Bridging the gap: organisational value frames and sustainable alliance portfolios

Tulin Dzhengiz; Tulin Dzhengiz

doi:10.3934/GF.2020018

Green Finance

2020, Volume 2, Issue 4: 323-343. doi: 10.3934/GF.2020018

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Review Special Issues

Bridging the gap: organisational value frames and sustainable alliance portfolios

Tulin Dzhengiz ^,

Department of Management, Aalto School of Business, Espoo, Finland

Research on sustainability-oriented partnerships focused either on inter-firm or cross-sector partnerships separately and often took the partnership as a level of analysis. As opposed to the partnership level, the firm-level analysis that investigates portfolios of sustainability-oriented partnerships were brought forward by only a few studies. By drawing on the literature of alliance portfolios, this paper builds the notion of "sustainable alliance portfolio" further to move scholarly attention towards the bigger picture of firms' partnership efforts for sustainability. Taking stock on the research that introduced organisational cognition to corporate sustainability which showed how partners' value frames co-evolve, converge, diverge or fuse over time; this paper theorises how business case and paradoxical frames impact the configuration, management and development of sustainable alliance portfolios. Overall, this paper bridges the gap between two constructs-organisational value frames and sustainable alliance portfolios- and offers propositions for future research to draw attention to the under-theorised portfolios of sustainability-oriented partnerships.

Keywords:

Citation: Tulin Dzhengiz. Bridging the gap: organisational value frames and sustainable alliance portfolios[J]. Green Finance, 2020, 2(4): 323-343. doi: 10.3934/GF.2020018

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Abstract

1. Introduction

More often, impaired mobility results in fewer opportunities for socializing, engaging in leisure pursuits and pursuing goals. Due to this, physically impaired people are at heightened risk of depression, anxiety, and social isolation. By providing an adapted aid to mobility, an electric powered wheelchair can significantly improve the quality of life of a physically disabled person ^[1,2].

The kinematic and dynamic modelling of electric powered wheelchair confirms its multivariate nonlinear nature ^[3]. It is an electromechanical system whose complete analysis calls together the main disciplines: mechanical, electrical, power electronics, automatic, computer science ^[4,5].

Several strategies have focused on the EPW's velocity and direction control as well as trajectory tracking using different kinds of motors, such as DC Motor, PMSM ^[6], brushless DC motors; e.g., adaptive controller, neural control techniques ^[7], robust controllers ^[8], sliding mode ^[9,10], backstepping ^[4], fuzzy logic ^[11,12], fuzzy sliding mode ^[13,14].

The main objective purpose of this paper is to use the Lagrangian method to model the electric powered wheelchair with permanents magnets synchronous motor as an actuator, due to some advantageous features, such as high efficiency, high torque to current ratio, low noise and robustness. After that, the application of the Fuzzy Sliding Mode Controller (FSMC), which combines the Sliding Mode Control algorithm (SMC) with a Fuzzy Logic Control scheme (FLC). It is well known that sliding mode control algorithm can give good transient performance and system robustness. However, these performances are obtained at the price of certain disadvantages: the appearance of the chattering phenomenon caused by the discontinuous part of the control which can have a harmful effect on the actuators ^[15,16]; the system is subjected at all times to a high control in order to ensure its convergence to the desired state and this is not desirable. Hence the need to incorporate a fuzzy controller in the sliding mode control in order to obtain a robust and smooth control ^[17,18]. This combination makes it possible to overcome the problems of the chattering of the sliding mode control and of deficit in tools for analyzing of the fuzzy logic control without forgetting the reduction in the number of fuzzy rules. Moreover, the salient advantages of each control strategy are kept during control action. For example, the fuzzy system is used to approximate unknown functions, while the sliding mode approach adds the possibility of establishing stable adaptation laws ^[19,20,21]. Therefore, as a type of robust controller with strong points, FSMC has been widely used in many application fields ^{[22,23,24,25]}.

After the introduction section, the document is structured as follows: Section 2 covers the dynamic modelling with Lagrangian method for electric powered wheelchair. After that, fuzzy sliding mode control applied to the global system is proposed in section 3. Simulations and result analysis are carried out in section 4. Finally, the conclusions are drawn in section 5.

2. Dynamic modelling of the EPW based on PMSM actuators

The dynamic model of the electric wheelchair is essential for the controller design and simulation analysis, it is determined according to the Lagrange method considering the different forces that affect its movement.

Lagrange dynamics approach is a very powerful method for formulating the motion equations of mechanical systems. This method is used to systematically derive the equations of motion by considering the kinetic and potential energies of the given system ^[10,26].

To analyze the motion of this system; a fixed coordinate O-xy has been assigned as shown in Figure 1.

Figure 1. Wheelchair model for going up a slope.

DownLoad: Full-Size Img PowerPoint

Table 1 shows the meaning of the terms used in this paper.

Table 1. Parameters of the EPW.

