Real Time Implementation of Type-2 Fuzzy Backstepping Sliding Mode Controller for Twin Rotor MIMO System (TRMS)

The autopilot of planes and helicopters was born with modern aviation and has evolved over time to meet increasingly restrictive needs. It can be used when the task to be performed is too repetitive or too difficult for the pilot. The control of the automatic control of the evolution of helicopters controlled by radio opens the way to applications in the fields of security (surveillance of the airspace, the urban and interurban traffic), the management of the natural risks (monitoring of volcano activity), the environment (measurement of air pollution, forest monitoring), for intervention in hostile environments (radioactive environments, demining of land without human intervention), management major infrastructures (dams, high-voltage lines, pipelines), agriculture (detection and treatment of infested crops) and aerial photography in film production [1, 2].

The TRMS (Twin Rotor MIMO System) is an aerodynamic physical system similar to a helicopter, designed for the development and implementation of new control laws. [3- 5] Several works have been done to control of the TRMS, the researchers have been developed the control strategies to solve the problems of this type of system. The sliding mode control, is a robust control in the presence of parametric variations, characterized by its robustness against nonlinearity and its efficiency in the rejection of disturbances, [6, 7], in [8] sliding mode control is designed for a linearized model of the TRMS system. The authors in [9], proposed an adaptive second-order sliding mode controller in objectives to stabilize and control the TRMS system in significant cross couplings, in [10], a model for the TRMS is used based an optimal linear quadratic regulator and a sliding mode controller. Real-time control of the vertical and horizontal positions of TRMS using a decentralized sliding mode controller is presented in [11].

The major disadvantage of sliding mode is the chattering phenomenon, which can damage the actuators by too frequent stress and impair the operation and performance of the system, in order to reduce these oscillations several solutions have been made, for example: use artificial intelligence tools, including fuzzy logic type 1 and type 2, in [2] authors presented the real time implementation of non linear observer-based on type-1 fuzzy sliding mode controller for TRMS system, the proposed control in this paper can be attenuating the chattering phenomena of the sliding mode control. In [12, 13] the type 2 fuzzy logic and adaptive type 2 fuzzy logic controllers are proposed to stabilize the TRMS Helicopter system, using two independent type-2 fuzzy controllers for the yaw and pitch angles of the TRMS, the performance of each control scheme is examined under a number of simulations.

In this work, the real time implementation of T2FBSMC approaches for a TRMS system is proposed. An interval type-2 fuzzy logic is used to solve the chattering problem due to the correction term. Compared to previous studies on sliding mode control [9, 11], the proposed control approach can reduce the chattering phenomenon and obtained a good dynamic response. Compared to boundary layer sliding mode control [2, 10] and higher order sliding mode control, the control approach can also reduce these oscillations. The contributions of this paper could be briefly summarized as follows.

1. An effective and robust controller is developed for TRMS system with external disturbances.
2. A fuzzy logic type 2 is used to reduce the chattering phenomenon.
3. The type 2 fuzzy logic inference mechanism is employed in the stability analysis.

The rest of the paper is organized as follows. Section 2 focuses on the nonlinear dynamic model of the 2-DOF helicopter (TRMS). Design of the T2FBSMC is highlighted in section 3 and 4. Simulation results and related discussions are given in section 5. The experimental results to validate the effectiveness of the proposed approach are presented in Section 6. Finally some conclusions are drawn in section 7.

