Time-varying Output Constraint Control of High-order Strict-feedback Systems Based on Fully Actuated System Approach
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摘要: 针对输出受不对称时变约束的不确定高阶严反馈系统, 提出一种基于全驱系统方法的高阶自适应动态面输出约束控制方法. 所研究的高阶严反馈系统, 每个子系统都是高阶形式, 通过非线性转换函数将原输出约束系统转换为新的无约束系统, 从而将原系统输出约束问题转化为新系统输出有界的问题. 进一步结合全驱系统方法和自适应动态面控制, 直接将每个高阶子系统作为一个整体进行控制器设计, 而不需要将其转化为一阶系统形式, 有效简化了设计步骤; 同时通过引入一系列低通滤波器来获得虚拟控制律的高阶导数, 以代替复杂的微分运算. 基于Lyapunov稳定性理论证明闭环系统所有信号是一致最终有界的, 系统输出在满足约束的条件下能有效跟踪期望的参考信号, 且可通过调整参数使得系统跟踪误差收敛到零附近的足够小的邻域内. 最后, 通过对柔性关节机械臂系统进行仿真, 验证了所提出控制方法的有效性.Abstract: A high-order adaptive dynamic surface output constraint control method based on fully actuated system approach is proposed for uncertain high-order strict-feedback systems with asymmetric time-varying output constraints. The high-order strict-feedback system studied in this paper, each subsystem is a high-order form. The original output constraint system is transformed into a new unconstrained system by nonlinear transformation function, so that the original system output constraint problem is transformed into the new system output bounded problem. Furthermore, combined with the fully actuated system approach and the adaptive dynamic surface control method, the controller is designed directly for each high-order subsystem as a whole without converting it into a first-order system form, which effectively reduces the design steps. At the same time, a series of low-pass filters are introduced to obtain the high-order derivative of the virtual control law to replace the complex differential operation. Based on Lyapunov theory, it is proved that all signals of the closed-loop system are uniformly ultimately bounded. The system output can effectively track the desired reference signal without violating the constraints, and the system tracking error can be converged to any small neighborhood of the origin by adjusting the parameters. Finally, the effectiveness of the control method is verified by simulating the flexible joint manipulator system.
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实际系统由于安全和性能方面的要求, 以及一些物理器件的限制, 会不可避免地受到各种约束条件的限制, 如果约束条件得不到满足, 系统性能将可能急剧下降甚至出现不稳定. 为满足实际应用需求, 有必要研究系统在不同约束条件下的控制问题. 常见的约束条件包括: 输出约束、状态约束和输入约束等, 其中输出约束是指系统输出需满足特定的约束条件. 现有的输出约束控制方法主要有模型预测控制[1]、参考轨线调节[2] 以及基于不变集的方法[3]等, 但这些方法均基于数值计算, 计算量较大. 与这些方法不同, 基于障碍Lyapunov函数(Barrier Lyapunov function, BLF)的设计是处理输出约束问题的一种常用的方法, 其基本思想是当自变量的值趋于某些区域边界时, BLF的值趋于无穷大, 通过保证BLF的有界性, 进而使系统输出满足约束条件[4]. 经过近些年的发展, 所研究的问题由对称常值输出约束问题[4]发展到非对称时变输出约束问题[5], 应用方面也包括在直升机系统[6]、航天器交会对接[7]等领域的实际应用. 然而, 大多数基于BLF的输出约束控制方法, 需要将输出约束问题转换为系统跟踪误差约束问题, 会造成系统输出初始值较为保守的问题.
针对上述问题, 有学者提出了基于非线性转换函数的输出约束控制方法, 将输出约束问题转换为非线性转换函数的有界性问题, 即无需将其转换为跟踪误差约束问题, 设计过程较为简单, 且系统输出初值可以在整个约束区域内选取, 放宽了对系统输出初值选取的要求[8-9]. 基于这一方法, 学者们研究了严反馈系统、切换系统、随机系统、纯反馈系统、MIMO (Multiple-input and multiple-output)系统等定常输出约束以及时变输出约束控制问题[8-14], 同时在应用方面也研究了四旋翼飞行器[15]、机器人[16]等实际系统的输出约束控制问题, 在这一领域取得了较多的研究成果.
然而, 上述方法是应用一阶系统方法(状态空间方法)进行研究的. 由于牛顿定律、拉格朗日定律、角动量定律等物理定律的应用, 大多数实际系统最初被建模为二阶或高阶形式. 文献[17-20]指出, 当采用一阶系统方法设计控制器时, 需要对二阶或高阶系统降阶, 将其转化为一阶系统, 这样会破坏系统的全驱特性、增大系统阶数、增加控制器的设计步骤. 针对这一问题, 有学者提出了全驱系统方法, 该方法直接对高阶系统设计控制器而不需要降阶, 通过设计伪线性状态反馈控制律, 将闭环系统转化为具有特定结构的线性定常系统, 并基于参数化设计方法配置闭环系统的特征多项式系数矩阵实现期望的系统特性, 从而满足系统控制需求[17-20].
目前, 基于全驱系统方法, 在自适应控制、最优控制、预测控制、故障诊断、亚全驱系统控制[21-27]等领域取得了较多的研究成果, 且都具有良好的控制效果. 同时全驱系统方法也出色应用到了航天器制导、控制, 以及飞行器制导控制一体化[28-30] 等领域. 尤其针对高阶严反馈系统, 文献[31-33]基于全驱系统方法提出高阶反步法, 不需要将高阶系统转化为多个一阶子系统, 所需推导的步骤少, 过程简洁高效. 同时, 在其基础上, 文献[21-22, 34]进一步研究了高阶严反馈系统动态面、指令滤波控制等问题. 上述工作取得了较多的研究成果, 但尚未考虑系统输出约束控制问题. 文献[35]研究了高阶严反馈系统的输出约束问题, 但其研究的是基于BLF的对称常值输出约束控制问题.
由于许多实际系统的状态参数和控制参数是时变的, 因此研究时变输出约束更具有一般性. 受上述研究工作的启发, 本文研究高阶严反馈系统的非对称时变输出约束控制问题, 旨在实现直接对高阶系统设计控制器, 而不需要将其转化为一阶系统. 本文提出的高阶自适应动态面输出约束控制方法具有以下特点:
1)不同于状态空间框架下一阶严反馈系统的输出约束问题, 本文针对高阶严反馈系统, 研究其输出约束问题. 基于全驱系统方法, 直接研究高阶系统, 不需要将其转化为一阶系统, 减少了控制器设计的步骤, 从而减轻计算负担.
2)与基于BLF的输出在对称常值约束下的控制问题相比, 本文在其基础上进一步研究输出在非对称时变约束下的控制问题. 非对称时变约束相比于对称常值约束更加复杂, 更具一般性. 且与基于BLF的方法相比, 本文基于非线性转换函数的方法能够将系统输出初值可行区间扩大为整个约束区间, 避免了系统输出初始值较为保守的问题.
本文结构安排如下: 第1节是问题描述, 介绍不确定高阶严反馈系统, 并给出一些必要的假设和引理; 第2节和第3节分别给出高阶自适应动态面输出约束控制器的设计和稳定性证明, 从理论上证明系统输出约束得到满足, 且闭环系统所有信号是一致最终有界的; 第4节基于柔性关节机械臂系统进行仿真, 验证本文所提出控制方法的有效性; 最后, 在第5节给出本文的结论.
为了简化表达, 定义一些符号如下: $ {\lambda _{\min }}({P_i}) $ 和$ {\lambda _{\max }}({P_i}) $分别表示矩阵$ {P_i} $的最小和最大特征值. 矩阵$ {P_i} > 0 $表示矩阵$ {P_i} $是一个正定对称矩阵. 以及
$$ {x^{\left( {0 \sim n} \right)}} = \left[ {\begin{array}{*{20}{c}} x\\ {\dot x}\\ \vdots \\ {{x^{\left( n \right)}}} \end{array}} \right] $$ $$ x_{i \sim j}^{\left( {0 \sim n} \right)} = \left[ {\begin{array}{*{20}{c}} {x_i^{\left( {0 \sim n} \right)}}\\ {x_{i + 1}^{\left( {0 \sim n} \right)}}\\ \vdots \\ {x_j^{\left( {0 \sim n} \right)}} \end{array}} \right],\;\;j \ge i $$ $$ {A^{0 \sim n - 1}} = \left[ {\begin{array}{*{20}{c}} {{A_0} \quad {A_1} \quad \cdots \quad {A_{n - 1}}} \end{array}} \right] $$ $$ {{\Phi }}\left( {{A^{0 \sim n - 1}}} \right) = \left[ {\begin{array}{*{20}{c}} 0&{\boldsymbol{I}}&{}&{}\\ {}&{}& \ddots &{}\\ {}&{}&{}&{\boldsymbol{I}}\\ { - {A_0}}&{ - {A_1}}& \cdots &{ - {A_{n - 1}}} \end{array}} \right] $$ 1. 问题描述
考虑含有模型不确定性的高阶严反馈系统[33]
$$ \left\{ \begin{aligned} x_1^{({m_1})} = \;&{f_{10}}(x_1^{(0 \sim {m_1} - 1)}){\text{ + }} \\ &{f_1}(x_1^{(0 \sim {m_1} - 1)}){\theta _1} + \\ &{g_1}(x_1^{(0 \sim {m_1} - 1)}){x_2} \\ x_2^{({m_2})} =\;& {f_{20}}(x_i^{(0 \sim {m_i} - 1)}{|_{i = 1 \sim 2}}){\text{ + }} \\&{f_2}(x_i^{(0 \sim {m_i} - 1)}{|_{i = 1 \sim 2}}){\theta _2} \,+ \\ & {g_2}(x_i^{(0 \sim {m_i} - 1)}{|_{i = 1 \sim 2}}){x_3} \\ &\qquad\qquad\vdots \\ x_n^{({m_n})} =\;& {f_{n0}}(x_i^{(0 \sim {m_i} - 1)}{|_{i = 1 \sim n}}){\text{ + }} \\ &{f_n}(x_i^{(0 \sim {m_i} - 1)}{|_{i = 1 \sim n}}){\theta _n} \,+ \\& {g_n}(x_i^{(0 \sim {m_i} - 1)}{|_{i = 1 \sim n}})u \end{aligned} \right. $$ (1) 其中, $ {x_i},i = 1,2, \cdots ,n $为系统状态量, ${m_i},i = 1, 2, \cdots ,n$为正整数, $ u $为系统输入, $ {f_{j0}}(x_i^{(0 \sim {m_i} - 1)}{|_{i = 1 \sim j}}) $和$ {g_j}(x_i^{(0 \sim {m_i} - 1)}{|_{i = 1 \sim j}}) $, $ j = 1,2, \cdots ,n $为两组已知的光滑函数, $ {\theta _i},i = 1,2, \cdots ,n $为未知的常数向量, $ {f_j}(x_i^{(0 \sim {m_i} - 1)}{|_{i = 1 \sim j}}) $为一组已知的函数向量. 系统输出$ y = {x_1} $需在时变且不对称开区间内运行, 即
$$ \begin{split} {x_1} \in {D_1} =\;& \left\{ \left( {{x_1},t} \right) \in {\bf{R}} \times {{\bf{R}}_ + }: \right.\\&\left.- {F_1}(t) < {x_1} < {F_2}(t) \right\} \end{split} $$ (2) 其中, $ {F_1}(t) $和$ {F_2}(t) $为已知时变约束边界函数.
注 1. 在工程实际中, 根据物理定律建模得到的系统模型一般是高阶的, 因此本文研究的高阶严反馈系统(1)相比于传统一阶形式的严反馈系统更贴近于物理实际. 同时, 全驱系统方法是基于系统的全驱特性直接对原始的高阶系统模型设计控制器, 而不需要将其转化为一阶形式. 针对高阶系统(1), 本文无需将其转化为一阶形式, 直接对每个高阶子系统设计控制器, 尤其对反步法而言, 更少的设计步骤更有利于降低算法的复杂度.
本文控制目标是: 设计高阶自适应动态面控制器使得系统(1)输出在满足约束条件(2)的情况下有效跟踪期望的参考信号, 闭环系统所有信号都是一致最终有界的, 且跟踪误差能够收敛到零附近的足够小的邻域内.
