Adaptive CFB Control for a Class of Nonlinear Systems with Intermittent Actuator Faults
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摘要: 控制系统的执行器在运行过程中经常发生各种各样不可预测的间歇性故障. 如何有效地处理这些故障仍然是控制领域的一个难题. 针对一类不确定严格反馈非线性系统, 提出一种自适应CFB (Command Filtered Backstepping) 控制方案解决了间歇性执行器故障的补偿问题. 利用神经网络逼近控制器中的未知函数, 并采用投影算子实时在线更新控制器中的估计参数使得参数估计随着故障次数的累积而不断增加的问题被消除. 考虑到未知参数间歇性跳变对系统稳定性的影响, 提出一种改进的Lyapunov函数分析了闭环系统的稳定性. 证明了所提出的控制方案能够保证所有闭环信号的有界性, 同时建立了跟踪误差与Lyapunov函数跳变幅度, 最小故障时间间隔, 设计参数之间的关系. 如果Lyapunov函数的跳变幅度越小以及两个连续故障之间的时间间隔越长, 系统的稳态跟踪指标越好. 通过迭代计算建立了暂态跟踪误差指标的均方根型界. 该界表明了通过选择恰当的设计参数, 可改善系统的暂态指标. 仿真结果表明了所提方案的有效性.
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关键词:
- 自适应控制 /
- Command filtered backstepping /
- 非线性系统 /
- 间歇性执行器故障
Abstract: Actuators of control systems frequently occur various unpredictable intermittent faults during operation. How to effectively handle such faults is still a difficult problem. In this paper, an adaptive command filtered backstepping (CFB) compensation control scheme is proposed for a class of uncertain strict-feedback nonlinear systems to address the issue of compensation for intermittent actuator faults. Neural networks are utilized in the controller to approximate unknown functions, and a smooth projection algorithm is adopted to update the estimated parameters in the controller such that the problem of parameter estimate increase with the accumulation of the number of faults is eliminated. Considering the effects of the intermittent jumps of unknown parameters on the system stability, a modified Lyapunov function is proposed to analyze to the stability of the close-loop system. It is proved that the proposed control scheme can guarantee the boundedness of all closed-loop signals and, the relationship among the tracking error, jumping amplitude of Lyapunov function, minimum fault time interval and design parameters can be established. It is shown that if the jumping amplitude of Lyapunov function is smaller and the time interval between two adjacent faults is longer, the system steady-state tracking performance is better. A root mean square type of bound for the transient tracking error performance is established by using iterative calculation to illustrate that the system transient performance is improved by appropriate choice of design parameters. Simulation results validate the effectiveness of the proposed scheme. -
图 1 控制结构图.
$x_{\mathrm{c}i}$ 和$\dot{x}_{\mathrm{c}i}$ ,$i = 2,\ldots,n$ , 为滤波器 (11) 的输出.$\alpha_i$ ,$i = 1,\ldots,n$ , 为式 (12)−(14) 中定义的虚拟控制律.$\xi_i$ ,$i = 1,\ldots,n-1$ 为式 (17) 中定义的滤波误差补偿信号.Fig. 1 Control block diagram.
$x_{\mathrm{c}i}$ and$\dot{x}_{\mathrm{c}i}$ for$i = 2, \ldots,n$ are the outputs of the filter (11).$\alpha_i$ for$i = 1,\ldots,n$ is virtual control law defined in (12)−(14).$\xi_i$ for$i = 1,\ldots, n-1$ is the compensating signal of the filtered error defined in (17).
图 2 (a) 输出
$y$ 和期望输出$y_{\mathrm{d}}$ ; (b) 跟踪误差$\bar{z}_1$ .Fig. 2 (a) Output
$y$ and desired reference$y_{\mathrm{d}}$ ; (b) Tracking error$\bar{z}_1$ .
图 4 命令滤波误差及其补偿信号. (a)
$x_{\mathrm{c}2}-\alpha_1$ ; (b)$x_{\mathrm{c}3}-\alpha_2$ ; (c)$\xi_1$ ; (d)$\xi_2$ .Fig. 4 Command filtered errors and their compensating signals. (a)
$x_{\mathrm{c}2}-\alpha_1$ ; (b)$x_{\mathrm{c}3}-\alpha_2$ ; (c)$\xi_1$ ; (d)$\xi_2$ .
图 5 自适应参数. (a)
$\|\hat{\pmb{\theta}}_1\|$ ; (b)$\|\hat{\pmb{\theta}}_2\|$ ; (c)$\|\hat{\pmb{\theta}}_3\|$ ; (d)$\hat{\rho}$ ; (e)$\hat{\beta}_1$ ; (f)$\hat{\beta}_2$ .Fig. 5 Adaptive parameters. (a)
$\|\hat{\pmb{\theta}}_1\|$ ; (b)$\|\hat{\pmb{\theta}}_2\|$ ; (c)$\|\hat{\pmb{\theta}}_3\|$ ; (d)$\hat{\rho}$ ; (e)$\hat{\beta}_1$ ; (f)$\hat{\beta}_2$ .
