Plant/controller co-design of motor driving systems based on finite-time filtering control
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摘要:
针对电机驱动系统进行了基于有限时间控制器的结构/控制一体化设计.针对电机驱动系统跟踪控制问题，采用有限时间收敛方法设计了跟踪控制器.考虑系统状态信息不可测的情况，设计了有限时间滤波控制器，在估计系统速度信息的同时实现了有限时间跟踪控制.为进一步提升系统控制性能，考虑结构与控制之间存在的耦合问题，对电机驱动系统进行结构/控制一体化设计.首先针对电机驱动系统设计了同时考虑结构优化和控制器优化的一体化性能指标.所设计一体化性能指标能够在满足控制性能要求的同时，得到所能驱动的最大负载.同时优化系统的结构参数与控制器参数能够使控制系统达到全局最优，从而取得良好的控制效果.随后，采用嵌套优化策略对电机驱动系统的一体化设计问题进行简化，采用自适应步长的布谷鸟搜索算法对控制器参数优化问题进行求解，得到了一体化最优解.通过数值仿真验证了所提方法的有效性.
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关键词:
- 电机驱动系统 /
- 有限时间控制 /
- 状态估计 /
- 结构/控制一体化设计 /
- 嵌套优化策略
Abstract:Recently, motor driving systems have been widely applied in the military and industries. Load tracking control is one of the commonly considered issues in such systems. In this study, a plant/controller co-design based on finite-time control was developed for the motor driving system. A finite-time convergent controller was also presented to address the tracking problem in the motor driving system. Because the system state was unknown, a filter was developed to estimate the velocity of the load. The overall system, including the tracking controller and filter, is proven to be finite-time stable. Hence, the upper bound of the convergence time can be determined. To enhance the control performance of the motor driving system, the coupling between plant and controller is considered and a co-design scheme was developed for the motor driving system. First, a combined performance index, which could indicate the largest load with satisfactory control performance, was established. Both the plant and controller parameters were considered in the developed performance index to simultaneously optimize the plant and controller. Through this optimization, the system-level optimality can be determined and a better control performance can be achieved. Moreover, a nested optimization strategy was adopted to simplify the co-design scheme and an adaptive cuckoo search algorithm was used to achieve the co-design result. Through the nested optimization scheme, the controller parameter is optimized in the inner loop and the plant parameter can be optimized in the outer loop. The cuckoo search algorithm exhibits a superior performance because it has fewer parameters that need to be tuned than most existing algorithms. Hence, the co-design problem can be simplified and resolved reliably using the proposed method. Contrastive simulation results indicates the efficacy of the proposed method.
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在常见的控制系统设计过程中，结构设计和控制器设计是分开进行的：首先进行结构设计，在固定的结构参数基础上再进行控制器设计. 由于控制系统中结构与控制之间存在着耦合[1]，控制器性能会受到结构参数的影响. 尤其对于电机驱动系统而言，控制性能不仅受控制器设计的影响，还会受到负载、齿隙和驱动电机型号等结构设计参数的影响[2]. 因此，为提高电机驱动系统的控制性能，需要同时考虑结构设计与控制器设计，从而得到控制系统的全局最优解.
随着对控制系统性能要求的不断提高，结构/控制一体化设计方法已经在一些领域得到了应用. 采用一体化设计方法，能够在考虑结构与控制器设计之间耦合的条件下得到系统的全局最优解，从而提升了系统的性能[3-4]. 为提升一种微型指向定位系统的性能，Wereley将结构参数与控制器参数同时考虑在同一个性能指标中，从而对二者进行同时优化[5]. 类似方法在空间飞行器天线系统设计[6]、柔性机械臂设计[7]压电结构设计[8]和风力涡轮机系统[9]等方面得到了应用，提升了系统的性能. Fathy等研究了结构设计与控制设计之间存在的耦合问题，并指出只有当满足一阶条件时，结构设计与控制器设计才能解耦[10]. 同时，即使结构优化问题与控制器优化问题分别是凸优化问题，由结构设计和控制器设计组成的一体化目标函数通常是一个非凸优化问题[11]. 嵌套优化策略能够将优化问题分为两个嵌套的自由化问题，并在交通运输规划[12]以及工程架构设计[13]等领域有广泛的应用. 采用嵌套优化策略，能够将一体化设计问题分为内、外两个嵌套在一起的优化环，从而降低了优化问题的复杂度，便于求解. 文献[14]中对比了嵌套优化、迭代优化和整体优化等优化策略，嵌套优化具有较低的计算量，能够高效地获得一体化设计的最优解. 因此，本文采用嵌套优化策略设计了电机驱动系统的一体化设计结构，在外环优化结构参数，内环优化控制器参数的策略下得到了一体化设计最优解.
由于电机驱动系统在实际应用中往往对动态性能与鲁棒性有较高的要求，因此，简单的PID控制器难以满足控制性能需求. 有限时间收敛理论能够使系统在一个时间上界内收敛，从而使得控制系统具有较好的动态性能[15]. 近年来，有限时间控制方法得到了广泛的研究，相继提出了有限时间自适应控制[16]，基于神经网络的有限时间控制[17]等方法. 采用有限时间收敛理论设计控制器，能够使系统获得较好的动态性能以及鲁棒性. 因此，本文采用有限时间收敛理论，设计电机驱动系统的跟踪控制器. 同时，考虑到实际系统中负载状态未知的问题，提出了加幂积分滤波器以估计负载的速度信息.
本文首先设计了电机驱动系统的有限时间滤波跟踪控制器，所设计控制器能够估计系统的速度信息，同时实现电机驱动系统的有限时间跟踪控制；然后，同时考虑结构参数优化和控制器参数优化，对电机驱动系统进行一体化设计，得到了电机驱动系统结构与控制的全局最优解. 最后，通过数值仿真验证了所提出算法的有效性.
