Data-driven Operating Monitoring for Coal-fired Power Generation Equipment: The State of the Art and Challenge
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摘要: 大容量、高参数、低能耗的百万千瓦超超临界机组是燃煤发电领域的重大装备, 已成为全国电力工业发展的主流方向, 其安全可靠运行对推动发电企业转型升级具有重要意义. 本文从分析以百万千瓦超超临界机组为代表的燃煤发电装备的本质特性出发, 揭示了其变负荷深度调峰导致的非平稳运行特性和全流程复杂耦合特性, 总结了燃煤发电过程区别于一般连续过程的问题, 指出了研究燃煤发电装备运行工况监控算法的必要性. 进而, 基于这些特性, 我们对面向燃煤发电装备工况监控的数据驱动算法近30年的发展进行了回顾和分析, 展示了算法发展的不同阶段. 在此基础上, 梳理了目前燃煤发电装备工况监控中存在的问题, 并进一步介绍了燃煤发电装备工况监控未来可能的发展方向.Abstract: As major equipment in coal-fired power generation, 1000 MW ultra-supercritical unit has advantages of large capacity, high parameter and low energy consumption, which has become the mainstream of the development of the power industry in China. Its safety and reliability in operation are of great significance to promote the transformation and upgrading of power generation enterprises. Starting from the analysis of essential characteristics of coal-fired power generation equipment, this article revealed the nonstationary operation characteristics caused by the variable load, deep peak shaving, and the complex correlation characteristics of the whole process. Then, it summarized the problems that the coal-fired power generation process is different from general continuous processes, and points out the necessity of studying monitoring algorithms for coal-fired power generation equipment. Furthermore, based on these characteristics, it reviewed and analyzed the development of the data-driven algorithms for coal-fired power generation equipment monitoring in the past 30 years, showing different stages of algorithm development. On this basis, this article presented the current problems in operation monitoring of coal-fired power generation equipment, and further introduced the possible development direction in the future.
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切换控制方法是解决一类难以建立精确数学模型的非线性系统的主要控制方法, 采用多模型切换控制方法的思想可追溯到20世纪70年代[1], 其特点是: 根据被控对象的不确定范围, 以多个模型来逼近对象的全局动态特性, 进而基于多个模型建立相应的控制器, 通过模型(控制器)的调度策略从而达到快速响应外界需要的目的. 自90年代以来, 该方法成为非线性系统控制领域的研究热点[2-5]. Chen和Narendra[6]针对一类零动态渐近稳定的非线性动态系统, 提出了一种带有非线性项估计与补偿的切换控制方法, 并取得了较好的控制效果. 沿着这种切换控制方法的设计思路, 文献[7-11]分别针对一类零动态不稳定的非线性系统, 采用不同的控制策略提出了非线性切换控制方法. 在上述文献中, 为了估计系统的未知非线性项, 分别采用BP (Back propagation)神经网络、高阶神经网络、神经模糊推理系统和模糊逻辑系统等智能工具对非线性项进行估计, 这种估计方法在估计非线性项时往往缺乏考虑非线性项中蕴含的可测数据信息, 造成部分可测数据信息也通过估计算法产生, 存在增大估计误差的可能性. 另外, 在设计线性控制器时, 以往的方法直接忽略非线性项, 并没有充分利用非线性的历史可测数据进行控制器设计, 造成有用数据丢失. 由于数据驱动控制方法是解决机理不明确或含不确定性机理模型的非线性系统建模与控制问题的有效方法[12], 其主要思想是直接利用被控对象的离线、在线数据来描述对象的运行规律和相关模式, 并结合反映系统参数、结构等数据, 实现非线性系统的预报、评价、调度、监控、诊断、决策、优化和控制等的各种期望功能[13-14]. 如文献[15]利用数据驱动方法, 提出了一种新的多状态空间模型状态估计方法. 文献[16]首次提出了一种基于数据驱动的非线性系统交替辨识算法. 另外, 近年来已有不少基于数据驱动的黑箱建模与控制方法(如迭代学习[17]、无模型自适应控制方法[18]和自适应动态规划方法[19-20]等)被提出. 文献[21]把基于模型的控制方法、数据驱动控制方法以及切换控制方法相结合, 优势互补, 提出了基于数据与虚拟未建模动态驱动的非线性切换控制方法, 但该方法在处理系统的虚拟未建模动态时仍然采用了智能估计算法, 该估计方法与文献[7-11]所采用的估计算法本质相同, 有待于进一步改进. 文献[22-24]提出了一种带死区的切换控制方法, 并采用了具有任意切换次数的控制策略. 文献[25-26]采用Backstepping 方法分别研究了一类具有下三角结构的非线性切换控制方法. 文献[27]提出了一种基于神经网络的非线性自适应输出反馈切换控制方法. 上述文献对特定的非线性系统都取得了较好的控制效果, 但仍然存在没有充分利用被控对象过程数据的缺陷. 文献[28-29]对于一类具有全状态约束的非线性系统, 提出一种具有随机切换的自适应智能控制方法, 为研究非线性系统提供了新思路.