为实现本文控制目标, 给出必要的假设以及引理. 首先, 系统(1)需要满足如下的全驱系统假设.
假设1[33].${g_j}(x_i^{(0\sim{m_i}-1)}{|_{i \;=\; 1\;\sim\; j}})\ne 0,\;j \;= \;1,\,2, \cdots , n$.
假设2[9, 13, 21]. 期望的参考信号$ {y_d}(t) $是光滑且可以得到的, 并满足
$$ \begin{split} &{y_d},{{\dot y}_d}, \cdots ,y_d^{({m_1} + 1)} \in {\Omega _0} = \\ &\qquad\bigg\{ {{{\left[ {\begin{array}{*{20}{c}} {{y_d}}&{{{\dot y}_d}}& \cdots &{y_d^{({m_1} + 1)}} \end{array}} \right]}^{\text{T}}}}| \\ & \qquad {y_d^2 + \dot y_d^2 + \cdots + {{(y_d^{({m_1} + 1)})}^2} \leq r_0^2} \bigg\} \end{split} $$ 其中, $ {r_0} $为一常数.
假设3[9, 13]. 约束边界函数$ {F_1}(t) $和$ {F_2}(t) $都是正的函数, 且存在常数$ {\varepsilon _F} > 0 $使得${F_2}(t) + {F_1}(t) \ge {\varepsilon _F} > 0$. 且存在满足不等式$ - {\underline \kappa _d}(t) < {\overline \kappa _d}(t) $的实变函数$ {\underline \kappa _d}(t) $, $ {\overline \kappa _d}(t) $; 以及满足不等式$ {F_2}(t) - {\overline \kappa _d}(t) \ge {\overline \varepsilon _d} $, $ {F_1}(t) - {\underline \kappa _d}(t) \ge {\underline \varepsilon _d} $的常数$ {\overline \varepsilon _d} $和$ {\underline \varepsilon _d} $. 同时, 定义集合${y_d}(t) \in {\Omega _d} = \left\{ {{y_d} \in {\bf{R}}\mid - {{\underline \kappa }_d}(t) \le {y_d} \le {{\overline \kappa }_d}(t)} \right\}$.
假设4[9, 13]. 约束边界函数$ {F_1}(t) $和$ {F_2}(t) $以及其导数$ F_1^{(i)} $和$ F_2^{(i)}\,(1 \le i \le n) $是连续且有界的.
注2. 假设1为全驱系统假设, 相比于传统一阶严反馈系统要求控制增益非零, 高阶严反馈系统也要求控制增益非零. 假设2要求期望的跟踪信号及其各阶导数是连续有界的, 在状态空间框架下的动态面控制方法同样要求跟踪信号及其各阶导数连续可导, 这个假设也是容易实现的. 假设3是为了保证约束上下边界不交叉, 且期望的跟踪信号在约束边界内, 这是符合工程实际的. 假设4要求约束边界函数及其各阶导数连续有界, 也是合理的.
本文中, 通过选取$ {A^{0 \sim n - 1}} $使得矩阵${{\Phi }}( {{A^{0 \sim n - 1}}} )$是Hurwitz的. 为了方便后续控制器设计, 给出如下引理.
引理 1[33]. 对任意$ \mu > 0 $, 存在$ {A^{0 \sim n - 1}} $使得
$$ \begin{equation} {\lambda _{\min }}\left[ {{{\Phi }}\left( {{A^{0 \sim n - 1}}} \right)} \right] < - \frac{\mu }{2} \end{equation} $$ (3) 进一步可得存在正定对称矩阵$ P\left( {{A^{0 \sim n - 1}}} \right) $, 使得
$$ \begin{split} &{\Phi ^{\text{T}}}\left( {{A^{0 \sim n - 1}}} \right)P\left( {{A^{0 \sim n - 1}}} \right) + \\ &\qquad P\left( {{A^{0 \sim n - 1}}} \right)\Phi \left( {{A^{0 \sim n - 1}}} \right) < - \mu P\left( {{A^{0 \sim n - 1}}} \right) \end{split} $$ (4) $$ \begin{equation} {\zeta _1} = \frac{{{x_1}}}{{\left( {{F_1}(t) + {x_1}} \right)\left( {{F_2}(t) - {x_1}} \right)}} \end{equation} $$ (5) 在系统输出初值在区间$ {D_1} $内确保$ {\zeta _1}\left( 0 \right) $有界的前提下, 只需设计控制器使得$ {\zeta _1} $在$ \left[ {0,\infty } \right) $内有界, 输出约束条件(2)就能始终得到满足.
对转换函数式(5)求导可得
$$ \begin{equation} {\dot \zeta _1} = {\sigma _1}{\dot x_1} + {\sigma _2} \end{equation} $$ (6) 其中
$$ \begin{equation} {\sigma _1} = \frac{{{F_1}(t){F_2}(t) + x_1^2}}{{{{\left( {{F_1}(t) + {x_1}} \right)}^2}{{\left( {{F_2}(t) - {x_1}} \right)}^2}}} \end{equation} $$ (7) $$ \begin{equation} {\sigma _2} = - \frac{{[ {{{\dot F}_1}{F_2} + {F_1}{{\dot F}_2} + ( {{{\dot F}_2} - {{\dot F}_1}} ){x_1}} ]{x_1}}}{{{{\left( {{F_1}(t) + {x_1}} \right)}^2}{{\left( {{F_2}(t) - {x_1}} \right)}^2}}} \end{equation} $$ (8) 定义
$$ \begin{equation} {x_0} = {\dot x_1} \end{equation} $$ (9) 将式(9)与式(6)结合可得
$$ \begin{equation} {\dot \zeta _1} = {\sigma _1}{x_0} + {\sigma _2} \end{equation} $$ (10) 进而将含有输出约束的原系统(1)转化为如下系统
$$ \begin{equation} \left\{ \begin{aligned} & {{\dot \zeta }_1} = {\sigma _1}{x_0} + {\sigma _2} \\ &x_0^{({m_1} - 1)} = {f_{10}}(x_1^{(0 \sim {m_1} - 1)}){\text{ + }} \\ &\;\;\,\qquad \qquad {f_1}(x_1^{(0 \sim {m_1} - 1)}){\theta _1} + \\ &\;\,\;\qquad \qquad {g_1}(x_1^{(0 \sim {m_1} - 1)}){x_2} \\ &x_2^{({m_2})} = {f_{20}}(x_i^{(0 \sim {m_i} - 1)}{|_{i = 1 \sim 2}})\;+ \\ &\qquad\;\;\; \quad {f_2}(x_i^{(0 \sim {m_i} - 1)}{|_{i = 1 \sim 2}}){\theta _2} \;+ \\ &\qquad \;\;\;\quad {g_2}(x_i^{(0 \sim {m_i} - 1)}{|_{i = 1 \sim 2}}){x_3} \\ &\qquad\qquad \qquad\qquad\vdots \\ &x_n^{({m_n})} = {f_{n0}}(x_i^{(0 \sim {m_i} - 1)}{|_{i = 1 \sim n}})\;+ \\ &\qquad \quad \;\;\; {f_n}(x_i^{(0 \sim {m_i} - 1)}{|_{i = 1 \sim n}}){\theta _n} \;+ \\ & \qquad \quad \; \;\; {g_n}(x_i^{(0 \sim {m_i} - 1)}{|_{i = 1 \sim n}})u \end{aligned} \right. \end{equation} $$ (11) 当系统输出初始值在$ {D_1} $内, 只需设计控制器使得系统(11)的第1个状态$ {\zeta _1} $有界, 则原系统(1)的输出约束就能得到满足[9, 13].
2. 控制器设计
针对输出受时变约束的不确定高阶严反馈系统(1), 基于全驱系统方法[21, 33], 设计高阶自适应动态面控制器, 主要步骤如下.
步骤1. 定义如下坐标变换
$$ \begin{equation} {z_0} = {\zeta _1} - {x_{0c}} \end{equation} $$ (12) 其中
$$ \begin{equation} {x_{0c}} = \frac{{{y_d}}}{{\left( {{F_1}(t) + {y_d}} \right)\left( {{F_2}(t) - {y_d}} \right)}} \end{equation} $$ (13) 在区间$ {\Omega _d} $内是一个连续的有界函数.
定义
$$ \begin{equation} {z_1} = {x_0} - {x_{1c}} \end{equation} $$ (14) 虚拟控制律$ {x_{1d}} $通过一系列低通滤波器[21]得到$ {x_{1c}} $
$$ \begin{equation} {\alpha _{1i}}{\dot \omega _{1i}} + {\omega _{1i}} = {\omega _{1(i - 1)}} ,i = 1,2, \cdots ,{m_1} - 1 \end{equation} $$ (15) 其中, $ {\omega _{10}} = {x_{1d}} $, $ {\omega _{1({m_1} - 1)}} = {x_{1c}} $, 设计的滤波参数${\alpha _{1i}} > 0 ,\; i = 1,2, \cdots ,{m_1} - 1$为常值, 初值${\omega _{10}}(0) = {\omega _{11}}(0) = \cdots = {\omega _{1({m_1} - 1)}}(0)$.
定义滤波误差为
$$ \begin{equation} {s_{1i}} = {\omega _{1i}} - {\omega _{1(i - 1)}},i = 1,2, \cdots ,{m_1} - 1 \end{equation} $$ (16) 可以得到
$$ \begin{equation} {x_0} = {x_{1d}} + {z_1} + \sum\limits_{i = 1}^{{m_1} - 1} {{s_{1i}}} \end{equation} $$ (17) 以及
$$ \left\{ \begin{aligned} &{{\dot s}_{11}} = - \frac{1}{{{\alpha _{11}}}}{s_{11}} - {{\dot x}_{1d}} \\ &{{\dot s}_{1i}} = - \frac{1}{{{\alpha _{1i}}}}{s_{1i}} + \frac{1}{{{\alpha _{1(i - 1)}}}}{s_{1(i - 1)}}, \\ & \quad\qquad\qquad i = 2,3, \cdots, {m_1} - 1 \end{aligned} \right. $$ (18) 选取Lyapunov函数为
$$ \begin{equation} {V_0} = \frac{1}{2}z_0^2 \end{equation} $$ (19) 对式(19)求导, 可得
$$ \begin{equation} {\dot V_0} = {z_0}{\dot z_0} = {z_0}\left( {{\sigma _1}{x_0} + {\sigma _2} - {\sigma _{d1}}{{\dot y}_d} - {\sigma _{d2}}} \right) \end{equation} $$ (20) 其中
$$ \begin{equation} {\sigma _{d1}} = \frac{{{F_1}{F_2} + y_d^2}}{{{{\left( {{F_1} + {y_d}} \right)}^2}{{\left( {{F_2} - {y_d}} \right)}^2}}} \end{equation} $$ (21) $$ \begin{equation} {\sigma _{d2}} = - \frac{{[ {{{\dot F}_1}{F_2} + {F_1}{{\dot F}_2} + ( {{{\dot F}_2} - {{\dot F}_1}} ){y_d}} ]{y_d}}}{{{{\left( {{F_1} + {y_d}} \right)}^2}{{\left( {{F_2} - {y_d}} \right)}^2}}} \end{equation} $$ (22) 取虚拟控制律为
$$ \begin{equation} {x_{1d}} = \frac{{ - {\sigma _2} + {\sigma _{d1}}{{\dot y}_d} + {\sigma _{d2}} - {k_0}{z_0}}}{{{\sigma _1}}} \end{equation} $$ (23) 其中, $ {k_0} > 0 $为一常数, 则可得
$$ \begin{equation} {\dot V_0} = - {k_0}z_0^2 + {\sigma _1}{z_0}\left( {{z_1} + \sum\limits_{i = 1}^{{m_1} - 1} {{s_{1i}}} } \right) \end{equation} $$ (24) 由式(14)可得
$$ \begin{equation} z_1^{\left( {0 \sim {m_1} - 2} \right)} = x_0^{\left( {0 \sim {m_1} - 2} \right)} - x_{1c}^{\left( {0 \sim {m_1} - 2} \right)} \end{equation} $$ (25) 定义
$$ \begin{equation} {z_2} = {x_2} - {x_{2c}} \end{equation} $$ (26) 虚拟控制律$ {x_{2d}} $通过一系列低通滤波器得到$ {x_{2c}} $
$$ \begin{equation} {\alpha _{2i}}{\dot \omega _{2i}} + {\omega _{2i}} = {\omega _{2(i - 1)}} , i = 1,2, \cdots ,{m_2} \end{equation} $$ (27) 其中, $ {\omega _{20}} = {x_{2d}} $, $ {\omega _{2{m_2}}} = {x_{2c}} $, 设计的滤波参数$ {\alpha _{2i}} > 0 , i = 1,2, \cdots ,{m_2} $为常值, 初值${\omega _{20}}\left( 0 \right) = {\omega _{21}}\left( 0 \right) = \cdots = {\omega _{2{m_2}}}\left( 0 \right)$.