1. 电机驱动系统模型
首先给出了电机驱动系统的数学模型. 电机驱动系统主要包括驱动电机和负载两部分，并由齿轮箱链接. 系统框图如图 1所示. 随后，为降低控制器设计难度，对电机驱动系统模型进行了合理简化.
电机驱动系统模型如下：
$$ \begin{array}{c} J_{\mathrm{m}} \ddot{\theta}_{\mathrm{m}}+b_{\mathrm{m}} \dot{\theta}_{\mathrm{m}}=u-\omega \\ J_{\mathrm{L}} \ddot{\theta}_{\mathrm{L}}+b_{\mathrm{L}} \dot{\theta}_{\mathrm{L}}=\omega \end{array} $$ (1) 式中，θm和θL分别为驱动电机位置和负载位置；Jm和JL分别为电机和负载的转动惯量；bm和bL分别为电机端和负载端的黏滞摩擦系数；u为系统控制输入；ω为传动力矩并且有如下形式：
$$ \omega=k f(\varDelta)+c f(\dot{\varDelta}) $$ (2) 式中，k为扭转系数；c为阻尼系数；Δ=θm－θL；f(·)为齿隙引起的非线性，并且可以由死区非线性代表：
$$ f(\cdot)=\left\{\begin{array}{ll} \varDelta-\delta, & \varDelta \geqslant \delta \\ 0, & |\varDelta|<\delta \\ \varDelta+\delta, & \varDelta \leqslant-\delta \end{array}\right. $$ (3) 式中，δ为齿隙宽度. 为方便控制器设计，假设齿轮箱两端力矩为线性关系[18]，可得如下简化模型：
$$ \left(n J_{\mathrm{m}}+J_{\mathrm{L}}\right) \ddot{\theta}_{\mathrm{L}}+\left(n b_{\mathrm{m}}+b_{\mathrm{L}}\right) \dot{\theta}_{\mathrm{L}}=u $$ (4) 式中，n为传动比系数. 选取系统状态变量x1=θm, x2= ${{{\dot{\theta }}}_{\text{m}}} $，式(4)可写为
$$ \left\{\begin{array}{l} \dot{x}_{1}=x_{2} \\ \dot{x}_{2}=\frac{1}{n J_{\mathrm{m}}+J_{\mathrm{L}}}\left[u-\left(n b_{\mathrm{m}}+b_{\mathrm{L}}\right) x_{2}\right] \\ y=x_{1} \end{array}\right. $$ (5) 本文将针对上述电机驱动系统，设计滤波器估计不可测状态变量x2；同时设计有限时间跟踪控制器，使得系统输出y能够跟踪期望信号yd. 考虑电机驱动系统结构与控制之间的耦合问题，进行结构/控制一体化设计，取得电机驱动系统全局最优参数.
2. 有限时间滤波控制器设计
本节设计了电机驱动系统的有限时间跟踪控制器. 考虑电机驱动系统速度信息未知的情况，设计了滤波器估计系统状态信息，并分析了控制系统的稳定性. 最后，对电机驱动系统进行结构/控制一体化设计.
2.1 控制器设计
引理1[19]对于系统(5)，如果存在连续正定函数V(x)满足
$$ \dot{V}(x) \leqslant-a V(x)^{b} $$ (6) 式中，a和0 < b < 1为正常数，则系统(5)为有限时间稳定，且收敛时间可通过式(7)计算.
$$ T_{\mathrm{c}} \leqslant \frac{1}{a(1-b)} V\left(x_{0}\right)^{1-b} $$ (7) 针对电机驱动系统模型，设计有限时间控制器：
$$ \begin{array}{c} u=\left(n J_{\mathrm{m}}+J_{\mathrm{L}}\right)\left(-k_{1} \operatorname{sig}\left[\operatorname{sig}\left(\dot{e}_{\mathrm{t}}\right)^{\alpha_{1}}+k_{2} e_{\mathrm{t}}\right]^{\alpha_{2}}+\right. \\ \left.n b_{\mathrm{m}} x_{2}+b_{\mathrm{L}} x_{2}\right) \end{array} $$ (8) 式中，et=y－yd为系统跟踪误差；1 < α1 < 2，0 < α2 < 2α1－1－1，k1, k2>0.
考虑电机驱动系统速度状态x2不可测，设计如下基于滤波器的控制器：
$$ \left\{\begin{array}{l} u=\left(n J_{\mathrm{m}}+J_{\mathrm{L}}\right)\left(-k_{1} \operatorname{sig}\left[\operatorname{sig}\left(\hat{x}_{2}-\dot{y}_{\mathrm{d}}\right)^{\alpha_{1}}+k_{2} e_{\mathrm{t}}\right]^{\alpha_{2}}+\right. \\ \ \ \ \ \ \ \ \ \ \left.n b_{\mathrm{m}} \hat{x}_{2}+b_{\mathrm{L}} \hat{x}_{2}\right) \\ \hat{x}_{2}=\dot{y}_{\mathrm{d}}-k_{3} \operatorname{sig}\left(\int\left(\hat{x}_{2}-\dot{y}_{\mathrm{d}}\right) \mathrm{d} t-e_{\mathrm{t}}\right)^{\frac{1}{\alpha_{1}}} \end{array}\right. $$ (9) 式中，k3>0，${{{\hat{x}}}_{2}} $为电机驱动系统负载速度x2的估计值.
2.2 稳定性分析
定理1 在所设计滤波控制器(9)作用下，系统(5)的跟踪误差能够在有限时间内收敛有界，收敛时间上界如式(10)所示.
$$ T_{\mathrm{c}}=\frac{2 V_{\mathrm{t}}(0)^{\frac{1-1 / \alpha_{1}}{2}}}{\lambda_{\mathrm{m}}\left(1-\frac{1}{\alpha_{1}}\right)} $$ (10) 式中，λm为正常数.
