Mode Control of Under-actuated Nonlinear Systems. International Journal of Automation and Computing, vol. 15, no. 3, pp. 364-376, 2018 doi:  10.1007/s11633-017-1108-5

doi: 10.1007/s11633-017-1108-5

Adaptive Fuzzy Sliding Mode Control of Under-actuated Nonlinear Systems

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    Author Bio:

    Amir Hossein Davaie Markazi received the Ph.D. degree in mechanical engineering from Mechanical Engineering Department, McGill university, Canada in 1995. He is a professor at Iran University of Since and Technology Iran. He is the Editor-in-Chief of International Journal of Engineering Science. His research interests include adaptive fuzzy siding mode control (AFSMC), control of chaotic systems, control of parallel robots, networked control systems and wavelet pattern recognition. E-mail: markazi@iust.ac.ir ORCID iD: 0000-0003-4077-2614

    Mohammad Maadani received the B. Sc. degree in mechanical engineering from Amirkabir University of Technology (Tehran Polytechnic), Iran in 2009, the M. Sc. degree in mechanical engineering from Iran University of Science and Technology, Iran in 2011. His research interests include adaptive control, nonlinear dynamics and control. E-mail: mohammad.maadani1@gmail.com

    Seyed Hassan Zabihifar received the B. Sc. and M. Sc. degrees in mechanical engineering and mechatronic respectively from Iran University of Since and Technology, Iran in 2012. He is currently a Ph.D. degree candidate in mechatronic and robotic systems in Bauman Moscow State Technical University (BMSTU), Russia. His research interests include nonlinear control, adaptive fuzzy sliding mode, adaptive neural network, and optimal control. Email: zabihifar@student.bmstu.ru (Corresponding author) ORCID iD: 0000-0002-4808-478X

    Nafiseh Doost-Mohammadi received the M. Sc. degree in mechanical engineering from Iran University of Since and Technology, Iran in 2011. Her research interests include nonlinear control, adaptive fuzzy sliding mode and fuzzy control systems. E-mail: nafise_doostmohamadi@yahoo.com

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出版历程
  • 收稿日期:  2017-05-28
  • 录用日期:  2017-10-25
  • 网络出版日期:  2018-01-09
  • 刊出日期:  2018-06-01

Adaptive Fuzzy Sliding Mode Control of Under-actuated Nonlinear Systems

doi: 10.1007/s11633-017-1108-5
    作者简介:

    Amir Hossein Davaie Markazi received the Ph.D. degree in mechanical engineering from Mechanical Engineering Department, McGill university, Canada in 1995. He is a professor at Iran University of Since and Technology Iran. He is the Editor-in-Chief of International Journal of Engineering Science. His research interests include adaptive fuzzy siding mode control (AFSMC), control of chaotic systems, control of parallel robots, networked control systems and wavelet pattern recognition. E-mail: markazi@iust.ac.ir ORCID iD: 0000-0003-4077-2614

    Mohammad Maadani received the B. Sc. degree in mechanical engineering from Amirkabir University of Technology (Tehran Polytechnic), Iran in 2009, the M. Sc. degree in mechanical engineering from Iran University of Science and Technology, Iran in 2011. His research interests include adaptive control, nonlinear dynamics and control. E-mail: mohammad.maadani1@gmail.com

    Seyed Hassan Zabihifar received the B. Sc. and M. Sc. degrees in mechanical engineering and mechatronic respectively from Iran University of Since and Technology, Iran in 2012. He is currently a Ph.D. degree candidate in mechatronic and robotic systems in Bauman Moscow State Technical University (BMSTU), Russia. His research interests include nonlinear control, adaptive fuzzy sliding mode, adaptive neural network, and optimal control. Email: zabihifar@student.bmstu.ru (Corresponding author) ORCID iD: 0000-0002-4808-478X

    Nafiseh Doost-Mohammadi received the M. Sc. degree in mechanical engineering from Iran University of Since and Technology, Iran in 2011. Her research interests include nonlinear control, adaptive fuzzy sliding mode and fuzzy control systems. E-mail: nafise_doostmohamadi@yahoo.com

English Abstract

Amir Hossein Davaie Markazi, Mohammad Maadani, Seyed Hassan Zabihifar and Nafiseh Doost-Mohammadi. Adaptive Fuzzy Sliding Mode Control of Under-actuated Nonlinear Systems. International Journal of Automation and Computing, vol. 15, no. 3, pp. 364-376, 2018 doi:  10.1007/s11633-017-1108-5
Citation: Amir Hossein Davaie Markazi, Mohammad Maadani, Seyed Hassan Zabihifar and Nafiseh Doost-Mohammadi. Adaptive Fuzzy Sliding Mode Control of Under-actuated Nonlinear Systems. International Journal of Automation and Computing, vol. 15, no. 3, pp. 364-376, 2018 doi:  10.1007/s11633-017-1108-5
    • Many of the existing methods for control of nonlinear systems need a rather accurate model of the plant. In recent decades, many control methods for less-known nonlinear systems have been developed[1, 2]. The well-known, sliding mode control (SMC) which is based on the theory of variable structure systems is a powerful method for control of uncertain nonlinear systems[3, 4]. In order to preserve the closed-loop stability of uncertain systems, conventional SMC methodology may potentially suffer from chattering in the control input signal, when the estimated bound of uncertainty is not small enough. Such a high-frequency chattering may damage the actuators and also excite the un-modeled high-frequency dynamics of the system which degrades the control performance and may even lead to instability[5].

      Fuzzy logic control (FLC) methodology has been widely considered as another alternative for coping with nonlinearities and unknown dynamics and external disturbances[6, 7]. Conventional FLC methods suffer from lack of systematic methods for incorporation of human knowledge into the rule base of a fuzzy inference system, for guaranteeing the closed-loop stability[8, 9].

      In order to exploit the best of the SMC and FLC methods, different combinations of those methods have been proposed in the literature[10, 11]. The objective of this new class of methods is to cope with uncertainty and external disturbances, and at the same time prevent the chattering phenomena, as much as possible.

      One such hybrid control approach is the so called, adaptive fuzzy sliding mode control (AFSMC). AFSMC is, in particular, suitable for systems with a rather large bound of uncertainty[12, 13]. In this approach, fuzzy control rules can be determined systematically and the asymptotic stability of the closed loop system can be guaranteed under certain conditions[14].

      In [15], an indirect AFSM control is proposed to strengthen the tracking control performance of a certain class of multiple-input multiple-output (MIMO) nonlinear uncertain systems. This indirect approach requires the so called multiple estimation algorithms. In [16], a new AFSM controller with a model predictor is proposed for a class of uncertain nonlinear systems with unknown constant input time delay. In [17], an adaptive type-2 fuzzy sliding mode control to tolerate actuator faults of unknown nonlinear systems is proposed while two adaptive type-2 fuzzy logic systems are used to approximate the unknown functions. They also considered that the G matrix is always non-zero. In [18], a stable adaptive fuzzy sliding mode controller is investigated for a class of uncertain underactuated nonlinear dynamic systems, where the underactuated system is decoupled into two subsystems. AFSM algorithm has also been applied in various practical nonlinear control systems such as in the control of MicroElectroMechanical Systems (MEMS) resonators[19], and some other applications[2022].

      The AFSMC technique presented in this paper can be considered as an important extension of the direct AFSMC method of [13] and [23], which were applicable to the affine form of unknown chaotic systems. In those papers, the closed-loop asymptotic stability was guaranteed only if the MIMO plant had a diagonal input matrix with non-zero entries. There are, however, many well-known under-actuated systems, such as robotic systems, mass-spring-damper systems, and electrical machines, to name a few. In this paper, the original AFSMC method is extended to the case of nonlinear affine systems in which some of the diagonal entries of the gain matrix could be zero.

      As a case study, the control of the well-known remote environmental monitoring units (REMUS) which is autonomous underwater vehicle (AUV) of [24] is considered. REMUS is a low-cost, modular vehicle with applications in autonomous docking and long-range oceanographic survey. A 6-DOF dynamic model is used for the simulation of the motions. The results of this case study reveal the effectiveness of the new AFSMC methodology for both diving and steering modes control, despite the existence of under-actuation structure in the plant model.

      The remainder of this paper is organized as follows: In Section 2, the most relevant features of the conventional SMC are introduced. The extensions of SMC and AFSMC methods to the case of under-actuated systems are elaborated in Section 3, respectively. In order to demonstrate the effectiveness of the proposed control scheme, an extensive case study on the control of REMUS AUV is presented in Section 4 and the simulation results are depicted in Section 5. Finally, conclusions are presented in Section 6.