图 6 (a) 输出
$y$ 和期望输出$y_{\mathrm{d}}$ ; (b) 跟踪误差$\bar{z}_1$ .Fig. 6 (a) Output
$y$ and desired reference$y_{\mathrm{d}}$ ; (b) Tracking error$\bar{z}_1$ .
图 8 命令滤波误差及其补偿信号. (a)
$x_{\mathrm{c}2}-\alpha_1$ ; (b)$x_{\mathrm{c}3}-\alpha_2$ ; (c)$\xi_1$ ; (d)$\xi_2$ .Fig. 8 Command filtered errors and their compensating signals. (a)
$x_{\mathrm{c}2}-\alpha_1$ ; (b)$x_{\mathrm{c}3}-\alpha_2$ ; (c)$\xi_1$ ; (d)$\xi_2$ .
图 9 自适应参数. (a)
$\|\hat{\pmb{\theta}}_1\|$ ; (b)$\|\hat{\pmb{\theta}}_2\|$ ; (c)$\|\hat{\pmb{\theta}}_3\|$ ; (d)$\hat{\rho}$ ; (e)$\hat{\beta}_1$ ; (f)$\hat{\beta}_2$ .Fig. 9 Adaptive parameters. (a)
$\|\hat{\pmb{\theta}}_1\|$ ; (b)$\|\hat{\pmb{\theta}}_2\|$ ; (c)$\|\hat{\pmb{\theta}}_3\|$ ; (d)$\hat{\rho}$ ; (e)$\hat{\beta}_1$ ; (f)$\hat{\beta}_2$ .
图 10 (a) 输出
$y$ 和期望输出$y_{\mathrm{d}}$ ; (b) 跟踪误差$\bar{z}_1$ .Fig. 10 (a) Output
$y$ and desired reference$y_{\mathrm{d}}$ ; (b) Tracking error$\bar{z}_1$ .
图 12 命令滤波误差及其补偿信号. (a)
$x_{\mathrm{c}2}-\alpha_1$ ; (b)$x_{\mathrm{c}3}-\alpha_2$ ; (c)$\xi_1$ ; (d)$\xi_2$ .Fig. 12 Command filtered errors and their compensating signals. (a)
$x_{\mathrm{c}2}-\alpha_1$ ; (b)$x_{\mathrm{c}3}-\alpha_2$ ; (c)$\xi_1$ ; (f)$\xi_2$ .
图 13 自适应参数. (a)
$\|\hat{\pmb{\theta}}_2\|$ ; (b)$\|\hat{\pmb{\theta}}_3\|$ ; (c)$\hat{\rho}$ ; (d)$\hat{\beta}_1$ 和$\hat{\beta}_2$ .Fig. 13 Adaptive parameters. (a)
$\|\hat{\pmb{\theta}}_2\|$ ; (b)$\|\hat{\pmb{\theta}}_3\|$ ; (c)$\hat{\rho}$ ; (d)$\hat{\beta}_1$ and$\hat{\beta}_2$ .
图 14 (a) 输出
$y$ 和期望输出$y_{\mathrm{d}}$ ; (b) 跟踪误差$\bar{z}_1$ .Fig. 14 (a) Output
$y$ and desired reference$y_{\mathrm{d}}$ ; (b) Tracking error$\bar{z}_1$ .
图 16 命令滤波误差及其补偿信号. (a)
$x_{\mathrm{c}2}-\alpha_1$ ; (b)$x_{\mathrm{c}3}-\alpha_2$ ; (c)$x_{\mathrm{c}4}-\alpha_3$ ; (d)$\xi_1$ ; (e)$\xi_2$ ; (f)$\xi_3$ .Fig. 16 Command filtered errors and their compensating signals. (a)
$x_{\mathrm{c}2}-\alpha_1$ ; (b)$x_{\mathrm{c}3}-\alpha_2$ ; (c)$x_{\mathrm{c}4}-\alpha_3$ ; (d)$\xi_1$ ; (e)$\xi_2$ ; (f)$\xi_3$ .
图 17 自适应参数. (a)
$\|\hat{\pmb{\theta}}_2\|$ ; (b)$\|\hat{\pmb{\theta}}_4\|$ ; (c)$\hat{\rho}$ ; (d)$\hat{\beta}_1$ 和$\hat{\beta}_2$ .Fig. 17 Adaptive parameters. (a)
$\|\hat{\pmb{\theta}}_2\|$ ; (b)$\|\hat{\pmb{\theta}}_4\|$ ; (c)$\hat{\rho}$ ; (d)$\hat{\beta}_1$ and$\hat{\beta}_2$ . -
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