Adaptive Predictive Proportional-integral-resonant Current Control for Permanent Magnet Synchronous Motors
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摘要: 考虑数字控制系统一个采样周期输入延时和驱动器功率管非线性特性的影响，为增强永磁同步电机（Permanent magnet synchronous motor，PMSM）电流环稳定性和提高电流控制精度，提出一种自适应预测比例-积分-谐振控制（Adaptive predictive proportional-integral-resonant，APPI-RES）策略.该方法能够在电机电阻和电感参数不确定的条件下，预测电流控制误差和未知周期电压扰动，将所得预测量执行反馈控制，实现了对系统输入延时和相电流谐波的有效补偿.最后，通过仿真分析验证了所提控制策略的有效性.
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关键词:
- 永磁同步电机 /
- 自适应预测比例-积分-谐振控制器 /
- 输入延时 /
- 参数不确定 /
- 相电流畸变
Abstract: In order to enhance the stability of current loop and to improve the current control accuracy, an adaptive predictive proportional-integral-resonant (APPI-RES) current control strategy is proposed, in which the one cycle delay of digital control system and the nonlinear characteristic of voltage source inverter (VSI) are considered. The proposed method can predict the control errors of currents, unknown constant and periodic voltage disturbances when the resistor and inductance of the motor are unknown. Then, the predicted variables are used to execute the feedback control to compensate the system input delay and the phase current harmonics effectively. At last, simulation results verify the feasibility of the control strategy. -
永磁同步电机(Permanent magnet synchronous motor, PMSM)功率密度高、电流输出转矩比小, 被广泛应用于航天、数控机床等高精度调速场合[1-3].交流调速系统通常采用速度外环级联电流内环的控制结构.其中, 电流控制性能决定了转矩控制精度和电机损耗.常规PID控制简单易行, 但对系统参数和外界扰动的鲁棒性较差.针对这一缺点, 文献[4-8]分别提出自适应PID控制[4]、滑模控制[5]、无差拍控制[6]、预测控制[7-8], 以提高电流控制精度和系统对电机参数、电压扰动的鲁棒性.
电机相电流谐波源可等效为周期电压扰动, 扰动电压通常由电压源逆变器非线性畸变、转子非正弦反电动势和磁饱和等因素引起[9].重复电流控制策略[10]可有效抑制周期扰动电压, 但当电机运行于低速时, 存储一个周期内的电流数据占用系统内存较多.迭代学习控制[11]通过在线反复修正相电流幅值和相位, 可完全消除谐波电流, 但计算量较大.比例-积分-谐振电流控制策略(Proportional-integral-resonant controller, PI-RES)[12]简单易行, 可有效抑制周期扰动电压, 已广泛应用于风力发电及交流调速领域.然而, 这种常规的PI-RES控制器只是内模控制器的等效并联形式, 并不是真正意义上的渐近稳定并联控制器.同时, 数字控制系统存在一个采样周期的输入延时, 当系统采样频率/电流输出频率($f_s$/$f_o$)小于10时, 电流将出现震荡失稳现象[13].文献[12]采用电流前馈补偿, 消除了电流超调, 但未对扰动进行预测补偿.文献[14]利用高阶滤波器最优逼近包含延时的系统误差模型, 但系统采样频率低于2.5 kHz时, 电流不稳定.文献[15]在电流环中加入了有源阻尼衰减相, 有效抑制了输入延时对解耦控制的影响.文献[16]采用二阶pade模型近似系统输入延时, 通过最优配置系统根轨迹法削弱了输入延时对电流控制的影响, 但未研究算法对模型参数的鲁棒性.
本文通过将多种PI-RES控制器化为统一的状态空间表达形式, 分析系统输入延时对电流控制稳定性的影响.针对系统输入延时和电流谐波问题, 提出一种自适应预测比例-积分谐振(Adaptive predictive proportional-integral-resonant, APPI-RES)电流控制策略.该方法可在电机电阻、电感参数未知的情况下预测电流指令、电压扰动和电流控制误差.通过执行预测量的反馈控制, 补偿系统输入延时和相电流谐波, 算法收敛性分析保证了所提控制器的渐近稳定性.最后, 通过仿真研究验证了所提算法的有效性.
1. 输入延时对传统PI-RES控制的影响分析
1.1 表贴式PMSM电气模型
逆变器驱动永磁同步电机相电流通常包含6$n \pm1$ ($n$为自然数)次谐波, 可将电流谐波源等效为频率为6$n \pm1$次周期扰动电压, 在同步旋转坐标系下该扰动电压频率为6$n$次[9].考虑控制器输入延时和周期扰动电压的同步旋转坐标系下表贴式PMSM电气模型:
$ \left\{ \begin{array}{ll} u_{dD}^* + u_d^{dis} = R{i_d} + L{{\dot i}_d} - {\omega _e}L{i_q}\\ u_{qD}^* + u_q^{dis} = R{i_q} + L{{\dot i}_q} + {\omega _e}L{i_d} + {\omega _e}{\varphi _f} \end{array} \right. $
(1) 式中, 下标$D$表示延时时间, $u_{dD}^{*}$、$u_{qD}^{*}$分别为$t-D$时刻$d$、$q$轴控制器输出电压指令, $u_{d}^{dis}$、$u_{q}^{dis}$分别为$t$时刻$d$、$q$轴各次谐波扰动电压的总和, $i_d$、$i_q$分别为$d$、$q$轴电流, $R$、$L$、$\omega _e$、$\varphi _f$分别为定子电阻、电感、转子电角速度和永磁体磁链.