Since the characteristic of TRMS is very complex in the nature, it would be convenient to design a controller for TRMS with the TRMS decoupled into horizontal and vertical subsystems by fixing the horizontal angle  a_h and posing  u_h=0 , from Eq. (30), it is easy to see that state equations with the state vector  X_v  for the vertical subsystem of the TRMS could be defined as:

${{X}_{v}}=\left[ {{x}_{1}},{{x}_{2}},{{x}_{3}} \right]=\left[ {{\alpha }_{v}},{{S}_{v}},{{u}_{vv}} \right]$ (31)

$\left\{ \begin{align}  & {{{\dot{x}}}_{1}}\ =\ \ {{x}_{2}} \\ & {{{\dot{x}}}_{2}}=\ \ \frac{1}{{{J}_{v}}}\left\{ {{l}_{m}}\ {{S}_{f}}\ {{F}_{v}}\ \left( {{\omega }_{m}}\ \left( {{x}_{3}} \right) \right)-{{K}_{v}}\left( {{x}_{2}} \right)\ +g\ \left( \left( A\ -\ B\  \right)\ \cos \ \ x{{\ }_{1}}-\ \ C\ \sin \ {{x}_{1}} \right) \right\} \\ & {{{\dot{x}}}_{3}}=\ \ \frac{1}{{{T}_{mr}}}\left( -{{x}_{3}}+{{K}_{mr}}\ {{u}_{v}} \right) \\ \end{align} \right.$ (32)

where,  u_v  is a control action of the vertical subsystem Likewise, we have the horizontal subsystem by posing  $α_v=α_v (0)=α_v0$ and $u_v =(0)$      

${{X}_{h}}=\left[ {{x}_{4}},{{x}_{5}},{{x}_{6}} \right]=\left[ {{\alpha }_{h}},{{S}_{h}},{{u}_{hh}} \right]$ (33)

$\left\{ \begin{align}  & {{{\dot{x}}}_{4}}=\ \ {{x}_{5}} \\ & {{{\dot{x}}}_{5}}=\ \ \frac{1}{{{J}_{h}}\left( {{\alpha }_{v0}} \right)}\ \left\{ {{l}_{t}}\ {{S}_{f}}\ {{F}_{h}}\ \left( {{\omega }_{t}} \right)\cos \ \alpha {}_{v0}-\ {{K}_{h}}\ {{x}_{5}} \right\} \\ & {{{\dot{x}}}_{6}}=\ \ \frac{1}{{{T}_{tr}}}\left( -{{x}_{6}}+{{K}_{tr}}\ {{u}_{h}} \right) \\ \end{align} \right.$ (34)

The parameters of the model are shown in Table 1.

Table 1. The parameters of the TRMS [15]

The advantage of control algorithms based on sliding mode techniques is its insensitivity to the model errors and parametric uncertainties, as well as the ability to globally stabilize the system in the presence of other disturbances.

The recursive nature of the proposed control design is similar to the standard backstepping methodology. However, the proposed control design uses backstepping to design virtual controllers with a zero-order sliding surface at each recursive step. The benefit of this approach is that each virtual controller can compensate for unknown bounded function which contains unmodelled dynamics and external disturbances [23]. 

The control algorithms based on sliding mode techniques suffers from a main disadvantage that is chattering effect, which is the high frequency oscillation of the controller output. To overcome this problem and in order to reduce the chattering phenomenon, an interval type-2 fuzzy system is used to approximate the hitting control term. The configuration of the proposed type-2 fuzzy backstepping sliding mode control (T2FBSMC) scheme is shown in Figure 7; it contains an equivalent control part and single input single output interval type-2 fuzzy logic.

7.png

Figure 7. Block diagram of the IT2FSMC

The equivalent control $u_eq$, is calculated in such a way as to have $\overset{.}{\mathop{\text{s}}}\,$. Then the hitting control is computed by:

${{u}_{r}}={{k}_{fs}}{{u}_{fs}}$ (44)

${{u}_{fs}}=T2FLC\ \left( s \right)$ (45)

where, $(k_{fs}>0)$ is the normalization factor of the output variable, and $u_{fs}$ is the output of the T2FLC, which is obtained by the normalized s.

The fuzzy type-2 membership functions of the input sliding surface(s), and the output discontinuous control ($u_{fs}$) sets are presented by Figure 8.

