2.1 Mathematical modeling of nonlinear quarter car suspension system
For control and optimization applications, a nonlinear quarter car suspension model [24, 26] is utilized. In this study, a quarter car is modeled as a nonlinearity with quadratic tire stiffness and cubic stiffness in the suspension spring, as illustrated in Fig. 1.
The governing equations of the system are defined as—
$$\left. {\begin{array}{*{20}l} {m_{{{\text{us}}}} \ddot{x}_{{{\text{us}}}} - c_{{\text{s}}} \left( {\dot{x}_{{\text{s}}} - \dot{x}_{{{\text{us}}}} } \right) - k_{{\text{s}}} \left( {x_{{\text{s}}} - x_{{{\text{us}}}} } \right) + k_{{{\text{tnl}}}} (x_{{{\text{us}}}} - x_{{\text{r}}} )^{2} - k_{{{\text{snl}}}} (x_{{\text{s}}} - x_{{{\text{us}}}} )^{3} + k_{{\text{t}}} \left( {x_{{{\text{us}}}} - x_{{\text{r}}} } \right) - f_{{{\text{flc}}}} = 0} \hfill \\ {m_{{\text{s}}} \ddot{x}_{{\text{s}}} + c_{{\text{s}}} \left( {\dot{x}_{{\text{s}}} - \dot{x}_{{{\text{us}}}} } \right) + k_{{\text{s}}} \left( {x_{{\text{s}}} - x_{{{\text{us}}}} } \right) + k_{{{\text{snl}}}} (x_{{\text{s}}} - x_{{{\text{us}}}} )^{3} + f_{{{\text{flc}}}} = 0} \hfill \\ \end{array} } \right\}$$
(1)
2.2 Fuzzy logic control of nonlinear quarter car
FLC is a heuristic-based human-in-loop control that uses rule-based information obtained from user experience [14,15,16]. Figure 2 represents a typical FLC structure with a 2-DoF quarter car. FLC has two inputs—sprung mass velocity and acceleration, and FLC force is the output.
In this study [24], FLC consists of seven membership functions each, for two inputs and an output with Mamdani fuzzy inference system and centroid defuzzification method. The FLC controller is optimized using the NSGA-II [27] algorithm having objective functions such as frequency-weighted RMS acceleration, VDV, RMS suspension travel, RMS tire deflection, and RMS control force. The design variables are scaling factors of input and output and the range of membership functions [24]. The rule base of the FLC force is represented in Fig. 3.
The critical decision in optimization problems is to select proper objective functions. The FLC is implemented to improve ride characterized by RMS frequency-weighted sprung mass acceleration, VDV, RMS suspension space deflection, RMS tire deflection, and RMS controller force. The objective functions are defined as follows—
Frequency-weighted RMS acceleration As per ISO 2631-1 (1997), RMS acceleration is given by
$${A}_{\mathrm{w}}={\left\{\frac{1}{T}{\int }_{0}^{T}{\left[{a}_{\mathrm{s}}(t)\right]}^{2}\mathrm{d}t\right\}}^\frac{1}{2}$$
(2)
According to ISO 2631-1 (1997), VDV assesses the cumulative effect of vibrations on the body; thus, it is a measure for whole body vibrations.
Vibration dose value (VDV) VDV is the fourth power of acceleration time histories. It is expressed by Eq. (3)—
$$\mathrm{VDV}={\left\{{\int }_{0}^{T}{\left[{a}_{\mathrm{w}}(t)\right]}^{4}\mathrm{d}t\right\}}^\frac{1}{4}$$
(3)
Suspension travel Suspension travel is characterized by the relative travel between the sprung and unsprung mass.
$${\mathrm{RMS}}\,{\text{Suspension}}\,{\text{Travel}}={\left\{\frac{1}{T}{\int }_{0}^{T}{\left[\left({x}_{\mathrm{s}}\left(t\right)-{x}_{\mathrm{us}}\left(t\right)\right)\right]}^{2}\mathrm{d}t\right\}}^\frac{1}{2}$$
(4)
Dynamic tire deflection Dynamic tire force is related to tire deflection. The following equation expresses the RMS of tire deflection—
$${\mathrm{RMS}}\,{\text{Tire}}\,{\text{Deflection}}={\left\{\frac{1}{T}{\int }_{0}^{T}{\left[\left({x}_{\mathrm{us}}\left(t\right)-{x}_{\mathrm{r}}\left(t\right)\right)\right]}^{2}\mathrm{d}t\right\}}^\frac{1}{2}$$
(5)
Control force is introduced as one of the objective functions to find the optimum control force to achieve ride comfort.
