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摘要: 随着信息物理系统（Cyber-physical system，CPS）融合深度和融合广度的不断增加，信息物理生产系统（Cyber-physical production system，CPPS）呈现出显著的社会化趋势.通过对信息物理生产系统相关技术的研究，分析了信息物理生产系统的社会化演进历程，建立了社会信息物理生产系统（Social cyber-physical production system，SCPPS）模型；根据人与智能体的信息物理交互行为差异，基于对人类社会行为特点的分析，类比研究了智能体社会与人类社会融合的广义互联社会特点；归纳出信息物理系统的七种交互模式及其在社会信息物理生产系统中的应用；总结出社会信息物理生产系统面临标准化、人性化和安全化的挑战问题.
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关键词:
- 信息物理系统 /
- 信息物理生产系统 /
- 社会信息物理生产系统 /
- 物联网 /
- 交互模式
Abstract: With the increasing fusion depth and breadth of cyber and physical spaces, the cyber-physical production system (CPPS) presents a significant social trend. In this paper, through investigations on related CPPS technologies, we expound the social evolution of CPPS, and establish a social cyber-physical production system (SCPPS) model. According to the difference of cyber-physical interaction between humans and agents, we derive the behavior characteristics of the networked society of agents and humans based on an analogical analysis of the characteristics of human society. We also discuss the cyber-physical interaction models and their application in SCPPS, as well as the challenges of SCPPS in terms of standardization, humanization and security. -
迭代学习控制技术问世于上世纪80年代初, 适用于重复作业对象的控制器设计, 可实现在整个作业区间上的零误差跟踪 [1]. 当系统在固定区间内重复运行时, 可学习的不确定性虽然随着时间的变化而变化, 但在各次运行中呈现相同的变化规律, 沿迭代轴来看, 同一时刻对应的不确定性为一常值. 由此, 可通过学习方法对其进行估计, 并根据误差不断修正控制输入. 这样, 经过足够多次迭代运行后, 可将闭环系统中可学习的不确定性予以完全补偿, 实现系统状态对参考信号在整个作业区间上的完全跟踪 [2-7]. 至今, 这种控制技术已应用于机械臂、磁盘驱动器和逆变电路等.
目前, 基于Lyapunov方法设计学习控制系统引起了人们的关注 [2-3]. 在设计学习控制系统过程中, 需要处理各种不确定性, 常见的有线性参数不确定、非线性参数不确定性 [8]和非参数不确定性 [9] 等. 线性参数不确定性又可分为固定常数 [10-11]、不随迭代次数变化的时变参数 [12], 以及随迭代次数变化的时变参数 [13]. 从已经发表的文献数量来看, 非线性参数不确定性和非参数不确定性方面的结果较少. 文献[14]利用界函数设计反馈项补偿非参数不确定性. 文献[15-16] 结合使用鲁棒方法与学习方法处理非参数不确定性. 傅里叶级数等逼近工具也可用于估计该类不确定 [17]. 文献[18]针对控制增益时变的非参数不确定系统, 基于Backstepping方法设计迭代学习控制系统. 文献[19]针对一类同含参数不确定性和非参数不确定性的 非线性系统, 分别提出准最优迭代学习控制算法和准最优重复学习控制算法.
在应用常规迭代学习控制算法时, 需要在每次迭代开始前进行严格初始定位, 以使系统初态与期望轨迹的起始点完全一致 [3]. 但在实际中, 受复位条件的限制, 系统存在非零误差初值. 因此, 研究适用于任意误差初值的迭代学习控制算法, 不仅具有理论意义, 还可拓宽迭代学习控制技术的应用范围. 针对连续系统的Lyapunov方法初值问题解决方案见文献[20-22]. 文献[20]提出了时变边界层解决方案. 其控制策略是: 经过足够多次迭代后, 闭环系统的滤波误差可以收敛到与迭代初值相关的时变死区中. 文献[21] 给出误差跟踪设计方法, 并将其与参考信号初始修正方法进行对比. 文献[22]研究非参数不确定系统的误差跟踪学习控制算法. 另外, 在参考信号光滑闭合场合, 可采用重复学习控制方法设 计控制器, 该法在运行过程中勿需停顿及复位 [2, 23-24].
设计自适应或自适应学习控制器时, 为了处理有界的不确定性, 常采用鲁棒方法予以处理. 根据界函数与符号函数设计反馈项可完全补偿不确定性, 但据此设计的控制器在实现时容易发生颤振现象. 为了克服这一不足, 可以采用饱和函数代替符号函数, 实现边界层外的切换控制和边界层内的线性反馈控制. 类似的方法还有单位向量连续化 [25]. 在一些场合, 例如根据反演方法设计控制器时, 为了设计上的方便, 可以采用双曲正切函数代替符号函数 [26]. 文献[27]利用双曲正切函数为严格反馈时变系统设计学习控制器.
为解决参数/非参数混合不确定系统的轨迹跟踪问题, 针对任意初态的非严格复位系统, 本文 提出基于滤波误差初始修正的自适应迭代学习控制方法. 在构造修正滤波误差后, 采用Lyapunov方法设计迭代学习控制器并进行性能分析, 利用鲁棒手段确保系统变量有界, 处理非参数不确定性后, 将所得的各未知时变参数合并为两个未知时变参数向量, 并通过学习方法分别予以估计. 经过足够多次迭代后, 闭环系统的修正滤波误差在整个作业区间收敛于零, 滤波 误差在预设的部分作业区间上收敛于零. 文中所给出的修正滤波误差构造方案, 具有构造简单实现方便的特点.
1. 问题描述
考虑有限时间区间$[0, T]$上重复运行的非线性不确定系统
$\begin{cases} \dot{x}_{ik}=x_{i+1k}, i=1, 2, \cdots, n-1\\dot{x}_{nk}=f( x_k, t)+g( x_k, t)u_k \end{cases}\label{01cssys} $
(1) 式中, $k=0, 1, 2, \cdots$为重复作业次数. ${x}_k=[x_{1k}$, ${x}_{2k}, \cdots, x_{nk}]^{\rm T}\in R^n$为状态向量, $u_k\in R$为控制输入, $f( x_k, t)$与$g( x_k, t)$为不确定性, 未知实数 $g( x_k, t)>0$. 参考信号为${x_d} = {[{x_d}, {{\dot x}_d}, {{\ddot x}_d}, \cdots , {x_d}^{(n - 1)}]^{\rm{T}}}, {x_d}^{(n)}$存在.
本文考虑系统状态初值$ x_k(0)$, $k=0, 1, 2, \cdots$, 任意情形下的学习控制算法, 拟构造修正滤波误差设计控制器, 在足够多次迭代后, 实现闭环系统的$ x_k$ 在部分作业区间上精确跟踪$ x_d$. 因控制器设计需要, 做如下假设.