In order to attenuate the chattering effect and handle the uncertainty of the six rotors helicopter, a type‐2 fuzzy controller has been used with single input and single output for each subsystem. Then, the input of the controller is the sliding surface and the output is the discontinuous control $u_{fs}$. All the membership functions of the fuzzy input variable are chosen to be triangular and trapezoidal for all upper and lower membership functions. The used labels of the fuzzy variable (surface) are: {negative medium (NM), negative big (NB), zero (ZE), positive medium (NM), positive big (PB)}.

The corrective control is decomposed into five levels represented by a set of linguistic variables: negative big (NB), negative medium (NM), zero (ZE), positive medium (PM) and positive big (PB). Table.2 presents the rules base which contains five rules.

The membership  functions  of  the  input  (sliding  surface)  and  output  (ufs)  has  been normalized in the interval [-1, 1], therefore: |{u_fs}|≤1.

8.png

Figure 8. Membership functions of input  s  and output  u_fs  [24]

Table 2. Fuzzy rules for type-2 FLCs [24]

$u_{fs}$ given in equation (30) satisfies the following condition:

$s{{u}_{fs}}=-{{K}^{+}}\left| s \right|$(46)

where , K⁺>0  is positive constant determined by a fuzzy type-2 inference system.       

Proof

The hitting control laws are computed by type-2 fuzzy logic inference using equations (42) and (43) and the iterative Karnik Mendel Algorithms presented in [16-22]. Where $α_i=[α_{ilow},α_{iup}]$ for i=[1,...,5] are the membership interval of rules 1 to 5 presented in Table 2. Moreover, $u_{fs} can be further analyzed as the following six conditions given thereafter. Only one of six conditions will occur for any value of the sliding surface according to Figure 3 [24].

Condition 1

Only rule 1 is activated ($s>0.5,\ \alpha {{}_{1}}=\left[ 0.8,1 \right],{{\alpha }_{j}}=\left[ 0,0 \right]$ for  $j=2,3,4,5$)

$u_fs=T2FLC (s)=(-0.8-1)/2=-0.9$ (47)

Condition 2

Rule 1 and 2 are activated ($0.25<s<0.5,\ \alpha {{}_{1}}=\left[ {{\alpha }_{1low}},{{\alpha }_{1up}} \right],\ \alpha {{}_{2}}=\left[ {{\alpha }_{2low}},{{\alpha }_{2up}} \right],{{\alpha }_{j}}=\left[ 0,0 \right]$ for  $j=3,4,5$)

and 

${{u}_{fs}}=T2FLC\ \left( s \right)=\frac{1}{2}\left( \frac{-0.8\ {{\alpha }_{1low}}-0.3{{\alpha }_{2up}}}{{{\alpha }_{1low}}+{{\alpha }_{2up}}}+\frac{-\ {{\alpha }_{1up}}-0.5{{\alpha }_{2low}}}{{{\alpha }_{1up}}+{{\alpha }_{2low}}} \right)$ (48)

Condition 3

Rule 2 and 3 are activated ($0<s<0.25,\ \alpha {{}_{2}}=\left[ {{\alpha }_{2low}},{{\alpha }_{2up}} \right],\ \alpha {{}_{3}}=\left[ {{\alpha }_{3low}},{{\alpha }_{3up}} \right],{{\alpha }_{j}}=\left[ 0,0 \right]$ for  $j=1,4,5$)

$0\le {{\alpha }_{2low}},{{\alpha }_{3low}}\le 0.8$ and $0\le \alpha {{}_{2up}},{{\alpha }_{3up}}\le 1$

${{u}_{fs}}=T2FLC\ \left( s \right)=\frac{1}{2}\left( \frac{-0.3\ {{\alpha }_{2low}}+0.1{{\alpha }_{3up}}}{{{\alpha }_{2low}}+{{\alpha }_{3up}}}+\frac{-\ 0.5{{\alpha }_{2up}}}{{{\alpha }_{2up}}+{{\alpha }_{3low}}} \right)$ (49)