综上, 本文在数据驱动控制方法、非线性切换控制方法的基础上, 提出了一种新的基于数据驱动的非线性自适应切换控制方法. 首先, 考虑到被控对象的非线性项历史数据可测, 本文把非线性项分解为前一时刻可测部分与未知增量的和, 并利用非线性项前一时刻的可测数据信息进行控制器设计, 克服了以往方法中没有充分利用非线性项可测数据信息, 造成有用数据信息丢失的不足. 在上述工作的基础上, 结合文献[9]和文献[21]中的未建模动态估计方法, 提出了一种未建模动态未知增量的估计算法, 在简化估计算法的同时, 提高估计精度. 其次, 分别设计了带有非线性项增量估计与补偿的非线性自适应控制器、带有非线性项前一拍数据补偿的非线性控制器和不带非线性补偿的线性自适应控制器, 三个自适应控制器采用改进文献[6]中的切换策略来协调控制被控对象. 这种控制方法结合了多模型切换控制的优势, 同时也充分利用了被控对象的大数据信息和知识, 通过补偿器的设计, 消除了非线性项对闭环系统的不利影响. 通过切换控制策略既保证了闭环系统的稳定性, 同时提高了闭环系统的瞬态性能. 在此基础上, 分析了闭环切换系统的稳定性和收敛性. 最后, 将所提的控制算法通过实验进行验证, 实验结果说明了该新型切换控制算法的有效性. 综上所述, 本文的主要创新点如下:
1) 针对一类难以建立精确数学模型的复杂非线性系统, 充分利用被控对象的过程大数据信息和有用知识, 提出了一种新的基于数据驱动的非线性自适应切换控制方法.
2) 利用数据驱动控制的思想, 将未建模动态的前一时刻可测数据用于控制器设计, 克服了常规切换控制算法没有充分利用数据, 造成有用数据丢失的不足.
3) 给出了所提的新型切换控制算法的设计方法, 并分析了闭环系统的稳定性和收敛性.
1. 控制问题描述
单输入单输出(Sinple-Input Sinple-Output, SISO)离散时间非线性动态系统可描述为:
$$ \begin{split} y(k + \tau ) = & f[y(k), \cdots ,y(k - {n_s} + 1),u(k),\\ &u(k - 1), \cdots ,u(k - {m_s})] \end{split} $$ (1) 其中,
$ \tau $ 为系统的时滞且$1 \le \tau < {n_s},\;$ $u(k) \in{\rm{{R} }}$ ,$y(k) \in {\rm{{R}}}$ , 分别为系统在时刻的输入和输出; 系统阶次$ {n_s} $ 和$ {m_s} $ 已知;$f( \cdot ) \in \rm{R}$ 是连续可微的非线性函数.为简单, 本文研究系统时滞
$ \tau = 1 $ 的一类非线性系统.假设原点
$ (u,y) = (0,0) $ 为上述非线性系统的平衡点, 将系统(1)在原点处Taylor展开得到如下等价模型:$$ \begin{split} y(k + 1) =\,& - {a_1}y(k) - \cdots - {a_{{n_s}}}y(k - {n_s} + 1) + \\ &{{b_0}u(k)\; \cdots + {b_{{m_s}}}u(k - {m_s}) + v[{{x}}(k)]} \end{split} $$ (2) 其中,
$ {a_i},(i = 1, \cdots ,{n_s}) $ ,$ {b_j},(j = 0, \cdots ,{m_s}) $ 为在工作点处的一阶Taylor系数, 分别为$$ \begin{split} & {a_{i} = -\left.\frac{\partial f\left[y(k-1), \cdots, u\left(k-1-n_{s}\right)\right]}{\partial y(k-i)}\right|_{y = 0 \atop u = 0}, }\\ & \qquad\qquad\qquad{ i = 0, \cdots, n_s } \end{split} $$ $$ \begin{split} & {b_{i} = -\left.\frac{\partial f\left[y(k-1), \cdots, u\left(k-1-m_{s}\right)\right]}{\partial u(k-1-j)}\right|_{y = 0 \atop u = 0},}\\ & \qquad\qquad {i = 0, \cdots, m_s } \end{split} $$ $ {{x}}(k) $ 为数据向量, 定义$$ \begin{split} & {{{x}}(k) = }\\ & {[y(k), \cdots ,y(k - {n_s} + 1),u(k), \cdots ,u(k - {m_s})]^{\rm T}} {} \end{split} $$ $ v[{{x}}(k)] $ 是由$ {{x}}(k) $ 的高阶项组成的光滑非线性函数, 称为未建模动态.注1. 若平衡点偏离或远离原点, 可用坐标变换将其移至该点.
