Predefined time tracking control of underactuated surface vessel with input saturation
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摘要:目的
为解决欠驱动水面舰艇(USV)在模型不确定性、强耦合特性和控制器输入饱和情况下的轨迹跟踪问题,提出基于输入饱和的USV预定义时间跟踪控制方法。
方法针对USV模型具有非零对角项和强耦合特性问题,引入坐标变换,将系统模型转变为斜对角形式; 将预定义时间性能函数与障碍Lyapunov函数(BLF)结合,保证瞬态和稳态的跟踪性能;利用自组织神经网络(SSNN)降低未知外部环境扰动和复杂的连续未知非线性项以及输入饱和产生的影响,以保证控制系统的跟踪精度,并且在线调整优化SSNN的神经元数目,减少控制系统的计算负担。
结果基于Lyapunov稳定性理论,证明了闭环系统在预定义时间内是有界稳定的,跟踪误差始终保持在约束范围内。
结论仿真结果表明,所提控制策略是有效的,其具有良好的跟踪性能。
Abstract:ObjectiveTo solve the trajectory tracking problem of underactuated surface vessels (USVs) under the condition of model uncertainty, strong coupling characteristics and controller input saturation, this study proposes a predefined time tracking control method for USVs based on input saturation.
MethodsDue to the non-zero diagonal terms and strong coupling characteristics of the USV model, coordinate transformation is introduced to transform the system model into a diagonal form. The predefined time performance function is combined with the barrier Lyapunov function (BLF) to ensure transient and stable tracking performance. Self-structuring neural networks (SSNN) are used to approximate unknown external disturbances and complex continuous unknown nonlinear terms, and deal with the impact of actuator saturation, thus ensuring the tracking performance of the control system. Moreover, the number of SSNN neurons can be adjusted online, reducing the computational burden on the control system.
ResultsBased on Lyapunov stability theory, it is proven that the closed-loop system is bounded stable in a predefined time, and the tracking error is always within the constraint range.
ConclusionThe simulation results show that the proposed control strategy is effective and has good tracking performance.
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0. 引 言
在海洋资源大开发的背景下,欠驱动水面舰艇(USV)被广泛应用于海上救援、海上侦察、目标搜索跟踪以及海洋环境调查等[1]。欠驱动水面舰艇的轨迹跟踪控制问题,多年来一直备受关注。USV动力学模型是高度非线性和强耦合的,复杂多变的海洋环境以及船舶模型的不确定性加大了控制器的设计难度,此外,USV的欠驱动特性对控制器设计提出了更高要求[2]。
面对复杂的海洋环境,欠驱动水面舰艇是否具有快速准确的跟踪性能,极大地影响了执行海上任务的准确性和自主性。基于船舶精确动态模型的假设,Wang等[3]实现了USV跟踪误差的渐近收敛。实际上,由于复杂的水动力矩和风效应,船舶模型是高度非线性的,获得精确的模型信息并非易事。考虑船舶模型的不确定性以及外部环境干扰,文献[4-8]保持了跟踪误差的一致最终有界性,尽管跟踪误差的收敛速度和残差集的大小可调整,但是无法提前确定跟踪误差的精度和收敛速度。文献[9-11]实现了跟踪误差以规定的速率落入预选区域,进一步提高了船舶的瞬态和稳态的跟踪精度;但由于指数型性能函数的收敛速度较慢,无法保证跟踪误差能够在预设时间内收敛至期望精度;实际上,精确的跟踪精度需要在预定义的时间内实现。文献[12-13]引入了障碍Lyapunov函数(BLF)以确保USV的跟踪误差满足预设的跟踪精度,但是系统误差是渐近收敛的。文献[14-16]引入了正切型和对数型BLF,以解决误差约束问题,并提出了有限时间稳定控制策略,然而,有限时间稳定会受初始状态的影响。Zhang等[17]基于对数型BLF函数,并且进一步考虑了欠驱动船舶控制系统的固定时间稳定特性,但是,其固定时间稳定过于保守。与此同时,不可忽视的事实是,对数型BLF会受到功能上的限制,而正切型BLF又会使控制器的设计过于复杂[18]。在现实中,USV的运动通常会受到执行器输出饱和的限制,因此,在发生执行器输出饱和的情况下,如何实现USV快速准确的跟踪是一个值得研究的问题。文献[19-20]在保证USV的瞬态和稳态跟踪精度的情况下,进一步考虑了执行器输出饱和问题。
鉴于难以通过流体动力学的方法获得精确的USV模型信息,为了逼近非线性阻尼项,张强等[21]利用径向基神经网络(radial basis function neural network, RBFNN)重构船舶的动力学模型不确定性项,实现了欠驱动船舶的有限时间跟踪控制。为估计外部环境扰动,文献[22-24]设计了扰动观测器。文献[25- 26]采用了自适应神经网络方案逼近未知非线性阻尼项以及外部环境扰动。然而,在神经网络补偿策略中,大量用于估计和识别的参数的在线更新,显著增加了控制系统的计算成本。文献[27]引入自组织模糊神经网络对船舶未知动态进行补偿,通过结构学习准则在线生成或修剪模糊规则,有效调节了控制系统的计算量,然而自组织模糊神经网络需构造复杂的模糊规则。RBFNN具有逼近不确定动态的特性,传统RBFNN具有固定的网络结构,难以有效处理复杂的未知时变动态,且过多的神经元额外增加了控制系统的计算负担。因此,将自组织准则与RBF结合值得进一步讨论。
基于上述讨论,针对欠驱动水面舰艇的轨迹跟踪控制问题,本文拟提出一种受模型不确定性和未知时变扰动影响下的自组织神经网络(self-structuring neural network,SSNN)预定义时间轨迹跟踪控制方法。与指数衰减型性能函数不同,本文将引入具有任意预定时间收敛的性能函数,为跟踪误差提供预定义的约束规范,BLF函数作为预定义约束的边界函数以满足规定的跟踪精度。在控制设计中结合动态面控制技术以避免对虚拟控制律求导产生的“计算爆炸问题”[28],从而提出自适应预定义时间滤波器。与此同时,还考虑USV的输入饱和问题,利用SSNN逼近船舶模型不确定性、外部环境时变扰动以及输入饱和带来的影响,并且可以在线调整神经元个数,优化神经网络结构,以此来降低系统的计算负担。针对欠驱动水面舰艇的轨迹跟踪控制,提出一种预定义时间稳定方法,来保证闭环控制系统在预定义时间内稳定并一致有界。通过构造Lyapunov函数进行预定义时间稳定性分析,结合期望轨迹进行仿真实验,以验证该控制方法的有效性和跟踪效果。
1. 预备知识和问题描述
1.1 预备知识
引理1:考虑非线性系统{\dot {\boldsymbol{x}}_0} = f\left( {t,{{\boldsymbol{x}}_0},{\boldsymbol{d}}} \right),其中{{\boldsymbol{x}}_0}为系统状态,d为不确定项;定义连续正定函数V\left( {{{\boldsymbol{x}}_0}} \right),设计参数0 < \gamma < 1,{T_{\text{c}}} > 0以及0 < \vartheta < \infty ,且满足以下条件[29]:
\dot V \leqslant - \frac{{\text{π}}}{{\gamma {T_{\text{c}}}}}\left( {{V^{1 - \gamma /2}} + {V^{1 + \gamma /2}}} \right) + \vartheta (1) 由上可知,{\dot {\boldsymbol{x}}_0} = f\left( {t,{{\boldsymbol{x}}_0},{\boldsymbol{d}}} \right)的轨迹是实际的预定义时间稳定,收敛区域由式(2)给出。
\left\{ {\mathop {\lim }\limits_{t \to {T'_{\text{c}}}} {{\boldsymbol{x}}_0}|V \leqslant \min \left\{ {{{\left( {\frac{{2\gamma {T_{\text{c}}}\vartheta }}{{\text{π}}}} \right)}^{\tfrac{2}{{2 - \gamma }}}},{{\left( {\frac{{2\gamma {T_{\text{c}}}\vartheta }}{{\text{π}}}} \right)}^{\tfrac{2}{{2 + \gamma }}}}} \right\}} \right\} (2) 式中,{T'_{\text{c}}}为建立的时间,满足{T'_{\text{c}}} < {T_{\max }} = \sqrt 2 {T_{\text{c}}},而{T_{{\text{max}}}}为上限。
引理2:对于任意定义的k > 0,h \geqslant 0以及y > 0,满足以下不等式[30]:
{h^k}\left( {y - h} \right) \leqslant \frac{1}{{1 + k}}\left( {{y^{1 + k}} - {h^{1 + k}}} \right) (3) 对于k > 1,h > 0以及y \leqslant h,满足
{\left( {h - y} \right)^k} \leqslant {y^k} - {h^k} (4) 引理3:任意\Im \in {\bold{{R}}},存在以下不等式[31]:
\left\{ \begin{aligned} & {\sum\limits_{i = 1}^m {\left| {{\Im _i}} \right|} ^{1 + p}} \geqslant {\left( {{{\sum\limits_{i = 1}^m {\left| \Im \right|} }^2}} \right)^{\left( {1 + p} \right)/2}},& 0 < p \leqslant 1 \\& {\sum\limits_{i = 1}^m {\left| {{\Im _i}} \right|} ^p} \geqslant {m^{1 - p}}{\left( {\sum\limits_{i = 1}^m {\left| \Im \right|} } \right)^p},& p > 1 \end{aligned} \right. (5) 引理4:对于任意\ell \in {\bold{{R}}^ + }以及X \in {\bold{{R}}},以下不等式成立[32]:
0 < \left| X \right| - X\tanh \left( {\frac{X}{\ell }} \right) \leqslant \kappa \ell (6) 式中, \kappa=0.278\ 5 ,且满足\kappa = {{\text{e}}^{ - \left( {\kappa + 1} \right)}}。
1.2 自组织神经网络(SSNN)
研究表明[33-34],在RBFNN网络中,神经元个数越多,对未知非线性函数的逼近效果越好。值得注意的是,不是所有神经元都是有效神经元,无效神经元不仅不能提高逼近性能,而且会给控制系统增加较大的计算成本。SSNN与RBFNN相比,有所不同,因为SSNN能够在线调整神经元个数,通过判断神经元的有效性,自主增加或删除神经元,从而能够有效地调整系统的计算量,并且可以获得良好的逼近性能。
SSNN逼近函数如下:
f\left( {\boldsymbol{x}} \right) = {{\boldsymbol{W}}^{\rm{T}}}{\boldsymbol{S}}\left( {\boldsymbol{x}} \right) + \varepsilon \left( {\boldsymbol{x}} \right) (7) {\boldsymbol{W}} = \arg {\text{ }}\min \left\{ {\mathop {\sup }\limits_{{\boldsymbol{x}} \in \bf{R}} \left| {f\left( {\boldsymbol{x}} \right) - {{\hat {\boldsymbol{W}}}^{\rm{T}}}{\boldsymbol{S}}\left( {\boldsymbol{x}} \right)} \right|} \right\} (8) 上式中:x表示SSNN的输入; \varepsilon \left( {\boldsymbol{x}} \right) 为逼近误差; {\boldsymbol{W}} \in {\bold{R}} ,为理想权重;SSNN神经元数N > 1, \hat {\boldsymbol{W}} 为W的估计, \boldsymbol{S}\left(\boldsymbol{x}\right)=[S_1\left(\boldsymbol{x}\right),S_2\left(\boldsymbol{x}\right),...,S_N\left(\boldsymbol{x}\right)]^{\rm{T}} ,为基函数向量, {S_i}\left( {\boldsymbol{x}} \right) 为高斯函数,表示为
{S_i}\left( {\boldsymbol{x}} \right) = \exp \left( { - \frac{{{{\left\| {{\boldsymbol{x}} - {{\boldsymbol{c}}_i}} \right\|}^2}}}{{b_i^2}}} \right),{\text{ }}i = 1,2, \ldots ,N (9) 式中: {b_i} 为高斯基函数的宽度; \boldsymbol{c}_i 为高斯基函数的中心向量。
假设1:SSNN网络的理想权值是有界的, \left\| {\boldsymbol{W}} \right\| \leqslant {{\boldsymbol{W}}^ * } ,其中 {{\boldsymbol{W}}^ * } 为一个正常数; \varepsilon \left( {\boldsymbol{x}} \right) 为逼近误差,满足 \varepsilon \left( {\boldsymbol{x}} \right) \leqslant {\varepsilon ^ * } ,其中 {\varepsilon ^ * } 为一个正常数。
定义具有最佳激活效果的神经元 {S_{\rm{M}}} = \mathop {\max }\limits_{1 \leqslant k \leqslant N} {S_k} ,SSNN的分裂阈值{\varUpsilon _{\text{s}}} \in \left( {0,1} \right),衰减阈值{\varUpsilon _{\rm{d}}} \in \left( {0,1} \right),并且{\varUpsilon _{\text{s}}} > {\varUpsilon _{\text{d}}}。神经元的分裂策略是通过判断具有最佳激活效果的神经元是否超过了预设阈值,若小于分裂阈值(即 {S_{\text{M}}} < {\varUpsilon _{\text{s}}} ),意味着神经元的激活效果没有达到理想值,那么就执行神经元分裂策略,以获得更好的逼近效果。新分裂的神经元由{S_{{\mathrm{new}}}}表示,新神经元的参数为
\left\{ \begin{gathered} {{\boldsymbol{c}}_{{\text{new}}}} = \frac{{{{\boldsymbol{x}}_{\rm{M}}} + {{\boldsymbol{c}}_{\rm{M}}}}}{2} \\ {b_{{\text{new}}}} = {b_{\rm{M}}} \\ {W_{{\text{new}}}} = 0 \\ \end{gathered} \right. (10) 式中: {{\boldsymbol{c}}_{{\text{new}}}} 为新神经元高斯基函数的中心向量; {b_{{\text{new}}}} 为新神经元高斯基函数的宽度; {{\boldsymbol{x}}_{\text{M}}} , {{\boldsymbol{c}}_{\text{M}}} 以及 {b_{\text{M}}} 为激活效果最好的参数; {W_{{\text{new}}}} 为新神经元权重的初始值。
神经元衰减参数定义为{I_{\rm{n}}},初始值为1。衰减阈值{\varUpsilon _{\text{d}}}的作用是确定是否执行神经元衰减策略,遵循以下规则:
{I_{\rm{n}}} = \left\{ \begin{gathered} \varLambda {I_{\rm{n}}},{\text{ }}{S_i} \leqslant {p_{\text{d}}} \\ 1,\quad{\text{ }}{S_i} > {p_{\text{d}}} \\ \end{gathered} \right.,{\text{ }}i = 1,2, \ldots ,N (11) 式中: \varLambda 为衰减系数; {p_{\text{d}}} 为自定义值;如果 {I_{\rm{n}}} \leqslant {\varUpsilon _{\text{d}}} ,触发神经元衰减策略,删除第n个神经元。