假设 1. 函数
$f( x_k, t)= \theta^{\rm T}(t) \varphi( x_k, t)+\Delta f( x_k, t) $
(2) 其中, $ \theta(t)\in R^m$为未知时变常数, $ \varphi( x_k, t)$为与$ \theta(t)$同维的连续向量, $\Delta f( x_k, t)$满足
$|\Delta f({\xi _1}, t) - \Delta f({\xi _2}, t)| \le {\alpha _f}({\xi _1}, {\xi _2}, t){\xi _1} - {\xi _2}, \forall {\xi _1} \in {R^n}, \forall {\xi _2} \in {R^n}$
此处, $\alpha_f(\cdot, \cdot, \cdot)$为非负连续函数.
假设 2. 函数$g(\cdot, \cdot)$满足
$|g({\xi _1}, t) - g({\xi _2}, t)| \le {\alpha _g}({\xi _1}, {\xi _2}, t){\xi _1} - {\xi _2}, \forall {\xi _1} \in {R^n}, \forall {\xi _2} \in {R^n}{\rm{ }}$
其中, $\alpha_g(\cdot, \cdot, \cdot)$为非负连续函数, 且 存在连续函数$g_{m}( x_k, t)$, 满足$0<g_{m}( x_k, t)\leq g( x_k, t)$.
假设 3. $\partial g({x_k}, t)\partial {x_k} < + \infty $.
参数不确定性和非参数不确定性是系统中常见的不确定性, 本文在假设性方面的要求较文献[28]低. 为叙述简便, 下文记$g_{m}( x_k, t)$, $ \varphi( x_k, t)$, $\Delta f( x_k, t)$, $g( x_k, t)$, $\Delta f( x_d, t)$, $g( x_d, t)$, $\alpha_f( x_k, x_d$, $t)$, $\alpha_g( x_k, x_d, t)$分别为$g_{mk}$, $ \varphi_k$, $\Delta f_k$, gk, $\Delta f_d$, gd, $\alpha_{fk}$, $\alpha_{gk}$. 在不引起歧义时, 函数的自变量t常被略去.
2. 滤波误差初始修正下的控制器设计
记
${e_k} = {[{e_{1k}}, {e_{2k}}, \cdots , {e_{nk}}]^{\rm{T}}} = {x_k} - {x_d}{s_k} = {c_1}{e_{1k}} + \cdots + {c_{n - 1}}{e_{n - 1k}} + {e_{nk}}$
(3) 选取合适的参数$c_1$, $\cdots$, $c_{n-1}$, 使得多项式$\Delta(p)=$ $p^{n-1}$ $+$ $c_{n-1}p^{n-2}+\cdots+c_2p+c_1$为Hurwitz多项式.
定义 1.
$s_{\phi k}=s_k-\phi(t)s_k(0) \label{sfkdef} $
(4) 其中, $\phi(t)$为一类连续可导的单调递减函数, 满足$\phi(0)=1$, $\phi(t)=0$ $(\forall t\in [t_1, T])$. 一种可选的$\phi(t)$ 构造方案为 {\small
$\phi (t) = \left\{ {\matrix{ {{{10{{({t_1} - t)}^3}} \over {t_1^3}} - {{15{{({t_1} - t)}^4}} \over {t_1^4}} + {{6{{({t_1} - t)}^5}} \over {t_1^5}}, } \hfill&{0 \le t \le {t_1}\;0, } \hfill&{{t_1}} \hfill&{ < t \le T\;} \hfill \cr } } \right.$
本文称$s_{\phi k}$为修正滤波误差, 其与误差$ e_k$的关系满足引理1.
引理 1. 对于任意$\mu>0$及取值任意的系统初态$ x_k(0)$, 不等式
$\int_0^t {{{\rm{e}}^{ - \lambda \tau }}} \mu |{s_{\phi k}}|{e_k}{\rm{d}}\tau \le \int_0^t {{{\rm{e}}^{ - \lambda \tau }}} {\beta _1}(\mu , t)s_{\phi k}^2{\rm{d}}\tau + \int_0^t {{{\rm{e}}^{ - \lambda \tau }}} {\beta _2}(\tau )\mu |{s_{\phi k}}|{\rm{d}}\tau $
成立. 其中, $\beta_1(\mu, t)=[\mu+t\|A\|{\rm e}^{\|A\| t}(1+\mu)^2]$, $\beta_2(t)$ $=$ $t\|A \|{\rm e}^{t\|A\|} (|s_k(0)|+\|{ e}_{k}(0)-bs_k(0)\|) +\|{ e}_{k}(0)$ $+$ $ b(\phi(t)-1) s_k(0)\|$, $ b=[0, 0, \cdots, 0, 1]^{\rm T}$,
$A = \left( {\matrix{ 0&1&0& \cdots &{} \cr 0&0&1& \cdots &{} \cr \vdots & \vdots & \vdots & \ddots & \vdots \cr 0&{ - {c_1}}&{ - {c_2}}& \cdots &{} \cr } } \right)$
(5) 证明 . 由式(4)知,
$\eqalign{ &{{\dot s}_{\phi k}} = {c^{\rm{T}}}{e_k} + {{\dot e}_{nk}} - \eta (t){s_k}(0) \cr &{{\dot e}_k} = A{e_k} + b({{\dot s}_{\phi k}} + \eta (t){s_k}(0)) \cr} $
(6) 式中, $ c=[0, c_1, c_2, \cdots, c_{n-1}]^{\rm T}$, $\eta(t)= \dfrac{{\rm d}\phi}{{\rm d}t}$. 对式(6)两边定积分, 可得
$\eqalign{ &{e_k} = \int_0^t A {e_k}{\rm{d}}\tau + b{s_{\phi k}} + {e_k}(0) - \cr &b{s_{\phi k}}(0) + b{s_k}(0)\int_0^t \eta (\tau ){\rm{d}}\tau = \int_0^t A {e_k}{\rm{d}}\tau + b{s_{\phi k}} + {e_k}(0) + b(\phi (t) - 1){s_k}(0) \cr} $
(7) 两边取范数
${e_k} \le \int_0^t {A} {e_k}{\rm{d}}\tau + |{s_{\phi k}}| + {e_k}(0) + b(\phi (t) - 1){s_k}(0)$
利用Bellman引理%
${e_k} \le A{{\rm{e}}^{tA}}\int_0^t | {s_{\phi k}}|{\rm{d}}\tau + |{s_{\phi k}}| + {\beta _2}$
(8) 在式(8)的两边同乘以${\rm e}^{-\lambda t}\mu|{s}_{\phi k}|$后取定积分, 根据柯西不等式可以推得
$\eqalign{ &\int_0^t {{{\rm{e}}^{ - \lambda \tau }}} \mu |{s_{\phi k}}|{e_k}{\rm{d}}\tau \le A{{\rm{e}}^{At}} \cr &\int_0^t {{{\rm{e}}^{ - \lambda \tau }}} \mu (\tau )|{s_{\phi k}}(\tau )|\int_0^v | {s_{\phi k}}(v)|{\rm{d}}v{\rm{d}}\tau + \cr &\int_0^t {{{\rm{e}}^{ - \lambda \tau }}} \mu s_{\phi k}^2{\rm{d}}\tau + \int_0^t {{{\rm{e}}^{ - \lambda \tau }}} {\beta _2}(\tau )\mu |{s_{\phi k}}|{\rm{d}}\tau \le \cr &A{{\rm{e}}^{At}}{[\int_0^t {{{\rm{e}}^{ - \lambda \tau }}} (1 + \mu (\tau ))|{s_{\phi k}}(\tau )|{\rm{d}}\tau ]^2} + \cr &\int_0^t {{{\rm{e}}^{ - \lambda \tau }}} \mu s_{\phi k}^2{\rm{d}}\tau + \int_0^t {{{\rm{e}}^{ - \lambda \tau }}} {\beta _2}(\tau )\mu |{s_{\phi k}}|{\rm{d}}\tau \le \cr &\int_0^t {{{\rm{e}}^{ - \lambda \tau }}} [\mu + tA{{\rm{e}}^{At}}{(1 + \mu )^2}]s_{\phi k}^2{\rm{d}}\tau + \int_0^t {{{\rm{e}}^{ - \lambda \tau }}} {\beta _2}(\tau )\mu |{s_{\phi k}}|{\rm{d}}\tau \cr} $
文给出了修正滤波误差与系统误差之间的不等式关系, 在下文的控制器设计和收敛性分析中, 将利用该不等式关系处理不确定性.