    • Consider the class of MIMO affine nonlinear systems

      $\left[\!\!\begin{array}[c]{c} y_{1}^{(r_{1})}\\\vdots\\ y_{m}^{(r_{m})}\end{array}\!\!\right] \!=\left[\!\!\begin{array}[c]{c} f_{1}(X)\\\vdots\\ f_{m}(X)\end{array}\!\right]\!+\!\left[\!\!\begin{array}[c]{ccc} g_{11} & 0 & 0\\0 & \ddots & 0\\0 & 0 & g_{mm}\end{array}\!\right] \left[\!\begin{array}[c]{c} u_{1}\\\vdots\\ u_{m}\end{array}\!\!\right].$

      (1)

      Using a more compact notation, (1) can be written as

      $Y^{(R)}=F(X)+GU$

      (2)

      where $X={[{y_{1}}, {\dot{y}_{1}}, \cdots, y_{1}^{({r_{1}}-1)}, \cdots, {y_{m}}, {\dot {y}_{m}}, \cdots, y_{m}^{({r_{m}}-1)}]^{\rm{T}}}$ is the vector of states which are assumed to be measurable. Furthermore, $U=[u_{1}, \cdots, u_{m}]^{\rm{T}}$ is the vector of control inputs, and $F(x)=[f_{1}(X), \cdots, f_{m}(X)]^{\rm{T}}$ is the vector of smooth functions of X. Also, $R=[{r_{1}}, \cdots, {r_{m}}]^{\rm{T}}$ is the vector of relative degrees for the m subsystems, where ${r_{1}}+ \cdots +{r_{m}=n}$ and n is the overall system order. Furthermore, it is assumed that the entries of the input gain matrix, G, are not known exactly. In presence of uncertainties, the description of system 1 can be modified as

      $Y^{(R)}=F(X)+GU + D$

      (3)

      in which $D=[{d_{1}}, \cdots, {d_{m}}]^{\rm{T}}$ is the vector of lumped uncertainties which are assumed to be of bounded norms, i.e., $|d_{i}|< \delta_{i}$ , $i=1, \cdots, m$ .

      The vector of outputs is defined as $Y=[{y_{1}}, \cdots, {y_{m}}]^{\rm{T}}$ . Let us consider the desired trajectory vector $Y_{d}=[{y_{1d}},$ $ \cdots, {y_{md}}]^{\rm{T}}$ . The tracking error is then defined as

      $\tilde{Y}=Y_{d}-Y=[\tilde{y}_{1}, \cdots, \tilde{y}_{m}]^{\rm{T}}.$

      (4)

      The objective is to design a control law so that $\tilde{y}_{i}$ converges to zero asymptotically.

    • For now, it is assumed that all the entries $g_{ii}, i=1, \cdots, m$ in (1) are non-zero. This assumption will be later on relaxed in Section 3. The design of sliding mode control includes two steps: Step one is to design a sliding surface, $s(x)$ , to represent the desired system dynamics, which is of a lower order than the given plant. The second step is to design a variable structure control u such that any state outside the switching surface is driven to reach the surface in finite time. On the sliding surface, the sliding mode takes place, following the desired system dynamics. In this way, the stability of trajectory on s is guaranteed[3].

      The m-dimensional vector of sliding surfaces are defined as

      $S=[{s_{1}}, \cdots, {s_{m}}]^{\rm{T}}=\left[\beta_1, \cdots, \beta_m \right]\left[\begin{array}{c}\tilde{Y}_1 \\\vdots \\\tilde{Y}_m\end{array}\right]$

      (5)

      where $\beta_{i}=[{\beta_{i1}}, \cdots, {\beta_{i(r_{i}-1)}}, 1]$ is vector of a Hurwitz polynomial coefficients and $\tilde{Y}_{i}=[\tilde{y}_{i}, \cdots, {\tilde{y}}_{i}^{(r_{i}-1)}]^{\rm{T}}$ is the states tracking error of the i-th subsystem.

      By time differentiation of the sliding surface, one gets

      $\begin{split}{{{\dot s}_i} = {\beta _i}{{\dot {\tilde Y}}_i}} = & \sum\limits_{j = 1}^{{r_i} - 1} {{\beta _{ij}}\tilde y_i^j} + \tilde y_i^{\left( {{r_i}} \right)}=\\ & {\sum\limits_{j = 1}^{{r_i} - 1} {{\beta _{ij}}\tilde y_i^j} + y_{di}^{({r_i})} - y_i^{({r_i})} = }\\& {{E_{\beta i}} + y_{di}^{({r_i})} - y_i^{({r_i})} = }\\ & {{E_{\beta i}} + y_{di}^{({r_i})} - {f_i}(x) - {g_{ii}}{u_i} - {d_i}}\end{split}$

      (6)

      in which

      ${E_{\beta i}} = \sum\limits_{j = 1}^{{r_{i}} - 1} { {{\beta_{ij}}\tilde y_{i}^{j}} }.$

      The sliding mode control law is defined as[13]

      $u_i = {u_i^{eq}}+ {u_i^{rb}}$

      (7)

      where the equivalent control law ${u_{i}^{eq}}$ can be obtained using the condition $\dot s_{i} = 0$ , i.e.,

      $\begin{split}{{\dot s}_i} =\, & {E_{\beta i}} + y_{di}^{({r_i})} - {f_i}(x) - {g_{ii}}u_i^{eq} = 0 \Rightarrow \\ & u_i^{eq}\left( {\frac{1}{{{g_{ii}}}}} \right)\left( {{E_{\beta i}} + y_{di}^{({r_i})} - {f_i}(x)} \right).\end{split}$

      (8)

      The robust controller ${u_i^{rb}}$ is used to overcome the uncertainties of the plant, ensuring a finite time reaching towards the sliding surface, where

      ${u_i^{rb}} =\left(\frac{1}{g_{ii}} \right)\nu_i.$

      (9)

      The control term $\nu_i$ is selected as

      ${\nu_{i}} = {\delta_{i}}{\mathop{\rm sgn}} ({s_{i}}) \Rightarrow \nu ={\left[{\Delta{\mathop{\rm sgn}} (S)} \right] ^{\rm{T}}}$

      (10)

      where ${\mathop{\rm sgn}} (S) = {[{\mathop{\rm sgn}} ({s_{1}}), \cdots\!,\; {\mathop{\rm sgn}} ({s_{m}})]^{\rm{T}}}$ and ${\mathop{\rm sgn}} (\cdot)$ is the sign function. Also, $\Delta=[\delta_{1}, \cdots\!,\; \delta_{m}]^{\rm{T}}$ . Substituting (7) into (6), and using (8) and (9) lead to

      $\begin{split}{{{\dot s}_i}} = & {E_{\beta i}} + y_{di}^{({r_i})} - {f_i}(x) - {g_{ii}}{u_i} - {d_i} = \\ & {{E_{\beta i}} + y_{di}^{({r_i})} - {f_i}(x) - {g_{ii}}\left( {u_i^{eq} + u_i^{rb}} \right) - {d_i} = }\\ & {{E_{\beta i}} + y_{di}^{({r_i})} - {f_i}(x) - \left( {{E_{\beta i}} + y_{di}^{({r_i})} - {f_i}(x) + {\nu _i}} \right) - {d_i} = }\\ & { - {d_i} - {\nu _i} = - {d_i} - {\delta _i}{\rm{sgn}}({s_i})}.\end{split}$

      (11)

      Now, the following Lyapunov candidate function is defined as

      ${L_{i}} = \frac{1}{2}s_{i}^{2}.$

      (12)

      The time derivative of (12) together with (11) yields

      $\begin{split}{{\dot L}_i} = & {s_i}{{\dot s}_i} = - {s_i}{d_i} - \left| {{s_i}} \right|{\delta _i} \le \left| {{s_i}} \right|\left| {{d_i}} \right| - \left| {{s_i}} \right|{\delta _i} = \\ & - \left| {{s_i}} \right|({\delta _i} - \left| {{d_i}} \right|) \le 0.\end{split}$

      (13)

      Therefore, the sliding mode control (7) can guarantee the stability of the MIMO system of (1) in the Lyapunov sense.

    • The sliding mode controller outlined in the previous section is only applicable to systems where all $g_{ii}$ in (1) are non-zero, i.e., every subsystem of (1) is directly affected by a specific control input channel. In this section, the sliding mode control is extended to the case of under-actuated systems and is combined with the adaptive fuzzy concept to drastically increase the robustness of the system in the presence of un-modeled dynamics.