利用单位后移算子
$ {z^{ - 1}} $ , 系统(2)可等价地写成如下形式:$$ A({z^{ - 1}})y(k + 1) = B({z^{ - 1}})u(k) + v[{{x}}(k)] $$ (3) 其中,
$ A({z^{ - 1}}) $ 和$ B({z^{ - 1}}) $ 是关于单位延迟算子$ {z^{ - 1}} $ 的多项式, 其阶次分别为$ {n_s} $ 和${m_s} ;\;$ $ A({z^{ - 1}}) $ 和$ B({z^{ - 1}}) $ 的具体表达式如下:$$ A({z^{ - 1}}) = 1 + {a_1}{z^{ - 1}} + \cdots + {a_{{n_s}}}{z^{ - {n_s}}} $$ $$ B({z^{ - 1}}) = {b_0} + {b_1}{z^{ - 1}} + \cdots + {b_{{m_s}}}{z^{ - {m_s}}} $$ 在上述非线性系统(3)中, 由于非线性项
$ v[{{x}}(k)] $ 未知, 因此, 基于此模型设计的控制器无法实现.由式(3)可知:
$$ v[{{x}}(k - 1)] = A({z^{ - 1}})y(k) - B({z^{ - 1}})u(k - 1) $$ (4) 从式(4)可以看出,
$ A({z^{ - 1}}) $ 和$ B({z^{ - 1}}) $ 可通过系统的历史数据信息获得, 从而可间接获得未建模动态前一拍的数据$ v[{{x}}(k - 1)] $ , 因此, 可充分利用$ v[{{x}}(k - 1)] $ 的可测数据信息进行控制器设计. 以往的方法没有考虑该历史数据信息, 从而造成有用数据丢失. 为此, 下面首先将非线性项等价地表示成如下形式:$$ v[{{x}}(k)] = v[{{x}}(k - 1)] + \Delta v[{{x}}(k)] $$ (5) 其中,
$ {\Delta = 1} - {z^{ - 1}} $ ,$$ \Delta v\left[ {{{x}}(k)} \right] = v\left[ {{{x}}(k)} \right] - v\left[ {{{x}}(k - 1)} \right] $$ 则系统(3)可进一步地表示为如下形式:
$$ \begin{split} A({z^{ - 1}})y(k + 1) =\,& B({z^{ - 1}})u(k) +\\ & {v[{{x}}(k - 1)] + {\Delta }v[{{x}}(k)]} \end{split} $$ (6) 假设1. 对任意的
$k ,\;$ 非线性项$ v[{{x}}(k)] $ 的增量$ \Delta v[{{x}}(k)] $ 全局有界, 即$$ \left| {\Delta v[{{x}}(k)]} \right| \le M $$ (7) 其中,
$ M > 0 $ 为已知正数.注2. 事实上, 假设1要求未建模动态的变化率有界, 实际物理系统, 如水箱液位控制系统、倒立摆控制系统, pendubot 欠驱动机械臂控制系统等, 由于受到传感器、执行器以及电子元件等物理特性的限制, 如限幅、饱和特性等, 系统的输入输出信号是有界的, 未建模动态也是有界的, 因此, 假设1是合理的.
控制目标为设计控制器使得:
1)闭环系统的输入输出信号有界, 即闭环系统BIBO (Bounded-input bounded-output)稳定.
2)系统输出渐近跟踪预先给定的有界参考信号的变化.
2. 控制器设计
2.1 系统已知时的控制器设计
由于系统运行的过程中不可避免地存在某种干扰的影响, 导致系统的工作点会偏离平衡点, 此时未建模动态如果过大将会对系统的控制精度以及稳定性和收敛性造成影响, 甚至导致系统不稳定, 因此, 须采用前馈补偿的策略抑制其对闭环系统的影响. 为此, 针对被控对象(6), 设计如图1所示的带有非线性项
$ v[{{x}}(k)] $ 前一拍数据及其未知增量补偿的非线性控制器.
图 1 带有$v[{{x}}(k)]$ 前一拍数据及其未知增量补偿的非线性控制器Fig. 1 Nonlinear controller with$v[{{x}}(k)]$ previous step data and its unknown incremental compensation由图1可知, 该非线性控制器
$ u(k) $ 可表示为$$\begin{split} & {H({z^{ - 1}})u(k) + G({z^{ - 1}})y(k) = }\\ & \quad {R({z^{ - 1}})w(k + 1) -}\\ & \quad {K({z^{ - 1}})\{ v[{{x}}(k - 1)] + {\Delta }v[{{x}}(k)]\}}\\ \end{split} $$ (8) 给控制器方程(8)乘以
$ B({z^{ - 1}}) $ , 可得:$$ \begin{split} & {H({z^{ - 1}})B({z^{ - 1}})u(k) + B({z^{ - 1}})G({z^{ - 1}})y(k) = }\\ & \quad {B({z^{ - 1}})R({z^{ - 1}})w(k + 1) -}\\ & \quad {B({z^{ - 1}})K({z^{ - 1}})\{ v[{{x}}(k - 1)] + {\Delta }v[{{x}}(k)]\} } \end{split} $$ (9) 由式(6)可得:
$$ \begin{split} B({z^{ - 1}})u(k) =\,& A({z^{ - 1}})y(k + 1) - v[{{x}}(k - 1)] -\\ & { \Delta v[{{x}}(k)]} \end{split} $$ (10) 将式(10)代入式(9), 整理可得闭环系统