定义滤波误差
$$ \begin{equation} {s_{2i}} = {\omega _{2i}} - {\omega _{2(i - 1)}},i = 1,2, \cdots ,{m_2} \end{equation} $$ (28) 可得
$$ \begin{equation} {x_2} = {x_{2d}} + {z_2} + \sum\limits_{i = 1}^{{m_2}} {{s_{2i}}} \end{equation} $$ (29) 以及
$$ \begin{equation} \left\{ \begin{aligned} &{{\dot s}_{21}} = - \frac{1}{{{\alpha _{21}}}}{s_{21}} - {{\dot x}_{2d}}\\ &{{\dot s}_{2i}} = - \frac{1}{{{\alpha _{2i}}}}{s_{2i}} + \frac{1}{{{\alpha _{2(i - 1)}}}}{s_{2(i - 1)}}, i = 2,3, \cdots, {m_2} \end{aligned} \right. \end{equation} $$ (30) 设$ {\hat{\theta} _1} $为$ {\theta _1} $的估计值, 定义估计误差为
$$ \begin{equation} {\tilde \theta _1}= {\hat{\theta} _1} - {\theta _1} \end{equation} $$ (31) 进一步可得
$$ \begin{equation} \begin{split} z_1^{\left( {{m_1} - 1} \right)} =\;& x_0^{({m_1} - 1)} - x_{1c}^{\left( {{m_1} - 1} \right)} = \\ &{f_{10}}{\text{ + }}{f_1}{{\hat{\theta} }_1} + {g_1}\left( {{x_{2d}} + {z_2} + \sum\limits_{i = 1}^{{m_2}} {{s_{2i}}} } \right) - \\ & {f_1}{{\tilde \theta }_1} - x_{1c}^{\left( {{m_1} - 1} \right)} \end{split} \end{equation} $$ (32) 取虚拟控制律为
$$ \begin{equation} \begin{split} {x_{2d}} = \;&\frac{1}{{{g_1}}}\left( { - {f_{10}} - {f_1}{{\hat{\theta} }_1}\; - } \right. \\ &\left. {A_1^{0 \sim {m_1} - 2}z_1^{\left( {0 \sim {m_1} - 2} \right)} + x_{1c}^{\left( {{m_1} - 1} \right)}} \right) \end{split} \end{equation} $$ (33) 其中, $ A_1^{0 \sim {m_1} - 2} $为待设计参数. 结合式(32)和式(33), 可得
$$ \begin{equation} \begin{split} &z_1^{\left( {{m_1} - 1} \right)} + A_1^{0 \sim {m_1} - 2}z_1^{\left( {0 \sim {m_1} - 2} \right)} = \\ &\qquad{g_1}\left( {{z_2} + \sum\limits_{i = 1}^{{m_2}} {{s_{2i}}} } \right) - {f_1}{{\tilde \theta }_1} \end{split} \end{equation} $$ (34) 定义
$$ \begin{equation} {V_{z1}} = {\left( {z_1^{\left( {0 \sim {m_1} - 2} \right)}} \right)^{\rm{T}}}{P_1}z_1^{\left( {0 \sim {m_1} - 2} \right)} \end{equation} $$ (35) 其中, 矩阵$ {P_1} $为引理1中描述的正定矩阵.
对式(35)求导, 结合式(34), 并根据引理1可得
$$ \begin{split} {{\dot V}_{z1}} = \;&{\left( {\dot z_1^{\left( {0 \sim {m_1} - 2} \right)}} \right)^{\text{T}}}{P_1}z_1^{\left( {0 \sim {m_1} - 2} \right)}{\text{ + }} \\ &{\left( {z_1^{\left( {0 \sim {m_1} - 2} \right)}} \right)^{\text{T}}}{P_1}\dot z_1^{\left( {0 \sim {m_1} - 2} \right)}{\text{ = }} \\ &{\left( {z_1^{\left( {0 \sim {m_1} - 2} \right)}} \right)^{\text{T}}}(\Phi _1^{\text{T}}{P_1} + {P_1}{\Phi _1})z_1^{\left( {0 \sim {m_1} - 2} \right)}\;- \end{split} $$ $$ \begin{split} & 2{\left( {z_1^{\left( {0 \sim {m_1} - 2} \right)}} \right)^{\text{T}}}{P_{1L}}{f_1}{{\tilde \theta }_1} + \\ &2{\left( {z_1^{\left( {0 \sim {m_1} - 2} \right)}} \right)^{\text{T}}}{P_{1L}}{g_1}({z_2} + \sum\limits_{i = 1}^{{m_2}} {{s_{2i}}} ) \leq \\ &- {\mu _1}{\left( {z_1^{\left( {0 \sim {m_1} - 2} \right)}} \right)^{\text{T}}}{P_1}z_1^{\left( {0 \sim {m_1} - 2} \right)} \;- \\ &2{\left( {z_1^{\left( {0 \sim {m_1} - 2} \right)}} \right)^{\text{T}}}{P_{1L}}{f_1}{{\tilde \theta }_1} \;+ \\ &2{\left( {z_1^{\left( {0 \sim {m_1} - 2} \right)}} \right)^{\text{T}}}{P_1}\left[ \begin{gathered} 0 \\ {g_1} \end{gathered} \right]\left( {{z_2} + \sum\limits_{i = 1}^{{m_2}} {{s_{2i}}} } \right) \end{split} $$ (36) 式中, 下标$L $表示相应矩阵的最后一列.
对式(31)求导, 可得
$$ \begin{equation} \dot{\tilde {\theta}} _1 = \dot{\hat{\theta}} _1 \end{equation} $$ (37) 定义
$$ \begin{equation} {V_{{\theta _1}}} = \tilde \theta _1^{\text{T}}{\tilde \theta _1} \end{equation} $$ (38) 以及
$$ \begin{equation} {V_1} = {V_{z1}} + {V_{\theta 1}} \end{equation} $$ (39) 对式(39)求导, 可得
$$ \begin{equation} \begin{split} {{\dot V}_1} \leq\;& - {\mu _1}{\left( {z_1^{\left( {0 \sim {m_1} - 2} \right)}} \right)^{\text{T}}}{P_1}z_1^{\left( {0 \sim {m_1} - 2} \right)}{\text{ + }} \\ & 2\tilde \theta _1^{\text{T}}\left[ {{{\dot {\hat{\theta}} }_1} - {{\left( {z_1^{\left( {0 \sim {m_1} - 2} \right)}} \right)}^{\text{T}}}{P_{1L}}{f_1}} \right] + \\ & 2{\left( {z_1^{\left( {0 \sim {m_1} - 2} \right)}} \right)^{\text{T}}}{P_1}\left[ \begin{gathered} 0 \\ {g_1} \end{gathered} \right]\left( {{z_2} + \sum\limits_{i = 1}^{{m_2}} {{s_{2i}}} } \right) \end{split} \end{equation} $$ (40) 取自适应律为
$$ \begin{equation} \dot{\hat{\theta}} _1{\rm{ = }}{\left( {z_1^{\left( {0 \sim {m_1} - 2} \right)}} \right)^{\rm{T}}}{P_{1L}}{f_1} - {\gamma _1}{\hat{\theta} _1} \end{equation} $$ (41) 其中, 常值$ {\gamma _1} > 0 $为设计参数, 进一步可得
$$ \begin{equation} \begin{split} &{{\dot V}_1} \leq - {\mu _1}{\left( {z_1^{\left( {0 \sim {m_1} - 2} \right)}} \right)^{\text{T}}}{P_1}z_1^{\left( {0 \sim {m_1} - 2} \right)} - 2{\gamma _1}\tilde \theta _1^{\text{T}}{{\hat{\theta} }_1} \;+ \\ &\qquad\quad 2{\left( {z_1^{\left( {0 \sim {m_1} - 2} \right)}} \right)^{\text{T}}}{P_1}\left[ \begin{gathered} 0 \\ {g_1} \end{gathered} \right]\left( {{z_2} + \sum\limits_{i = 1}^{{m_2}} {{s_{2i}}} } \right) \\[-15pt] \end{split} \end{equation} $$ (42) 步骤 i (${\boldsymbol{i = 2,3, \cdots ,n - 1 }}$).
