工程科学与技术   2017, Vol. 49 Issue (5): 164-170

New Exponential Reaching Law Sliding Mode Control for the Pure Feedback Nonlinear Systems with Mismatched Uncertainties
DIAN Songyi, LI Yinfeng, PU Ming, CHEN Lin, HE Quanlin
School of Electrical Eng. and Info., Sichuan Univ., Chengdu 610065, China
Abstract: In order to solve the output problem of a class of non-matching uncertain nonlinear systems represented by gas metal arc welding (GMAW),a sliding mode control method based on variable power law was proposed.Firstly,the first derivative of the output of system with non-matching uncertainty was obtained by using the sliding mode differentiator.Since the terminal sliding mode is stable in a finite time interval,the method has high estimation precision and the advantage that the speed the estimation error convergence is fast.Then,the paper put forward a new kind of exponential reaching law,and proved that under the same gain,the approach speed was faster than the existing various kinds of reaching laws.Furthermore,the approach speed of the proposed reaching law can be adaptively controlled,which not only ensured the system trajectory in the global scope to reach the sliding surface in a limited time,but also avoided the chattering around the sliding surface.Finally,the simulation results of sliding mode control method with the variable power reaching law and ones with traditional reaching laws are compared in controlling the arc length of non-interfered GMAW.Then,the tracking effect of arc length were compared and the steady-state errors were analyzed.The results showed that the sliding mode method with the variable power reaching law and variable structure can effectively improve the fastness of the system convergence.In conclusion,the sliding mode control method had strong robustness to the non-matching uncertain nonlinear system.
Key words: mismatched    uncertainties    pure feedback    sliding mode

1 问题描述 1.1 电流模型

 $I = \frac{1}{{\tau s + 1}}{I_{\rm{g}}}$ (1)

 $\dot I = - \frac{1}{\tau }I + \frac{1}{\tau }{I_{\rm{g}}}$ (2)

 ${\dot l_{\rm{s}}} = {v_{\rm{e}}} - {v_{\rm{m}}} + {\dot l_{{\rm{ct}}}}$ (3)
 图1 焊接过程简化模型图 Fig. 1 Simplified model of the welding process

 ${l_{{\rm{ct}}}} = {l_{\rm{s}}} + {l_{{\rm{arc}}}}$ (4)

 ${\dot l_{{\rm{arc}}}} = {v_{\rm{m}}} - {v_{\rm{e}}}$ (5)

 ${v_{\rm{m}}} = {m_{\rm{1}}}I + {m_{\rm{2}}}{l_{\rm{s}}}{I^2}$ (6)

 ${\dot l_{{\rm{arc}}}} = {m_1}I + {m_2}{I^2}\left( {{l_{{\rm{ct}}}} - {l_{{\rm{arc}}}}} \right) - {v_{\rm{e}}}$ (7)

 \left\{ \begin{aligned}{{\dot x}_1} = &{m_1}{x_2} + {m_2}{x_2}^2\left( {{l_{{\rm{ct}}}} - {x_1}} \right) - {v_{\rm{e}}} + {d_1}\left( t \right),\\{{\dot x}_2} = &- \frac{1}{\tau }{x_2} + \frac{1}{\tau }u + {d_2}\left( t \right)\end{aligned} \right. (8)

 \left\{ \begin{aligned}&{{\dot x}_1} = f\left( {{x_1},{x_2}} \right) + {d_1}\left( t \right),\\&{{\dot x}_2} = g\left( {{x_1},{x_2}} \right) + l\left( {{x_1},{x_2}} \right)u + {d_2}\left( t \right),\\&y = {x_1}\end{aligned} \right. (9)

1.2 控制器的设计

 $\dot e = {\dot x_{\rm{d}}} - {\dot x_1}$ (10)

 $\beta = ke + \dot e$ (11)