证明：选取李雅普诺夫函数Vt1=et2. 定义中间变量$\xi =-{{k}_{2}}\text{sig}{{({{e}_{t}})}^{\frac{1}{{{\alpha }_{1}}}}} $，则对Vt1求导可得
$$ \begin{array}{c} \dot{V}_{\operatorname{t1}}=2 e_{\operatorname{t}} \xi+2 e_{\operatorname{t}}\left(\dot{e}_{\operatorname{t}}-\xi\right) \leqslant \\ -2 k_{2}\left|e_{\operatorname{t}}\right|^{\frac{1}{\alpha_{1}}+1}+2\left|e_{\operatorname{t}}\right|\left|\dot{e}_{\operatorname{t}}-\xi\right| \leqslant \\ -2 k_{2}\left|e_{\operatorname{t}}\right|^{\frac{1}{\alpha_{1}}+1}+2^{2-\frac{1}{\alpha_{1}}}\left|e_{\operatorname{t}}\right|\left|\operatorname{sig}\left(\dot{e}_{\operatorname{t}}\right)^{\alpha_{1}}-\operatorname{sig}(\xi)^{\alpha_{1}}\right| \end{array} $$ (11) 由于$ 2^{2-\frac{1}{\alpha_{1}}}\left|e_{\mathrm{t}}\right|\left|\operatorname{sig}\left(\dot{e}_{\mathrm{t}}\right)^{\alpha_{1}}-\operatorname{sig}(\xi)^{\alpha_{1}}\right| \leqslant 2^{2-\frac{1}{\alpha_{1}}} \alpha_{1}$$\left(\alpha_{1}+1\right)^{-1} \quad\left|e_{1}\right|^{1+\frac{1}{\alpha_{1}}}+2^{2-\frac{1}{\alpha_{1}}} \quad\left(\alpha_{1}+1\right)^{-1} $$ \left|\operatorname{sig}\left(\dot{e}_{\mathrm{t}}\right)^{\alpha_{1}}-\operatorname{sig}(\xi)^{\alpha_{1}}\right|^{1+\frac{1}{\alpha_{1}}}$，式(11)可进一步写为
$$ \begin{array}{c} \dot{V}_{\mathrm{t} 1} \leqslant-2 k_{2}\left|e_{\mathrm{t}}\right|{\alpha}_{\bar{\alpha}}+1+2^{2-\frac{1}{\alpha_{1}}} \alpha_{1}\left(\alpha_{1}+1\right)^{-1}\left|e_{\mathrm{t}}\right|^{1+\frac{1}{\alpha_{1}}}+ \\ 2^{2-\frac{1}{\alpha_{1}}}\left(\alpha_{1}+1\right)^{-1}\left|\operatorname{sig}\left(\dot{e}_{\mathrm{t}}\right)^{\alpha_{1}}-\operatorname{sig}(\xi)^{\alpha_{1}}\right|^{1+\frac{1}{\alpha_{1}}} \end{array} $$ (12) 定义滤波器估计误差$ e_{1}=\operatorname{sig}\left(x_{2}\right)^{\alpha_{1}}-\operatorname{sig}\left(\hat{x}_{2}\right)^{\alpha_{1}}$，选取李雅普诺夫函数Vt2=el2. 对Vt2求导得
$$ \begin{array}{c} \dot{V}_{12}=2 e_{1} \dot{e}_{1}=2 e_{1} \alpha_{1}\left|x_{2}\right|^{\alpha_{1}-1} \dot{x}_{2}-2 e_{1} k_{3}^{\alpha_{1}}\left(x_{2}-\hat{x_{2}}\right)= \\ 2 e_{1} \alpha_{1}\left|x_{2}\right|^{\alpha_{1}-1} \frac{1}{n J_{\mathrm{m}}+J_{\mathrm{L}}}\left[u-\left(n b_{\mathrm{m}}+b_{\mathrm{L}}\right) x_{2}\right]- \\ 2 e_{1} k_{3}^{\alpha_{1}}\left(x_{2}-\hat{x}_{2}\right) \end{array} $$ (13) 根据式(9)所提出的滤波器有：
$$ \begin{array}{c} \dot{V}_{12}=2 e_{1} \alpha_{1}\left|x_{2}\right|^{\alpha_{1}-1} \frac{1}{n J_{\mathrm{m}}+J_{\mathrm{L}}}\left[u-\left(n b_{\mathrm{m}}+b_{\mathrm{L}}\right) x_{2}\right]- \\ 2 e_{1} k_{3}^{\alpha_{1}}\left\{x_{2}-\operatorname{sig}\left[\operatorname{sig}\left(x_{2}\right)^{\alpha_{1}}-e_{1}\right]^{\frac{1}{\alpha_{1}}}\right\} \leqslant \\ 2 \alpha_{1}\left|e_{1}\right|\left|x_{2}\right|^{\alpha_{1}-1} \frac{n b_{\mathrm{m}}+b_{\mathrm{L}}}{n J_{\mathrm{m}}+J_{\mathrm{L}}} x_{2}+2 \alpha_{1}\left|e_{1}\right|\left|x_{2}\right|^{\alpha_{1}-1} \frac{|u|}{n J_{\mathrm{m}}+J_{\mathrm{L}}}- \\ 2 e_{1} k_{3}^{\alpha_{1}}\left\{x_{2}-\operatorname{sig}\left[\operatorname{sig}\left(x_{2}\right)^{\alpha_{1}}-e_{1}\right]^{\frac{1}{\alpha_{1}}}\right\} \end{array} $$ (14) 由于