    • Consider an under-actuated system with m outputs and m–1 inputs, in the form of

      $\begin{split}\left[\begin{array}[c]{c} z_{1}^{(r_{z})}\\x_{1}^{(r_{x})}\\\hline y_{3}^{(r_{3})}\\\vdots\\y_{m}^{r_{m}}\end{array}\right] = & \left[\begin{array}[c]{c} f_ {1}+g_{1} {x_1}\\f_{2}(Z_{1}, X_{1})+d_{x}\\\hline f_{3}(X)\\\vdots\\f_{m}(X)\end{array}\right] + \\& \left[\begin{array}[c]{c|ccc}0 & & & \\g_{22} & & & \\\hline& g_{33} & & \\& & \ddots & \\& & & g_{mm}\end{array}\right] \left[\begin{array}[c]{c} u_{2}\\\vdots\\u_{m}\end{array}\right]\end{split}$

      (14)

      where ${f_ {1}}$ and ${g_{1}}$ are smooth functions of $z_{1}$ , and ${x_1}$ can be considered as the input of this subsystem. This is a modification of (1), in which ${g}_{11}=0$ and ${z}_{1}$ and ${x}_{1}$ are the first and second outputs of the system, respectively. Consider the first subsystem of (4), where ${z}_{1}$ and its derivatives are not directly affected by the control input, and instead are indirectly controlled through the second output, ${x}_{1}$ , justifying the name under-actuated. Here, $ {Z_{1}}={[{z_{1}}, \cdots\!,\; z_{1}^{\left( {{r_{z}}-1}\right) }]^{\rm{T}}}$ , ${X_{1}}={[{x_{1}}, \cdots\!,\; x_{1}^{\left( {{r_{x}}-1}\right) }]^{\rm{T}}}$ , $X = \left[ {{Z_1},} \right.$ $ {\left. {{X_1},{y_3},{{\dot y}_3}, \cdots ,y_3^{({r_3} - 1)}, \cdots\! ,\;{y_m},{{\dot y}_m}, \cdots\! ,\;y_m^{({r_m} - 1)}} \right]^{\rm{T}}}$ and the output vector is defined as ${Y}={[{z_{1}}, x_1, y_3, \cdots\!,\; y_m}]^{\rm{T}}$ . Also, ${d_{x}}$ is the lumped uncertainties of the second subsystem, which is assumed to be of bounded norms, i.e., $|d_{x}| < \delta_{2}$ .

    • The sliding surface for under-actuated system must be defined first. For this purpose, the output error is defined as

      ${\tilde{z}_{1}}={z_{1d}}-z{_{1}}$

      (15)

      where $z_{1d}$ and $\tilde{z}_{1}$ are the desired values for ${z}_{1}$ and its tracking error, respectively.

      The primary sliding surface is constructed for the first part of state variables as

      ${s_{z}}={\lambda_{z}}{\tilde{Z}_{1}}$

      (16)

      where ${\lambda_{z}}=\left[{{\lambda_{z1}}, \cdots\!,\; {\lambda_{z({r_{z}}-1)}}, 1}\right] $ is a vector of Hurwitz polynomial coefficients and ${\tilde{Z}_{1}}={[{\tilde{z}_{1}}, \cdots\!,\; \tilde{z}_{1}^{\left( {{r_{z}}-1}\right) }]^{\rm{T}}}$ is the vector of states tracking errors. Define, ${Z_{1d}}={[{{z}_{1d}}, \cdots\!,\; z_{1d}^{\left( {{r_{z}}-1}\right) }]^{\rm{T}}}$ as the desired trajectory vector for ${Z_{1}}$ . Suppose the first subsystem of (14) can be written in the following form:

      $z_{1}^{(r_{z})}=f_ {1}^{*}+g_{1} {\tilde x_1}$

      (17)

      where ${f_ {1}^{*}}$ is smooth function of $z_{1}$ , and ${\tilde x_1}$ can be considered as the input of this subsystem. Suppose the system of (17) can be stabilized by a smooth state feedback control law $\tilde{x}_{1}={\varphi_{1}}({Z_{1}})$ , with ${{\varphi_{1}}({Z_{1d}})=0}$ , i.e., the origin of

      $z_{1}^{(r_{z})}=f_ {1}^{*}+g_{1} {{\varphi_{1}}({Z_{1}})}$

      (18)

      is asymptotically stable. Suppose that we know a smooth and positive definite Lyapunov function $V_{1}({Z_{1}})$ that satisfies the inequality.

      $\frac{\partial V_{1}}{\partial {{z}_{1}}}[f({{z}_{1}})+g({{z}_{1}})\phi({{z}_{1}})]\le -W({{z}_{1}}).$

      (19)

      Inspired by the well-known back-stepping technique, (18) is re-written in the tracking error form. By adding and subtracting $g_{1}{\varphi_{1}({Z_{1}})}$ on the right-hand side of (17), one can obtain the equivalent representation

      $z_{1}^{(r_{z})}=f_ {1}^{*}+g_{1} {{\varphi_{1}}({Z_{1}})}+g_{1} ({\tilde x_1}-{{\varphi_{1}}({Z_{1}})}).$

      (20)

      To backstep, the following change of variables

      ${e_{1}}=\tilde{x}_{1}-{\varphi_{1}}({Z_{1}})$

      (21)

      results in the system of

      $z_{1}^{(r_{z})}=f_ {1}^{*}+g_{1} {{\varphi_{1}}({Z_{1}})}+g_{1}{e_{1}}.$

      (22)

      In (22), $e_{1}$ can be viewed as the input of system and the output has an asymptotically stable origin when the input is zero. The derivative of $e_{1}$ can be written as

      $\dot{e}_{1}={\tilde x}^{(1)}_{1}-\dot{\varphi}_{1}(Z_{1}).$

      (23)

      In (23), ${\tilde x}_{1}^{(1)}$ acts as a virtual input for state variable $e_1$ .

      Suppose the system of (23) can be stabilized by a smooth state feedback control law $\tilde{x}_{1}^{(1)}={\varphi_{2}}(Z_{1}, X_{1})$ , with ${{\varphi_{2}}(Z_{1d}, X_{1d})=0}$ , i.e., the origin of

      $\dot{e}_{1}={\varphi_{2}}(Z_{1}, X_{1})-\dot{\varphi}_{1}(Z_{1})$

      (24)

      is asymptotically stable. Also, ${X_{1d}}={[{{x}_{1d}}, \cdots, x_{1d}^{\left( {{r_{x}}-1}\right) }]^{\rm{T}}}$ is the desired trajectory vector for ${X_{1}}$ . In order to preserve the closed-loop stability, the function ${\varphi_{2}}(Z_{1}, X_{1})$ must be selected such that the rate of change of the following Lyapunov function remains negative. Using ${V_{{e_{1}}}}=V_{1}+\frac{1}{2}e_{1}^{2}$ as a Lyapanov function candidate, we obtain

      $\begin{split}& {{V_{{e_1}}} = {V_1} + \frac{1}{2}{{({e_1})}^2}}\\& {{{\dot V}_{{e_1}}} = {{\dot V}_1} + {e_1}{{\dot e}_1} = }\\& \quad\quad{{{\dot V}_1} + {e_1}(\tilde x_1^{(1)} - {{\dot \varphi }_1}({Z_1})) = }\\& \quad\quad{\frac{{\partial {V_1}}}{{\partial {z_1}^{({r_z} - 1)}}}{{({z_1})}^{({r_z})}} + {e_1}({\varphi _2}) - {e_1}{{\dot \varphi }_1}({Z_1}) = }\\& \quad\quad{\frac{{\partial {V_1}}}{{\partial {z_1}^{({r_z} - 1)}}}(f_1^* + {g_1}{\varphi _1}({z_1}) + {g_1}{e_1}) + {e_1}({\varphi _2}) - {e_1}{{\dot \varphi }_1}({Z_1}) \le }\\& \quad\quad- W({z_1}) + \frac{{\partial {V_1}}}{{\partial {z_1}^{({r_z} - 1)}}}({g_1}{e_1}) + {e_1}({\varphi _2}) - {e_1}{{\dot \varphi }_1}({Z_1}).\end{split}$

      (25)

      By having $ \frac{\partial V_{1}}{\partial {{z}_{1}}^({r}_{z}-1)}(g_{1})+ (\varphi_2)-\dot{\varphi}_{1}(Z_{1}) = -{K_{1}}e_{1}$ , we obtain

      ${{V}_{e_{1}}} \le -W({{z}_{1}})+-{K_{1}}(e_{1})^2$

      (26)

      in which ${K_{1}}$ is a positive constant. By choosing $\varphi_2$ as follows, the rate of change of the Lyapunov function remains negative.