Condition 4

Rule 3 and 4 are activated ($-0.25<s<0,\ \alpha {{}_{3}}=\left[ {{\alpha }_{3low}},{{\alpha }_{3up}} \right],\ \alpha {{}_{4}}=\left[ {{\alpha }_{4low}},{{\alpha }_{4up}} \right],{{\alpha }_{j}}=\left[ 0,0 \right]$ for  $j=1,2,5$)

$0\le {{\alpha }_{3low}},{{\alpha }_{4low}}\le 0.8$ and  $0\le \alpha {{}_{4up}},{{\alpha }_{4up}}\le 1$

${{u}_{fs}}=T2FLC\ \left( s \right)=\frac{1}{2}\left( \frac{0.1{{\alpha }_{3low}}+0.5\ {{\alpha }_{4up}}}{{{\alpha }_{3low}}+{{\alpha }_{4up}}}+\frac{\ 0.3{{\alpha }_{4low}}}{{{\alpha }_{3up}}+{{\alpha }_{4low}}} \right)$ (50)

Condition 5

Rule 4 and 5 are activated ($-0.5<s<-0.25,\ \alpha {{}_{4}}=\left[ {{\alpha }_{4low}},{{\alpha }_{4up}} \right],\ \alpha {{}_{5}}=\left[ {{\alpha }_{5low}},{{\alpha }_{5up}} \right],{{\alpha }_{j}}=\left[ 0,0 \right]$ for  $j=1,2,3$)

$0\le {{\alpha }_{4low}},{{\alpha }_{4low}}\le 0.8$ and  $0\le \alpha {{}_{5up}},{{\alpha }_{5up}}\le 1$

${{u}_{fs}}=T2FLC\ \left( s \right)=\frac{1}{2}\left( \frac{0.5\ {{\alpha }_{4low}}+{{\alpha }_{5up}}}{{{\alpha }_{4low}}+{{\alpha }_{5up}}}+\frac{0.3\ {{\alpha }_{4up}}+0.8{{\alpha }_{5low}}}{{{\alpha }_{4up}}+{{\alpha }_{5low}}} \right)$ (51)

Condition 6

Only rule 5 is activated ($s<-0.5,\ \alpha {{}_{5}}=\left[ 0.8,1 \right],{{\alpha }_{j}}=\left[ 0,0 \right]$ for $j=1,2,3,4$)

${{u}_{fs}}=IT2FLC\ \left( s \right)=\frac{1+0.8}{2}=0.9$ (52)

According to six possible conditions shown in (33)-(38) we conclude

$s{{u}_{fs}}=s\,T2FLC\ \left( s \right)=-{{K}^{+}}\left| s \right|$ (53)

with:

${{K}^{+}}=\left\{ \begin{align}  & 0.9\quad if\quad s>0.5\,ands<-0.5 \\ & \left| \frac{1}{2}\left( \frac{-0.8\ {{\alpha }_{1low}}-0.3{{\alpha }_{2up}}}{{{\alpha }_{1low}}+{{\alpha }_{2up}}}+\frac{-\ {{\alpha }_{1up}}-0.5{{\alpha }_{2low}}}{{{\alpha }_{1up}}+{{\alpha }_{2low}}} \right) \right|\quad \quad if\quad 0.25<s<0.5 \\ & \left| \frac{1}{2}\left( \frac{-0.3\ {{\alpha }_{2low}}+0.1{{\alpha }_{3up}}}{{{\alpha }_{2low}}+{{\alpha }_{3up}}}+\frac{-\ 0.5{{\alpha }_{2up}}}{{{\alpha }_{2up}}+{{\alpha }_{3low}}} \right) \right|\quad \quad \quad \quad if\quad \ \ 0<s<0.25 \\ & \left| \frac{1}{2}\left( \frac{0.1{{\alpha }_{3low}}+0.5\ {{\alpha }_{4up}}}{{{\alpha }_{3low}}+{{\alpha }_{4up}}}+\frac{\ 0.3{{\alpha }_{4low}}}{{{\alpha }_{3up}}+{{\alpha }_{4low}}} \right) \right|\quad \quad \quad \quad \ \ if\quad -0.25<s<0 \\ & \left| \frac{1}{2}\left( \frac{0.5\ {{\alpha }_{4low}}+{{\alpha }_{5up}}}{{{\alpha }_{4low}}+{{\alpha }_{5up}}}+\frac{0.3\ {{\alpha }_{4up}}+0.8{{\alpha }_{5low}}}{{{\alpha }_{4up}}+{{\alpha }_{5low}}} \right) \right|\quad \quad \quad if-0.5<s<-0.25 \\ \end{align} \right.$ (54)