定义
$$ \begin{equation} {z_{i + 1}} = {x_{i + 1}} - {x_{\left( {i + 1} \right)c}} \end{equation} $$ (43) 其中, 虚拟控制律$ {x_{\left( {i + 1} \right)d}} $通过一系列低通滤波器得到$ {x_{\left( {i + 1} \right)c}} $
$$ \begin{split} &{\alpha _{\left( {i + 1} \right)j}}{{\dot \omega }_{\left( {i + 1} \right)j}} + {\omega _{\left( {i + 1} \right)j}} = {\omega _{\left( {i + 1} \right)\left( {j - 1} \right)}}, \\ &\qquad\qquad\qquad\qquad\qquad j = 1,2, \cdots ,{m_{i + 1}} \end{split} $$ (44) 其中, $ {\omega _{(i + 1)0}} = {x_{(i + 1)d}} $, $ {\omega _{(i + 1){m_{i + 1}}}} = {x_{(i + 1)c}} $, 设计的滤波参数$ {\alpha _{\left( {i + 1} \right)j}} > 0 , j = 1,2, \cdots ,{m_{i + 1}} $为常值, 初值${\omega _{\left( {i + 1} \right)0}}(0) \;=\; {\omega _{\left( {i + 1} \right)1}}(0) = \cdots = {\omega _{\left( {i + 1} \right){m_{i + 1}}}}(0)$. 定义滤波误差为
$$ \begin{equation} {s_{\left( {i + 1} \right)j}} = {\omega _{\left( {i + 1} \right)j}} - {\omega _{\left( {i + 1} \right)(j - 1)}} ,j = 1,2, \cdots ,{m_{i + 1}} \end{equation} $$ (45) 进一步可得
$$ \begin{equation} {x_{i + 1}} = {x_{\left( {i + 1} \right)d}} + {z_{i + 1}} + \sum\limits_{j = 1}^{{m_{i + 1}}} {{s_{\left( {i + 1} \right)j}}} \end{equation} $$ (46) 以及
$$ \begin{equation} \left\{ \begin{aligned} {{\dot s}_{\left( {i + 1} \right)1}} = \;&- \frac{1}{{{\alpha _{\left( {i + 1} \right)1}}}}{s_{\left( {i + 1} \right)1}} - {{\dot x}_{\left( {i + 1} \right)d}}\\ {{\dot s}_{\left( {i + 1} \right)j}} =\;& - \frac{1}{{{\alpha _{\left( {i + 1} \right)j}}}}{s_{\left( {i + 1} \right)j}} + \frac{1}{{{\alpha _{\left( {i + 1} \right)(j - 1)}}}}{s_{\left( {i + 1} \right)(j - 1)}}, \\ & \qquad \qquad\qquad j = 2,3, \cdots,{m_{i + 1}} \end{aligned} \right. \end{equation} $$ (47) 设$ {\hat{\theta} _i} $为对$ {\theta _i} $的估计, 定义估计误差为
$$ \begin{equation} {\tilde \theta _i}={\hat{\theta} _i} - {\theta _i} \end{equation} $$ (48) 进一步可得
$$ \begin{equation} \begin{split} &z_i^{\left( {{m_i}} \right)} = x_i^{\left( {{m_i}} \right)} - x_{ic}^{\left( {{m_i}} \right)} = \\ &\qquad{f_{i0}}{\text{ + }}{f_i}{{\hat{\theta} }_i} + {g_i}\left( {{x_{\left( {i + 1} \right)d}} + {z_{i + 1}} + \sum\limits_{j = 1}^{{m_{i + 1}}} {{s_{\left( {i + 1} \right)j}}} } \right) - \\ &\qquad {f_i}{{\tilde \theta }_i} - x_{ic}^{\left( {{m_i}} \right)} \\[-10pt] \end{split} \end{equation} $$ (49) 取虚拟控制律为
$$ \begin{equation} \begin{split} {x_{\left( {i + 1} \right)d}} =\;& \frac{1}{{{g_i}}}\left( { - {f_{i0}} - {f_i}{{\hat{\theta} }_i}\; - } \right. \\ &A_i^{0 \sim {m_i} - 1}z_i^{\left( {0 \sim {m_i} - 1} \right)} + x_{ic}^{\left( {{m_i}} \right)} \Big) \end{split} \end{equation} $$ (50) 其中, $ A_i^{0 \sim {m_i} - 1} $为设计参数. 进一步可得
$$ \begin{equation} \begin{split} &z_i^{\left( {{m_i}} \right)} + A_i^{0 \sim {m_i} - 1}z_i^{\left( {0 \sim {m_i} - 1} \right)} = \\ &\qquad{g_i}\left( {{z_{i + 1}} + \sum\limits_{j = 1}^{{m_{i + 1}}} {{s_{\left( {i + 1} \right)j}}} } \right) - {f_i}{{\tilde \theta }_i} \end{split} \end{equation} $$ (51) 定义
$$ \begin{equation} {V_{zi}} = {\left( {z_i^{\left( {0 \sim {m_i} - 1} \right)}} \right)^{\rm{T}}}{P_i}z_i^{\left( {0 \sim {m_i} - 1} \right)} \end{equation} $$ (52) 对式(52)求导, 可得
$$ \begin{split} {{\dot V}_{zi}} \leq \;& - {\mu _i}{\left( {z_i^{\left( {0 \sim {m_i} - 1} \right)}} \right)^{\text{T}}}{P_i}z_i^{\left( {0 \sim {m_i} - 1} \right)}\; - \\ &2{\left( {z_i^{\left( {0 \sim {m_i} - 1} \right)}} \right)^{\text{T}}}{P_{iL}}{f_i}{{\tilde \theta }_i} \;+ \\ &2{\left( {z_i^{\left( {0 \sim {m_i} - 1} \right)}} \right)^{\text{T}}}{P_i}\left[ \begin{gathered} 0 \\ {g_i} \end{gathered} \right]\left( {{z_{i + 1}} + \sum\limits_{j = 1}^{{m_{i + 1}}} {{s_{\left( {i + 1} \right)j}}} } \right) \end{split} $$ (53) 对式(48)求导, 可得
$$ \begin{equation} \dot{\tilde {\theta}} _i = \dot{\hat{\theta}} _i \end{equation} $$ (54) 定义
$$ \begin{equation} {V_{{\theta _i}}} = \tilde \theta _i^{\text{T}}{\tilde \theta _i} \qquad \end{equation} $$ (55) $$ \begin{equation} {V_i} = {V_{zi}} + {V_{\theta i}} \; \end{equation} $$ (56) 取自适应律为
$$ \begin{equation} \dot{\hat{\theta}} _i = {\left( {z_i^{\left( {0 \sim {m_i} - 1} \right)}} \right)^{\rm{T}}}{P_{iL}}{f_i} - {\gamma _i}{\hat{\theta} _i} \end{equation} $$ (57) 其中, $ {\gamma _i} > 0 $为设计参数. 进一步可得
$$ \begin{split} {{\dot V}_i} \leq\;& - {\mu _i}{\left( {z_i^{\left( {0 \sim {m_i} - 1} \right)}} \right)^{\text{T}}}{P_i}z_i^{\left( {0 \sim {m_i} - 1} \right)} - 2{\gamma _i}\tilde \theta _i^{\text{T}}{{\hat{\theta} }_i} \;+ \\ &2{\left( {z_i^{\left( {0 \sim {m_i} - 1} \right)}} \right)^{\text{T}}}{P_i}\left[ \begin{gathered} 0 \\ {g_i} \end{gathered} \right]\left( {{z_{i + 1}} + \sum\limits_{j = 1}^{{m_{i + 1}}} {{s_{\left( {i + 1} \right)j}}} } \right) \end{split} $$ (58) 步骤 n.
设$ {\hat{\theta} _n} $为对$ {\theta _n} $的估计, 定义估计误差为
$$ \begin{equation} {\tilde \theta _n}={\hat{\theta} _n} - {\theta _n} \end{equation} $$ (59) 进一步可得
$$ \begin{equation} \begin{split} z_n^{({m_n})} =\;& x_n^{({m_n})} - x_{nc}^{({m_n})} = \\ &{f_{n0}} + {f_n}{{\hat{\theta} }_n} + {g_n}u - {f_n}{{\tilde \theta }_n} - x_{nc}^{({m_n})} \end{split} \end{equation} $$ (60) 取控制律为
$$ \begin{split} u = \;&\frac{1}{{{g_n}}}\left( { - {f_{n0}} - {f_n}{{\hat{\theta} }_n} \;- } \right. \\ &\left. {A_n^{0 \sim {m_n} - 1}z_n^{\left( {0 \sim {m_n} - 1} \right)} + x_{nc}^{\left( {{m_n}} \right)}} \right) \end{split} $$ (61) 其中, $ A_n^{0 \sim {m_n} - 1} $为设计参数, 可得
$$ \begin{equation} z_n^{\left( {{m_n}} \right)} + A_n^{0 \sim {m_n} - 1}z_n^{\left( {0 \sim {m_n} - 1} \right)} = - {f_n}{\tilde \theta _n} \end{equation} $$ (62) 定义
$$ \begin{equation} {V_{zn}} = {\left( {z_n^{\left( {0 \sim {m_n} - 1} \right)}} \right)^{\rm{T}}}{P_n}z_n^{\left( {0 \sim {m_n} - 1} \right)} \end{equation} $$ (63) 对式(63)求导, 可得
$$ \begin{equation} \begin{split} {{\dot V}_{zn}} \leq \;& - {\mu _n}{\left( {z_n^{\left( {0 \sim {m_n} - 1} \right)}} \right)^{\text{T}}}{P_n}z_n^{\left( {0 \sim {m_n} - 1} \right)}\; - \\ & 2{\left( {z_n^{\left( {0 \sim {m_n} - 1} \right)}} \right)^{\text{T}}}{P_{nL}}{f_n}{{\tilde \theta }_n} \end{split} \end{equation} $$ (64) 对式(59)求导, 可得
$$ \begin{equation} \dot{\tilde {\theta}} _n = \dot{\hat{\theta}} _n \end{equation} $$ (65) 定义
$$ \begin{equation} {V_{{\theta _n}}} = \tilde \theta _n^{\text{T}}{\tilde \theta _n} \end{equation} $$ (66) 以及
$$ \begin{equation} {V_n} = {V_{zn}} + {V_{\theta n}} \end{equation} $$ (67) 取自适应律为
$$ \begin{equation} \dot{\hat{\theta}} _n = {\left( {z_n^{\left( {0 \sim {m_n} - 1} \right)}} \right)^{\rm{T}}}{P_{nL}}{f_n} - {\gamma _n}{\hat{\theta} _n} \end{equation} $$ (68) 其中, $ {\gamma _n} > 0 $为设计参数, 可得
$$ \begin{equation} {\dot V_n} \leq - {\mu _n}{\left( {z_n^{\left( {0 \sim {m_n} - 1} \right)}} \right)^{\text{T}}}{P_n}z_n^{\left( {0 \sim {m_n} - 1} \right)} - 2{\gamma _n}\tilde \theta _n^{\text{T}}{\hat{\theta} _n} \end{equation} $$ (69) 3. 稳定性证明
结合第2节中的控制器设计过程, 给出定理1, 并根据Lyapunov理论给出其证明.
定理 1. 考虑满足假设1 ~ 4下的具有时变输出约束的高阶严反馈系统(1). 由虚拟控制律(23), (33), (50), 控制律(61), 自适应律(41), (57), (68), 组成的高阶自适应动态面输出约束控制, 能够使闭环系统所有信号是一致最终有界的, 系统输出满足时变约束条件, 并能有效跟踪期望的参考信号, 且跟踪误差可通过调整参数收敛到零附近足够小的邻域内.
证明.
定义
$$ \begin{equation} {\bar V_n} = \sum\limits_{i = 0}^n {{V_i}} \end{equation} $$ (70) 则可以得到
$$ \begin{split} {\dot{ \bar {V}}_n} \leq \;& - {k_0}z_0^2 + {\sigma _1}{z_0}\left( {{z_1} + \sum\limits_{i = 1}^{{m_1} - 1} {{s_{1i}}} } \right) - \\ &{\mu _1}{\left( {z_1^{\left( {0 \sim {m_1} - 2} \right)}} \right)^{\text{T}}}{P_1}z_1^{\left( {0 \sim {m_1} - 2} \right)} - \sum\limits_{i = 1}^n {2{\gamma _i}\tilde \theta _i^{\text{T}}{{\hat{\theta} }_i} \;- } \\ &\sum\limits_{i = 2}^n {\left( {{\mu _i}{{\left( {z_i^{\left( {0 \sim {m_i} - 1} \right)}} \right)}^{\text{T}}}{P_i}z_i^{\left( {0 \sim {m_i} - 1} \right)}} \right)} + \\ &\sum\limits_{i = 2}^{n - 1} 2{{\left( {z_i^{\left( {0 \sim {m_i} - 1} \right)}} \right)}^{\text{T}}}{P_i}\left[ \begin{gathered} 0 \\ {g_i} \end{gathered} \right]\times\\ &\left( {{z_{i + 1}} + \sum\limits_{j = 1}^{{m_{i + 1}}} {{s_{\left( {i + 1} \right)j}}} } \right) + \\ &2{\left( {z_1^{\left( {0 \sim {m_1} - 2} \right)}} \right)^{\text{T}}}{P_1}\left[ \begin{gathered} 0 \\ {g_1}\end{gathered} \right]\left( {{z_2} + \sum\limits_{i = 1}^{{m_2}} {{s_{2i}}} } \right) \\[-15pt]\end{split} $$ (71) 由假设3以及式(7)可知, $ {\sigma _1} > 0 $, 根据Young＇s不等式可得