$$ \begin{array}{c} 2 \alpha_{1}\left|e_{1}\right|\left|x_{2}\right|^{\alpha_{1}-1} \frac{|u|}{n J_{\mathrm{m}}+J_{\mathrm{L}}}=\\ \frac{2 \alpha_{1}\left|e_{1}\right|}{n J_{\mathrm{m}}+J_{\mathrm{L}}}\left(\left|x_{2}\right|^{1-\alpha_{1}}\right)^{\frac{1}{1-\alpha_{1}}-1}|u| \leqslant \\ \frac{2 \alpha_{1}^{2}}{n J_{\mathrm{m}}+J_{\mathrm{L}}}\left|e_{1}\right|\left|x_{2}\right|+\frac{2 \alpha_{1}\left(1-\alpha_{1}\right)}{n J_{\mathrm{m}}+J_{\mathrm{L}}}\left|e_{1}\right||u|^{\frac{1}{1-\alpha_{1}}} \end{array} $$ (15) 同时
$$ \begin{array}{c} \left|x_{2}\right|\left|e_{1}\right| \leqslant \frac{k_{2}}{1+\alpha_{1}}\left|x_{1}\right|^{1+\frac{1}{\alpha_{1}}}+\frac{2^{1-\frac{1}{\alpha_{1}}}}{1+\alpha_{1}} \mid \operatorname{sig}\left(x_{2}\right)^{\alpha_{1}}- \\ \left.\operatorname{sig}(\xi)^{\alpha_{1}}\right|^{1+\frac{1}{\alpha_{1}}}+\frac{k_{2}+2^{1-\frac{1}{\alpha_{1}}}}{1+\frac{1}{\alpha_{1}}}\left|e_{1}\right|^{1+\frac{1}{\alpha_{1}}} \end{array} $$ (16) 并且
$$ \begin{array}{c} \left|e_{1}\right||u|^{\frac{1}{1-\alpha_{1}}}=\gamma_{1}\left|\operatorname{sig}\left(x_{2}\right)^{\alpha_{1}}-\operatorname{sig}(\xi)^{\alpha_{1}}-e_{1}\right|^{\frac{1}{\alpha_{1}}}\left|e_{1}\right| \leqslant \\ \gamma_{1}\left|e_{1}\right|\left|\operatorname{sig}\left(x_{2}\right)^{\alpha_{1}}-\operatorname{sig}(\xi)^{\alpha_{1}}\right|^{\frac{1}{\alpha_{1}}}+\gamma_{1}\left|e_{1}\right|^{1+\frac{1}{\alpha_{1}}} \leqslant \\ \frac{\gamma_{1}}{1+\alpha_{1}}\left|\operatorname{sig}\left(x_{2}\right)^{\alpha_{1}}-\operatorname{sig}(\xi)^{\alpha_{1}}\right|^{\frac{1}{\alpha_{1}}+1}+\frac{\gamma_{1}\left(1+2 \alpha_{1}\right)}{1+\alpha_{1}}\left|e_{1}\right|^{1+\frac{1}{\alpha_{1}}} \end{array} $$ (17) 式中，$ \gamma_{1}=2^{1-\frac{1}{\alpha_{1}}} \alpha_{1}\left(1+\alpha_{1}\right)^{-1}$，所以，式(15)可进一步写为
$$ 2 \alpha_{1}\left|e_{1}\right|\left|x_{2}\right|^{\alpha_{1}-1} \frac{|u|}{n J_{\mathrm{m}}+J_{\mathrm{L}}} \leqslant \frac{1}{n J_{\mathrm{m}}+J_{\mathrm{L}}}\left(\gamma_{2}\left|x_{1}\right|^{1+\frac{1}{\alpha_{1}}}+\right.$ $\left.\gamma_{3}\left|e_{1}\right|^{1+\frac{1}{\alpha_{1}}}+\gamma_{4}\left|\operatorname{sig}\left(x_{2}\right)^{\alpha_{1}}-\operatorname{sig}(\xi)^{\alpha_{1}}\right|^{1+\frac{1}{\alpha_{1}}}\right) $$ (18) 式中，$ \gamma_{2}=2 \alpha_{1} \alpha_{2} \frac{\left(\alpha_{1}-1\right)}{\left(\alpha_{1}+1\right)}, \quad \gamma_{3}=2 \alpha_{1} \alpha_{2}$$ \frac{\left(\alpha_{1}-1\right)\left(2^{2-\frac{1}{\alpha_{1}}}+\alpha_{2}\right)}{\left(\frac{1}{\alpha_{1}}+1\right)}+2 \alpha_{1} \gamma_{1} \frac{\left(2-\alpha_{1}\right)\left(2 \alpha_{1}+1\right)}{\left(\alpha_{1}+1\right)}$$\gamma_{4}=2^{2-\frac{1}{\alpha_{1}}} \alpha_{1} \frac{\left(\alpha_{1}-1\right)}{\left(\alpha_{1}+1\right)}+2 \alpha_{1} \gamma_{1} \frac{\left(2-\alpha_{1}\right)}{\left(\alpha_{1}+1\right)} $.