      $\varphi_2 = -\frac{\partial V_{1}}{\partial {{z}_{1}}^({r}_{z}-1)}(g_{1})+\dot{\varphi}_{1}(Z_{1}) -{K_{1}}e_{1}.$

      (27)

      Similarly, by adding and subtracting ${\varphi_{2}}$ on the right-hand side of (13), one can obtain the equivalent representation

      $\dot{e}_{1}=({\tilde x}^{(1)}_{1}-{\varphi_{2}})+{\varphi_{2}}-\dot{\varphi}_{1}(Z_{1}).$

      (28)

      The change of variables

      ${e_{2}}={\tilde x}^{(1)}_{1}-{\varphi_{2}}({Z_{1}}, {X_{1}})$

      (29)

      results in the system of

      $\dot{e}_{1}={e_{2}}+{\varphi_{2}}({Z_{1}}, {X_{1}})-\dot{\varphi}_{1}(Z_{1}).$

      (30)

      In (30), $e_{2}$ can be viewed as the input of system and the output has an asymptotically stable origin when the input is zero. This procedure will be continued for other state variable $x_{1}^{(j)}$ up to $j={r_{x}}-1$ . Eventually, the last subsystem change of variable, $e_{(r_{x})}$ will actually define the sliding surface for the given subsystem, namely,

      ${s_{1}}=\tilde{x}_{1}^{({r_{x}}-1)}-{\varphi_{({r_{x}})}}({Z_{1}}, X_{1}).$

      (31)

      By differentiating ${s_{1}}$ , one gets

      $\begin{split}{{\dot{s}}_{1}}= & \tilde{x}_{1}^{({r_{x}})}-\frac{{\partial{\varphi_{(r_{x})}}}}{{\partial X_{1}}}(\dot{X}_{1})-\frac{{\partial{\varphi_{({r_{x}})}}}}{{\partial{Z_{1}}}}({{\dot{Z}}_{1}})=\\ & x_{1d}^{({r_{x}})}-x_{1}^{({r_{x}})}-\frac{{\partial{\varphi_{1({r_{x}})}}}}{{\partial X_{1}}}(\dot{X}_{1})-\frac{{\partial{\varphi_{1({r_{x}})}}}}{{\partial{Z_{1}}}}({{\dot{Z}}_{1}})=\\ & x_{1d}^{({r_{x}})}-x_{1}^{({r_{x}})}-{E_{x}}-{E_{z}}=\\ & x_{1d}^{({r_{x}})}-{f}_{2}({Z_{1}}, {X_{1}})-{g_{22}}{u_{2}}-{d_{x}}-{E_{x}}-{E_{z}}\end{split}$

      (32)

      where ${E_{x}}=\frac{{\partial{\varphi_{(r_{x})}}}}{{\partial X_{1}}}(\dot{X}_{1})$ and ${E_{z}}=\frac{{\partial{\varphi_{({r_{x}})}}}}{{\partial{Z_{1}}}}({\dot{Z}_{1}})$ . The equivalent control law, $u_{2}^{eq}$ , can then be obtained from ${\dot{s}_{1}}=0$ , as

      $u_{2}^{eq}={\left( {{g_{22}}}\right) ^{-1}}\left( {x_{1d}^{({r_{x}})}-{E_{x}}-{E_{z}}-{f_{2}}({Z_{1}}, {X_{1}})}\right).$

      (33)

      The sliding mode control law is defined as

      $u_2 = {u_2^{eq}}+ {u_2^{rb}}.$

      (34)

      The robust controller ${u_{2}^{rb}}$ is used to overcome the uncertainties of the plant, ensuring a finite time reaching towards the sliding surface, where

      ${u_2^{rb}} = {g_{22}^{ - 1}} \nu_{2}.$

      (35)

      The control term $\nu_{2}$ is selected as

      ${\nu_{2}} = {\delta_{2}}{\mathop{\rm sgn}} ({s_{1}}).$

      (36)

      This leads to

      ${{\dot s}_{1}}=- {d_{x}} - {\nu_{2}} = - {d_{x}} - {\delta_{2}}{\mathop{\rm sgn}} ({s_{1}}).$

      (37)

      Now, the following Lyapunov candidate function is defined:

      ${L_{1}} = \frac{1}{2}s_{1}^{2}.$

      (38)

      The time derivative of (38) together with (37) yields

      $\begin{split}{{\dot L}_{1}}=\; & {s_{1}}{{\dot s}_{1}} = - {s_{1}}{d_{x}} - \left|{{s_{1}}} \right| {\delta_{2}} \le\left| {{s_{1}}} \right| \left| {{d_{x}}} \right| - \left| {{s_{1}}} \right| {\delta_{2}}=\\ & - \left| {{s_{1}}} \right| ({\delta_{2}} - \left| {{d_{x}}} \right| ) \le 0.\end{split}$

      (39)

      Therefore, the sliding mode control (34) can guarantee the stability of the first subsystem of (14) in the Lyapunov sense.

      In the remainder of this paper, $S={\left[{{s_{1}}, \cdots, {s_{m}}}\right]^{\rm{T}}}$ is supposed to be the vector of sliding surface for system of (14).

    • By combining the fuzzy and SMC approaches, the control effort can be defined to be a nonlinear function of the deviations from the sliding surface. The advantage is that, unlike the conventional SMC approach, the control input is not computed from the plant dynamic equations directly. The fuzzy sliding mode controller is actually a fuzzy logic controller, for which the inputs are the deviations from the sliding surface and its time derivative and the output is the control command.

      For this purpose, for the i-th subsystem of (14), a Takagi-Sugeno (TS) fuzzy system with the output $u_{i}^{fuz}$ and the fuzzy IF-THEN rules are considered, with,

      Rule r: If ${s_{i}}$ is $A_{i}^{r}$ , then $u_{i}^{fuz}=b_{i}^{r}$ , $r=1, \cdots, {n_{r}}$ , where $b_{i}^{r}$ is the fuzzy singleton for the output of the r-th rule, and $A_{i}^{r}$ is a fuzzy set characterized by a Gaussian membership function as

      ${\mu_{A_{i}^{r}}}({s_{i}})=\exp\left[{-{{\left( {\frac{{{s_{i}}-c_{i}^{r}}}{{\sigma_{i}^{r}}}}\right) }^{2}}}\right].$

      (40)

      Here, $c_{i}^{r}$ and $\sigma_{i}^{r}$ are the center and width of the membership functions, respectively. Using singleton fuzzifier, product inference, and center average defuzzifier, the output of the fuzzy system is obtained as

      $u_{i}^{fuz} = \frac{{\sum\limits_{r = 1}^{{n_{r}}} {b_{i}^{r}{\mu_{A_{i}^{r}}}({s_{i}})} }}{{\sum\limits_{r = 1}^{{n_{r}}} {{\mu_{A_{i}^{r}}}({s_{i}})} }}.$

      (41)

      By defining the firing strength of the r-th rule as

      $w_{i}^{r} = \frac{{{\mu_{A_{i}^{r}}}({s_{i}})}}{{\sum\limits_{r =1}^{{n_{r}}} {{\mu_{A_{i}^{r}}}({s_{i}})} }}, \;\;\;\;\;\;\;\;\;r =1, \cdots, {n_{r}}$

      (42)

      the output of fuzzy system can be written as

      $u_{i}^{fuz}({s_{i}}, {b_{i}}) = b_{i}^{\rm{T}}{w_{i}}$

      (43)

      where

      ${w_{i}} = {\left[{w_{i}^{1}, \cdots, w_{i}^{{n_{r}}}} \right] ^{\rm{T}}}, \quad\quad {b_{i}} ={\left[{b_{i}^{1}, \cdots\!,\; b_{i}^{{n_{r}}}} \right] ^{\rm{T}}}.$

      For nominal case, i.e., when accurate mathematical model of the system is available, the output of fuzzy controller for a system with m inputs $S = {\left[{{s_{1}}, \cdots\!,\; {s_{m}}} \right] ^{\rm{T}}}$ and m outputs $u_{1}^{fuz}, \cdots\!,\; u_{m}^{fuz}$ is denoted as