$$ \begin{equation} {\sigma _1}{z_0}\left( {{z_1} + \sum\limits_{i = 1}^{{m_1} - 1} {{s_{1i}}} } \right) \le \frac{{{\sigma _1}}}{2}\left( {z_1^2+{m_1}z_0^2+\sum\limits_{i = 1}^{{m_1} - 1} {s_{1i}^2} } \right) \end{equation} $$ (72) 结合式(71)和式(72), 可得
$$ \begin{equation} \begin{split} {\dot{ \bar {V}}_n} \leq \;& - \left( {{k_0} - \frac{{{m_1}{\sigma _1}}}{2}} \right)z_0^2 \;+ \\ & \frac{{{\sigma _1}}}{{2{\lambda _{\min }}\left( {{P_1}} \right)}}{\left( {z_1^{\left( {0 \sim {m_1} - 2} \right)}} \right)^{\text{T}}}{P_1}z_1^{\left( {0 \sim {m_1} - 2} \right)} \;+ \\ & \frac{{{\sigma _1}}}{2}\sum\limits_{i = 1}^{{m_1} - 1} {s_{1i}^2} - {\mu _1}{\left( {z_1^{\left( {0 \sim {m_1} - 2} \right)}} \right)^{\text{T}}}{P_1}z_1^{\left( {0 \sim {m_1} - 2} \right)} \;- \\ & \sum\limits_{i = 1}^n {2{\gamma _i}\tilde \theta _i^{\text{T}}{{\hat{\theta} }_i}} \;-\\ &\sum\limits_{i = 2}^n {\left( {{\mu _i}{{\left( {z_i^{\left( {0 \sim {m_i} - 1} \right)}} \right)}^{\text{T}}}{P_i}z_i^{\left( {0 \sim {m_i} - 1} \right)}} \right)} \; + \\ & \sum\limits_{i = 2}^{n - 1} 2{{\left( {z_i^{\left( {0 \sim {m_i} - 1} \right)}} \right)}^{\text{T}}}{P_i}\left[ \begin{gathered} 0 \\ {g_i} \end{gathered} \right]\times\\ &\left( {{z_{i + 1}} + \sum\limits_{j = 1}^{{m_{i + 1}}} {{s_{\left( {i + 1} \right)j}}} } \right) + \\ & 2{\left( {z_1^{\left( {0 \sim {m_1} - 2} \right)}} \right)^{\text{T}}}{P_1}\left[ \begin{gathered} 0 \\ {g_1} \end{gathered} \right]\left( {{z_2} + \sum\limits_{i = 1}^{{m_2}} {{s_{2i}}} } \right) \\[-18pt] \end{split} \end{equation} $$ (73) 且有如下不等式成立
$$ \begin{equation} \begin{split} &{\left( {z_i^{\left( {0 \sim {m_i} - 1} \right)}} \right)^{\text{T}}}{P_i}\left[ \begin{gathered} 0 \\ {g_i} \end{gathered} \right]\left( {{z_{i + 1}} + \sum\limits_{j = 1}^{{m_{i + 1}}} {{s_{\left( {i + 1} \right)j}}} } \right) = \\ &\;\;\;{\left( {z_i^{\left( {0 \sim {m_i} - 1} \right)}} \right)^{\text{T}}}P_i^{\frac{1}{2}}P_i^{\frac{1}{2}}\left[ \begin{gathered} 0 \\ {g_i} \end{gathered} \right]\left( {{z_{i + 1}} + \sum\limits_{j = 1}^{{m_{i + 1}}} {{s_{\left( {i + 1} \right)j}}} } \right) \leq \\ &\;\;\;\frac{1}{2}\left( {{m_{i + 1}} + 1} \right){\left\| {P_i^{\frac{1}{2}}z_i^{\left( {0 \sim {m_i} - 1} \right)}} \right\|^2}\; + \\ &\;\;\;\frac{1}{2}g_i^2{\left\| {P_i^{\frac{1}{2}}} \right\|^2}\left( {z_{i + 1}^2 + \sum\limits_{j = 1}^{{m_i} + 1} {s_{\left( {i + 1} \right)j}^2} } \right) \leq \\ &\;\;\;\frac{1}{2}\left( {{m_{i + 1}} + 1} \right){\left( {z_i^{\left( {0 \sim {m_i} - 1} \right)}} \right)^{\text{T}}}{P_i}z_i^{\left( {0 \sim {m_i} - 1} \right)}\; + \\ &\;\;\;\frac{1}{2}{\lambda _{\max }}\left( {{P_i}} \right)g_i^2\left( {{{\left\| {z_{i + 1}^{\left( {0 \sim {m_{i + 1}} - 1} \right)}} \right\|}^2} + \sum\limits_{j = 1}^{{m_i} + 1} {s_{\left( {i + 1} \right)j}^2} } \right) \end{split} \end{equation} $$ (74) 定义
$$ \begin{equation} {V_{s1}} = \frac{1}{2}\sum\limits_{i = 1}^{{m_1} - 1} {s_{1i}^2} \end{equation} $$ (75) 以及
$$ \begin{equation} {V_{si}} = \frac{1}{2}\sum\limits_{j = 1}^{{m_i}} {s_{ij}^2} , i = 2,3, \cdots ,n \end{equation} $$ (76) 进一步可得
$$ \begin{equation} \left\{ \begin{aligned} &{{\dot V}_{s1}} \leq - \frac{1}{{2{\alpha _{11}}}}s_{11}^2 - \sum\limits_{i = 2}^{{m_1} - 2} {\left[ {\left( {\frac{1}{{2{\alpha _{1i}}}} - \frac{1}{{2{\alpha _{1\left( {i - 1} \right)}}}}} \right)s_{1i}^2} \right] - } \\ &\qquad\left( {\frac{1}{{{\alpha _{1\left( {{m_1} - 1} \right)}}}} - \frac{1}{{2{\alpha _{1\left( {{m_1} - 2} \right)}}}}} \right)s_{1\left( {{m_1} - 1} \right)}^2 - {s_{11}}{{\dot x}_{1d}} \\ &{{\dot V}_{si}} \leq - \frac{1}{{2{\alpha _{i1}}}}s_{i1}^2 - \sum\limits_{j = 2}^{{m_i} - 1} {\left[ {\left( {\frac{1}{{2{\alpha _{ij}}}} - \frac{1}{{2{\alpha _{i\left( {j - 1} \right)}}}}} \right)s_{ij}^2} \right] - } \\ &\qquad\left( {\frac{1}{{{\alpha _{i{m_i}}}}} - \frac{1}{{2{\alpha _{i\left( {{m_i} - 1} \right)}}}}} \right)s_{i{m_i}}^2 - {s_{i1}}{{\dot x}_{id}} , \\&\qquad \qquad\qquad \qquad \qquad\qquad \quad \quad\quad i = 2,3, \cdots ,n \end{aligned} \right. \end{equation} $$ (77) 定义
$$ \begin{equation} V = {\bar V_n} + \sum\limits_{i = 1}^n {{V_{si}}} \end{equation} $$ (78) 且结合如下不等式
$$ \begin{equation} - 2{\gamma _i}\tilde \theta _i^{\text{T}}{\theta _i} < {\gamma _i}\tilde \theta _i^{\text{T}}{\tilde \theta _i} + {\gamma _i}\theta _i^{\text{T}}{\theta _i} , i = 1,2, \cdots ,n \end{equation} $$ (79) 可得
$$ \begin{equation} \begin{split} \dot V \leq \;& - \left( {{k_0} - \frac{{{m_1}{\sigma _1}}}{2}} \right)z_0^2 \;+ \\ &\frac{{{\sigma _1}}}{{2{\lambda _{\min }}\left( {{P_1}} \right)}}{\left( {z_1^{\left( {0 \sim {m_1} - 2} \right)}} \right)^{\text{T}}}{P_1}z_1^{\left( {0 \sim {m_1} - 2} \right)}\; + \\ &\frac{{{\sigma _1}}}{2}\sum\limits_{i = 1}^{{m_1} - 1} {s_{1i}^2} - {\mu _1}{\left( {z_1^{\left( {0 \sim {m_1} - 2} \right)}} \right)^{\text{T}}}{P_1}z_1^{\left( {0 \sim {m_1} - 2} \right)} \;- \\ &\sum\limits_{i = 1}^n {{\gamma _i}\tilde \theta _i^{\text{T}}{{\tilde \theta }_i}} + \sum\limits_{i = 1}^n {{\gamma _i}\theta _i^{\text{T}}{\theta _i}} - \sum\limits_{i = 2}^n {{s_{i1}}} {{\dot x}_{id}}\; - \\ &\sum\limits_{i = 2}^n {\left( {{\mu _i}{{\left( {z_i^{\left( {0 \sim {m_i} - 1} \right)}} \right)}^{\text{T}}}{P_i}z_i^{\left( {0 \sim {m_i} - 1} \right)}} \right)} \; - \\ &\sum\limits_{i = 2}^n {\frac{1}{{2{\alpha _{i1}}}}} s_{i1}^2\; + \\ &\left( {{m_2} + 1} \right){\left( {z_1^{\left( {0 \sim {m_1} - 2} \right)}} \right)^{\text{T}}}{P_1}z_1^{\left( {0 \sim {m_1} - 2} \right)} - {s_{11}}{{\dot x}_{1d}} \;+ \\ &\sum\limits_{i = 2}^{n - 1} {\left( {{m_{i + 1}} + 1} \right){{\left( {z_i^{\left( {0 \sim {m_i} - 1} \right)}} \right)}^{\text{T}}}{P_i}z_i^{\left( {0 \sim {m_i} - 1} \right)}} \;+ \\ &{\lambda _{\max }}\left( {{P_1}} \right)g_1^2\left( {{{\left\| {z_2^{\left( {0 \sim {m_2} - 1} \right)}} \right\|}^2} + \sum\limits_{j = 1}^{{m_2}} {s_{2j}^2} } \right) + \\ &\sum\limits_{i = 2}^{n - 1} {\lambda _{\max }}\left( {{P_i}} \right)g_i^2\;\times\\ &\left( {{{\left\| {z_{i + 1}^{\left( {0 \sim {m_{i + 1}} - 1} \right)}} \right\|}^2} + \sum\limits_{j = 1}^{{m_i} + 1} {s_{\left( {i + 1} \right)j}^2} } \right) - \end{split} \end{equation} $$ $$ \begin{equation} \begin{split} &\qquad\frac{1}{{2{\alpha _{11}}}}s_{11}^2 - \sum\limits_{i = 2}^{{m_1} - 2} {\left[ {\left( {\frac{1}{{2{\alpha _{1i}}}} - \frac{1}{{2{\alpha _{1\left( {i - 1} \right)}}}}} \right)s_{1i}^2} \right]} \; - \\ &\qquad\left( {\frac{1}{{{\alpha _{1\left( {{m_1} - 1} \right)}}}} - \frac{1}{{2{\alpha _{1\left( {{m_1} - 2} \right)}}}}} \right)s_{1\left( {{m_1} - 1} \right)}^2 \;- \\ &\qquad\sum\limits_{i = 2}^n {\sum\limits_{j = 2}^{{m_i} - 1} {\left[ {\left( {\frac{1}{{2{\alpha _{ij}}}} - \frac{1}{{2{\alpha _{i\left( {j - 1} \right)}}}}} \right)s_{ij}^2} \right]} } \;- \\ &\qquad\sum\limits_{i = 2}^n {\left( {\frac{1}{{{\alpha _{i{m_i}}}}} - \frac{1}{{2{\alpha _{i\left( {{m_i} - 1} \right)}}}}} \right)s_{i{m_i}}^2} \\[-15pt] \end{split} \end{equation} $$ (80) 经计算可知, $ {\dot x_{id}}, \forall i \in \left\{ {1,2,3, \cdots ,n} \right\} $是变量$ {F_1} $, $ {F_2} $, $ {\dot F_1} $, $ {\dot F_2} $, $ {z_0} $, $ z_j^{\left( {0 \sim {m_j} - 1} \right)}, j = 1,2, \cdots ,i - 1 $, ${s_{11}}, \cdots, {s_{1({m_1} - 1)}},\;{s_{21}}, \cdots ,{s_{2{m_2}}}, \cdots ,\;{s_{i1}}, \cdots ,{s_{i{m_i}}}$, $ y_d^{(0 \sim {m_i} + 1)} $, $ {\tilde \theta _j}, j = 1,2, \cdots ,i - 1 $的函数, 且依赖于设计参数${\alpha _{11}}, \cdots, {\alpha _{1({m_1} - 1)}},{\alpha _{21}}, \cdots ,{\alpha _{2{m_2}}}, \cdots ,{\alpha _{i1}}, \cdots ,{\alpha _{i{m_i}}}$, $ A_j^{\left( {0 \sim {m_j} - 1} \right)} $, $ {P_j}, j = 1,2, \cdots ,i - 1 $, 以及$ {k_0} $. 因此, 存在非负连续函数$ {\eta _i} $, 它是上述变量的函数, 且依赖于上述设计参数, 使得$ \left| {{{\dot x}_{id}}} \right| \le {\eta _i} $.