SSNN算法流程图如图1所示,函数\hat {\boldsymbol{F}}为逼近的未知函数 f\left( {\boldsymbol{x}} \right) = {{\boldsymbol{W}}^{\rm{T}}}{\boldsymbol{S}}\left( {\boldsymbol{x}} \right) + \varepsilon \left( {\boldsymbol{x}} \right) 的近似。当未知非线性函数更复杂时,需要添加更有效的神经元,并删除无效的神经元。通过增大参数{\varUpsilon _{\text{s}}}、减小参数 {\varUpsilon _{\text{d}}} ,来增加有效神经元数量,以实现更好的逼近效果,而且不会给控制系统带来过多的计算成本。此外,可以减少{\varUpsilon _{\text{s}}}并增加 {\varUpsilon _{\text{d}}} ,在不影响逼近效果的情况下删除更多无效神经元,减少控制系统的计算成本。
1.3 USV数学模型
针对USV的数学模型,先通过坐标变换解决质量惯性矩阵中非对角项引起的强耦合问题。欠驱动水面舰艇的运动学和动力学模型如下[35]:
\left\{\begin{gathered}\dot{\boldsymbol{\eta}}=\boldsymbol{J}\left(\psi\right)\boldsymbol{v} \\ \boldsymbol{M}\dot{\boldsymbol{v}}=-\boldsymbol{C}\left(\boldsymbol{v}\right)\boldsymbol{v}-\boldsymbol{D}\left(\boldsymbol{v}\right)\boldsymbol{v}+\boldsymbol{\tau}_{\text{d}}+\boldsymbol{\tau}_{\text{c}} \\ \end{gathered}\right. (12) {\boldsymbol{J}}(\psi ) = \left[ {\begin{array}{*{20}{c}} {\cos (\psi )}&{ - \sin (\psi )}&0 \\ {\sin (\psi )}&{\cos (\psi )}&0 \\ 0&0&1 \end{array}} \right] 其中,
{\boldsymbol{M}} = \left[ {\begin{array}{*{20}{c}} {{m_{11}}}&0&0 \\ 0&{{m_{22}}}&{{m_{23}}} \\ 0&{{m_{32}}}&{{m_{33}}} \end{array}} \right] {\boldsymbol{C}}({\boldsymbol{v}}) = \left[ {\begin{array}{*{20}{c}} 0&0&{{C_{13}}} \\ 0&0&{{C_{23}}} \\ {{C_{31}}}&{{C_{32}}}&0 \end{array}} \right] {\boldsymbol{D}}({\boldsymbol{v}}) = \left[ {\begin{array}{*{20}{c}} {{d_{11}}}&0&0 \\ 0&{{d_{22}}}&{{d_{23}}} \\ 0&{{d_{32}}}&{{d_{33}}} \end{array}} \right] 式中: {\boldsymbol{\eta }}= {\left[ {x,y,\psi } \right]^{\text{T}}} ,为USV的位置和偏航角, \left( {x,y} \right) 为位置坐标, \psi 为偏航角; {\boldsymbol{J}}(\psi ) 为大地坐标系与船体坐标系之间的旋转矩阵; {\boldsymbol{v}} = {[u,\upsilon ,r]^{\text{T}}} ,为速度矢量, u,\upsilon ,r 分别为前进速度,横漂速度以及偏航方向的角速度;M为质量惯性矩阵; {\boldsymbol{C}}({\boldsymbol{v}}) 为科氏力向心矩阵; {\boldsymbol{D}}({\boldsymbol{v}}) 为水动力阻尼系数矩阵;{m_{11}} = {m_0} - {X_{\dot u}}, {m_{22}} = {m_0} - {Y_{\dot \upsilon }} , {m_{23}} = {m_0}{{\boldsymbol{x}}_g} - {Y_{\dot r}},{m_{33}} = {I_z} - {N_{\dot r}},{d_{11}} = - {X_u} - {X_{\left| u \right|u}}\left| u \right| - {X_{uuu}}{u^2}, {d_{22}} = - {Y_\upsilon } - {Y_{\left| \upsilon \right|\upsilon }}\left| \upsilon \right| - {Y_{\left| r \right|\upsilon }}\left| r \right|, {d_{23}} = - {Y_r} - {Y_{\left| \upsilon \right|r}}\left| \upsilon \right| - {Y_{\left| r \right|r}}\left| r \right|, {d_{32}} = - {N_\upsilon } - {N_{\left| \upsilon \right|\upsilon }}\left| \upsilon \right| - {N_{\left| r \right|}}_\upsilon \left| r \right| , {d_{33}} = - {N_r} - {N_{\left| \upsilon \right|r}}\left| \upsilon \right| - {N_{\left| r \right|r}}\left| r \right|,其中{m_0}为USV的质量,{X_{\dot u}}, {Y_{\dot \upsilon }}, {Y_{\dot r}}和{N_{\dot r}}为附加质量,{{\boldsymbol{x}}_g}为USV重心和船体坐标系原点的偏差值,{I_z}为偏航方向上的转动惯量,X\left( \cdot \right),Y\left( \cdot \right)和N\left( \cdot \right)为前进、横漂和偏航的线性和二次水动力阻尼系数; {{\boldsymbol{\tau}} _{\rm{d}}} = {\left[ {{\tau _{{\text{d}}u}},{\tau _{{\text{d}}\upsilon }},{\tau _{{\text{d}}r}}} \right]^{\text{T}}} ,为外部时变扰动矢量; {{\boldsymbol{\tau}} _{\text{c}}} = {[{\tau _{{\text{c}}u}},0,{\tau _{{\text{c}}r}}]^{\text{T}}} ,为输入饱和下的控制输入,执行器输出饱和定义如下:
{\tau _{{\text{c}}\iota }} = \left\{ \begin{gathered} \tau _\iota ^ + ,{\text{ }}{\tau _\iota } > \tau _\iota ^ + \\ {\tau _\iota },{\text{ }}\tau _\iota ^ - \leqslant {\tau _\iota } \leqslant \tau _\iota ^ + \\ \tau _\iota ^ - ,{\text{ }}{\tau _\iota } < \tau _\iota ^ + \\ \end{gathered} \right. (13) 其中: {\tau _{\iota} } (\iota = u,{\text{ }}r),为无输入饱和的控制指令; \tau _\iota ^ + 和 {\tau _\iota ^ -} 分别为输入饱和的上界和下界。
由于饱和模型式(13)不能直接用于反步法设计,为此定义一个如式(14)所示平滑形式的模型来描述非对称饱和非线性模型[36]。
{\tau _{{\text{c}}\iota }} = {\tau _{{\text{Mc}}\iota }} \times {\text{erf}}\left( {\frac{{\sqrt {\text{π}} }}{{2{\tau _{{\text{Mc}}\iota }}}}{\tau _\iota }} \right) (14) 其中:
{\tau _{{\text{Mc}}\iota }} = \left( {\tau _\iota ^ + + \tau _\iota ^ - } \right)/2 + \left( {\left( {\tau _\iota ^ + - \tau _\iota ^ - } \right)/2} \right){\text{sign}}\left( {{\tau _\iota }} \right) ,\;\iota = u, r {\text{erf}}( {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{{\boldsymbol{x}}} } ) = \dfrac{2}{{\sqrt {\text{π}} }}\displaystyle\int_0^{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{x} } {{{\text{e}}^{ - {t^2}}}} {\mathrm{d}}t 式中: \text{sign}(·) 为标准的符号函数; \text{erf}(·) 为高斯误差函数。通过调整 \tau _\iota ^ + 和 \tau _\iota ^ - 可以得到对称和非对称的饱和模型。当 \left| {\tau _\iota ^ + } \right| = \left| {\tau _\iota ^ - } \right| 时,可以得到对称的饱和模型;当 \left| {\tau _\iota ^ + } \right| \ne \left| {\tau _\iota ^ - } \right| 时,则为非对称的饱和模型。图2 表示了饱和模型式(13)是平滑形式的饱和限制,其中, \tau _\iota ^ + = 8 , \tau _\iota ^ - = - 4 ,输入信号为 {\tau _{\text{Mc}}} = 25\sin (1.5t){\text{ N}} 。
进一步地,考虑输入饱和误差{\boldsymbol{\varDelta}} = {\boldsymbol{\tau}} - {{\boldsymbol{\tau}} _{\text{c}}},\tau 为无输入饱和下的控制输入,则动力学模型可改写为
{\boldsymbol{M}}\dot {\boldsymbol{v}} = - {\boldsymbol{C}}\left( {\boldsymbol{v}} \right){\boldsymbol{v}} - {\boldsymbol{D}}\left( {\boldsymbol{v}} \right){\boldsymbol{v}} + {{\boldsymbol{\tau}} _{\text{d}}} + {\boldsymbol{\tau}} - {\boldsymbol{\varDelta}} (15) 其中, {\boldsymbol{\tau}} = {\left[ {{{\tau} _u},0,{{\tau} _r}} \right]^{\rm{T}}} , {\boldsymbol{\varDelta}} = {\left[ {{\varDelta _u},0,{\varDelta _r}} \right]^{\rm{T}}} 。
在质量惯性矩阵M的影响下,横漂和偏航速度会共同影响控制输入,从而加大了控制器的设计难度。为此,引入坐标变换解决USV耦合问题,其坐标变换描述如下[34]:
\left\{ \begin{gathered} \bar x = x + \chi \cos (\psi ) \\ \bar y = y + \chi \sin (\psi ) \\ \bar \upsilon = \upsilon + \chi r \\ \end{gathered} \right. (16) 式中, \chi = {m_{23}}/{m_{22}} 。
引入坐标变换式(16)后,USV的数学模型可以转化为
\left\{ \begin{gathered} \dot {\bar {\boldsymbol{\eta}}} = {\boldsymbol{J}}\left( \psi \right)\bar {\boldsymbol{v}} \\ \dot {\bar {\boldsymbol{v}}} = {\boldsymbol{f}} + {\boldsymbol{d}} + {{\boldsymbol{\tau}} ^\prime } - {\boldsymbol{\varDelta}} ' \\ \end{gathered} \right. (17) 其中,
\bar{\boldsymbol{ \eta}} = {[\bar x{\text{,}}\bar y{\text{,}}\psi ]^{\rm{T}}} ,\;\bar {\boldsymbol{v}} = {[u,\bar \upsilon ,r]^{\rm{T}}} ,\;{\boldsymbol{ f }}= {[{f_1},{f_2},{f_3}]^{\rm{T}}} {{\boldsymbol{\tau}} ^\prime } = {[{\zeta _u}{{{\tau}} _u},0,{\zeta _r}{{{\tau}} _r}]^{\text{T}}} ,\; {{\boldsymbol{\varDelta}} ^\prime } = {[{\zeta _u}{\varDelta _u},0,{\zeta _r}{\varDelta _r}]^{\rm{T}}} ,\; {\boldsymbol{d}} ={[{d_u},{d_v},{d_r}]^{\rm{T}}} {d_u} = {\xi _u}{{{\tau}} _{{\text{d}}u}} ,\; {d_\upsilon } = {\xi _\upsilon }{{{\tau}} _{{\text{d}}\upsilon }} ,\; {d_r} = {\xi _r}\left( {{{{\tau}} _{{\text{d}}r}} - \chi {{{\tau}} _{{\text{d}}\upsilon }}} \right) \left\{ \begin{aligned} & {f_1} = ({m_{22}}(\bar \upsilon - \chi r)r + {m_{23}}{r^2} - {d_{11}}u)/{m_{11}} \\& {f_2} = ( - {m_{11}}ur - {d_{22}}(\bar \upsilon - \chi r) - {d_{23}}r)/{m_{22}} \\& {f_3} = (({m_{11}}{m_{22}} - m_{22}^2)u(\bar \upsilon - \chi r) + \left( {{m_{11}}{m_{32}}} - \right. \\& \qquad \left. { {m_{23}}{m_{22}}} \right)ur - ({d_{33}}r + {d_{32}}(\bar \upsilon - \chi r)){m_{22}} + \\& \qquad ({d_{23}}r + {d_{22}}(\bar \upsilon - \chi r)){m_{23}})/\nabla \end{aligned}\right. (18) 式中, {\boldsymbol{f}} = {[{f_1},{f_2},{f_3}]^{\rm{T}}} ,为USV模型的不确定性项, \nabla = {m_{22}}{m_{33}} - {m_{23}}{m_{32}} 。
假设2:外部扰动 {\boldsymbol{d }}= {[{d_u},{d_v},{d_r}]^{\rm{T}}} 有界, {d_i} \leqslant {d_{i{\text{max}}}} ,i = u,v,r。
假设3:期望位置{{\boldsymbol{\eta}} _{\text{d}}} = {[{x_{\text{d}}},{\text{ }}{y_{\text{d}}},{\text{ }}{\psi _{\text{d}}}]^{\text{T}}}以及{\dot {\boldsymbol{\eta}} _{\text{d}}}是有界的。
图3中, \left( {x,y} \right) 为USV实际位置, \left( {{x_{\rm{d}}},{y_{\rm{d}}}} \right) 为期望位置, \psi 为USV实际航向角, {\psi _{\rm{d}}} 为期望航向角, {\psi _{\rm{e}}} 为航向角误差。
定义经过坐标转换后的跟踪误差:
\left\{ \begin{gathered} {{\bar x}_{\rm{e}}} = {x_{\rm{d}}} - \bar x,{\text{ }}{{\bar y}_{\text{e}}} = {y_{\rm{d}}} - \bar y \\ {\psi _{\rm{e}}} = {\psi _{\rm{d}}} - \psi ,{\text{ }}\bar E = \sqrt {\bar x_{\rm{e}}^2 + \bar y_{\rm{e}}^2} \\ \end{gathered} \right. (19) 式中: \left( {{{\bar x}_{\rm{e}}},{\text{ }}{{\bar y}_{\text{e}}}} \right) 为坐标转换后的位置跟踪误差;\bar E为视线(LOS)制导范围。期望航向角 {\psi _{\rm{d}}} 定义如下[18]:
\left\{ \begin{gathered} {\psi _{\rm{e}}} = \arctan 2({{\tilde y}_{\rm{e}}},{{\tilde x}_{\rm{e}}}) \\ {\psi _{\rm{d}}} = \psi + {\psi _{\rm{e}}} \\ \end{gathered} \right. (20) 其中,
