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Systematic Design Method of Disturbance Observer Guaranteeing Closed-loop System's Robust Stability

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The sufficient condition on the robust stability against model uncertainties in the feedback system with disturbance observer (DOB) is studied. Weighting functions satisfying this condition are consequently selected such that robust stability to plant s parameters variation is guaranteed by the disturbance observer designed with standard H∞ disturbance observer design method. Selection methods of frequency weighting functions are proposed concerning about the robustness design specifications and structural restrictions in designing the H∞ DOB. Trade-off between peek magnitude in cutoff frequency of Q-filter and disturbance depressing performance is optimized in designing low-pass filter of disturbance observer via selections of the weighting functions. Therefore, the resultant disturbance observer has the optimal disturbance rejection performance with satisfactory peak magnitude. Experimental results show the proposed method could guarantee robust stability of closed-loop system while providing design freedom for other available design specifications.
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38 1 自动化学报 Vol. 38, No. 1
2012 1ACTA AUTOMATICA SINICA January, 2012
保证闭环系统鲁棒稳定性的干扰观测器系统性设计方法
尹正男 1,2苏剑波 1高秀行 1,3
摘 要 研究带有干扰观测器 (Disturbance observer, DOB) 的反馈控制系统对模型不确定性鲁棒稳定的充分条件,在此基
础上,选取满足此充分条件的加权函数,使得标准 H干扰观测器设计方法保证对受控对象参数变化的鲁棒稳定性.提出了在
H干扰观测器设计中兼顾鲁棒性设计指标和结构约束的频率加权函数的选取方法.利用加权函数选取的自由度,在干扰观
测器低通滤波器设计中,实现 Q滤波器在截止频率上的高峰幅度与干扰抑制性能之间的最佳折中,使得干扰观测器在满足其
幅度指标的条件下,具有最优干扰抑制性能.实验结果表明该方法保证了闭环反馈系统的鲁棒稳定性,同时,具有实现其他设
计指标的自由度.
关键词 鲁棒运动控制,干扰观测器,H标准控制问题,低通滤波器, 2 自由度控制系统
DOI 10.3724/SP.J.1004.2012.00012
Systematic Design Method of Disturbance Observer Guaranteeing
Closed-loop System0s Robust Stability
YIN Zheng-Nan1,2SU Jian-Bo1GAO Xiu-Xing1,3
Abstract The sufficient condition on the robust stability against model uncertainties in the feedback system with
disturbance observer (DOB) is studied. Weighting functions satisfying this condition are consequently selected such
that robust stability to plant0s parameters variation is guaranteed by the disturbance observer designed with standard
Hdisturbance observer design method. Selection methods of frequency weighting functions are proposed concerning
about the robustness design specifications and structural restrictions in designing the HDOB. Trade-off between peek
magnitude in cut-off frequency of Q- filter and disturbance depressing performance is optimized in designing low-pass filter
of disturbance observer via selections of the weighting functions. Therefore, the resultant disturbance observer has the
optimal disturbance rejection performance with satisfactory peak magnitude. Experimental results show the proposed
method could guarantee robust stability of closed-loop system while providing design freedom for other available design
specifications.
Key words Robust motion control, disturbance observer (DOB), Hstandard control problem, low-pass filter, two
degrees of freedom control system
基于干扰观测器 (Disturbance observer, DOB)
的鲁棒控制[1] ,因为其实现简单,且对受控对象参数
变动的鲁棒性,而广泛应用在受非线性和不确定
性影响的高性能鲁棒运动控制领域中,例如机器人
关节运动控制、硬盘磁头定位控制系统等[26].
DOB 系统中,干扰抑制性能和鲁棒性等主要一些
能取决于其低通滤波器 (Q滤波器), DOB
收稿日期 2011-07-04 录用日期 2011-09-20
Manuscript received July 4, 2011; accepted September 20, 2011
国家自然科学基金重点项目 (60935001) 资助
Supported by Key Project of National Natural Science Foun-
dation of China (60935001)
本文责任编委 贾英民
Recommended by Associate Editor JIA Ying-Min
1. 上海交通大学电子信息与电气工程学院 上海 200240, 中国 2.
日成综合大学电子与自动化系 平壤 526-890, 朝鲜 3. 金策工业综合
大学自动化系 平壤 999093, 朝鲜
1. Electronic Information and Electric Engineering School,
Shanghai Jiao Tong University, Shanghai 200240, P.R. China
2. Electronics and Automation Faculty of Kim Il Sung Uni-
versity, Pyongyang 526-890, D. P. R. of Korea 3. Department
of Automation of Kim Chaek Industrial University, Pyongyang
999093, D. P. R. of Korea
统设计的关键是 Q滤波器的设计.