In Figure 7 the control law is computed by:

$u={{u}_{eq}}+{{u}_{r}}={{u}_{eq}}+{{k}_{fs}}{{u}_{fs}}$  (55)

Then sliding condition can be rewritten as follow:

$s\dot{s}=-{{k}_{fs}}{{K}^{+}}\left| s \right|\ <0$  (56)

Or

$\dot{s}=-{{k}_{fs}}T2FLC\ \left( s \right)$ (57)

To simplify the synthesis of our controller, we divide the TRMS model into two subsystem models, vertical and horizontal, with 1 degree of freedom each; the coupling effect is considered as uncertainties. We propose to use a decentralized control, that is, from each subsystem model, a T2FBSMC is designed and then the resulting controllers are used to control the TRMS (Figure 9).

Let's start with the vertical subsystem whose model is obtained by the equation (32):

$\left\{ \begin{align}  & \dot{x}{{\ }_{1}}\ =\ \ {{x}_{2}} \\ & {{{\dot{x}}}_{2}}=\ \ {{f}_{v}}\left( {{x}_{3}} \right)-{{b}_{v}}\ {{x}_{2}}+{{g}_{v}}\left( {{x}_{1}} \right) \\ & {{{\dot{x}}}_{3}}=\ \ -{{C}_{v\ }}{{x}_{3}}+{{d}_{v}}\ {{u}_{v}} \\ \end{align} \right.$ (58)

where

$\left\{ \begin{align}  & \dot{x}{{\ }_{1}}\ =\ \ {{x}_{2}} \\ & {{{\dot{x}}}_{2}}=\ \ {{f}_{v}}\left( {{x}_{3}} \right)-{{b}_{v}}\ {{x}_{2}}+{{g}_{v}}\left( {{x}_{1}} \right) \\ & {{{\dot{x}}}_{3}}=\ \ -{{C}_{v\ }}{{x}_{3}}+{{d}_{v}}\ {{u}_{v}} \\ \end{align} \right.$ (59)

To determine the control law, we proceed in three steps as follows:

Step 1

We define the first error variable ${{z}_{1v}}$ as the tracking error, such as:

${{z}_{1v}}={{x}_{1}}-{{x}_{1d}}={{\alpha }_{v}}-{{\alpha }_{vd}}$ (60)

We define the first function of Lyapunov $V_1v$ by:

${{V}_{1v}}=\frac{1}{2}z_{1v}^{2}$ (61)

The time derivative of (61) is obtained by:

${{\dot{V}}_{1v}}={{z}_{1v}}\ {{\dot{z}}_{1v}}={{z}_{1v}}\left( {{x}_{2}}-{{{\dot{x}}}_{1d}} \right)$ (62)

There is no control input in (38). By letting  $x_2$ be the virtual control, the desired virtual control is defined as:

${{\left( {{x}_{2}} \right)}_{d}}=-{{c}_{1v}}{{z}_{1v}}+{{\dot{x}}_{1d}}$ (63)

where, $c_1v$ is a positive constant for increasing the convergence speed of the vertical angle tracking loop.