$$\begin{split} & {\left[A\left(z^{-1}\right) H\left(z^{-1}\right)+z^{-1} B\left(z^{-1}\right) G\left(z^{-1}\right)\right]y(k+1)}=\\ & \quad {B({z^{ - 1}})R({z^{ - 1}})w(k + 1) + [H({z^{ - 1}}) - }\\ & \quad {B({z^{ - 1}})K({z^{ - 1}})]{\kern 1pt} \{ v[{{x}}(k - 1)] + \Delta v[{{x}}(k)]\}}\\[-10pt] \end{split} $$ (11) 分析闭环系统方程(11)可以看出, 适当选择
$ H({z^{ - 1}}),G({z^{ - 1}}),R({z^{ - 1}}) $ 可以使闭环系统的特征多项式$ \dfrac{{B({z^{ - 1}})R({z^{ - 1}})}}{{H({z^{ - 1}})A({z^{ - 1}}) + {z^{ - 1}}B({z^{ - 1}})\bar G({z^{ - 1}})}} $ 的稳态增益为1. 通过选择$ K({z^{ - 1}}) $ 可以使$ H({z^{ - 1}}) $ 与$B({z^{ - 1}}) K({z^{ - 1}})$ 的差尽可能的小, 以减小未建模动态$ v[{{x}}(k)] $ 对被控对象输出的影响, 从而使被控对象的输出尽可能地跟踪理想输出.针对控制器(8), 选择控制器加权多项式
$ H({z^{ - 1}}) $ 和$ G({z^{ - 1}}) $ 的原则是在保证闭环系统稳定的前提下, 使闭环系统的输出$ y(k + 1) $ 能够很好地跟踪$ w(k + 1) $ , 即:$$ {A({z^{ - 1}})H({z^{ - 1}}) + {z^{ - 1}}B({z^{ - 1}})G({z^{ - 1}}) \ne 0,} \;\;{\left| z \right| \ge 1} $$ (12) 且
$$ {\mathop {\lim }\limits_{k \to \infty } {e_g}(k + 1) = \mathop {\lim }\limits_{k \to \infty } [y(k + 1) - w(k + 1)]} { = 0 } $$ (13) 其中,
$ {e_g}(k + 1) $ 为系统的跟踪误差, 定义为:$$ {e_g}(k + 1) = y(k + 1) - w(k + 1) $$ 从闭环系统方程(11)还可以看出, 设计补偿器
$ K({z^{ - 1}}) $ 是非线性控制器的关键, 如果系统是零动态渐近稳定的, 则选择补偿器的形式为:$$ K({z^{ - 1}}) = \frac{H({z^{ - 1}})} {B({z^{ - 1}})} $$ (14) 如果系统是零动态不稳定的, 则选择补偿器
$ K({z^{ - 1}}) $ 为使得$ H({z^{ - 1}}) $ 与$ B({z^{ - 1}})K({z^{ - 1}}) $ 近似相等的最小二乘解.对上述非线性控制器(8), 如果未建模动态的变化不是十分剧烈, 此时非线性项的增量
$ \Delta v[{{x}}(k)] $ 比较小, 可将其忽略, 直接采用前一时刻未建模动态的可测信息补偿当前时刻的未建模动态, 因此, 设计非线性控制器为$$ \begin{split} & {H({z^{ - 1}})u(k) + G({z^{ - 1}})y(k) = }\\ & \quad {R({z^{ - 1}})w(k + 1) - K({z^{ - 1}})v[{{x}}(k - 1)]} \end{split} $$ (15) 如果被控对象在工作点处运行时, 那么线性化的模型将能表示整个非线性模型的主体部分, 此时可设计不带补偿的控制器作用于被控对象. 故, 将非线性控制器(8)中的
$ v[{{x}}(k)] $ 完全忽略, 直接设计不带非线性项补偿的线性控制器:$$ {H({z^{ - 1}})u(k) + G({z^{ - 1}})y(k) =} {R({z^{ - 1}})w(k + 1)} $$ (16) 2.2 自适应控制算法
考虑到系统的阶次
$ {n_s} $ ,$ {m_s} $ 和时滞$ \tau $ 已知, 但组成$ A({z^{ - 1}}) $ ,$ B({z^{ - 1}}) $ 的参数未知, 由式(6)以及$ A({z^{ - 1}}) $ ,$ B({z^{ - 1}}) $ 多项式的定义, 利用后移算子$ {z^{ - 1}} $ 可将式(6)改写成如下等价形式:$$ \begin{split} & {y(k) = - {a_1}y(k - 1) - {a_2}y(k - 2) - \cdots - }\\ & \quad {{a_{{n_s}}}y(k - {n_s}) + {b_1}u(k - 1)+ {b_2}u(k - 2) + }\\ & \quad {\cdots - {b_{{m_s}}}u(k - {m_s}) + v[{{x}}(k - 1)] + \Delta v[{{x}}(k)]}=\\ & \quad { [ - y(k - 1), \cdots , - y(k - 2),u(k - 1), \cdots ,}\\ & \quad {u(k - {m_s})] {[{a_1}, \cdots ,{a_{{n_s}}},{b_1}, \cdots ,{b_{{m_s}}}]^{\rm T}} +}\\ & \quad {v[{{x}}(k - 1)] + \Delta v[{{x}}(k - 1)]}=\\ & \quad { {{\varphi }}{(k - 1)^{\rm T}}{{\theta }} + v[{{x}}(k - 1)] +}\\ & \quad {\Delta v[{{x}}(k - 1)]}\\[-10pt] \end{split} $$ (17) 其中,
$$ \begin{split} {{\varphi }}^{\rm T}{(k - 1)} =& [ - y(k - 1), \cdots , - y(k - 2), \\ & u(k - 1), \cdots ,u(k - {m_s})] \end{split} $$ $$ {{{\theta }}^{\rm T}} = [{a_1}, \cdots ,{a_{{n_s}}},{b_1}, \cdots ,{b_{{m_s}}}] $$ 2.2.1 线性自适应控制算法
