The optimal control of semi-active suspension based on improved particle swarm optimization

2.1. The filter white noise road input

Based on the known power spectral density of the road surface, the time-domain model of the road surface was established by the filter white noise [4] method, as follows:

where $z_g\left(t\right)$ is the road displacement, $f_0$ is the lower cut-off frequency, $G_q\left(n_0\right)$ is the road roughness coefficient, $u$ is the vehicle speed, $w\left(t\right)$ is the filtered white noise.

2.2. The semi-active suspension model establishment

A reasonable assumption for the suspension system was made to establish the model of the quarter vehicle semi-active suspension system which had two degree of freedom, shown as in Fig. 1.

Fig. 1The model of the quarter vehicle semi-active suspension system

According to Newton second law, the motion differential equation of the semi-active suspension system which had two degree of freedom can be expressed as:

A dynamics model for two degree of freedom suspension system, the state variables of the system was $X=[{\dot{z}}_1,{\dot{z}}_2,z_1,z_2,z_g]$, the input were the controller’s control force $U_a$ and the road disturbance $z_g$, made $U=[U_a,z_g]$.

The established state equation of the system was: $\dot{X}=AX+BU+DW$:

3.1. The optimal controller establishment

The target of the optimal control of semi-active suspension was to make the vehicle getting higher ride comfort and smoothness. Therefore, the linear quadratic comprehensive performance index functional of semi-active suspension was defined as:

where $q_1$ is the weighting coefficient of body acceleration, $q_2$ is the weighting coefficient of suspension working space, and $q_3$ is the weighting coefficient of tires dynamic displacement, and the form of matrix was:

The $Q$ and $R$ in the upper formula are refer to the weighted matrices of the state variables and the control variables respectively. The different values of $Q$ and $R$ allowed the different weighting coefficients to be added to the different performance indexes in the function. After the calculation, the values of $Q$, $R$, and $N$ were respectively as follows:

In the optimal controller of semi-active suspension, the state variable weighting coefficient matrix $Q$ and the control variable weighted coefficient matrix $R$ are determined by the major experts who are known the prior knowledge. It was difficult to obtain the global optimal LQR controller. For semi-active suspension system of the vehicle, an important problem of the optimal control was determining the reasonable weighted coefficient value of the performance function. In order to control the weighted coefficient matrix better, this paper optimized the weighting coefficient $q_1$, $q_2$ and $q_3$ of the optimal controller according to the particle swarm optimization, and then changed the state variable weighted coefficient matrix and the control variable weighted coefficient matrix, and obtained the different optimal control feedback gain matrix, and designed an optimal controller of semi-active suspension which could meet the requirements of riding comfort and smoothness [5].

3.2. The improved particle swarm optimization

In particle swarm optimization, each particle dynamically adjusted its speed and position according to its own flight experience and the group flight experience. Besides, each particle position represented the feasible solution of the optimization problem [6]. Given a D-dimensional searched space with $n$ particles, the iteration step was $j_{step}$, the position and velocity of the $i$th particle were $X_i=$ ($X_{1,j}$, $X_{2,j}$, …, $X_{d,j}$) and $V_i=$ ($V_{1,j}$, $V_{2,j}$, …, $V_{d,j})$ respectively. In each iteration, the particle dynamically adjusted its velocity vector to adjust its position according to the individual extreme $P_i=$ ($P_{1,j}$, $P_{2,j}$, …, $P_{d,j}$) and the global extreme $P_g=$ ($P_{1,j}$,$P_{2,j}$, …, $P_{d,j}$). The traditional particle swarm optimization algorithm was prone to premature convergence. In order to ensure the stability of the algorithm, this paper improved the particle swarm optimization, and introduced the contraction factor $\chi$ to ensure the convergence of the particle update [7]. Then the evolution equation of the velocity and position of the $i$th particle was as follows:

In the formula, $c_1$ and $c_2$ are acceleration factors, $r_1$ and $r_2$ are the random number uniformly distributed between the sum of the [0 1], $j_{step}$ is the number of current iterations, $\chi$ is the contraction factor:

3.3. The optimal controller of the improved particle swarm optimization

The particle swarm optimization controller was proposed in this paper, the weighting coefficient was continuously improved according to the particle swarm optimization, so the feedback gain matrix $K$ was constantly updated. In order to call the optimal controller weighted matrix easily by the particle swarm program, the optimal controller was written in the MATLAB S-function. Part of the programs were follows as:

function [sys, x0, str, ts]n= mdl Initialize Sizes;

sizes = sim sizes;

sizes.Num Cont States = 0;

sizes.Num Disc States = 0;

sizes.Num Out puts = 1;

sizes.Num Inputs = 5;

sizes.Dir Feed through = 1;

sizes.Num Sample Times = 1;

[K, S, E] = lqr(A, B, Q, R, N);

sys = – K * u.

This paper selected fitness function as:

where $a_j$, $a_{j-1}$ is the body acceleration in step $j$ and step $j-1$; $SW{S}_j$, $SW{S}_{j-1}$ is the suspension working space in step $j$ and step $j-1$; $DT{D}_j$, $DT{D}_{j-1}$ is the Tires Dynamic Displacement in step $j$ and step $j-1$. The model parameters of vehicle semi-active suspension system were shown in Table 1.

The simulation semi-active suspension model was established that based on the optimal control strategy of improved particle swarm optimization was shown in Fig. 2.

The program flowchart of the optimal controller based on IPSO is shown in Fig. 3.

The test vehicle passed the B level road surface with the uniform speed of 20 m/s, the lower cut-off frequency was 0.01 Hz, the white noise power was 20 dB, the solver was a fixed step ode4 solver [8], the simulation time was 10s. The optimal controller parameters were as follows: $q_1=$ 2808, $q_2=$ 614, $q_3=$ 32, $K=$1.0×10⁵×[0.78, –0.20, 4.30, –5.6, 1.67]^T.

Fig. 2The semi-active suspension model based on the optimal control strategy of improved particle swarm optimization

Fig. 3The program flowchart of the optimal controller based on IPSO