令$i_{d}^{*}$、$i_{q}^{*}$别为$d$、$q$轴参考电流指令, ${\dot i}_{d}^{*}$、${\dot i}_{q}^{*}$为参考电流指令的导数, 采用$i_d^*=0$控制, 由式(1)得表贴式PMSM的电流误差空间模型:
$ \dot z = Az - Bu_D^* - Bd $
(2) 式中
$ z = \left[ {\begin{array}{*{20}{c}} {{z_d}}\\ {{z_q}} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} {i_d^* - {i_d}}\\ {i_q^* - {i_q}} \end{array}} \right], u_D^* = \left[ {\begin{array}{*{20}{c}} {u_{dD}^*}\\ {u_{qD}^*} \end{array}} \right]\\ A = \left[ {\begin{array}{*{20}{c}} { - \frac{R}{L}}&{{\omega _e}}\\ { - {\omega _e}}&{ - \frac{R}{L}} \end{array}} \right], B = \left[ {\begin{array}{*{20}{c}} {\frac{1}{L}}&0\\ 0&{\frac{1}{L}} \end{array}} \right]\\ {d_1} = \left[ {\begin{array}{*{20}{c}} {u_d^{dis}}\\ {u_q^{dis}} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} {\sum\limits_{n = 1}^\infty {d_{6n}^d} }\\ {\sum\limits_{n = 1}^\infty {d_{6n}^q} } \end{array}} \right], d = {d_0} + {d_1}\\ {d_0} = \left[ {\begin{array}{*{20}{c}} 0\\ { - {\omega _e}{\varphi _f}} \end{array}} \right] - {B^{ - 1}}\left[ {\begin{array}{*{20}{c}} {\dot i_d^*}\\ {\dot i_q^*} \end{array}} \right] + {B^{ - 1}}A\left[ {\begin{array}{*{20}{c}} {i_d^*}\\ {i_q^*} \end{array}} \right]. $
当电机运行于稳态时, $d_0$为恒定值, $d_1$为三角函数的级数和, 因此扰动$d$可表示为
$ d\left( t \right) = \sum\limits_{n = 0}^\infty {{d_n}\left( t \right)} = \sum\limits_{n = 0}^\infty {A_{6n}^{dq}{f_n}\left( t \right)} = \sum\limits_{n = 0}^\infty {{\theta _{dq6n}}{f_{mn}}\left( t \right)} $
(3) 式中
$ A_{6n}^{dq} = \left[{\begin{array}{*{20}{c}} {A_{6n}^d}&0\\ 0&{A_{6n}^q} \end{array}} \right], {\theta _{dq6n}} = \left[{\begin{array}{*{20}{c}} {{\theta _{ds6n}}}&{{\theta _{dc6n}}}\\ {{\theta _{qs6n}}}&{{\theta _{qc6n}}} \end{array}} \right]\\ {\theta _{ds6n}} = A_{6n}^d\sin \left( {\varphi _{6n}^d} \right), {\theta _{dc6n}} = A_{6n}^d\cos \left( {\varphi _{6n}^d} \right)\\ {\theta _{qs6n}} = A_{6n}^q\sin \left( {\varphi _{6n}^q} \right), {\theta _{qc6n}} = A_{6n}^q\cos \left( {\varphi _{6n}^q} \right)\\ {f_n}\left( t \right) = \left[{\begin{array}{*{20}{c}} {\cos \left( {6n\omega _e^*t-\varphi _{6n}^d} \right)}\\ {\cos \left( {6n\omega _e^*t-\varphi _{6n}^q} \right)} \end{array}} \right]\\ {f_{mn}}\left( t \right) = \left[{\begin{array}{*{20}{c}} {\sin \left( {6n\omega _e^*t} \right)}\\ {\cos \left( {6n\omega _e^*t} \right)} \end{array}} \right].\\ $
其中, $\omega _e^*$为给定转子电角速度指令.
1.2 传统PI-RES控制器的等效状态空间形式
传统PI-RES控制策略通过在系统谐振频率处并联谐振器, 可实现无静差抑制系统周期扰动[9].这里为分析方便, 令式(3)中$n=0\sim2, $则所需设计的PI-RES控制器可表示为[9, 12, 16]
$ \begin{align} C\left( s \right) =\, & {k_p} + \frac{{{k_i}}}{s} + \frac{{{a_2}{s^2} + {a_1}s + {a_0}}}{{{s^2} + \omega _0^2}} +\nonumber\\& \frac{{{b_2}{s^2} + {b_1}s +{b_0}}}{{{s^2} + \omega _1^2}} \end{align} $
(4) 式中, $s$为拉氏复变量, $\omega _0=6\omega_e^*$, $\omega_1=12\omega_e^*$, $k_p$、$k_i$、$a_0\sim a_2$、$b_0\sim b_2$分别为调节器参数.将式(4)中各项分式通分得:
$ C\left( s \right) = {k_\varepsilon } + \frac{{{k_4}{s^4} + {k_3}{s^3} + {k_2}{s^2} + {k_1}{s^1} + {k_0}}}{{s\left( {{s^2} + \omega _0^2} \right)\left( {{s^2} + \omega _1^2} \right)}} $
(5) 式中
$ {\lambda _a} = {a_0} - {a_2}\omega _0^2, {\lambda _b} = {b_0} - {b_2}\omega _1^2, {k_\varepsilon } = {k_p} + {a_2} + {b_2}\\ {k_0} = {k_i}\omega _0^2\omega _1^2, {k_1} = {\lambda _a}\omega _1^2 + {\lambda _b}\omega _0^2, {k_4} = {a_1} + {b_1} + {k_i}\\ {k_3} = {\lambda _a} + {\lambda _b}, {k_2} = \left( {{k_i} + {b_1}} \right)\omega _0^2 + \left( {{k_i} + {a_1}} \right)\omega _1^2. $
由文献[17]知, 式(5)为内模控制器, 系统模态可表示为
$ \begin{align} P\left( \sigma \right) = \sigma \left( {{\sigma ^2} + \omega _0^2} \right)\left( {{\sigma ^2} + \omega _1^2} \right) \end{align} $
(6) 式中, 为微分算子.