由式(17)可知, 当采用线性控制器时, 系统的参数辨识方程定义为
$$ {\hat y_1}(k + 1) = {{\varphi }}^{\rm T}{(k)}{{\hat{{\theta }}}_1}(k) $$ (18) 其中,
$$ \begin{split} {{\hat{{\theta }}}_1^{\rm T}}{(k)} =\,&[ - {a_{1,1}}(k), \cdots , - {a_{1, {n_s}}}(k),\\ & {{b_{1,1}}(k), \cdots ,{b_{1,{m_s}}}(k)]} \end{split} $$ (19) 表示在
$ k $ 时刻对参数$ {{\theta }} $ 的估计. 采用下面改进的投影算法[6, 30]进行辨识:$$ {{\hat{{\theta }}}_1}(k) = {{\hat{{\theta }}}_1}(k - 1) + \frac{{{\mu _1}(k){{\varphi }}(k - 1){e_1}(k)}}{{1 + {{{\varphi }}^{\rm T}}(k - 1){{\varphi }}(k - 1)}} $$ (20) $$ {\mu _1}(k) = \left\{ {\begin{array}{*{20}{l}} {1,}&{{\mkern 1mu} |{e_1}(k)| > 2M}\\ {0,}&{\text{其他}} \end{array}} \right. $$ (21) $$ {e_1}(k) = y(k) - {\hat y_1}(k) $$ (22) 由式(16)可知, 线性自适应控制器通过下式获得:
$$ \begin{split} & {{\hat H_1}(k,{z^{ - 1}})u(k) + {\hat G_1}(k,{z^{ - 1}})y(k) =}\\ & \quad {{\hat R_1}(k,{z^{ - 1}})w(k + 1)} \end{split} $$ (23) 其中,
$ {\hat H_1}(k,{z^{ - 1}}) $ 、$ {\hat G_1}(k,{z^{ - 1}}) $ 和$ {\hat R_1}(k,{z^{ - 1}}) $ 通过下式在线计算:$$ \begin{split} & {{\hat H_1}(k,{z^{ - 1}}){\hat A_1}(k,{z^{ - 1}}) + {z^{ - 1}}{\hat B_1}(k,{z^{ - 1}}){\hat G_1}(k,{z^{ - 1}})}\ne 0,\\ & \quad {\left| z \right| \ge 1}\\[-10pt] \end{split} $$ (24) $$\begin{split} &{{\hat R}_1}(k,{z^{ - 1}}) \!=\\ &\dfrac{{{\hat H}_1}(k,{z^{ \!-\! 1}}){{\hat A}_1}(k,{z^{ \!-\! 1}}) \!+\! {z^{ \!-\! 1}}{{\hat B}_1}(k,{z^{ \!-\! 1}}){{\hat G}_1}(k,{z^{ \!-\! 1}})}{{{\hat B}_1}(k,{z^{ \!-\! 1}})} \end{split} $$ (25) 2.2.2 带有
$ v[{{x}}(k - 1)] $ 补偿的的非线性自适应控制算法系统(17)的参数辨识方程定义为
$$ {\hat y_2}(k + 1) = {{\varphi }}^{\rm T}{(k)}{{\hat{{\theta }}}_2}(k) + v[{{x}}(k - 1)] $$ (26) 其中,
$$ \begin{split} {{\hat{{\theta }}} _2^{\rm T}}{(k)} =\,& [ - {a_{2,1}}(k), \cdots , - {a_{2,{n_s}}}(k), \\ &{{b_{2,1}}(k), \cdots ,{b_{2,{m_s}}}(k)]} \end{split} $$ (27) 表示在
$ k $ 时刻对参数$ {{\theta }} $ 的估计. 采用类似于(18) ~ (22)的辨识算法[6, 30]得到, 不同的是辨识误差为$$ {e_2}(k) = y(k) - {\hat y_2}(k) $$ (28) 由式(15)和辨识模型(26)可知, 非线性自适应控制器通过下式获得:
$$ \begin{split} & {{\hat H_2}(k,{z^{ - 1}})u(k) + {\hat G_2}(k,{z^{ - 1}})y(k) =}\\ & \quad {{\hat R_2}(k,{z^{ - 1}})w(k + 1) -}\\ & \quad {{\hat K_2}(k,{z^{ - 1}})v[{{x}}(k - 1)]} \end{split} $$ (29) 其中,
$ {\hat H_2}(k,{z^{ - 1}}) $ 、$ {\hat G_2}(k,{z^{ - 1}}) $ 和$ {\hat R_2}(k,{z^{ - 1}}) $ 是非线性自适应控制器参数, 采用类似于式(24) ~ (25)的方式计算. 补偿器$ {\hat K_2}(k,{z^{ - 1}}) $ 通过下式在线计算:$$ {\hat K_2}(k,{z^{ - 1}}) =\frac {{{\hat H}_2}(k,{z^{ - 1}})} {{{\hat B}_2}(k,{z^{ - 1}})} $$ (30) 2.2.3 带有非线性项未知增量估计的非线性自适应控制算法
由式(17)可知, 带有非线性项增量
$ \Delta v[{{x}}(k)] $ 估计的参数辨识方程为$$ \begin{split} \hat y(k + 1) =\,& {{\varphi }}^{\rm T}{(k)}{{\hat{{\theta}}}} _{\bf{3}}(k) + v[{{x}}(k - 1)] + \\] & {\Delta \hat v[{{x}}(k)]} \end{split} $$ (31) 其中,