式(14)中，$ \alpha_{1}\left|e_{1}\right|\left|x_{2}\right|^{\alpha_{1}-1} \frac{n b_{\mathrm{m}}+b_{\mathrm{L}}}{n J_{\mathrm{m}}+J_{\mathrm{L}}} x_{2}$满足
$$ \begin{array}{c} 2{\alpha _1}\left| {{e_1}} \right|{\left| {{x_2}} \right|^{{\alpha _1} - 1}}\frac{{n{b_{\rm{m}}} + {b_{\rm{L}}}}}{{n{J_{\rm{m}}} + {J_{\rm{L}}}}}{x_2} \le \\ \frac{{{\alpha _1}\left| {{e_1}} \right|\left( {n{b_{\rm{m}}} + {b_{\rm{L}}}} \right)}}{{n{J_{\rm{m}}} + {J_{\rm{L}}}}}{\left| {{x_2}} \right|^{{\alpha _1} - 1 + m}} + \\ \frac{{{\alpha _1}\left| {{e_1}} \right|\left( {n{b_{\rm{m}}} + {b_{\rm{L}}}} \right)}}{{n{J_{\rm{m}}} + {J_{\rm{L}}}}}{\left| {{x_2}} \right|^{{\alpha _1} - 1 + n}} \le \\ \begin{array}{*{20}{c}} {\frac{{n{b_{\rm{m}}} + {b_{\rm{L}}}}}{{n{J_{\rm{m}}} + {J_{\rm{L}}}}}\left( {{2^{{\alpha _1} - 2 + m}}k_2^{{\alpha _1} - 1 + m}{{\left| {{e_{\rm{t}}}} \right|}^{\frac{{{\alpha _1} - 1 + m}}{{{\alpha _1}}}}}\left| {{e_1}} \right| + } \right.}\\ {\left. {{2^{{\alpha _1} - 2 + m + \left( {{\alpha _1} - 1 + m} \right)\left( {1 - \frac{1}{{{\alpha _1}}}} \right)}}\left| {{e_1}} \right|\left| {{\mathop{\rm sig}\nolimits} {{\left( {{x_2}} \right)}^{{\alpha _1}}} - {\mathop{\rm sig}\nolimits} {{(\xi )}^{{\alpha _1}}}} \right|} \right) + }\\ {\frac{{n{b_{\rm{m}}} + {b_{\rm{L}}}}}{{n{J_{\rm{m}}} + {J_{\rm{L}}}}}\left( {k_2^{{\alpha _1} - 1 + n}{{\left| {{e_{\rm{t}}}} \right|}^{\frac{{{\alpha _1} - 1 + m}}{{{\alpha _1}}}}}\left| {{e_1}} \right| + } \right.} \end{array}\\ \begin{array}{*{20}{c}} {\left. {{2^{\left( {{\alpha _1} - 2 + m} \right)\left( {1 - \frac{1}{{{\alpha _1}}}} \right)}}\left| {{e_1}} \right|\left| {{\mathop{\rm sig}\nolimits} {{\left( {{x_2}} \right)}^{{\alpha _1}}} - {\mathop{\rm sig}\nolimits} {{(\xi )}^{{\alpha _1}}}} \right|} \right) \le }\\ {{\gamma _1}{{\left| {{e_{\rm{t}}}} \right|}^{\frac{{m - 1}}{{{\alpha _1}}} + 2}} + {\gamma _2}{{\left| {{\mathop{\rm sig}\nolimits} {{\left( {{x_2}} \right)}^{{\alpha _1}}} - {\mathop{\rm sig}\nolimits} {{(\xi )}^{{\alpha _1}}}} \right|}^{\frac{{m - 1}}{{{\alpha _1}}} + 2}} + } \end{array}\\ \begin{array}{*{20}{c}} {{\gamma _3}{{\left| {{e_1}} \right|}^{\frac{{m - 1}}{{{\alpha _1}}} + 2}} + {\gamma _4}{{\left| {{e_{\rm{t}}}} \right|}^{\frac{{n - 1}}{{{\alpha _1}}} + 2}} + }\\ {{\gamma _5}{{\left| {{\mathop{\rm sig}\nolimits} {{\left( {{x_2}} \right)}^{{\alpha _1}}} - {\mathop{\rm sig}\nolimits} {{(\xi )}^{{\alpha _1}}}} \right|}^{\frac{{n - 1}}{{{\alpha _1}}} + 2}} + {\gamma _6}{{\left| {{e_1}} \right|}^{\frac{{n - 1}}{{{\alpha _1}}} + 2}}} \end{array} \end{array} $$ (19) 式中，$ \gamma_{5}=2 k_{3}^{\alpha_{1}} k_{2}^{\alpha_{1}+1} \varepsilon^{2}, \gamma_{6}=2 k_{3}^{\alpha_{1}} 2^{\left(\alpha_{1}+1\right)}\left(1-\frac{1}{\alpha_{1}}\right) \varepsilon^{2}$，参数m满足m+α1－1∈(0, 1)，n满足n+α1－1 < 1. 存在常数ε>0，使得2elk3α1(x2－sig(sig(x2)α1－el)$ ^{\frac{1}{{{\alpha }_{1}}}}$)满足：