\left[ \begin{gathered} {{\tilde x}_{\rm{e}}} \\ {{\tilde y}_{\rm{e}}} \\ \end{gathered} \right] = \left[ {\begin{array}{*{20}{c}} {\cos \psi }&{\sin \psi } \\ { - \sin \psi }&{\cos \psi } \end{array}} \right]\left[ \begin{gathered} {{\bar x}_{\rm{e}}} \\ {{\bar y}_{\rm{e}}} \\ \end{gathered} \right] (21) \mathrm{atan}2({\tilde{y}}_{\rm{e}},{\tilde{x}}_{\rm{e}})=\left\{ \begin{aligned} & \mathrm{arctan}\left(\frac{{\tilde{y}}_{\rm{e}}}{{\tilde{x}}_{\rm{e}}}\right), & {\tilde{x}}_{\rm{e}} > 0\\[-2pt]& \mathrm{arctan}\left(\frac{{\tilde{y}}_{\rm{e}}}{{\tilde{x}}_{\rm{e}}}\right)+{\text{π}} , & {\tilde{x}}_{\rm{e}} < 0,\; {\tilde{y}}_{\rm{e}}\ge 0\\[-2pt]& \mathrm{arctan}\left(\frac{{\tilde{y}}_{\rm{e}}}{{\tilde{x}}_{\rm{e}}}\right)-{\text{π}} , & {\tilde{x}}_{\rm{e}} < 0,\; {\tilde{y}}_{\rm{e}} < 0\\[-2pt]& +\frac{\text{π}}{2}, & {\tilde{x}}_{\rm{e}}=0,\; {\tilde{y}}_{\rm{e}} > 0\\[-2pt]& -\frac{\text{π}}{2}, & {\tilde{x}}_{\rm{e}}=0,\; {\tilde{y}}_{\rm{e}} < 0\\[-2pt]& 不定义, & {\tilde{x}}_{\rm{e}}=0,\; {\tilde{y}}_{\rm{e}}=0 \end{aligned}\right. (22) 2. 控制器设计
2.1 预定义时间状态约束控制器
大多数性能约束问题通过指数衰减型性能函数来解决,以保证跟踪误差限制在预设范围内。然而,指数衰减型性能函数只能保证跟踪误差在无限时间内收敛至预设范围内,不能保证在预设的时间达到设定的跟踪性能。为了获得更好的跟踪性能,本文引入了预定义时间性能函数,保证跟踪误差在预定义的时间内达到期望的跟踪精度。预定义时间性能函数定义如下[37]:
{\sigma _i} = \left\{ \begin{gathered} {\sigma _{i0}}{{\text{e}}^{1 - \tfrac{{{T_{\text{h}}}}}{{{T_{\text{h}}} - t}}}} + {\sigma _{i\infty }},\;\;t \geqslant {T_{\text{h}}} \\ {\sigma _{i\infty }}, \quad\qquad\qquad t \geqslant {T_{\text{h}}} \\ \end{gathered} \right. (23) 式中: {\sigma _{i0}} > {\sigma _{i\infty }} > 0 \left( {i = u,r} \right) ; {T_{\text{h}}} 为自由设定的时间值。因此,跟踪误差满足以下条件:
\left\{ \begin{gathered} \bar E < {\sigma _u} \\ - {\sigma _r} < {\psi _{\text{e}}} < {\sigma _r} \\ \end{gathered} \right. (24) 为了得到更好的跟踪性能,为此构造以下时变BLF函数[38]:
{V_u} = \frac{{\sigma _u^2\bar E}}{{\sigma _u^2 - {{\bar E}^2}}},{\text{ }}{V_r} = \frac{{\sigma _r^2{\psi _{\rm{e}}}}}{{\sigma _r^2 - \psi _{\rm{e}}^2}} (25) 考虑以下Lyapunov函数:
{V_1} = \frac{1}{2}V_u^2 + \frac{1}{2}V_r^2 (26) 则 {V_1} 的一次导数为
\begin{split} & \qquad\quad{{\dot V}_1} = {V_u}{{\dot V}_u} + {V_r}{{\dot V}_r} = \\[-2pt] & \frac{{\sigma _u^2\bar E}}{{\sigma _u^2 - {{\bar E}^2}}}\frac{{( {\sigma _u^4 + \sigma _u^2{{\bar E}^2}} )\dot {\bar E} - 2{\sigma _u}{{\dot \sigma }_u}{{\bar E}^3}}}{{{{( {\sigma _u^2 - {{\bar E}^2}} )}^2}}} + \\[-2pt] & \;\; \frac{{\sigma _r^2{\psi _{\rm{e}}}}}{{\sigma _r^2 - \psi _{\rm{e}}^2}}\frac{{( {\sigma _r^4 + \sigma _r^2\psi _{\rm{e}}^2} ){{\dot \psi }_{\rm{e}}} - 2{\sigma _r}{{\dot \sigma }_r}\psi _{\rm{e}}^3}}{{{{( {\bar \sigma _r^2 - \psi _{\rm{e}}^2} )}^2}}} = \\[-2pt] & \;\; \frac{{\sigma _u^2\bar E}}{{\sigma _u^2 - {{\bar E}^2}}}( {( {\sigma _u^4 + \sigma _u^2{{\bar E}^2}} )( {{{\dot x}_{\rm{d}}}\cos ( {{\psi _{\rm{d}}}} )} } + \\[-2pt] & \;\;\; {{{\dot y}_{\text{d}}}\sin ( {{\psi _{\rm{d}}}} ) - u\cos ( {{\psi _{\rm{e}}}} ) - \bar \upsilon \sin ( {{\psi _{\rm{e}}}} )} ) - \\[-2pt] & \;\;\qquad\quad {2{\sigma _u}{{\dot \sigma }_u}{{\bar E}^3}} )/{( {\sigma _u^2 - {{\bar E}^2}} )^2} + \\[-2pt] & \frac{{\sigma _r^2{\psi _{\rm{e}}}}}{{\sigma _r^2 - \psi _{\rm{e}}^2}}\frac{{( {\sigma _r^4 + \sigma _r^2\psi _{\rm{e}}^2} )( {{{\dot \psi }_{\rm{d}}} - r} ) - 2{\sigma _r}{{\dot \sigma }_r}\psi _{\rm{e}}^3}}{{{{( {\sigma _r^2 - \psi _{\rm{e}}^2} )}^2}}} \end{split} (27) 设计虚拟控制律为
\left\{\begin{aligned} & {a_u} = \Bigg( {{{\dot x}_{\rm{d}}}\cos ( {{\psi _{\rm{d}}}} ) + {{\dot y}_{\rm{d}}}\sin ( {{\psi _{\rm{d}}}} ) - \bar \upsilon \sin ( {{\psi _{\rm{e}}}} )} - \\[-2pt]& \qquad \frac{{2{\sigma _u}{{\dot \sigma }_u}{{\bar E}^3}}}{{( {\sigma _u^4 + \sigma _u^2{{\bar E}^2}} )}} + \frac{{{k_1}{{( {\sigma _u^2 - {{\bar E}^2}} )}^3}\bar E}}{{( {\sigma _u^6 + \sigma _u^4{{\bar E}^2}} )}} + \\[-2pt]& \qquad {\frac{{{{( {\sigma _u^2 - {{\bar E}^2}} )}^2}{\text{π}}}}{{( {\sigma _u^4 + \sigma _u^2{{\bar E}^2}} )\gamma {T_u}}}( {V_u^{1 - \gamma } + V_u^{1 + \gamma }} )} \Bigg)/\cos ( {{\psi _{\rm{e}}}} ) \\[-2pt]& {a_r} = {{\dot \psi }_{\rm{d}}} - \frac{{2{\sigma _r}{{\dot \sigma }_r}\psi _{\rm{e}}^3}}{{( {\sigma _r^4 + \sigma _r^2\psi _{\rm{e}}^2} )}} + \frac{{{k_2}( {\sigma _r^2 - \psi _{\text{e}}^2} )}}{{\sigma _r^2}}{\psi _{\rm{e}}} + \\[-2pt]& \qquad \frac{{{{( {\sigma _r^2 - \psi _{\rm{e}}^2} )}^2}{\text{π}}}}{{( {\sigma _r^4 + \sigma _r^2\psi _{\rm{e}}^2} )\gamma {T_r}}}( {V_r^{1 - \gamma } + V_r^{1 + \gamma }} ) \end{aligned}\right. (28) 式中: {k_1} , {k_2} , \alpha ,\beta 为正常数,其中 0 < \alpha \leqslant 1 , 0 < \beta \leqslant 1 ,设计参数0 < \gamma < 1,{T_\iota } > 0,\iota = u,r。
为避免虚拟控制信号求导出现的“计算爆炸”问题,通常采用传统一阶滤波器,即 {t_\iota }{\dot a_{f\iota }} = {a_{{\text{c}}\iota }} - {a_{{\text{f}}\iota }} 。(其中, {t_\iota } 为滤波器增益, {a_{{\text{c}}\iota }} 为滤波器输入信号, {a_{{\text{f}}\iota }} 为滤波后的输入信号,\iota = u,r)。而本文中提出的是一种自适应预定义时间滤波器,可避免出现“微分爆炸”问题,并采用自适应律来估计虚拟控制输入导数的未知上界。
假设4:自适应预定义时间滤波器输入信号的导数( {\dot a_{{\text{f}}\iota }} , \iota = u,r)是连续的,且 \left| {{{\dot a}_{{\text{f}}\iota }}} \right| \leqslant {\bar \gamma _\iota } , {\bar \gamma _\iota } 为未知的正常数。
自适应预定义时间滤波器为
{T_{{\text{f}}\iota }}{\dot a_{{\text{f}}\iota }} = \frac{{\text{π}}}{\gamma }\left( {{\xi _\iota }^{1 - \gamma } + {\xi _\iota }^{1 + \gamma }} \right) + {T_{{\text{f}}\iota }}{\hat {\bar \gamma} _\iota }\tanh \left( {\frac{{{\xi _\iota }}}{{{\mathchar'26\mkern-10mu\lambda _{i\iota }}}}} \right) + {a_{m\iota }} (29) {\dot {\hat {\bar \gamma}} _\iota } = - \frac{{\text{π}}}{{\gamma {T_{\gamma \iota }}}}\left( {\frac{{2 - \gamma }}{2}{{\hat {\bar \gamma} }_\iota }^{1 - \gamma } + \frac{{2 + \gamma }}{2}{{\hat {\bar \gamma} }_\iota }^{1 + \gamma }} \right) + {\xi _\iota }\tanh \left( {\frac{{{\xi _\iota }}}{{{\varepsilon _{0\iota }}}}} \right) (30) 其中,
\iota = u,r, {a_{mu}} = \dfrac{{\left( {\sigma _u^4 + \sigma _u^2{{\bar E}^2}} \right){V_u}}}{{{{\left( {\sigma _u^2 - {{\bar E}^2}} \right)}^2}}} , {a_{mr}} = \dfrac{{\left( {\sigma _r^4 + \sigma _r^2\psi _{\text{e}}^2} \right){V_r}}}{{{{\left( {\sigma _r^2 - \psi _{\text{e}}^2} \right)}^2}}} 式中: {T_{{\text{f}}\iota }} 和 {T_{\gamma \iota }} 为预设时间参数。
定义速度误差以及滤波误差如下:
\begin{split} & {u_{\text{e}}} = {a_{{\text{f}}u}} - u,{\text{ }}{r_{\text{e}}} = {a_{{\text{f}}r}} - r \\& {\xi _u} = {a_u} - {a_{{\text{f}}u}},{\text{ }}{\xi _r} = {a_r} - {a_{{\text{f}}r}} \end{split} (31) 构造如下Lyapunov函数:
{V_{\text{f}}} = \frac{1}{2}\xi _u^2 + \frac{1}{2}\xi _r^2 + \frac{1}{2}{\tilde {\bar \gamma}} _u^2 + \frac{1}{2}{\tilde {\bar \gamma}} _r^2 (32) 式(32)关于时间的导数为
\begin{gathered} {{\dot V}_{\text{f}}} = {\xi _u}{{\dot \xi }_u} + {\xi _r}{{\dot \xi }_r} + {{{\tilde {\bar \gamma}} }_u}{{\dot {\tilde {\bar \gamma}} }_u} + {{{\tilde {\bar \gamma}} }_r}{{\dot {\tilde {\bar \gamma}} }_r} = \\ {\xi _u}\Bigg({{\bar \gamma }_u} - T_{{\text{f}}u}^{ - 1}\Bigg(\frac{{\text{π}}}{\gamma }\left( {{\xi _u}^{1 - \gamma } + {\xi _u}^{1 + \gamma }} \right) + {T_{{\text{f}}u}}{{\hat {\bar \gamma} }_u}\tanh \left( {\frac{{{\xi _u}}}{{{\varepsilon _{0u}}}}} \right) + \\ {a_{mu}}\Bigg)\Bigg) + {\xi _r}\Bigg({{\bar \gamma }_r} - T_{{\text{f}}r}^{ - 1}\Bigg(\frac{{\text{π}}}{\gamma }\left( {{\xi _r}^{1 - \gamma } + {\xi _r}^{1 + \gamma }} \right) + \end{gathered} \begin{split} & \quad\;\;\;{T_{{\text{f}}r}}{{\hat {\bar \gamma} }_r}\tanh \left( {\frac{{{\xi _r}}}{{{\varepsilon _{0r}}}}} \right) + {a_{mr}}\Bigg)\Bigg) + {{{\tilde {\bar \gamma}} }_u}{{\dot {\tilde {\bar \gamma}} }_u} + {{{\tilde {\bar \gamma}} }_r}{{\dot {\tilde {\bar \gamma}} }_r} \leqslant \\&\quad \;\; {\xi _u}\left( { - T_{{\text{f}}u}^{ - 1}\left( {\frac{{\text{π}}}{\gamma }\left( {{\xi _u}^{1 - \gamma } + {\xi _u}^{1 + \gamma } + {a_{mu}}} \right)} \right)} \right) + \left| {{\xi _u}} \right|{{\bar \gamma }_u} - \\&\quad \qquad {\xi _u}{{\bar \gamma }_u}\tanh \left( {\frac{{{\xi _u}}}{{{\varepsilon _{0u}}}}} \right) + {{{\tilde {\bar \gamma}} }_u}{\xi _u}\tanh \left( {\frac{{{\xi _u}}}{{{\varepsilon _{0u}}}}} \right) + \\&\quad\;\;\; {\xi _r}\left( { - T_{{\text{f}}r}^{ - 1}\left( {\frac{{\text{π}}}{\gamma }\left( {{\xi _r}^{1 - \gamma } + {\xi _r}^{1 + \gamma }} \right) + {a_{mr}}} \right)} \right) + \left| {{\xi _r}} \right|{{\bar \gamma }_r} - \\&\quad {\xi _r}{{\bar \gamma }_r}\tanh \left( {\frac{{{\xi _r}}}{{{\varepsilon _{0r}}}}} \right) + {{{\tilde {\bar \gamma}} }_r}{\xi _r}\tanh \left( {\frac{{{\xi _r}}}{{{\varepsilon _{0r}}}}} \right) + {{{\tilde {\bar \gamma}} }_u}{{\dot {\tilde {\bar \gamma}} }_u} + {{{\tilde {\bar \gamma}} }_r}{{\dot {\tilde {\bar \gamma}} }_r} \leqslant \\&\quad\;\; - \frac{{\text{π}}}{{\gamma {T_{{\text{f}}u}}}}\left( {{\xi _u}^{2 - \gamma } + {\xi _u}^{2 + \gamma }} \right) - \frac{{\text{π}}}{{\gamma {T_{{\text{f}}r}}}}\left( {{\xi _r}^{2 - \gamma } + {\xi _r}^{2 + \gamma }} \right) + \\&\quad {{{\tilde {\bar \gamma}} }_u}({\xi _u}\tanh \left( {\frac{{{\xi _u}}}{{{\varepsilon _{0u}}}}} \right) - {{\dot {\hat {\bar \gamma}} }_u}) + {{{\tilde {\bar \gamma}} }_r}\Bigg({\xi _r}\tanh \left( {\frac{{{\xi _r}}}{{{\varepsilon _{0r}}}}} \right) - {{\dot {\hat {\bar \gamma}} }_r}\Bigg) - \\&\quad \qquad\qquad\qquad {\xi _u}{a_{mu}} - {\xi _r}{a_{mr}} \end{split} (33) 根据引理2,以下不等式成立成立。