$ {\hat{{\theta }}}_3 $ 表示在$ k $ 时刻对参数$ {{\theta }} $ 的估计. 定义为$$ \begin{split} {{\hat{{\theta }}} _3^{\rm T}}{(k)} =\,& [ - {a_{3,1}}(k), \cdots , - {a_{3,{n_s}}}(k),\\ & {{b_{3,1}}(k), \cdots ,{b_{3,{m_s}}}(k)]} \end{split} $$ (32) 采用类似于
$ {\hat{{\theta }}} _1 $ 的辨识算法来估计$ {\hat{{\theta }}} _3 $ , 其中$$ {e_3}(k) = y(k) - {\hat y_3}(k) $$ (33) $ \Delta \hat v[{{x}}(k)] $ 是$ \Delta v[{{x}}(k)] $ 的估计值.采用文献[6]和文献[21]提出的估计算法对
$ \Delta v[{{x}}(k)] $ 进行估计. 于是, 由式(8)可知, 非线性自适应控制器为:$$ \begin{split} & {{\hat H_3}(k,{z^{ - 1}})u(k) + {\hat G_3}(k,{z^{ - 1}})y(k) = }\\ & \quad {{\hat R_3}(k,{z^{ - 1}})w(k + 1) -}\\ & \quad {{\hat K_3}(k,{z^{ - 1}})\left\{ {v[{{x}}(k - 1)] + \Delta \hat v[{{x}}(k)]} \right\}} \end{split} $$ (34) 其中,
$ {\hat H_3}(k,{z^{ - 1}}) $ 、$ {\hat G_3}(k,{z^{ - 1}}) $ 、${\hat R_3}(k,{z^{ - 1}})$ 和${\hat K_3} (k, \;{z^{ - 1}})$ 是非线性自适应控制器参数, 采用类似于式(24)~(25)的方式计算.$\Delta \hat v[{{x}}(k)] $ 的值通过估计算法获得.2.3 切换控制
上述线性自适应控制器(23)、非线性自适应控制器(29)以及带有非线性项估计和补偿的非线性自适应控制器(34) 通过下面的切换函数来协调控制被控对象(7), 从而形成一种闭环切换控制系统. 由式(22), (28)和(33)可知, 模型误差越小, 那么以该模型所设计的控制器将会使得闭环系统的输出与理想输出之间的跟踪误差越小, 同时也表明该控制器能更快收敛到以实际被控对象模型为控制器设计模型所设计的控制器. 因此将模型误差
$ {e_1}(k) $ 、${e_2}(k) \;$ 和$ {e_3}(k) $ 引入切换机制, 在某一时刻, 选择较小切换指标所对应的控制器作用于被控对象. 为此, 提出如图2所示的由$ {C_1} $ (线性自适应控制器),$ {C_2} $ (带有非线性项前一拍补偿的非线性自适应控制器),$ {C_3} $ (带有非线性项增量估计器和补偿器的非线性自适应控制器)和切换函数组成的切换控制方法, 其中,$ {C_1} $ 、$ {C_2} $ 和$ {C_3} $ 分别由(23)、(29)和(34)决定.图2中的切换函数为:
$$ \begin{split} {J_j}[{{ e}_j}(k)] =\,& \sum\limits_{l = 1}^k {\frac{{{\mu _j}(l)[{ e}_j^2(l) - 4{M^2}]}}{{2[1 + {{\varphi }}^{\rm T}{{(l - 1)}}{{\varphi }}(l - 1)]}}} +\\ &{c\sum\limits_{l = k - N + 1}^k {[1 - {\mu _j}(l)]{e}_j^2(l} ) } \end{split} $$ (35) $$ {\mu _j}(k) = \left\{ \begin{array}{*{20}{l}} \begin{aligned} & {1,}\quad {|{{ e}_j}(k)| > 2M}\\ & {0,}\quad {\text{其他}} \end{aligned}&{j{\rm{ = 1,2,3}}} \end{array} \right. $$ (36) 其中,
$ {\mu _j}(k),\;(j = 1,2,3) $ 表示死区函数; 当$ j = 1 $ 时,${{ e}_1}(k)$ 表示采用模型(18)时的误差;$ j = 2 $ 时,${{ e}_2}(k)$ 表示采用非线性模型(26)时的误差;$ j = 3 $ 时,${{ e}_3}(k)$ 表示采用非线性模型(31)时的误差.$ N $ 是正整数,$ c \ge 0 $ 是一个常数.任意时刻
$k ,\;$ 切换机制选择最小的切换函数所对应的控制器来控制系统, 即$$ {J^*}(k) ={\rm{ {min}}}\,[{J_j}[{{e}_j}(k)]],j = 1,2,3 $$ (37) 如果
${J^*}(k) = {J_1}[{{ e}_1}(k)]$ , 选择控制器(23);如果
${J^*}(k) = {J_2}[{{ e}_2}(k)]$ , 选择控制器(29);如果
${J^*}(k) = {J_3}[{{ e}_3}(k)]$ , 选择控制器(34).注3. 整个切换函数分为两部分, 第一部分为
$\displaystyle\sum\nolimits_{l = 1}^k {\dfrac{{{\mu _j}(l)[{ e}_j^2(l) - 4{M^2}]}}{{2[1 + {{\varphi }}^{\rm T}{{(l - 1)}}{{\varphi }}(l - 1)]}}}$ , 用来区分不同信号的增长速度, 可以用来保证闭环系统的鲁棒性; 第二部分为$c\displaystyle\sum\nolimits_{l = k - N + 1}^k {[1 - {\mu _j}(l)]{ e}_j^2(l} )$ , 是有限时间内辨识误差的累积, 这部分主要用于提高系统的性能. 因此, 该切换机制在保证闭环系统稳定的同时提高系统的性能.3. 稳定性和收敛性分析
下面给出运用上述非线性自适应控制器设计算法时闭环切换系统的稳定性和收敛性分析.