      ${u^{fuz * }} = {\left[{u_{1}^{fuz * }({s_{1}}, b_{1}^{*} ), \cdots\!,\; u_{m}^{fuz *}({s_{m}}, b_{m}^{*} )} \right] ^{\rm{T}}}.$

      (44)

      The ideal controller is obtained as

      ${u^{*} } = {u^{fuz * }}(S, {B^{*} }) + \Xi= {\rm{diag}}({B^{ * T}}W) + \Xi$

      (45)

      where

      $W = {\left[{w_{1}^{}, \cdots\!,\; w_{m}^{}} \right] ^{\rm{T}}}, \quad\quad {B^{*} }^{\rm{T}} = {[b_{1}^{*},\cdots\!, b_{m}^{*}]^{\rm{T}}}$

      and $\Xi= {\left[{\xi_{1}^{}, \cdots, \xi_{m}^{}} \right] ^{\rm{T}}}$ is the approximation error or the uncertainty which is assumed to be bounded, $\left| {{\xi_{i}}} \right| < {\kappa_{i}}$ . Also $B^{*}$ is the optimal parameter vector

      ${B^ * } \buildrel \Delta \over = {\arg \min _B}\left\{ {\left| {{\rm diag}({B^{\rm T}}W) - {u^*}} \right|} \right\}.$

      (46)

      In practice, the entries of the optimal parameter vector $b_{i}^{*}$ and the uncertainty or approximation error bounds ${K} = {\left[{{\kappa_{1}}, \cdots, {\kappa_{m}}} \right] ^{\rm{T}}}$ may be unknown. Denoting the estimation of this uncertainty bounds as $\hat{K}$ , the estimation error is defined as

      $\tilde{{K}}(t) = {K} - \hat{K}(t).$

      (47)

      The output of fuzzy system to approximate the ideal controller can be rewritten as

      $\hat u_{i}^{fuz}({s_{i}}, {\hat b_{i}}) = \hat b_{i}^{\rm{T}}{w_{i}}, \;\;\;\;\;\;\;\;\;\;\;k = 1, 2, \cdots\!,\; m$

      (48)

      Here, $\hat b_{i}^{}$ is the estimation of $b_{i}^{*}$ , thus the control law can be represented as

      ${u_{i}} = \hat u_{i}^{fuz}({s_{i}}, {\hat b_{i}}) + u_{i}^{rb}({s_{i}}), \;\;\;\;\;\;\;k = 1, 2, \cdots, m$

      (49)

      where $u_{i}^{rb}$ is employed to compensate the difference between the fuzzy controller and the ideal one. By substituting the matrix form of (49) into (1), one gets

      ${y^{(r)}} = F(x) + G\left[{{{\hat u}^{fuz}} + {u^{rb}}} \right].$

      (50)

      By defining the approximation errors as

      ${\tilde u^{fuz}} = {u^{*} } - {\hat u^{fuz}}, \;\;\tilde B = {B^{*} } -\hat B$

      (51)

      and by using (45), (48) and (51), one gets

      $\tilde u_{}^{fuz} = {\rm{diag}}({\tilde B^{\rm{T}}}W) + \Xi.$

      (52)

      Multiplying (8) with ${g_{ii}}$ , added to (50) and using (6), it turns out for the i-th subsystem that

      $\begin{split} & {{\dot s}_{i}} = {g_{ii}}\left( {u_{i}^{*} - {{\hat u}_{i}}^{fuz} - {u_{i}}^{rb}} \right) \to\\ & \dot S = G\left[{{{\tilde u}^{fuz}}-{u^{rb}}} \right] = G\left({{\rm diag}({{\tilde B}^{\rm{T}}}W) + \Xi- {u^{rb}}} \right).\end{split}$

      (53)

      In the case of underactuated variable in first subsystem of (14), multiplying (33) with ${g_{22}}$ , added to (50) and using (32), provide the same results as (53).

      The closed-loop stability of the AFSM controlled system is proved in the sequel.

      Theorem 1. Consider system of (1) and the control law given by (49), where

      1) The fuzzy controller is tuned by the adaptive law

      $\dot{\hat{B}} = - \dot{\tilde{ B}} = {\alpha_{1}}W(S)$

      (54)

      2) The switching part of the control input is obtained from

      ${u^{rb}} = {\rm{diag}}(\hat{{K}}){\mathop{\rm sgn}} (G){\mathop{\rm sgn}}(S(t))$

      (55)

      3) The estimated value of uncertainty bound is adaptively tuned according to

      $\dot{\hat{{{K}}}} = - \dot{\tilde{{{ K}}}} = {\alpha_{2}}{\mathop{\rm sgn}} (G)\left| {S(t)} \right|.$

      (56)

      ${\alpha_{1}}$ and ${\alpha_{2}}$ are pre-selected positive values for adaption rates. Then, the tracking error converges to zero asymptotically.

      Proof. By choosing the following Lyapunov candidate function

      $\begin{split} & V = \sum\limits_{i = 1}^{m} {{V_{i}}}\\ & {V_{i}}({s_{i}}, {{\tilde b}_{i}}, {{\tilde\kappa}_{i}}) = \frac{1}{2}s_{i}^{2}+ \frac{1}{{2{\alpha_{1}}}}{g_{ii}}\tilde b_{i}^{\rm{T}}{{\tilde b}_{i}} + \frac{1}{{2{\alpha_{2}}}}{g_{ii}}({{\tilde\kappa}_{i}})({{\tilde\kappa}_{i}})\end{split}$

      (57)

      and by differentiating (57) with respect to time, and employing (53)–(56), it can be shown that

      $\begin{split} & {{\dot V}_{i}}({s_{i}}, {{\tilde b}_{i}}, {{\tilde\kappa}_{i}}) = {s_{i}}{{\dot s}_{i}} + \frac{1}{{{\alpha_{1}}}} {{g_{ii}}} \tilde b_{i}^{\rm{T}}{{\dot{ \tilde{ b}}}_{i}} + \frac{1}{{{\alpha_{2}}}} {{g_{ii}}}({{\tilde\kappa}_{i}})({{\dot{\tilde{\kappa}} }_{i}})=\\ & \quad {s_{i}}{g_{ii}}\left({\tilde b_{i}^{\rm{T}}{w_{i}} + {\xi_{i}}-{u_{i}}^{rb}} \right) \!+\! \frac{1}{{{\alpha_{1}}}} {{g_{ii}}} \tilde b_{i}^{\rm{T}}{{\dot{ \tilde{ b}}}_{i}} \!+\!\frac{1}{{{\alpha_{2}}}} {{g_{ii}}} ({{\tilde\kappa}_{i}})({{\dot{\tilde{ \kappa}}}_{i}})=\\ & \quad {{g_{ii}}} \tilde b_{i}^{\rm{T}}\left( \! {{s_{i}}{w_{i}} + \frac{{{{\dot{\tilde{ b}}}_{i}}}}{{{\alpha_{1}}}}} \!\right) \!+\! {s_{i}}{g_{ii}}\left(\! {{\xi_{i}} - u_{i}^{rb}} \!\right) \!+\! \frac{1}{{{\alpha_{2}}}}{{g_{ii}}} ({{\tilde\kappa}_{i}})({{\dot{\tilde{\kappa}}}_{i}})\!=\\ & \quad {s_{i}}{g_{ii}}{\xi_{i}} \!-\! {s_{i}}{g_{ii}}{{\hat\kappa}_{i}}{\mathop{\rm sgn}} \left( {{g_{ii}}} \right){\mathop{\rm sgn}} \left( {{s_{i}}} \right) \!-\! \frac{1}{{{\alpha_{2}}}}\left| {{g_{ii}}} \right| ({{\tilde\kappa}_{i}}){\alpha_{2}}\left|{{s_{i}}} \right| =\\& \quad {s_{i}}{g_{ii}}{\xi_{i}} - \left| {{s_{i}}}\right| \left| {{g_{ii}}} \right| \left( {{{\hat\kappa}_{i}} +{{\tilde\kappa}_{i}}} \right) = {s_{i}}{g_{ii}}{\xi_{i}} - \left| {{s_{i}}}\right| \left| {{g_{ii}}} \right| \left( {{\kappa_{i}}} \right) =\\ & \quad{s_{i}}{g_{ii}}{\xi_{i}} - \left| {{s_{i}}}\right| \left| {{g_{ii}}} \right| \left( {{\kappa_{i}}} \right)\le\left( {\left| {{s_{i}}} \right| \left| {{g_{ii}}} \right| {\xi_{i}} -\left| {{s_{i}}} \right| \left| {{g_{ii}}} \right| \left( {{\kappa_{i}}}\right) } \right)= \\ & \quad - \left( {\left| {{s_{i}}} \right| \left|{{g_{ii}}} \right| \left( {{\kappa_{i}} - \left| {{\xi_{i}}} \right| }\right) } \right) \le0.\end{split}\!\!\!\!\!$