Symbol	Description	Value
$v$	Longitudinal velocity of the EPW	$m/s$
${\alpha }_{r}, {\alpha }_{l}$	Rotational angle of the right/left wheel	$rad$
${\alpha }_{mr}, {\alpha }_{ml}$	Rotational angle of the right/left motor	$rad$
${C}_{emr}, {C}_{eml}$	Electromagnetic torques of the right/left motor	$N.m$
${C}_{Fr}, {C}_{Fl}$	Torque applied on the right/left wheel	$N.m$
${C}_{r}, {C}_{l}$	Load torques required of the right/left motor	$N.m$
${\varOmega }_{r}, {\varOmega }_{l}$	Angular rotor velocities of the right/left motor	$rad/s$
${I}_{d}, {I}_{q}$	d- and q-axis stator currents	$A$
${V}_{d}, {V}_{q}$	d- and q-axis stator voltages	$V$
$M$	EPW with operator mass	$210 kg$
${m}_{w}$	Mass of driving wheel	$2 kg$
$L$	Distance between the two driving wheels	$0.57 m$
$l$	Length of the EPW	$0.87 m$
$R$	Radius of the driving wheel	$0.17 m$
$J$	Moment of inertia of the EPW	$16.08kg.{m}^{2}$
${J}_{w}$	Moment of inertia of the driving wheel	$0.0289kg.{m}^{2}$
$g$	Acceleration due to gravity	$9.81m/{s}^{2}$
$\psi$	Slope angle	$\%$
${f}_{w}$	Viscous friction coefficient of the wheel	$0.008N.m.s/rad$
$\sigma$	Gear ratio	$0.03$
${J}_{a}$	Moment of inertia of the motor	$0.0008 kg.{m}^{2}$
${f}_{v}$	Viscous friction coefficient of the motor	$0.00005 N.m.s/rad$
${R}_{s}$	Per phase stator resistance	$2.56 \varOmega$
${L}_{d}$	d-axis stator inductances	$0.0064 H$
${L}_{q}$	q-axis stator inductances	$0.0056 H$
${\phi }_{f}$	Permanent magnet flux	$0.06 Wb$
$P$	Number of pairs of poles	$4$
${P}_{n}$	Rated power	$400 W$
${N}_{n}$	Rated speed	$3000 rpm$
${V}_{n}$	Rated voltage	$120 V$
${I}_{n}$	Rated current	$2 A$
${Cem}_{n}$	Rated torque	$1.27 N.m$

| Show Table

DownLoad: CSV

The last two wheels are powered independently by two PMSM which their outputs are the right and left torque.

We start by establishing the equations of motion of the right and left motors, given by:

$\left\{ {\begin{array}{*{20}{c}} {{J_a}\frac{{d{\varOmega _r}}}{{dt}} + {f_v}{\varOmega _r} + {C_r} = {C_{emr}}}\\ {{J_a}\frac{{d{\varOmega _l}}}{{dt}} + {f_v}{\varOmega _l} + {C_l} = {C_{eml}}} \end{array}} \right.$

(1)

Where:

${C_{em}} = P\left[ {\left( {{L_d} - {L_q}} \right){I_d}{I_q} + {\varphi _f}{I_q}} \right]$

(2)

The nonlinear Park model of PMSM is defined in a rotor d-q reference frame by the following expression:

$\left\{ {\begin{array}{*{20}{c}} {\frac{{d{I_d}}}{{dt}} = - \frac{{{R_s}}}{{{L_d}}}{I_d} + P\varOmega \frac{{{L_q}}}{{{L_d}}}{I_q} + \frac{1}{{{L_d}}}{V_d}}\\ {\frac{{d{I_q}}}{{dt}} = - P\varOmega \frac{{{L_d}}}{{{L_q}}}{I_d} - \frac{{{R_s}}}{{{L_q}}}{I_q} - P\varOmega \frac{{{\varphi _f}}}{{{L_q}}} + \frac{1}{{{L_q}}}{V_q}} \end{array}} \right.$

(3)

The Lagrange equations of the right and left wheel can be written in the following form:

$\left\{ {\begin{array}{*{20}{c}} {\frac{d}{{dt}}\left( {\frac{{\partial L}}{{\partial {{\dot \alpha }_r}}}} \right) - \frac{{\partial L}}{{\partial {\alpha _r}}} = {C_{Fr}} - {f_w}{{\dot \alpha }_r}}\\ {\frac{d}{{dt}}\left( {\frac{{\partial L}}{{\partial {{\dot \alpha }_l}}}} \right) - \frac{{\partial L}}{{\partial {\alpha _l}}} = {C_{Fl}} - {f_w}{{\dot \alpha }_l}} \end{array}} \right.$

(4)

Where $L = K-U$ is the Lagrangian function, $K$ and $U$ are the kinetic and potential energies of the system respectively.

$\left\{ \begin{align} & {{{\ddot{\alpha }}}_{r}}\left( \left( {{m}_{w}}+\frac{M}{4} \right){{R}^{2}}+{{J}_{w}}+J\frac{{{R}^{2}}}{{{L}^{2}}} \right)+{{{\ddot{\alpha }}}_{l}}\left( \frac{M}{4}-\frac{J}{{{L}^{2}}} \right){{R}^{2}}+\left( {{m}_{w}}+\frac{M}{2} \right)gRsin~\psi = {{C}_{Fr}}-{{f}_{w}}{{{\dot{\alpha }}}_{r}} \\ & {{{\ddot{\alpha }}}_{l}}\left( \left( {{m}_{w}}+\frac{M}{4} \right){{R}^{2}}+{{J}_{w}}+J\frac{{{R}^{2}}}{{{L}^{2}}} \right)+{{{\ddot{\alpha }}}_{r}}\left( \frac{M}{4}-\frac{J}{{{L}^{2}}} \right){{R}^{2}}+\left( {{m}_{w}}+\frac{M}{2} \right)gRsin~\psi = {{C}_{Fl}}-{{f}_{w}}{{{\dot{\alpha }}}_{l}} \\ \end{align} \right.$

(5)

Taking all these relations to introduce the motors dynamics in the total dynamic that includes the two-driver wheel as well as the total mass, the parameter which connects them is called gear reduction ratio noted by $\sigma$ such as:

$\left\{ {\begin{array}{*{20}{c}} {{{\dot \varOmega }_r} = \frac{1}{\sigma }{{\ddot \alpha }_r}}\\ {{{\dot \varOmega }_l} = \frac{1}{\sigma }{{\ddot \alpha }_l}} \end{array}} \right.$

(6)

$\left\{ {\begin{array}{*{20}{c}} {{\varOmega _r} = \frac{1}{\sigma }{{\dot \alpha }_r}}\\ {{\varOmega _l} = \frac{1}{\sigma }{{\dot \alpha }_l}} \end{array}} \right.$

(7)

$\left\{ {\begin{array}{*{20}{c}} {{C_r} = \sigma {C_{Fr}}}\\ {{C_l} = \sigma {C_{Fl}}} \end{array}} \right.$

(8)

The system has two degrees of freedom $\left[{\alpha }_{r}, {\alpha }_{l}\right]$ , where their stored values are the displacements ${S}_{r}$ and ${S}_{l}$ such as:

$\left\{ {\begin{array}{*{20}{c}} {{S_r} = R{\alpha _r}}\\ {{S_l} = R{\alpha _l}} \end{array}} \right.$

(9)

Then (1) becomes:

$\left\{ {\begin{array}{*{20}{c}} {\frac{1}{R}{J_e}\left( {\begin{array}{*{20}{c}} {{{\ddot S}_r}}\\ {{{\ddot S}_l}} \end{array}} \right) + \frac{1}{R}{C_e}\left( {\begin{array}{*{20}{c}} {{{\dot S}_r}}\\ {{{\dot S}_l}} \end{array}} \right) = \left( {\begin{array}{*{20}{c}} {{C_{emr}}}\\ {{C_{eml}}} \end{array}} \right) + T}\\ {y = \left( {\begin{array}{*{20}{c}} v\\ w \end{array}} \right) = \left[ {\begin{array}{*{20}{c}} {R/2R/2}\\ { - R/LR/L} \end{array}} \right]\left( {\begin{array}{*{20}{c}} {{{\dot \alpha }_r}}\\ {{{\dot \alpha }_l}} \end{array}} \right)} \end{array}} \right.$

(10)

Where $T = -\sigma \left(\frac{M}{2}+{m}_{w}\right)gRsin\psi$ , ${J}_{e} = \left[\begin{array}{c}a b\\ b a\end{array}\right]$ , ${C}_{e} = \left[\begin{array}{c}c0\\ 0c\end{array}\right]$ .

With $a = \frac{{J}_{a}}{\sigma }+\sigma \left\{{J}_{w}+\left(\frac{M}{4}+{m}_{w}\right){R}^{2}+{\left(\frac{R}{L}\right)}^{2}J\right\}$ , $b = \sigma {R}^{2}\left(\frac{M}{4}-\frac{J}{{L}^{2}}\right)$ , $c = \frac{1}{\sigma }{f}_{v}+\sigma {f}_{w}$ .

Finally, the nonlinear global model of the EPW is as follow:

$\left\{ \begin{matrix} {{{\dot{x}}}_{1}} = {{x}_{2}} \\ {{{\dot{x}}}_{2}} = {{l}_{1}}{{x}_{2}}+{{l}_{2}}{{x}_{4}}+{{y}_{1}}P\left[ \left( {{L}_{d}}-{{L}_{q}} \right){{x}_{5}}{{x}_{7}}+{{\varphi }_{f}}{{x}_{7}} \right]+{{y}_{2}}p\left[ \left( {{L}_{d}}-{{L}_{q}} \right){{x}_{6}}{{x}_{8}}+{{\varphi }_{f}}{{x}_{8}} \right]+{{b}_{1}}T \\ {{{\dot{x}}}_{3}} = {{x}_{4}} \\ {{{\dot{x}}}_{4}} = {{l}_{3}}{{x}_{2}}+{{l}_{4}}{{x}_{4}}+{{y}_{3}}P\left[ \left( {{L}_{d}}-{{L}_{q}} \right){{x}_{5}}{{x}_{7}}+{{\varphi }_{f}}{{x}_{7}} \right]+{{y}_{4}}P\left[ \left( {{L}_{d}}-{{L}_{q}} \right){{x}_{6}}{{x}_{8}}+{{\varphi }_{f}}{{x}_{8}} \right]+{{b}_{2}}T \\ {{{\dot{x}}}_{5}} = -\frac{{{R}_{s}}}{{{L}_{d}}}{{x}_{5}}+\frac{P{{L}_{q}}}{\sigma R{{L}_{d}}}{{x}_{2}}{{x}_{7}}+\frac{1}{{{L}_{d}}}{{V}_{dr}} \\ {{{\dot{x}}}_{6}} = -\frac{{{R}_{s}}}{{{L}_{d}}}{{x}_{6}}+\frac{P{{L}_{q}}}{\sigma R{{L}_{d}}}{{x}_{4}}{{x}_{8}}+\frac{1}{{{L}_{d}}}{{V}_{dl}} \\ {{{\dot{x}}}_{7}} = -\frac{P{{L}_{d}}}{\sigma R{{L}_{q}}}{{x}_{2}}{{x}_{5}}-\frac{{{R}_{s}}}{{{L}_{q}}}{{x}_{7}}-\frac{P{{\varphi }_{f}}}{\sigma R{{L}_{q}}}{{x}_{2}}+\frac{1}{{{L}_{q}}}{{V}_{qr}} \\ {{{\dot{x}}}_{8}} = -\frac{P{{L}_{d}}}{\sigma R{{L}_{q}}}{{x}_{4}}{{x}_{6}}-\frac{{{R}_{s}}}{{{L}_{q}}}{{x}_{8}}-\frac{P{{\varphi }_{f}}}{\sigma R{{L}_{q}}}{{x}_{4}}+\frac{1}{{{L}_{q}}}{{V}_{ql}} \\ \end{matrix} \right.$

(11)

Where $x = {\left[{S}_{r}{\dot{S}}_{r}{S}_{l}{\dot{S}}_{l}{I}_{dr}{I}_{dl}{I}_{qr}{I}_{ql}\right]}^{T}$ , ${B}_{v} = {\left[0{b}_{1}0{b}_{2}\right]}^{T}$ , $u = {\left[{V}_{dr}{V}_{dl}{V}_{qr}{V}_{ql}\right]}^{T}$ , $V = T$ .

