Adaptive Control for High-order Nonlinear Feedforward Systems With Input and State Delays
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Abstract: This paper considers the adaptive stabilization of a class of high-order uncertain nonlinear feedforward systems. The growth condition on nonlinearities is further relaxed by allowing not only input and state delays but also the unknown growth rate. By a combined method of adaptive technique, dynamic gain control approach and adding a power integrator technique, a state feedback controller is designed to guarantee that all signals are bounded, the equilibrium point of the closed-loop system is globally stable and the original state converges to zero.
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Key words:
- Adaptive control /
- dynamic gain /
- high-order nonlinear feedforward systems /
- input and state delays
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电熔镁砂结构致密、熔点高、抗氧化、绝缘性强, 是制造、冶金、化工、电气设备、航天工业等行业所需耐火材料的主要原料[1].电熔镁砂以菱镁矿石为原矿, 采用我国特有的埋弧方式的电熔镁炉进行熔炼, 菱镁矿石熔化所需温度在2 850 ℃以上, 远高于炼钢电弧炉所需的1 700 ℃, 需要采用埋弧方式.熔炼过程中控制系统通过调整三相电极与熔池之间的距离, 来控制三相电极电流跟踪熔化电流, 使之产生电弧, 通过电弧放热使炉内原矿受热熔化形成熔液, 边熔化边加料, 当熔池升高到炉口上表面时熔炼结束, 经过冷却结晶后生成成品.
电熔镁炉是一种典型的高耗能设备, 每熔炼一炉大约耗电40 000千瓦时, 电能成本占整个生产成本的60 %以上.所以电熔镁炉的运行目标是将单吨合格产品所消耗的电能, 即单吨能耗, 控制在目标值范围内并使其尽可能小.只有将电极电流控制在熔化电流范围内, 才能保证产品质量合格[1-2].只有将电极电流稳定控制在最佳熔化电流上才能保证单吨能耗最小[2].
目前针对电弧炉的电流控制研究大都集中在采用开弧方式的炼钢电弧炉电流控制上, 例如, 文献[3]针对炼钢过程呈非线性, 采用熔炼过程的脉冲响应模型作为电流预测模型, 提出了模型算法控制并进行了仿真实验.文献[4]针对电弧参数随着炼钢炉温度变化而慢时变且温度难以在线测量的问题, 采用在线辨识电流模型参数提出了电极电流温度权重自适应控制器.文献[5]针对电弧炉熔炼过程呈非线性且具有时变特性的问题, 采用电流跟踪误差调整电流设定值来抵偿电弧炉特性的变化, 提出了自调整模型算法控制.文献[6]针对电弧炉输入具有死区特性、输出为非线性且运行过程受约束条件限制的问题, 将电流模型在工作点附近线性化, 提出了模型预测综合控制算法.
针对电熔镁砂熔炼过程电极电流控制问题, 文献[7]提出了一种基于神经网络的电熔镁炉智能控制系统, 并进行了仿真验证.针对参数未知的被控对象, 文献[8]和文献[9]提出了自校正PID控制算法, 通过在线辨识模型参数来校正PID控制器的参数.针对非线性、大时延及参数时变的复杂过程, 文献[10-11]提出了专家PID控制算法, 利用专家系统和规则推理来调节PID控制器参数而使其具有自适应能力, 文献[12]和文献[13]采用误差信号的非线性映射提出了非线性PID控制算法.由于电熔镁炉的电流模型参数埋弧电阻率、熔池电阻率与熔池高度是未知非线性函数并随熔炼过程变化和原矿变化发生未知随机变化, 导致熔炼过程始终处于动态变化之中, 上述文献[3-13]所述控制方法和PID控制器的积分作用失效, 无法将电极电流控制在目标值范围内.
文献[14]和文献[15]针对PID控制器积分器失效问题, 将被控对象模型描述成线性模型加高阶非线性项的形式, 通过对线性模型设计一步最优PI控制器、对高阶非线性项设计前一拍高阶非线性项补偿器而得到基于高阶非线性项补偿的一步最优PI控制器, 并分别应用于热交换过程和混合选别浓密过程的跟踪控制, 取得了较好的效果.本文在文献[14]和文献[15]基础上, 利用电熔镁炉运行在工作点附近的特点, 将电熔镁砂熔炼过程用线性模型和未知高阶非线性项来描述, 采用线性模型设计PID控制器, 设计消除前一时刻高阶非线性项和其变化率的补偿器, 提出了一种针对电熔镁砂熔炼过程电极电流控制的带输出补偿的PID控制器, 仿真实验和工业应用表明当电熔镁砂熔炼过程动态特性发生变化时, 所提控制方法无需参数辨识可将电流控制在目标值范围内.
1. 控制问题描述
1.1 电熔镁砂熔炼过程简介
如图 1所示, 电熔镁炉熔炼系统由电流控制系统、三个交流电机和三根电极组成的电极移动系统、原矿仓和电振给料机组成的加料系统、供电系统和电熔镁炉构成.
熔炼过程首先由加料系统向炉内加入菱镁矿石, 通过供电系统向三相电极供电, 产生电弧.原矿吸收电弧放出的热量熔化, 形成熔池.电流控制系统通过电极移动系统调节电极与熔池之间的距离, 进而控制阻抗使三相电极平均电流跟踪熔化电流设定值.由于熔化温度高, 因此采用埋弧方式.三相电极埋在原矿之中, 边熔化边加料, 随着原矿的不断加入和熔化, 熔池增高, 当达到炉口上表面时, 熔炼过程结束.使用小车将炉体拖离熔炼工位, 进行自然冷却并破碎, 得到电熔镁砂产品.