为分析方便, 暂不考虑系统输入延时, 即$D = 0$, $u_D^* = {u^*}$.令$\varepsilon = P\left( \sigma \right)z$为扩展误差空间状态分量, 由式(2)、(6)建立扩展电流误差空间方程[17]:
$ \dot e = Fe - G\eta $
(7) 式中
$ F = \left[{\begin{array}{*{20}{c}} 0&I&0&0&0&0\\ 0&0&I&0&0&0\\ 0&0&0&I&0&0\\ 0&0&0&0&I&0\\ 0&{-\omega _0^2\omega _1^2I}&0&{-\left( {\omega _0^2 + \omega _1^2} \right)I}&0&I\\ 0&0&0&0&0&A \end{array}} \right]\\ {G^{\rm T}} = \left[{\begin{array}{*{20}{c}} 0&0&0&0&0&{{B^{\rm T}}} \end{array}} \right], \eta = P\left( \sigma \right){u^*}\\ {e^{\rm T}} = \left[{\begin{array}{*{20}{c}} {{z^{\rm T}}}&{{{\dot z}^{\rm T}}}&{{{\ddot z}^{\rm T}}}&{{z^{(3){\rm T}}}}&{{z^{(4){\rm T}}}}&{{\varepsilon ^{\rm T}}} \end{array}} \right].\\ $
其中, ${z^{\left(i \right)}}$表示$z$的$i$阶导数, 系数0为2阶零矩阵, $I$为2阶单位矩阵.由式(7)知, 只需设计 状态反馈矩阵${K_r}$使${F_c} = F -G{K_r}$为Hurwitz矩阵, 即可得到渐近稳定的内模控制器.由式(6)知, 双输入系统(7)的解耦PI-RES 控制器所对应的的状态反馈矩阵为
$ \begin{align} {K_r} = \left[{\begin{array}{*{20}{c}} {{K_0}}&{{K_1}}&{{K_2}}&{{K_3}}&{{K_4}}&{{K_\varepsilon }} \end{array}} \right] \end{align} $
(8) 式中
$ {K_0} = \left[{\begin{array}{*{20}{c}} {k_0^d}&0\\ 0&{k_0^q} \end{array}} \right], {K_1} = \left[{\begin{array}{*{20}{c}} {k_1^d}&0\\ 0&{k_1^q} \end{array}} \right]\\ {K_2} = \left[{\begin{array}{*{20}{c}} {k_2^d}&0\\ 0&{k_2^q} \end{array}} \right], {K_3} = \left[{\begin{array}{*{20}{c}} {k_3^d}&0\\ 0&{k_3^q} \end{array}} \right]\\ {K_4} = \left[{\begin{array}{*{20}{c}} {k_4^d}&0\\ 0&{k_4^q} \end{array}} \right], {K_\varepsilon } = \left[{\begin{array}{*{20}{c}} {k_\varepsilon ^d}&{{\omega _e}L}\\ {-{\omega _e}L}&{k_\varepsilon ^q} \end{array}} \right]\\ $
其中, 上标$d$、$q$分别对应直、交轴内模控制器参数, 由系统矩阵$A$知, ${\omega _e}L$为耦合项, 因此, ${K_\varepsilon }$中${\omega _e}L$为电流解耦项.
注. 由式(5)~(8)知, PI-RES控制器需要合理选取并联谐振器的参数以保证系统渐近稳定.因此, PI-RES控制器并不是真正的并联型渐近稳定控制器.