 $\dot \beta = k\dot e + {\ddot x_{\rm{d}}} - \left( {\frac{{\partial f\left( {{x_1},{x_2}} \right)}}{{\partial {x_1}}}{{\dot x}_1} + \frac{{\partial f\left( {{x_1},{x_2}} \right)}}{{\partial {x_2}}}{{\dot x}_2} + {{\dot d}_1}\left( t \right)} \right)$ (12)

 \begin{aligned}[b]\dot \beta = &k\dot e + {{\ddot x}_{\rm{d}}} - \frac{{\partial f\left( {{x_1},{x_2}} \right)}}{{\partial {x_1}}}{{\dot x}_1} - {\rm{ }}\frac{{\partial f\left( {{x_1},{x_2}} \right)}}{{\partial {x_2}}} \times \\&\left( {g\left( {{x_1},{x_2}} \right) + l\left( {{x_1},{x_2}} \right)u + {d_2}\left( t \right)} \right) - {{\dot d}_1}\left( t \right)\end{aligned} (13)

 $\dot \beta = \varXi \left( \beta \right) = \left\{ {\begin{array}{*{20}{c}}{\!\!\!\! - c{{\left| \beta \right|}^{{\gamma _1}\left| \beta \right|}}{\mathop{\rm sgn}} \left( \beta \right),\left| \beta \right| \ge 1;}\\[8pt]{\!\!\!\!\!\!\!\!\!\! - c{{\left| \beta \right|}^{{\gamma _2}}}{\mathop{\rm sgn}} \left( \beta \right),\left| \beta \right| < 1}\end{array}} \right.$ (14)

 $V\left( \beta \right) = \frac{1}{2}{\beta ^2}$ (15)

 $\dot V\left( \beta \right) = \beta \dot \beta$ (16)

 \begin{aligned}\dot V\left( \beta \right) = &\beta \varXi \left( \beta \right) = \\&\left\{ {\begin{array}{*{20}{c}}{\!\!\!\!\! - \left( {{c_1} + {c_2}} \right)\beta {{\left| \beta \right|}^{{\gamma _1}\left| \beta \right|}}{\mathop{\rm sgn}} \left( \beta \right),{\rm{ }}\left| \beta \right| > 1;}\\{\!\!\!\!\!\!\!\! - \left( {{c_1} + {c_2}} \right)\beta {{\left| \beta \right|}^{{\gamma _2}}}{\mathop{\rm sgn}} \left( \beta \right),{\rm{ }}\left| \beta \right| < 1}\end{array}} \right. = \\&\left\{ {\begin{array}{*{20}{c}}{ \!\!\!\!\! - \left( {{c_1} + {c_2}} \right)\left| \beta \right|{{\left| \beta \right|}^{{\gamma _1}\left| \beta \right|}},{\rm{ }}\left| \beta \right| > 1};\\{\!\!\!\!\!\!\!\! - \left( {{c_1} + {c_2}} \right)\left| \beta \right|{{\left| \beta \right|}^{{\gamma _2}}},{\rm{ }}\left| \beta \right| < 1}\end{array}} \right.\end{aligned} (17)

$\beta$ ≠0时，易知 $- \left( {{c_1} \!+\! {c_2}} \right)\left| \beta \right|{\left| \beta \right|^{{\gamma _1}\left| \beta \right|}} \!<\! 0$ $- \left( {{c_1} \!+\! {c_2}} \right)\left| \beta \right|{\left| \beta \right|^{{\gamma _2}}} \!<\! 0$ ，即 $\dot V\left( \beta \right) < 0$ ；当 $\beta$ =0时， $- \left( {{c_1} + {c_2}} \right)\left| \beta \right|{\left| \beta \right|^{{\gamma _2}}} = 0$ $\dot V\left( \beta \right) = 0$

 \begin{aligned}[b]{t_{\beta 1}} = &\int_{{\mathop{\rm sgn}} \left( \beta \right)}^{\beta \left( 0 \right)} {\frac{{{\rm{d}}\beta }}{{\left( {{c_1} + {c_2}} \right){{\left| \beta \right|}^{{\gamma _1}\left| \beta \right|}}{\mathop{\rm sgn}} \left( \beta \right)}}}= \\ &\left\{ {\begin{array}{*{20}{c}}{\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \int_1^{\beta \left( 0 \right)} {\displaystyle\frac{{{\rm{d}}\beta }}{{\left( {{c_1} + {c_2}} \right){\beta ^{{\gamma _1}\beta }}}}} ,{\rm{ }}\beta > 1;}\\[8pt]{\!\!\!\! \int_{ - 1}^{\beta \left( 0 \right)} {\displaystyle\frac{{{\rm{d}}\beta }}{{ - \left( {{c_1} + {c_2}} \right){{\left| \beta \right|}^{{\gamma _1}\left| \beta \right|}}}}} ,\beta < - 1}\end{array}} \right.=\\ &\int_1^{\left| {\beta \left( 0 \right)} \right|} {\frac{{{\rm{d}}\beta }}{{\left( {{c_1} + {c_2}} \right){\beta ^{{\gamma _1}\beta }}}}} \end{aligned} (18)