Q滤波器的实现最常采用传统滤波器,But-
terworth 模型、二项系数模型[79].此外,也有通
过对传通滤波器模型系数的调整进行设计[1011],
是这种滤波器形式对模型参数变动的鲁棒性能及干
扰抑制性能较差.因此,基于鲁棒控制理论的干扰
观测器设计理论和方法逐渐成为研究热点,尤其是
H控制理论.基于 H控制理论的 DOB 设计将干
扰和传感器噪声抑制性能、鲁棒稳定性等反馈系统
的所有性能表达为一些闭环传递函数的 H范数,
并进一步分析其性能或优化 Q滤波器.
文献 [1214] 通过将闭环系统的控制性能定义
为一些传递函数的 H范数,然后利用数值计算方
法求出其最优解.文献 [12] 虽然保证闭环系统对受
控对象参数变化的鲁棒稳定性,但无法确定干扰观
测器需要构造, ,阶次内模阶 .
文献 [13] 缺乏对模型变化的鲁棒性,文献 [14] 采用
先设计后降阶的方法,这样得到的 Q滤波器结果不
是原模型的最优解.然这些方法能够满足对模
1期 尹正男等:保证闭环系统鲁棒稳定性的干扰观测器系统型设计方法 13
不确定性的鲁棒稳定性或者干扰抑制性能等控制指
,但是它们只能通过数值计算求解,缺乏全局性和
收敛性以及求解过程的系统性.
在基于 H范数优化的 DOB 设计中,无法使
用类似 H标准控制问题的系统性求解算法,因为
DOB 设计问题通常含有比较复杂的阶次和结构条
,不符合标准问题的前提条件.包括整个滤波器
阶次、相对阶次、内模阶次等在内的阶次要求是求
解优化问题的难点.文献 [15] 提出了标准 H干扰
观测器的系统性设计方法,采用一种虚拟开环系统
的受控对象和控制器概念,消除相对阶次约束并将
Q滤波器设计问题转换为无约束的标准控制问题.
此方法使用标准设计问题框架内的完整的求解方法,
因此其设计过程既直观又有系统性.而且,通过调
DOB 局部反馈回路的混合灵敏度函数,直接形成
Q滤波器的具体频率特性,能满足类似于滤波器频
率特性高峰要求的其他设计指标.但在其优化评价
函数中,只考虑到对 DOB 本身反馈系统,而不是整
个闭环系统对模型参数变动的鲁棒稳定性.因此,
H干扰观测器设计方法,需要更加深入关于反
馈系统鲁棒稳定性的分析和为其实现的研究,从而
不仅可以得到最优化过程的系统性和全局性,还保
证了整个反馈系统的鲁棒稳定性.
通过对上述 Q滤波器设计方法的分析,本文将
上述两种设计方法结合起来,索保证整体闭环系
统鲁棒稳定性的系统性设计方法.了实现两种
法的结合,文解了如下问: 1) 提出种新
鲁棒稳定性充分条件,将闭环系统的鲁棒稳定性反
映在 DOB 的设计方法当; 2) 建立不仅反映此鲁
棒稳定性条件和干扰抑制性能,还反映 DOB
特定阶次及构造约束的 H加权函数的选取方法;
3) 用加权函数构成的自由度改善干扰和噪声抑制性
能的方.为优化设计的评价函数采用整体反馈系
统传递函数的 H,保证闭环控制系统对模型
不确定性的鲁棒稳定性;求解方法采用基于虚拟开
环系统的 Q滤波器设计方法,保证求解的系统性和
全局最优性.本文结构安排如下:首先,简单介绍两
HDOB 设计方法并分析其特点和缺陷.其次,
提出一种考虑闭环控制系统鲁棒稳定性的充分条件,
并分析其保守性.在此基础上,提出加权函数的新选
取方法,使标准 H干扰观测器设计问题的评价函
数能够反映整个闭环系统的鲁棒稳定性.
1基于 H
H
H
范数优化的干扰观测器设计方法
与问题
基于 H范数优化的 DOB 设计方法大致可分
I型和 II 型两种, I 是基于数值计H范数
优化的设计; II 型是基于标准 H控制问题的设计.
利用 H控制理论设计干扰观测器需要把干扰对控
制系统的影响表达为 H范数评价函数,然后通过
优化其评价函数来设计 Q滤波器.因此,不同方法
的差别在于控制目标的实现性能,以及优化问题的
求解效率.