With ${l}_{1} = {l}_{4} = -\frac{ac}{{a}^{2}-{b}^{2}}$ , ${l}_{2} = {l}_{3} = \frac{bc}{{a}^{2}-{b}^{2}}$ , ${y}_{1} = {y}_{4} = \frac{aR}{{a}^{2}-{b}^{2}}$ , ${y}_{2} = {y}_{3} = -\frac{bR}{{a}^{2}-{b}^{2}}$ , ${b}_{1} = {b}_{2} = \frac{R}{a+b}$ .

The model obtained is multivariable (MIMO), nonlinear and strongly coupled.

Since both rear wheels are driven by two motors, the speed of each driving wheel is controlled independently. The electronic differential is therefore used to provide the required torque and speed references for each wheel. The slip on the rear wheels is ignored, so the speed of the wheels can be defined as a function of the radius of the wheels ^[27,28,29].

Figure 2 shows the steering left of the EPW.

Figure 2. EPW design model during steering.

DownLoad: Full-Size Img PowerPoint

The longitudinal speed of each wheel drive can be expressed as:

$\left\{ {\begin{array}{*{20}{c}} {{V_l} = {w_V}\left( {r - \frac{L}{2}} \right)}\\ {{V_r} = {w_V}\left( {r + \frac{L}{2}} \right)} \end{array}} \right.$

(12)

Where $r = \frac{l}{tan\delta }$ and $\delta$ is the steering angle, the angular speeds are:

$\left\{ {\begin{array}{*{20}{c}} {{w_l} = \frac{{l - \left( {\frac{L}{2}} \right)tan\delta }}{l}{w_V}}\\ {{w_r} = \frac{{l + \left( {\frac{L}{2}} \right)tan\delta }}{l}{w_V}} \end{array}} \right.$

(13)

If the steering angle $\delta >0$ , the EPW drives left and if $\delta <0$ , the EPW drives right. If $\delta = 0$ , the EPW drives straight ahead (longitudinal movement).

${w}_{V}$ is the center angular speed, expressed by:

${w_V} = \frac{{{w_l} + {w_r}}}{2}$

(14)

3. Fuzzy sliding mode control of the global system

The Sliding Mode Control is a very popular strategy for control of nonlinear uncertain systems, with a very large frame of applications fields. Due to the use of the discontinuous function, its main features are the robustness of closed-loop system and the finite-time convergence. Such technique consists on establishing a switching surface depending on existence and convergence laws, and then compels the dynamic behavior of the system to bring back toward this surface and slides around it till attaining the equilibrium state. This action is realized using a control law $U$ as defined in (15) ^[30,31,32]. So, to tuning a sliding mode controller, requires to:

• Choose the sliding surface $S = 0$ ,

• Determine the existence and convergence conditions,

• Establish the appropriate control law which can be able to force the trajectory to match the manifold $S = 0$ and keep it switching around this surface.

$U = {U_{eq}} + {U_c}$

(15)

To find the sliding mode as given by (16), a general form that can be adopted ^[33]:

$S\left( x \right) = {\left( {\frac{d}{{dt}} + \lambda } \right)^{r - 1}}e\left( x \right)$

(16)

Where $e\left(x\right)$ is the difference between the controlled variable and its reference, $\lambda \in {\mathbb{R}}_{+}^{*}$ and $r$ is the relative degree of derivation number to be applied to the output signal $x$ to generate explicitly the control component.

If $S$ move towards $0$ , then $e$ and its derivative are also lending to $0$ .

Concerning the convergence condition, it is recommended to define a scalar function which makes the surface $S = 0$ attractive and invariant. To do we can choose the Lyapunov function defined as:

$V\left( x \right) = \frac{1}{2}{S^2}\left( x \right) \Rightarrow \dot V\left( x \right) = S\left( x \right)\dot S\left( x \right)$

(17)

To insure the attractive phenomenon of the regulated variable toward the desired trajectory (reference), this function must realize:

$\left\{ {\begin{array}{*{20}{c}} {V\left( x \right) > 0}\\ {\dot V\left( x \right) < 0} \end{array}} \right.$

(18)

In other words, we have ^[34]:

$S\left( x \right)\dot S\left( x \right) < 0$

(19)

We will then analyze the behavior of our system controlled by sliding mode.

Let's put the error: $e = {\left[{e}_{r}, {e}_{l}\right]}^{T} = {\left[{S}_{r}-{S}_{rref}, {S}_{l}-{S}_{lref}\right]}^{T}$ .