1.2 熔炼过程电极电流动态模型
三相电极电流动态模型以三相电机转动方向与频率$u_i(t)$为输入, 以三相电极电流$y_i(t)$为输出.三相电极电流$y_i(t)$与工作电阻$R_i(t)$之间的关系为:
$ \begin{equation} y_i(t) = \frac{U}{\sqrt{3}R_i(t)} \end{equation} $
(1) 其中, $i=1, 2, 3$分别表示A、B、C三相电极, $U$为熔炼电压, $R_i(t)$可由如下公式表示[16-17]:
$ \begin{equation} R_i(t)=R_{iarc}(t)+R_{ipool}(t) \end{equation} $
(2) 其中, $R_{iarc}(t)$为埋弧电阻, $R_{ipool}(t)$为熔池电阻.将埋弧等效为弧柱来计算埋弧电阻[17], 埋弧弧柱长度$L_{iarc}(t)$为:
$ \begin{align} L_{iarc}(t) =\,&h_{ielec}(t)-h_{ipool}(B_1, B_2, y_i)=\nonumber\\ &\int_0^t\omega_i(\tau)r_d{\rm d}\tau-h_{ipool}(B_1, B_2, y_i) \end{align} $
(3) 其中, $h_{ielec}(t)$为电极高度, $h_{ipool}(B_1, B_2, y_i)$为未知非线性函数, 表示熔池高度, 其取值随原矿颗粒长度$B_1$、原矿杂质成分$B_2$和电极电流$y_i$的变化而变化. $r_d$为升降机构的等效齿轮半径, $\omega_i(\tau)~({\rm rad/s})$为升降电机转动角速度, 其中$\tau$为运行时间.又升降变频电机转速$n_i(t)~({\rm r/min})$与变频电机转动方向与频率$u_i(t)~({\rm Hz})$之间的关系如下:
$ \begin{equation} n_i(t)=\frac{60(1-s)u_i(t)}{p} \end{equation} $
(4) 其中, $p$为电机的极对数, $s$为转差率.因此电弧长度式(3)可表示为:
$ \begin{align} L_{iarc}(t) =\, &\int_0^t\frac{2{\rm \pi}}{60}\times\frac{60(1-s)u_i (\tau)}{p}r_d{\rm d}\tau-\nonumber\\ &h_{ipool}(B_1, B_2, y_i)=\nonumber\\ &\int_0^t\frac{2{\rm \pi}(1-s)u_i(\tau)}{p}r_d{\rm d}\tau-\nonumber\\ &h_{ipool}(B_1, B_2, y_i) \end{align} $
(5) 埋弧电阻$R_{iarc}(t)$和熔池电阻$R_{ipool}(t)$[17]分别如式(6)和式(7)所示:
$ \begin{align} R_{iarc}(t) =\, &f_1(B_1, B_2)\frac{L_{iarc}(t)}{{\rm \pi} r_{iarc}^2}=\frac{f_1(B_1, B_2)}{{\rm \pi} r_{iarc}^2}\times\nonumber\\ & \Bigg[\int_0^t\frac{2{\rm \pi}(1-s)u_i(\tau)}{p}r_d{\rm d}\tau-\nonumber\\ &\quad h_{ipool}(B_1, B_2, y_i) \Bigg] \end{align} $
(6) $ \begin{equation} R_{ipool}(t)=\frac{f_2(B_1, B_2)}{{\rm \pi} d}\left[ 1-\frac{d}{2h_{ipool}(B_1, B_2, y_i)}\right] \end{equation} $
(7) 其中, $f_1(B_1, B_2)$和$f_2(B_1, B_2)$分别表示埋弧电阻率和熔池电阻率, 均为随$B_1$和$B_2$变化而变化的未知非线性函数; $r_{iarc}$为埋弧等效弧柱半径; $d$为电极直径.由式(6)和式(7)得${\rm d}R_{iarc}(t)/{\rm d}t$和${\rm d}R_{ipool}(t)/{\rm d}t$如下:
$ \begin{equation} \left\{ \begin{array}{l}%array 中lrc表示各列内容的居左、居中、居右 \frac{{\rm d}R_{iarc}(t)}{{\rm d}t}=\\ \qquad \frac{f_1(B_1, B_2)}{{\rm \pi} r_{iarc}^2}\left[\frac{2{\rm \pi}(1-s) u_i(t)}{p}r_d-\dot{h}_{ipool}(B_1, B_2, y_i)\right]\\ \frac{{\rm d}R_{ipool}(t)}{{\rm d}t}=\frac{f_2(B_1, B_2)}{2{\rm \pi} h_{ipool}^2(B_1, B_2, y_i)}\dot{h}_{ipool}(B_1, B_2, y_i) \end{array} \right. \end{equation} $
(8) 其中, $\dot{h}_{ipool}(B_1, B_2, y_i)$表示熔池高度变化率, 为未知非线性函数.由式(1)得:
$ \begin{equation} R_i(t) = \frac{U}{\sqrt{3}y_i(t)} \end{equation} $
(9) 对式(9)中的工作电阻$R_i(t)$求导得:
$ \begin{equation} \frac{{\rm d}R_i(t)}{{\rm d}t}=-\frac{U}{\sqrt{3}y_i^2(t)}\frac{{\rm d}y_i(t)}{{\rm d}t} \end{equation} $
(10) 由式(2)得:
$ \begin{equation} \frac{{\rm d}R_i(t)}{{\rm d}t}=\frac{{\rm d}R_{iarc}(t)}{{\rm d}t}+\frac{{\rm d}R_{ipool}(t)}{{\rm d}t} \end{equation} $
(11) 由式(8)、式(10)和式(11)可得:
$ \begin{align} &-\frac{U}{\sqrt{3}y_i^2(t)}\frac{{\rm d}y_i(t)}{{\rm d}t}= \frac{f_1(B_1, B_2)}{{\rm \pi} r_{iarc}^2} \times \nonumber\\ &\qquad\left[\frac{2{\rm \pi}(1-s)u_i(t)}{p}r_d-\dot{h}_{ipool}(B_1, B_2, y_i)\right]+\nonumber\\ &\qquad\frac{f_2(B_1, B_2)}{2{\rm \pi} h_{ipool}^2(B_1, B_2, y_i)}\dot{h}_{ipool}(B_1, B_2, y_i) \end{align} $
(12) 将模型参数$f_1(B_1, B_2)$、$f_2(B_1, B_2)$、$h_{ipool}(B_1, $$B_2, y_i)$和$\dot{h}_{ipool}(B_1, B_2, y_i)$简写为$f_1(\cdotp)$、$f_2(\cdotp)$、$h_{ipool}(\cdotp)$和$\dot{h}_{ipool}(\cdotp)$, 于是可得如式(13)所示的三相电极电流动态模型.