\left\{\begin{aligned} & {{{\tilde {\bar \gamma}} }_\iota }\hat {\bar \gamma} _\iota ^{1 - \gamma } \leqslant \frac{1}{{2 - \gamma }}\left( {2\bar \gamma _\iota ^{2 - \gamma } - {\tilde {\bar \gamma}} _\iota ^{2 - \gamma }} \right) \\& {{{\tilde {\bar \gamma}} }_\iota }\hat {\bar \gamma} _\iota ^{1 + \gamma } \leqslant \frac{1}{{2 + \gamma }}\left( {2\bar \gamma _\iota ^{2 + \gamma } - {\tilde {\bar \gamma}} _\iota ^{2 + \gamma }} \right) \end{aligned} \right. (34) 将式(30)代入式(33),结合式(34),根据引理4,式(33)可以化为
\begin{split} & \quad\quad{{\dot V}_{\text{f}}} = {\xi _u}{{\dot \xi }_u} + {\xi _r}{{\dot \xi }_r} + {{{\tilde {\bar \gamma}} }_u}{{\dot {\tilde {\bar \gamma}} }_u} + {{{\tilde {\bar \gamma}} }_r}{{\dot {\tilde {\bar \gamma}} }_r} \leqslant \\& - \frac{{\text{π}}}{{\gamma {T_{{\text{f}}u}}}}\left( {{{\left( {\xi _u^2} \right)}^{1 - \tfrac{\gamma }{2}}} + {{\left( {\xi _u^2} \right)}^{1 + \tfrac{\gamma }{2}}}} \right) - \frac{{\text{π}}}{{\gamma {T_{{\text{f}}r}}}}\left( {{{\left( {\xi _r^2} \right)}^{1 - \tfrac{\gamma }{2}}}} +\right. \\& \quad \left. { {{\left( {\xi _r^2} \right)}^{1 + \tfrac{\gamma }{2}}}} \right) - \frac{{\text{π}}}{{\gamma {T_{\gamma u}}}}{\left( {{\tilde {\bar \gamma}} _u^2} \right)^{1 - \tfrac{\gamma }{2}}} - \frac{{\text{π}}}{{\gamma {T_{\gamma u}}}}{\left( {{\tilde {\bar \gamma}} _u^2} \right)^{1 + \tfrac{\gamma }{2}}} - \\& \frac{{\text{π}}}{{\gamma {T_{\gamma r}}}}{\left( {{\tilde {\bar \gamma}} _r^2} \right)^{1 - \tfrac{\gamma }{2}}} - \frac{{\text{π}}}{{\gamma {T_{\gamma r}}}}{\left( {{\tilde {\bar \gamma}} _r^2} \right)^{1 + \tfrac{\gamma }{2}}} + \frac{{\text{π}}}{{\gamma {T_{\gamma u}}}}{(\bar \gamma _u^2)^{1 - \tfrac{\gamma }{2}}} + \\& \frac{{\text{π}}}{{\gamma {T_{\gamma u}}}}{(\bar \gamma _u^2)^{1 + \tfrac{\gamma }{2}}} + \frac{{\text{π}}}{{\gamma {T_{\gamma r}}}}{(\bar \gamma _r^2)^{1 - \tfrac{\gamma }{2}}} + \frac{{\text{π}}}{{\gamma {T_{\gamma r}}}}{(\bar \gamma _r^2)^{1 + \tfrac{\gamma }{2}}} - \\&\qquad \quad {\xi _r}{a_{mr}} - {\xi _r}{a_{mu}} + k_1'{\mathchar'26\mkern-10mu\lambda _1}{{\bar \gamma }_u} + k_2'{\mathchar'26\mkern-10mu\lambda _2}{{\bar \gamma }_r} \leqslant\\& \qquad\qquad - \frac{{\text{π}}}{{\gamma {T_{\text{f}}}}}\left( {V_{\text{f}}^{1 - \tfrac{\gamma }{2}} + V_{\text{f}}^{1 + \tfrac{\gamma }{2}}} \right) + {c_1} \end{split} (35) 式中: {T_{\text{f}}} = {T_{{\text{f}}\iota }} = {T_{\gamma \iota }} > 0 ,\iota = u,r,k_1' = k_2' = 0.278{\text{ }}5。
\begin{split} & {c_1} = \frac{{\text{π}}}{{\gamma {T_{\gamma u}}}}{(\bar \gamma _u^2)^{1 - \tfrac{\gamma }{2}}} + \frac{{\text{π}}}{{\gamma {T_{\gamma u}}}}{(\bar \gamma _u^2)^{1 + \tfrac{\gamma }{2}}} + \frac{{\text{π}}}{{\gamma {T_{\gamma r}}}}{(\bar \gamma _r^2)^{1 - \tfrac{\gamma }{2}}} + \\ &\quad \frac{{\text{π}}}{{\gamma {T_{\gamma r}}}}{(\bar \gamma _r^2)^{1 + \tfrac{\gamma }{2}}} - {\xi _u}{a_{mu}} - {\xi _r}{a_{mr}} + k_1'{\mathchar'26\mkern-10mu\lambda _1}{{\bar \gamma }_u} + k_2'{\mathchar'26\mkern-10mu\lambda _2}{{\bar \gamma }_r} \end{split} (36) 根据引理1,闭环控制系统的所有滤波信号将会在预定义时间( {T_{{\text{fc}}}} = \sqrt 2 {T_{\text{f}}} )内收敛到一个邻域内:
\left\{ {\mathop {\lim }\limits_{t \to {{T'}_{\text{c}}}} x|V \leqslant \min \left\{ {{{\left( {\frac{{2\gamma {T_{\text{f}}}{c_1}}}{{\text{π}}}} \right)}^{\tfrac{2}{{2 - \gamma }}}},{{\left( {\frac{{2\gamma {T_{\text{f}}}{c_1}}}{{\text{π}}}} \right)}^{\tfrac{2}{{2 + \gamma }}}}} \right\}} \right\} (37) 2.2 预定义时间自组织神经网络
为解决未知连续的非线性部分和外部环境干扰对控制器性能的影响,引入了自组织神经网络逼近策略:
\left\{ \begin{gathered} {F_1} = {\boldsymbol{W}}_1^{\rm{T}}{{\boldsymbol{S}}_1}({{\boldsymbol{Z}}_1}) + {\varepsilon _1}({{\boldsymbol{Z}}_1}) \\ {F_2} = {\boldsymbol{W}}_2^{\rm{T}}{{\boldsymbol{S}}_2}({{\boldsymbol{Z}}_2}) + {\varepsilon _2}({{\boldsymbol{Z}}_2}) \\ \end{gathered} \right. (38) 式中: {{\boldsymbol{Z}}_1} = {[u,{a_u}]^{\rm{T}}} 和 {{\boldsymbol{Z}}_2} = {[r,{a_r}]^{\rm{T}}} ,均为神经网络输入信号; {{\boldsymbol{S}}_1}({{\boldsymbol{Z}}_1}) 和 {{\boldsymbol{S}}_2}({{\boldsymbol{Z}}_2}) ,均为神经网络自适应律的中心函数; {{\boldsymbol{W}}_i} = {\text{arg min}}\left\| {{{\boldsymbol{f}}_i} - {{\boldsymbol{W}}_i}{{\boldsymbol{S}}_i}} \right\| ,为理想的权重矩阵; {\varepsilon _i} 为神经网络逼近误差。
设计逼近函数为
{\hat {\boldsymbol{F}}_i} = \hat {\boldsymbol{W}}_i^{\rm{T}}{{\boldsymbol{S}}_i}({{\boldsymbol{Z}}_i}) (39) 权重的误差矩阵为
{\tilde {\boldsymbol{W}}_i} = {{\boldsymbol{W}}_i} - {\hat {\boldsymbol{W}}_i} (40) 估计误差为
{\tilde \hbar _i} = {\hbar _i} - {\hat \hbar _i} (41) 式中, {\hbar _i} 为估计神经网络逼近误差的上界,定义 {\hbar _i} = \max \left\{ {\left| {{\varepsilon _i}} \right|} \right\} ,i = 1,2。
对式(31)中的 {u_{\text{e}}} 和 {r_{\text{e}}} 求导,得到
\left\{ \begin{aligned} & {{\dot u}_{\text{e}}} = {{\dot a}_{{\text{f}}u}} - \dot u = \\ &\qquad {{\dot a}_{{\text{f}}u}} - \left( {{f_1} + {\zeta _u}{{\boldsymbol{\tau}} _{{\rm d}u}} - {\zeta _u}{\varDelta _u}} \right) - {\xi _u}{{\boldsymbol{\tau}} _u} \\ & {{\dot r}_{\text{e}}} = {{\dot a}_{{\text{f}}r}} - \dot r =\\ &\qquad {{\dot a}_{{\text{f}}r}} - \left( {{f_3} + {\zeta _r}({{\boldsymbol{\tau}} _{{\text{d}}r}} - \chi {{\boldsymbol{\tau}} _{{\text{d}}v}} - {\varDelta _r})} \right) - {\xi _r}{{\boldsymbol{\tau}} _r} \end{aligned}\right. (42) 对于未知动态 {F_1} = {f_1} + {\zeta _u}{{\boldsymbol{\tau}} _{{\text{d}}u}} - {\zeta _u}{\varDelta _u} 和 {F_2} = {f_3} + {\zeta _r}({{\boldsymbol{\tau}} _{{\text{d}}r}} - \chi {{\boldsymbol{\tau}} _{{\text{d}}v}}) - {\zeta _r}{\varDelta _r} ,包括由于输入饱和所产生的误差,未知连续的非线性部分以及外部环境干扰,利用自组织神经网络进行逼近。
因此,实际控制律设计如式(43)所示。
\left\{ \begin{aligned} & {{\boldsymbol{\tau}} _u} = \left( {{{\dot a}_{{\text{f}}u}} + \frac{{\left( {\bar \sigma _u^4 + \bar \sigma _u^2{{\bar E}^2}} \right){V_u}\cos {\psi _{\rm{e}}}}}{{{{\left( {\bar \sigma _u^2 - {{\bar E}^2}} \right)}^2}}} + {k_u}{u_{\rm{e}}} - \hat {\boldsymbol{W}}_1^{\rm{T}}{{\boldsymbol{S}}_1}({{\boldsymbol{Z}}_1}) + } \right. \\& \qquad \left. {{{\hat \hbar }_1}\tanh \left(\frac{{{u_{\rm{e}}}}}{{{\mathchar'26\mkern-10mu\lambda _1}}}\right) + \frac{{\text{π}}}{{\gamma {T_{{\text{c}}u}}}}\left( {u_{\rm{e}}^{1 - \gamma } + u_{\rm{e}}^{1 + \gamma }} \right)} \right)/{\zeta _u} \\& {{\boldsymbol{\tau}} _r} = \left( {{{\dot a}_{{\text{f}}r}} + \frac{{\left( {\bar \sigma _r^4 + \bar \sigma _r^2\psi _{\text{e}}^2} \right){V_r}}}{{{{\left( {\bar \sigma _r^2 - \psi _{\text{e}}^2} \right)}^2}}} + {k_r}{r_{\rm{e}}} - \hat {\boldsymbol{W}}_2^{\rm{T}}{{\boldsymbol{S}}_2}({{\boldsymbol{Z}}_2}) + } \right. \\& \qquad \left. {{{\hat \hbar }_2}\tanh \left(\frac{{{r_{\rm{e}}}}}{{{\mathchar'26\mkern-10mu\lambda _2}}}\right) + \frac{{\text{π}}}{{\gamma {T_{{\text{c}}r}}}}\left( {r_{\rm{e}}^{1 - \gamma } + r_{\rm{e}}^{1 + \gamma }} \right)} \right)/{\zeta _r} \end{aligned}\right. (43) 自适应律设计为
\left\{ \begin{gathered} {{\dot {\hat {\boldsymbol{W}}}}_1} = - {\varGamma _1}\left( {{{\boldsymbol{S}}_1}({{\boldsymbol{Z}}_1}){u_{\rm{e}}} + \frac{{2{\text{π}}}}{{\gamma {T_{{\text{w}}u}}}}{{\hat {\boldsymbol{W}}}_1}} \right) \\ {{\dot {\hat {\boldsymbol{W}}}}_2} = - {\varGamma _2}\left( {{{\boldsymbol{S}}_2}({{\boldsymbol{Z}}_2}){r_{\rm{e}}} + \frac{{2{\text{π}}}}{{\gamma {T_{{\text{w}}r}}}}{{\hat {\boldsymbol{W}}}_2}} \right) \\ \end{gathered} \right. (44) \left\{ \begin{aligned} & {{\dot {\hat \hbar} }_1} = \\& {\varUpsilon _1}\left( {{u_{\rm{e}}}\tanh \left( {\frac{{{u_{\rm{e}}}}}{{{\mathchar'26\mkern-10mu\lambda _1}}}} \right) - \frac{{\left( {2 - \gamma } \right){\text{π}}}}{{\gamma {T_{{\text{h}}u}}}}\hat \hbar _1^{1 - \gamma } - \frac{{\left( {2 + \gamma } \right){\text{π}}}}{{\gamma {T_{{\text{h}}u}}}}\hat \hbar _1^{1 + \gamma }} \right) \\& {{\dot {\hat \hbar} }_2} = \\& {\varUpsilon _2}\left( {{r_{\rm{e}}}\tanh \left( {\frac{{{r_{\rm{e}}}}}{{{\mathchar'26\mkern-10mu\lambda _2}}}} \right) - \frac{{\left( {2 - \gamma } \right){\text{π}}}}{{\gamma {T_{{\text{h}}r}}}}\hat \hbar _2^{1 - \gamma } - \frac{{\left( {2 + \gamma } \right){\text{π}}}}{{\gamma {T_{{\text{h}}r}}}}\hat \hbar _2^{1 + \gamma }} \right) \end{aligned}\right. (45) 式中: {\varGamma _1} , {\varGamma _2} , {\varUpsilon _1} 和 {\varUpsilon _2} 为自适应律增益。
3. 预定义时间稳定性分析
考虑以下Lyapunov函数:
\begin{split} & {V_3} = {V_1} + {V_{\text{f}}} + \frac{1}{2}u_{\text{e}}^2 + \frac{1}{2}r_{\text{e}}^2 + \frac{1}{2}(\varGamma _1^{ - 1}\tilde {\boldsymbol{W}}_1^{\rm{T}}{{\tilde {\boldsymbol{W}}}_1}) + \\&\quad \frac{1}{2}(\varGamma _2^{ - 1}\tilde {\boldsymbol{W}}_2^{\rm{T}}{{\tilde {\boldsymbol{W}}}_2}) + \frac{1}{2}(\varUpsilon _1^{ - 1}\tilde \hbar _1^2) + \frac{1}{2}(\varUpsilon _2^{ - 1}\tilde \hbar _2^2) \end{split} (46) {V_3} 的导数为
\begin{split} & \qquad\qquad{{\dot V}_3} = {{\dot V}_1} + {{\dot V}_{\text{f}}} + {u_{\rm{e}}}{{\dot u}_{\rm{e}}} + {r_{\rm{e}}}{{\dot r}_{\rm{e}}} + \varGamma _1^{ - 1}\tilde {\boldsymbol{W}}_1^{\text{T}}{{\dot {\tilde {\boldsymbol{W}}}}_1} + \varGamma _2^{ - 1}\tilde {\boldsymbol{W}}_2^{\text{T}}{{\dot {\tilde {\boldsymbol{W}}}}_2} + \varUpsilon _1^{ - 1}{{\tilde \hbar }_1}{{\dot {\hat \hbar} }_1} + \varUpsilon _2^{ - 1}{{\tilde \hbar }_1}{{\dot {\hat \hbar} }_2} \leqslant\\& \qquad\quad - {k_1}{{\bar E}^2} - {k_2}\psi _{\text{e}}^2 - \frac{{\text{π}}}{{\gamma {T_u}}}\left( {{{\left( {V_u^2} \right)}^{1 - \tfrac{\gamma }{2}}} + {{\left( {V_u^2} \right)}^{1 + \tfrac{\gamma }{2}}}} \right) - {k_u}u_{\rm{e}}^2 - {k_r}r_{\rm{e}}^2 - \frac{{\text{π}}}{{\gamma {T_r}}}\left( {{{\left( {V_r^2} \right)}^{1 - \tfrac{\gamma }{2}}} + {{\left( {V_r^2} \right)}^{1 + \tfrac{\gamma }{2}}}} \right) - \\& \frac{{\text{π}}}{{\gamma {T_{{\text{c}}u}}}}\left( {u_{\rm{e}}^{1 - \tfrac{\gamma }{2}} + u_{\rm{e}}^{1 + \tfrac{\gamma }{2}}} \right) - \frac{{\text{π}}}{{\gamma {T_{{\text{c}}r}}}}\left( {r_{\rm{e}}^{1 - \tfrac{\gamma }{2}} + r_{\rm{e}}^{1 + \tfrac{\gamma }{2}}} \right) - \frac{{\text{π}}}{{\gamma {T_{{\text{f}}u}}}}\left( {{{\left( {\xi _u^2} \right)}^{1 - \tfrac{\gamma }{2}}} + {{\left( {\xi _u^2} \right)}^{1 + \tfrac{\gamma }{2}}}} \right) - \frac{{\text{π}}}{{\gamma {T_{{\text{f}}r}}}}\left( {{{\left( {\xi _r^2} \right)}^{1 - \tfrac{\gamma }{2}}} + {{\left( {\xi _r^2} \right)}^{1 + \tfrac{\gamma }{2}}}} \right) - \\& \qquad \frac{{\text{π}}}{{\gamma {T_{\gamma u}}}}{\left( {{\tilde {\bar \gamma}} _u^2} \right)^{1 - \tfrac{\gamma }{2}}} - \frac{{\text{π}}}{{\gamma {T_{\gamma u}}}}{\left( {{\tilde {\bar \gamma}} _u^2} \right)^{1 + \tfrac{\gamma }{2}}} - \frac{{\text{π}}}{{\gamma {T_{\gamma r}}}}{\left( {{\tilde {\bar \gamma}} _r^2} \right)^{1 - \tfrac{\gamma }{2}}} - \frac{{\text{π}}}{{\gamma {T_{\gamma r}}}}{\left( {{\tilde {\bar \gamma}} _r^2} \right)^{1 + \tfrac{\gamma }{2}}} - {u_{\rm{e}}}\tilde {\boldsymbol{W}}_1^{\rm{T}}{{\boldsymbol{S}}_1}({{\boldsymbol{Z}}_1}) - {r_{\rm{e}}}\tilde {\boldsymbol{W}}_2^{\rm{T}}{{\boldsymbol{S}}_2}({{\boldsymbol{Z}}_2}) + \\& \left( {\left| {{u_{\rm{e}}}} \right|{\hbar _1} - {u_{\rm{e}}}{{\hat \hbar }_1}\tanh \left( {\frac{{{u_{\rm{e}}}}}{{{\mathchar'26\mkern-10mu\lambda _1}}}} \right)} \right) + \left( {\left| {{r_{\rm{e}}}} \right|{\hbar _2} - {r_{\rm{e}}}{{\hat \hbar }_2}\tanh \left( {\frac{{{r_{\rm{e}}}}}{{{\mathchar'26\mkern-10mu\lambda _2}}}} \right)} \right) - \varGamma _1^{ - 1}\tilde {\boldsymbol{W}}_1^{\rm{T}}{{\dot {\hat {\boldsymbol{W}}}}_1} - \varGamma _2^{ - 1}\tilde {\boldsymbol{W}}_2^{\rm{T}}{{\dot {\hat {\boldsymbol{W}}}}_2} - \varUpsilon _1^{ - 1}{{\tilde \hbar }_1}{{\dot {\hat \hbar} }_1} - \varUpsilon _2^{ - 1}{{\tilde \hbar }_2}{{\dot {\hat \hbar} }_2} + \\& \qquad\qquad\qquad \frac{{\text{π}}}{{\gamma {T_{\gamma u}}}}{(\bar \gamma _u^2)^{1 - \tfrac{\gamma }{2}}} + \frac{{\text{π}}}{{\gamma {T_{\gamma u}}}}{(\bar \gamma _u^2)^{1 + \tfrac{\gamma }{2}}} + \frac{{\text{π}}}{{\gamma {T_{\gamma r}}}}{(\bar \gamma _r^2)^{1 - \tfrac{\gamma }{2}}} + \frac{{\text{π}}}{{\gamma {T_{\gamma r}}}}{(\bar \gamma _r^2)^{1 + \tfrac{\gamma }{2}}} + k_1'{\mathchar'26\mkern-10mu\lambda _1}{{\bar \gamma }_u} + k_2'{\mathchar'26\mkern-10mu\lambda _2}{{\bar \gamma }_r} \end{split} (47) 根据引理4,结合式(41),式(47)可转化为
\begin{gathered} {{\dot V}_3} \leqslant - {k_1}{{\bar E}^2} - {k_2}\psi _{\rm{e}}^2 - {k_u}u_{\rm{e}}^2 - {k_r}r_{\rm{e}}^2 - \frac{{\text{π}}}{{\gamma {T_u}}}\left( {{{\left( {V_u^2} \right)}^{1 - \tfrac{\gamma }{2}}} + {{\left( {V_u^2} \right)}^{1 + \tfrac{\gamma }{2}}}} \right) - \frac{{\text{π}}}{{\gamma {T_r}}}\left( {{{\left( {V_r^2} \right)}^{1 - \tfrac{\gamma }{2}}} + {{\left( {V_r^2} \right)}^{1 + \tfrac{\gamma }{2}}}} \right) - \\ \frac{{\text{π}}}{{\gamma {T_{{\text{c}}u}}}}\left( {u_{\rm{e}}^{1 - \tfrac{\gamma }{2}} + u_{\rm{e}}^{1 + \tfrac{\gamma }{2}}} \right) - \frac{{\text{π}}}{{\gamma {T_{{\text{c}}r}}}}\left( {r_{\rm{e}}^{1 - \tfrac{\gamma }{2}} + r_{\rm{e}}^{1 + \tfrac{\gamma }{2}}} \right) - \frac{{\text{π}}}{{\gamma {T_{{\text{f}}u}}}}\left( {{{\left( {\xi _u^2} \right)}^{1 - \tfrac{\gamma }{2}}} + {{\left( {\xi _u^2} \right)}^{1 + \tfrac{\gamma }{2}}}} \right) - \frac{{\text{π}}}{{\gamma {T_{{\text{f}}r}}}}\left( {{{\left( {\xi _r^2} \right)}^{1 - \tfrac{\gamma }{2}}} + {{\left( {\xi _r^2} \right)}^{1 + \tfrac{\gamma }{2}}}} \right) - \\ \frac{{\text{π}}}{{\gamma {T_{\gamma u}}}}{\left( {{\tilde {\bar \gamma}} _u^2} \right)^{1 - \tfrac{\gamma }{2}}} - \frac{{\text{π}}}{{\gamma {T_{\gamma u}}}}{\left( {{\tilde {\bar \gamma}} _u^2} \right)^{1 + \tfrac{\gamma }{2}}} - \frac{{\text{π}}}{{\gamma {T_{\gamma r}}}}{\left( {{\tilde {\bar \gamma}} _r^2} \right)^{1 - \tfrac{\gamma }{2}}} - \frac{{\text{π}}}{{\gamma {T_{\gamma r}}}}{\left( {{\tilde {\bar \gamma}} _r^2} \right)^{1 + \tfrac{\gamma }{2}}} + \varGamma _1^{ - 1}\tilde {\boldsymbol{W}}_1^{\rm{T}}( - {{\dot {\hat {\boldsymbol{W}}}}_1} - {\varGamma _1}{u_{\rm{e}}}{{\boldsymbol{S}}_1}\left( {{{\boldsymbol{Z}}_1}} \right)) + \end{gathered} \begin{split} & \quad \varGamma _2^{ - 1}\tilde {\boldsymbol{W}}_2^{\rm{T}}( - {{\dot {\hat {\boldsymbol{W}}}}_2} - {\varGamma _2}{r_{\rm{e}}}{{\boldsymbol{S}}_2}\left( {{{\boldsymbol{Z}}_2}} \right)) + \varUpsilon _1^{ - 1}{{\tilde \hbar }_1}\left( {{\varUpsilon _1}{u_{\text{e}}}\tanh \left( {\frac{{{u_{\rm{e}}}}}{{{\lambda _1}}}} \right) - {{\dot {\hat \hbar} }_1}} \right) + \varUpsilon _2^{ - 1}{{\tilde \hbar }_2}\left( {{\varUpsilon _2}{r_{\rm{e}}}\tanh \left( {\frac{{{r_{\rm{e}}}}}{{{\lambda _2}}}} \right) - {{\dot {\hat \hbar} }_2}} \right) + \\ & \frac{{\text{π}}}{{\gamma {T_{\gamma u}}}}{(\bar \gamma _u^2)^{1 - \tfrac{\gamma }{2}}} + \frac{{\text{π}}}{{\gamma {T_{\gamma u}}}}{(\bar \gamma _u^2)^{1 + \tfrac{\gamma }{2}}} + \frac{{\text{π}}}{{\gamma {T_{\gamma r}}}}{(\bar \gamma _r^2)^{1 - \tfrac{\gamma }{2}}} + \frac{{\text{π}}}{{\gamma {T_{\gamma r}}}}{(\bar \gamma _r^2)^{1 + \tfrac{\gamma }{2}}} + k_1'{\mathchar'26\mkern-10mu\lambda _1}{{\bar \gamma }_u} + k_2'{\mathchar'26\mkern-10mu\lambda _2}{{\bar \gamma }_r} + k_3'{\mathchar'26\mkern-10mu\lambda _3}\left| {{\hbar _1}} \right| + k_4'{\mathchar'26\mkern-10mu\lambda _4}\left| {{\hbar _2}} \right| \end{split} (48) 将式(44)和式(45)代入式(48),可得
\begin{split} & \qquad{{\dot V}_3} \leqslant - {k_1}{{\bar E}^2} - {k_2}\psi _{\rm{e}}^2 - {k_u}u_{\rm{e}}^2 - {k_r}r_{\rm{e}}^2 - \frac{{\text{π}}}{{\gamma {T_u}}}\left( {{{\left( {V_u^2} \right)}^{1 - \tfrac{\gamma }{2}}} + {{\left( {V_u^2} \right)}^{1 + \tfrac{\gamma }{2}}}} \right) - \frac{{\text{π}}}{{\gamma {T_r}}}\left( {{{\left( {V_r^2} \right)}^{1 - \tfrac{\gamma }{2}}} + {{\left( {V_r^2} \right)}^{1 + \tfrac{\gamma }{2}}}} \right) - \\& \qquad\qquad\qquad \frac{{\text{π}}}{{\gamma {T_{{\text{c}}u}}}}\left( {u_{\rm{e}}^{1 - \tfrac{\gamma }{2}} + u_{\rm{e}}^{1 + \tfrac{\gamma }{2}}} \right) - \frac{{\text{π}}}{{\gamma {T_{{\text{c}}r}}}}\left( {r_{\rm{e}}^{1 - \tfrac{\gamma }{2}} + r_{\rm{e}}^{1 + \tfrac{\gamma }{2}}} \right) - \frac{{\text{π}}}{{\gamma {T_{{\text{f}}u}}}}\left( {{{\left( {\xi _u^2} \right)}^{1 - \tfrac{\gamma }{2}}} + {{\left( {\xi _u^2} \right)}^{1 + \tfrac{\gamma }{2}}}} \right) - \\& \qquad \frac{{\text{π}}}{{\gamma {T_{{\text{f}}r}}}}\left( {{{\left( {\xi _r^2} \right)}^{1 - \tfrac{\gamma }{2}}} + {{\left( {\xi _r^2} \right)}^{1 + \tfrac{\gamma }{2}}}} \right) - \frac{{\text{π}}}{{\gamma {T_{\gamma u}}}}{\left( {{\tilde {\bar \gamma}} _u^2} \right)^{1 - \tfrac{\gamma }{2}}} - \frac{{\text{π}}}{{\gamma {T_{\gamma u}}}}{\left( {{\tilde {\bar \gamma}} _u^2} \right)^{1 + \tfrac{\gamma }{2}}} - \frac{{\text{π}}}{{\gamma {T_{\gamma r}}}}{\left( {{\tilde {\bar \gamma}} _r^2} \right)^{1 - \tfrac{\gamma }{2}}} - \frac{{\text{π}}}{{\gamma {T_{\gamma r}}}}{\left( {{\tilde {\bar \gamma}} _r^2} \right)^{1 + \tfrac{\gamma }{2}}} - \\& \frac{{\text{π}}}{{\gamma {T_{{\text{w}}u}}}}\tilde {\boldsymbol{W}}_1^{\rm{T}}{{\tilde {\boldsymbol{W}}}_1} + \frac{{\text{π}}}{{\gamma {T_{{\text{w}}u}}}}\tilde {\boldsymbol{W}}_1^{\rm{T}}{{\boldsymbol{W}}_1} - \frac{{\text{π}}}{{\gamma {T_{{\text{w}}r}}}}\tilde {\boldsymbol{W}}_2^{\rm{T}}{{\tilde {\boldsymbol{W}}}_2} + \frac{{\text{π}}}{{\gamma {T_{{\text{w}}r}}}}\tilde {\boldsymbol{W}}_2^{\rm{T}}{{\boldsymbol{W}}_2} + \frac{{\text{π}}}{{\gamma {T_{{\text{h}}u}}}}{{\tilde \hbar }_1}\left( {\hat \hbar _1^{1 - \gamma } + \hat \hbar _1^{1 + \gamma }} \right) + \frac{{\text{π}}}{{\gamma {T_{{\text{h}}r}}}}{{\tilde \hbar }_2}\left( {\hat \hbar _2^{1 - \gamma } + \hat \hbar _2^{1 + \gamma }} \right) + \\& \qquad \frac{{\text{π}}}{{\gamma {T_{\gamma u}}}}{(\bar \gamma _u^2)^{1 - \tfrac{\gamma }{2}}} + \frac{{\text{π}}}{{\gamma {T_{\gamma u}}}}{(\bar \gamma _u^2)^{1 + \tfrac{\gamma }{2}}} + \frac{{\text{π}}}{{\gamma {T_{\gamma r}}}}{(\bar \gamma _r^2)^{1 - \tfrac{\gamma }{2}}} + \frac{{\text{π}}}{{\gamma {T_{\gamma r}}}}{(\bar \gamma _r^2)^{1 + \tfrac{\gamma }{2}}} + k_1'{\mathchar'26\mkern-10mu\lambda _1}{{\bar \gamma }_u} + k_2'{\mathchar'26\mkern-10mu\lambda _2}{{\bar \gamma }_r} + k_3'{\mathchar'26\mkern-10mu\lambda _3}\left| {{\hbar _1}} \right| + k_4'{\mathchar'26\mkern-10mu\lambda _4}\left| {{\hbar _2}} \right| \end{split} (49) 根据引理2,满足以下不等式:
\left\{ \begin{gathered} {{\tilde \hbar }_\iota }\hat \hbar _\iota ^{1 - \gamma } \leqslant \frac{1}{{2 - \gamma }}(2\hbar _\iota ^{2 - \gamma } - \tilde \hbar _\iota ^{2 - \gamma }) \\ {{\tilde \hbar }_\iota }\hat \hbar _\iota ^{1 + \gamma } \leqslant \frac{1}{{2 + \gamma }}(2\hbar _\iota ^{2 + \gamma } - \tilde \hbar _\iota ^{2 + \gamma }) \\ \end{gathered} \right. (50) 根据杨氏不等式,以下不等式成立:
\left\{ \begin{aligned} & \frac{{2{\text{π}}}}{{\gamma {T_{{\text{w}}u}}}}\tilde {\boldsymbol{W}}_1^{\rm{T}}{{\boldsymbol{W}}_1} \leqslant \frac{{\text{π}}}{{\gamma {T_{{\text{w}}u}}}}\tilde {\boldsymbol{W}}_1^{\rm{T}}{{\tilde {\boldsymbol{W}}}_1} + \frac{{\text{π}}}{{\gamma {T_{{\text{w}}u}}}}{\boldsymbol{W}}_1^{\rm{T}}{{\boldsymbol{W}}_1} \\& \frac{{2{\text{π}}}}{{\gamma {T_{{\text{w}}r}}}}\tilde {\boldsymbol{W}}_2^{\rm{T}}{{\boldsymbol{W}}_2} \leqslant \frac{{\text{π}}}{{\gamma {T_{{\text{w}}r}}}}\tilde {\boldsymbol{W}}_2^{\rm{T}}{{\tilde {\boldsymbol{W}}}_2} + \frac{{\text{π}}}{{\gamma {T_{{\text{w}}r}}}}{\boldsymbol{W}}_2^{\rm{T}}{{\boldsymbol{W}}_2} \end{aligned}\right. (51) 则式(49)可以化为
\begin{split} & \quad\;{{\dot V}_3} \leqslant - {k_1}{{\bar E}^2} - {k_2}\psi _{\rm{e}}^2 - {k_u}u_{\rm{e}}^2 - {k_r}r_{\rm{e}}^2 - \frac{{\text{π}}}{{\gamma {T_u}}}\left( {{{\left( {V_u^2} \right)}^{1 - \tfrac{\gamma }{2}}} + {{\left( {V_u^2} \right)}^{1 + \tfrac{\gamma }{2}}}} \right) - \frac{{\text{π}}}{{\gamma {T_r}}}\left( {{{\left( {V_r^2} \right)}^{1 - \tfrac{\gamma }{2}}} + {{\left( {V_r^2} \right)}^{1 + \tfrac{\gamma }{2}}}} \right) - \\& \;\;\; \frac{{\text{π}}}{{\gamma {T_{{\text{c}}u}}}}\left( {u_{\rm{e}}^{1 - \tfrac{\gamma }{2}} + u_{\rm{e}}^{1 + \tfrac{\gamma }{2}}} \right) - \frac{{\text{π}}}{{\gamma {T_{{\text{c}}r}}}}\left( {r_{\rm{e}}^{1 - \tfrac{\gamma }{2}} + r_{\rm{e}}^{1 + \tfrac{\gamma }{2}}} \right) - \frac{{\text{π}}}{{\gamma {T_{{\text{f}}u}}}}\left( {{{\left( {\xi _u^2} \right)}^{1 - \tfrac{\gamma }{2}}} + {{\left( {\xi _u^2} \right)}^{1 + \tfrac{\gamma }{2}}}} \right) - \frac{{\text{π}}}{{\gamma {T_{{\text{f}}r}}}}\left( {{{\left( {\xi _r^2} \right)}^{1 - \tfrac{\gamma }{2}}} + {{\left( {\xi _r^2} \right)}^{1 + \tfrac{\gamma }{2}}}} \right) - \\& \qquad \frac{{\text{π}}}{{\gamma {T_{\gamma u}}}}{\left( {{\tilde {\bar \gamma}} _u^2} \right)^{1 - \tfrac{\gamma }{2}}} - \frac{{\text{π}}}{{\gamma {T_{\gamma u}}}}{\left( {{\tilde {\bar \gamma}} _u^2} \right)^{1 + \tfrac{\gamma }{2}}} - \frac{{\text{π}}}{{\gamma {T_{\gamma r}}}}{\left( {{\tilde {\bar \gamma}} _r^2} \right)^{1 - \tfrac{\gamma }{2}}} - \frac{{\text{π}}}{{\gamma {T_{\gamma r}}}}{\left( {{\tilde {\bar \gamma}} _r^2} \right)^{1 + \tfrac{\gamma }{2}}} - \frac{{\text{π}}}{{\gamma {T_{{\text{w}}u}}}}\tilde {\boldsymbol{W}}_{\text{1}}^{\text{T}}{{\tilde {\boldsymbol{W}}}_1} - \frac{{\text{π}}}{{\gamma {T_{{\text{w}}r}}}}\tilde {\boldsymbol{W}}_2^{\text{T}}{{\tilde {\boldsymbol{W}}}_2} - \\& \frac{{\text{π}}}{{\gamma {T_{{\text{h}}u}}}}\left( {{{\left( {\tilde \hbar _1^2} \right)}^{1 - \tfrac{\gamma }{2}}} + {{\left( {\tilde \hbar _1^2} \right)}^{1 + \tfrac{\gamma }{2}}}} \right) - \frac{{\text{π}}}{{\gamma {T_{{\text{h}}r}}}}\left( {{{\left( {\tilde \hbar _2^2} \right)}^{1 - \tfrac{\gamma }{2}}} + {{\left( {\tilde \hbar _2^2} \right)}^{1 + \tfrac{\gamma }{2}}}} \right) + \frac{{2{\text{π}}}}{{\gamma {T_{{\text{f}}u}}}}{(\bar \gamma _u^2)^{1 - \tfrac{\gamma }{2}}} + \frac{{2{\text{π}}}}{{\gamma {T_{{\text{f}}u}}}}{(\bar \gamma _u^2)^{1 + \tfrac{\gamma }{2}}} + \frac{{2{\text{π}}}}{{\gamma {T_{{\text{f}}r}}}}{(\bar \gamma _r^2)^{1 - \tfrac{\gamma }{2}}} + \\& \qquad \frac{{2{\text{π}}}}{{\gamma {T_{{\text{f}}r}}}}{(\bar \gamma _r^2)^{1 + \tfrac{\gamma }{2}}} + k_1'{\mathchar'26\mkern-10mu\lambda _1}{{\bar \gamma }_u} + k_2'{\mathchar'26\mkern-10mu\lambda _2}{{\bar \gamma }_r} + k_3'{\mathchar'26\mkern-10mu\lambda _3}\left| {{\hbar _1}} \right| + k_4'{\mathchar'26\mkern-10mu\lambda _4}\left| {{\hbar _2}} \right| + \frac{{2{\text{π}}}}{{\gamma {T_{{\text{h}}u}}}}\left( {\hbar _1^{2 - \gamma } + \hbar _1^{2 + \gamma }} \right) + \frac{{2{\text{π}}}}{{\gamma {T_{{\text{h}}r}}}}\left( {\hbar _2^{2 - \gamma } + \hbar _2^{2 + \gamma }} \right) + \\& \qquad\qquad\qquad\qquad\qquad\qquad \frac{{\text{π}}}{{2\gamma {T_{{\text{w}}u}}}}{\boldsymbol{W}}_1^{\text{T}}{{\boldsymbol{W}}_1} + \frac{{\text{π}}}{{2\gamma {T_{{\text{w}}r}}}}{\boldsymbol{W}}_{\text{2}}^{\text{T}}{{\boldsymbol{W}}_2} \leqslant\\& - \frac{{\text{π}}}{{\gamma {T_u}}}V_u^2 - \frac{{\text{π}}}{{\gamma {T_r}}}V_r^2 - \frac{{\text{π}}}{{\gamma {T_{{\text{f}}u}}}}\xi _u^2 - \frac{{\text{π}}}{{\gamma {T_{{\text{f}}r}}}}\xi _r^2 - \frac{{\text{π}}}{{\gamma {T_{{\text{f}}u}}}}{\tilde {\bar \gamma}} _u^2 - \frac{{\text{π}}}{{\gamma {T_{{\text{f}}r}}}}{\tilde {\bar \gamma}} _r^2 - {k_u}u_{\rm{e}}^2 - {k_r}r_{\rm{e}}^2 - \frac{{\text{π}}}{{\gamma {T_{{\text{w}}u}}}}\tilde {\boldsymbol{W}}_1^{\text{T}}{{\tilde {\boldsymbol{W}}}_1} - \frac{{\text{π}}}{{\gamma {T_{{\text{w}}r}}}}\tilde {\boldsymbol{W}}_{\text{2}}^{\text{T}}{{\tilde {\boldsymbol{W}}}_2}- \\& \frac{{\text{π}}}{{\gamma {T_{{\text{h}}u}}}}\tilde \hbar _1^2 - \frac{{\text{π}}}{{\gamma {T_{{\text{h}}r}}}}\tilde \hbar _2^2 + \frac{{2{\text{π}}}}{{\gamma {T_{{\text{f}}u}}}}{(\bar \gamma _u^2)^{1 - \tfrac{\gamma }{2}}} + \frac{{2{\text{π}}}}{{\gamma {T_{{\text{f}}u}}}}{(\bar \gamma _u^2)^{1 + \tfrac{\gamma }{2}}} + \frac{{2{\text{π}}}}{{\gamma {T_{{\text{f}}r}}}}{(\bar \gamma _r^2)^{1 - \tfrac{\gamma }{2}}} + \frac{{2{\text{π}}}}{{\gamma {T_{{\text{f}}r}}}}{(\bar \gamma _r^2)^{1 + \tfrac{\gamma }{2}}} + k_1'{\mathchar'26\mkern-10mu\lambda _1}{{\bar \gamma }_u} + k_2'{\mathchar'26\mkern-10mu\lambda _2}{{\bar \gamma }_r} + \\& k_3'{\mathchar'26\mkern-10mu\lambda _3}\left| {{\hbar _1}} \right| + k_4'{\mathchar'26\mkern-10mu\lambda _4}\left| {{\hbar _2}} \right| + \frac{{2{\text{π}}}}{{\gamma {T_{{\text{h}}u}}}}\left( {\hbar _1^{2 - \gamma } + \hbar _1^{2 + \gamma }} \right) + \frac{{2{\text{π}}}}{{\gamma {T_{{\text{h}}r}}}}\left( {\hbar _2^{2 - \gamma } + \hbar _2^{2 + \gamma }} \right) + \frac{{\text{π}}}{{\gamma {T_{{\text{w}}u}}}}{\boldsymbol{W}}_1^{\text{T}}{{\boldsymbol{W}}_1} + \frac{{\text{π}}}{{\gamma {T_{{\text{w}}r}}}}{\boldsymbol{W}}_2^{\text{T}}{{\boldsymbol{W}}_2} \leqslant {\rho _0}{V_3} + {\varDelta _0} \end{split} (52) 将式(52)两边同时积分,满足以下不等式:
0 \leqslant {V_3} \leqslant \left( {{V_3}\left( 0 \right) - \frac{{{\varDelta _0}}}{{{\rho _0}}}} \right){{\text{e}}^{ - {k_0}t}} + \frac{{{\varDelta _0}}}{{{\rho _0}}} (53) 可见, {V_3} 显然是有界的,这意味着跟踪误差保持一致有界。因此,进一步假设总是有一个正常数 {\nabla _{{\text{M}}i}} 且满足以下不等式 \tilde {\boldsymbol{W}}_i^{\rm{T}}{\tilde {\boldsymbol{W}}_i} \leqslant {\nabla _{{\text{M}}i}} ,i = 1,2。