      (58)

      Let us define

      $\Gamma (t) = \sum\limits_{i = 1}^m {\left( {\left| {{s_i}} \right|\left| {{g_{ii}}} \right|\left( {{\kappa _i} - \left| {{\xi _i}} \right|} \right)} \right)} \le - \dot V.$

      (59)

      Integration of this equation leads to

      $\int_{0}^{t} {\Gamma(\tau)} {\rm{d}}\tau\le V(S(0), \tilde B, \tilde K) - V(S(t), \tilde B, \tilde K)$

      (60)

      where $V(S(0), \tilde B, \tilde K)$ is bounded and $V(S(t), \tilde B, \tilde K)$ is at least non-increasing. Therefore,

      $\int_{0}^{t} {\Gamma(\tau)} {\rm{d}}\tau\le\infty.$

      (61)

      By exploiting the semi-negativeness of ${\dot V}$ from (58) and by considering the fact that, absolute functions are uniformly continuous, one concludes, from Barbalat Lemma, that

      $\mathop {\lim }\limits_{t \to\infty} \Gamma(t) = 0.$

      (62)

      As a result, when $t \to\infty$ , the sliding surface $S(t) \to0$ uniformly, and hence the asymptotic stability is guaranteed.

    • In order to demonstrate the feasibility of the proposed approach in previous section, AFSMC is applied to the plant of an autonomous underwater vehicle (AUV). The 6 degree of freedom (DOF) dynamic equations of motion of an AUV can be written as

      $M\dot\nu+ C(\nu)\nu+ D(\nu)\nu+ g(\eta) = \tau.$

      (63)

      Here,

      $\begin{split}& \eta={\left[{x, y, z, \phi, \theta, \psi}\right] ^{\rm{T}}}\\& \nu={\left[{u, v, w, p, q, r}\right] ^{\rm{T}}}\\& \tau={\left[{X, Y, Z, K, M, N}\right] ^{\rm{T}}}\\& M={M_{RB}}+{M_{A}}\\& \rm{{C(}}\nu\rm{{)=}}{\rm{{C}}_{RB}}\rm{{(}}\nu\rm{{)}}+{\rm{{C}}_{A}}\rm{{(}}\nu\rm{{)}}\end{split}$

      where $\eta$ denotes the position and orientation of the vehicle with respect to the inertial or earth-fixed reference frame, $\nu$ is the translational and rotational velocities of the vehicle with respect to the body-fixed reference frame, and $\tau$ includes the total forces and moments acting on the vehicle with respect to the body-fixed reference frame. $\tau$ contains the propulsion forces and moments, which can be expressed in the same way as environmental forces acting on the vehicle. Also, ${M_{RB}}$ is the rigid body inertia matrix and ${M_{A}}$ is the added inertia matrix, ${C_{RB}}(\nu)$ is the rigid body Coriolis and centripetal matrix, also ${C_{A}}(\nu)$ is added hydrodynamic Coriolis and centripetal matrix, $D(\nu)$ contains all the hydrodynamic damping forces acting on the ocean vehicle throughout its mission and $g(\eta)$ illustrates gravitational and buoyant forces and moments. In hydrodynamics terminology, the gravitational and buoyant forces are called restoring forces. The position and orientation of the vehicle should be described relative to inertial reference frame while the linear and angular velocities of the vehicle should be expressed in the body-fixed coordinate system[25].

      By retaining the acceleration terms of (63) on the left-hand side, one gets

      $\begin{split}& M\dot\nu = - C\left( \nu\right) \nu- D\left(\nu\right) \nu- g\left( \eta\right) + \tau\\& \dot\nu = {M^{ - 1}}\left[{-C\left( \nu\right) \nu-D\left( \nu\right) \nu-g\left( \eta\right) } \right] + {M^{ -1}}\tau.\end{split}$

      (64)

      Based on the vehicle equations of motion presented in Appendix 7, the corresponding terms in $\tau$ vector of (64) can be written as

      $\begin{split}\tau = & \left[{\begin{array}{*{20}{c}}{{X_{prop}}}\\{{Y_{uu\delta r}}{u_0}^2{\delta _r}\;\;}\\{{Z_{uu{\delta _s}}}{u_0}^2{\delta _s}}\\{{K_{prop}}}\\{{M_{uu{\delta _s}}}{u_0}^2{\delta _s}}\\{{N_{uu{\delta _r}}}{u_0}^2{\delta _r}}\end{array}} \right] = \\& \left[{\begin{array}{*{20}{c}}{{X_{prop}}}\\0\\0\\{{K_{prop}}}\\0\\0\end{array}} \right] + \left[{\begin{array}{*{20}{c}}0 & 0\\{{Y_{uu\delta r}}{u_0}^2} & 0\\0 & {{Z_{uu{\delta _s}}}{u_0}^2}\\0 & 0\\0 & {{M_{uu{\delta _s}}}{u_0}^2}\\{{N_{uu{\delta _r}}}{u_0}^2} & 0\end{array}} \right] \times \left[{\begin{array}{*{20}{c}}{{\delta _r}}\\{{\delta _s}}\end{array}} \right]\end{split}$

      (65)

      where ${{\delta_{r}}}$ and ${{\delta_{s}}}$ are control actuators of the vehicle and ${{u_{0}}}$ is the constant surge velocity. In compact form, (64) can be written as

      $\dot{\nu}=f+G\left[{\begin{array}[c]{c}{{\delta_{r}}}\\{{\delta_{s}}}\end{array}}\right].$

      (66)

      The constant coefficient matrix of G and F are

      $G\!=\!{M^{-1}}\left[\!\!{\begin{array}[c]{cc}0 & 0\\{{Y_{uu\delta r}}{u_{0}}^{2}} & 0\\0 & {{Z_{uu{\delta_{s}}}}{u_{0}}^{2}}\\0 & 0\\0 & {{M_{uu{\delta_{s}}}}{u_{0}}^{2}}\\{{N_{uu{\delta_{r}}}}{u_{0}}^{2}} & 0\end{array}}\!\!\right] \!=\!\left[{\!\!\begin{array}[c]{cc}{{g_{11}}} & {{g_{12}}}\\{{g_{21}}} & {{g_{22}}}\\{{g_{31}}} & {{g_{32}}}\\{{g_{41}}} & {{g_{42}}}\\{{g_{51}}} & {{g_{52}}}\\{{g_{61}}} & {{g_{62}}}\end{array}}\!\!\right]$

      (67)

      $\begin{split}f= & \left[{\begin{array}[c]{c}{{f_{1}}}\\{{f_{2}}}\\{{f_{3}}}\\{{f_{4}}}\\{{f_{5}}}\\{{f_{6}}}\end{array}}\right] =\\& {M^{-1}}\left( {-C(\nu)\nu-D(\nu)\nu-g(\eta)}\right) +{M^{-1}}\left[{\begin{array}[c]{c}{{X_{prop}}}\\0\\0\\{{K_{prop}}}\\0\\0\end{array}}\right].\end{split}$

      (68)
    • Pitch mode is actually the diving motion of vehicle in vertical plane. In a pure vertical motion, one could eliminate and nullify some unrelated state variables, i.e., roll angle $(\phi)$ , yaw angle $(\psi)$ , and sway velocity (v). Therefore, depth (z), pitch angle $(\theta)$ , and pitch angular velocity (q), would be as the main state variables for this mode. The equations of diving motion can then be written as

      $\begin{split}& \dot{z} =\left( {-\sin\theta}\right) {u_{0}}+\left( {\cos\theta}\right) w\\& \dot{\theta} =q\\& \left[{\begin{array}[c]{c}{\dot{w}}\\{\dot{q}}\end{array}}\right] =\left[{\begin{array}[c]{c}{{f_{3}}}\\{{f_{5}}}\end{array}}\right] +\left[{\begin{array}[c]{c}{{g_{32}}}\\{{g_{52}}}\end{array}}\right] \left[{{\delta_{s}}}\right].\end{split}$

      (69)

      In this subsystem, depth z, is the output.