1.1 基于数值计算 H
H
H
范数优化的设计 (I )
1显示的控制系统由两个回路组成,即由标
称逆模P1
n(s)和低通滤波器 Q(s)组成的 DOB
内环回路和一般反馈控制器 C(s)的外环回路.在模
型匹配时 (Pn=P), 系统的输入输出方程为
y=Pn(s)C(s)
1 + Pn(s)C(s)r+Pn(s)(1 Q(s))
1 + Pn(s)C(s)d+
Pn(s)C(s) + Q(s)
1 + Pn(s)C(s)ξ(1)
式中 r,d,ξy分别为参考输入、干扰、检测噪声
和输出信号.
1带有 DOB 的反馈控制系统
Fig. 1 Feedback control system with DOB
由上式可知,为了保证闭环系统在模型匹配时
的标称稳定性,Q(s)必须稳定.与此同时,系统对可
能出现的模型参数变化一定保持鲁棒稳定性.设受
控对象含有如下乘型不确定性.
P(s) = Pn(s)(I+ ∆(s)) (2)
DOB 设计 的一 [12, 16] 提出
DOB Q滤波器对这种不确定性的鲁棒稳定性条
.由图 1的框图可知,整个闭环系统对此模型误差
的鲁棒稳定性充分条件为满足下式
¯σ(∆(jω)T(jω)) <1,ω(3)
或者,对满足 ¯σ(∆(jω)) <|WT(jω)|,ωWT(s)
kWTTk<1, T (s) = Pn(s)C(s) + Q(s)
1 + Pn(s)C(s)(4)
其中,T(s)为全闭环系统的补灵敏度函数.(1)
的右边最后一项可,T(s),噪声抑制性能
14 自动化学报 38
.应注意的是鲁棒稳定性不仅取决于 DOB 的低通
滤波器 Q(s) , 还取决于一般反馈控制器 C(s), 这是
H控制框架内很难得DOB 的系统性设计方
法的原因之一.另一方面,由式 (1) 右边第二项可知,
为抑制干扰 d对闭环系统输出 y的影,必须在
频上尽量减小灵敏度函数 S(s)
min
Q(s)kWSSk<1, S(s) = 1Q(s)
1 + Pn(s)C(s)(5)
对满足
¯¯¯¯WS(jω)1
(1 + Pn(jω)C(jω))¯¯¯¯<|WSD (jω)|,ω
WSD(s), 满足式 (5) 的充分条件为
min
Q(s)kWSDSDOB k<1, SDOB(s) = 1Q(s) (6)
其中,SDOB(s)DOB 反馈系统的灵敏度函数,
(4) 和式 (6) 表明,满足闭环系统鲁棒稳定性和最优
干扰抑制性能条件的 DOB 设计评价函数为
min
Q(s)°
°
°
°
°"WSD(s)SDOB (s)
WT(s)T(s)#°
°
°
°
°
<1 (7)
2H(7) 的广义受控对象和
优化变量 Q(s)的连接框图.基于 H范数优化的
DOB 设计常使用 LMI 数值计算算法求解[1214].
方法很容易实现闭环系统的鲁棒稳定性,但由于
用数值计算算法,其预备流程比较复杂,无法保证解
的全局最优性,而且无法实现 Q滤波器的构造约
.文献 [12] 提供了满足局部鲁棒稳定性式 (3)
(4) Q滤波器无法满足整体反馈系统鲁棒稳
定性的实例.
2DOB 设计的 H控制问题
Fig. 2 Hcontrol problem for DOB design
1.2 基于标准 H
H
H
控制问题的设计 (II )
考察 DOB 内环回路,探讨 Q滤波器的通用设
计方.在不考虑外环反馈系统时,从干扰 d
测噪ξ到对象输出 y的传递函数分别等于 DOB
局部系统的灵敏度函数 SDOB(s)和补灵敏度函数
TDOB(s)=1SDOB(s) = Q(s). ,
局部系统对模型误差 ∆(s)的鲁棒稳定性充分条件
kWTD(s)TDOB (s)k<1, WTD(s)
¯σ(∆(jω)) <|WTD(jω)|,ω的上限加权函数.因此,
设计 DOB 的评价函数为
max γ, min
Q(s)RH
Q(s)k°
°
°
°
°"γ·WSDSDOB
WTDTDOB #°
°
°
°
°
<1 (8)
式中
k=nF(s)|F(s) = N(s)
D(s),
deg(D)deg(N)ko(9)
是满足 Q滤波器相对阶次条件 k的有理传递函数
集合.文献 [15] 通过变换把含有阶次约束的优化问
(8) 转换为标准 H控制问题,这样,(8) 就可
进一步转换为无约束优化问题:
max γ, min
˜
K(s)°
°
°
°
°"γ·WSD(1 + e
Pe
K)1
WTD e
Pe
K(1 + e
Pe
K)1#°
°
°
°
°
<1
(10)
式中,e
P(s), e
K(s)Q(s)有关的虚拟开环系统的受
控对象和控制器,此问题因为不带任何阶次约束并
满足标准问题前提条件,采用系统性的求解算法可
以得到最Q滤波器.此设计方法不仅可实现阶
次条件、相对阶次 件、内模阶次件等 DOB
Q滤波器的所有阶次条件要求,其系统性的求解方
法能保证求解过程的全局最优和收敛.