\begin{gathered} {{\dot V}_3} \leqslant - {k_1}{{\bar E}^2} - {k_2}\psi _{\rm{e}}^2 - {k_u}u_{\rm{e}}^2 - {k_r}r_{\rm{e}}^2 - \frac{{\text{π}}}{{\gamma {T_u}}}\left( {{{\left( {V_u^2} \right)}^{1 - \tfrac{\gamma }{2}}} + {{\left( {V_u^2} \right)}^{1 + \tfrac{\gamma }{2}}}} \right) - \frac{{\text{π}}}{{\gamma {T_r}}}\left( {{{\left( {V_r^2} \right)}^{1 - \tfrac{\gamma }{2}}} + {{\left( {V_r^2} \right)}^{1 + \tfrac{\gamma }{2}}}} \right) - \frac{{\text{π}}}{{\gamma {T_{{\text{c}}u}}}}\left( {u_{\rm{e}}^{1 - \tfrac{\gamma }{2}} + u_{\rm{e}}^{1 + \tfrac{\gamma }{2}}} \right) -\\ \frac{{\text{π}}}{{\gamma {T_{{\text{c}}r}}}}\left( {r_{\rm{e}}^{1 - \tfrac{\gamma }{2}} + r_{\rm{e}}^{1 + \tfrac{\gamma }{2}}} \right) - \frac{{\text{π}}}{{\gamma {T_{{\text{f}}u}}}}\left( {{{\left( {\xi _u^2} \right)}^{1 - \tfrac{\gamma }{2}}} + {{\left( {\xi _u^2} \right)}^{1 + \tfrac{\gamma }{2}}}} \right) - \frac{{\text{π}}}{{\gamma {T_{{\text{f}}r}}}}\left( {{{\left( {\xi _r^2} \right)}^{1 - \tfrac{\gamma }{2}}} + {{\left( {\xi _r^2} \right)}^{1 + \tfrac{\gamma }{2}}}} \right) - \frac{{\text{π}}}{{\gamma {T_{\gamma u}}}}{\left( {{\tilde {\bar \gamma}} _u^2} \right)^{1 - \tfrac{\gamma }{2}}} - \frac{{\text{π}}}{{\gamma {T_{\gamma u}}}}{\left( {{\tilde {\bar \gamma}} _u^2} \right)^{1 + \tfrac{\gamma }{2}}} - \\ \frac{{\text{π}}}{{\gamma {T_{\gamma r}}}}{\left( {{\tilde {\bar \gamma}} _r^2} \right)^{1 - \tfrac{\gamma }{2}}} - \frac{{\text{π}}}{{\gamma {T_{\gamma r}}}}{\left( {{\tilde {\bar \gamma}} _r^2} \right)^{1 + \tfrac{\gamma }{2}}} - \frac{{\text{π}}}{{\gamma {T_{{\text{w}}u}}}}\tilde {\boldsymbol{W}}_1^{\rm{T}}{{\tilde {\boldsymbol{W}}}_1} - \frac{{\text{π}}}{{\gamma {T_{{\text{w}}r}}}}\tilde {\boldsymbol{W}}_2^{\rm{T}}{{\tilde {\boldsymbol{W}}}_2} - \frac{{\text{π}}}{{\gamma {T_{{\text{w}}u}}}}\left( {{{\left( {\tilde {\boldsymbol{W}}_1^{\rm{T}}{{\tilde {\boldsymbol{W}}}_1}} \right)}^{1 - \tfrac{\gamma }{2}}} + {{\left( {\tilde {\boldsymbol{W}}_1^{\rm{T}}{{\tilde {\boldsymbol{W}}}_1}} \right)}^{1 + \tfrac{\gamma }{2}}}} \right) - \\ \frac{{\text{π}}}{{\gamma {T_{{\text{w}}r}}}}\left( {{{\left( {\tilde {\boldsymbol{W}}_2^{\text{T}}{{\tilde {\boldsymbol{W}}}_2}} \right)}^{1 - \tfrac{\gamma }{2}}} + {{\left( {\tilde {\boldsymbol{W}}_2^{\text{T}}{{\tilde {\boldsymbol{W}}}_2}} \right)}^{1 + \tfrac{\gamma }{2}}}} \right) + \frac{{\text{π}}}{{\gamma {T_{{\text{w}}u}}}}{\left( {\tilde {\boldsymbol{W}}_1^{\text{T}}{{\tilde {\boldsymbol{W}}}_1}} \right)^{1 - \tfrac{\gamma }{2}}} + \frac{{\text{π}}}{{\gamma {T_{{\text{w}}u}}}}{\left( {\tilde {\boldsymbol{W}}_1^{\text{T}}{{\tilde {\boldsymbol{W}}}_1}} \right)^{1 + \tfrac{\gamma }{2}}} + \frac{{\text{π}}}{{\gamma {T_{{\text{w}}r}}}}{\left( {\tilde {\boldsymbol{W}}_2^{\text{T}}{{\tilde {\boldsymbol{W}}}_2}} \right)^{1 - \tfrac{\gamma }{2}}} + \frac{{\text{π}}}{{\gamma {T_{{\text{w}}r}}}}{\left( {\tilde {\boldsymbol{W}}_2^{\text{T}}{{\tilde {\boldsymbol{W}}}_2}} \right)^{1 + \tfrac{\gamma }{2}}}- \end{gathered} \begin{split} & \frac{{\text{π}}}{{\gamma {T_{{\text{h}}u}}}}\left( {{{\left( {\tilde \hbar _1^2} \right)}^{1 - \tfrac{\gamma }{2}}} + {{\left( {\tilde \hbar _1^2} \right)}^{1 + \tfrac{\gamma }{2}}}} \right) - \frac{{\text{π}}}{{\gamma {T_{{\text{h}}r}}}}\left( {{{\left( {\tilde \hbar _2^2} \right)}^{1 - \tfrac{\gamma }{2}}} + {{\left( {\tilde \hbar _2^2} \right)}^{1 + \tfrac{\gamma }{2}}}} \right) + \frac{{2{\text{π}}}}{{\gamma {T_{{\text{f}}u}}}}{(\bar \gamma _u^2)^{1 - \tfrac{\gamma }{2}}} + \frac{{2{\text{π}}}}{{\gamma {T_{{\text{f}}u}}}}{(\bar \gamma _u^2)^{1 + \tfrac{\gamma }{2}}} + \frac{{2{\text{π}}}}{{\gamma {T_{{\text{f}}r}}}}{(\bar \gamma _r^2)^{1 - \tfrac{\gamma }{2}}} + \frac{{2{\text{π}}}}{{\gamma {T_{{\text{f}}r}}}}{(\bar \gamma _r^2)^{1 + \tfrac{\gamma }{2}}} + \\ &\quad k_1'{\mathchar'26\mkern-10mu\lambda _1}{{\bar \gamma }_u} + k_2'{\mathchar'26\mkern-10mu\lambda _2}{{\bar \gamma }_r} + k_3'{\mathchar'26\mkern-10mu\lambda _3}\left| {{\hbar _1}} \right| + k_4'{\mathchar'26\mkern-10mu\lambda _4}\left| {{\hbar _2}} \right| + \frac{{2{\text{π}}}}{{\gamma {T_{{\text{h}}u}}}}\left( {\hbar _1^{2 - \gamma } + \hbar _1^{2 + \gamma }} \right) + \frac{{2{\text{π}}}}{{\gamma {T_{{\text{h}}r}}}}\left( {\hbar _2^{2 - \gamma } + \hbar _2^{2 + \gamma }} \right) + \frac{{\text{π}}}{{2\gamma {T_{{\text{w}}u}}}}{\boldsymbol{W}}_1^{\text{T}}{{\boldsymbol{W}}_1} + \frac{{\text{π}}}{{2\gamma {T_{{\text{w}}r}}}}{\boldsymbol{W}}_{\text{2}}^{\text{T}}{{\boldsymbol{W}}_2} \end{split} (54) 事实上,若有a > 0,则满足以下不等式:
{a^{1 - \tfrac{\gamma }{2}}} - a \leqslant {\left( {\frac{{2 - \gamma }}{2}} \right)^{\tfrac{{2 - \gamma }}{\gamma }}} - {\left( {\frac{{2 - \gamma }}{2}} \right)^{\tfrac{2}{\gamma }}} = {\nabla _1} (55) 式(54)实现了闭环系统的渐近收敛。进一步考虑闭环系统的预定义时间收敛,结合式(55),式(54)可以改写为
\begin{split} & {{\dot V}_3} \leqslant - {k_1}{{\bar E}^2} - {k_2}\psi _{\rm{e}}^2 - {k_u}u_{\rm{e}}^2 - {k_r}r_{\rm{e}}^2 - \frac{{\text{π}}}{{\gamma {T_u}}}\left( {{{\left( {V_u^2} \right)}^{1 - \tfrac{\gamma }{2}}} + {{\left( {V_u^2} \right)}^{1 + \tfrac{\gamma }{2}}}} \right) - \frac{{\text{π}}}{{\gamma {T_r}}}\left( {{{\left( {V_r^2} \right)}^{1 - \tfrac{\gamma }{2}}} + {{\left( {V_r^2} \right)}^{1 + \tfrac{\gamma }{2}}}} \right) - \frac{{\text{π}}}{{\gamma {T_{{\text{c}}u}}}}\left( {u_{\rm{e}}^{1 - \tfrac{\gamma }{2}} + u_{\rm{e}}^{1 + \tfrac{\gamma }{2}}} \right) - \\&\quad \frac{{\text{π}}}{{\gamma {T_{{\text{c}}r}}}}\left( {r_{\rm{e}}^{1 - \tfrac{\gamma }{2}} + r_{\rm{e}}^{1 + \tfrac{\gamma }{2}}} \right) - \frac{{\text{π}}}{{\gamma {T_{{\text{f}}u}}}}\left( {{{\left( {\xi _u^2} \right)}^{1 - \tfrac{\gamma }{2}}} + {{\left( {\xi _u^2} \right)}^{1 + \tfrac{\gamma }{2}}}} \right) - \frac{{\text{π}}}{{\gamma {T_{{\text{f}}r}}}}\left( {{{\left( {\xi _r^2} \right)}^{1 - \tfrac{\gamma }{2}}} + {{\left( {\xi _r^2} \right)}^{1 + \tfrac{\gamma }{2}}}} \right) - \frac{{\text{π}}}{{\gamma {T_{\gamma u}}}}{\left( {{\tilde {\bar \gamma}} _u^2} \right)^{1 - \tfrac{\gamma }{2}}} - \frac{{\text{π}}}{{\gamma {T_{\gamma u}}}}{\left( {{\tilde {\bar \gamma}} _u^2} \right)^{1 + \tfrac{\gamma }{2}}} - \\& \qquad \frac{{\text{π}}}{{\gamma {T_{\gamma r}}}}{\left( {{\tilde {\bar \gamma}} _r^2} \right)^{1 - \tfrac{\gamma }{2}}} - \frac{{\text{π}}}{{\gamma {T_{\gamma r}}}}{\left( {{\tilde {\bar \gamma}} _r^2} \right)^{1 + \tfrac{\gamma }{2}}} - \frac{{\text{π}}}{{{{\gamma}}{T_{{\text{w}}u}}}}\left( {{{\left( {\tilde {\boldsymbol{W}}_1^{\rm{T}}{{\tilde {\boldsymbol{W}}}_1}} \right)}^{1 - \tfrac{\gamma }{2}}} + {{\left( {\tilde {\boldsymbol{W}}_1^{\rm{T}}{{\tilde {\boldsymbol{W}}}_1}} \right)}^{1 + \tfrac{\gamma }{2}}}} \right) - \frac{{\text{π}}}{{\gamma {T_{{\text{w}}r}}}}\left( {{{\left( {\tilde {\boldsymbol{W}}_2^{\rm{T}}{{\tilde {\boldsymbol{W}}}_2}} \right)}^{1 - \tfrac{\gamma }{2}}} + {{\left( {\tilde {\boldsymbol{W}}_2^{\rm{T}}{{\tilde {\boldsymbol{W}}}_2}} \right)}^{1 + \tfrac{\gamma }{2}}}} \right) - \\& \quad\quad\quad \frac{\pi }{{\gamma {T_{{\text{h}}u}}}}\left( {{{\left( {\tilde \hbar _1^2} \right)}^{1 - \tfrac{\gamma }{2}}} + {{\left( {\tilde \hbar _1^2} \right)}^{1 + \tfrac{\gamma }{2}}}} \right) - \frac{{\text{π}}}{{\gamma {T_{{\text{h}}r}}}}\left( {{{\left( {\tilde \hbar _2^2} \right)}^{1 - \tfrac{\gamma }{2}}} + {{\left( {\tilde \hbar _2^2} \right)}^{1 + \tfrac{\gamma }{2}}}} \right) + \frac{{2{\text{π}}}}{{\gamma {T_{\gamma u}}}}{(\bar \gamma _u^2)^{1 - \tfrac{\gamma }{2}}} + \frac{{2{\text{π}}}}{{\gamma {T_{\gamma u}}}}{(\bar \gamma _u^2)^{1 + \tfrac{\gamma }{2}}} + \frac{{2{\text{π}}}}{{\gamma {T_{\gamma r}}}}{(\bar \gamma _r^2)^{1 - \tfrac{\gamma }{2}}} + \\& \quad\quad\quad\quad \frac{{2{\text{π}}}}{{\gamma {T_{\gamma r}}}}{(\bar \gamma _r^2)^{1 + \tfrac{\gamma }{2}}} + k_1'{\mathchar'26\mkern-10mu\lambda _1}{{\bar \gamma }_u} + k_2'{\mathchar'26\mkern-10mu\lambda _2}{{\bar \gamma }_r} + k_3'{\mathchar'26\mkern-10mu\lambda _3}\left| {{\hbar _1}} \right| + k_4'{\mathchar'26\mkern-10mu\lambda _4}\left| {{\hbar _2}} \right| + \frac{{2{\text{π}}}}{{\gamma {T_{{\text{h}}u}}}}\left( {\hbar _1^{2 - \gamma } + \hbar _1^{2 + \gamma }} \right) + \frac{{2{\text{π}}}}{{\gamma {T_{{\text{h}}r}}}}\left( {\hbar _2^{2 - \gamma } + \hbar _2^{2 + \gamma }} \right) + \\& \quad\quad\quad\quad\quad \frac{{\text{π}}}{{\gamma {T_{{\text{w}}u}}}}{\boldsymbol{W}}_1^{\text{T}}{{\boldsymbol{W}}_1} + \frac{{\text{π}}}{{\gamma {T_{{\text{w}}r}}}}{\boldsymbol{W}}_{\text{2}}^{\text{T}}{{\boldsymbol{W}}_2} + {\nabla _{{\text{M}}1}} + {\nabla _{{\text{M}}2}} + {\nabla _1} \leqslant - \frac{{\text{π}}}{{\gamma {T_{\text{c}}}}}\left( {{V_3}^{1 - \tfrac{\gamma }{2}} + {V_3}^{1 + \tfrac{\gamma }{2}}} \right) + \vartheta \end{split} (56) 其中,