1.3 输入延时对电流控制的影响
由第1.2节分析可知, 考虑系统输入延时后, 电流解耦PI-RES控制器所对应的状态反馈为
$ {\eta _D} = {K_r}{e_D} $
(9) 式中, ${e_D} = \lambda \left( t \right)e$为$t-D$时刻误差状态, $e$为$t$时刻误差状态, $\lambda \left( t \right)$为等效延时系数矩阵.由式(9)得闭环误差空间系统为
$ \dot e = {F_{c1}}\left( t \right)e = \left( {F - G{K_r}\lambda \left( t \right)} \right)e $
(10) 式中, ${F_{c1}}\left( t \right) = F - G{K_r}\lambda \left( t \right)$为扩展电流误差空间系统矩阵, $\lambda \left( t \right)$为对应维数的分块对角时变等效延时矩阵, 其对角线元素所构成的矩阵为
$ \begin{align} \varsigma \left( t \right) =\, & \left[{\begin{array}{*{20}{c}} {{\lambda _0}\left( t \right)}&{{\lambda _1}\left( t \right)}&{{\lambda _2}\left( t \right)\ \cdots} \end{array}} \right.\nonumber\\ & \left. {\begin{array}{*{20}{c}} {{\lambda _3}\left( t \right)}&{{\lambda _4}\left( t \right)}&{{\lambda _\varepsilon }\left( t \right)} \end{array}} \right] \end{align} $
(11) 式中, ${\lambda _0}$、${\lambda _1}$、${\lambda _2}$、${\lambda _3}$和${\lambda _\varepsilon }$均为对角矩阵.
由式(8)~(11)知, 考虑系统输入延时后的反馈矩阵为
$ \begin{align} {K_\lambda }\left( t \right) =\, & \left[{\begin{array}{*{20}{c}} {{K_0}{\lambda _0}\left( t \right)}&{{K_1}{\lambda _1}\left( t \right)}&{{K_2}{\lambda _2}\left( t \right)\ \cdots} \end{array}} \right.\nonumber\\ & \left. {\begin{array}{*{20}{c}} {{K_3}{\lambda _3}\left( t \right)}&{{K_4}{\lambda _4}\left( t \right)}&{{K_\varepsilon }{\lambda _\varepsilon }\left( t \right)} \end{array}} \right] \end{align} $
(12) 由式(11)和(12)知, 当$\lambda \left( t \right)$对角线元素为1时, 系统矩阵${F_{c1}}\left( t \right) = {F_c}$, 即输入延时对系统无影响.当$\lambda \left( t \right)$对角线元素不为1时, 系统矩阵${F_{c1}}\left( t \right) \ne{F_c}$, 即输入延时对闭环系统的动态性能和稳定性产生了影响.例如, 由式(8)和(9)知, ${k_\varepsilon}$对应比例反馈系数, 当${k_\varepsilon}$取值大于电机时间常数且${\lambda _\varepsilon }\left( t \right)$的系数小于零时, 系统将发散.因此, 需消除由输入延时引起的系统失稳隐患.
2. APPI-RES电流控制器设计
为消除系统输入延时对电流环稳定性的影响, 实现低${f_s}/{f_o}$工况下的渐近稳定并联谐振电流控制, 需要对系统状态和扰动进行预测.
2.1 扰动预测
考虑电机电阻、电感参数未知, 但其变化的界已知时, 设计系统(2)的自适应谐振扰动观测器:
$ \begin{align} \dot {\hat z} = \hat A\hat z - \hat Bu_D^* + G\tilde z - \hat B\sum\limits_{n = 0}^\infty {{{\hat \theta }_{dq6n}}{f_{mn}}} \end{align} $
(13) 式中, $\hat z\left( t \right)$为$z\left( t \right)$的观测值, ${\hat \theta _{dq6n}}$为${\theta _{dq6n}}$的估计值, $G$为对角反馈阵, 满足${A_G} = A - G$为Hurwitz阵, $\hat A$、$\hat B$分别为$A$、$B$的观测值.实际应用中, 通常不能准确获得表贴式PMSM的电阻和电感参数, 这里假设系统矩阵参数满足: $ - R/L = a \in \left[{{a_{\min }}, {a_{\max }}} \right], 1/L = b \in \left[{{b_{\min }}, {b_{\max }}} \right]$, ${a_{\min }}$、${a_{\max }}$、${b_{\min }}$、${b_{\max }}$为已知常数.式(13)的输入和系数矩阵满足:
$ \begin{align} \left\{\begin{array}{c} \hat A = \left[{\begin{array}{*{20}{c}} {\hat a}&{{\omega _e}}\\ {-{\omega _e}}&{\hat a} \end{array}} \right], \hat B = \left[{\begin{array}{*{20}{c}} {\hat b}&0\\ 0&{\hat b} \end{array}} \right]\\ u_D^* = u_{0D}^* - \hat d, \hat d = \sum\limits_{n = 0}^\infty {{{\hat \theta }_{dq6n}}{f_{mn}}} \end{array}\right. \end{align} $
(14) 式中, $u_{0D}^* = u_0^*\left( {t - D} \right)$为状态反馈分量, $\hat d$为$d$的观测值.
将式(2)和(13)做差得:
$ \begin{align} \dot {\tilde z} = {A_G}\tilde z + \tilde a\hat z - \tilde bu_{0D}^* - B\sum\limits_{n = 0}^\infty {{{\tilde \theta }_{dq6n}}{f_{mn}}} \end{align} $
(15) 式中, $\tilde z = z - \hat z$, ${\tilde \theta _{dq6n}} = {\theta _{dq6n}} - {\hat \theta _{dq6n}}$, $\tilde a = a - \hat a$, $\tilde b = b - \hat b$.