对$V_{1k}=\dfrac{1}{2g_k}{\rm e}^{-\lambda t}s_{\phi k}^2$求关于时间的导数, 得
$\eqalign{ &{{\dot V}_{1k}} = - \lambda {1 \over {2{g_k}}}{{\rm{e}}^{ - \lambda t}}s_{\phi k}^2 + {{\rm{e}}^{ - \lambda t}}[ - {1 \over {2g_k^2}}{{\dot g}_k}s_{\phi k}^2 + {s_{\phi k}}g_k^{ - 1}({c^{\rm{T}}}{e_k} - {x_d}^{(n)} - \eta (t){s_k} \cr &(0) + {\theta ^{\rm{T}}}{\varphi _k}) + {s_{\phi k}}(g_k^{ - 1}\Delta {f_k} + {u_k})] \cr} $
根据假设1和假设2, 可以推出
$\begin{align} &{{s}_{\phi k}}{{({{g}_{k}})}^{-1}}\Delta {{f}_{k}}={{s}_{\phi k}}[g_{d}^{-1}\Delta {{f}_{d}}+g_{d}^{-1}(\Delta {{f}_{k}}-\Delta {{f}_{d}})+ \\ &g_{k}^{-1}g_{d}^{-1}({{g}_{d}}-{{g}_{k}})\Delta {{f}_{d}}+g_{k}^{-1}g_{d}^{-1}({{g}_{d}}-{{g}_{k}})(\Delta {{f}_{k}}-\Delta {{f}_{d}})] \\ &\le {{s}_{\phi k}}g_{d}^{-1}\Delta {{f}_{d}}+|{{s}_{\phi k}}|g_{d}^{-1}{{\alpha }_{fk}}{{e}_{k}}+|{{s}_{\phi k}}|g_{k}^{-1}g_{d}^{-1}|\Delta {{f}_{d}}|{{\alpha }_{gk}}{{e}_{k}}+|{{s}_{\phi k}}| \\ &g_{k}^{-1}g_{d}^{-1}{{\alpha }_{fk}}{{e}_{k}}{{\alpha }_{gk}}{{e}_{k}} \\ \end{align}$
和
${{s}_{\phi k}}{{({{g}_{k}})}^{-1}}({{c}^{\text{T}}}{{e}_{k}}+{{\theta }^{\text{T}}}{{\varphi }_{k}}-{{x}_{d}}^{(n)}-\eta (t){{s}_{k}}(0))$
$\eqalign{ &{s_{\phi k}}{({g_k})^{ - 1}}({c^{\rm{T}}}{e_k} + {\theta ^{\rm{T}}}{\varphi _k} - {x_d}^{(n)} - \eta (t){s_k}(0)) \le {s_{\phi k}}g_d^{ - 1} \cr &({c^{\rm{T}}}{e_k} + {\theta ^{\rm{T}}}{\varphi _k} - {x_d}^{(n)} - \eta (t){s_k}(0)) + \cr &|{s_{\phi k}}|{({g_k}{g_d})^{ - 1}}{e_k}{\alpha _{gk}}|{c^{\rm{T}}}{e_k} + {\theta ^{\rm{T}}}{\varphi _k} - {x_d}^{(n)} - \eta (t){s_k}(0)| \cr} $
结合以上三式, 有
$\eqalign{ &{V_{1k}} \le \int_0^t {{{\rm{e}}^{ - \lambda \tau }}} [ - {\lambda \over {2{g_k}}}s_{\phi k}^2 - {1 \over {2g_k^2}}{{\dot g}_k}s_k^2 + \cr &_{\phi k}({p^{\rm{T}}}{\psi _{1k}} + {u_k})]{\rm{d}}\tau + \int_0^t {{{\rm{e}}^{ - \lambda \tau }}} {\rho _k}|{s_{\phi k}}|{e_k}{\rm{d}}\tau \cr} $
(9) 其中,
$\eqalign{ &{\rho _k} = g_d^{ - 1}{\alpha _{fk}} + g_{mk}^{ - 1}g_d^{ - 1}|\Delta {f_d}|{\alpha _{gk}} + g_{mk}^{ - 1}g_d^{ - 1}{\alpha _{fk}}{\alpha _{gk}}{e_k} \cr & + {({g_{mk}}{g_d})^{ - 1}}{\alpha _{gk}}|{c^{\rm{T}}}{e_k} - {x_d}^{(n)} - \eta (t){s_k}(0)| + {({g_{mk}}{g_d})^{ - 1}}{\alpha _{gk}} \cr &\theta {\varphi _k}\;p = {\left( {\matrix{ {g_d^{ - 1}\Delta {f_d}}&{g_d^{ - 1}}&{g_d^{ - 1}{\theta ^{\rm{T}}}\;} \cr } } \right)^{\rm{T}}}\;{\psi _{1k}} = \cr &1{c^{\rm{T}}}{e_k} - {x_d}^{(n)} - \eta (t){s_k}(0){\varphi _k}^{\rm{T}} \cr} $
根据引理1,
$\eqalign{ &\int_0^t {{{\rm{e}}^{ - \lambda \tau }}} {\rho _k}|{s_{\phi k}}|{e_k}{\rm{d}}\tau \le \int_0^t {{{\rm{e}}^{ - \lambda \tau }}} {\beta _1}({\rho _k}, t)s_{\phi k}^2{\rm{d}}\tau + \cr &\int_0^t {{{\rm{e}}^{ - \lambda \tau }}} {\beta _2}(\tau ){\rho _k}|{s_{\phi k}}|{\rm{d}}\tau = \int_0^t {{{\rm{e}}^{ - \lambda \tau }}} {\beta _1}({\rho _k}, t)s_{\phi k}^2{\rm{d}}\tau + \cr &\int_0^t {{{\rm{e}}^{ - \lambda \tau }}} |{s_{\phi k}}|{\vartheta ^{\rm{T}}}(\tau ){\psi _{2k}}{\rm{d}}\tau \cr} $
成立. 其中
