In the present work, Pareto multiobjective approach-based PID controller design methodology is proposed for improving the vehicle active suspension system dynamics. The PID controller gains are tuned using the grasshopper optimization technique. Furthermore, the Pareto multiobjective-based methodology is compared with the genetic algorithm-based conventional weighted multiobjective function.

The following equations define the objectives functions suspension deflection, tyre deflection, sprung mass acceleration, and eigenvalue stability. The role of the active suspension system is to improve ride comfort by minimizing the acceleration and the tyre and suspension deflections.

The suspension deflection (*Z*_{SD}) is defined as the difference between the vertical displacement of the sprung mass and the unsprung mass.

$$Z_{{{\text{SD}}}} (k) = \left( {X_{1} (k) - X_{2} (k)} \right)$$

(20)

The excessive vertical movement of the vehicle wheel results in hard impact with body of the vehicle. To maintain quality ride and conformability, the suspension deflections of the vehicle during the bump road disturbance period should be minimized as little as possible. The performance index for minimizing suspension deflection *J*_{SD} can be expressed by the following equation.

$$J_{{{\text{SD}}}} = \sum\limits_{k = 1}^{N} {\left( {Z_{{{\text{SD}}}} (k)} \right)^{2} }$$

(21)

*J*_{SD} is the performance index of the suspension deflection and can be calculated by the sum of *N* samples considered during the period crossing the bump road disturbance.

Excessive tyre deflections lead to poor contact between the tyre and the road surface (for the tyre extended) and hence a reduced ability to control the vehicle, for example, during braking. The tyre deflection (*Z*_{TD}) is the difference between the vertical displacement of unsprung mass and road input.

$$Z_{{{\text{TD}}}} (k) = \left( {X_{2} (k) - R_{{\text{d}}} (k)} \right)$$

(22)

The following equation shows the performance index for minimizing the tyre deflection.

$$J_{{{\text{TD}}}} = \sum\limits_{k = 1}^{N} {\left( {Z_{{{\text{TD}}}} (k)} \right)^{2} }$$

(23)

*J*_{TD} is the performance index of the tyre deflection calculated from the *N* samples considering road disturbance input.

The sprung mass acceleration can be calculated from the following expression.

$$\ddot{Z}_{{{\text{SM}}}} (k) = \frac{{(X_{3} (k) - X_{3} (k - 1))}}{T}$$

(24)

The performance objective for minimizing the vertical acceleration of the sprung mass can be expressed by the following equation.

$$J_{{{\text{VA}}}} = \sum\limits_{k = 1}^{N} {\left( {\ddot{Z}_{{{\text{SM}}}} (k)} \right)^{2} }$$

(25)

*J*_{VA} is the sum of the acceleration samples calculated over *N* samples.

### Conventional weighted multiobjective function

For formulating a multiobjective function, all the above objectives are normalized with respect to passive suspension system performance. The normalized conventional weighted multiobjective [35] function (*J*_{C}) can be formulated as follows:

$$J_{{\text{C}}} = W_{1} \, \frac{{J_{{{\text{SD}}}}^{A} }}{{J_{{{\text{SD}}}}^{P} }} + W_{2} \, \frac{{J_{{{\text{TD}}}}^{A} }}{{J_{{{\text{TD}}}}^{P} }} + W_{3} \, \frac{{J_{{{\text{VA}}}}^{A} }}{{J_{{{\text{VA}}}}^{P} }}.$$

(26)

In the above expression, *W*_{1}, *W*_{2} and *W*_{3} are weighting factors for the three objective functions considered and can be selected based on the importance given to the objective functions. In the present case, equal importance is considered for all the objectives. The \(J_{{{\text{VA}}}}^{A}\) is the objective function for acceleration in the case of an active suspension system, and \(J_{{{\text{VA}}}}^{P}\) is the objective function for vehicle acceleration in the case of a passive suspension system.

### Pareto-based multiobjective function formation

An active suspension system is a multiobjective problem where suspension deflection and ride quality are equally important. In addition, the system's stability with a controller should also be satisfactory for the successful function of the active suspension system. In the present case, the objectives are suspension deflection, vehicle acceleration and tyre deflection. In addition to the three objectives mentioned, it is also essential to consider the stability of the system. The system is said to be stable if all the eigenvalues lie on the *s* plane’s left half. Furthermore, the real values of the eigenvalues must be far from the origin. Hence the following objective function is considered to improve the stability of the vehicle’s active suspension system with a PID controller.

$$\zeta = \max ({\text{real}}(\lambda_{i} );\;i = 1,2, \ldots n$$

(27)

$$J_{{{\text{EV}}}} = \left\{ {\begin{array}{*{20}l} {\left| \zeta \right|} \hfill & {{\text{if}}\;\zeta < 0} \hfill \\ 0 \hfill & {{\text{if}}\;\zeta > 0} \hfill \\ \end{array} } \right.$$

(28)

where \(\lambda_{i}\) is the *i*th eigenvalue of the system. *J*_{EV} is the eigenvalue-based objective function to improve the stability of the system. A solution is said to be Pareto optimal if and only if all the objectives are improved compared to the solution obtained in the previous iteration [36].

The Pareto objective set can be defined as

$$J_{{\text{P}}}^{m} = \, \left[ {J_{{{\text{SD}}}}^{m} ,\;J_{{{\text{TD}}}}^{m} ,\;J_{{{\text{VA}}}}^{m} ,\;J_{{{\text{EV}}}}^{m} } \right].$$

(29)

According to Pareto optimality, the (*m* + 1)th solution vector \(J_{{\text{P}}}^{m + 1}\) is a better solution than the *m*th solution vector if and only if at least one of the Pareto objectives related to (*m* + 1)th solution vector must be improved over the *m*th solution vector \(J_{{\text{P}}}^{m}\) while the others retain their advantage. The Pareto optimality conditions on four objectives considered are shown by the following equations.

If *m* is the iteration number, the solution \(J_{{\text{P}}}^{m + 1}\) is Pareto optimal only when all the following conditions are satisfied.

$$J_{{{\text{SD}}}}^{m + 1} \le J_{{{\text{SD}}}}^{m}$$

(30)

$$J_{{{\text{TD}}}}^{m + 1} \le J_{{{\text{TD}}}}^{m}$$

(31)

$$J_{{{\text{VA}}}}^{m + 1} \le J_{{{\text{VA}}}}^{m}$$

(32)

$$J_{{{\text{EV}}}}^{m + 1} \ge J_{{{\text{EV}}}}^{m}$$

(33)

The Pareto objectives for suspension deflection, tyre deflection and vehicle acceleration must be less than the previous generation's best values before moving to next generation while the eigenvalue-based objective must be greater than the previous generation's best value to improve the system's stability.