Optimal Design of a Proportional-Derivative State Feedback Controller Based on Meta-Heuristic Optimization for a Quarter Car Suspension System

In order to examine the impact of the design of a state feedback controller on the active suspension system on the quarter car, an accurate mathematical model of the system based on the state-space approach must be established. The two degrees of freedom (2DOF) active suspension system that is considered in this study is illustrated in Figure 1 [9] where m_s (Kg) is the sprung mass (chassis) and m_u (Kg) is the unsprung mass (wheel). An active suspension system can be represented as a passive suspension system with an actuator. The passive suspension system consists of a spring with a stiffness factor (k_s) N/m and a damper with a damping factor (b_s) Ns/m. A hydraulic actuator with force (F_a) N is added in parallel to the passive suspension. The tire has a stiffness factor (k_t) N/m. The state variables of the system are the displacement (x₁) m and (x₂) m and velocity $\left(\dot{\mathrm{x}}_{1}\right) \frac{\mathrm{m}}{\mathrm{s}}$ and $\left(\dot{\mathrm{x}}_{2}\right) \frac{\mathrm{m}}{\mathrm{s}}$ of the sprung and unsprung mass respectively. The vertical displacement of the road (d) m is represented the road disturbance.

1.png

Figure 1. 2DOF active suspension system of the quarter car

Based on Newton’s laws, the equations of motion for the sprung mass are given by [8]:

$m_{s} \times \ddot{x}_{1}=-\left(k_{s} \times\left(x_{1}-x_{2}\right)\right)-\left(b_{s} \times\left(\dot{x}_{1}-\dot{x}_{2}\right)\right)+ \mathrm{F}_{\mathrm{a}}$              (1)

Rearrange of Eq. (1) yields:

$\ddot{\mathrm{x}}_{1}=-\left(\frac{\mathrm{k}_{\mathrm{s}}}{\mathrm{m}_{\mathrm{s}}} \times \mathrm{x}_{1}\right)+\left(\frac{\mathrm{k}_{\mathrm{s}}}{\mathrm{m}_{\mathrm{s}}} \times \mathrm{x}_{2}\right)-\left(\frac{\mathrm{b}_{\mathrm{s}}}{\mathrm{m}_{\mathrm{s}}} \times \dot{\mathrm{x}}_{1}\right)+\left(\frac{\mathrm{b}_{\mathrm{s}}}{\mathrm{m}_{\mathrm{s}}} \times \dot{\mathrm{x}}_{2}\right)+\left(\frac{1}{\mathrm{~m}_{\mathrm{s}}} \times \mathrm{F}_{\mathrm{a}}\right)$                   (2)

In the same way, the equations of motion for the unsprung mass are given by [8]:

$\mathrm{m}_{\mathrm{u}} \times \ddot{\mathrm{x}}_{2}=\left(\mathrm{k}_{\mathrm{s}} \times\left(\mathrm{x}_{1}-\mathrm{x}_{2}\right)\right)+\left(\mathrm{b}_{\mathrm{s}} \times\left(\dot{\mathrm{x}}_{1}-\dot{\mathrm{x}}_{2}\right)\right)-\left(\mathrm{k}_{\mathrm{t}} \times\left(\mathrm{x}_{2}-\mathrm{d}\right)\right)-\mathrm{F}_{\mathrm{a}}$                   (3)

Rearrange of Eq. (3) yields:

$\ddot{\mathrm{x}}_{2}=\left(\frac{\mathrm{k}_{\mathrm{s}}}{\mathrm{m}_{\mathrm{u}}} \times \mathrm{x}_{1}\right)-\left(\left(\frac{\mathrm{k}_{\mathrm{s}}}{\mathrm{m}_{\mathrm{u}}}+\frac{\mathrm{k}_{\mathrm{t}}}{\mathrm{m}_{\mathrm{u}}}\right) \times \mathrm{x}_{2}\right)+\left(\frac{\mathrm{b}_{\mathrm{s}}}{\mathrm{m}_{\mathrm{u}}} \times\right.\left.\dot{\mathrm{x}}_{1}\right)-\left(\frac{\mathrm{b}_{\mathrm{s}}}{\mathrm{m}_{11}} \times \dot{\mathrm{x}}_{2}\right)+\left(\frac{\mathrm{k}_{\mathrm{t}}}{\mathrm{m}_{\mathrm{u}}} \times \mathrm{d}\right)-\left(\frac{1}{\mathrm{~m}_{\mathrm{u}}} \times \mathrm{F}_{\mathrm{a}}\right)$             (4)

Based on Eq. (2) and Eq. (4) the state variable equations of the active quarter car suspension system are given by [8]:

$\left[\begin{array}{l}\dot{\mathrm{x}}_{1} \\ \ddot{\mathrm{x}}_{1} \\ \dot{\mathrm{x}}_{2} \\ \ddot{\mathrm{x}}_{2}\end{array}\right]=\left[\begin{array}{cccc}0 & 1 & 0 & 0 \\ -\frac{\mathrm{k}_{\mathrm{s}}}{\mathrm{m}_{\mathrm{s}}} & -\frac{\mathrm{b}_{\mathrm{s}}}{\mathrm{m}_{\mathrm{s}}} & \frac{\mathrm{k}_{\mathrm{s}}}{\mathrm{m}_{\mathrm{s}}} & \frac{\mathrm{b}_{\mathrm{s}}}{\mathrm{m}_{\mathrm{s}}} \\ 0 & 0 & 0 & 1 \\ \frac{\mathrm{k}_{\mathrm{s}}}{\mathrm{m}_{\mathrm{u}}} & \frac{\mathrm{b}_{\mathrm{s}}}{\mathrm{m}_{\mathrm{u}}} & -\frac{\left(\mathrm{k}_{\mathrm{s}}+\mathrm{k}_{\mathrm{t}}\right)}{\mathrm{m}_{\mathrm{u}}} & -\frac{\mathrm{b}_{\mathrm{s}}}{\mathrm{m}_{\mathrm{u}}}\end{array}\right] \times\left[\begin{array}{l}\mathrm{x}_{1} \\ \dot{\mathrm{x}}_{1} \\ \mathrm{x}_{2} \\ \dot{\mathrm{x}}_{2}\end{array}\right]+ \left[\begin{array}{c}0 \\ \frac{1}{\mathrm{~m}_{\mathrm{s}}} \\ 0 \\ -\frac{1}{\mathrm{~m}_{\mathrm{v}}}\end{array}\right] \times \mathrm{F}_{\mathrm{a}}+\left[\begin{array}{c}0 \\ 0 \\ 0 \\ \frac{\mathrm{k}_{\mathrm{t}}}{\mathrm{m}_{\mathrm{u}}}\end{array}\right] \times \mathrm{d}$              (5)

$\mathrm{y}=\left[\begin{array}{llll}1 & 0 & 0 & 0\end{array}\right] \times\left[\begin{array}{l}\mathrm{x}_{1} \\ \dot{\mathrm{x}}_{1} \\ \mathrm{x}_{2} \\ \dot{\mathrm{x}}_{2}\end{array}\right]$                (6)

As mention above, the PDSF controller has tuning parameters that need to be adjusted to obtain the desired performance.  Instead of using the trial and error method which is considered a time-consuming method, the tuning process in this study is formulated as an optimization problem. Then, two meta-heuristic optimizations, Bees Algorithm (BA) and Grey Wolf Optimization (GWO) are utilized to solve the problem.

4.1 Bees Algorithm

The Bees Algorithm (BA) is a swarm-based optimization algorithm developed by Pham et al. in 2005 [13]. The algorithm mimics the way that the honey bee when it is foraging for food. It consists of two methods of search: local search and random search. The optimization procedure of the BA begins with randomly initialized sites of a population of N scout bees within the search space as given in Eq. (10) [14]:

$p_{i}=p_{1}+\left(r \times\left(p_{u}-p_{1}\right)\right)$             (10)

where, p_i is the initial site for each individual in the algorithm. i is the counter (i=1,2,3,…N). r is a round value generated between [0,1]. p_l and p_u are the lower and upper bounds of the search space of the problem.

Then, the objective value of each scout bee is evaluated. The next step is to select an elite bee (eli) based on the best objective value was found by the scout bees. After that, a number of sites (m) are selected to perform the local search. A local search towards the site of the elite bee (p^eli) is performed based on the patch size (h) as given in Eq. (11) [14]:

$p_{i}^{\text {new }}=p_{i}^{\text {old }}+h \times\left(p^{\text {eli }}-p_{i}^{\text {old }}\right)$                 (11)

where, $p_{i}^{\text {new }}$ and $p_{i}^{\text {old }}$ are the new site and old site of the bee i respectively. The remaining bees (N-m) assign to search randomly as given in Eq. (10). The new population is evaluated and greedy selection is used for the best solution. The pseudo code of the BA is given in Figure 2.

2.png

Figure 2. Pseudo code of the BA

BA has been used by many authors as a controller-tuning technique. For example, Bilgic et al. [15] demonstrated the efficiency of the tuning of the LQR controller by using BA to stabilize an inverted pendulum. Moreover, Hameed et al. [16] presented a comparative study of tuning the parameters of the PID controller using BA and Firefly Algorithm (FA). Two different transfer functions were used in the study. Both algorithms achieve good dynamic performance when the system is subject to a step input.

