Robust H_2 / H_∞ Multi Objective Controller Design with Takagi-Sugeno Fuzzy Model for a Mobile Two-Wheeled Inverted Pendulum

الموضوعات : مهندسی هوشمند برق

Davood Allahverdy ¹ (Science and Research Branch, Islamic Azad University, Tehran, IRAN)
Ahmad Fakharian ² (Faculty of Electrical, Biomedical and Mechatronics Engineering, Qazvin Branch, Islamic Azad University)

تاريخ الإرسال : 25 السبت , صفر, 1443 تاريخ التأكيد : 03 الثلاثاء , جمادى الأولى, 1443 تاريخ الإصدار : 28 الثلاثاء , رجب, 1443

الکلمات المفتاحية: Robust H_2/H_∞ Multi Objective Controller, linear Matrix Inequalities, Takagi-Sugeno Fuzzy Model, Mobile Two-Wheeled Inverted Pendulum,

ملخص المقالة :

In this study, a robust H_2/H_∞ multi-objective state-feedback controller and tracking design are presented for a mobile two-wheeled inverted pendulum (MTWIP). The proposed control has to track the desired angular velocity while keeping the mobile two-wheeled inverted pendulum balanced. First, error of output states are added to the dynamic of system for better tracking control. And uncertainties of parameters are defined by affine parameters. Next, Takagi-Sugeno (T-S) fuzzy model is used for estimating the uncertainty of nonlinear model parameters. Robust H_2/H_∞ controller is designed and analyzed for each local linear subsystem of mobile two-wheeled inverted pendulum by using a linear matrix inequalities method. To sum up, in order to calculate the whole dynamic of system from each local linear subsystem, weight average defuzzifer method is used and the total controller is designed and analyzed according to parallel distribute compensation. The simulation indicate that the proposed scheme has high accuracy, robustness, good tracking, fast transient responses and lower control effort for a mobile two-wheeled inverted pendulum despite the uncertainties and external disturbance.

نص كامل:



Robust  /  Multi Objective Controller Design with Takagi-Sugeno Fuzzy Model for a Mobile Two-Wheeled Inverted Pendulum

D. Allahverdy¹ and A. Fakharian²

¹Science and Research Branch, Islamic Azad University, Tehran, IRAN, Email: [e-mail]

²Faculty of Electrical, Biomedical and Mechatronics Engineering, Qazvin Branch, Islamic Azad University, Qazvin, Iran, Corresponding Author Email: [e-mail]

 

 

Abstract

In this study, a robust / multi-objective state-feedback controller and tracking design are presented for a mobile two-wheeled inverted pendulum (MTWIP). The proposed control has to track the desired angular velocity while keeping the mobile two-wheeled inverted pendulum balanced. First, error of output states are added to the dynamic of system for better tracking control. And uncertainties of parameters are defined by affine parameters. Next, Takagi-Sugeno (T-S) fuzzy model is used for estimating the uncertainty of nonlinear model parameters. Robust / controller is designed and analyzed for each local linear subsystem of mobile two-wheeled inverted pendulum by using a linear matrix inequalities method. To sum up, in order to calculate the whole dynamic of system from each local linear subsystem, weight average defuzzifer method is used and the total controller is designed and analyzed according to parallel distribute compensation. The simulation indicate that the proposed scheme has high accuracy, robustness, good tracking, fast transient responses and lower control effort for a mobile two-wheeled inverted pendulum despite the uncertainties and external disturbance.

 

Key Words: Robust /Multi Objective Controller, linear Matrix Inequalities ,Takagi-Sugeno Fuzzy Model, Mobile Two-Wheeled Inverted Pendulum

1. Introduction

     In the past few years, researchers have started to pay more attention to MTWIP, for this reason that it has been used in all aspects of our routine lives. The longitudinal model of MTWIP is unstable, nonlinear, and under actuated so it is very difficult to design the controller in order to control angular velocity while keeping the MTWIP balanced despite the parameters uncertainties.