建立观测误差系统(15)的Lyapunov函数:
$ \begin{align} &V\left( {\tilde z, \tilde a, \tilde b, {{\tilde \theta }_{dq6n}}} \right) = \dfrac{1}{{2\gamma }}\left( {{{\tilde a}^2} + {{\tilde b}^2}} \right)+\nonumber\\ &\qquad \dfrac{1}{2}\left| b \right|\sum\limits_{n = 0}^\infty {{\rm tr}\left( {\tilde \theta _{dq6n}^ {\rm T}\Gamma _{6n}^{ - 1}{{\tilde \theta }_{dq6n}}} \right)} + \dfrac{1}{2}{{\tilde z}^{\rm T}}P\tilde z \end{align} $
(16) 式中, ${\Gamma _{6n}}$为正定对称矩阵, $\alpha$、$\gamma$为正常数, tr$\left( \cdot \right)$为求矩阵迹运算符号, $P$为正定对称矩阵, 满足:
$ \begin{align} A_G^{\rm T}P + P{A_G} = - Q < 0 \end{align} $
(17) 式中, $Q$为正定对称矩阵.为简化运算, 取$P = {I_{2 \times 2}}$, ${I_{2 \times 2}}$为2阶单位阵.
对Lyapunov函数(16)求导得:
$ \begin{align} &\dot V\left( {\tilde z, \tilde a, \tilde b, {{\tilde \theta }_{dq6n}}} \right) = \tilde a\left( {{{\tilde z}^{\rm T}}\hat z + {\gamma ^{ - 1}}\dot {\tilde a}} \right)+\nonumber\\ &\qquad {\rm tr}\left( {\sum\limits_{n = 0}^\infty {\tilde \theta _{dq6n}^ {\rm T}\left( {b\tilde zf_{mn}^{\rm T} + \left| b \right|\Gamma _ {6n}^{ - 1}{{\dot {\tilde \theta} }_{dq6n}}} \right)} } \right)- \nonumber\\ &\qquad \tilde b\left( {{{\tilde z}^{\rm T}}u_{0D}^* - {\gamma ^{ - 1}}\dot {\tilde b}} \right) - \frac{1}{2}{{\tilde z}^{\rm T}}Q\tilde z \end{align} $
(18) 若保证$\tilde a\left( {{{\tilde z}^{\rm T}}\hat z + {\gamma ^{ - 1}}\dot {\tilde a}} \right) - \tilde b\left( {{{\tilde z}^{\rm T}}u_{0D}^* - {\gamma ^{ - 1}}\dot {\tilde b}} \right) \le 0$, 参数$\hat a$和$\hat b$的辨识就不会影响参数${\hat \theta _{dq6n}}$的辨识.因此, 引入标准投影算子:
$ \begin{align} \begin{array}{l} {{\mathop{\rm Proj}\nolimits} _{\hat \varepsilon }}\left( {\gamma \kappa } \right) = \left\{ {\begin{array}{ll} {0, }&{\hat \varepsilon > {\varepsilon _{\max }}, \ \gamma \kappa > 0}\\ {0, }&{\hat \varepsilon < {\varepsilon _{\min }}, \ \gamma \kappa < 0}\\ {\gamma \kappa, }&\mbox{其他} \end{array}} \right.\\ \qquad \left( {\varepsilon - \hat \varepsilon } \right)\left( {{\gamma ^{ - 1}}{{{\mathop{\rm Proj}\nolimits} }_{\hat \varepsilon }}\left( {\gamma \kappa } \right) - \kappa } \right) \le 0 \end{array} \end{align} $
(19) 式中, $\hat \varepsilon$为$\varepsilon$的观测值, ${\varepsilon _{\max }}$、${\varepsilon _{\min }}$和$\gamma$为已知常数, $\kappa$为已知变量.
选择自适应率:
$ \begin{align} \left\{ \begin{array}{l} \dot {\hat a} = {{\mathop{\rm Proj}\nolimits} _{\hat a}}\left[{\gamma {{\hat z}^{\rm T}}\tilde z} \right]\\ \dot {\hat b} = {{\mathop{\rm Proj}\nolimits} _{\hat b}}\left[{-\gamma u_{0D}^{*{\rm T}}\tilde z} \right]\\ {{\dot {\hat \theta} }_{dq6n}} = - {\rm sgn}\left( b \right){\Gamma _{6n}}\tilde z{f_{mn}^{\rm T}} \end{array} \right. \end{align} $
(20) 式中, sgn$\left( \cdot \right)$为符号函数.
由式(17)~(20)知, , 当$t \to \infty$时, ${\hat \theta _{dq6n}} \to {\theta _{dq6n}}$, 即观测系统渐近稳定.