    • Yaw mode describes the motion of vehicle in the horizontal plane. Again, for the pure horizontal motion, some of the state variables, such as depth (z), pitch angle $(\theta)$ , heave velocity (w) and pitch angular velocity (q), can be ignored. The remaining states would then be, lateral absolute position (y), yaw angle $(\psi)$ , sway velocity (v), and yaw angular velocity (r), being the main state variables for this mode. Based on these assumptions, the equation of motion for yaw mode would be

      $\begin{split}& \dot{y} =\left( {\sin\psi}\right) {u_{0}}+\left( {\cos\psi}\right) v\\& \dot{\psi} =r\\& \left[{\begin{array}[c]{c}{\dot{v}}\\{\dot{r}}\end{array}}\right] =\left[{\begin{array}[c]{c}{{f_{2}}}\\{{f_{6}}}\end{array}}\right] +\left[{\begin{array}[c]{c}{{g_{21}}}\\{{g_{61}}}\end{array}}\right] \left[{{\delta_{r}}}\right].\end{split}$

      (70)

      In this subsystem, the yaw angle, $\psi$ , is the output.

      Both pitch and yaw mode subsystems can be further simplified, by assuming small perturbations in $\theta$ . In the following section, AFSM controllers are designed for diving and steering mode subsystems.

    • For simplicity, it is assumed that the heave velocity (w) is small during the diving phase[26].

      The equations of diving mode then are reduced to

      $\begin{split}& \dot z = - (\theta){u_{0}}\\& \dot\theta= q\\& \dot q = {f_{5}} + \left( {{g_{52}}} \right) \left[{{\delta_{s}}} \right].\end{split}$

      (71)

      It is noted that in the above equations, z is indirectly controlled through the variable $\theta$ , which is directly actuated by the input ${\delta_{s}}$ . The method proposed in Section 3 is used for defining the sliding surfaces. The relative degree of this subsystem is 2 and the tracking errors are defined as

      $\begin{split}& \tilde z = {z_{d}} - z\\& \tilde\theta= {\theta_{d}} - \theta\\& \tilde q = {q_{d}} - q.\end{split}$

      (72)

      In the case of REMUS AUV example, $z_{d}$ would be a positive constant scalar, $\theta_{d}=0$ and $q_{d}=0$ . Based on the methodology described in Section 3.2, ${{\tilde\theta}={\varphi_{1}}(z)}$ is chosen such that the time derivative of the following Lyapunov function would be negative definite

      ${V_{z}} = \frac{1}{2}s_{z}^{2} = \frac{1}{2}{\tilde z^{2}}$

      (73)

      i.e.,

      ${\dot V_{z}} = \tilde z\left( {{{\dot z}_{d}} - \dot z} \right) = \tilde z({\dot z_{d}} + {u_{0}}\theta) = \tilde z({\dot z_{d}} + {u_{0}}\left({{\theta_{d}} - {\varphi_{1}}} \right) ).$

      (74)

      Since ${{{\dot z}_{d}}}=0$ and ${{\theta_{d}}=0}$ , by defining ${\varphi_{1}}(z)={k_{1}}\tilde z$ , where ${k_{1}} > 0$ , the derivative of the Lyapunov function would be negative. Consider the first dynamic error as

      ${e_{1}} = \tilde\theta- {\varphi_{1}}(z).$

      (75)

      Again, the function $\tilde q = {\varphi_{2}}(z, \theta)$ is chosen such that the time derivative of the following Lyapunov function would be negative definite

      ${V_{e1}} = \frac{1}{2}e_{1}^{2}$

      (76)

      i.e.,

      ${\dot V_{e1}} = {e_{1}}({\dot e_{1}})$

      (77)

      and

      ${\dot e_{1}} = \dot{\tilde\theta}- {k_{1}}\dot{\tilde z}= \tilde q -{k_{1}}\dot{\tilde z}.$

      (78)

      In order to make the Lyapunov function negative, the function of ${\varphi_{2}}\left( {z, \theta} \right) $ is determined as

      ${\varphi_{2}}\left( {z, \theta} \right) = - {k_{2}}{e_{1}} + {k_{1}}\dot{\tilde z}$

      (79)

      where ${k_{2}} > 0$ .

      The final dynamic error is the sliding surface of the underactuated variable configuration which must be adopted for control of diving mode:

      $\begin{split}{s_{p}} = & {e_{2}} = \tilde q - {\varphi_{2}}\left( {z, \theta} \right) = \\ & \tilde q + {k_{2}}{e_{1}} - {k_{1}} \dot{\tilde z}=\\ & \tilde q + {k_{2}}\tilde\theta- {k_{2}}{k_{1}}\tilde z - {k_{1}}\dot{\tilde z}=\\ & \tilde q + {k_{2}}\tilde\theta- {k_{2}}{k_{1}}\tilde z - {k_{1}}{u_{0}}\theta=\\ & \tilde q + {k_{2}}\tilde\theta- {k_{2}}{k_{1}}\tilde z + {k_{1}}{u_{0}}\tilde\theta=\\ & \tilde q + \left( {{k_{2}} + {k_{1}}{u_{0}}} \right) \tilde\theta- {k_{2}}{k_{1}}\tilde z\end{split}$

      (80)

      where ${s_{p}}$ is the sliding surface used in control of depth in pitch mode.

    • In this mode, it can be assumed that $\dot{y}$ during the steering motion is small. Therefore, the steering equations of motion of (70) can be written as

      $\begin{split}& \dot{\psi}=r\\& \left[{\begin{array}[c]{c}{\dot{v}}\\{\dot{r}}\end{array}}\right] =\left[{\begin{array}[c]{c}{{f_{2}}}\\{{f_{6}}}\end{array}}\right] +\left[{\begin{array}[c]{c}{{g_{21}}}\\{{g_{61}}}\end{array}}\right] \left[{{\delta_{r}}}\right].\end{split}$

      (81)

      By combining these two equations, one gets

      $\left[{\begin{array}[c]{c}{\dot{v}}\\{\ddot{\psi}}\end{array}}\right] =\left[{\begin{array}[c]{c}{{f_{2}}}\\{{f_{6}}}\end{array}}\right] +\left[{\begin{array}[c]{c}{{g_{21}}}\\{{g_{61}}}\end{array}}\right] \left[{{\delta_{r}}}\right].$

      (82)

      If the distance between center of pressure of control fin and the vehicle′s center of gravity is denoted by $d_{0}$ , the relation between the sway and heading angle rate can be approximately written as

      $v = {d_{0}}\dot\psi$

      (83)

      i.e., (82) can be re-written as

      $\begin{split}& {d_{0}}\ddot\psi= {f_{2}} + ({g_{21}}){\delta_{r}}\\& \ddot\psi= {f_{6}} + ({g_{61}}){\delta_{r}}.\end{split}$

      (84)

      Therefore,

      $\ddot\psi= \frac{{\left( {{f_{2}} + {f_{6}}} \right) }}{{\left( {{d_{0}} +1} \right) }} + \frac{{({g_{21}} + {g_{61}})}}{{\left({{d_{0}} + 1} \right) }}{\delta_{r}}.$

      (85)

      This final equation is a single-input single-output equation, and the conventional sliding surface, $s_{y}$ , for this mode is

      ${s_{y}}=\left[{\begin{array}[c]{cc}\lambda & 1\end{array}}\right] \left[{\begin{array}[c]{c}{\tilde{\psi}}\\{\dot{\tilde{\psi}}}\end{array}}\right] =\left[{\begin{array}[c]{cc}\lambda & 1\end{array}}\right] \left[{\begin{array}[c]{c}{\tilde{\psi}}\\{\tilde{r}}\end{array}}\right] =\lambda\tilde{\psi}+\tilde{r}.$

      (86)

      where ${s_{y}}$ is the sliding surface used in control of vehicle in yaw mode.

    • In this section, the effectiveness of the proposed AFSMC method is assessed in controlling the 6 DOF dynamic model of the REMUS AUV. Technical details and performance testing results of REMUS are introduced in [24]. The dynamic equations of REMUS are used to study the performance of the proposed AFSMC algorithm in diving and steering modes of motion. In the first part of this section, disturbances are not taken into account but in the second part, they are considered in the form of ocean currents. Furthermore, the effect of un-modeled dynamics on the closed-loop performance of the system is studied.

      In order to demonstrate the statistic and accidental characteristics of the ocean currents, a first order Gaussian-Markov process is considered, which could appear in any direction. In particular, the following equation is considered:

      ${\dot V_{c}}(t) + {\mu_{0}}{V_{c}}(t) = \varpi(t)$

      (87)

      where $V_{c}$ is the earth-referenced velocity of the current and $\varpi$ is the vector of a white noise signal. In order to confine the magnitude of the ocean currents, a saturation block is used, as in Fig. 1.