对任意$ V(0) > 0 $, 集合
$$ \begin{equation} \begin{split} &{{\Omega }} = \left\{ {{{\left. {{z_0},z_1^{\left( {0 \sim {m_1} - 2} \right)},z_i^{\left( {0 \sim {m_i} - 1} \right)}} \right|}_{i = 2 \sim n}},} \right.\\ &\qquad\;{s_{11}}, \cdots, {s_{1({m_1} - 1)}},{s_{21}}, \cdots , {s_{2{m_2}}}, \cdots ,{s_{i1}}, \cdots ,\\ &\qquad \left. { {s_{i{m_i}}}, {{\tilde \theta }_i}, i = 1, \cdots ,n \mid V \le V(0)} \right\}\\[-12pt] \end{split} \end{equation} $$ (81) 是一个紧集, 由于${[ {{y_d} \quad {{\dot y}_d} \quad\cdots \quad y_d^{\left( {{m_1} + 1} \right)}} ]^{\rm{T}}} \in {{{\Omega }}_0}$, 所以, ${\eta _i}, \forall i \in \left\{ {1,2, \cdots ,n} \right\}$在$ {{\Omega }} \times {{{\Omega }}_0} $上有最大值, 将其定义为$ {\bar \eta _i} $. 因此在$ {{\Omega }} $上, 可得
$$ \begin{split} &\dot V \leq - \left( {{k_0} - \frac{{{m_1}{\sigma _1}}}{2}} \right)z_0^2 \;-\\ &\left( {{\mu _1} - \frac{{{\sigma _1}}}{{2{\lambda _{\min }}\left( {{P_1}} \right)}} - {m_2} - 1} \right) \times \\ &{\left( {z_1^{\left( {0 \sim {m_1} - 2} \right)}} \right)^{\text{T}}}{P_1}z_1^{\left( {0 \sim {m_1} - 2} \right)} - \sum\limits_{i = 2}^{n - 1} {\Big( {{\mu _i} - {m_{i + 1}} \;- } } \\ &1 - {\frac{{{\lambda _{\max }}\left( {{P_{i - 1}}} \right)}}{{{\lambda _{\min }}\left( {{P_i}} \right)}}g_{i - 1}^2} \Big){\left( {z_i^{\left( {0 \sim {m_i} - 1} \right)}} \right)^{\text{T}}}{P_i}z_i^{\left( {0 \sim {m_i} - 1} \right)}\; - \\ &\left( {{\mu _n} - \frac{{{\lambda _{\max }}\left( {{P_{n - 1}}} \right)}}{{{\lambda _{\min }}\left( {{P_n}} \right)}}g_{n - 1}^2} \right){\left( {z_n^{\left( {0 \sim {m_n} - 1} \right)}} \right)^{\text{T}}}\times\\ &{P_n}z_n^{\left( {0 \sim {m_n} - 1} \right)} \;- \\ &\left( {\frac{1}{{2{\alpha _{11}}}} - \frac{{{\sigma _1}}}{2} - \frac{{\bar \eta _1^2}}{{4\bar \varepsilon }}} \right)s_{11}^2 - \sum\limits_{i = 2}^n {\left( {\frac{1}{{2{\alpha _{i1}}}} - \frac{{\bar \eta _i^2}}{{4\bar \varepsilon }}} \right)} s_{i1}^2 \;- \\ &\sum\limits_{i = 2}^{{m_1} - 2} {\left( {\frac{1}{{2{\alpha _{1i}}}} - \frac{1}{{2{\alpha _{1\left( {i - 1} \right)}}}} - \frac{{{\sigma _1}}}{2}} \right)s_{1i}^2} \; - \\ &\left( {\frac{1}{{{\alpha _{1\left( {{m_1} - 1} \right)}}}} - \frac{1}{{2{\alpha _{1\left( {{m_1} - 2} \right)}}}} - \frac{{{\sigma _1}}}{2}} \right)s_{1\left( {{m_1} - 1} \right)}^2 \;- \end{split} $$ $$ \begin{split} &\sum\limits_{i = 2}^n {\left( {\frac{1}{{{\alpha _{i{m_i}}}}} - \frac{1}{{2{\alpha _{i\left( {{m_i} - 1} \right)}}}} - {\lambda _{\max }}\left( {{P_{i - 1}}} \right)g_{i - 1}^2} \right)s_{i{m_i}}^2} \; - \\ &\sum\limits_{i = 2}^n \sum\limits_{j = 2}^{{m_i} - 1} \Bigg[ \Bigg( \frac{1}{{2{\alpha _{ij}}}} - \frac{1}{{2{\alpha _{i\left( {j - 1} \right)}}}} \;-\\ & {\lambda _{\max }}\left( {{P_{i - 1}}} \right)g_{i - 1}^2 \Bigg)s_{ij}^2 \Bigg] -\sum\limits_{i = 1}^n {{\gamma _i}\tilde \theta _i^{\text{T}}{{\tilde \theta }_i}} + \sum\limits_{i = 1}^n {\left( {{\gamma _i}\theta _i^{\text{T}}{\theta _i} + \bar \varepsilon } \right)} \end{split} $$ (82) 其中, 常数$ \bar \varepsilon > 0 $, 且有如下不等式在推导过程中被用到
$$ \begin{equation} \sum\limits_{i = 1}^n {\left| {{s_{i1}}} \right|} {\bar \eta _i} \le \sum\limits_{i = 1}^n {\frac{{\bar \eta _i^2}}{{4\bar \varepsilon }}s_{i1}^2} + n\bar \varepsilon \end{equation} $$ (83) 以及
$$ \begin{split} &\sum\limits_{i = 1}^{n - 1} {{\lambda _{\max }}\left( {{P_i}} \right)} g_i^2{\left\| {z_{i + 1}^{\left( {0 \sim {m_{i + 1}} - 1} \right)}} \right\|^2} \leq \\ &\qquad\mathop \sum \limits_{i = 1}^{n - 1} \frac{{{\lambda _{\max }}\left( {{P_i}} \right)}}{{{\lambda _{\min }}\left( {{P_{i + 1}}} \right)}}g_i^2{\left( {z_{i + 1}^{\left( {0 \sim {m_{i + 1}} - 1} \right)}} \right)^{\text{T}}}\;\times\\ &\qquad{P_{i + 1}}z_{i + 1}^{\left( {0 \sim {m_{i + 1}} - 1} \right)} = \\ &\qquad\mathop \sum \limits_{i = 2}^n \frac{{{\lambda _{\max }}\left( {{P_{i - 1}}} \right)}}{{{\lambda _{\min }}\left( {{P_i}} \right)}}g_{i - 1}^2{\left( {z_i^{\left( {0 \sim {m_i} - 1} \right)}} \right)^{\text{T}}}{P_i}z_i^{\left( {0 \sim {m_i} - 1} \right)} \end{split} $$ (84) 因此, 只需设计参数满足
$$ \begin{equation} \left\{ \begin{aligned} &2{k_0} - {m_1}{\sigma _1} \geq \kappa ;\;\; {\mu _1} - \frac{{{\sigma _1}}}{{2{\lambda _{\min }}\left( {{P_1}} \right)}} - {m_2} - 1 \geq \kappa ; \\ &{\mu _i} - {m_{i + 1}} - 1 - \frac{{{\lambda _{\max }}\left( {{P_{i - 1}}} \right)}}{{{\lambda _{\min }}\left( {{P_i}} \right)}}g_{i - 1}^2 \geq \kappa , \\ &\qquad\qquad\qquad\qquad\qquad \quad\qquad i = 2,3, \cdots ,n - 1 ; \\ &{\mu _n} - \frac{{{\lambda _{\max }}\left( {{P_{n - 1}}} \right)}}{{{\lambda _{\min }}\left( {{P_n}} \right)}}g_{n - 1}^2 \geq \kappa ; \\ &{\gamma _i} \geq \kappa , \;i = 1,2, \cdots ,n ; \\ & \frac{1}{{2{\alpha _{11}}}} - \frac{{{\sigma _1}}}{2} - \frac{{\bar \eta _1^2}}{{4\bar \varepsilon }} \geq \kappa ; \\ &\frac{1}{{2{\alpha _{1i}}}} - \frac{1}{{2{\alpha _{1\left( {i - 1} \right)}}}} - \frac{{{\sigma _1}}}{2} \geq \kappa ,\; i = 2,3, \cdots ,{m_1} - 2 ; \\ &\frac{1}{{{\alpha _{1\left( {{m_1} - 1} \right)}}}} - \frac{1}{{2{\alpha _{1\left( {{m_1} - 2} \right)}}}} - \frac{{{\sigma _1}}}{2} \geq \kappa ; \\ &\frac{1}{{2{\alpha _{i1}}}} - \frac{{\bar \eta _i^2}}{{4\bar \varepsilon }} \geq \kappa , \quad\quad i = 2,3, \cdots ,n ; \\ &\frac{1}{{{\alpha _{i{m_i}}}}} - \frac{1}{{2{\alpha _{i\left( {{m_i} - 1} \right)}}}} - {\lambda _{\max }}\left( {{P_{i - 1}}} \right)g_{i - 1}^2 \geq \kappa , \\ &\;\quad \qquad \qquad \qquad\quad \qquad \quad\quad i = 2,3, \cdots ,n ; \\ &\left( {\frac{1}{{2{\alpha _{ij}}}} - \frac{1}{{2{\alpha _{i\left( {j - 1} \right)}}}} - {\lambda _{\max }}\left( {{P_{i - 1}}} \right)g_{i - 1}^2} \right)y_{ij}^2 \geq \kappa , \\ &\qquad\qquad\quad\;\; i = 2,3, \cdots ,n, j = 2,3, \cdots ,{m_i} - 1 \end{aligned} \right. \end{equation} $$ (85) 其中, $ \kappa > 0 $为一常数. 进一步可得
$$ \begin{equation} \dot V \le - \kappa V + \varepsilon \end{equation} $$ (86) 其中
$$ \begin{equation} \varepsilon = \sum\limits_{i = 1}^n {\left( {{\gamma _i}\theta _i^{\text{T}}{\theta _i} + \bar \varepsilon } \right)} \end{equation} $$ (87) 对式(86), 若选取参数满足$\kappa \ge {\varepsilon }/{{V(0)}}$, 则$ \dot V \le 0 $. 所以对任意$ t \ge 0 $, 有$ V(t) \le V(0) $成立. 因此可知$ {{\Omega }} $是一个不变集. 根据比较原理[36]可得
$$ \begin{equation} 0 \le V(t) \le \frac{\varepsilon }{\kappa } + \left(V(0) - \frac{\varepsilon }{\kappa }\right){{\rm{e}}^{ - \kappa t}}, \quad\forall t \ge 0 \end{equation} $$ (88) 因此, 可得$ V(t) $是一致最终有界的. 进一步可知, 坐标转换误差$ {z_i},i = 0,1, \cdots ,n $, 滤波误差${s_{11}}, \cdots, {s_{1({m_1} - 1)}}, {s_{21}}, \cdots ,{s_{2{m_2}}}, \cdots ,{s_{i1}}, \cdots ,{s_{i{m_i}}}$以及参数估计误差$ {\tilde \theta _i}, i = 1, \cdots ,n $是一致最终有界的, 即闭环系统所有信号都是一致最终有界的. 且由于${\lim _{t \to \infty }}V(t) \to {\varepsilon }/{\kappa }$, 而$ \varepsilon $不依赖于$ \kappa $, 所以可通过调整参数使得${\varepsilon }/{\kappa }$足够小. 由于$ {z_0} = {\zeta _1} - {x_{0c}} $, 且$ {x_{0c}} $在$ {\Omega _d} $内有界, 所以可知$ {\zeta _1} $有界, 从而$ {x_1} \in {D_1} $, 即输出约束始终得到满足. 定义跟踪误差为
$$ \begin{equation} e = {x_1} - {y_d} \end{equation} $$ (89) 由式(5), (12)和(13), 可得
$$ \begin{equation} e = {\delta _0}{z_0} \end{equation} $$ (90) 其中
$$ \begin{equation} {\delta _0} = \frac{{\left( {{F_1} + {x_1}} \right)\left( {{F_2} - {x_1}} \right)\left( {{F_1} + {y_d}} \right)\left( {{F_2} - {y_d}} \right)}}{{{F_1}{F_2} + {x_1}{y_d}}} \end{equation} $$ (91) 根据上述分析可知, 存在两个正常数$ {\overline \delta _0} $和$ {\underline \delta _0} $, 使得$ 0 < {\underline \delta _0} \le {\delta _0} \le {\overline \delta _0} < +\infty $. 进一步由$ {z_0} $一致最终有界可知, $ e $是一致最终有界的.
□ 注 3. 由上述分析可以看出, 本文不需要将高阶严反馈系统转化为一阶形式, 直接对其设计控制器, 会简化一些步骤. 且可通过调整设计参数使得$ {\varepsilon }/{\kappa } $足够小以提高跟踪精度, 然而这样会增大控制输入, 因此在实际应用中, 应考虑跟踪精度和控制输入之间的权衡.