$$\mathrm{RMS }f= {\left\{\frac{1}{T}{\int }_{0}^{T}{\left[\left({f}_{\mathrm{FLC}}\left(t\right)\right)\right]}^{2}\mathrm{d}t\right\}}^\frac{1}{2}$$
(6)
2.2.1 Optimization of fuzzy logic control using NSGA-II algorithm
FLC with 2 input–1 output functions is initialized with trapezoidal membership functions having range [− 1 1] and is multiplied with scaling factors. Also, inputs sprung mass velocity (error, e), mass acceleration (change in error, ec), and control output (f) are scaled using scaling factors ke, kec, and kf, respectively. The search space for the input–output membership functions and scaling factors is
$$\begin{aligned}&{\mathrm{Membership}}\, {\text{Functions}}\,{\text{for}}\,{\text{Input}}\,1\in \left[1, 10\right],\,{\mathrm{Membership}}\,{\text{Functions}}\,{\text{for}}\,{\text{Input}}\,2\in \left[1, 10\right],\\ &\quad {\mathrm{Membership}}\,{\text{Functions}}\, {\text{for}}\,{\text{Output}}\in \left[1, 10\right],\,{\mathrm{ke}}\in \left[-5, 5\right],\,{\mathrm{kec}}\in \left[-5, 5\right],\,{\mathrm{kf}}\in \left[0, 25\right]\,\,[18]\end{aligned}$$
The FLC is optimized by implementing the objective functions represented by Eqs. (2)–(7) and the search spaces defined above for the design variables. The optimized values are then further implemented using MATLAB/Simulink® and the quarter car model, and time domain results are obtained.
The optimization problem involves objectives such as ride comfort, health criterion, and stability; thus, problem becomes multi-objective optimization (MOO). The MOO problem consists of multiple solutions, thus forming a Pareto-front. NSGA-II is implemented [24] due to the ability of an algorithm to handle multi-domains as it handles noisy and multi-modal complex and discontinues functions. NSGA-II supports parallel computing, and a non-dominated sorting algorithm reduces computing complexities. NSGA-II supports diversity by crowding distance and uniform spread operators [27]. Figure 4 represents the flowchart of the NSGA-II algorithm.
2.3 Optimization of passive suspension system to mimic FLC
In this study, optimization of passive suspension to mimic the FLC active suspension system is implemented by incorporating results obtained by Nagarkar et al. [24]. An NSGA-II algorithm is used to optimize the controller parameters for optimal results [24]. As a result, the suspension system is optimized to mimic the active control system, and outcomes of the NSGA-II optimized FLC are implemented. The FLC active suspension system is designed to minimize the objective functions such as RMS fflc, VDV, frequency-weighted RMS acceleration, RMS suspension travel, and RMS tire deflection.
The passive suspension system under optimization study is considered to have the same suspension travel and velocity as an active suspension system. The actuator force generated by FLC active system is deemed equal to spring and damper forces generated by the initial passive suspension system. The methodology is explained in Fig. 5.