定理1. 当被控对象(7)满足如下条件1) ~ 3)时具有稳定性和收敛性.
1)系统(1)的零动态渐近稳定;
2)非线性项满足假设条件1;
3)根据式(23) ~ (25)和(30)给出的参数选择方案来适当选择自适应控制器(23)、(29)和(34)中的参数加权多项式
$ {\hat H_i}(k,{z^{ - 1}}) $ 、$ {\hat G_i}(k,{z^{ - 1}}) $ 、$ {\hat R_i}(k,{z^{ - 1}}) $ 和$ {\hat K_i}(k,{z^{ - 1}}) $ ,$ i = 1,2,3 $ ,$ ({\hat K_1}(k,{z^{ - 1}}) = 0) $ . 则把辨识算法(18) ~ (22), 非线性项增量的估计算法以及线性自适应控制器(23)、带有非线性项前一拍补偿的非线性自适应控制器(29)、非线性自适应控制器(34)作用于被控对象(7), 并采用本文的切换控制算法时, 闭环切换系统具有如下特性:1) 闭环系统BIBO稳定, 即:
$$ \left| {u(k)} \right| < \infty ,{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \left| {y(k)} \right| < \infty $$ 2) 当
$ k \to \infty $ 时, 系统的跟踪误差有界. 即:$$ \left| {{e_g}(k)} \right| \le \delta $$ 其中, 定义系统的跟踪误差为:
$$ {{e_g}(k) = y(k) - w(k), }\;{\delta \ge 0} $$ 证明. 采用类似于文献[9]与文献[21]的方法可得结论(1) ~ (2)成立, 这里不再赘述.
4. 水箱液位控制系统实验
将本文所提的方法应用在多功能过程控制实验平台上进行水箱的液位控制实验. 如图3所示, 控制目标是通过调节水泵电压占空比, 使得水箱的液位输出值保持在预设定的高度. 实验中, 液位的指定高度为8 cm. 我们针对水箱液位系统进行物理实验, 采样周期为0.1秒. 选择切换函数中的参数
$ N = 2 $ ,$ c = 1 $ . 实验结果如图3 ~ 6 所示.
图 4 采用本文方法时水箱液位的实际响应曲线(输出y)Fig. 4 The actual response curve of tank level by the proposed method (output y)
图 5 采用本文切换控制方法时水箱液位的控制输入uFig. 5 The actual input of tank level by the proposed method in this paper (input u)图4为采用本文切换控制方法时水箱液位的实际响应曲线. 由图中可以看出, 系统的跟踪特性较好, 所设计的控制器可以达到控制目标. 由于实际系统是动态变化, 可能受到一些高频干扰或者噪声的影响, 所以系统的跟踪误差收敛到原点的小邻域内. 图5为相对应的同一时间范围内的系统输入的情况. 图6为控制器的切换序列, 从图中可以看出, 大部分时间是非线性自适应控制器在工作, 说明带有补偿的非线性自适应控制器能有效抑制未建模动态对闭环系统的影响.
5 结论
本文针对一类离散时间非线性动态系统, 提出了一种新的非线性自适应切换控制算法, 该算法在设计控制器时, 充分利用了非线性项的历史数据设计了非线性项前一拍补偿的非线性自适应控制器以及带有非线性项增量估计器和补偿器的非线性自适应控制器, 通过切换机制来协调控制系统. 理论分析表明本文提出的控制方法不仅具有稳定性和收敛性, 而且使得闭环系统具有良好的动静态特性. 通过水箱液位控制系统的物理实验, 实验结果进一步验证了所提方法的有效性和优越性.