\left\{ \begin{aligned} & {T_{\text{c}}} = \\ & \max \left\{ {{T_u},{\text{ }}{T_r},{\text{ }}{T_{{\text{f}}u}},{\text{ }}{T_{{\text{f}}r}},{\text{ }}{T_{\gamma u}},{\text{ }}{T_{\gamma r}},{\text{ }}{T_{{\text{w}}u}},{\text{ }}{T_{{\text{w}}r}},{\text{ }}{T_{{\text{h}}u}},{\text{ }}{T_{{\text{h}}r}}} \right\} \\ & \vartheta = \frac{{2{\text{π}}}}{{\gamma {T_{\gamma u}}}}{(\bar \gamma _u^2)^{1 - \tfrac{\gamma }{2}}} + \frac{{2{\text{π}}}}{{\gamma {T_{\gamma u}}}}{(\bar \gamma _u^2)^{1 + \tfrac{\gamma }{2}}} + \frac{{2{\text{π}}}}{{\gamma {T_{\gamma r}}}}{(\bar \gamma _r^2)^{1 - \tfrac{\gamma }{2}}} +\\ & \qquad \frac{{2{\text{π}}}}{{\gamma {T_{\gamma r}}}}{(\bar \gamma _r^2)^{1 + \tfrac{\gamma }{2}}} + k_1'{\mathchar'26\mkern-10mu\lambda _1}{{\bar \gamma }_u} + k_2'{\mathchar'26\mkern-10mu\lambda _2}{{\bar \gamma }_r} + k_3'{\mathchar'26\mkern-10mu\lambda _3}\left| {{\hbar _1}} \right| +\\ &\qquad k_4'{\mathchar'26\mkern-10mu\lambda _4}\left| {{\hbar _2}} \right| + \frac{{2{\text{π}}}}{{\gamma {T_{{\text{h}}u}}}}\left( {\hbar _1^{2 - \gamma } + \hbar _1^{2 + \gamma }} \right) + \\ &\qquad \frac{{2{\text{π}}}}{{\gamma {T_{{\text{h}}r}}}}\left( {\hbar _2^{2 - \gamma } + \hbar _2^{2 + \gamma }} \right) + \frac{{\text{π}}}{{\gamma {T_{{\text{w}}u}}}}{\boldsymbol{W}}_1^{\text{T}}{{\boldsymbol{W}}_1} +\\ &\qquad \frac{{\text{π}}}{{\gamma {T_{{\text{w}}r}}}}{\boldsymbol{W}}_{\text{2}}^{\text{T}}{{\boldsymbol{W}}_2} + {\nabla _{{\text{M}}1}} + {\nabla _{{\text{M}}2}} + {\nabla _1} \end{aligned}\right. (57) 根据引理1,在控制系统中,所有的信号都是预定义时间收敛到靠近原点的区域集合
\left\{ {\mathop {\lim }\limits_{t \to {T'_{\text{c}}}} x|V \leqslant \min \left\{ {{{\left( {\frac{{2\gamma {T_{\text{c}}}\vartheta }}{{\text{π}}}} \right)}^{\tfrac{2}{{2 - \gamma }}}},{{\left( {\frac{{2\gamma {T_{\text{c}}}\vartheta }}{{\text{π}}}} \right)}^{\tfrac{2}{{2 + \gamma }}}}} \right\}} \right\} 收敛时间满足:
{T_{\text{N}}} \leqslant {T_{\max }} = \sqrt 2 {T_{\text{c}}} (58) 4. 仿真结果
圆形期望轨迹:
\left\{ \begin{gathered} {{\boldsymbol{x}}_{\rm{d}}} = 80 + 50\sin (0.05t) \\ {y_{\rm{d}}} = 80 - 50\cos (0.05t) \\ \end{gathered} \right. (59) 在圆形轨迹下,选取4种初始状态:
状态1:{{\boldsymbol{\eta}}_0} = {[75,{\text{ 28}},{\text{ 0}}]^{\text{T}}},{{\boldsymbol{v}}_0} = {[0,{\text{ 0}},{\text{ 0}}]^{\text{T}}}
状态2:{{\boldsymbol{\eta}}_0} = {[77,{\text{ 27}},{\text{ 0}}{\text{.2}}]^{\text{T}}},{{\boldsymbol{v}}_0} = {[0.01,{\text{ 0}},{\text{ 0}}]^{\text{T}}}
状态3: {{\boldsymbol{\eta}}_0} = {[76,{\text{ 33}},{\text{ 0}}{\text{.1}}]^{\text{T}}},{{\boldsymbol{v}}_0} = {[0.02,{{ - 0}}{\text{.01}},{\text{ 0}}]^{\text{T}}}
状态4: {{\boldsymbol{\eta}}_0} = {[83,{\text{ 34}},{{ - 0}}{\text{.8}}]^{\text{T}}},{{\boldsymbol{v}}_0} = {[0.02,{\text{ 0}},{\text{ 0}}{\text{.01}}]^{\text{T}}}
非对称输入饱和设置为:
{\boldsymbol{\tau}} _u^ + = 500{\text{ N}} , {\boldsymbol{\tau}} _u^ - = - 400{\text{ N}}
{\boldsymbol{\tau}} _r^ + = 50{\text{ N}} \cdot {\text{m}} , {\boldsymbol{\tau}} _r^ - = - 40{\text{ N}} \cdot {\text{m}}
设置外部环境时变干扰为:
\begin{split} &\;\\[-10pt]& \left\{ \begin{aligned} & {{\boldsymbol{\tau}} _{{\text{d}}u}} = - 14 + 6\sin (0.5t)\cos (0.5t) -\\ &\qquad 8\sin (0.5t)\sin (t) - 4\cos (0.5t) \\ & {{\boldsymbol{\tau}} _{{\text{d}}v}} = 5\sin (0.1t) + 2\cos (0.3t) \\ & {{\boldsymbol{\tau}} _{{\text{d}}r}} = 6\cos (0.3t)\sin (1.1t) + 3\sin (0.5t) + 5\cos (0.2t) \end{aligned} \right. \end{split} (60) 在仿真实验中, 表1为所选择USV的动力学模型参数,本文所设置的控制参数如表2所示。
表 1 USV的动力学模型参数Table 1. The parameters of dynamic USV model参数 数值 参数 数值 参数 数值 参数 数值 {{{m_0}} /{{\text{kg}}}} 23.800 0 {X_{| u |u}}/( {{\text{kg}} \cdot {{\text{m}}^{ - 1}}} ) −1.327 4 {N_\upsilon }/( {{\text{kg}} \cdot {\text{m}} \cdot {{\text{s}}^{ - 1}}} ) 0.105 2 {N_{\dot r}}/( {{\text{kg}} \cdot {{\text{m}}^2}} ) −1.000 0 {{{{\boldsymbol{x}}_g}} / {\text{m}}} 0.046 0 {X_{uuu}}/( {{\text{kg}} \cdot {\text{s}} \cdot {{\text{m}}^{ - 2}}} ) −5.866 4 {{{N_{| \upsilon |\upsilon }}} / {{\text{kg}}}} 5.043 7 {Y_{| \upsilon |r}}/{\text{kg}} −0.845 0 {{{I_z}} / {( {{\text{kg}} \cdot {{\text{m}}^{\text{2}}}} )}} 1.760 0 {Y_\upsilon }/( {{\text{kg}} \cdot {{\text{s}}^{ - 1}}} ) −0.861 2 {N_{| r |\upsilon }}/( {{\text{kg}} \cdot {\text{m}}} ) 0.130 0 {X_u}/( {{\text{kg}} \cdot {{\text{s}}^{ - 1}}} ) −0.725 5 {X_{\dot u}}/{\text{kg}} −2.000 0 {Y_{| \upsilon |\upsilon }}/( {{\text{kg}} \cdot {{\text{m}}^{ - 1}}} ) −36.282 3 {N_r}/( {{\text{kg}} \cdot {{\text{m}}^2} \cdot {s^{ - 1}}} ) −1.900 0 {Y_{| r |r}}/( {{\text{kg}} \cdot {\text{m}}} ) −3.450 0 {Y_{\dot \upsilon }}/{\text{kg}} −10.000 0 {Y_{| r |\upsilon }}/( {{\text{kg}} \cdot {{\text{m}}^{ - 1}}} ) −0.805 0 {N_{| \upsilon |r}}/( {{\text{kg}} \cdot {\text{m}}} ) 0.080 0 {Y_{\dot r}}/( {{\text{kg}} \cdot {\text{m}}} ) 0 {Y_r}/( {{\text{kg}} \cdot {\text{m}} \cdot {{\text{s}}^{ - 1}}} ) 0.107 9 {N_{| r |r}}/( {{\text{kg}} \cdot {{\text{m}}^2}} ) −0.750 0 表 2 控制参数Table 2. Control parameters参数 数值 参数 数值 {\sigma _{u0}} 8.00 {\sigma _{u\infty }} 0.10 {\sigma _{r0}} 5.00 {\sigma _{r\infty }} 0.05 {k_1} 0.50 {k_2} 1.65 {k_u} 0.50 {k_r} 12.00 {\varUpsilon }_{1} 25.00 {\varUpsilon }_{2} 25.00 {\mathchar'26\mkern-10mu\lambda }_{1} 0.50 {\mathchar'26\mkern-10mu\lambda }_{2} 0.50 {T_{\text{h}}}/s 5.00 预定义时间参数: \gamma = 0.35 , {T_\iota } = 6{\text{ s}} , {T_{{\text{c}}\iota }} = 6{\text{ s}} , {T_{{\text{w}}\iota }} = 6{\text{ s}} , {T_{{\text{h}}\iota }} = 6{\text{ s}} , {T_{{\text{f}}\iota }} = 6{\text{ s}} , {T_{\gamma \iota }} = 6{\text{ s}} , \iota =u,r 。
SSNN网络参数为: {\varGamma _1} = 250 , {\varGamma }_{2}=250 。SSNN权值估计 {\hat{\boldsymbol{W}}}_{1} 和 {\hat{\boldsymbol{W}}}_{2} 的初始值为 {\hat {\boldsymbol{W}}_1} = {\hat {\boldsymbol{W}}_2} = 0 。SSNN初始神经元个数为N = 41,SSNN高斯函数中心均匀分布在 \left[ { - 8,{\text{ 8}}} \right] ,宽度为0.85。分裂阈值为 {\varUpsilon }_{\text{s}}=0.8 ,衰减阈值为 {\varUpsilon }_{\text{d}}=0.1 , {p}_{\text{d}}=0.3 。通过比较SSNN和RBFNN的性能,RBFNN使用与SSNN相同的参数。
根据仿真结果,USV不同初始位置的轨迹跟踪效果如图4所示。图5~图6为相应的跟踪误差。图4~图6的结果表明,尽管初始位置不同,所设计的控制系统具有预定义时间收敛能力,且预定义时间性能函数可以确保跟踪误差在规定时间({T_{\text{h}}} = 5{\text{ s}})内进入预定范围,从而确保了瞬态和稳态跟踪性能。
图7中给出了自组织神经网络的逼近效果,{F_1}和{F_2}分别为浪涌和偏航方向上的不确定性项,而{\hat {\boldsymbol{F}}_1}和{\hat {\boldsymbol{F}}_2}为SSNN逼近输出项,可以看出SSNN具有良好的逼近能力。图8显示了SSNN神经元个数的变化,神经元初始数量为N = 41,SSNN最终的神经元个数分别稳定在27和26。图9为SSNN权重估计的2−范数,图中SSNN权重的估计2−范数在3 s左右收敛,SSNN是有效的。结合图7~图9, SSNN具有良好的逼近效果,SSNN神经元个数可以在线调整,神经元个数分别减少了14和15个,有效减少了系统的计算成本。
为了说明自组织神经网络的优越性,SSNN初始神经元个数设为 N=41 ,最终稳定的神经元个数分别是27和26,因此选择41和27个神经元的RBFNN进行对比。RBFNN的参数设置为与SSNN相等的增益,高斯函数中心和宽度来进行比较。图10为不同神经网络的控制输入,图中显示了在0~6 s内使用SSNN的策略的控制输入具有较为稳定平滑的曲线,而使用RBFNN策略的控制输入变化较大,并且具有突变。图11为RBFNN和SSNN逼近误差对比效果图,与具有41和27个神经元的RBFNN相比,SSNN的逼近误差更小;从逼近效果上分析,SSNN具有更好的逼近效果,并且SSNN神经元个数可以在线调整,有效减少闭环控制系统的计算负担。
5. 结 语
本文提出了自组织神经网络预定义时间轨迹跟踪控制方案,解决了欠驱动舰艇存在未知的时变扰动、模型不确定性以及输入饱和问题,提高了闭环控制系统的跟踪精度以及系统误差的收敛速度。利用自组织神经网络补偿了未知动态、外部环境时变扰动以及输入饱和产生的影响。通过与RBFNN对比,进一步验证了SSNN方案的优越性。通过Lyapunov理论证明了欠驱动水面舰艇的轨迹跟踪控制系统的所有信号都是有界的,并且系统误差都是在预定义时间内收敛至小范围内。下一步将就如何减少控制系统的通信负担方面进行研究。
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表 1 USV的动力学模型参数
Table 1 The parameters of dynamic USV model
参数 数值 参数 数值 参数 数值 参数 数值 {{{m_0}} /{{\text{kg}}}} 23.800 0 {X_{| u |u}}/( {{\text{kg}} \cdot {{\text{m}}^{ - 1}}} ) −1.327 4 {N_\upsilon }/( {{\text{kg}} \cdot {\text{m}} \cdot {{\text{s}}^{ - 1}}} ) 0.105 2 {N_{\dot r}}/( {{\text{kg}} \cdot {{\text{m}}^2}} ) −1.000 0 {{{{\boldsymbol{x}}_g}} / {\text{m}}} 0.046 0 {X_{uuu}}/( {{\text{kg}} \cdot {\text{s}} \cdot {{\text{m}}^{ - 2}}} ) −5.866 4 {{{N_{| \upsilon |\upsilon }}} / {{\text{kg}}}} 5.043 7 {Y_{| \upsilon |r}}/{\text{kg}} −0.845 0 {{{I_z}} / {( {{\text{kg}} \cdot {{\text{m}}^{\text{2}}}} )}} 1.760 0 {Y_\upsilon }/( {{\text{kg}} \cdot {{\text{s}}^{ - 1}}} ) −0.861 2 {N_{| r |\upsilon }}/( {{\text{kg}} \cdot {\text{m}}} ) 0.130 0 {X_u}/( {{\text{kg}} \cdot {{\text{s}}^{ - 1}}} ) −0.725 5 {X_{\dot u}}/{\text{kg}} −2.000 0 {Y_{| \upsilon |\upsilon }}/( {{\text{kg}} \cdot {{\text{m}}^{ - 1}}} ) −36.282 3 {N_r}/( {{\text{kg}} \cdot {{\text{m}}^2} \cdot {s^{ - 1}}} ) −1.900 0 {Y_{| r |r}}/( {{\text{kg}} \cdot {\text{m}}} ) −3.450 0 {Y_{\dot \upsilon }}/{\text{kg}} −10.000 0 {Y_{| r |\upsilon }}/( {{\text{kg}} \cdot {{\text{m}}^{ - 1}}} ) −0.805 0 {N_{| \upsilon |r}}/( {{\text{kg}} \cdot {\text{m}}} ) 0.080 0 {Y_{\dot r}}/( {{\text{kg}} \cdot {\text{m}}} ) 0 {Y_r}/( {{\text{kg}} \cdot {\text{m}} \cdot {{\text{s}}^{ - 1}}} ) 0.107 9 {N_{| r |r}}/( {{\text{kg}} \cdot {{\text{m}}^2}} ) −0.750 0 表 2 控制参数
Table 2 Control parameters
参数 数值 参数 数值 {\sigma _{u0}} 8.00 {\sigma _{u\infty }} 0.10 {\sigma _{r0}} 5.00 {\sigma _{r\infty }} 0.05 {k_1} 0.50 {k_2} 1.65 {k_u} 0.50 {k_r} 12.00 {\varUpsilon }_{1} 25.00 {\varUpsilon }_{2} 25.00 {\mathchar'26\mkern-10mu\lambda }_{1} 0.50 {\mathchar'26\mkern-10mu\lambda }_{2} 0.50 {T_{\text{h}}}/s 5.00 -
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