$$ \begin{array}{c} -2 e_{1} k_{3}^{\alpha_{1}}\left\{x_{2}-\operatorname{sig}\left[\operatorname{sig}\left(x_{2}\right)^{\alpha_{1}}-e_{1}\right]^{\frac{1}{\alpha_{1}}}\right\}= \\ 2 k_{3}^{\alpha_{1}}\left|e_{1}\right|^{1+\frac{1}{\alpha_{1}}}\left\{-\operatorname{sig}\left[\frac{\operatorname{sig}\left(x_{2}\right)^{\alpha_{1}}}{e_{1}}\right]^{\frac{1}{\alpha_{1}}}-\right. \\ \left.\operatorname{sig}\left[1-\frac{\operatorname{sig}\left(x_{2}\right)^{\alpha_{1}}}{e_{1}}\right]^{\frac{1}{\alpha_{1}}}\right\} \leqslant \\ 2 k_{3}^{\alpha_{1}} \varepsilon^{2}\left|x_{2}\right|^{1+\alpha_{1}}-k_{3}^{\alpha_{1}} \varepsilon^{1-\frac{1}{\alpha_{1}}}\left|e_{1}\right|^{1+\frac{1}{\alpha_{1}}}\left(2+2^{1+\frac{1}{\alpha_{1}}}\right) \leqslant \\ 2 k_{3}^{\alpha_{1}} k_{2}^{\alpha_{1}+1} \varepsilon^{2}\left|x_{1}\right|^{1+\frac{1}{\alpha_{1}}}+2 k_{3}^{\alpha_{1}} \varepsilon^{2} 2^{\left(1-\frac{1}{\alpha_{1}}\right)\left(1+\alpha_{1}\right)} \cdot \\ \left|\operatorname{sig}\left(x_{2}\right)^{\alpha_{1}}-\operatorname{sig}(\xi)^{\alpha_{1}}\right|^{1+\frac{1}{\alpha_{1}}}- \\ k_{3}^{\alpha_{1}} \varepsilon^{1-\frac{1}{\alpha_{1}}}\left|e_{1}\right|^{1+\frac{1}{\alpha_{1}}}\left(2+2^{1+\frac{1}{\alpha_{1}}}\right) \end{array} $$ (20) 综合式(14)，(18)~(20)可得
$$ \begin{array}{c} \dot{V}_{\operatorname{t2}} \leqslant \gamma_{5}\left|x_{1}\right|^{1+\frac{1}{\alpha_{1}}}+\gamma_{6}\left|\operatorname{sig}\left(x_{2}\right)^{\alpha_{1}}-\operatorname{sig}(\xi)^{\alpha_{1}}\right|^{1+\frac{1}{\alpha_{1}}}- \\ k_{3}^{\alpha_{1}} \varepsilon^{1-\frac{1}{\alpha_{1}} }\left|e_{1}\right|^{1+\frac{1}{\alpha_{1}}}\left(2+2^{1+\frac{1}{\alpha_{1}}}\right)+\gamma_{3}\left|e_{1}\right|^{1+\frac{1}{\alpha_{1}}}+ \\ \gamma_{1}\left|e_{t}\right|^{\frac{m-1}{\alpha_{1}}+2}+\gamma_{2}\left|\operatorname{sig}\left(x_{2}\right)^{\alpha_{1}}-\operatorname{sig}(\xi)^{\alpha_{1}}\right|^{\frac{m-1}{\alpha_{1}}+2}+ \\ \gamma_{3}\left|e_{1}\right|^{\frac{m-1}{\alpha_{1}}+2}+\gamma_{4}\left|e_{t}\right|^{\frac{n-1}{\alpha_{1}}+2}+ \\ \gamma_{5}\left|\operatorname{sig}\left(x_{2}\right)^{\alpha_{1}}-\operatorname{sig}(\xi)^{\alpha_{1}}\right|^{\frac{n-1}{\alpha_{1}}+2}+\gamma_{6}\left|e_{1}\right|^{\frac{n-1}{\alpha_{1}}+2} \end{array} $$ (21) 定义李雅普诺夫函数$V = {V_{{\rm{t1}}}} + {V_{{\rm{t2}}}} + \int_{{k_4}}^{{{\dot e}_{\rm{t}}}} {{\rm{sig}}} {[{\rm{sig}}{\left( \zeta \right)^{{\alpha _1}}} - {\rm{sig}}{({k_4})^{{\alpha _1}}}]^{2 - \frac{1}{{{\alpha _1}}}}}{\rm{d}}\zeta $，其中${k_4} = - {k_2}{\rm{sig}}{({e_{\rm{t}}})^{\frac{1}{{{\alpha _1}}}}} $. 对V求导可得
$$ \begin{array}{c} \dot{V}=\dot{V}_{\operatorname{t1}}+\dot{V}_{\operatorname{t2}}+\dot{V}_{\mathrm{t} 3}=-\lambda_{1}\left|e_{\mathrm{t}}\right|^{1+\frac{1}{\alpha_{1}}}- \\ \lambda_{2}\left|\operatorname{sig}\left(x_{2}\right)^{\alpha_{1}}-\operatorname{sig}(\xi)^{\alpha_{1}}\right|^{1+\frac{1}{\alpha_{1}}}-\lambda_{3}\left|e_{1}\right|^{1+\frac{1}{\alpha_{1}}}+ \\ \gamma_{1}\left|e_{\mathrm{t}}\right|^{\frac{m-1}{\alpha_{1}}+2}+\gamma_{2}\left|\operatorname{sig}\left(x_{2}\right)^{\alpha_{1}}-\operatorname{sig}(\xi)^{\alpha_{1}}\right|^{\frac{m-1}{\alpha_{1}}+2}+ \\ \gamma_{3}\left|e_{1}\right|^{\frac{m-1}{\alpha_{1}}+2}+\gamma_{4}\left|e_{\mathrm{t}}\right|^{\frac{n-1}{\alpha_{1}}+2}+ \\ \gamma_{5}\left|\operatorname{sig}\left(x_{2}\right)^{\alpha_{1}}-\operatorname{sig}(\xi)^{\alpha_{1}}\right|^{\frac{n-1}{\alpha_{1}}+2}+\gamma_{6}\left|e_{1}\right|^{\frac{n-1}{\alpha_{1}}+2} \end{array} $$ (22) 其中λ1，λ2，λ3>0，且通过调节控制器式(9)的参数使得λ1，λ2和λ3满足：
