A Multiple-Model Based Adaptive Actuator Failure Compensation Scheme for Nonlinear Systems With Dynamic Mutations

• WEN Li-Yan^{1, 2 ,}  , 
• TAO Gang^{2 ,}  , 
• JIANG Bin^{1 ,}  , 
• YANG Jie^{1 ,} 

• 1.

  College of Automation Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China

• 2.

  Department of Electrical and Computer Engineering, University of Virginia, Charlottesville 22903, USA

Author Bio:

WEN Li-Yan　 Associate professor at the College of Automation Engineering, Nanjing University of Aeronautics and Astronautics (NUAA). She received her Ph.D. degree from NUAA in 2016. Her research interest covers adaptive control, adaptive fault isolation, and fault tolerant control and their applications. Corresponding author of this paper

TAO Gang　Professor at University of Virginia, USA. His research interest covers adaptive control and its applications

JIANG Bin　Professor at the College of Automation Engineering, Nanjing University of Aeronautics and Astronautics. His research interest covers fault diagnosis and fault tolerant control and their applications

YANG Jie　Master student at the College of Automation Engineering, Nanjing University of Aeronautics and Astronautics. His research interest covers adaptive control and its application



¹ 针对单输入单输出的非线性系统: $ \dot{{\boldsymbol{x}}} = f({\boldsymbol{x}})+g({\boldsymbol{x}}){\boldsymbol{u}}+ p({\boldsymbol{x}}){\boldsymbol{d}}, $ $y = $$ h({\boldsymbol{x}}) , $ 若对所有的$ {\boldsymbol{x}}\in {\bf{R}}^{{n}} ,$ $ L_gL_f^kh({\boldsymbol{x}}) = 0, $ $ \forall k = 0, 1, 2, \cdots, $ $ \rho-2 ,$ 且$ L_gL_f^{\rho-1}h({\boldsymbol{x}})\neq 0, $ 则此非线性系统的控制相对阶为$ \rho ;$ 若对所有的$ {\boldsymbol{x}} \in {\bf{R}}^{{n}} ,$ $ L_pL_f^kh({\boldsymbol{x}}) = 0, $ $ \forall k = 0, 1, 2, \cdots, \nu-2, $ 且$ L_pL_f^{\nu-1}h({\boldsymbol{x}})\neq $$ 0 ,$ 则此非线性系统的扰动相对阶为$ \nu .$ 关于李导数的定义参见脚注2.² 若$ f({\boldsymbol{x}})\in {\bf{R}}^n, $ $ g({\boldsymbol{x}})\in {\bf{R}}^{n}, $ $ h({\boldsymbol{x}})\in {\bf{R}}, $ 定义李导数: $ L_fh({\boldsymbol{x}}) = $$ \dfrac{\partial h({\boldsymbol{x}})}{\partial {\boldsymbol{x}}}f({\boldsymbol{x}}) = \dfrac{\partial h({\boldsymbol{x}})}{\partial x_1}f_1+\cdots+\dfrac{\partial h(x)}{\partial x_n}f_{{n}} $, 且有$ L_f^0h({\boldsymbol{x}}) = h({\boldsymbol{x}}) $, $ L_f^{i+1}h({\boldsymbol{x}}) =L_f(L_f^ih({\boldsymbol{x}})) = \dfrac{\partial L_f^ih({\boldsymbol{x}})}{\partial {\boldsymbol{x}}}f({\boldsymbol{x}}) $. 类似地, $ L_gL_fh ({\boldsymbol{x}}) = $$ \dfrac{\partial L_fh({\boldsymbol{x}})}{\partial {\boldsymbol{x}}}g({\boldsymbol{x}}) $, $ L_gL_f^ih({\boldsymbol{x}}) = \dfrac{\partial L_f^ih({\boldsymbol{x}})}{\partial {\boldsymbol{x}}}g({\boldsymbol{x}}) $.
　
³ 在自适应控制系统中, 平均小(Small in the mean)是很常用的概念, 其是保证闭环系统稳定的充分条件.
⁴ $ \|{\boldsymbol{q}}\|_t $为截断$ L^{\infty} $范数, 即: $ \sup_{\tau\leq t}\|{\boldsymbol{q}}(\tau)\| $.⁵ 若对所有的$ t\geq 0 $, 有$ |\dot{{\boldsymbol{z}}}(t)|\leq k_1\|{\boldsymbol{z}}_t\|_{\infty}+k_2 $, 其中$ k_1\geq 0 $, $ k_2\geq0 $, 则信号$ {\boldsymbol{z}}(t) $是正则的.