Compensation of Wave Actuator Dynamics for Nonlinear Systems
Compensation of Wave Actuator Dynamics for Nonlinear Systems
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DOI：
10.1109/tac.2014.2309057
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发表时间：
2014-03
影响因子：
6.8
通讯作者：
N. Bekiaris-Liberis;M. Krstić
中科院分区：
文献类型：
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作者：
N. Bekiaris-Liberis;M. Krstić
对于几类偏微分方程组，在线性情况下已解决了PDE-ODE串级的镇定问题，而在非线性情况下，只解决了传输/延迟PDE的镇定问题，即补偿非线性对象输入的任意延迟。受海洋钻井工程应用的启发，我们解决了具有一般非线性常微分方程组的波动偏微分方程级联的镇定问题。由于任意增长的非线性的存在和波动偏微分方程组的时间可逆性，以及基于Lyapunov泛函或显式解的变元的可能性，几种稳定性分析方法是可能的。我们给出了一般非线性常微分方程组的H_2×H_1和C_1×C_0范数以及线性常微分方程组的H_1×L_2范数下的稳定性结果。我们将波PDE-ODE的一般设计专门化到波PDE级联的情况，该波PDE的未受控端不驱动ODE，而是受非线性Robin边界条件(“非线性弹簧”，如钻井中的摩擦定律)控制。这是第一个包含超线性增长的非配置不稳定非线性的波动方程的全局镇定结果。我们给出了两个数值例子，一个是非线性常微分方程组，另一个是在波动偏微分方程组的非受控边界处有一个非线性弹簧。
The problem of stabilization of PDE-ODE cascades has been solved in the linear case for several PDE classes, whereas in the nonlinear case the problem has been solved only for the transport/delay PDE, namely for compensation of an arbitrary delay at the input of a nonlinear plant. Motivated by a specific engineering application in off-shore drilling, we solve the problem of stabilization of the cascade of a wave PDE with a general nonlinear ODE. Due to the presence of nonlinearities of arbitrary growth and the time-reversibility of the wave PDE, and due to the possibility of using arguments based on Lyapunov functionals or explicit solutions, several stability analysis approaches are possible. We present stability results in the H2 × H1 and C1 × C0 norms for general nonlinear ODEs, as well as in the H1 × L2 norm for linear ODEs. We specialize our general design for wave PDE-ODE cascades to the case of a wave PDE whose uncontrolled end does not drive an ODE but is instead governed by a nonlinear Robin boundary condition (a “nonlinear spring,” as in the friction law in drilling). This is the first global stabilization result for wave equations that incorporate non-collocated destabilizing nonlinearities of superlinear growth. We present two numerical examples, one with a nonlinear ODE and one with a nonlinear spring at the uncontrolled boundary of the wave PDE.