     Sliding mode control (SMC) is a well known control strategy for nonlinear system despite the parameters uncertainties which has robust feature to cope with uncertainties on the sliding surface. Although the technique has good robustness properties, pure sliding mode control presents a drawback that includes chattering [1], [2].  In [3], Sliding-Mode Velocity Control is designed which this nonlinear control strategy could provide robust feature to fast convergence of system dynamics while dealing with parameters uncertainties. In [4], in order to eliminate chattering a nonlinear disturbance observer is designed for MTWIP. In [5], advanced sliding mode control is used for inverted pendulum which terminated sliding mode control has better performance in comparison with integral sliding mode control. Cascade adaptive fuzzy sliding-mode control for nonlinear two-axis inverted-pendulum is used in [6]. Cascade control structures are often implemented in [7], [8]. The chattering phenomena can be removed by designing sliding mode control strategy with high order surface which also improves the accuracy of the system [9]. In [10], in order to deal with chattering of sliding surface, nonlinear sliding is used for MTWIP in the presence of uncertainty. The mentioned control strategies can reduce the chattering of control input and cope with the uncertainties. 

     In recent years, neural network and fuzzy control have been used in order to regulate the body angle through wheel velocity and track velocity through body angle set points. Adaptive fuzzy controllers have also been used at varying capacities to eliminate this problem [11], [12]. In [13], [14], a type-1 (T1) FLS is designed for MTWIP with T-S fuzzy model and a type 2 (T2) T-S fuzzy method is designed and proposed for balancing and position control of inverted pendulum [15]-[18]. From mentioned references, it’s clear that T2 T-S fuzzy has better performance in order to cope with uncertainties in comparison with (T1) T-S fuzzy method. Radial basis function (RBF) neural network controller is proposed as a robust controller for MTWIP [19]. A rule-based neural controller for inverted pendulum system is presented in [20] and in [21], for learning a control strategy of inverted pendulum, neural networks is used. The mentioned control methods, can guarantee that MTWIP being in balance when the angular velocity is tracking the set point.

     The main contribution of this paper is about designing robust / multi objective controller with T-S fuzzy model for MTWIP in the presence of uncertainty and external disturbance. On the other hand, is used for tracking the desired values and robustness of the system and  is employed for linear quadratic Gaussian aspects. For fast transient response and lower overshoot, pole placement is employed. Several robust controller are designed based on linear matrix inequality (LMI) [22]-[24], but in this paper some states which consist of output errors are augmented to the system [25],[27] and by T-S fuzzy model, each local linear subsystem model of the nonlinear dynamic of MTWIP is represented. T-S fuzzy model employs fuzzy IF-THEN rules which give us linear models of the nonlinear system dynamic [27], [28]. For each local linear model, by using LMI method, a robust multi controller is designed. Eventually, the overall fuzzy model is obtained based on weighed of linear subsystems and in order to designed total controller, parallel distributed compensation is used. This scheme has shown effectiveness of the control strategy, high accuracy, robustness, good tracking for angular velocity while keeping MTWIP balanced and fast transient responses for a MTWIP despite the uncertainties and external disturbance.

     This paper is categorized as follows: section 2 presents primary description for T-S fuzzy model and /multi objective controller, the dynamic of MTWIP dynamic is given in section 3, augmented states and affine parameters are presented in section 4, MTWIP T-S fuzzy model is given in section 5, the results of the simulation are included in section 6, and in the end, section 7 presents the conclusion.

2. Primary Description

2. 1. Takagi-Sugeno Fuzzy Model

     T-S fuzzy model is a very powerful method in order to approximate highly accuracy model based on input - output data. T-S fuzzy model is a simple and less computational method based on weighting of parameters which are achieved according to some update fuzzy laws. T-S fuzzy weights are achieved based on the membership function, and in this paper, Gaussian membership function is used. T–S fuzzy model rules with parameters uncertainties in the mentioned control strategy framework can be described as follow:

IF  is  and…and  is  THEN

= [[+] x(t) + [+] w(t) + [+] u(t)], x(0) = 0

= [[+] x(t) + [+] w(t) +    [+] u(t)]                                                 (1)

= [[+] x(t) + [+] w(t) + [+] u(t)]

i=1, 2, 3… r

Where the fuzzy set is addressed with, r is the number of fuzzy rules and (t) → (t) is the fuzzy variables. The structure of whole linear system which is achieved by weight average defuzzifer method is shown in Figure (1).