$$ \begin{array}{c} \lambda_{1}<2 k_{2}-\left(2^{2-\frac{1}{\alpha_{1}}} \alpha_{1}\left(\alpha_{1}+1\right)^{-1}+2^{2-\frac{1}{\alpha_{1}}}\left(\alpha_{1}+1\right)^{-1}+\right. \\ \left.2 k_{3}^{\alpha_{1}} k_{2}^{\alpha_{1}+1} \varepsilon^{2}+2\left(2+2^{\left(1-\frac{1}{\alpha_{1}}\right)\left(1+\alpha_{1}\right)}\right) k_{3}^{\alpha_{1}} \varepsilon^{2}\right) \\ \lambda_{2}<k_{1}-\left[2^{2-\frac{1}{\alpha_{1}}}\left(\alpha_{1}+1\right)^{-1}+\gamma_{4}+2 k_{3}^{\alpha_{1}} \varepsilon^{2} 2^{\left(1-\frac{1}{\alpha_{1}}\right)\left(1+\alpha_{1}\right)}\right] \\ \lambda_{3}<k_{3}^{\alpha_{1}} \varepsilon^{1-\frac{1}{\alpha_{1}}}\left(2+2^{1+\frac{1}{\alpha_{1}}}\right)-\gamma_{3} \end{array} $$ (23) 取σ1，σ2，σ3∈(0, 1)，定义集合
$$ \begin{array}{c} \varXi=\left\{e_{\mathrm{t}}, e_{1}: \gamma_{1}\left|e_{\mathrm{t}}\right|^{\frac{m-1}{\alpha_{1}}+2}+\right.\\ \gamma_{2} \left|\operatorname{sig}\left(x_{2}\right)^{\alpha_{1}}-\operatorname{sig}(\xi)^{\alpha_{1}}\right|^{\frac{m-1}{\alpha_{1}}+2}+\\ \gamma_{3}\left|e_{1}\right|^{\frac{m-1}{\alpha_{1}}+2}+\gamma_{4}\left|e_{\mathrm{t}}\right|^{\frac{n-1}{\alpha_{1}}+2}+\\ \gamma_{5} \left|\operatorname{sig}\left(x_{2}\right)^{\alpha_{1}}-\operatorname{sig}(\xi)^{\alpha_{1}}\right|^{\frac{n-1}{\alpha_{1}}+2}+\\ \gamma_{6}\left|e_{1}\right|^{\frac{n-1}{\alpha_{1}}+2} \leqslant \lambda_{1}\left|e_{\mathrm{t}}\right|^{1+\frac{1}{\alpha_{1}}}+\\ \left.\lambda_{2}\left|\operatorname{sig}\left(x_{2}\right)^{\alpha_{1}}-\operatorname{sig}(\xi)^{\alpha_{1}}\right|^{1+\frac{1}{\alpha_{1}}}+\lambda_{3}\left|e_{1}\right|^{1+\frac{1}{\alpha_{1}}}\right\} \end{array} $$ (24) 则当et和el在集合$ \mathit{\Xi }$之外时，式(22)可以写作如下形式：
$$ \begin{array}{c} \dot{V}_{\mathrm{t}} \leqslant-\left(1-\sigma_{1}\right) \lambda_{1}\left|e_{\mathrm{t}}\right|^{1+\frac{1}{\alpha_{1}}}-\left(1-\sigma_{2}\right) \lambda_{2} \mid \operatorname{sig}\left(x_{2}\right)^{\alpha_{1}}- \\ \left.\operatorname{sig}(\xi)^{\alpha_{1}}\right|^{1+\frac{1}{\alpha_{1}}}-\left(1-\sigma_{3}\right) \lambda_{3}\left|e_{1}\right|^{1+\frac{1}{\alpha_{1}}} \end{array} $$ (25) 将$ |{e_{\rm{t}}}| = V_{{\rm{t1}}}^{\frac{1}{2}}$和$ |{e_{\rm{l}}}| = V_{{\rm{t2}}}^{\frac{1}{2}}$代入上式可得
$$ {\dot V_{\rm{t}}} \le - {\lambda _4}V_{{\rm{t1}}}^{\frac{{1 + 1/{\alpha _1}}}{2}} - {\lambda _5}V_{{\rm{t2}}}^{\frac{{1 + 1/{\alpha _1}}}{2}} \le - {\lambda _{\rm{m}}}V_{\rm{t}}^{\frac{{1 + 1/{\alpha _1}}}{2}} $$ (26) 式中，λm=min{λ1, λ2}.
根据引理1，式(26)符合式(6)所给出的形式. 因此，跟踪误差与观测误差能够在有限时间内收敛. 收敛时间可由式(10)计算得到.
证明完毕.
2.3 结构/控制一体化设计
由于电机驱动系统中结构参数的设计会影响到控制性能，基于前文所设计的有限时间跟踪控制器，同时考虑结构参数设计与控制器参数选取，对电机驱动系统进行结构/控制一体化设计. 首先建立了电机驱动系统一体化设计的目标函数；然后采用嵌套优化策略，将所得一体化优化问题分解为两个镶嵌在一起的自由化问题，从而简化了一体化问题的求解；最后，采用自适应步长的布谷鸟搜索算法得到一体化设计的最优解
考虑系统(5)，选取齿轮箱的齿轮间隙作为结构优化变量；选取所设计有限时间跟踪控制器的参数k1，k2和k3作为控制器优化变量. 一体化性能指标设计为如下形式：
$$ \begin{array}{l} F_{\mathrm{y}}=\min \left\{\omega_{1} f_{1}+\omega_{2} f_{2}\right\} \\ \text { s.t. } h(\delta)=0, g(\delta) \leqslant 0 \end{array} $$ (27) 式中，ω1和ω1为权值系数；h(δ)=0为包括系统动态在内的等式约束；g(δ)≤0为不等式约束，本文以齿隙作为结构优化变量，有不等式约束：Jd≤JL≤Ju，Jd为负载转动惯量约束最小值，Ju为约束最大值；$ {f_1} = \frac{1}{{{J_{\rm{L}}}}}$为系统负载的转动惯量；f2=tr为系统的上升时间. 因此，式(27)的物理意义是在调节控制器参数使系统的上升时间最小，同时驱动获得控制系统所能驱动的最大负载.
为简化式(27)，从而获得电机驱动系统一体化设计的最优解，本文采用嵌套式优化策略设计如下的一体化设计结构.