由式(3)、(14)、(20)得扰动观测值:
$ \begin{align} \hat d = \sum\limits_{n = 0}^\infty {{{\hat \theta }_{dq6n}}{f_{mn}}} \end{align} $
(21) 频率已知、幅值和初始相位未知的正弦函数满足:
$ \begin{align} v\left( {t + D} \right) = 2\cos \left( {\omega D} \right)v\left( t \right) - v\left( {t - D} \right) \end{align} $
(22) 当$n \ne 0$时, 由式(21)和(22)得扰动预测值:
$ \begin{align} {\hat d_{6n}}\left( {t + D} \right) = 2\cos \left( {6n\omega _e^*D} \right){\hat d_{6n}}\left( t \right) - {\hat d_{6n}}\left( {t - D} \right) \end{align} $
(23) 当$n = 0$时, 上式可简化为
$ \begin{align} {\hat d_0}\left( {t + D} \right) = {\hat d_0}\left( t \right) \end{align} $
(24) 2.2 状态预测
当$t$时刻误差系统状态$z\left( t \right)$已知时, $t+D$时刻误差系统状态可表示为[9]
$ \begin{align} z\left( {t + D} \right) = \mu \left( {t + D} \right) - \delta \left( {t + D} \right) \end{align} $
(25) 式中
$ \begin{align} \mu \left( {t + D} \right) = {{\rm e}^{AD}}z\left( t \right) \end{align} $
(26) $ \begin{align} \sigma \left( t \right) = {u^*}\left( t \right) + d\left( {t + D} \right) \end{align} $
(27) $ \begin{align} \delta \left( {t + D} \right) = \int_{t - D}^t {{{\rm e}^{A\left( {t - \tau } \right)}}B\sigma \left( \tau \right){\rm d}\tau } \end{align} $
(28) 其中, $\mu \left( {t + D} \right)$为$t$时刻零输入响应, 为$t$时刻零状态响应.进一步, 将式(26)~(28)表示为状态空间形式:
$ \begin{align} \left\{ \begin{array}{l} \dot \mu \left( {t + D} \right) = A\mu \left( {t + D} \right) - {{\rm e}^{AD}}B\sigma \left( {t - D} \right)\\ \mu \left( D \right) = {{\rm e}^{AD}}z\left( 0 \right)\\ \dot \delta \left( {t + D} \right) = A\delta \left( {t + D} \right) + B\sigma \left( t \right) - \\ \quad {{\rm e}^{AD}}B\sigma \left( {t - D} \right)\\ \delta \left( D \right) = {0_{2 \times 1}} \end{array} \right. \end{align} $
(29) 式中, $\delta \left( D \right)$、为系统初始状态.
由式(2)、(14)知, 系统矩阵及其观测矩阵满足:
$ \begin{align} {{\rm e}^{\hat AD}}A = A{{\rm e}^{\hat AD}} \end{align} $
(30) 利用第2.1节所得系统参数和扰动的估计值得$t+D$时刻误差系统状态的预测值:
$ \begin{align} {z_p}\left( {t + D} \right) = \hat \mu \left( {t + D} \right) - \hat \delta \left( {t + D} \right) \end{align} $
(31) 式中
$ \begin{align} \hat \mu \left( {t + D} \right) = \, &{{\rm e}^{\hat AD}}z\left( t \right) \end{align} $
(32) $ \begin{align} \dot {\hat \mu} \left( {t + D} \right) = \, &A\hat \mu \left( {t + D} \right) - {{\rm e}^{\hat AD}}B\sigma \left( {t - D} \right)+\nonumber\\ & \dot {\hat A}D\hat \mu \left( {t + D} \right)\nonumber\\ \hat \mu \left( D \right) =\, & {{\rm e}^{\hat AD}}z\left( 0 \right)\nonumber\\ \dot {\hat \delta} \left( {t + D} \right) =\, & \hat A\hat \delta \left( {t + D} \right) + \hat Bu_0^*\left( t \right) -\nonumber\\ & {{\rm e}^{\hat AD}}\hat Bu_0^*\left( {t - D} \right)\nonumber\\ \hat \delta \left( D \right) =\, & {0_{2 \times 1}}\nonumber\\ {{\dot z}_p}\left( {t + D} \right) = \, &A{z_p}\left( {t + D} \right) - Bu_0^*\left( t \right)+\nonumber\\ & \tilde A\hat \delta \left( {t + D} \right) + \tilde Bu_0^*\left( t \right)+\nonumber\\ & {{\rm e}^{\hat AD}}\hat Bu_0^*\left( {t - D} \right) + \dot {\hat A}D\hat \mu \left( {t + D} \right) -\nonumber\\ & {{\rm e}^{\hat AD}}\left( {B\tilde d\left( t \right) + \tilde Bu_0^*\left( {t - D} \right)} \right) \end{align} $
(33) 2.3 稳定性分析
由式(2)、(25)、(29)、(31)、(33)得系统状态预测误差:
$ \begin{align} {{\dot {\tilde z}}_p}\left( {t + D} \right)& = A{{\tilde z}_p}\left( {t + D} \right) - B\tilde d\left( {t + D} \right)+\nonumber\\ &\, {{\rm e}^{\hat AD}}B\tilde d\left( t \right) + {{\rm e}^{\hat AD}}\tilde Bu_0^*\left( {t - D} \right) -\nonumber\\ & \left( {\tilde A\hat \delta \left( {t + D} \right) + \tilde Bu_0^*\left( t \right)} \right) - \dot {\hat A}D\hat \mu \left( {t + D} \right) \end{align} $
(34) 式中, ${\tilde z_p}\left( {t + D} \right) = z\left( {t + D} \right) - {z_p}\left( {t + D} \right)$为系统状态预测误差.由式(15)、(18)、(20)知谐振观测系统渐近稳定, 因此, 观测误差系统(15)中$\mathop {\lim}\nolimits_{t \to \infty } \tilde A\hat z\left( t \right) - \tilde Bu_0^*\left( {t - D} \right) = {0_{2 \times 1}}$, $\mathop {\lim}\nolimits_{t \to \infty } \tilde z\left( t \right) = {0_{2 \times 1}}$.进而, 可得:
$ \begin{align} \mathop {\lim}\limits_{t \to \infty } \tilde Az\left( t \right) - \tilde Bu_0^*\left( {t - D} \right) = {0_{2 \times 1}} \end{align} $
(35) 由式(30)、(31)、(34)、(35)得:
$ \begin{align} &\mathop {\lim }\limits_{t \to \infty } {{\rm e}^{\hat AD}}\tilde Bu_0^*\left( {t - D} \right) - \left( {\tilde A\hat \delta \left( {t + D} \right) + \tilde Bu_0^*\left( t \right)} \right) = \nonumber\\ &\qquad - \tilde A{{\tilde z}_p}\left( {t + D} \right) \end{align} $
(36) 因此, 当$t \to \infty $时, 式(34)可化为
$ \begin{align} {{\dot {\tilde z}}_p}\left( {t + D} \right) =\, & \hat A{{\tilde z}_p}\left( {t + D} \right) - B\tilde d\left( {t + D} \right)+\nonumber\\ & {{\rm e}^{\hat AD}}B\tilde d\left( t \right) - \dot {\hat A}D\hat \mu \left( {t + D} \right) \end{align} $
(37) 由第2.1节分析可知:
$ \begin{align} &\mathop {\lim}\limits_{t \to \infty } \tilde d\left( {t + D} \right) = {0_{2 \times 1}}\nonumber\\ &\qquad \mathop {\lim}\limits_{t \to \infty } \tilde d\left( t \right) = {0_{2 \times 1}}\nonumber\\ &\qquad \mathop {\lim}\limits_{t \to \infty } \dot {\hat A} = {0_{2 \times 2}}\nonumber\\ &\qquad {\lambda _i}\left( {\hat A} \right) < 0, \quad i = 1, 2 \end{align} $
(38) 式中, ${\lambda _i}$为$\hat A$的特征值.由式(37)和(38)得:
$ \begin{align} \mathop {\lim}\limits_{t \to \infty } {\tilde z_p}\left( {t + D} \right) = {0_{2 \times 1}} \end{align} $
(39) 设计控制器:
$ \begin{align} {u^*}\left( t \right) = K{z_p}\left( {t + D} \right) - \hat d\left( {t + D} \right) \end{align} $
(40) 式中, 状态反馈增益矩阵$K$保证渐近稳定.闭环系统可表示为
$ \begin{align} \dot z\left( t \right) = {A_K}z\left( t \right) - BK{\tilde z_p}\left( t \right) - B\tilde d\left( t \right) \end{align} $
(41) 因此, 由式(38)~(40)知, 式(41)为渐近稳定系统.执行APPI-RES电流控制策略的步骤如下:
步骤1. 根据系统模型(2)建立自适应谐振扰动观测系统(13), $t$时刻参数自适应率和电压扰动分别为式(20)和(21).
步骤2. 根据式(23)和(24)预测系统扰动, 求得$t+D$时刻系统扰动.
步骤3. 利用步骤1中参数自适应率和步骤2中的扰动预测值, 根据(31)~(33)预测系统(2)在$t+D$时刻的状态.
注. 系统未知参数慢变的假设条件可保证未知参数在$t$到$t+D$时间内不变.因此, 由步骤1中系统未知参数自适应率(20)所求得$t$时刻系统估计参数可用于$t+D$时刻系统状态的预测.
步骤4. 根据(34)~(39)的稳定性证明可得$t$时刻系统渐近稳定控制器(40).
基于APPI-RES电流控制策略的PMSM系统框图如图 1所示.
图 1 采用APPI-RES电流控制器的PMSM驱动系统框图Fig. 1 Block diagram of PMSM driving system using APPI-RES current controller3. 仿真研究
为验证APPI-RES电流控制器的有效性, 在Matlab/Simulink仿真环境下搭建了永磁同步电机调速系统仿真平台.仿真中, 永磁同步电机参数如下:定子电阻$R = 2.93\, \Omega$, 定子$d$轴和$q$轴上的电感${L_d} = {L_q} = 7$ mH, 永磁体磁链${\varphi _f} = 0.125$ Wb, 转子磁极对数$p = 2$, 转动惯量$J = 1.16 \times {10^{ - 4}}\, ~{\rm kg \cdot {m^2}}$, 粘滞摩擦系数$F = 2.1 \times {10^{ - 4}}\, {\rm N \cdot m \cdot s}$.系统其他参数如下:母线电压$V_{dc}=400$ V, 死区时间$3\, {\rm\mu s}$, 电流环采样时间$100\, {\rm\mu s}$, 速度环采样时间$200\, {\rm\mu s}$.仿真验证中, 速度环所用PI调节器的比例系数${K_{ps}} = 0.5$, 积分器系数${K_{is}} = 10$, 电流环PI-RES控制器形式为: ${k_{pc}} + {k_{ic}}s + \sum_{n = 1}^6 {\frac{{{k_{resn}}s}}{{{s^2} + {{\left( {6n\omega _e^*} \right)}^2}}}}$.