$ \begin{align} &\frac{{\rm d}y_i(t)}{{\rm d}t}=-\frac{\sqrt{3}y_i^2(t)}{U}\times\nonumber\\ &\qquad\left\{ \frac{f_1(\cdotp)}{{\rm \pi} r_{iarc}^2}\left[ \frac{2{\rm \pi}(1-s)u_i(t)}{p}r_d-\dot{h}_{ipool}(\cdotp)\right]\right.+\nonumber\\ &\qquad\left. \frac{f_2(\cdotp)}{2{\rm \pi} h_{ipool}^2(\cdotp)}\dot{h}_{ipool}(\cdotp)\right\} \end{align} $
(13) 式(13)表明电流动态模型具有强非线性, 模型参数$f_1(\cdotp)$、$f_2(\cdotp)$、$h_{ipool}(\cdotp)$和$\dot{h}_{ipool}(\cdotp)$为随熔炼过程和原矿颗粒长度及杂质成分的变化而变化的非线性函数, 由于熔炼过程中电极电流动态特性始终处于变化之中, 使控制器的积分作用失效, 因此难以采用基于参数估计的自适应控制方法和常规PID控制方法将电极电流控制在目标值范围内, 导致单吨能耗高.
1.3 控制问题描述
电熔镁砂熔炼过程中, 存在使得单吨能耗最小的最佳熔化电流值, 只有三相电极电流平均值很好地跟踪熔化电流最佳设定值$y_{sp}(k)$, 即在所有运行时间内将熔炼过程的三相电极电流平均值$y(k)$与设定值$y_{sp}(k)$的跟踪误差$e(k)$控制在目标值范围内, 才能将单吨能耗控制在目标值范围内.因此当熔池高度和所加原矿的颗粒长度及杂质成分发生变化时, 必须设计一个控制器, 使得:
$ \begin{equation} \left| e(k)\right|=\left| y_{sp}(k)-y(k)\right|<\delta, \quad 0<k<\infty \end{equation} $
(14) 且使电机转动方向与频率$u_i(k)$的波动尽可能小, 即:
$ \begin{equation} u_{{\rm min}}<u_i(k)<u_{{\rm max}}, \quad i=1, 2, 3 \end{equation} $
(15) 式中, $\delta$为跟踪误差$e(k)$的上限值, $u_{{\rm max}}$、$u_{{\rm min}}$为电机转动方向与频率$u_i(k)$波动的上下界, 保证实际熔化电流尽可能为最佳熔化电流.
2. 带输出补偿的PID控制方法
2.1 控制策略
由于电熔镁炉运行在工作点附近, 因此可以将电极电流动态模型式(13)表示为线性模型和未知高阶非线性项之和的形式[18], 首先采用欧拉法离散化模型(13)如下:
$ \begin{align} &y_i(k+1)=H\left[u_i(k), y_i(k)\right]=y_i(k)-\nonumber\\ &\qquad\delta_t\frac{\sqrt{3}y_i^2(k)}{U}\left\{ \frac{f_1(\cdotp)}{{\rm \pi} r_{iarc}^2}\left[\frac{2{\rm \pi} (1-s)u_i(k)}{p}r_d\right. \right.-\nonumber\\ &\qquad \left.\dot{h}_{ipool}(\cdotp)\right]\left. +\frac{f_2(\cdotp)}{2{\rm \pi} h_{ipool}^2(\cdotp)}\dot{h}_{ipool}(\cdotp)\right\} \end{align} $
(16) 其中$\delta_t$为采样时间.将电极电流模型式(16)在工作点$(u_{i0}, y_{i0})$附近Taylor展开, 其一阶Taylor系数为:
$ \begin{align} &\left. \frac{{\rm \partial}H\left[u_i(k), y_i(k)\right]}{{\rm \partial}y_i(k)}\right|_{\substack{u_i(k)=u_{i0}\\y_i(k)=y_{i0}}}=\nonumber\\ &\qquad\quad 1-\delta_t\frac{2\sqrt{3}y_{i0}}{U}\left\{ \frac{f_1(\cdotp)}{{\rm \pi} r_{iarc}^2}\left[ \frac{2{\rm \pi}(1-s)u_{i0}}{p}r_d\right. \right.-\nonumber\\ &\qquad\quad\left.\dot{h}_{ipool}(\cdotp)\right]\left. +\frac{f_2(\cdotp)}{2{\rm \pi} h_{ipool}^2(\cdotp)}\dot{h}_{ipool}(\cdotp)\right\}\nonumber\\ &\left. \frac{{\rm \partial}H\left[u_i(k), y_i(k)\right]}{{\rm \partial}u_i(k)} \right|_{\substack{u_i(k)=u_{i0}\\y_i(k)=y_{i0}}}= -\delta_t\frac{\sqrt{3}y_{i0}^2}{U}\times\nonumber\\ &\qquad\quad \frac{f_1(\cdotp)}{{\rm \pi} r_{iarc}^2}\times \frac{2{\rm \pi}(1-s)}{p}r_d \end{align} $
(17) 令