 ${t_{{\rm{\beta 2}}}} = \int_0^1 {\frac{{{\rm{d}}\beta }}{{\left( {{c_1} + {c_2}} \right){\beta ^{{\gamma _2}}}}}} ,\left| \beta \right| < 1$ (19)
 ${t_{\rm{\beta }}} = {t_{{\rm{\beta 1}}}} + {t_{{\rm{\beta 2}}}}$ (20)

 ${t_{{c}}} = \int_0^{\left| {\beta \left( 0 \right)} \right|} {\frac{{{\rm{d}}\beta }}{{{c_1}{\beta ^{\gamma {}_1}} + {c_2}{\beta ^{\gamma {}_2}}}}} = \frac{{1 + \beta {{\left( 0 \right)}^{1 - {\gamma _1}}}}}{{{c_1}\left( {{\gamma _1} - 1} \right)}} + \frac{1}{{{c_2}\left( {1 - {\gamma _1}} \right)}}$ (21)

 ${t_{{{c}}_2}} = \int_0^1 {\frac{{{\rm{d}}\beta }}{{{c_1}{\beta ^{\gamma {}_1}} + {c_2}{\beta ^{\gamma {}_2}}}}} > {t_{{\rm{\beta }}_2}} = \int_0^1 {\frac{{{\rm{d}}\beta }}{{\left( {{c_1} + {c_2}} \right){\beta ^{{\gamma _2}}}}}}$ (22)

 ${t_{{{c}}_1}} = \int_1^{\left| {\beta \left( 0 \right)} \right|} {\frac{{{\rm{d}}\beta }}{{{c_1}{\beta ^{\gamma {}_1}} + {c_2}{\beta ^{\gamma {}_2}}}}} > {t_{{\rm{\beta 1}}}} = \int_1^{\left| {\beta \left( 0 \right)} \right|} {\frac{{{\rm{d}}\beta }}{{\left( {{c_1} + {c_2}} \right){\beta ^{{\gamma _1}\beta }}}}}$ (23)

 \begin{aligned}[b]{t_{{c}}} = &{t_{{{c}}_1}} + {t_{{{c}}_2}} = \int_0^{\left| {\beta \left( 0 \right)} \right|} {\frac{{{\rm{d}}\beta }}{{{c_1}{\beta ^{\gamma {}_1}} + {c_2}{\beta ^{\gamma {}_2}}}}}= \\ &\frac{{1 + \beta {{\left( 0 \right)}^{1 - {\gamma _1}}}}}{{{c_1}\left( {{\gamma _1} - 1} \right)}} + \frac{1}{{{c_2}\left( {1 - {\gamma _1}} \right)}} > {t_{\rm{\beta }}} = {t_{{\rm{\beta 1}}}} + {t_{{\rm{\beta 2}}}}\end{aligned} (24)

 \begin{aligned}[b]\varXi \left( \beta \right) = &k\dot e + {{\ddot x}_{\rm{d}}} - \frac{{\partial f\left( {{x_1},{x_2}} \right)}}{{\partial {x_1}}}{{\dot x}_1} - {{\dot d}_1}\left( t \right) - \\&\frac{{\partial f\left( {{x_1},{x_2}} \right)}}{{\partial {x_2}}}\left( {g\left( {{x_1},{x_2}} \right) + l\left( {{x_1},{x_2}} \right)u + {d_2}\left( t \right)} \right)\end{aligned} (25)

 $u = \frac{{\left[ {\varUpsilon - {{\dot d}_1}\left( t \right)/\displaystyle\frac{{\partial f\left( {{x_1},{x_2}} \right)}}{{\partial {x_2}}} - {d_2}\left( t \right)} \right]}}{{l\left( {{x_1},{x_2}} \right)}}$ (26)