$\begin{align} &\vartheta (t)=[g_{d}^{-1}{{l}_{1}}{{\beta }_{5}}, g_{d}^{-1}|\Delta {{f}_{d}}|{{l}_{1}}{{\beta }_{5}}, g_{d}^{-1}\| \\ &\theta \|{{l}_{1}}{{\beta }_{5}}, g_{d}^{-1}{{\beta }_{5}}, g_{d}^{-1}|\Delta {{f}_{d}}|{{\beta }_{5}}, g_{d}^{-1}\|\theta \|{{\beta }_{5}}|{{]}^{\text{T}}} \\ &{{\psi }_{2k}}=\left[ {{l}_{2}}\frac{{{\beta }_{3}}}{{{\beta }_{5}}}, g_{mk}^{-1}{{\alpha }_{gk}}\frac{{{\beta }_{3}}}{{{\beta }_{5}}}, g_{mk}^{-1}{{\alpha }_{gk}}\|{{\varphi }_{k}}\|\frac{{{\beta }_{3}}}{{{\beta }_{5}}}, \right. \\ &{{\left. {{l}_{2}}\frac{{{\beta }_{4}}}{{{\beta }_{5}}}, g_{mk}^{-1}{{\alpha }_{gk}}\frac{{{\beta }_{4}}}{{{\beta }_{5}}}, g_{mk}^{-1}{{\alpha }_{gk}}\|{{\varphi }_{k}}\|\frac{{{\beta }_{4}}}{{{\beta }_{5}}} \right]}^{\text{T}}} \\ &\ {{l}_{1}}=t\|A\|{{\text{e}}^{t\|A\|}}\ \\ &{{l}_{2}}={{\alpha }_{fk}}+g_{mk}^{-1}{{\alpha }_{fk}}{{\alpha }_{gk}}\|{{e}_{k}}\|+g_{mk}^{-1}{{\alpha }_{gk}}|{{c}^{\text{T}}}{{e}_{k}}-\eta (t){{s}_{k}}(0)-{{x}_{d}}^{(n)}| \\ \end{align}$
其中, $\beta_1$与$\beta_2$的含义见引理1, $\beta_3(t)=(|s_k(0)|$ $+$ $\|{ e}_{k}(0)$ $-$ $ bs_k(0)\|)$, $\beta_4= \|{ e}_{k}(0)+ b(\phi(t)-1) s_k(0)\|$, $\beta_5$ $>$ $0$ 为设计参数.
将上式的结果应用于式(9), 有
$\begin{align} &{{V}_{1k}}\le \int_{0}^{t}{{{\text{e}}^{-\lambda \tau }}}[\frac{-\lambda }{2{{g}_{k}}}s_{\phi k}^{2}\frac{1}{2g_{k}^{2}}{{{\dot{g}}}_{k}}s_{k}^{2}+{{s}_{\phi k}}({{p}^{\text{T}}}{{\psi }_{1k}}+{{u}_{k}})] \\ &\text{d}\tau +\int_{0}^{t}{{{\text{e}}^{-\lambda \tau }}}{{\beta }_{1}}({{\rho }_{k}}, t)s_{\phi k}^{2}\text{d}\tau +\int_{0}^{t}{|}{{s}_{\phi k}}|{{\vartheta }^{\text{T}}}{{\psi }_{2k}}\text{d}\tau \\ \end{align}$
(10) 由此, 设计控制律
$\begin{align} &{{u}_{k}}=-p_{k}^{\text{T}}{{\psi }_{1k}}-\vartheta _{k}^{\text{T}}{{\psi }_{2k}}\text{tanh}({{\gamma }_{1}}(k+1)\times \\ &(k+2)\vartheta _{k}^{\text{T}}{{\psi }_{2k}}{{s}_{\phi k}})-\frac{{{s}_{\phi k}}\|{{\psi }_{1k}}\|}{\varepsilon }-\frac{{{s}_{\phi k}}}{\varepsilon {{g}_{mk}}}(1+|{{c}^{\text{T}}}{{e}_{k}}|+\|{{\varphi }_{k}}\|+{{\alpha }_{fk}}\|{{e}_{k}}\|) \\ \end{align}$
(11) 其中,
$\begin{align} & {{p}_{k}}=\text{sat}({{{\hat{p}}}_{k}}) \\ & {{{\hat{p}}}_{k}}=\text{sat}({{{\hat{p}}}_{k-1}})+{{\gamma }_{2}}{{s}_{\phi k}}{{\psi }_{1k}}, {{{\hat{p}}}_{-1}}=0 \\ \end{align}$
(12) $\begin{align} & {{\vartheta }_{k}}=\text{sat}({{{\hat{\vartheta }}}_{k}}) \\ & {{{\hat{\vartheta }}}_{k}}=\text{sat}({{{\hat{\vartheta }}}_{k}}-1)+{{\gamma }_{3}}|{{s}_{\phi k}}|{{\psi }_{2k}}, {{{\hat{\vartheta }}}_{-1}}=0 \\ \end{align}$
(13) 式中, $ p_{k}$为对$ p$的估计, $ \vartheta_k$为对$ \vartheta$的估计, $\gamma_1>0$, $\gamma_2$ $>$ $0$, $\gamma_3>0$, $\varepsilon\gg 0$. 对于$\hat a \in R$,
$\text{sat}(\hat{a})=\left\{ \begin{align} &\bar{a}, |\hat{a}|\text{sgn}(\hat{a})>\bar{a} \\ &\hat{a}, 其他 \\ \end{align} \right.\ $
(14) $\bar{a}$为对应的限幅, 对于向量$\hat { a}$, ${\rm sat}(\hat { a})$ 表示对向量中的各元素进行上述限幅运算.
在上文的设计过程中, 系统中原有的参数不确定性$ \theta$和非参数不确定性$\Delta f_k$, gk被转化为新的参数不确定性$ p$, $ \vartheta$, 即非参数不确定性的补偿问题被转化为未知时变参数的估计问题. 利用学习方法估计线性时变参数不确定性的已有成果较多, 可以采用部分限幅学习、完全限幅学习或无限幅学习的方法对未知时变参数进行估计. 本文采用的是完全限幅学习方案.
3. 收敛性分析
闭环系统具有的稳定性与收敛性方面的性质可总结为定理1.
定理1. 在$ x_k(0)$, $k=0, 1, 2, \cdots$, 取值任意的情况下, 将控制律(11)施加于系统(1), 足够多次迭代后, 可使sk在区间$[t_1, T]$上收敛于零, 即
$\lim_{k\rightarrow +\infty} s_k(t)=0, t\in [t_1, T] $
(15) 并保证闭环系统所有信号有界.