$ \begin{align} &a_{i1}=-\left. \frac{{\rm \partial}H\left[ u_i(k), y_i(k)\right]}{{\rm \partial}y_i(k)}\right|_{\substack{u_i(k)=u_{i0}\\y_i(k)=y_{i0}}}\nonumber\\ &b_{i0}=\left. \frac{{\rm \partial}H\left[ u_i(k), y_i(k)\right]}{{\rm \partial}u_i(k)}\right|_{\substack{u_i(k)=u_{i0}\\y_i(k)=y_{i0}}} \end{align} $
(18) 因此电极电流模型式(16)在工作点$(u_{i0}, y_{i0})$附近的Taylor展开式为:
$ \begin{align} y_i(k+1)=\, &H\left[u_{i0}, y_{i0}\right]-a_{i1}\left[y_i(k)-y_{i0}\right]+\nonumber\\ &b_{i0}\left[u_i(k)-u_{i0}\right]+R_2 \end{align} $
(19) 其中,
$ \begin{align} R_2=\, &\frac{1}{2}\left\{ \left. \frac{{\rm \partial}H\left[ u_i(k), y_i(k) \right]}{{\rm \partial}u_i(k){\rm \partial}u_i(k)}\right|_{\substack{u_i(k)= u_{i0}+\ell\left[ u_i(k)-u_{i0}\right]\\y_i(k)=y_{i0}+\ell\left[y_i(k)-y_{i0} \right]}}\times\right.\nonumber\\ &\left[u_i(k)-u_{i0}\right]^2+\nonumber\\ &2\left. \frac{{\rm \partial}H\left[u_i(k), y_i(k)\right]} {{\rm \partial}u_i(k){\rm \partial}y_i(k)}\right|_{\substack{u_i(k)= u_{i0}+\ell\left[u_i(k)-u_{i0}\right]\\y_i(k)=y_{i0}+\ell\left[ y_i(k)- y_{i0}\right]}}\times\nonumber\\ &\left[u_i(k)-u_{i0}\right]\left[y_i(k)-y_{i0}\right]+\nonumber\\ &\left. \frac{{\rm \partial}H\left[u_i(k), y_i(k)\right]}{{\rm \partial}y_i(k){\rm \partial}y_i(k)}\right|_{\substack{u_i(k)= u_{i0}+\ell\left[u_i(k)-u_{i0}\right]\\y_i(k)=y_{i0}+\ell \left[y_i(k)-y_{i0}\right]}}\times\nonumber\\ &\left. \left[y_i(k)-y_{i0}\right]^2\right\}, 0<\ell<1 \end{align} $
(20) 将式(19)进行变换可得由确定线性模型与高阶非线性项组成的电极电流动态模型:
$ \begin{align} &A_i(z^{-1})y_i(k + 1) = B_i(z^{-1})u_i(k)+v_i(k)\nonumber\\ &A_i(z^{-1})=1+a_{i1}z^{-1}, B_i(z^{-1})=b_{i0}, \nonumber\\ &i=1, 2, 3 \end{align} $
(21) 使用实际过程数据, 采用递推最小二乘和神经网络交替辨识方法[19]确定$a_{i1}$、$b_{i0}$, 且知$v_i(k)$有界, 被控对象式(21)为最小相位系统. $a_{i1}$和$b_{i0}$的辨识误差由$v_i(k)$来描述, 通过设计$v_i(k)$的补偿器来消除采用$a_{i1}$和$b_{i0}$设计控制器对控制效果的影响.
由于$v_i(k)$在$k$时刻未知, 因此可将$v_i(k)$表示为前一时刻高阶非线性项$v_i(k-1)$与其变化率$\Delta v_i(k)$之和的形式, 即:
$ \begin{equation} v_i(k)=v_i(k-1)+\Delta v_i(k) \end{equation} $
(22) 由式(21)可知:
$ \begin{align} v_i(k-1)=\, &y_i(k)+A_i^*(z^{-1})y_i(k)-\nonumber\\ &B_i(z^{-1})u_i(k-1)=\nonumber\\ &y_i(k)-y_i^*(k) \end{align} $
(23) 式中, $y_i^*(k)=-a_{i1}y_i(k-1)+ b_{i0}u_i(k-1)$为电极电流控制器驱动模型[20].
将式(22)代入式(21), 于是可得电极电流模型为:
$ \begin{align} &A_i(z^{-1})y_i(k+1)=B_i(z^{-1})u_i(k)+\nonumber\\ &\qquad v_i(k-1)+\Delta v_i(k) \end{align} $
(24) 采用模型(24)中的确定线性部分可以设计PID控制器, 由式(23)可知, 前一时刻高阶非线性项$v_i(k-1)$可以精确获得, 因此可以设计消除其影响的控制器, 虽然高阶非线性项变化率$\Delta v_i(k)$未知, 但可以通过设计消除跟踪误差$e_i(k)$的补偿器来消除$\Delta v_i(k)$的影响, 将上述补偿器产生的补偿信号$u_{i2}(k)$、$u_{i3}(k)$叠加到PID控制器的输出$u_{i1}(k)$, 带输出补偿的PID控制器如图 2所示.