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图 1 发电过程能量转化简图(虚线框内为热力系统的能量转化过程)
Fig. 1 Energy conversion diagram of power generation process (Energy conversion process of thermal system is shown in the dotted box)
图 2 百万千瓦超超临界机组非平稳运行轨迹示意图
Fig. 2 Schematic of non-stationary operation trajectory of 1000 MW ultra-supercritical unit
图 3 百万千瓦超超临界机组全流程运行示意图
Fig. 3 Schematic of the full-condition operation of 1000 MW ultra-supercritical unit
图 4 磨煤机磨辊磨损与空预器磨损的故障表征示意图
Fig. 4 Diagram of fault characterization of coal mill roller wear and air preheater wear
图 5 条件轴数据重组(从时间轴非平稳数据到条件轴不同条件片)
Fig. 5 Rearranging data according to condition axis (from nonstationary data on time axis to different condition slices on condition axis)
图 6 不同运行条件上的过程动态变化示意图
Fig. 6 Illustration of process dynamics under different operating conditions
图 7 发电设备运行工况监控方法发展趋势示意图
Fig. 7 Schematic of development trend of operation monitoring methods of power generation equipment
表 1 基于解析模型和数据驱动的发电装备工况监控方法总结
Table 1 A comparing summary between analytical-model-based methods and data-driven methods for condition monitoring of power generation equipment
类型 方法 原理 优点 缺点 应用对象举例 基于解
析模型数学模型
的方法建立精确的数学模型和可观测输入输出量构造残差信号来反映装备期望行为与实际运行模式间是否一致 1) 可靠, 精确
2) 模型通用性强
3) 机理解释性强1) 领域知识需求高
2) 模型参数辨识难
3) 复杂对象耗时长1) 基于多模型状态估计的除氧器状态监测和故障诊断[39]
2) 基于观测器残差模型的加热器性能监测[40-41]
3) 基于简化数学模型的回热加热器在线工况监控[42]等数据
驱动多元统计
的方法对历史过程数据进行统计分析, 比较正常样本估算得到的监控指标置信限和每个样本的监控统计量以确定当前样本的运行状态 1) 无需假设或对参数进行经验估计
2) 降维能力强
3) 算法解释性强, 参数易调整.1) 处理非高斯、多模态、非线性数据时, 效果较差
2) 忽视数据微小特征对结果的影响1) 基于主元分析相关的电厂状态监测[45-47]
2) 基于向量自回归模型的设备故障预测[51-52]
3) 基于潜空间投影的运行过程性能评估和状态监测[53, 55]等人工智能
的方法利用人工智能算法模拟和实现人类的思维和行为, 自动完成工况监控过程 1) 实时数据分析, 减少人工干预
2) 强大的非线性表达能力和自适应学习能力1) 黑箱模型, 参数和模型不易理解
2) 对数据质量和规模要求高1) 基于混合优化递归神经网络的热力系统实时预测[66]
2) 基于人工神经网络和最优变焦搜索的加热器故障诊断[67]等
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表 2 三类非平稳分析方法对比总结
Table 2 Comparison and summary of three types of non-stationary analysis methods
类型 方法 优点 缺点 应用实例 典型非平稳分析方法 信号处理方法 能处理非平稳非线性信号 应用对象局限于高频振动信号 1) 经验模态分解[85] 处理非线性非平稳时间序列
2) 应用小波变换[75] 对齿轮箱振动信号进行故障诊断自适应策略 快速适应新模式, 计算效率高 无法有效区分正常的变化和缓变故障 1) 应用递归PCA[34] 进行自适应过程监测
2) 应用递归指数慢特征分析[84] 进行自适应过程监测基于协整分析的方法 模型数量少、有效时间长 应用对象局限于存在协整关系的变量 1) 协整分析结合慢特征分析[96] 进行全工况过程监测
2) 稀疏协整分析[99] 进行分布式过程监测时间驱动的多模式分析方法 统计检验或平稳性指标判断法 计算效率高 模式划分粗糙, 未考虑多变量间的关系 1) 利用统计检验[104] 确定稳态工况
2) 利用稳定性因子[105] 进行模式划分特性变化度量与模式划分策略 自动划分模式 模式数量大且冗余; 在线工况确认难 文献[8, 57, 106−107] 指出可以根据过程变量相关关系的变化反推过程特性的变化, 从而将运行状态分为不同模式 高斯混合模型聚类方法 拟合能力强, 自动聚类 需预先指定模式数量 利用高斯混合模型 (Gaussian mixed model, GMM) 进行多工况划分进而进行故障诊断[108] 软过渡方法 模式划分更合理, 监测模型更灵敏 模式划分结果复杂, 可解释性较差 1) 建立了一种软过渡PCA模型[109] 进行过程监测
2) 文献 [110] 进一步发展了软时段过渡方法条件驱动的多模式分析方法 有序条件模式划分方法 从条件轴出发, 消除了非平稳特性的影响, 抓住了工况变化的本质 硬划分, 在模式边界处的样本归属可能不够合理; 同时对过渡过程进行了合并简化处理 文献 [114] 提出了多条件模式的表征理论方法, 有利于不同模式内关键信息的分析与提取, 显著提高了模型的精度, 并在百万千瓦超超临界机组上进行了验证
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表 3 动态潜投影建模方法比较
Table 3 Comparison of dynamic latent projecting methods
方法 目标 模型参数 优化解 静动信息
是否分离是否实现降维 优点 缺点 动态主元分析 DPCA[133] 最大化潜变量的方差 1个时延参数、潜变量个数 特征根分解1次 否 否, 潜变量维度随时延增加, 可能会大于原始数据维度 求解简单, 传统方法可直接使用 1) 无法实现动态与静态信息的分离
2) 所提取潜变量易受静态信息主导动态潜变量模型 DLV[135] 最大化潜变量线性组合的方差 1个时延参数、潜变量个数 迭代求解, 每次找到一个潜变量 否 是 具有降维能力 1) 无法实现动态与静态信息的分离
2) 所提取潜变量易受静态信息主导动态内部主元分析DiPCA[136] 最大化潜变量与其预测值的协方差 1个时延参数、潜变量个数 迭代求解, 每次找到一个潜变量 是 是 具有降维能力, 且实现了动静态信息的分开监测 未直接按照时序性强弱提取潜变量 状态空间模型CVA[137] 最大化潜变量的时序相关性 2个时延参数、潜变量个数 奇异值分解1次 是 否, 潜变量维度随时延增加, 可能会大于原始数据维度 求解简单, 实现了动静态信息的分开监测 数据共线性时求解不稳定 慢特征分析SFA[139] 最大化潜变量的变化速度, 即一阶自相关性 潜变量个数 特征根分解2次 是 是 具有降维能力, 实现对数据变化速度的表征 只关注了数据的一阶时序相关性
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