二阶弹簧阻尼系统粒子群优化

下面是二阶弹簧阻尼系统粒子群优化的Python代码示例：

import numpy as np

# 定义目标函数（稳态误差）
def objective(c, k):
    m = 1.0
    omega = np.sqrt(k/m)
    zeta = c/(2*np.sqrt(m*k))
    e = 1/(1+(omega**2)*(1-zeta**2))
    return e

# 定义粒子群优化函数
def pso(obj_func, dim, n_particles, n_iter, lb, ub):
    # 初始化粒子群
    x = np.random.uniform(lb, ub, (n_particles, dim))
    v = np.zeros((n_particles, dim))
    p = x.copy()
    pbest = np.zeros(n_particles)
    gbest = np.zeros(dim)
    gbest_fit = np.inf

    # 迭代优化
    for i in range(n_iter):
        # 更新速度和位置
        r1 = np.random.rand(n_particles, dim)
        r2 = np.random.rand(n_particles, dim)
        v = 0.5*v + 2*r1*(p-x) + 2*r2*(gbest-x)
        x = x + v

        # 边界处理
        x = np.clip(x, lb, ub)

        # 更新个体最优解和全局最优解
        for j in range(n_particles):
            fitness = obj_func(*x[j])
            if fitness < pbest[j]:
                pbest[j] = fitness
                p[j] = x[j].copy()
            if fitness < gbest_fit:
                gbest_fit = fitness
                gbest = x[j].copy()

    return gbest, gbest_fit

# 调用粒子群优化函数求解最优解
c_best, k_best = pso(objective, dim=2, n_particles=50, n_iter=100, lb=0, ub=10)
print("最优解：c = {:.4f}, k = {:.4f}".format(c_best, k_best))
print("最小稳态误差：{:.4f}".format(objective(c_best, k_best)))

其中，objective函数定义了目标函数（稳态误差），pso函数实现了粒子群优化算法，c_best和k_best分别表示最优解的阻尼系数和弹簧劲度系数，最小稳态误差即为优化结果。