 $u = \frac{{\left[ {\hat \varUpsilon - \varPsi /\displaystyle\frac{{\partial f\left( {{x_1},{x_2}} \right)}}{{\partial {x_2}}} - \varPhi } \right]}}{{l\left( {{x_1},{x_2}} \right)}}$ (27)

 $\left\{ \!\!\!\!\! \begin{array}{c}{{\dot {\textit{z}}}_0} = {v_0} = - {\lambda _{0,1}}\left( {{{\textit{z}}_0} - f\left( t \right)} \right) - {\lambda _{0,2}}{\left| {{{\textit{z}}_0} - f\left( t \right)} \right|^{n/n + 1}}\times \\\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! {\mathop{\rm sgn}} \left( {{{\textit{z}}_0} - f\left( t \right)} \right) + {{\textit{z}}_1};{\rm{ }}\\ \vdots \\\ {{\dot {\textit{z}}}_i} = {v_i} = - {\lambda _{i,1}}({{\textit{z}}_i} - {v_{i - 1}}) - {\lambda _{i,2}}{\left| {{{\textit{z}}_i} - {v_{i - 1}}} \right|^{\left( {n - i} \right)/\left( {n - i + 1} \right)}}\times\\\quad\quad\quad {\mathop{\rm sgn}} \left( {{{\textit{z}}_i} - {v_{i - 1}}} \right) + {{\textit{z}}_{i + 1}},i = 1,2, \cdots ,n - 1;\\ \vdots \\\!\!{{\dot {\textit{z}}}_n} = {v_n} = - {\lambda _{n,1}}({{\textit{z}}_n} - {v_{n - 1}}) - {\lambda _{n,2}}{\left| {{{\textit{z}}_{n - 1}} - {v_{n - 1}}} \right|^{q/p}} \times\\ \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! {\mathop{\rm sgn}} \left( {{{\textit{z}}_n} - {v_{n - 1}}} \right){\rm{ }}\end{array} \right.$ (28)

 $r\left( {{B_i}} \right) = \frac{{{L^{\left( {n + 1} \right)p/q}}}}{{{\varLambda _i}{\varTheta _i}}} + {M_i}\varepsilon \buildrel \Delta \over = r{\left( {{B_i}} \right)_1} + r{\left( {{B_i}} \right)_2}$ (29)

 \left\{ \begin{aligned}{\varLambda _i} = &{\left( {{\lambda _{n,1}} + {\lambda _{n,2}}} \right)^{\left( {n + 1} \right)p/q}}{\left( {{\lambda _{n - 1,1}} + {\lambda _{n - 1,2}}} \right)^{n + 1}} \times \\& \cdots \times {\left( {{\lambda _{i,1}} + {\lambda _{i,2}}} \right)^{\left( {n + 1} \right)/\left( {n - i} \right)}},\\{\varTheta _i} = &{\left( {{\lambda _{i - 1,1}} + {\lambda _{i - 1,2}}} \right)^{i/n - i + 1}}{\left( {{\lambda _{i - 2,1}} + {\lambda _{i - 2,2}}} \right)^{\left( {i - 1} \right)/\left( {n - i + 2} \right)}} \times \\& \cdots \times {\left( {{\lambda _{0,1}} + {\lambda _{0,2}}} \right)^{1/n}},\\{M_i} =& 2\left( {{\lambda _{i - 1,1}} + {\lambda _{i - 1,2}}} \right) \times \cdots \times \left( {{\lambda _{0,1}} + {\lambda _{0,2}}} \right)\end{aligned} \right.\!\!\!\!\!\! (30)

 (31)
 (32)

 ${\ddot x_1} = \frac{{{\rm{d}}f\left( {{x_1},{x_2}} \right)}}{{{\rm{d}}t}} + {\dot d_1}\left( t \right)$ (33)

 (34)

 $\varPhi = - {\eta _{{{d}}_1}}{\mathop{\rm sgn}} \left( {\hat \beta } \right)$ (35)
 $\varPsi = - {\eta _2}{\mathop{\rm sgn}} \left( {\hat \beta } \right){\mathop{\rm sgn}} \left( {\frac{{\partial f\left( {{x_{1}},{x_{2}}} \right)}}{{\partial {x_2}}}} \right)$ (36)
2 稳定性分析