As a result, Eq. (7) gives the generated suspension force for the passive system as follows:
$${f}_{\mathrm{p}}={k}_{\mathrm{s}}\left({x}_{\mathrm{s}}-{x}_{\mathrm{us}}\right)+{c}_{\mathrm{s}}\left({\dot{x}}_{\mathrm{s}}{-\dot{x}}_{\mathrm{us}}\right)+{k}_{\mathrm{snl}}({x}_{\mathrm{s}}-{x}_{\mathrm{us}}{)}^{3}$$
(7)
Let x_1 be suspension travel and \(\dot{{x}}\_1\) be suspension velocity, Eq. (7) becomes
$${f}_{\mathrm{p}}={k}_{\mathrm{s}}(x\_1)+{c}_{\mathrm{s}}\left(\dot{x}\_1\right)+{k}_{\mathrm{snl}}(x\_1{)}^{3}$$
(8)
In an active system using FLC, the input control force is fflc. The optimal parameters of a passive suspension system are obtained by the least square technique, where a square of force error is minimized, i.e.,
$$\underbrace {{\text{min }}}_{{k_{{\text{s}}} , c_{{\text{s}}} }}\mathop \sum \limits_{t} \left[ {f_{{{\text{flc}}}} \left( t \right) - f_{{\text{p}}} \left( t \right)} \right]^{2}$$
(9)
Partially differentiating Eq. (9) with respect to ks and cs and equating it to zero to obtain optimal suspension parameters. Thus, for optimal ks and cs we have
$$\frac{\partial }{{\partial k}_{\mathrm{s}}} \sum_{t}{\left[{f}_{\mathrm{flc}}\left(t\right)-{f}_{\mathrm{p}}\left(t\right)\right]}^{2}{|}_{{k}_{\mathrm{s}}={{k}_{\mathrm{s}}}_{\mathrm{opt}}}= 0$$
(10)
$$\frac{\partial }{{\partial c}_{\mathrm{s}}} \sum_{t}{\left[{f}_{\mathrm{flc}}\left(t\right)-{f}_{\mathrm{p}}\left(t\right)\right]}^{2} {|}_{{c}_{\mathrm{s}}={{c}_{\mathrm{s}}}_{\mathrm{opt}}}= 0$$
(11)
Solving Eqs. (10) and (11) by substituting fp from Eq. (3), we have
$$\begin{aligned}&{{k}_{\mathrm{s}}}_{\mathrm{opt}} \sum ({x\_1)}^{2}+{{c}_{\mathrm{s}}}_{\mathrm{opt}} \sum \left[(\dot{x}\_1).(x\_1)\right]\\ &\quad =\sum \left[{f}_{\mathrm{flc}} . (x\_1)\right] -{k}_{\mathrm{snl}} \sum ({x\_1)}^{4}\end{aligned}$$
(12)
and
$$\begin{aligned}&{{k}_{\mathrm{s}}}_{\mathrm{opt}} \sum \left[\left({x}_{1}\right).(\dot{x}\_1)\right]+{{c}_{\mathrm{s}}}_{\mathrm{opt}} \sum {(\dot{x}\_1)}^{2}\\ &\quad =\sum \left[{f}_{\mathrm{flc}} . (\dot{x}\_1)\right] -{k}_{\mathrm{snl}} \sum \left[(\dot{x}\_1).({x\_1)}^{3}\right]\end{aligned}$$
(13)
Rearranging Eqs. (12) and (13) in matrix form,
$$\begin{aligned}&\left[\begin{array}{cc}\sum {\left(x\_1\right)}^{2}& \sum \left[(\dot{x}\_1).(x\_1)\right]\\ \sum \left[(\dot{x}\_1).(x\_1)\right]& \sum {(\dot{x}\_1)}^{2}\end{array}\right]\left\{\begin{array}{c}{{k}_{\mathrm{s}}}_{\mathrm{opt}}\\ {{c}_{\mathrm{s}}}_{\mathrm{opt}}\end{array}\right\}\\ &\quad = \left[\begin{array}{c}\sum \left[{f}_{\mathrm{flc}} . (x\_1)\right]-{k}_{\mathrm{snl}}\sum ({x}^{4}) \\ \sum \left[{f}_{\mathrm{flc}} . (\dot{x}\_1)\right]-{k}_{\mathrm{snl}}\sum \left[(\dot{x}\_1).({x\_1})^{3}\right] \end{array}\right] \end{aligned}$$
(14)
Thus, we have
$$\begin{aligned}& \left\{\begin{array}{c}{{k}_{\mathrm{s}}}_{\mathrm{opt}}\\ {{c}_{\mathrm{s}}}_{\mathrm{opt}}\end{array}\right\}\\ & \quad= {\left[\begin{array}{cc}\sum {\left(x\_1\right)}^{2}& \sum \left[(\dot{x}\_1).(x\_1)\right]\\ \sum \left[(\dot{x}\_1).(x\_1)\right]& \sum {(\dot{x}\_1)}^{2}\end{array}\right]}^{-1}\\ &\qquad \left[\begin{array}{c}\sum \left[{f}_{\mathrm{flc}} . (x\_1)\right]-{k}_{\mathrm{snl}}\sum ({x}^{4}) \\ \sum \left[{f}_{\mathrm{flc}} . (\dot{x}\_1)\right]-{k}_{\mathrm{snl}}\sum \left[(\dot{x}\_1).({x\_1)}^{3}\right]\end{array}\right]\end{aligned}$$
(15)