证明 1) 系统变量的有界性 对Lyapunov函数$V_{2k}=\dfrac{1}{2}s_{\phi k}^2$求导
$\begin{align} &{{{\dot{V}}}_{2k}}={{s}_{\phi k}}({{c}^{\text{T}}}{{e}_{k}}-{{x}_{d}}^{(n)}-\eta {{s}_{k}}(0)+{{\theta }^{\text{T}}}{{\varphi }_{k}}+{{f}_{k}}+{{g}_{k}}{{u}_{k}})\le \\ &|{{s}_{\phi k}}||{{c}^{\text{T}}}{{e}_{k}}|+|{{s}_{\phi k}}|(|{{x}_{d}}^{(n)}|+|{{f}_{d}}|+|\eta {{s}_{k}}(0)|)+|{{s}_{\phi k}}| \\ &\|\theta \|\|{{\varphi }_{k}}\|+|{{s}_{\phi k}}|{{\alpha }_{fk}}\|{{e}_{k}}\|+|{{s}_{\phi k}}|{{g}_{k}}\|{{p}_{k}}\|\|{{\psi }_{1k}}\| \\ &-\frac{{{g}_{k}}s_{\phi k}^{2}}{\varepsilon {{g}_{mk}}}(1+|{{c}^{\text{T}}}{{e}_{k}}|+\|{{\varphi }_{k}}\|+{{\alpha }_{fk}}\|{{e}_{k}}\|)-{{g}_{k}}\frac{s_{\phi k}^{2}\|{{\psi }_{1k}}\|}{\varepsilon } \\ &-{{s}_{\phi k}}\vartheta _{k}^{\text{T}}{{\psi }_{2k}}\text{tanh}({{\gamma }_{1}}(k+1)(k+2)\vartheta _{k}^{\text{T}}{{\psi }_{2k}}{{s}_{\phi k}}) \\ \end{align}$
(16) 当$|s_{\phi k}|\geq \varepsilon $时,
$|{{s}_{\phi k}}||{{c}^{\text{T}}}{{e}_{k}}|-\frac{{{g}_{k}}s_{\phi k}^{2}}{\varepsilon {{g}_{mk}}}|{{c}^{\text{T}}}{{e}_{k}}|\le 0$
(17) $|{{s}_{\phi k}}|{{\alpha }_{fk}}\|{{e}_{k}}\|-\frac{{{g}_{k}}s_{\phi k}^{2}}{\varepsilon {{g}_{mk}}}{{\alpha }_{fk}}\|{{e}_{k}}\|\le 0$
(18) 当$|{{s}_{\phi k}}|\ge \varepsilon (|{{x}_{d}}^{(n)}|+|{{f}_{d}}|+|\eta {{s}_{k}}(0)|)$时,
$|{{s}_{\phi k}}|(|{{x}_{d}}^{(n)}|+|{{f}_{d}}|)-\frac{{{g}_{k}}s_{\phi k}^{2}}{\varepsilon {{g}_{mk}}}\le 0$
(19) 当$|s_{\phi k}|\geq \varepsilon \| p_{k}\| $时,
$|s_{\phi k}|g_k \| p_{k}\| \| \psi_{1k} \| -g_k\frac{s_{\phi k}^2\| \psi_{1k}\|}{\varepsilon }\leq 0 $
(20) 当$|s_{\phi k}|\geq \varepsilon \| \theta\| $时,
$|s_{\phi k}| \| \theta\| \| \varphi_k\| -\frac{g_k s_{\phi k}^2}{\varepsilon g_{mk}}\| \varphi_k\|\leq 0 $
(21) 综合式(16) $\sim$ (21), 可知当
$|{{s}_{\phi k}}|\ge \max (\varepsilon , \varepsilon (|{{x}_{d}}^{(n)}|+|{{f}_{d}}|+|\eta {{s}_{k}}(0)|), \varepsilon \|{{p}_{k}}\|, \varepsilon \|\theta \|)$
(22) 时,
${{{\dot{V}}}_{2k}}\le -{{s}_{\phi k}}\vartheta _{k}^{\text{T}}{{\psi }_{2k}}\text{tanh}({{\gamma }_{1}}(k+1)(k+2)\vartheta _{k}^{\text{T}}{{\psi }_{2k}}{{s}_{\phi k}})$
根据双曲函数的性质可知
${{\gamma }_{1}}(k+1)(k+2){{s}_{\phi k}}\vartheta _{k}^{\text{T}}{{\psi }_{2k}}\times \text{tanh}({{\gamma }_{1}}(k+1)(k+2)\vartheta _{k}^{\text{T}}{{\psi }_{2k}}{{s}_{\phi }}k)\ge 0$
结合以上两式, 可以推出满足条件(22)时
$\dot V_{2k}\leq 0 $
(23) 由此可以得到$s_{\phi k}$的有界性, 在此基础上, 结合饱和函数的性质, 易得其他变量也为有界.
2) 误差的收敛性 选择Lyapunov泛函
${{L}_{k}}={{V}_{1k}}+\frac{1}{2{{\gamma }_{2}}}\int_{0}^{t}{{{\text{e}}^{-\lambda \tau }}}\tilde{p}_{k}^{\text{T}}{{{\tilde{p}}}_{k}}\text{d}\tau \frac{1}{2{{\gamma }_{3}}}\int_{0}^{t}{{{\text{e}}^{-\lambda \tau }}}\tilde{\vartheta }_{k}^{\text{T}}{{{\tilde{\vartheta }}}_{k}}\text{d}\tau $
(24) 式中, $\tilde{ p}_{k}= p-p_{k}$, $\tilde{ \vartheta}_{k}= \vartheta-\vartheta_{k}$.