4.2 Grey Wolf Optimization

Grey Wolf Optimization (GWO) is a swarm-based meta-heuristic optimization technique proposed by Mirjalili et al. in 2014. GWO simulates the social predatory behavior of gray wolves. Mirjalili et al. [17] categorized the individuals of gray wolves into four levels. The first category is the alpha wolf (α), these wolves either males or females are responsible to make decisions and lead the pack. The second category is the beta wolf (β), these wolves are lower level than the alphas which provide consultants to the alphas. The third category is the delta wolf (δ), these wolves submit information to alphas and betas. The last category is the omega wolf (ω) [18]. The simulation of the hunting process in GWO can be divided into three processes which are named: searching for prey, encircling prey and attacking prey. The general mathematical model of the movement of wolves towards the prey is formulated as follows [19]:

$C_{\mathrm{wo}}=2 \times r$             (12)

$A_{w o}=\left(2 \times a_{w o} \times r\right)-a$                  (13)

$\mathrm{D}_{\text {wo }}=\left|\mathrm{C}_{\text {wo }} \times \mathrm{W}_{\text {wop }}(\mathrm{t})-\mathrm{W}_{\text {wo }}(\mathrm{t})\right|$               (14)

$\mathrm{W}_{\mathrm{wo}}(\mathrm{t}+1)=\mathrm{W}_{\mathrm{wop}}(\mathrm{t})-\left(\mathrm{A}_{\mathrm{wo}} \times \mathrm{D}_{\mathrm{wo}}\right)$                  (15)

where, a_wo is a coefficient its value is linearly decreased from 2 to 0 for each iteration. C_wo, A_wo and D_wo are coefficients their value are calculated as given in Eq. (12), Eq. (13) and Eq. (14) respectively. r is a random value generated between [0,1]. t refers to the current iteration in the algorithm. W_wop and W_wo are the position of the prey and the wolf respectively.

The hunting process is guided by the first three categories of wolves (α, β and δ). Hence, all wolves should adjust their position and movement based on the information that is provided by these three categories of wolves. The mathematical model of the hunting process can be represented as follows [17]:

The movement of individual wolf with respect to alphas α :

$\mathrm{D}_{\mathrm{wo \alpha}}=\left|\left(\mathrm{C}_{\text {wo1 }} \times \mathrm{W}_{\mathrm{wo \alpha}}\right)-\mathrm{W}_{\mathrm{wo}}(\mathrm{t})\right|$              (16)

W_wo1=W_woα-(A_wo1×D_woα)               (17)

The movement of individual wolf with respect to betas β:

$\mathrm{D}_{\mathrm{wo} \beta}=\left|\left(\mathrm{C}_{\mathrm{wo} 2} \times \mathrm{W}_{\mathrm{wo} \beta}\right)-\mathrm{W}_{\mathrm{wo}}(\mathrm{t})\right|$                 (18)

$W_{\mathrm{wo} 2}=W_{\mathrm{wo} \beta}-\left(A_{\mathrm{wo} 2} \times \mathrm{D}_{\mathrm{wo} \beta}\right)$                 (19)

The movement of individual wolf with respect to deltas δ:

$\mathrm{D}_{\mathrm{wo \delta}}=\left|\mathrm{C}_{\mathrm{wn} 3} \cdot \mathrm{W}_{\mathrm{wo \delta}}-\mathrm{W}_{\mathrm{wo}}(\mathrm{t})\right|$                 (20)

$W_{\mathrm{WO} 3}=W_{\mathrm{wo} \delta}-\left(A_{\mathrm{wo} 3} \times \mathrm{D}_{\mathrm{wo} \delta}\right)$                (21)

The new position of individual wolf:

$W_{\text {wo }}{ }^{N e w}(t+1)=\frac{W_{\text {wo1 }}+W_{\text {W02 }}+W_{\text {wo3 }}}{3}$               (22)

where, W_woα, W_{woβ and} W_δwo are the best solution found by alpha, beta and delta respectively. The Pseudo code of the GWO algorithm is given in Figure 3. In the context of control engineering, it has been observed that the GWO is successfully handling the tuning process of the parameters of controllers in a number of papers such as for Proportional-Integral (PI) controller [19], Proportional-Integral-Derivative (PID) [20], and Fractional Order PID (FOPID) controller [21].

3.png

Figure 3. Pseudo code of the GWO

In this paper, a Proportional-Derivative State Feedback (PDSF) controller based on two meta-heuristic optimizations named Bees Algorithm (BA) and Grey Wolf Optimization (GWO) is successfully designed for active suspensions systems of a quarter car. The problem of finding the optimal value of the feedback gain matrix of the PDSF controller to make the system reach stabilization and reduce the oscillation is formulated as an optimization problem. Both controllers, BA-PDSF and GWO-PDSF, were able to stabilize the suspension system effectively in comparison with the passive suspension system. However, the simulation results show that the BA-PDSF was able to reduce the TIAE in comparison with the GWO-PDSF by 14.3%. These results show that adopting BA as a tuning technique for PDSF controller is more promising as compared with the results obtained from the GWO algorithm. This work could be extended by utilizing another optimization technique to handle the tuning process.