3.1 小谐振增益下的PI-RES与APPI-RES电流控制对比仿真研究
根据前文的分析, 小谐振器增益下, 1个采样周期的延时不会使系统震荡.仿真中, 只注入VSI死区产生5、7、11、13等奇次谐波电流.电机低速轻载运行时, VSI非线性畸变电压使相电流畸变较严重.因此仿真中电机负载0.1 Nm, 给定转速300 r/min, 模拟低速轻载运行工况.分别采用PI、PI-RES、APPI-RES电流控制器.其中, PI-RES控制器参数为: ${k_{pc}} = 20$、${k_{ic}} = 4\, 500$、${k_{resn}} = 2000$, $n$ = 1~6. APPI-RES控制器参数为:式(13)中反馈增益阵$G$为对角阵, 取${G_{11}} = {G_{22}} = 10\, 000$, 式(20)中$\gamma = 15\, 000$、${\Gamma _{6n}}$为对角阵, 取${\Gamma _{6n11}} = {\Gamma _{6n22}} = 10\, 000$, 式(40)中$K$为对角阵, 取${K_{11}} = {K_{22}} = 20$.如图 2和3所示, PI电流控制器抑制VSI死区电压的能力较差, 相电流中含有大量的$6n \pm 1$次谐波, 电流畸变率(Total harmonic distorition, THD)为22.79 %. PI-RES和APPI-RES控制器所并联的谐振器数量可根据电流控制精度和运算速度折中选取.这里选择并联谐振器数量为6, 即谐振器只能消除$36 \pm 1$次之前的谐波电流.由图 4~7知, PI-RES和APPI-RES两种电流控制器都能很好地抑制$6n \pm 1$次谐波电流, THD分别为1.85 %和1.56 %.由第1.2节、第1.3节分析可知, 在小谐振器增益情况下PI-RES控制器可保证系统渐近稳定, 且仿真所取延时时间较小, 扰动电压在延时时间内变化小, 因此, PI-RES控制器能够实现较高精度的电流控制.仿真结果同样验证了第2.2节中对所提APPI-RES电流控制策略理论分析的有效性.
3.2 大谐振增益下的PI-RES与APPI-RES电流控制对比仿真研究
级联型PMSM调速系统要求电流环响应速度远快于速度环响应速度.为了更快、更准控制电机电流, 需要增大电流环控制器增益系数.这里, PI-RES控制器的参数中${k_{pc}}$和${k_{ic}}$取值与第3.1节相同, 但取${k_{resn}}=4\, 000$, $n=1$~6. PI-RES控制器中谐振器增益增大后, 导致其输出电压振荡, 进而引起电流、电磁转矩和转速的振荡, 如图 8~11所示.为了验证APPI-RES控制器对输入延时的补偿效果和大范围参数不确定的鲁棒性, 增大APPI-RES控制器的谐振器增益并将电机电阻和电感设置为标称值的3倍和2倍.由第2.1节分析知, 所提参数自适应辨识算法需已知未知参数的界, 这里假设系统未知参数$\hat a$、$\hat b$分别满足: $\hat a \in \left[{-800, -50} \right]、\hat b \in \left[{50, 500} \right]$.所选未知参数的界应满足: $\hat a$的界恒为负, $\hat b$的界恒为正.这里所选$\hat a$、$\hat b$的界只为验证所提算法的有效性, 即界的选取不唯一. APPI-RES控制器参数选为:式(13)中反馈增益阵$G$为对角阵, 取${G_{11}} = {G_{22}} = 15\, 000$, 式(20)中$\gamma = 25\, 000$, ${\Gamma _{6n}}$为对角阵, 取${\Gamma _{6n11}} = {\Gamma _{6n22}} = 18\, 000$, 式(40)中$K$为对角阵, 取${K_{11}} = {K_{22}} = 60$.由图 12~14可见, 当$t = 0.3$ s时负载由0.1 Nm突变为0.2 Nm, 速度PI控制器级联APPI-RES电流器的控制策略可有效抑制电压扰动, 消除谐波电流, 实现了快速、准确的电流、转矩和转速控制. 图 16为$t$时刻$q$轴电流控制误差${z_q}$与$t- D$时刻${z_q}$的预测值${z_{qp}}$的仿真结果.可见, 所提预测算法能够实现渐近稳定的电流误差预测.然而, 在电机起动过程中, 预测值与真实值符号相反, 即控制器执行短时正反馈, 这造成了起动过程中的短时电压振荡现象, 如图 15所示.
图 16 $q$轴电流误差及其预测值(APPI-RES)Fig. 16 Current error and its predictive value of $q$ axis (APPI-RES)4. 结论
1) 在${f_s}/{f_o}$工况下, PI-RES电流控制器受系统输入延时的影响, 将出现电流振荡或不稳定现象.基于电流误差系统状态空间模型给出了常规PI-RES电流控制器设计方法, 并分析了输入延时降低解耦性能和使系统失稳的本质原因.
2) 提出APPI-RES电流控制策略, 实现对系统输入延时的补偿和扰动电压的抑制.该方法可在系统参数未知的情况下观测系统扰动和预测系统状态.
3) 所提方法是一种真正意义上的渐近稳定并联型控制策略, 实际应用中可通过电流控制精度和计算时间要求这种选择并联核函数的数量.
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图 1 采用APPI-RES电流控制器的PMSM驱动系统框图
Fig. 1 Block diagram of PMSM driving system using APPI-RES current controller
图 16 $q$轴电流误差及其预测值(APPI-RES)
Fig. 16 Current error and its predictive value of $q$ axis (APPI-RES)
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