﻿ 一类反馈型非线性系统的跟踪控制<sup>*</sup>
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Tracking control for a class of nonlinear systems in feedback form
YU Jianghang, XU Jun, HUANG Yuke
School of Aerospace Engineering, Beijing Institute of Technology, Beijing 100081, China
Received: 2018-11-22; Accepted: 2019-02-16; Published online: 2019-02-26 13:42
Corresponding author. XU Jun, E-mail: xujun2324@x263.net
Abstract: In order to achieve the control of a class of nonlinear systems in feedback form, the system is studied. First, according to LaSalle's invariance principle, the convergence of a class of autonomous systems is proved. The error function is introduced, and the Lyapunov function of the error function is used to find the controller which makes the error function asymptotically stable. Then, according to the lemma, the trajectories tracked by the system states are all converged, so that the system states are bounded and the output of the system converges to input. The condition and the proof of the stability of the closed-loop system are given. Finally, an example of longitudinal dynamics of an fixed-wing aircraft flight control system is presented, and the controller is designed according to the proposed method. The simulation is verified under the Simulink module of MATLAB. The results show that, for step signals and sinusoidal signals, the proposed controller can enable the pitch angle of aircraft to quickly converge the tracking command.
Keywords: nonlinear systems     feedback     Lyapunov methods     system stability     flight control systems

1 问题的提出

 (1)

 (2)
 (3)

2 控制器的设计

 (4)

1)

2) 设DcR上一闭区间，当x1Dc, x2, x3R时，ϕ1ϕ2有界。

 (5)

 (6)

1)

2) 设DcR上一闭区间，当x1Dcx2, x3, x4R时，ϕ1ϕ2ϕ3有界。

 (7)

 (8)

 (9)

 (10)

1) M的所有特征值均大于0。

2) M的特征空间维数等于n

3) 存在常数cf，使得任意xRn，都有‖f(x)‖≤cf

 (11)

 (12)

 (13)

 (14)

 (15)

V(x)对t求导：

 (16)

Mn个特征值为λ1, λ2, …, λn，由于M的特征空间维数为n，所以有n个线性无关的特征向量。因此对于任意非零向量xRn，总是存在n个实数c1, c2, …, cn，使得

 (17)

 (18)

x‖→∞时，‖β‖为实数，λmin(M)＞0，因此Vmin(x)→∞。Vmin(x)是径向无界的，对于Vmax(x)也有相同的结果。又由式(15)，可以得出V(x)也是径向无界的。所以对于任意一个l＞0，集合Ω={xRn|V(x)≤l}都是有界的。根据Ω的定义，它又是一个闭集，故Ω是一个紧集。设，则。而就是系统的平衡点，故E为不变集。其最大不变集J等于自身。根据引理1，集合Ω是关于系统的正向不变的紧集，在Ω内满足≤0。集合，其最大不变集J=E。又V(x)径向无界，故对于任意的初始状态x(0)，当t→∞时，轨迹x(t)都将趋于集合J。也就是系统的解x(t)→M-1f(x)。而‖f(x)‖有界，因此‖x(t)‖在全局上都满足有界。         证毕

 (19)

 (20)

 (21)

V(e)求导，并且代入式(4)、式(19)得

 (22)

 (23)

 (24)

 (25)

V(e)径向无界，则由误差函数e构成的系统全局渐近稳定。也就有当t→∞，有e0，即x→[r α1 α2 α3]T。注意，此时已有x1有界。若能证明函数α1α2α3t→∞时收敛，则有系统镇定，所有状态均有界。

 (26)

3 算例

 (27)
 (28)

 (29)

 (30)

 (31)

x=[θ α q]Tu=δe，系统可以写为

 (32)

 (33)

 (34)

 (35)

1) 阶跃信号

 图 1 阶跃信号下俯仰角变化曲线 Fig. 1 Pitch angle change curve under step signal

 图 2 阶跃信号下系统状态变化曲线 Fig. 2 System states change curves under step signal

2) 正弦信号

 图 3 正弦信号下俯仰角变化曲线 Fig. 3 Pitch angle change curve under sinusoidal signal

 图 4 正弦信号下系统状态变化曲线 Fig. 4 System states change curves under sinusoidal signal

4 结论

1) 控制器能够保证闭环系统的稳定性，所有状态全部有界。

2) 控制器可以有效的完成跟踪任务，通过调整设计参数，跟踪误差可以收敛于满足需要的小范围内。

3) 系统输出的收敛速度非常快。

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#### 文章信息

YU Jianghang, XU Jun, HUANG Yuke

Tracking control for a class of nonlinear systems in feedback form

Journal of Beijing University of Aeronautics and Astronsutics, 2019, 45(7): 1444-1450
http://dx.doi.org/10.13700/j.bh.1001-5965.2018.0688