前文已证闭环系统变量均为有界, 结合假设3, 可知取足够大的$\lambda$, 由式(10)和式(11), 可得
$\begin{align} &{{V}_{1k}}\le \int_{0}^{t}{{{\text{e}}^{-\lambda \tau }}}[{{s}_{\phi k}}\tilde{p}_{k}^{\text{T}}{{\psi }_{1k}}+|{{s}_{\phi k}}|\tilde{\vartheta }_{k}^{\text{T}}{{\psi }_{2k}}+ \\ &|{{s}_{\phi k}}|\vartheta _{k}^{\text{T}}{{\psi }_{2k}}-{{s}_{\phi k}}\vartheta _{k}^{\text{T}}{{\psi }_{2k}}\times \text{tanh}({{\gamma }_{1}}(k+1)(k+2)\vartheta _{k}^{\text{T}}{{\psi }_{2k}}{{s}_{\phi k}})]\text{d}\tau \\ \end{align}$
对于$a\in R$, $\varepsilon>0$, 不等式$0\le |a|-a\text{tanh}({{a}_{\varepsilon }})\le {{\delta }_{\varepsilon }}$成立 [26-27], 式中$\delta$为一常数, 满足$\delta ={{\text{e}}^{-}}(\delta +1)$, 其值为0.2785. 于是, 可知
$\begin{align} &|{{s}_{\phi k}}|\vartheta _{k}^{\text{T}}{{\psi }_{2k}}-{{s}_{\phi k}}\vartheta _{k}^{\text{T}}{{\psi }_{2k}}\text{tanh}({{\gamma }_{1}}(k+1)\times (k+2)\vartheta _{k}^{\text{T}}{{\psi }_{2k}}{{s}_{\phi k}}) \\ &\le \delta {{\gamma }_{1}}(k+1)(k+2) \\ \end{align}$
(25) 成立. 据此, 可以推出
${{V}_{1k}}\le \int_{0}^{t}{{{\text{e}}^{-\lambda \tau }}}({{s}_{\phi k}}\tilde{p}_{k}^{\text{T}}{{\psi }_{1k}}+|{{s}_{\phi k}}|\tilde{\vartheta }_{k}^{\text{T}}{{\psi }_{2k}})\text{d}\tau +\int_{0}^{t}{{{\text{e}}^{-\lambda \tau }}}{{\gamma }_{1}}(k+1)(k+2)\text{d}\tau $
(26) 由式(26)及式(24), 知
$\begin{align} &{{L}_{k}}-{{L}_{k-1}}\le \int_{0}^{t}{{{\text{e}}^{-\lambda \tau }}}({{s}_{\phi k}}\tilde{p}_{k}^{\text{T}}{{\psi }_{1k}}+|{{s}_{\phi k}}|\tilde{\vartheta }_{k}^{\text{T}}{{\psi }_{2k}})\text{d}\tau + \\ &\int_{0}^{t}{{{\text{e}}^{-\lambda \tau }}}\delta {{\gamma }_{1}}(k+1)(k+2)\text{d}\tau -{{V}_{1k-1}}+\frac{1}{2{{\gamma }_{2}}}\int_{0}^{t}{{{\text{e}}^{-\lambda \tau }}}(\tilde{p}_{k}^{\text{T}}{{{\tilde{p}}}_{k}}-\tilde{p}_{k-1}^{\text{T}}{{{\tilde{p}}}_{k-1}})\text{d}\tau + \\ &\frac{1}{2{{\gamma }_{3}}}\int_{0}^{t}{{{\text{e}}^{-\lambda \tau }}}(\tilde{\vartheta }_{k}^{\text{T}}{{{\tilde{\vartheta }}}_{k}}-\tilde{\vartheta }_{k-1}^{\text{T}}{{{\tilde{\vartheta }}}_{k-1}})\text{d}\tau \\ \end{align}$
(27) 利用学习律(12)和(13), 分别可以推出
$\begin{align} &\frac{1}{2{{\gamma }_{2}}}\left( \tilde{p}_{k}^{\text{T}}{{{\tilde{p}}}_{k}}-\tilde{p}_{k-1}^{\text{T}}{{{\tilde{p}}}_{k-1}} \right)+{{s}_{\phi k}}\tilde{p}_{k}^{\text{T}}{{\psi }_{1k}}\le \\ &-\frac{1}{{{\gamma }_{2}}}(p-{{p}_{k{{)}^{\text{T}}}(p}}_{k}-{{p}_{k-1}})+{{s}_{\phi k}}\tilde{p}_{k}^{\text{T}}{{\psi }_{1k}}= \\ &\frac{1}{{{\gamma }_{2}}}{{(p-{{p}_{k}})}^{\text{T}}}(-{{p}_{k}}+{{p}_{k-1}}+{{\gamma }_{2}}{{s}_{\phi k}}{{\psi }_{1k}})= \\ &\frac{1}{{{\gamma }_{2}}}{{(p-\text{sat}({{{\hat{p}}}_{k}}))}^{\text{T}}}({{{\hat{p}}}_{k}}-\text{sat}({{{\hat{p}}}_{k}}))\le 0 \\ \end{align}$
(28) 及
$\begin{align} &\frac{1}{2{{\gamma }_{3}}}\left( \tilde{\vartheta }_{k}^{\text{T}}{{{\tilde{\vartheta }}}_{k}}-\tilde{\vartheta }_{k-1}^{\text{T}}{{{\tilde{\vartheta }}}_{k-1}} \right)+|{{s}_{\phi k}}|{{{\tilde{\vartheta }}}_{k}}{{\psi }_{2k}}\le -\frac{1}{{{\gamma }_{3}}}{{({{\vartheta }_{k}}-{{\vartheta }_{k-1}})}^{\text{T}}}{{{\tilde{\vartheta }}}_{k}}+ \\ &|{{s}_{\phi k}}|{{{\tilde{\vartheta }}}_{k}}{{\psi }_{2k}}=\frac{1}{{{\gamma }_{3}}}{{(-{{\vartheta }_{k}}+{{\vartheta }_{k-1}}+{{\gamma }_{3}}|{{s}_{\phi k}}|)}^{\text{T}}}{{{\tilde{\vartheta }}}_{k}}{{\psi }_{2k}}= \\ &\frac{1}{{{\gamma }_{3}}}{{({{{\hat{\vartheta }}}_{k}}-\text{sat}({{{\hat{\vartheta }}}_{k}}))}^{\text{T}}}(\vartheta -\text{sat}({{{\hat{\vartheta }}}_{k}}))\le 0 \\ \end{align}$
(29) 将式(28)和式(29)的结果应用于式(27)
${{L}_{k}}-{{L}_{k-1}}\le {{\int }_{0}}^{t}{{\text{e}}^{-\lambda \tau }}\delta {{\gamma }_{1}}(k+1)(k+2)\text{d}\tau -{{V}_{1k-1}}$
进一步地
${{L}_{k}}\le {{L}_{0}}+\int_{0}^{t}{{{\text{e}}^{-\lambda \tau }}}\sum\limits_{i=1}^{k}{\delta {{\gamma }_{1}}}(i+1)(i+2)\text{d}\tau -\sum\limits_{i=0}^{k-1}{\left( \frac{1}{2{{g}_{i}}}{{\text{e}}^{-\lambda t}}s_{\phi i}^{2} \right)}$
由于$L_0$为非负有界量, 且
$\begin{align} &\underset{k\to +\infty }{\mathop{\lim }}\,\int_{0}^{t}{{{\text{e}}^{-\lambda \tau }}}\sum\limits_{i=1}^{k}{\delta {{\gamma }_{1}}}(i+1)(i+2)\text{d}\tau =\frac{\delta }{{{\gamma }_{1}}\lambda }\left( 1-{{\text{e}}^{-\lambda t}} \right) \\ &\underset{k\to +\infty }{\mathop{\lim }}\,\left( \frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\cdots - \right.\left. \frac{1}{k+1}+\frac{1}{k+1}-\frac{1}{k+2} \right)<{{\frac{\delta }{2{{\gamma }_{1}}\lambda }}_{k}} \\ \end{align}$
(30) 是有界的, 故根据数列收敛的必要性, 可知
$\lim_{k\rightarrow +\infty}s_{\phi k}=0 $
(31) 至此, 根据$s_{\phi k}$的定义, 可得
$\lim_{k\rightarrow +\infty} s_k(t)= 0, t\in [t_1, T] $
上文给出了基于滤波误差初始修正的自适应迭代学习控制方法, 适用于系统初态任意情形. 经过足够多次迭代后, 籍由$s_{\phi k}$在整个作业区间收敛于零, 实现了sk 在预设的部分作业区间收敛于零.