Model 1

Weighted

              Rule 1                                                    

Input                                    .                            .                               Output

Weighted

Model p

                               .                  .

                   Rule p

Figure 1. Total T-S Fuzzy Model

The overall fuzzy model considered as:

= (v(t))[[+] x(t) + [+] w(t) + [+] u(t)], x(0)=0

= (v(t))[[+] x(t) + [+] w(t) + [+] u(t)]  (2)

= (v(t)) [[+] x(t) + [+] w(t) + [+] u(t)]

Where , , , , , , ,  and  refer to uncertainties matrices in the system and the fuzzy-mean formula is considered as:

(v(t)) =                                                  (3)

=                                               (4)

2. 2.  /  Multi Objective Controller

    In this section, according to Figure (2), which is entitled control structure, the main aim of proposed controller is to design u=kx that:

P(s)

           W

                                                                          

k

                       u x

 

Figure 2. Control Structure

(a) Maintain the root mean square (RMS) gain of transfer function between w to  in the limited value >0.

(b) Maintain the RMS gain of transfer function between w to  in the limited value >0.

(c) Places the closed loop poles in the LMI region.

Controller design based on LMI formulation is given the linear system as following:

=Ax+w +u

= x+w +u                                  (5)

= x+w +u

With replacing the u=kx in (5), closed-loop system model is defined as:

= (A+k) x +w

= (+k) x +w                               (6)

= +k) x +w

RMS gain of (transfer function between w to) will not go higher than, if and only if matrix inequality (7) is feasible:

 <0

>0 (7)

RMS gain of (transfer function between w to ) will not go higher than , if and only if  matrix inequalities (8) is feasible:

 >0

 <0

Trace (Q) <                                                        (8)

Consider LMI domain  as:

=

L==                                                  (9)

M =

Closed-loop poles will be located in domain, if

and only if there exists matrix inequality (10) so that

 <0,                                      (10)

    Eventually, according to minimization of the three sets of condition elements which consist of Q, , , and , k is calculated. In order to minimization, LMI approach is used which LMI formulation is achieved by replacing Y=kX in the mentioned conditions. In order for designing the total fuzzy controller, PDC technique is used which is shown in the Figure (3).

 

Weighted

              Rule 1                                                    

Input .                            .                               Output

Weighted

                               . .

                   Rule p

Figure 3. Overall T-S Fuzzy Controller

Each  (controller gain) for each local linear subsystem is obtained as:

IF  is  and…and  is  THEN

u(t) =x(t), i=1,2,3….r                                       (11)

Next, the total fuzzy controller according to PDC technique is obtained as:

u (t) =x(t)                                              (12)

Where  is the local controller gain for each local linear subsystem and the u denote to total fuzzy controller for the overall fuzzy system.

3. MTWIP Model

      In this paper, the model for longitudinal dynamic of MTWIP is considered from [29], which the longitudinal model of MTWIP is unstable, nonlinear, and under actuated. The MTWIP system is shown in Figure (4), wheel radius is addressed by r,  is rotation angle of wheel,  refers to inclination angle of inverted pendulum and l is the length between the center of wheel and the center of the inverted pendulum gravity.

Figure 4. The MTWIP System

The dynamic of MTWIP can be expressed by:

= 

(13)

Where

 =, ψ = , = (+, =, = , = +, =, = 2

Where u is the input signal as rotation torque which is generated by motor coaxial with the wheel,  and respectively denote to the masses of pendulum body and wheel, ,  refer to the moments of the body inertia for Y and Z axes, ,  are the moments of wheel inertia about its axis and diameter, ,  are the resistances in the driving system and ground. The parameter of the robot has been tabulated in Table 1.