 (37)

 (38)

 (39)

 \begin{aligned}[b] \dot \beta \!= & \frac{{\partial f\left( {{x_1},{x_2}} \right)}}{{\partial {x_2}}}\left( {\! - {\eta _2}{\mathop{\rm sgn}} \!\left( \beta \right){\mathop{\rm sgn}} \left( {\frac{{\partial f\left( {{x_1},{x_2}} \right)}}{{\partial {x_2}}}} \right) - {d_2}\left( t \right)} \right)\! + \\&\left( { - {\eta _{{{d}}1}}{\mathop{\rm sgn}} \left( \beta \right) - {{\dot d}_1}\left( t \right)} \right) + \varXi \left( \beta \right) + \varOmega \end{aligned}\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! (40)

 \begin{aligned}[b]\beta \dot \beta = & - \beta \frac{{\partial f\left( {{x_1},{x_2}} \right)}}{{\partial {x_2}}}\left( {{\eta _2}{\mathop{\rm sgn}} \left( \beta \right){\mathop{\rm sgn}} \left( {\frac{{\partial f\left( {{x_1},{x_2}} \right)}}{{\partial {x_2}}}} \right) + {d_2}\left( t \right)} \right) - \\&\beta \left( {{\eta _{{{d}}1}}{\mathop{\rm sgn}} \left( \beta \right) + {{\dot d}_1}\left( t \right)} \right) + \beta \varXi \left( \beta \right) + \beta \varOmega \end{aligned} (41)

 \begin{aligned}[b]\beta \dot \beta = &- {\eta _2}\left| \beta \right|\left| {\frac{{\partial f\left( {{x_1},{x_2}} \right)}}{{\partial {x_2}}}} \right| - \beta \frac{{\partial f\left( {{x_1},{x_2}} \right)}}{{\partial {x_2}}}{d_2}\left( t \right) - \\&{\eta _{{{d}}1}}\left| \beta \right| - \beta {{\dot d}_1}\left( t \right) + \beta \varXi \left( \beta \right) + \beta \varOmega \end{aligned} (42)

 \begin{align}& {\text 即：} - {\eta _2}\left| \beta \right|\left| {\displaystyle\frac{{\partial f\left( {{x_1},{x_2}} \right)}}{{\partial {x_2}}}} \right| \pm {d_2}\left( t \right)\beta \displaystyle\frac{{\partial f\left( {{x_1},{x_2}} \right)}}{{\partial {x_2}}} < 0{\text{，}}\\& \quad\quad\quad - {\eta _{{{d}}1}}\left| \beta \right| \pm {\dot d_1}\left( t \right)\beta < 0\end{align} (43)

 $\beta \varXi \left( \beta \right) < 0$ (44)

 $\beta < \varTheta \left( \varOmega \right)$ (45)

 $ke + \dot e < \varTheta \left( \varOmega \right)$ (46)

 $\dot e < \varTheta \left( \varOmega \right) - ke$ (47)

 $\dot V = e\dot e$ (48)

 $\dot V = e\dot e = - k{e^2} + e\varTheta \left( \varOmega \right) \le - k{e^2} + \left| e \right|\left| {\varTheta \left( \varOmega \right)} \right|$ (49)

3 仿真验证

 $\left\{\!\!\!\! {\begin{array}{*{20}{c}}{{{\dot x}_1} = {m_1}{x_2} + {m_2}{x_2}^2\left( {{l_{{\rm{ct}}}} - {x_1}} \right) - {v_{\rm{e}}} + {d_1}\left( t \right),}\\[8pt]\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! { {{\dot x}_2} = - \displaystyle\frac{1}{\tau }{x_2} + \frac{1}{\tau }u + {d_2}\left( t \right)}\end{array}} \right.$ (50)

 图2 两种趋近律电弧长度跟踪 Fig. 2 Arc lergth tracking results of the two reaching laws

 图3 两种趋近律动态性能对比 Fig. 3 Dynamic performance comparison of the two reaching laws

 图4 两种趋近律的跟踪误差 Fig. 4 Tracking error of two kinds of reaching law

 图5 两种趋近律的控制器输入 Fig. 5 Two controller inputs for the reaching law

4 结　论

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