2.2 控制器设计
2.2.1 PID控制器和前一时刻高阶非线性项补偿器设计
带输出补偿的PID控制器为:
$\begin{equation} u_i(k)=u_{i1}(k)+u_{i2}(k)+u_{i3}(k) \end{equation} $
(25) 以式(24)的确定线性部分模型设计的PID控制律为:
$ \begin{equation} H_i(z^{-1})u_{i1}(k)=G_i(z^{-1})e_i(k) \end{equation} $
(26) 式中, $H_i(z^{-1})=1-z^{-1}$, $G_i(z^{-1})= g_{i0}+ g_{i1}z^{-1}+g_{i2}z^{-2}$, $g_{i0}$、$g_{i1}$和$g_{i2}$为PID控制参数, $e_i(k)$为跟踪误差, 即: $e_i(k)=y_{sp}(k)-y_i(k)$.
前一时刻高阶非线性项$v_i(k-1)$补偿器为:
$ \begin{equation} u_{i2}(k)=-K_i(z^{-1})v_i(k-1) \end{equation} $
(27) 式中, $K_i(z^{-1})$为补偿器的参数.
采用一步最优前馈控制律来设计$G_i(z^{-1})$和$K_i(z^{-1})$的参数, 将式(26)中的$u_{i1}(k)$和式(27)中的$u_{i2}(k)$代入式(25)中得到$u_{i}(k)$为:
$ \begin{align} &H_i(z^{-1})u_i(k)=G_i(z^{-1})\left[y_{sp}(k)-y_i(k)\right]-\nonumber\\ &\qquad H_i(z^{-1})K_i(z^{-1})v_i(k-1)+H_i(z^{-1})u_{i3}(k) \end{align} $
(28) 引入下列性能指标[14]:
$ \begin{align} J=\, &\left[P_i(z^{-1})y_i(k+1)-R_i(z^{-1})y_{sp}(k)+\right.\nonumber\\ &Q_i(z^{-1})u_i(k)+\overline{K}_i(z^{-1})v_i(k-1)-\nonumber\\ &\left.H_i(z^{-1})u_{i3}(k)\right]^2 \end{align} $
(29) 式中, $P_i(z^{-1})$、$R_i(z^{-1})$、$Q_i(z^{-1})$和$\overline{K}_i(z^{-1})$均是关于$z^{-1}$的加权多项式.
引入广义输出$\phi_i(k+1)$为:
$\begin{equation} \phi_i(k+1)=P_i(z^{-1})y_i(k+1) \end{equation} $
(30) 定义广义理想输出$\phi_i^*(k+1)$为:
$ \begin{align} \phi_i^*(k+1)=\, &R_i(z^{-1})y_{sp}(k)-\nonumber\\ &Q_i(z^{-1})u_i(k)-\overline{K}_i(z^{-1})v_i(k) \end{align} $
(31) 定义式(29)中的$P_i(z^{-1})$为:
$ \begin{equation} P_i(z^{-1})=A_i(z^{-1})+z^{-1}G_i(z^{-1}) \end{equation} $
(32) 由式(24)和式(32)可得:
$ \begin{align} P_i(z^{-1})y_i(k+1)=\, &G_i(z^{-1})y_i(k)+B_i(z^{-1})u_i(k)+\nonumber\\ &v_i(k-1)+\Delta v_i(k) \end{align} $
(33) 将式(33)代入式(29), 使$J$最小可得带有前一时刻高阶非线性项$v_i(k-1)$补偿的一步最优前馈控制律为:
$ \begin{align} &\left[B_i(z^{-1})+Q_i(z^{-1})\right]u_i(k)=\nonumber\\ &\qquad R_i(z^{-1})y_{sp}(k)-G_i(z^{-1})y_i(k)-\nonumber\\ &\qquad\left[1+\overline{K}_i(z^{-1})\right]v_i(k-1)+H_i(z^{-1})u_{i3}(k) \end{align} $
(34) 由式(28)和式(34)可得$Q_i(z^{-1})$、$R_i(z^{-1})$和$\overline{K}_i(z^{-1})$为:
$ \begin{equation} \left\{ \begin{array}{l} Q_i(z^{-1})=H_i(z^{-1})-B_i(z^{-1})\\ R_i(z^{-1})=G_i(z^{-1})\\ \overline{K}_i(z^{-1})=H_i(z^{-1})K_i(z^{-1})-1 \end{array} \right. \end{equation} $
(35) 将式(34)和式(35)代入电极电流模型式(24)中得到电极电流闭环系统方程为:
$ \begin{align} &\left[A_i(z^{-1})H_i(z^{-1})+z^{-1}B_i(z^{-1})G_i(z^{-1})\right]\times\nonumber\\ &\qquad y_i(k+1)=B_i(z^{-1})G_i(z^{-1})y_{sp}(k)+\nonumber\\ &\qquad B_i(z^{-1})H_i(z^{-1})u_{i3}(k)+H_i(z^{-1})\times\nonumber\\ &\qquad \left[1-B_i(z^{-1})K_i(z^{-1})\right]v_i(k-1)+\nonumber\\ &\qquad H_i(z^{-1})\Delta v_i(k) \end{align} $
(36) 选择$G_i(z^{-1})$的参数$g_{i0}$、$g_{i1}$和$g_{i2}$使式(36)所示闭环系统稳定, 即:
$ \begin{equation} A_i(z^{-1})H_i(z^{-1})+z^{-1}B_i(z^{-1})G_i(z^{-1})\neq 0, |z|>1 \end{equation} $