4. 仿真分析
为验证所提算法的有效性, 选取柔性关节机械臂系统对本文算法进行仿真验证, 机械臂系统模型为[37-38]
$$ \begin{equation} \left\{ \begin{aligned} &I_Q\ddot q + MgL\sin q + Kq + {\Delta _1} = K\theta \\ &J\ddot \theta + K\left( {\theta - q} \right) + {\Delta _2} = \tau \end{aligned} \right. \end{equation} $$ (92) 其中, $q \in {{\bf{R}}^n}$表示连杆位置, 其一阶导数和二阶导数分别代表连杆的速度和加速度, $ I_Q \in {{\bf{R}}^{n \times n}} $是连杆转动惯量; $ M \in {{\bf{R}}^n} $和$ L \in {{\bf{R}}^n} $分别为连杆质量和长度; $ g $为重力加速度; $ J \in {{\bf{R}}^{n \times n}} $是电机转动惯量; $ \theta ,\dot \theta ,\ddot \theta \in {{\bf{R}}^n} $表示转子的角位移、角速度和角加速度; $ K \in {{\bf{R}}^{n \times n}} $表示等效弹簧弹性系数; $ \tau \in {{\bf{R}}^n} $为电机控制力矩; $ {\Delta _1} $和$ {\Delta _2} $表示系统模型不确定性, 其表达式为${\Delta _1} = 0.1(MgL\sin q + Kq)$和$ {\Delta _2} = 0.1K( \theta - q) $. 机械臂模型参数[37]为 $I_Q = 1 \,( {\rm{kg}} \cdot {{\rm{m}}^{\rm{2}}})$, $J = 1 \,( {\rm{kg}} \cdot {{\rm{m}}^{\rm{2}}} )$, $ M = 0.5 \ {\rm{kg}} $, $ L = 1 \ {\rm{m}} $, $K = 40\ ( {\rm{N}} \cdot {\rm{m}} \cdot {\rm{ra}}{{\rm{d}}^{ - 1}} )$, $g = $ $ 10 \ ( {\rm{m}} \cdot {{\rm{s}}^{{\rm{ - 2}}}} )$.
定义$ {x_1} = q $, $ {x_2} = \theta $, $ u = \tau $, 可得
$$ \begin{equation} \left\{ \begin{aligned} &{{\ddot x}_1} = {f_{10}}\left( {{x_1},{{\dot x}_1}} \right) + {f_1}\left( {{x_1},{{\dot x}_1}} \right){\theta _1}+{g_1}\left( {{x_1},{{\dot x}_1}} \right){x_2}\\ &{{\ddot x}_2} = {f_{20}}\left( {x_{1 \sim 2}^{\left( {0 \sim 1} \right)}} \right) + {f_2}\left( {x_{1 \sim 2}^{\left( {0 \sim 1} \right)}} \right){\theta _2}+{g_2}\left( {x_{1 \sim 2}^{\left( {0 \sim 1} \right)}} \right)u \end{aligned} \right. \end{equation} $$ (93) 其中, $ {g_1} = 40 $, $ {g_2} = 1 $, $ {f_1} = {f_{10}} = - 5\sin {x_1} - 40{x_1} $, $ {f_2} = {f_{20}} = 40{x_1} - 40{x_2} $. $ {\theta _1} $和$ {\theta _2} $为系统未知的模型不确定参数, 其真实值为0.1.
为验证本文方法的有效性, 首先将本文方法与不考虑输出约束的自适应动态面控制方法[39]进行对比仿真. 令$ {x_1} = q $, $ {x_2} = \dot q $, $ {x_3} = \theta $, $ {x_4} = \dot \theta $, 将机械臂系统(92)转化为如下一阶系统的形式, 如式(94)所示, 再对其设计自适应动态面控制器.
$$ \begin{equation} \left\{ \begin{aligned} &{{\dot x}_1} = {x_2} \\ &{{\dot x}_2} = - \frac{{MgL}}{I_Q}\sin {x_1} - \frac{K}{I_Q}{x_1} - \frac{{{\Delta _1}}}{I_Q} + \frac{K}{I_Q}{x_3} \\ &{{\dot x}_3} = {x_4} \\ &{{\dot x}_4} = - \frac{K}{J}\left( {{x_3} - {x_1}} \right) - \frac{{{\Delta _2}}}{J} + \frac{1}{J}u \end{aligned} \right. \end{equation} $$ (94) 情况 1. 期望的跟踪信号为$ {y_d} = 0.5\sin (0.5t) $, 约束边界为$ {F_1} = 0.65 + 0.2\sin (t) $、$ {F_2} = 0.7 + 0.2\sin (t) $, 其他相关参数选为$ {\alpha _{11}} = 0.04 $, $ {\alpha _{21}} = 0.03 $, $ {\alpha _{22}} = 0.01 $, $ {a_1} = 10 $, $ {P_1} = 1 $, $ a_2^{0 \sim 1} = \left[ {40, 10} \right] $, $ {\gamma _1} = 10 $, $ {\gamma _2} = 10 $, ${P_2} = \left[ {\begin{aligned} {2.1750}\;\;{0.0125}\\ {0.0125}\;\;{0.0512} \end{aligned}} \right]$. 系统初值选为 ${x_1}(0) = 0.6$、$ {x_2}(0) = {x_3}(0) = {x_4}(0) = 0 $, ${\hat{\theta} _1}( 0 )=0$, ${\hat{\theta} _2}( 0 )=0$. 仿真时的相关参数和初值, 二者保持一致, 将本文方法得到的系统输出记为$ {x_1} $、将不考虑输出约束的自适应动态面控制方法得到的系统输出记为$ {x_1}\;{\text{wu}} $, 得到仿真结果如图1所示.
情况 2. 增加更为复杂的约束边界进行仿真对比, 期望的跟踪信号为$ {y_d} = \cos (2t) + \sin (0.5t) $, 约束边界为$ {F_1} = 1.01 \times {2^{ - 2t}} + 1 - \cos (2t) - \sin (0.5t) $, $ {F_2} = 0.56 \times {2^{ - 1.2t}} + 0.84 + \cos (2t) + \sin (0.5t) $, 系统初值设置为$ {x_1}(0) = 2 $, $ {x_2}(0) = 1 $, $ {x_3}(0) = {x_4}(0) = 0.5 $. 其他相关参数及记法与情况1一致, 得到系统输出仿真结果如图2所示.
由图1和图2可知, 系统本身存在输出约束, 但是自适应动态面控制器不考虑输出约束问题, 就无法始终保持系统输出在约束范围内. 从仿真结果来看其中存在系统输出超出约束边界的情形, 这对于实际工程比如无人车系统、隧道无人机系统等, 超出约束边界就意味着无人车冲出跑道、无人机撞到隧道边界, 出现事故. 而本文算法能够有效使系统输出始终保持在约束范围内, 不违反约束条件.
为说明本文算法的先进性, 将本文算法与基于非对称障碍Lyapunov函数(BLF), 即参考文献[5]中的方法以及传统基于非线性映射函数(Nonlinear mapping, NM), 即参考文献[8-9]中的方法, 在情况1和情况2两种情况下进行对比仿真, 相关参数和初值保持不变. 将对比的两种方法简称为一阶BLF方法和一阶NM方法. 将本文方法仿真结果记为$ {x_{1,h}} $, $ {e_h} $, $ {\hat{\theta} _1}(h) ,{\hat{\theta} _2}(h)$, $ {u_h} $; 将一阶BLF方法仿真结果记为$ {x_{1,f1}} $, $ {e_{f1}} $, $ {\hat{\theta} _1}(f1),{\hat{\theta} _2}(f1) $, $ {u_{f1}} $; 将一阶NM方法仿真结果记为$ {x_{1,f2}} $, $ {e_{f2}} $, $ {\hat{\theta} _1}(f2),{\hat{\theta} _2}(f2) $, $ {u_{f2}} $. 情况1的仿真结果如图3 ~ 7所示; 情况2的仿真结果如图8 ~ 12所示.
上述仿真结果中, 图3、图8和图4、图9分别为系统输出跟踪结果和跟踪误差, 可知三种控制方法下的系统输出均能有效地跟踪期望信号, 且系统输出不违反约束条件, 始终保持在约束范围内; 相比于一阶BLF方法和一阶NM方法, 在相同的控制参数及相同的初值条件下, 本文方法收敛速度和精度都略优. 图5、图6和图10、图11为系统参数估计结果, 可知三种控制方法下的参数估计值均能有效收敛到参数真实值附近, 且本文方法收敛精度和收敛速度更好. 图7和图12为系统控制输入, 可知三种控制方法的输入均在合理范围内, 相比于一阶BLF方法和一阶NM方法, 本文方法控制输入初值更小.
图 5 参数估计${\hat{\theta} _1}(h)$, ${\hat{\theta} _1}(f1)$, ${\hat{\theta} _1}(f2)$(情况1)Fig. 5 Parameter estimation ${\hat{\theta} _1}(h)$, ${\hat{\theta} _1}(f1)$, and ${\hat{\theta} _1}(f2)$(Case 1)
图 6 参数估计${\hat{\theta} _2}(h)$, ${\hat{\theta} _2}(f1)$, ${\hat{\theta} _2}(f2)$(情况1)Fig. 6 Parameter estimation ${\hat{\theta} _2}(h)$, ${\hat{\theta} _2}(f1)$, and ${\hat{\theta} _2}(f2)$(Case 1)
图 10 参数估计${\hat{\theta} _1}(h)$, ${\hat{\theta} _1}(f1)$, ${\hat{\theta} _1}(f2)$(情况2)Fig. 10 Parameter estimation ${\hat{\theta} _1}(h)$, ${\hat{\theta} _1}(f1)$, and ${\hat{\theta} _1}(f2)$(Case 2)
图 11 参数估计${\hat{\theta} _2}(h)$, ${\hat{\theta} _2}(f1)$, ${\hat{\theta} _2}(f2)$(情况2)Fig. 11 Parameter estimation ${\hat{\theta} _2}(h)$, ${\hat{\theta} _2}(f1)$, and ${\hat{\theta} _2}(f2)$(Case 2)注 4. 对于反步法/动态面控制方法而言, 随着系统阶数的增多, 控制器设计步骤也会增多, 不仅会增加控制器设计的复杂度、引起微分爆炸问题, 还会增加中间步骤的误差. 而本文方法直接对高阶系统设计控制器, 不仅省去了将高阶系统转换为一阶系统这一步骤, 还能减少控制器设计步骤, 从而有效减弱上述问题.
为进一步说明本文方法的优势, 将本文方法与上述一阶BLF方法以及一阶NM方法在相同的仿真参数下, 并在同一台计算机同一个仿真软件下运行50次, 比较三者在情况1和情况2两种情况下的运行时间, 结果如表1所示. 可知相比于一阶NM方法和一阶BLF方法, 由于本文方法设计步骤更少, 所需仿真时间更少, 具有更高的效率.
表 1 三种算法运算时间对比(s)Table 1 Comparison of operation time of three algorithms (s)本文方法 一阶BLF方法 一阶NM方法 情况1 44.380383 58.879631 49.213382 情况2 35.173545 47.965324 41.693068 由上述仿真结果可知, 本文所提出的算法不需要将原始系统转化为4个一阶子系统, 设计步骤更少, 能使系统始终满足输出约束要求. 相比于一阶系统方法, 本文方法在收敛精度、收敛速度方面略优, 控制输入初值更小、仿真时间更少.
注 5. 一阶BLF方法、一阶NM方法以及本文方法均能有效处理输出约束问题, 对应的文献从理论方面说明了三种方法能确保闭环系统所有信号一致最终有界, 输出约束能始终满足, 而且本文的仿真也证实了三种方法都能有效处理输出约束问题. 从仿真结果来看, 本文方法的收敛精度、收敛速度略优于一阶BLF方法和一阶NM方法, 性能指标方面并没有很突出的优势. 这是因为本文的优势主要在于能减少设计步骤、降低算法设计复杂度, 从而减弱反步法/动态面控制方法的微分爆炸问题, 这对于反步法/动态面控制是很有意义的. 从仿真结果中也能证实本文方法控制输入初值更小、算法运算时间更少, 具有更高的效率.
5. 结束语
针对不确定高阶严反馈系统, 基于全驱系统方法, 本文提出一种高阶自适应动态面时变输出约束控制方法. 首先通过非线性转换函数将原输出约束系统转换为无约束的系统, 进而可通过确保新系统的有界性使得原系统输出约束得到满足. 然后基于全驱系统方法, 在高阶动态面控制方法的框架下直接对每个高阶子系统设计控制器, 而不需要将其转化为一阶系统形式, 减少了一些步骤; 通过引入一系列低通滤波器来获得虚拟控制律的高阶导数, 进而将复杂的高阶求导运算转化为简单的代数运算, 更加简洁高效. 通过Lyapunov稳定性理论证明闭环系统所有信号都是一致最终有界的, 系统输出约束条件始终得到满足, 且可通过调整参数使得系统跟踪误差收敛到原点的任意小邻域内. 最后通过对实际的机械臂系统进行仿真, 验证了本文所提控制方法的有效性.