Now, the virtual control is $x_2$ where the second error tracking is defined by:

${{z}_{2v}}={{x}_{2}}-{{\left( {{x}_{2}} \right)}_{d}}={{x}_{2}}+{{c}_{1v}}{{z}_{1v}}-{{\dot{x}}_{1d}}$ (64)

Step 2

The augmented Lyapunov function for the second step is given by:

${{V}_{2v}}=\frac{1}{2}z_{1v}^{2}+\frac{1}{2}z_{2v}^{2}$ (65)

The derivative of (65) is given by:      

${{\dot{V}}_{2v}}=-{{c}_{1v}}z_{1v}^{2}+{{z}_{2v}}{{\dot{z}}_{2v}}$ (66)

Substituting  ${{\dot{z}}_{2v}}$ in (66), we obtain:

${{\dot{V}}_{2v}}={{z}_{1v}}\ {{\dot{z}}_{1v}}+{{z}_{2v}}\ \left( {{f}_{v}}\left( {{x}_{3}} \right)-{{b}_{v}}{{x}_{2}}+{{g}_{v}}\left( {{x}_{1}} \right)+{{c}_{1v}}\left( {{x}_{2}}-\dot{x}_{1d}^{{}} \right)-\ddot{x}_{1d}^{{}} \right)$ (67)

By letting  ${{f}_{v}}\left( {{x}_{3}} \right)+{{g}_{v}}\left( {{x}_{1}} \right)$ be the virtual control, the desired virtual control in the second step is defined as:

${{\left( {{f}_{v}}\text{ }\left( {{x}_{3}}\text{ } \right)+{{g}_{v}}\text{ }\left( {{x}_{1}}\text{ } \right) \right)}_{d}}=-{{c}_{2v}}\text{ }{{z}_{2v}}+{{b}_{v}}\ {{x}_{2}}-{{c}_{1v}}\text{ }\left( {{x}_{2}}-{{x}_{1d}} \right)+\ddot{x}{}_{1d}$         

with 

$\left( {{c}_{2v}}>0 \right)$ (68)

Now, the virtual control is $\left( {{c}_{2v}}>0 \right)$ where the sliding surface is defined in the third step by:

${{s}_{v}}={{z}_{3v}}=\left( {{f}_{v}}\left( {{x}_{3}} \right)+{{g}_{v}}\left( {{x}_{1}} \right) \right)-{{\left( {{f}_{v}}\left( {{x}_{3}} \right)+{{g}_{v}}\left( {{x}_{1}} \right) \right)}_{d}}$

$={{f}_{v}}\left( {{x}_{3}} \right)+{{g}_{v}}\left( {{x}_{1}} \right)+{{c}_{2v}}\ {{z}_{2v}}-{{\ddot{x}}_{1d}}+{{c}_{1v}}\left( {{x}_{2}}-{{{\dot{x}}}_{1d}} \right)-{{b}_{v}}\ {{x}_{2}}$ (69)

Step 3

The augmented Lyapunov function for the third step is given by:

${{V}_{3v}}=\frac{1}{2}z_{1v}^{2}+\frac{1}{2}z_{2v}^{2}+\frac{1}{2}s_{v}^{2}$ (70)

The time derivative of (69) is given by:

${{\dot{V}}_{3v}}=z_{1v}^{{}}\dot{z}_{1v}^{{}}+z_{2v}^{{}}\dot{z}_{2v}^{{}}+{{s}_{v}}{{\dot{s}}_{v}}$ (71)

Substituting $s ̇_v$ in (71), we obtain:

${{\dot{V}}_{3v}}=-{{c}_{1v}}\,z_{1v}^{2}-{{c}_{2v}}\,z_{2v}^{2}+{{s}_{v}}\ \left( \frac{\partial \ {{f}_{v}}\left( {{x}_{3}} \right)\ }{\partial {{x}_{3}}}\dot{x}{{}_{3}}+\frac{\partial \ {{g}_{v}}\left( {{x}_{1}} \right)\ }{\partial \ {{x}_{1}}}{{{\dot{x}}}_{1}}+{{c}_{2v}}\left( {{{\dot{x}}}_{2}}+{{c}_{1\ v}}\ \left( {{x}_{2}}-{{{\dot{x}}}_{1\ d}} \right)-{{{\ddot{x}}}_{\ 1\ d}} \right)-{{{}}_{1d}}+{{c}_{1v}}\ \left( {{{\dot{x}}}_{2}}-\ddot{x}{}_{1d} \right)-{{b}_{v}}\ {{{\dot{x}}}_{2}} \right)$ (72)

The chosen low for the attractive surface is the time derivative of (69) stratifying the necessary condition of sliding $(s_v s ̇_v )<0$ obtained in (57):

${{\dot{s}}_{v}}=-{{k}_{fsv}}T2FLC\ \left( {{s}_{v}} \right)$ (73)

As for the proposed T2FBSMC approach, the control input  $u_v$ is extracted:

${{u}_{v}}=\frac{1}{{{d}_{v}}\frac{\partial \ {{f}_{v}}\left( {{x}_{3}} \right)}{\partial {{x}_{3}}}}\left[ \begin{align}  & {{C}_{v}}\frac{\partial {{f}_{v}}\left( {{x}_{3}} \right)}{\partial {{x}_{3}}}{{x}_{3}}-{{c}_{\ 2v}}\left( {{{\dot{x}}}_{2}}+c{{\ }_{1v}}\ \left( {{x}_{2}}-{{{\dot{x}}}_{1d}} \right)-{{{\ddot{x}}}_{1d}} \right)+{{{}}_{1d}}+\ b{{\ }_{v}}\ \left( {{f}_{v}}\left( {{x}_{3}} \right)-{{b}_{\ v}}\ {{x}_{\ 2}}+g{{\ }_{v}}\ \left( x{{\ }_{1}} \right) \right) \\ & -\frac{\partial {{g}_{v}}\left( {{x}_{1}} \right)}{\partial {{x}_{1}}}{{x}_{2}}-{{c}_{1v}}\ \left( {{{\dot{x}}}_{2}}-\ddot{x}{}_{1d} \right)-{{k}_{fsv}}T2FLC\ \left( {{s}_{v}} \right) \\\end{align} \right]$ (74)

After the design of the proposed T2FBSMC, we must find a control law to reach it and stay thereafter:

${{u}_{v}}={{u}_{eqv}}+{{u}_{rv}}$ (75)

where, $u_eqv$ is the equivalent control. It makes the derivative of the sliding surface equal to zero to stay on the sliding surface. $u_rv$ is the hitting control.

The condition to stay on the sliding surface is $s ̇_v=0$; therefore, the equivalent control is:

${{u}_{eqv}}=\frac{1}{{{d}_{v}}\frac{\partial \ {{f}_{v}}\left( {{x}_{3}} \right)}{\partial {{x}_{3}}}}\left[ {{C}_{v}}\frac{\partial {{f}_{v}}\left( {{x}_{3}} \right)}{\partial {{x}_{3}}}{{x}_{3}}-{{c}_{2v}}\left( {{{\dot{x}}}_{2}}+{{c}_{1v}}\ \left( {{x}_{2}}-{{{\dot{x}}}_{1d}} \right)-{{{\ddot{x}}}_{1d}} \right)+{{{}}_{1d}}+\ b{{\ }_{v}}\ \left( {{f}_{v}}\left( {{x}_{3}} \right)-{{b}_{\ v}}\ {{x}_{\ 2}}+g{{\ }_{v}}\ \left( x{{\ }_{1}} \right) \right)-... \right.$

$\left. \frac{\partial {{g}_{v}}\left( {{x}_{1}} \right)}{\partial {{x}_{1}}}{{x}_{2}}-{{c}_{1v}}\ \left( {{{\dot{x}}}_{2}}-\ddot{x}{}_{1d} \right) \right]$(76)

${{u}_{rv}}=-\,\left( \frac{{{k}_{fsv}}}{{{d}_{v}}\frac{\partial \ {{f}_{v}}\left( {{x}_{3}} \right)}{\partial {{x}_{3}}}} \right)T2FLC\ \left( {{s}_{v}} \right)$ (77)

where $k_{fsv}$ is a positive constant.