应该注意的是, 严格复位即$ x_k(0)= x_d(0)$, $k=$ $0, 1, 2, \cdots$ 情形, 属本文所考虑情形的一种特殊情况. 在此情况下, $s_k(0)=0$, $ e_k(0)=0$, $s_{\phi k}=s_k$, 引理1结论中的$\beta_2=0$, 控制律(11)中的$ \psi_{2k}= 0$. 引理1的结论及控制律(11)分别退化为
$\int_{0}^{t}{{{\text{e}}^{-\lambda \tau }}}{{\mu }_{k}}|{{s}_{\phi k}}|\|{{e}_{k}}\|\text{d}\tau \le \int_{0}^{t}{{{\text{e}}^{-\lambda \tau }}}{{\beta }_{1}}({{\mu }_{k}}, t)s_{\phi k}^{2}\text{d}\tau $
及
${{u}_{k}}=-p_{k}^{\text{T}}{{\psi }_{1k}}-\frac{{{s}_{k}}\|{{\psi }_{1k}}\|}{\varepsilon }-\frac{{{s}_{k}}}{\varepsilon {{g}_{mk}}}(1+|{{c}^{\text{T}}}{{e}_{k}}|+\|{{\varphi }_{k}}\|+{{\alpha }_{fk}}\|{{e}_{k}}\|)$
(32) 其中, $ p_{k}$的取值同式(12). 相应的收敛性质可总结为定理2.
定理2. 在$ x_k(0)= x_d(0)$, $k=0, 1, 2, \cdots$ 情形下, 将控制律(32)施加于系统(1), 足够多次迭代后, 可使sk在整个作业区间$[0, T]$上收敛于零, 即
$\lim_{k\rightarrow + \infty} s_k(t)= 0, t\in [0, T] $
(33) 并保证闭环系统中的所有信号有界.
容易看出, 修正滤波误差的构造方法简单. 由上文的设计和分析过程可以看出, 采用滤波误差初始修正方法设计控制器, 勿需进行分类讨论, 由此具有使用上的便捷性. 经过足够多次迭代后, 可实现滤波误差在预设的部分作业区间收敛于零.
4. 仿真算例
考虑如下倒立摆系统
$\begin{align} &{{{\dot{x}}}_{1k}}={{x}_{2k}}dot{{x}_{2k}}=g\sin {{x}_{1k}}-\frac{mlx_{2k}^{2}\cos {{x}_{1k}}\sin {{x}_{1k}}}{{{m}_{c}}+m} \\ &l\left( \frac{4}{3}-\frac{m{{\cos }^{2}}{{x}_{1k}}}{{{m}_{c}}+m} \right)+\frac{\cos {{x}_{1k}}}{{{m}_{c}}+m}l\left( \frac{4}{3}-\frac{m\text{co}{{\text{s}}^{2}}{{x}_{1k}}}{{{m}_{c}}+m} \right){{u}_{k}} \\ \end{align}$
(34) 这里, $x_{1k}$和$x_{2k}$分别是倒立摆的角位移与角速度, $x_{1k}(0)=1.2+0.1(r_1-0.5)$, $x_{2k}(0)=$ $-0.2$ $+$ $0.05r_2$, $r_1$和$r_2$均为0 $\sim$ 1之间的随机数. $g=9.8 {\rm m}/{\rm s}^2$为重力加速度, $m_c=1$ kg是小车的质量, $m=0.1$ kg为摆的质量, $l=0.5$ m为摆长的一半, uk为小车的推力. 参考信号是$[{\cos}(\pi t), -\pi{\sin} (\pi t)]^{\rm T}$. $[x_{1 k}(0), x_{2k}(0)]^{\rm T}\neq[1, 0]^{\rm T}$. 视 $g\sin {{x}_{1k}}-\frac{mlx_{2k}^{2}\cos {{x}_{1k}}\sin {{x}_{1k}}}{{{m}_{c}}+m}l(\frac{4}{3}-m{{\cos }^{2}}{{x}_{1k}}{{m}_{c}}+m)$和 $\frac{\cos {{x}_{1k}}}{{{m}_{c}}+m}l(\frac{4}{3}-m{{\cos }^{2}}{{x}_{1k}}{{m}_{c}}+m)$分别为$\Delta f_k$和gk. 考虑到实际系统存在的多种不确定性和扰动, 设各参数与标称值存在$\pm 40 % $的偏差, 取
$\begin{align} &{{g}_{mk}}=\frac{|{{\cos }_{1k}}|}{1.44+0.23\cos _{1k}^{2}}{{\alpha }_{fk}}= \\ &\frac{ml\max (|{{x}_{2k}}|, |{{{\dot{x}}}_{d}}|){{m}_{c}}+m}{l}{{\left( \frac{4}{3}-\frac{m}{{{m}_{c}}+m} \right)}^{2}} \\ &+g+ml\max (x_{2k}^{2}, \dot{x}_{d}^{2})2({{m}_{c}}+m)l\left( \frac{4}{3}-\frac{m}{{{m}_{c}}+m} \right)+ \\ &\max (x_{2k}^{2}, \dot{x}_{d}^{2})2({{m}_{c}}+m)l{{\left( \frac{4}{3}-\frac{m}{{{m}_{c}}+m} \right)}^{2}}\times \frac{m}{{{m}_{c}}+m}{{\alpha }_{gk}}= \\ &\frac{1}{{{m}_{c}}+m}l\left( \frac{4}{3}-\frac{m}{{{m}_{c}}+m} \right)+\frac{\frac{1}{{{m}_{c}}+m}}{l{{\left( \frac{4}{3}-\frac{m}{{{m}_{c}}+m} \right)}^{2}}} \\ \end{align}$
采用控制律(11)及相应学习律进行仿真. 仿真参数取 $\gamma_1=1$, $\gamma_2=5$, $\gamma_3=0.01$, $\varepsilon=100$, $\bar{p}=20$, $\bar{ \vartheta}$ $=$ $80$, $c_1=10$, $T=2$, $t_1=0.3$, $\beta_5=10$, $\beta_1$, $\beta_2$, $\beta_3$, $\beta_4$的含义见前文. 采用第2节给出的方案构造$\phi(t)$. 迭代30 次后, 仿真结果如图 1 $\sim$ 图 6所示. 图 1和图 2是第30次迭代时的系统状态情况. 图 3和图 4分别是第30次迭代过程中的状态误差和控制输入情况. 图 5是在$k=1, 5, 10, 20, 30$ 等次迭代过程中的滤波误差情况, 可以看出, 经过足够多次迭代后, 可实现滤波误差sk在$[t_1, T]$上的取值为零. 图 6是Jk的收敛过程, 在该图中, $J_{k}= \max_{t\in[0, T]} |s_{\phi k}(t)| $.