(37) 由式(36)可知, 为实现对$v_i(k-1)$的动态和静态补偿, 选择$K_i(z^{-1})$使$1- B_i(z^{-1})K_i(z^{-1})=0$, 即:
$ \begin{equation} K_i(z^{-1})=\frac{1}{B_i(z^{-1})}=k_{vi0} \end{equation} $
(38) 于是式(36)为:
$ \begin{align} &A_i(z^{-1})H_i(z^{-1})y_i(k+1)=\nonumber\\ &\qquad B_i(z^{-1})G_i(z^{-1})e_i(k)+\nonumber\\ &\qquad B_i(z^{-1})H_i(z^{-1})u_{i3}(k)+H_i(z^{-1})\Delta v_i(k) \end{align} $
(39) 2.2.2 高阶非线性项变化率补偿器设计
虽然高阶非线性项变化率$\Delta v_i(k)$未知, 但其造成的跟踪误差$e_i(k)$已知, 因此以消除跟踪误差$e_i(k)$为目标, 设计补偿器$u_{i3}(k)$, 将式(39)两边同时减$A_i(z^{-1})H_i(z^{-1})y_{sp}(k+1)$, 可以将式(39)表示为以$e_i(k+1)$为输出, 以$u_{i3}(k)$为输入的系统, 即:
$\begin{align} &\left[A_i(z^{-1})H_i(z^{-1})+z^{-1}B_i(z^{-1})G_i(z^{-1})\right]\times\nonumber\\ &\qquad e_i(k+1)=-B_i(z^{-1})H_i(z^{-1})u_{i3}(k)-\nonumber\\ &\qquad H_i(z^{-1})\Delta v_i(k)+A_i(z^{-1})H_i(z^{-1})y_{sp}(k+1) \end{align} $
(40) 为了尽可能地消除$e_i(k+1)$, 引入一步最优调节律[18]设计$u_{i3}(k)$, 引入下列性能指标:
$ \begin{equation} J' ={\rm min}\left[e_i(k+1)\right]^2 \end{equation} $
(41) 引入Diophantine方程:
$ \begin{align} &A_i(z^{-1})H_i(z^{-1})+z^{-1}B_i(z^{-1})G_i(z^{-1})+\nonumber\\ &\qquad z^{-1}G_i'(z^{-1})=1 \end{align} $
(42) 由式(42)可得$G_i'(z^{-1})$为:
$ \begin{align} G_i'(z^{-1})=\, &A_i(z^{-1})-B_i(z^{-1})G_i(z^{-1})-\nonumber\\ & a_{i1}=g_{i0}'+g_{i1}'z^{-1}+g_{i2}'z^{-2} \end{align} $
(43) 其中, $g_{i0}'=1-b_{i0}g_{i0}-a_{i1}$, $g_{i1}'= a_{i1}-b_{i0}g_{i1}$, $g_{i2}'=-b_{i0}g_{i2}$.
将式(42)代入式(40)中得:
$ \begin{align} &e_i(k+1)=G_i'(z^{-1})e_i(k)-\nonumber\\ &\qquad B_i(z^{-1})H_i(z^{-1})u_{i3}(k)-H_i(z^{-1})\Delta v_i(k)+\nonumber\\ &\qquad A_i(z^{-1})H_i(z^{-1})y_{sp}(k+1) \end{align} $
(44) 由式(44)可知, 跟踪误差的一步最优预报$e_i^*(k+1/k)$为:
$ \begin{align} &e_i^*(k+1/k)=G_i'(z^{-1})e_i(k)-\nonumber\\ &\qquad B_i(z^{-1})H_i(z^{-1})u_{i3}(k)-\nonumber\\ &\qquad H_i(z^{-1})\Delta v_i(k-1)+\nonumber\\ &\qquad A_i(z^{-1})H_i(z^{-1})y_{sp}(k+1) \end{align} $
(45) 令$e_i^*(k+1/k)=0$, 可得补偿信号$u_{i3}(k)$为:
$ \begin{align} &u_{i3}(k)=\frac{1}{H_i(z^{-1})B_i(z^{-1})}G_i'(z^{-1})e_i(k)-\nonumber\\ &\qquad \frac{1}{B_i(z^{-1})}\Delta v_i(k-1)+\nonumber\\ &\qquad\frac{A_i(z^{-1})}{B_i(z^{-1})}y_{sp}(k+1) \end{align} $
(46) 式中, 高阶非线性项变化率补偿器参数$G_i'(z^{-1})$由式(43)获得.
2.3 带输出补偿的PID控制算法
电熔镁砂熔炼过程带输出补偿的PID控制算法实现步骤如下:
1) 采用实际熔炼过程输入输出数据, 利用递推最小二乘和神经网络交替辨识电极电流控制器设计模型式(24)的参数$a_{i1}$和$b_{i0}$;
2) 由式(37)确定PID控制器参数$g_{i0}$、$g_{i1}$和$g_{i2}$;
3) 由式(38)确定前一时刻高阶非线性项$v_i(k-1)$补偿器参数$k_{vi0}$;
4) 由式(43)确定前一时刻高阶非线性项变化率$\Delta v_i(k)$补偿器参数$g_{i0}'$、$g_{i1}'$和$g_{i2}'$;
5) 采集输入输出数据, 求出跟踪误差$e_{i}(k)$, 并由式(23)求出前一时刻高阶非线性项$v_i(k-1)$, 由式(22)求出前一时刻高阶非线性项变化率$\Delta v_i(k-1)$;
6) 由式(26)求出PID控制器输出$u_{i1}(k)$, 由式(27)求出前一时刻高阶非线性项补偿器输出$u_{i2}(k)$, 由式(46)求出前一时刻高阶非线性项变化率补偿器输出$u_{i3}(k)$;
7) 由式(25)求出带输出补偿的PID控制器输出$u_{i}(k)$, 加到电熔镁炉被控对象上;
8) $t=k+1$, 返回步骤5).
3. 仿真验证
首先将本文所提电熔镁砂熔炼过程带输出补偿的PID控制算法进行仿真实验研究, 以验证其有效性和实用性.