Similar to the vertical subsystem case, the control law  $u_h$ for the horizontal subsystem is:

${{u}_{h}}={{u}_{eqh}}+{{u}_{rh}}$ (78)

$u{{}_{h}}\ =\ \frac{1}{{{d}_{h}}\frac{\partial {{f}_{h}}\left( {{x}_{6}} \right)}{\partial {{x}_{6}}}}\,\,\left[ {{C}_{h}}\frac{\partial {{f}_{h}}\left( x{{}_{6}} \right)}{\partial {{x}_{6}}}{{x}_{6}}-{{c}_{\ 2h}}\ \left( {{{\dot{x}}}_{5}}+c{{\ }_{1h}}\ \left( {{x}_{5}}-{{{\dot{x}}}_{4d}} \right)-{{{\ddot{x}}}_{4d}} \right)+{{}_{4d}}\ +b{{\ }_{h}}\ \left( {{f}_{h}}\left( {{x}_{6}} \right)-{{b}_{\ h}}\ {{x}_{\ 5}} \right)-{{c}_{1\ h}}\ \left( {{{\dot{x}}}_{5}}-\ddot{x}\ {}_{4d} \right)-... \right.$

$\left. {{k}_{fsh}}T2FLC\ \left( {{s}_{h}} \right) \right]$(79)

With:

\[\left( {{c}_{1h}},{{c}_{2h\text{ }}} \right)>0\text{ }and\text{ }{{s}_{h}}={{z}_{3h}}={{f}_{h}}\left( {{x}_{6}} \right)+{{c}_{2h}}{{z}_{2h}}-\ddot{x}{{\text{ }}_{4d}}+{{c}_{1h}}\left( {{x}_{5}}-{{{\dot{x}}}_{4d}} \right)-{{b}_{h}}\ {{x}_{5}}\]

The equivalent control for the horizontal subsystem is:

$u{{}_{eqh}}\ =\ \frac{1}{{{d}_{h}}\frac{\partial {{f}_{h}}\left( {{x}_{6}} \right)}{\partial {{x}_{6}}}}\,\,\left[ {{C}_{h}}\frac{\partial {{f}_{h}}\left( x{{}_{6}} \right)}{\partial {{x}_{6}}}{{x}_{6}}-{{c}_{2h}}\ \left( {{{\dot{x}}}_{5}}+c{{\ }_{1h}}\ \left( {{x}_{5}}-{{{\dot{x}}}_{4d}} \right)-{{{\ddot{x}}}_{4d}} \right)+{{{}}_{4d}}\ +b{{\ }_{h}}\ \left( {{f}_{h}}\left( {{x}_{6}} \right)-{{b}_{\ h}}{{x}_{\ 5}} \right)-{{c}_{1\ h}}\ \left( {{{\dot{x}}}_{5}}-\ddot{x}\ {}_{4d} \right) \right]$ (80)

and for the following corrective control

${{u}_{rh}}\ =-\ \left( \frac{{{k}_{fsh}}}{{{d}_{h}}\frac{\partial {{f}_{h}}\left( {{x}_{6}} \right)}{\partial {{x}_{6}}}} \right)\,\,T2FLC\ \left( {{s}_{h}} \right)$ (81)

The bloc diagram of the proposed T2FBSMC is shown in Figure 9.

9.png

Figure 9. Block diagram of the proposed T2FBSMC