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图 5 参数估计${\hat{\theta} _1}(h)$, ${\hat{\theta} _1}(f1)$, ${\hat{\theta} _1}(f2)$(情况1)
Fig. 5 Parameter estimation ${\hat{\theta} _1}(h)$, ${\hat{\theta} _1}(f1)$, and ${\hat{\theta} _1}(f2)$(Case 1)
图 6 参数估计${\hat{\theta} _2}(h)$, ${\hat{\theta} _2}(f1)$, ${\hat{\theta} _2}(f2)$(情况1)
Fig. 6 Parameter estimation ${\hat{\theta} _2}(h)$, ${\hat{\theta} _2}(f1)$, and ${\hat{\theta} _2}(f2)$(Case 1)
图 10 参数估计${\hat{\theta} _1}(h)$, ${\hat{\theta} _1}(f1)$, ${\hat{\theta} _1}(f2)$(情况2)
Fig. 10 Parameter estimation ${\hat{\theta} _1}(h)$, ${\hat{\theta} _1}(f1)$, and ${\hat{\theta} _1}(f2)$(Case 2)
图 11 参数估计${\hat{\theta} _2}(h)$, ${\hat{\theta} _2}(f1)$, ${\hat{\theta} _2}(f2)$(情况2)
Fig. 11 Parameter estimation ${\hat{\theta} _2}(h)$, ${\hat{\theta} _2}(f1)$, and ${\hat{\theta} _2}(f2)$(Case 2)
表 1 三种算法运算时间对比(s)
Table 1 Comparison of operation time of three algorithms (s)
本文方法 一阶BLF方法 一阶NM方法 情况1 44.380383 58.879631 49.213382 情况2 35.173545 47.965324 41.693068 
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[1] Limon D, Alvarado I, Alamo T, Camacho E F. MPC for tracking piecewise constant references for constrained linear systems. Automatica, 2008, 44(9): 2382-2387. doi: 10.1016/j.automatica.2008.01.023 [2] Kogiso K, Hirata K. Reference governor for constrained systems with time-varying references. Robotics and Autonomous Systems, 2009, 57(3): 289-295. doi: 10.1016/j.robot.2008.10.015 [3] Hatanaka T, Takaba K. Computations of probabilistic output admissible set for uncertain constrained systems. Automatica, 2008, 44(2): 479-487. doi: 10.1016/j.automatica.2007.03.031 [4] Tee K P, Ge S S, Tay E H. Barrier Lyapunov functions for the control of output-constrained nonlinear systems. Automatica, 2009, 45(4): 918-927. doi: 10.1016/j.automatica.2008.11.017 [5] Tee K P, Ren B, Ge S S. Control of nonlinear systems with time-varying output constraints. Automatica, 2011, 47(11): 2511-2516. doi: 10.1016/j.automatica.2011.08.044 [6] Ouyang Y, Dong L, Xue L, Sun C. Adaptive control based on neural networks for an uncertain 2-DOF helicopter system with input deadzone and output constraints. IEEE/CAA Journal of Automatica Sinica, 2019, 6(3): 807-815. doi: 10.1109/JAS.2019.1911495 [7] Sun L. Saturated adaptive output-constrained control of cooperative spacecraft rendezvous and docking. IEEE/CAA Journal of Automatica Sinica, 2019, 6(6): 1462-1470. [8] 郭涛, 王丁磊, 王爱民. 基于非线性映射的约束系统自适应反推控制. 自动化学报, 2013, 39(9): 1558-1563.Guo T, Wang D L, Wang A M. Adaptive backstepping control for constrained systems using nonlinear mapping. Acta Automatica Sinica, 2013, 39(9): 1558-1563. [9] Zhao K, Song Y D. Removing the feasibility conditions imposed on tracking control designs for state-constrained strict-feedback systems. IEEE Transactions on Automatic Control, 2018, 64(3): 1265-1272. [10] Ling S, Wang H, Liu P X. Adaptive tracking control of high-order nonlinear systems under asymmetric output constraint. Automatica, 2020, 122: 109281. doi: 10.1016/j.automatica.2020.109281 [11] Liu Y L, Ma H J, Ma H. Adaptive fuzzy fault-tolerant control for uncertain nonlinear switched stochastic systems with time-varying output constraints. IEEE Transactions on Fuzzy Systems, 2018, 26(5): 2487-2498. doi: 10.1109/TFUZZ.2018.2814596 [12] Cui R H, Xie X J. Output feedback stabilization of stochastic planar nonlinear systems with output constraint. Automatica, 2022, 143: 110471. doi: 10.1016/j.automatica.2022.110471 [13] Zhao K, Song Y D, Meng W C, Philip C L, Chen L. Low-cost approximation-based adaptive tracking control of output-constrained nonlinear systems. IEEE Transactions on Neural Networks and Learning Systems, 2020, 32(11): 4890-4900. [14] Kong L H, He W, Liu Z J, Yu X B, Silvestre C. Adaptive tracking control with global performance for output-constrained MIMO nonlinear systems. IEEE Transactions on Automatic Control, 2023, 68(6): 3760-3767. doi: 10.1109/TAC.2022.3201258 [15] Zhao Z J, Zhang J, Liu Z J, Mu C X, Hong K S. Adaptive neural network control of an uncertain 2-DOF helicopter with unknown backlash-like hysteresis and output constraints. IEEE Transactions on Neural Networks and Learning Systems, 2023, 34(12): 10018-10027. doi: 10.1109/TNNLS.2022.3163572 [16] 陈正升, 程玉虎, 王雪松. 考虑输出约束的冗余驱动绳索并联机器人预设性能控制. 自动化学报, 2022, 48(7): 1704-1717.Chen Z S, Cheng Y H, Wang X S. Prescribed performance control of redundantly-actuated cable driving parallel robots subjected to output constraint. Acta Automatica Sinica, 2022, 48(7): 1704-1717. [17] 段广仁. 高阶系统方法——I. 全驱系统与参数化设计. 自动化学报, 2020, 46(7): 1333-1345.Duan G R. High-order system approaches: I. Fully-actuated systems and parametric designs. Acta Automatica Sinica, 2020, 46(7): 1333-1345. [18] 段广仁. 高阶系统方法——Ⅱ. 能控性与全驱性. 自动化学报, 2020, 46(8): 1571-1581.Duan G R. High-order system approaches: II. Controllability and full-actuation. Acta Automatica Sinica, 2020, 46(8): 1571-1581. [19] 段广仁. 高阶系统方法——Ⅲ. 能观性与观测器设计. 自动化学报, 2020, 46(9): 1885-1895.Duan G R. High-order system approaches: III. Observability and observer design. Acta Automatica Sinica, 2020, 46(9): 1885-1895. [20] Duan G R. High-order fully actuated system approaches: Part I. Models and basic procedure. International Journal of Systems Science, 2021, 52(2): 422-435. doi: 10.1080/00207721.2020.1829167 [21] Hou M Z, Shi W R, Fang L Y, Duan G R. Adaptive dynamic surface control of high-order strict feedback nonlinear systems with parameter estimations. Science China Information Sciences, 2023, 66(5): 159203. doi: 10.1007/s11432-021-3488-6 [22] Liu W Z, Duan G R, Hou M Z. High-order command filtered adaptive backstepping control for second-and high-order fully actuated strict-feedback systems. Journal of the Franklin Institute, 2023, 360(6): 3989-4015. doi: 10.1016/j.jfranklin.2023.01.014 [23] Duan G R. High-order fully actuated system approaches: part VIII. Optimal control with application in spacecraft attitude stabilisation. International Journal of Systems Science, 2021, 53(1): 54-73. [24] Zhang D W, Liu G P, Cao L. Constrained cooperative control for high-order fully actuated multiagent systems with application to airbearing spacecraft simulators. IEEE/ASME Transactions on Mechatronics, 2023, 28(3): 1570-1581. doi: 10.1109/TMECH.2022.3223927 [25] Wang X B, Duan G R. High-order fully actuated system approaches: Model predictive control with applications to under-actuated systems. Journal of the Franklin Institute, 2023, 360(10): 6953-6975. doi: 10.1016/j.jfranklin.2023.05.009 [26] Cai M, He X, Zhou D H. An active fault tolerance framework for uncertain nonlinear high-order fully-actuated systems. Automatica, 2023, 152: 110969. doi: 10.1016/j.automatica.2023.110969 [27] Duan G R. Sub-stability and sub-stabilization: Control of sub-fully actuated systems. IEEE Transactions on Cybernetics, 2023, 53(11): 7309-7322. doi: 10.1109/TCYB.2023.3242277 [28] Chen S W, Wang W, Fan J F, Ji Y. Impact angle constraint guidance law using fully-actuated system approach. Aerospace Science and Technology, 2023, 136: 108220. doi: 10.1016/j.ast.2023.108220 [29] 刘明, 范睿超, 邱实, 曹喜滨. 基于全驱系统理论的航天器姿轨预设性能控制. 航空学报, DOI: 10.7527/S1000-6893.2023.28313Liu Ming, Fan Rui-Chao, Qiu Shi, Cao Xi-Bin. Spacecraft attitude-orbit prescribed performance control based on fully actuated system approach. Acta Aeronautica et Astronautica Sinica, DOI: 10.7527/S1000-6893.2023.28313 [30] 张宏岩, 王伟, 陈仕伟, 纪毅, 刘佳琪. 基于全驱系统方法的制导控制一体化设计. 航空学报, DOI: 10.7527/S1000-6893.2023.28891Zhang Hong-Yan, Wang Wei, Chen Shi-Wei, Ji Yi, Liu Jia-qi. Integrated guidance and control design based on fully actuated system method. Acta Aeronautica et Astronautica Sinica, DOI: 10.7527/S1000-6893.2023.28891 [31] Duan G R. High-order fully actuated system approaches: Part II. Generalized strict-feedback systems. International Journal of Systems Science, 2021, 52(3): 437-454. doi: 10.1080/00207721.2020.1829168 [32] Duan G R. High-order fully actuated system approaches: Part III. Robust control and high-order backstepping. International Journal of Systems Science, 2021, 52(5): 952-971. doi: 10.1080/00207721.2020.1849863 [33] Duan G R. High-order fully actuated system approaches: Part IV. Adaptive control and high-order backstepping. International Journal of Systems Science, 2021, 52(5): 972-989. doi: 10.1080/00207721.2020.1849864 [34] Liu W Z, Duan G R, Hou M Z. Concurrent learning adaptive command filtered backstepping control for high-order strict-feedback systems. IEEE Transactions on Circuits and Systems I: Regular Papers, 2023, 70(4): 1696-1709. doi: 10.1109/TCSI.2023.3234573 [35] Xiao Y Q, Cai G B, Duan G R. High-order adaptive dynamic surface control for output constrained nonlinear systems based on fully actuated system approach. International Journal of Systems Science, 2024, 55(3): 482-498. doi: 10.1080/00207721.2023.2277708 [36] Khalil H K. Nonlinear Systems 3rd Edition. Upper Saddle River: Prentice Hall, 2002. [37] 陈强, 丁科新, 南余荣. 带有输出约束的柔性关节机械臂预设性能自适应控制. 控制与决策, 2021, 36(2): 387-394.Chen Q, Ding K X, Nan Y R. Prescribed performance adaptive control of flexible-joint manipulators with output constraints. Control and Decision, 2021, 36(2): 387-394. [38] Liu X, Yang C G, Chen Z G, Wang M, Su C Y. Neuro-adaptive observer based control of flexible joint robot. Neuro Computing, 2018, 275: 73-82. [39] 顾玮. 不确定柔性关节机器人自适应跟踪控制方法研究. 自动化应用, 2016, 8: 87-89. doi: 10.3969/j.issn.1674-778X.2016.06.043Gu W. Research on adaptive tracking control method of uncertain flexible joint robot. Automation Applications, 2016, 8: 87-89. doi: 10.3969/j.issn.1674-778X.2016.06.043 期刊类型引用(0)
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