      Figure 1.  Implementation of Gaussian-Markov[26] to model ocean currents

      Ocean currents are defined in the earth-based coordinate and could be transformed to the body fixed coordinates. One example of the induced disturbance on the system is shown in Fig. 2.

      Figure 2.  Typical ocean current velocity disturbance

    • The initial conditions for roll and surge are taken as

      $\begin{split}& {\phi_{0}} = 5\,\deg\\& {u_{0}} = 1.54\;{\rm{m/s}}\end{split}$

      and other initial conditions are considered to be zero. The desired depth and heading angle are

      $\begin{split}& {z_{d}} = 10\; {\rm{m}}\\& {\psi_{d}} = 30\,\deg.\end{split}$

      Adaptation rates for diving AFSM controller are selected as ${\alpha_{1}} = 10$ and ${\alpha_{2}} = 8$ and for steering controller as ${\alpha_{1}} = 9$ and ${\alpha_{2}} = 8$ .

      The initial values for the output membership functions, for both controllers are arbitrarily selected as

      ${\hat B_{i}} = \left[{\rm{{-0}}\rm{{.5;-0}}\rm{{.25;-0}}\rm{{.05;0}}\rm{{.05;0}}\rm{{.25;0}}\rm{{.5}}} \right], i =1, \cdots\!,\; 7.$

      In Figs. 3 and 4, the performance of the vehicle in vertical and horizontal modes is depicted.

      Figure 3.  AFSM control of depth–nominal case

      Figure 4.  AFSM control of heading angle–nominal case

      The movements of control fins are shown in Figs. 5 and 6.

      Figure 5.  Movements of top and bottom control fins–nominal case

      Figure 6.  Movements of right and left control fins–nominal case

    • To examine the effect of un-modeled dynamics, hydrodynamic coefficients of control fins are increased by 50 percent. The desired values for depth and heading angle are set as before, i.e.,

      $\begin{split}& {z_{d}} = 10\;{\rm{m}}\\& {\psi_{d}} = 30\,\deg.\end{split}$

      In Figs. 7 and 8, the performance of the vehicle under AFSM control in the presence of un-modeled dynamics and random disturbances described by (87) is depicted, for vertical and horizontal modes, respectively.

      Figure 7.  AFSM control of depth under disturbance and un-modeled dynamics

      Figure 8.  AFSM control of heading angle with disturbances and un-modeled dynamics

      The movements of the control fins under disturbance and un-modeled dynamics are shown in Figs. 9 and 10, respectively.

      Figure 9.  Movements of top and bottom control fins with disturbances and un-modeled dynamics

      Figure 10.  Movements of right and left control fins with disturbances and un-modeled dynamics

      It is clearly seen that the vehicle has been tracking the desired output variables very well. Figs. 7 and 8 show that there is no significant chattering after the vehicle reaches to the neighborhood of the desired values. However, Figs. 3 and 7 depict that the vehicle experiences a small steady state error in the control of depth. This phenomenon can be attributed to the cross-coupling between yaw and pitch modes, which was ignored at the design stage.

      By observing Figs. 4 and 8, in the very initial seconds, the adaptive control tries to recognize the system and determine sliding control bounds. Hence the system is not completely stable and then after it becomes stable.

      For better comprehension of the vehicle behavior, a 3D graph of vehicle trajectory is depicted in Fig. 11.

      Figure 11.  Vehicle trajectory in 3D view

    • The AFSMC strategy reduces the dependency of the control approach to the plant model and is robust under plant uncertainties and exogenous disturbances. Such advantages are obtained by using a central fuzzy controller, which is continuously tuned by observing the deviations from the sliding surface, and also by continuously estimating the bound of uncertainties by an adaptive approach. The concept of AFSMC was extended to the case of under-actuated systems, by successive application of a particular back-stepping approach. As a challenging example, a highly nonlinear, under-actuated and multi-mode control of the REMUS AUV was considered. Control design was performed by decoupling the vertical and horizontal motions. Simulation studies revealed, the exceptional performance of the proposed AFSMC under the effects of disturbance and un-modeled dynamics.

      Two separate controllers were designed for vehicle′s diving (pitch) and steering (yaw) modes of motion. It was shown that by decoupling the vertical and horizontal equations, only a small amount of error appeared in the control of depth for both nominal and perturbed systems.

    • The REMUS nonlinear dynamic equations of motion in six degrees of freedom are as follows:

      $\hspace{-18pt}\begin{split}& {(m - {X_{\dot u}})\dot u + m{z_g}\dot q - m{y_g}\dot r = }\\&\quad {X_{HS}} + {X_{u\left| u \right|}}u\left| u \right| + ({X_{wq}} - m)wq + \\&\quad ({X_{qq}} + m{x_g}){q^2} + ({X_{vr}} + m)vr + ({X_{rr}} + m{x_g}){r^2}-\\&\quad {m{y_g}pq - m{z_g}pr + {X_{prop}}}\end{split}$

      (88)

      $\hspace{-82pt}\begin{split}& {(m - {Y_{\dot v}})\dot v - m{z_g}\dot p + (m{x_g} - {Y_{\dot r}})\dot r = }\\& \quad{Y_{HS}} + {Y_{v\left| v \right|}}v\left| v \right| + {Y_{r\left| r \right|}}r\left| r \right| + m{y_g}{r^2} + \\& \quad({Y_{ur}} - m)ur + ({Y_{wp}} + m)wp + \\& \quad{({Y_{pq}} - m{x_g})pq + {Y_{uv}}uv + m{y_g}{p^2} + }\\& \quad{m{z_g}qr + {Y_{uu\delta r}}{u^2}\delta r}\end{split}$

      (89)

      $\hspace{-30pt}\begin{split}& (m - {Z_{\dot w}})\dot w + m{y_{g}}\dot p - (m{x_{g}} + {Z_{\dot q}})\dot q = \\& \quad{Z_{HS}} + {Z_{w\left| w \right| }}w\left| w \right| + {Z_{q\left| q\right| }}q\left| q \right| + ({Z_{uq}} + m)uq + \\& \quad ({Z_{vp}} - m)vp + ({Z_{rp}} - m{x_{g}})rp + {Z_{uw}}uw + \\& \quad m{z_{g}}({p^{2}}+ {q^{2}}) - m{y_{g}}qr + {Z_{uu{\delta_{s}}}}{u^{2}}{\delta_{s}}\end{split}$

      (90)

      $\hspace{-25pt}\begin{split}& - m{z_{g}}\dot v + m{y_{g}}\dot w + ({I_{xx}} - {K_{\dot p}})\dot p = \\& \quad {K_{HS}} + {K_{p\left| p \right| }}p\left| p \right| - ({I_{zz}} - {I_{yy}})qr + m(uq - vp) - \\& \quad m{z_{g}}(wp - ur) + {K_{prop}}\end{split}$

      (91)

      $\hspace{-34pt}\begin{split} & m{z_{g}}\dot u - (m{x_{g}} + {M_{\dot w}})\dot w + ({I_{yy}} - {M_{\dot q} })\dot q =\quad\quad\quad\;\;\\ & \quad {M_{HS}} + {M_{w\left| w \right| }}w\left| w \right| + {M_{q\left| q \right| }}q\left| q \right| + \\ & \quad ({M_{uq}} - m{x_{g}})uq + ({M_{vp}} + m{x_{g}})vp + \\ & \quad[{M_{rp}}-({I_{xx}}-{I_{zz}})]rp + m{z_{g}}(vr - wq) + \\ & \quad{M_{uw}}uw + {M_{uu{\delta_{s}}}}{u^{2}}{\delta_{s}}\end{split}$

      (92)

      $\hspace{-24pt}\begin{split}& { - m{y_g}\dot u + (m{x_g} - {N_{\dot v}})\dot v + ({I_{zz}} - {N_{\dot r}})\dot r = }\\& \quad {{N_{HS}} + {N_{v\left| v \right|}}v\left| v \right| + {N_{r\left| r \right|}}r\left| r \right| + ({N_{ur}} - m{x_g})ur + }\\& \quad {({N_{wp}} + m{x_g})wp + [{N_{pq}} - ({I_{yy}} - {I_{xx}})]pq}-\\& \quad {m{y_g}(vr - wq) + {N_{uv}}uv + {N_{uu{\delta _r}}}{u^2}{\delta _r}.}\end{split}$

      (93)
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