1.3 对设计方法的比较分析及问题设定
上面讨论的每个设计方法都有自己的特征. I
可以反映闭环反馈系统的鲁棒稳定性,但只能依靠
数值计算方法求解,即以 Q滤波器传递函数的所
有系数或其状态空间表现的系数矩阵作为优化变量,
用反复计算进行优化.因此,缺乏求解过程的系统
,而且无法保证解的全局性和收敛性.另一方面,
II 型虽只考虑 DOB 内环系统的性能,但其求解很有
系统性,且能考虑阶次约束等不同设计指标,尤其是
Q滤波器的阶次、相对阶次、内模阶次等一些阶次
约束与求解过程很容易融合,而这些约束在 I型设
计中无法考虑.1显示两种方法的比较.
由表 1可知,除了第一个指标之外,其他指标 II
型均优于 I.因此,我们可知,II 型设计方法满
I型的设计评价函数,则设计结果符合表 1的所有
指标.这样,可得到把两种设计方法结合起来的新的
DOB 设计方法:I型设计问题出发,通过恰当的
1期 尹正男等:保证闭环系统鲁棒稳定性的干扰观测器系统型设计方法 15
1 I 型和 II HDOB 设计方法的比较
Table 1 Comparison of type I and type II design methods of HDOB
III
闭环系统鲁棒稳定性 可保证 只能保证 DOB 局部回路鲁棒稳定性
优化算法 只能用数值计算算法 保证系统性的优化算法
给定的 Q滤波器的阶次条件 无法实现 可实现
Q滤波器的频率响应直接成形 无法直接成形 可直接成形
数学处理把它转换为 II 型问题,再用 II 型问题的系
统性设计方法求解最优解.
2考虑到闭环系统鲁棒稳定性的评价函数
I型问题II 型问题之间的差异在于评价函数
的差异.由式 (7) 和式 (8) , I 型的评价函数由
闭环系统的灵敏度函数 S(s)和补灵敏度函数 T(s)
组成,II DOB 局部反馈系统的灵敏度函数
SDOB(s)和补灵敏度函数 TDOB (s)组成.因此, I
问题转换为 II 型问题的关键在于寻找把闭环系统灵
敏度函数所构成的评价函数 (7) 表达为 DOB 局部
系统灵敏度函数所构成的评价函数 (8) 的方法.
里给出一种带有 DOB 的闭环系统鲁棒稳定性充分
条件,在此基础上解决两种评价函数之间的转换.
2.1 满足闭环系统鲁棒稳定性的评价函数
2.1.1 鲁棒稳定性充分条件
I型问题II 型问题在鲁棒稳定性方面有本质
上的差异,但形式上, I 型问题是可以向 II 型转化的.
由闭环系统鲁棒稳定性充分条件 (4), 可以得到不等
:
|L(jω)+Q(jω)|<|W1
T(jω)(1+L(jω))|,ω(11)
其中,L(s) = Pn(s)C(s)是反馈系统在标称状态
开环回路传递函数.根据复数绝对值的性质,上式左
边满足
|L(jω) + Q(jω)|<|L(jω)|+|Q(jω)|,ω(12)
因此,如式 (13) 成立,(11) 一定成立
|Q(jω)|<|W1
T(jω)(1 + L(jω))|−|L(jω)|,ω
(13)
只要 Q滤波器满足此不等式,包括 DOB 内环回路
和反馈外环回路在内的闭环控制系统就一定稳定,
即式 (13) 是闭环系统鲁棒稳定的充分性判据条件.
但必须满足下面的几个条件.首先,不等式 (13)
右边必须大于 0,
E(ω) = |W1
T(jω)(1 + L(jω))|−|L(jω)|>0,ω
(14)
注意:如果反馈系统满足普遍意义上的鲁棒稳
定性,(14) 一定满足.换言之,在不带 DOB 的情
况下对受控对象乘型不确定性保证鲁棒稳定性的系
统一定满足式 (14). 出证,考察一般馈系
(不带 DOB 的系统)的鲁棒稳定性充分条件:
kWT(s)T0(s)k<1
|WT(jω)L(jω)(1 + L(jω))1|<1,ω(15)
其中,T0(s) = L(s)(1 + L(s))1为不带 DOB
闭环系统补灵敏度函数.容易判断 式 (15) 与 式 (14)
是等价的.