Table 1. Coefficients Value

State variables are selected as:

 X== [θ,, φ, ]                        

The following state variables illustrate the MTWIP state-space, which there are 2 output states such as angle of inverted pendulumand angular velocity.

=                                                                                   (15)

=( +)+                                                                         (16)

=                                                                     (17)

=( +) +                                                                         (18)

4. Augmented States

     In this paper, we aimed to design  /  multi objective controller that improve robust feature of MTWIP despite the uncertainties of parameters and external disturbance. The proposed controller should guarantee the stability of closed loop system and forces the states of output errors to zero. So that the angel of inverted pendulum and angular velocity track the desired commanded values  and. For achieving higher accurate performance, states of output errors should be minimized which 2 extra states are added to the MTWIP system. Therefore, in order for achieving mentioned goal, the integral of  and  are selected as extra states variables.

=dt, =-                                      (19)

=dt, =-                                    (20)

To sum up, the total state vector is considered as:

= -, = -

X =                                    (21)

 

 

4. 1. Affine Parameter

Affine parameters are described as:

 

=+

+….+                                                 (22)

B(p) =

C(p) =

D(p) =

In (22), w is an external input which is selected as:

w =                                                             (23)

= , =                        (24)

A =                                                     (25)

Where

 

, , ,

It’s clear that equation (25) consists of nonlinear element which in order for linearization the nonlinear system, T-S fuzzy modeling is used. The T-S fuzzy model approach use IF-THEN rules for achieving higher accuracy with updating laws.

5. MTWIP Fuzzy Modeling

     According to equation (25), two fuzzy rules are designed for linearization of nonlinear parameters as: (t) =(t).

Where  is fuzzy variable. In order to design the membership function, minimum and maximum of the fuzzy variable selected as:

Min ((t)) = 0, Max ((t)) =                                  (26)

Two membership functions  and  which represent are shown as follow:

(t) =(t) =.+.0                 (27)

The membership functions should satisfy fuzzy mathematic as:

+= 1                                                 (28)

According to equation (27) and (28), the membership functions are obtained as:

=, =                                         (29)

Figure 5. The Membership Function

To achieve the two local linear subsystem which only matrix A varied in each subsystem, the two rules are considered as follow:

Rule 1

IF  is THEN A=

Rule 2

IF  is THEN A=

In order to design mentioned control strategy for each linear subsystem, the two fuzzy rules are considered as follow:

Rule 1

IF  is THEN u(t) =

Rule 2

IF  is THEN u(t) =

Where the local controller gains are addressed by

  and .

Finally, the total fuzzy model and the overall fuzzy controller are obtained by weight average defuzzifer method based on equation (2) and (12). The closed-loop state-space feedback system according to overall fuzzy model and total fuzzy controller is given as equation (34):

=                                              (30)

=                                               (31)

=                               (32)

=                                            (33)

=+                   (34)

     In the mentioned control structure, first, in order to incline accurate tracking of angular velocity  and angular of inverted pendulum, two extra sates which consist of outputs error are augmented to the dynamic of the system. According to nonlinear dynamic of MTWIP, two local linear subsystems are achieved by using fuzzy rules. Then, for each local linear subsystem, by using LMI method, a  /  controller is calculated. To sum up, in order to calculate the whole system from each local linear subsystem, weight average defuzzifer method is used which it utilized instead of nonlinear system and the total controller is designed based on parallel distribute compensation which both have been shown in Figure (1) and (3).

6. Simulation

     In this simulation case, the performance of the mentioned method (a robust / multi-objective state-feedback controller) for MTWIP is compared with integral sliding mode controller in [3]. The main problem of MTWIP is the stability of the system during motion. The main goal of proposed method is to control angular velocity while keeping the MTWIP balanced despite the parameters uncertainties and external disturbance.

At first, affine parameters and MTWIP T–S fuzzy model for the completed dynamic of the MTWIP at the  rule are calculated as follows:

=

=

C =, D =

LMI region for placing closed-loop system poles in the mentioned region and guarantee the lower rate of damping is considered as Figure (6). In addition, four parameters in order to limiting / cost function is selected as:  [= [0 0 1 1].