The model of the difference between the reference and the real trajectory, considering synchronous motors with non-salient poles ( $L_d = L_q$ ), is:

$\left( {\begin{array}{*{20}{c}} {{{\dddot e}_r}} \\ {{{\dddot e}_l}} \end{array}} \right) = R{{\left[ \begin{matrix} a~~~~~~b \\ b~~~~~~~a \\ \end{matrix} \right]}^{-1}}\left( \left( \begin{matrix} {{V}_{q}}_{r}-{{R}_{s}}{{I}_{q}}_{r}-\frac{p{{\varphi }_{f}}}{\sigma R}{{{\dot{S}}}_{r}}-\frac{p{{L}_{d}}}{\sigma R}{{{\dot{S}}}_{r}}{{I}_{d}}_{r} \\ {{V}_{q}}_{l}-{{R}_{s}}{{I}_{q}}_{l}-\frac{p{{\varphi }_{f}}}{\sigma R}{{{\dot{S}}}_{l}}-\frac{p{{L}_{d}}}{\sigma R}{{{\dot{S}}}_{l}}{{I}_{d}}_{l} \\ \end{matrix} \right)-\frac{{{L}_{q}}}{Rp{{\varphi }_{f}}}\left[ \begin{matrix} c~~~~~~0 \\ 0~~~~~~~c \\ \end{matrix} \right]\left( \begin{matrix} {{{\ddot{S}}}_{r}} \\ {{{\ddot{S}}}_{l}} \\ \end{matrix} \right) \right) -\\ \left( {\begin{array}{*{20}{c}} {{{\dddot S}_{rref}}} \\ {{{\dddot S}_{lref}}} \end{array}} \right)$

(20)

Consider the following surfaces in the state space:

$\left( {\begin{array}{*{20}{c}} {{S_r}}\\ {{S_l}} \end{array}} \right) = \left( {\begin{array}{*{20}{c}} {k_1^2{e_r} + 2{k_1}{{\dot e}_r} + {{\ddot e}_r}}\\ {k_2^2{e_l} + 2{k_2}{{\dot e}_l} + {{\ddot e}_l}} \end{array}} \right)$

(21)

$\left( {\begin{array}{*{20}{c}} {{{\dot S}_r}} \\ {{{\dot S}_l}} \end{array}} \right) = \left( {\begin{array}{*{20}{c}} {k_1^2{{\dot e}_r} + 2{k_1}{{\ddot e}_r} + {{\dddot e}_r}} \\ {k_2^2{{\dot e}_l} + 2{k_2}{{\ddot e}_l} + {{\dddot e}_l}} \end{array}} \right)$

(22)

Replacing (20) in (22), we'll have:

$\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\left( {\begin{array}{*{20}{c}} {{{\dot S}_r}} \\ {{{\dot S}_2}} \end{array}} \right) = \\ \frac{Rp{{\varphi }_{f}}}{{{L}_{q}}}{{\left[ \begin{matrix} a~~~~~~b \\ b~~~~~~~a \\ \end{matrix} \right]}^{-1}}\left( \left( \begin{matrix} {{V}_{q}}_{r}-{{R}_{s}}{{I}_{q}}_{r}-\frac{p{{\varphi }_{f}}}{\sigma R}{{{\dot{S}}}_{r}}-\frac{p{{L}_{d}}}{\sigma R}{{{\dot{S}}}_{r}}{{I}_{d}}_{r} \\ {{V}_{q}}_{l}-{{R}_{s}}{{I}_{q}}_{l}-\frac{p{{\varphi }_{f}}}{\sigma R}{{{\dot{S}}}_{l}}-\frac{p{{L}_{d}}}{\sigma R}{{{\dot{S}}}_{l}}{{I}_{d}}_{l} \\ \end{matrix} \right)-\frac{{{L}_{q}}}{Rp{{\varphi }_{f}}}\left[ \begin{matrix} c~~~~~~0 \\ 0~~~~~~~c \\ \end{matrix} \right]\left( \begin{matrix} {{{\ddot{S}}}_{r}} \\ {{{\ddot{S}}}_{l}} \\ \end{matrix} \right) \right)+ \\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \left[ \begin{matrix} k_{1}^{2}~~~~~~0 \\ 0~~~~~~~k_{2}^{2} \\ \end{matrix} \right]\left( \begin{matrix} {{{\dot{S}}}_{r}}-{{{\dot{S}}}_{rref}} \\ {{{\dot{S}}}_{l}}-{{{\dot{S}}}_{lref}} \\ \end{matrix} \right)+\left[ \begin{matrix} 2{{k}_{1}}~~~~~~0 \\ 0~~~~~~~2{{k}_{2}} \\ \end{matrix} \right] \left( {\begin{array}{*{20}{c}} {{{\ddot S}_r} - {{\ddot S}_{rref}}} \\ {{{\ddot S}_l} - {{\ddot S}_{lref}}} \end{array}} \right) - \left( {\begin{array}{*{20}{c}} {{{\dddot S}_{rref}}} \\ {{{\dddot S}_{lref}}} \end{array}} \right)$

(23)

So, the equivalent controls ${{V}_{q}}_{r\_eq}$ and ${{V}_{q}}_{l\_eq}$ , which allow the sliding phenomenon to appear by bringing the state of the system back to the sliding surface defined by ${\dot{S}}_{r, l} = 0$ , are calculated as follows:

$\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\left( {\begin{array}{*{20}{c}} {{V_q}_{{r_{eq}}}} \\ {{V_q}_{{l_{eq}}}} \end{array}} \right) = \\ \frac{{{L}_{q}}}{Rp{{\varphi }_{f}}}\left[ \begin{matrix} a~~~~~~b \\ b~~~~~~~a \\ \end{matrix} \right] \left( {\left( {\begin{array}{*{20}{c}} {{{\dddot S}_{rref}}} \\ {{{\dddot S}_{lref}}} \end{array}} \right) - \left[ {\begin{array}{*{20}{c}} {k_1^2\;\;\;\;\;\;0} \\ {0\;\;\;\;\;\;k_2^2} \end{array}} \right]\left( {\begin{array}{*{20}{c}} {{{\dot S}_r} - {{\dot S}_{rref}}} \\ {{{\dot S}_l} - {{\dot S}_{lref}}} \end{array}} \right) - \left[ {\begin{array}{*{20}{c}} {2{k_1}\;\;\;\;\;\;0} \\ {0\;\;\;\;\;\;2{k_2}} \end{array}} \right]\left( {\begin{array}{*{20}{c}} {{{\ddot S}_r} - {{\ddot S}_{rref}}} \\ {{{\ddot S}_l} - {{\ddot S}_{lref}}} \end{array}} \right)} \right) + \\ \ \ \ \ \ \frac{{{L}_{q}}}{Rp{{\varphi }_{f}}}\left[ \begin{matrix} c~~~~~~0 \\ 0~~~~~~~c \\ \end{matrix} \right]\left( \begin{matrix} {{{\ddot{S}}}_{r}} \\ {{{\ddot{S}}}_{l}} \\ \end{matrix} \right)+\left( \begin{matrix} {{R}_{s}}{{I}_{q}}_{r} \\ {{R}_{s}}{{I}_{q}}_{l} \\ \end{matrix} \right)+\left[ \begin{matrix} \frac{p{{\varphi }_{f}}+p{{L}_{d}}{{I}_{d}}_{r}}{\sigma R}~~~~~~0 \\ 0~~~~~~~~~\frac{p{{\varphi }_{f}}p{{L}_{d}}{{I}_{d}}_{l}}{\sigma R} \\ \end{matrix} \right]\left( \begin{matrix} {{{\dot{S}}}_{r}} \\ {{{\dot{S}}}_{l}} \\ \end{matrix} \right)$

(24)

If the system states have not reached the sliding surfaces, the equivalent controls must be reinforced by another so-called robust control defined by:

$\left( {\begin{array}{*{20}{c}} {{V_q}_{r\_c}} \\ {{V_q}_{l\_c}} \end{array}} \right) = - \left( {\begin{array}{*{20}{c}} {{U_1}sgn\left( {{S_r}} \right)} \\ {{U_2}sgn\left( {{S_l}} \right)} \end{array}} \right)$

(25)

Where:

$sgn\left( S \right) = \left\{ \begin{matrix} -1, ~~if~S\le 0 \\ 1, ~~~~~if~S > 0 \\ \end{matrix} \right.$

(26)

The global control is defined by:

$\left( \begin{matrix} {{V}_{q}}_{r} \\ {{V}_{q}}_{l} \\ \end{matrix} \right) = \left( \begin{matrix} {{V}_{q}}_{r\_eq}+{{V}_{q}}_{r\_c} \\ {{V}_{q}}_{l\_eq}+~{{V}_{q}}_{l\_c} \\ \end{matrix} \right)$

(27)

By replacing (24) and (27) in (23) we find:

$\left( \begin{matrix} {{{\dot{S}}}_{r}} \\ {{{\dot{S}}}_{l}} \\ \end{matrix} \right) = \frac{Rp{{\varphi }_{f}}}{{{L}_{q}}}{{\left[ \begin{matrix} a~~~~~~b \\ b~~~~~~~a \\ \end{matrix} \right]}^{-1}}\left( \begin{matrix} {{V}_{q}}_{r\_c} \\ {{V}_{q}}_{l\_c} \\ \end{matrix} \right)$

(28)

The attractiveness condition expressed by (19) becomes:

$\left( \begin{matrix} {{S}_{r}}{{{\dot{S}}}_{r}} \\ {{S}_{l}}{{{\dot{S}}}_{l}} \\ \end{matrix} \right) < 0\Rightarrow \left( \frac{Rp{{\varphi }_{f}}}{{{L}_{q}}}{{\left[ \begin{matrix} a~~~~~~b \\ b~~~~~~~a \\ \end{matrix} \right]}^{-1}}\left( \begin{matrix} {{S}_{r}} \\ {{S}_{l}} \\ \end{matrix} \right)\left( \begin{matrix} {{V}_{q}}_{r\_c} \\ {{V}_{q}}_{l\_c} \\ \end{matrix} \right) \right) < 0$

(29)

In order to satisfy this condition, the sign of $\left(\begin{array}{c}{{V}_{q}}_{r\_c}\\ {{V}_{q}}_{l\_c}\end{array}\right)$ must be opposite to that of $\frac{R{p\phi }_{f}}{{L}_{q}}{\left[\begin{array}{c}a b\\ b a\end{array}\right]}^{-1}\left(\begin{array}{c}{S}_{r}\\ {S}_{l}\end{array}\right)$ .