本文与文献[22]采用仿真模型相同, 在上文的仿真中, 参数学习律的增益取值为5, 而在文献[22]的仿真中, 学习律的增益取值为30. 对比之下, 本文所提方法在学习增益较小的情况下, 仍具有较快的误差收敛速度.
为了进一步说明本文所提设计方法的有效性, 下面采用文献[20]所提的时变边界层方案解决相同问题, 迭代学习模糊控制器取
$u_k=u_{1k}+u_{2k} $
(35) 其中,
$\begin{align} &{{u}_{1k}}=w_{fk}^{\text{T}}(t){{z}_{f}}({{x}_{k}})+w_{gk}^{\text{T}}(t){{z}_{g}}({{x}_{k}}){{\varpi }_{k}}- \\ &\text{sat}\left( \frac{{{s}_{k}}}{{{v}_{k}}} \right){{\epsilon }_{k}}(1+{{\varpi }_{k}}){{u}_{2k}}=-{{\gamma }_{4}}{{s}_{vk}}z_{f}^{\text{T}}({{x}_{k}}){{z}_{f}}({{x}_{k}})- \\ &{{\gamma }_{5}}{{s}_{vk}}z_{g}^{\text{T}}({{x}_{k}}){{z}_{g}}({{x}_{k}})\varpi _{k}^{2}-{{\gamma }_{6}}{{s}_{vk}}{{(1+{{\varpi }_{k}})}^{2}}{{\varpi }_{k}}= \\ &-{{c}_{1}}{{e}_{2k}}+{{{\ddot{x}}}_{d}}-{{\gamma }_{7}}{{s}_{vk}}\ {{w}_{fk}}={{{\hat{w}}}_{fk}}=\text{sat}({{{\hat{w}}}_{fk-1}})-{{\gamma }_{8}}{{s}_{vk}}{{z}_{f}}({{x}_{k}}), {{{\hat{w}}}_{f-1}}=0 \\ &{{w}_{gk}}=\text{sat}({{{\hat{w}}}_{gk}}){{{\hat{w}}}_{gk}}=\text{sat}({{{\hat{w}}}_{gk-1}})-{{\gamma }_{9}}{{s}_{vk}}{{z}_{g}}({{x}_{k}}), \\ &{{{\hat{w}}}_{g-1}}=0{{n}_{k}}=\text{sat}({{\epsilon }_{k}}=\text{sat}({{{\hat{\epsilon }}}_{k}})+{{\gamma }_{10}}{{s}_{vk}}(1+|{{\omega }_{k}}|), {{{\hat{\epsilon }}}_{-1}}=0 \\ \end{align}$
(36) 式中, $s_{vk} =s_k-v_k{\rm sat}({s_k}/{v_k})$, $v_k=|c_1e_{1k}(0)+e_{2k}(0)|{\rm e}^{-\sigma t}$, $\sigma>0$.
模糊基函数$ z_{f}( x_k)$与$ z_{g}( x_k)$ 均采用下述属函数构造
$\begin{align} &{{\mu }_{1}}({{x}_{i}})=1+{{\text{e}}^{5({{x}_{i}}+2)}} \\ &{{\mu }_{2}}({{x}_{i}})={{\text{e}}^{-{{({{x}_{i}}+1.5)}^{2}}}} \\ &{{\mu }_{3}}({{x}_{i}})={{\text{e}}^{-{{({{x}_{i}}+0.5)}^{2}}}} \\ &{{\mu }_{4}}({{x}_{i}})={{\text{e}}^{-{{({{x}_{i}}-0.5)}^{2}}}} \\ &{{\mu }_{5}}({{x}_{i}})={{\text{e}}^{-{{({{x}_{i}}-1.5)}^{2}}}} \\ &{{\mu }_{6}}({{x}_{i}})=1+{{\text{e}}^{-5({{x}_{i}}-2)}} \\ \end{align}$
$i=1, 2$. 具体构造方法见文献[20]的第2节. 不难看出, $ z_{f}( x_k)$与$ z_{g}( x_k)$ 均为36维的向量.
仿真参数取 $\gamma_4=\gamma_5=\gamma_6=\gamma_7=\gamma_8=\gamma_9=5$, $\gamma_{10}$ $=$ $0.01$, $\bar{w}_f=100$, $\bar{w}_g=100$, $\bar{\epsilon}=30$, $\sigma=5$, $c_1$的取值同前. 迭代30 次后, 仿真结果如图 7 $\sim$ 图 11所示. 图 7和图 8是第30次迭代时的系统状态情况. 图 9和图 10分别是第30次迭代过程中的状态误差和控制输入情况. 图 11是Jk的收敛过 程, 在该图中, $J_{k}= \max_{t\in[0, T]} |s_{v k}(t)| $.
对比图 1和图 2与图 7和图 8, 可以看出 本文提出的基于滤波误差初始修正学习控制方法和文献[20]所提的基于时变边界层的模糊学习控制方法, 均可用于解决迭代学习控制的初值问题, 实现系统状态对参考信号在部分作业区间上的精确跟踪. 在作业周期的后半段, 本文所提方法具有较好的误差收敛性能, 由图 3可见, 误差曲线在作业周期后半段几乎完全为零, 且曲线几乎没有波动; 而在图 9中, 误差曲线在作业区间后半段存在一定幅度的波动. 该现象产生的原因与修正滤波误差/时变边界层的构造方式有关. $s_{\phi k}(t)=0$, $t\in [0, T]$ 蕴含
$s_k(t)=0, t\in [t_1, T] $
(37) 但$s_{v k}(t)=0$, $t\in [0, T]$ 则意味着$s_k(t)\leq |s_k(0)|{\rm e}^{-\sigma t}$, $t\in [0, T]$. 对比图 7和图 11可以看出, 本文所提方法在不使用高增益反馈的情况下, 仍然具有较快的误差收敛速度和较好的控制精度.
仿真结果表明, 利用本文给出的滤波误差初始修正方法设计学习控制器, 可用于解决参数/非参数不确定学习控制系统的初值问题. 使用这种方法进行控制器设计时, 可以比较方便地构造出修正滤波误差, 由其设计的控制器具有较快的误差收敛速度, 能够获得较好的控制精度. 修正滤波误差的使用较为方便. 上述结果说明了本文所提控制方法的有效性.
5. 结论
本文提出基于滤波误差初始修正的自适应迭代学习控制方法, 解决参数/非参数混合不确定系统在任意初态情形下的轨迹跟踪问题. 这种设计方法处理非参数不确定性后, 将系统中原有的非参数不确定性补偿问题转化为线性时变参数估计问题, 达到简化设计的目的. 文中所构造的修正滤波误差兼具构造和使用方面的便捷性. 仿真结果表明, 对比已有主流方法, 本文所提方案可获得较快的误差收敛速度和较好的控制精度.
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图 4 人类和智能体的信息物理系统对比图
Fig. 4 Cyber-physical interaction system comparison between human and agent
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Minds and Machines, 2017, 27(4):625-638 doi: 10.1007/s11023-017-9418-5 期刊类型引用(24)
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