 

 

 

 

      

 Figure 6.The Pole Placement Region

Control gain matrices for each local linear system, the total fuzzy / controller matrix  which are calculated by PDC technique, and the overall fuzzy model matrix  which is obtained by weighted average defuzzifer are shown as follows:

=, =

=

=

     The simulation consists of three subsections. In the first subsection, simulations are considered without any external disturbance and parameters variation. The simulation results for first part are shown in figs (7)-(9). In this study, desired value for angular velocity and angle of inverted pendulum are selected as = 5 rad/s and = 0 rad.

Figure 7. Angular Velocity without External Disturbance

Figure 8. Angle of Inverted Pendulum without External Disturbance

Figure 9. Control Input

     Figure (7) shows the result of the angular velocity of the proposed method in comparison with [5]. From this Figure, it’s crystal clear that, the robust / multi objective controller has good performance, higher accuracy, better transient responses, and lower fluctuation in order to achieve the desired value.

     From Figure (8) which is entitled angle of inverted pendulum, it can be concluded that the mentioned controller achieves good performance of keeping the inverted pendulum at balance point in comparison with [5]. Another positive outcome from this Figure is that, the robust designed method controller is very stable.

     Figure (9) shows the control input which demonstrates that mentioned controller has lower control effort and oscillation in comparison with the reference method. Numerical index is given in table 2 which shows sum of squared errors and sum of squared control signal for the proposed method in comparison with [5]. The numerical data confirms that proposed method has better performance and fewer control cost.

 Table 2. Numerical Index

Parameter	Value
(kg)	0.14
(kg)	2.58
(kg.)	0.00014
(kg.)	0.00054
(kg.)	0.00128
(kg.)	0.00128
	0.001
	0.01
l (m)	0.0622
b (m)	0.15
r (m)	1.00

 

    In the second subsection, for justifying the robust feature of the mentioned method, the simulations are considered with external disturbance which is applied with step command in a moment after transient phase for angular velocity and angle of inverted pendulum. The desired values are same as part one.

Figure 10. Angular Velocity with External Disturbance

Figure 11. Angle of Inverted Pendulum with External Disturbance

     Figure (10) shows the angular velocity response of the robust designed method and integral sliding mode method [5] to desired value despite the external disturbance. From Figure (10), it’s clear that proposed method is robust controller which has lower settling time and better tracking performance without any overshoot.

     The main problem for MTWIP is balancing during motion. Angel of inverted pendulum with external disturbance is shown in Figure (11) which can be seen that the robust /controller has fewer overshoot and better performance despite the external disturbance in comparison with [3] and it’s clear that peak to peak disturbance rejection value for the designed controller is fewer than reference paper.

     In the third subsection, the body inertia moments for Y and Z axes  and , the moments of wheel inertia about its axis and diameter  and , and the resistances in the driving system and ground and  are varied between 20%. The desired values are same as part one. In this sub section, angel of inverted pendulum responses are considered for nominal structure and when the mentioned parameters in the main dynamic are varied.

 

Figure 12. Angle of Inverted Pendulum with Parameters Variation

     From Figure (12) it can be concluded that the mentioned control strategy has good robustness when the parameters of dynamic are varied in a wide range.

7. Conclusion

     As conclusion, a robust / multi objective controller has been implemented and analyzed for angular velocity and angle of inverted pendulum tracking of MTWIP system in the presence of uncertainties. Error of output states was added to the dynamic for improving control performance. Then, the T–S fuzzy modeling was used for estimating uncertain nonlinear system. After that, according to each local linear model, by using linear matrix inequalities method, a robust / state feedback controller was designed by using a linear matrix inequalities method. Finally, in order to obtain the whole system from each local linear subsystem, weight average defuzzifer method was used and the total controller was designed based on PDC. In comparatively with [5], a robust proposed scheme has less control effort, less fluctuation, better transient responses, and has better performance to keep angle of inverted pendulum and angular velocity on desired values despite the parameters uncertainties and external disturbance.

 

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