The global control is finally given by:

$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \left( \begin{matrix} {{V}_{q}}_{r} \\ {{V}_{q}}_{l} \\ \end{matrix} \right) = \\ \frac{{{L}_{q}}}{Rp{{\varphi }_{f}}}\left[ \begin{matrix} a~~~~~~b \\ b~~~~~~~a \\ \end{matrix} \right] \left( {\left( {\begin{array}{*{20}{c}} {{{\dddot S}_{rref}}} \\ {{{\dddot S}_{lref}}} \end{array}} \right) - \left[ {\begin{array}{*{20}{c}} {k_1^2\;\;\;\;\;0} \\ {0\;\;\;\;\;k_2^2} \end{array}} \right]\left( {\begin{array}{*{20}{c}} {{{\dot S}_r} - {{\dot S}_{rref}}} \\ {{{\dot S}_l} - {{\dot S}_{lref}}} \end{array}} \right) - \left[ {\begin{array}{*{20}{c}} {2{k_1}\;\;\;\;\;0} \\ {0\;\;\;\;\;2{k_2}} \end{array}} \right]\left( {\begin{array}{*{20}{c}} {{{\ddot S}_r} - {{\ddot S}_{rref}}} \\ {{{\ddot S}_l} - {{\ddot S}_{lref}}} \end{array}} \right)} \right) + \\ \ \ \ \ \ \ \ \ \ \frac{{{L}_{q}}}{Rp{{\varphi }_{f}}}\left[ \begin{matrix} c~~~~~~0 \\ 0~~~~~~~c \\ \end{matrix} \right]\left( \begin{matrix} {{{\ddot{S}}}_{r}} \\ {{{\ddot{S}}}_{l}} \\ \end{matrix} \right)+\left( \begin{matrix} {{R}_{s}}{{I}_{q}}_{r} \\ {{R}_{s}}{{I}_{q}}_{l} \\ \end{matrix} \right)+\left[ \begin{matrix} \frac{p{{\varphi }_{f}}+p{{L}_{d}}{{I}_{d}}_{r}}{\sigma R}~~~~~~0 \\ 0~~~~~~~\frac{p{{\varphi }_{f}}p{{L}_{d}}{{I}_{d}}_{l}}{\sigma R} \\ \end{matrix} \right]\left( \begin{matrix} {{{\dot{S}}}_{r}} \\ {{{\dot{S}}}_{l}} \\ \end{matrix} \right)-\left( \begin{matrix} {{U}_{1}}sgn\left( {{S}_{r}} \right) \\ {{U}_{2}}sgn\left( {{S}_{l}} \right) \\ \end{matrix} \right)$

(30)

However, the main drawback of the sliding mode control is undesirable oscillations with finite amplitude and frequency due to the presence of unmodeled dynamics or discrete time implementation. This destructive phenomenon, so called "chattering", may lower control accuracy or incur unwanted wear of mechanical components.

Here, a hybridization methodology between fuzzy logic and sliding mode control is applied, the controller resulting from this combination is called "Fuzzy Sliding Mode Controller (FSMC)", This one presents the same structure of SMC, apart from the 2nd term ${U}_{c}$ , which will be replaced by a fuzzy controller. For this, the term $-Usgn\left(S\right)$ can be replaced by a fuzzy controller. This controller has an input and an output, and the rule base is used to establish a connection between $S$ and ${U}_{c}$ . This is interpreted by rules of the form: $IF-THEN$ as shown in Table 2.

Table 2. Rule base of FLC.

$IF$	$S$	is	NB	$THEN$	${U}_{c}$	is	VB (Very Big)
$IF$	$S$	is	NS	$THEN$	${U}_{c}$	is	B (Big)
$IF$	$S$	is	Z	$THEN$	${U}_{c}$	is	M (Medium)
$IF$	$S$	is	PS	$THEN$	${U}_{c}$	is	S (Small)
$IF$	$S$	is	PB	$THEN$	${U}_{c}$	is	VS (Very Small)

| Show Table

DownLoad: CSV

The fuzzy controller having an input $S$ of five membership functions and an output ${U}_{c}$ of five membership functions which are shown in Figure 3.

Figure 3. Membership functions of: (a) Input

${S}_{r}, {S}_{l}$ , (b) Output

${V}_{{q}_{r\_c}}, {V}_{{q}_{l\_c}}$ .

DownLoad: Full-Size Img PowerPoint

The second input of the system ${V}_{d}$ is determined using the PMSM vector control to eliminate the nonlinearity of the system by maintaining the current ${I}_{d}$ to zero and ${V}_{d}$ takes its corresponding value as follow:

${{V}_{d}} = -p\varOmega {{L}_{q}}{{I}_{q}}$

(31)

Finally, the block diagram for the FSMC is illustrated in Figure 4.

Figure 4. Block diagram of the FSMC for the EPW control.

DownLoad: Full-Size Img PowerPoint

4. Simulation results and discussion

First, we start by generating the reference trajectory of the right and left wheel in order to assess the robustness of the trajectory tracking of the controller. The use of polynomial form (as set point) is a very practical tool for calculating trajectory.

The point-to-point trajectory between ${S}^{i}$ and ${S}^{f}$ is determined by the following equations:

$S\left( t \right) = {{S}^{i}}+r\left( t \right)D \text{ for } 0\le t\le {{t}_{f}}$

(32)

$\dot{S}\left( t \right) = \dot{r}\left( t \right)D$

(33)

With $D = {S}^{f}-{S}^{i}$ ; the boundary conditions of the interpolation function $r\left(t\right)$ are given by: $r\left(0\right) = 0$ and $r\left({t}_{f}\right) = 1$ .

Several interpolation functions can provide a trajectory, such as polynomial interpolation (linear interpolation, third degree polynomials (cubic) and fifth degree polynomials (quintic)), bang-bang acceleration profile and trapeze velocity profile ^[35]. The most frequently encountered polynomial interpolation method is the interpolation by the fifth degree polynomials, this method ensures a regular and continuous trajectory in displacements, velocities and accelerations.

To allow quintic to represent a complete trajectory, the trajectory needs to be separated into several phases (acceleration, steady state and deceleration). Each phase will be represented by a quintic equation ^[36,37]. To achieve the smooth transition from one phase to another, the boundary conditions for the initial and final position, velocity and acceleration must be satisfied.