3.1 被控对象仿真模型
将式(13)所示的电极电流被控对象动态模型表示成如下形式:
$ \begin{equation} \dot{y}_i(t)=\frac{\sqrt{3}}{{\rm \pi}}F_i(\cdotp)y_i^2(t)-2\sqrt{3}Q_i(\cdotp)u_i(t)y_i^2(t) \end{equation} $
(47) 其中,
$ \begin{equation} \left\{ \begin{array}{l}%array 中lrc表示各列内容的居左、居中、居右 F_i(\cdotp)=\left[\dfrac{f_1(\cdotp)}{r_{iarc}^2}-\dfrac{f_2(\cdotp)} {2h_{ipool}^2(\cdotp)}\right]\dfrac{\dot{h}_{ipool}(\cdotp)}{U}\\[2mm] Q_i(\cdotp)=\dfrac{f_1(\cdotp)~(1-s)r_d}{Ur_{iarc}^2p} \end{array} \right. \end{equation} $
(48) 采用欧拉法将电极电流动态模型式(47)离散化, 使用实际工业过程中大量的电极电流和电机转动方向与频率数据, 采用递推最小二乘和神经网络交替辨识方法[19]对电流模型的参数$F_i(\cdotp)$、$Q_i(\cdotp)$及建模误差$\Delta y_i(k)$进行辨识, 于是电极电流仿真模型如下:
$ \begin{align} &y_i(k+1)=y_i(k)+\delta_t\frac{\sqrt{3}}{{\rm \pi}}\hat{F}_iy_i^2(k)-\nonumber\\ &\qquad \delta_t2\sqrt{3}\hat{Q}_iu_i(k)y_i^2(k)+\Delta \hat{y}_i(k) \end{align} $
(49) 其中, $\delta_t$为采样时间, $\hat{F}_i$、$\hat{Q}_i$和$\Delta \hat{y}_i(k)$通过辨识得到, 采用式(49)进行仿真实验.
3.2 控制目标及控制器参数选择
控制目标可表示为:
$ \begin{equation} \left|e(k)\right|=\left|y_{sp}(k)-y(k)\right|<2\, 000, 0<k<\infty \end{equation} $
(50) 其中, 设定值$y_{sp}(k)=15\, 300$ A, 电极电流$y_1(k)$和控制量$u_1(k)$的约束如下:
$ \begin{equation} \begin{array}{c} 12\, 000<y_1(k)<17\, 000\\ -20<u_1(k)<20\\ \end{array} \end{equation} $
(51) 电极电流控制器设计模型参数为:
$ \begin{align} &A_1(z^{-1})=1-1.0019z^{-1}\nonumber\\ &\qquad B_1(z^{-1})=-0.454 \end{align} $
(52) 由式(37)、式(38)和式(43)确定带输出补偿的PID控制器参数为:
$ \begin{equation} \left\{ \begin{array}{l} G_1(z^{-1})=-1.295+1.82z^{-1}-0.56z^{-2}\\ K_1(z^{-1})=-2.2026\\ G_1'(z^{-1})=1.414-0.1756z^{-1}-0.2542z^{-2}\\ \end{array} \right. \end{equation} $
(53) 常规PID控制器参数与$G_1(z^{-1})$参数相同.
3.3 仿真结果
采用递推最小二乘和神经网络交替辨识所得$F_1(\cdotp)$、$Q_1(\cdotp)$的估计值为$\hat{F}_1=-1.344\times10^{-4}$、$\hat{Q}_1=7.059\times10^{-4}$, 采用式(49)并叠加上如图 3所示的随机噪声信号$noise_1(k)$作为被控对象仿真模型, 将本文提出的控制算法与常规PID控制算法进行仿真对比实验.
仿真对比实验结果如图 4所示.
图 4 采用常规PID控制算法和本文控制算法时电极电流$y_1$的控制效果Fig. 4 The control effects of electrode current $y_1$ using traditional PID control algorithm and the control algorithm of this paper由图 4可以看出, 采用常规PID控制算法时, 电极电流跟踪误差绝对值存在超出跟踪误差上限的情况, 而采用本文算法能够在所有运行时间将电极电流跟踪误差控制在工艺要求的范围内.
采用如式(54)和式(55)所示的性能评价指标均方误差(Mean squared error, MSE)[21]和误差绝对值积分(Integrated absolute error, IAE)[22]对图 4所示的控制效果进行比较, 结果见表 1.
$ \begin{equation} {\rm MSE}=\frac{1}{N}\sum\limits_{k=1}^N\left[y_{sp}(k)-y(k)\right]^2 \end{equation} $
(54) $ \begin{equation} {\rm IAE}=\sum\limits_{k=1}^N\left|y_{sp}(k)-y(k)\right| \end{equation} $
(55) 表 1 采用PID控制器和本文所述控制器控制电流$y_1$时的性能评价表Table 1 The performance evaluating table of current $y_1$ controlled with PID controller and the proposed controller in this paperMSE IAE PID控制器 $2.3386\times10^6$ $0.6431\times10^6$ 本文所述控制器 $0.4502\times10^6$ $0.2787\times10^6$ 降低 $80.75\, \%$ $56.66\, \%$ 由表 1可以看出, 采用PID控制算法时, 电极电流的MSE为$2.3386\times10^6$, 而采用本文算法时, 电极电流的MSE为$0.4502\times10^6$, 降低了$80.75\, \%$; 采用PID控制算法时, 电极电流的IAE为$0.6431\times10^6$, 而采用本文算法时, 电极电流的IAE为$0.2787\times10^6$, 降低了$56.66\, \%$.
上述仿真结果表明本文所述方法优于常规PID控制方法, 可以将电熔镁砂熔炼过程电极电流控制在目标范围内.
4. 工业应用
将本文提出的带输出补偿的PID控制算法进行了工业应用, 以验证其有效性和实用性.