外优化层：
$$ \begin{aligned} J_{\mathrm{L}}^{*}=& \operatorname{argmin}\left\{\frac{1}{J_{\mathrm{L}}}+f_{2}^{*}\left(J_{\mathrm{L}}\right)\right\} \\ & \mathrm{s.t.}\ \ J_{\mathrm{d}} \leqslant J_{\mathrm{L}} \leqslant J_{\mathrm{u}}, \end{aligned} $$ (28) 内优化层：
$$ f_{2}^{*}\left(J_{\mathrm{L}}\right)=\min f_{2}\left(J_{\mathrm{L}}\right) $$ (29) 结合式(28)和式(29)，外优化层进行结构优化；内优化层进行控制器优化. 因此，采用上述一体化设计结构，结构参数与控制器参数能够同时得到优化，从而得到了电机驱动系统的最优设计. 同时，采用嵌套式优化结构能够简化一体化设计的优化问题求解，并且是一体化设计过程具有更加明确的物理意义.
在内优化层，为优化控制器参数，采用自适应步长的布谷鸟搜索(CS)算法对f2进行求解. 布谷鸟搜索算法一种结合了布谷鸟巢寄生性和Lévy flights模式的启发式群体智能搜索算法，通过随机游走的方式搜索得到一个最优的鸟窝来孵化自己的鸟蛋，可以通过较少的算法参数达到一种高效的寻优模式[20]. 布谷鸟搜索算法具有较好的可移植性，并且鲁棒性强[21]，为得到更好的收敛性，对其搜索步长做出改进. 布谷鸟搜索的第j个位置更新律为：
$$ \boldsymbol{\kappa}_{j}^{(t+1)}=\boldsymbol{\kappa}_{j}^{(t)}+\hbar \oplus L() $$ (30) 其中L()=t－为Lévy随机路径；⊕为点对点乘法；ħ为自适应步长：
$$ \hbar=\hbar_{\min }+\left(\hbar_{\max }-\hbar_{\min }\right) \varGamma_{j} $$ (31) $$ \begin{array}{c} \\ \varGamma_{j}=\frac{\left\|\boldsymbol{\kappa}_{j}-\boldsymbol{\kappa}_{\mathrm{be}}\right\|}{\varGamma_{\max }} \end{array} $$ (32) 式中，κbe为位置的历史最优值；Γmax为Γj的历史最大值. 布谷鸟搜索算法的步长可以根据式(31)调节，提高了算法的计算效率. 采用上述群体智能搜索算法，能够高效地得到内优化层的优化结果，进而得到电机及驱动系统的一体化设计结果.
3. 仿真结果与分析
为验证所提算法的有效性，选取yd=2sin(t)作为参考信号. 电机驱动系统参数如表 1所示. 采用所提出的电机驱动系统结构/控制一体化设计方法，Jd=0.05kg·m2, Ju=1.10kg·m2，得到最优结构参数JL=0.41kg·m2和控制器参数k1=1.26, k2=3.15.
表 1 电机驱动系统参数Table 1. System parameters of the motor driving systembL/
(Nm·s·rad－1)bm/
(Nm·s·rad－1)k/
(Nm·rad－1)δ/rad Jm/
(kg·m2)1.2 1.3 56 0.2 0.028 采用上述参数与参考信号，能够得到系统的跟踪控制结果如图 2~4所示.
从上述仿真结果中可以看出，本文所提出的跟踪控制器能够驱动负载跟踪期望信号. 在幅值为2rad的参考信号下，最大跟踪误差不超过0.1rad. 为进一步验证本文设计控制器性能，将初始误差设为3rad，仿真结果在图 2~4中给出. 从仿真结果中可以看出，在存在较大初始误差的条件下，本文所设计控制器仍能够保证电机驱动系统良好的控制性能，跟踪误差在0.1rad以内. 因此，本文设计的有限时间滤波控制器能够较好的解决电机驱动系统的跟踪控制问题.
为验证本文所提出一体化设计方法的有效性，选取yd=5rad的参考信号，设置不同的结构参数JL，仿真结果如图 5所示.
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图 5 驱动不同负载的控制性能比较Figure 5. Tracking performance of the motor driving system with different plant parameters从图 5中可以看出，改变负载的大小会影响系统的控制性能. 采用一体化设计得到的负载能够获得更好的控制效果. 而选取其他的结构参数会破坏系统的全局最优，从而导致系统控制性能下降. 经过一体化设计，可知系统的驱动能力. 所以在驱动更大负载的条件下，控制性能会受到影响.
4. 结论
为提升电机驱动系统的控制性能，考虑了结构与控制之间的耦合问题，提出了基于有限时间滤波控制的结构/控制一体化设计方法. 首先，为解决电机驱动系统的跟踪控制问题，设计了有限时间滤波控制器. 所提出的控制器能够在估计系统中负载速度信息的同时完成有限时间跟踪控制. 随后，考虑结构参数会对控制性能产生影响，提出了结构/控制一体化设计方法. 提出的一体化性能指标能够在满足控制性能要求的同时，优化得到系统所能驱动的最大负载，从而提升了控制系统的性能. 为便于一体化问题求解，本文引入嵌套优化策略，使得一体化设计结构物理意义明确，求解复杂度降低. 在此基础上，通过自适应补偿的布谷鸟搜索算法得到了一体化最优解. 仿真结果表明了所提出方法的有效性.
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图 5 驱动不同负载的控制性能比较
Figure 5. Tracking performance of the motor driving system with different plant parameters
表 1 电机驱动系统参数
Table 1 System parameters of the motor driving system
bL/
(Nm·s·rad－1)bm/
(Nm·s·rad－1)k/
(Nm·rad－1)δ/rad Jm/
(kg·m2)1.2 1.3 56 0.2 0.028 
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