The boundary conditions are as follows: $S\left(0\right) = {S}^{i}$ , $S\left({t}_{f}\right) = {S}^{f}$ , $\dot{S}\left(0\right) = 0$ , $\dot{S}\left({t}_{f}\right) = 0$ , $\ddot{S}\left(0\right) = 0$ , $\ddot{S}\left({t}_{f}\right) = 0$ .

And, using the following polynomial form:

$S(t)=a_0+a_1 t+a_2 t^2+a_3 t^3+a_4 t^4+a_5 t^5$

(34)

The expression (34) can also be written under the form (32) or (33) with the following interpolation function:

$r(t)=10{{(\frac{t}{{{t}_{f}}})}^{3}}-15{{(\frac{t}{{{t}_{f}}})}^{4}}+6{{(\frac{t}{{{t}_{f}}})}^{5}}$

(35)

For a displacement of $0 m$ to $17.8 m$ in $11.5 s$ which correspond to a variable velocity up to $2.3m/s$ , we have opted for the trajectories shown in by performing a slope variation from $\psi = 0 \%$ to $\psi = 10 \%$ in the time interval $t = 4.5 s$ to $t = 5.5 s$ , and a direction change to the right from $\delta = 0\mathrm{ }°$ to $\delta = 0.1\mathrm{ }°$ between $t = 6s$ and $t = \mathrm{ }7.5s$ achieved by the electronic differential. During this revolving, the wheels don't turn at the same velocity. Indeed, the left wheel travels more distance than the right wheel.

Figure 5. Reference trajectories of the right/left wheel: (a) Displacement, (b) Velocity, (c) Acceleration.

DownLoad: Full-Size Img PowerPoint

To evaluate the performance of the fuzzy sliding mode control applied to the EPW driven by two PMSMs with perfect voltage inverter, we simulate motion and velocity tracking in the presence of disturbances, we also show the evolution of electrical and mechanical quantities.

The consideration of this performance can be summarized in the following two cases:

4.1. Trajectory tracking

The and represent, respectively the displacement and velocity trajectory tracking of the right and left drive wheel, we note that the measured trajectories perfectly follow the references. This is verified by observing the tracking error curves ${(S}_{{r, l}_{ref}}-{S}_{r, l})$ and ${(\dot{S}}_{{r, l}_{ref}}-{\dot{S}}_{r, l})$ , given in Figure 8 and Figure 9. The velocities increase to a maximum value which remains maintained during the steady state and then returns to zero, which corresponds to the final state.

Figure 6. Displacement of the right/left wheel.

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Figure 7. Velocity of the right/left wheel.

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Figure 8. Displacement error of the right/left wheel.

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Figure 9. Velocity error of the right/left wheel.

DownLoad: Full-Size Img PowerPoint

shows the evolution of the electromagnetic torque ${C}_{emr, l\mathrm{ }}$ of the both PMSM, they increase to a maximum value, then they return to a very small positive value which remains constant in steady state, however, they react in case of slope variation, then they go to a negative minimum value and return to zero at the end. These torques are directly proportional to the stator quadrature currents ${I}_{qr, l}$ given in . The stator direct currents ${I}_{dr, l}$ are maintained at zero by the vector control as shown in . The direct and quadrature voltage inputs of both PMSM $({V}_{dr, l}, {V}_{qr, l})$ and voltage feeding $\left({V}_{ar, l}\right)$ do not exceed their nominal values as shown in Figure 13, Figure 14 and Figure 15 respectively. Likewise, the turning and slope variation have no effect on all quantities of the system.

Figure 10. Electromagnetic torque of the right/left motor.

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Figure 11. q-axis stator current of the right/left motor.

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Figure 12. d-axis stator current of the right/left motor.

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Figure 13. d-axis stator voltage of the right/left motor.

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Figure 14. q-axis stator voltage of the right/left motor.

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Figure 15. Voltage feeding of the right/left motor.

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4.2. Robustness test

By performing for the same trajectory a variation of –50% of the global moment of inertia. The evolution of the electrical and mechanical quantities clearly shows the robustness of this control.

The results obtained show the efficiency and robustness of the Sliding Mode Controller of the global system (EPW+PMSM), with better tracking, fast response, without overshoot.

5. Conclusions

In this study, a dynamic modelling of EPW using PMSM as an actuator according to the Lagrange method is considered. It is an electromechanical, multivariable, nonlinear and strongly coupled system, hence the necessity to introduce the robust controllers.

A nonlinear control by the fuzzy sliding mode controller was established to track the trajectory of the EPW. This technique based on the hybridization between fuzzy logic and sliding mode control provides a robust and smooth control with very satisfactory results for stabilization and precision.

The simulation results show the evolution of electrical and mechanical quantities, which are around their nominal values. These different results obtained confirm the feasibility of the fuzzy sliding mode controller of the global system (EPW+PMSM), with better tracking, fast response, without overshoot.

An experimental implementation of this controller is targeted in future work.

Conflict of interest

The authors declare that there is no conflict of interest in this paper.

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Bridging the gap: organisational value frames and sustainable alliance portfolios

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2. Dynamic modelling of the EPW based on PMSM actuators

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4.1. Trajectory tracking

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Bridging the gap: organisational value frames and sustainable alliance portfolios

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Abstract

1. Introduction

2. Dynamic modelling of the EPW based on PMSM actuators

3. Fuzzy sliding mode control of the global system

4. Simulation results and discussion

4.1. Trajectory tracking

4.2. Robustness test

5. Conclusions

Conflict of interest

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