4.1 电熔镁炉应用对象描述
中国辽宁省某电熔镁砂厂的实际电熔镁炉如图 5所示.该厂生产设备和工艺参数如表 2所示.
表 2 生产设备和工艺参数Table 2 Parameters of production equipment and technology参数 数值 电极直径 250 mm 电极长度 1 500 mm 炉体直径 2.5 m 熔炼电压 100 $\sim$ 150 V 熔炼时间 10 h 电熔镁砂熔炼过程控制系统硬件平台如图 6所示, 由人机交互平台、德国Siemens公司的S7-300PLC控制系统、传感器等组成.所开发的人机交互界面如图 7所示.
根据工艺要求, 电熔镁炉实际熔炼过程的电流控制目标可表示为:
$ \begin{equation} \left|e(k)\right|=\left|y_{sp}(k)-y(k)\right|<2\, 000, 0<k<\infty \end{equation} $
(56) 其中, 设定值$y_{sp}(k)=15\, 300$ A, 电极电流$y_i(k)$和控制量$u_i(k)$的约束如式(57)所示.
$ \begin{equation} \begin{array}{c} 12\, 000<y_i(k)<17\, 000, \\ -20<u_i(k)<20, \\ i=1, 2, 3 \end{array}\end{equation} $
(57) 4.2 控制器与补偿器参数选择
电极电流控制回路采样周期为1 s, 控制参数为: $g_{i0}=-1.4$, $g_{i1}=1.62$, $g_{i2}=-0.51$, $k_{vi0}=-2.35$, $g_{i0}'=1.5$, $g_{i1}'=-0.2$, $g_{i2}'=-0.27$.
4.3 应用效果分析
该厂某熔炼过程采用常规PID算法时的控制效果如图 8所示. $00:04$开始, 由于原矿性质变化导致该熔炼过程动态特性变化, 可以看出此时常规PID控制效果不佳. $00:32$开始, 将控制算法改为本文所提算法后电流控制效果如图 9所示.可以看出, 在熔炼工况相同的情况下, 本文算法能够明显减小电流波动.
图 8 采用常规PID控制算法时电熔镁炉三相电极电流平均值$y$的控制效果Fig. 8 The control effects of the average value $y$ of three phase electrode currents of fused magnesium furnace using traditional PID control algorithm
图 9 采用带输出补偿的PID控制算法时电熔镁炉三相电极电流平均值$y$的控制效果Fig. 9 The control effects of the average value $y$ of three phase electrode currents of fused magnesium furnace using PID control algorithm with output compensation将常规PID控制方法与本文所述方法的电极电流控制效果用性能指标MSE、IAE进行对比, 结果如表 3所示.采用常规PID控制器时三相电极电流平均值的MSE和IAE为$1.3083\times10^6$和$1.3503\times10^6$, 而采用本文所述带输出补偿的PID控制器时三相电极电流平均值的MSE和IAE为$0.4260\times10^6$和$0.7743\times10^6$, 分别降低了$67.44\, \%$和$42.66\, \%$.
表 3 采用常规PID控制器和本文所述带输出补偿的PID控制器时三相电极电流平均值$y$的性能评价表Table 3 The performance evaluating table of the average value $y$ of three phase electrode currents using traditional PID controller and the proposed PID controller with output compensation in this paperMSE IAE 常规PID $1.3083\times10^6$ $1.3503\times10^6$ 本文方法 $0.4260\times10^6$ $0.7743\times10^6$ 降低 $67.44\, \%$ $42.66\, \%$ 引入式(58)所示的性能指标超区间绝对误差累积和:
$ \begin{equation} \sum\limits_{k=1}^{N}\left\{\left|y_{sp}(k)-y(k)\right|-\varphi\Big|\left|y_{sp}(k)-y(k)\right|\geq\varphi\right\} \end{equation} $
(58) 式中, $\varphi$为误差波动允许上限值, 即$\varphi=2\, 000$ A.
经计算, 常规PID控制时三相电极电流平均值超区间绝对误差累积和为$3.3819\times10^4$, 而本文所述算法为0.
采用常规PID算法和本文所提算法时, 三相电极电流跟踪误差平均值的经验概率分布如图 10所示.可以看出, 采用常规PID算法时三相电极电流跟踪误差平均值(A)超出其上下限$\left[-2\, 000, ~2\, 000\right]$的比例为$7.29\, \%$, 而采用本文所提算法时为0.综上, 本文算法的控制效果明显改善, 满足工艺要求.
图 10 采用常规PID控制算法和本文控制算法时三相电极电流跟踪误差平均值的经验概率分布Fig. 10 Experienced probability distribution of the average value of three phase electrode currents$'$ tracking errors using traditional PID control algorithm and the control algorithm of this paper通过上述对比分析不难看出, 当熔炼过程动态特性变化时, 本文算法的控制效果优于常规PID控制算法, 这必然有利于降低产品单吨能耗.经过统计, 常规PID控制时产品单吨能耗平均值为2 459 kwh/t, 而本文所述算法控制时产品单吨能耗平均值为2 412 kwh/t, 降低了1.91$\, \%$.
5. 结论
针对电熔镁炉三相电极电流处于动态之中导致PID的积分器失效问题, 本文提出了一种电熔镁砂熔炼过程带输出补偿的PID控制器.该控制器由前一时刻高阶非线性项补偿器、消除其变化率补偿器和基于确定线性模型设计的常规PID控制器组成.仿真和工业应用结果表明, 当电极电流模型参数发生未知随机变化时, 所提出的控制方法无需参数估计可将三相电极电流平均值控制在目标值范围内.本文所提的带输出补偿的PID控制器设计方法对难以采用常规PID控制的复杂工业过程的控制器设计具有参考价值.
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