其次,应考虑此鲁棒稳定性判定的保守性.因为
新判定法是引入式 (12) 而建立的,所以其保守性取
决于式 (16) C(ω)在每个频率上大小:
C(ω) = |L(jω)|+|Q(jω)| − |L(jω) + Q(jω)|(16)
频率函数 C(ω)在给定频率 ω0上的大小最终取决于
L(jω0)Q(jω0)之间的相位差.在一般控制系统
的情况下,随着频率增加,这两个复数轨迹的相位靠
,尤其是在对鲁棒稳定性很重要的截止 (Cut-off)
频率以上的带宽上.3显示 Q(s)L(s)在现实
控制问题中的相位特性比.图中的相位特性比
显示,相位误差在全频带很小,守性也很.
而且,Q滤波器设计得越逼近最优,保守性就会
越小.选取稳定的加权函数 WTD(s)满足
|W1
TD(jω)|< E(ω),ω(17)
(13) 的鲁棒稳定性充分条件可改为
|Q(jω)|<|W1
TD(jω)|,ω
kWTD(s)Q(s)k<1 (18)
应该注意的是,(18) II 型问题的鲁棒稳定性条
件的形式完全相同.
16 自动化学报 38
(a) 尼奎斯特轨迹
(a) Nyquist locuses
(b) 频率 相位响应
(b) Frequency-phase responses
3Q(s)L(s)的频率相位特性比较
Fig. 3 Comparison of frequency-phase characteristics of
Q(s) and L(s)
2.1.2 干扰抑制性能
由式 (1) 可知,干扰 d对闭环系统的影响取决于
闭环系统的灵敏度函数 S(s), 因此,必须用式 (5)
(6) 来优d起作用的频带 (低频)S(s)
.注意:(5) 虽然是从闭环系统灵敏度优化观点
出发的,但是式 (6) 已经变成为 DOB 局部系统灵敏
度优化问题的形式.因此,灵敏度及干扰抑制的优化
问题可设定为:
max γ, min
Q(s)kγ·WSD(s)(1 Q(s))1k<1 (19)
2.1.3 评价函数
把式 (18) (19) 结合,可以获得 DOB 设计
价函数:
max γ, min
Q(s)°
°
°
°
°"γ·WSD(1 Q)
WTD Q#°
°
°
°
°
=
max γ, min
Q(s)°
°
°
°
°"γ·WSD SDOB
WTD TDOB#°
°
°
°
°
<1
(20)
(20) 的评价函数与式 (8) 一样,DOB
部系统的灵敏度函数和补灵敏度函数来定义的,
味着,I型评价函数出发的问题完全变为 II
题的.因此可以采用 II 型问题的系统性求解
算法设计满足闭环系统鲁棒稳定性和干扰抑制性能
DOB. 特别说明,DOB Q波器设计,
必然出现相对阶次条件匹配问题,Q(s)一定满足
Q(s)k.此条件因标称受控对象逆模型的物理实
现要求而产生.此外,含内模阶次条件[17] 等约束造
成求解复杂的问题在 II 型问题的求解算法中得以解
.
2.2 设计步骤
由上分析,提出保证闭环系统鲁棒稳定性的
DOB 系统性设计步骤:
步骤 1. 按照参照轨迹跟踪性能指标,首先设计
反馈控制器 C(s). 在设计中,般不考虑干扰抑制
性能,因为系统具有 2自由度控制功能,并且抗干扰
性能由 DOB 实现.
步骤 2. 按照设计指标定义 DOB 计评价函
(7), 即选取 I型评价函数中的加权函数 WSD(s)
WT(s).
步骤 3. 按照式 (14), 构成频率函数 E(ω).
4. 选取满足式 (17) 的稳定加权函数
WTD(s), 定义 II 型评价函数 (20). 在此过程中,
可以反映 DOB 设计的其他指标,比如 kQ(s)k
k1Q(s)k类似的指标.这种指标与干扰抑制以及
检测噪声有比较明显的关系.3节显示与此有关
的内容和实例.
步骤 5. 对此 II 型评价函数,用已开发的系统
性求解算法求解最优 Q滤波器.
关于5的设计算法通过简单的准备过程,
最终落到标准 H控制器设计问题,然后可使用标
准问题的求解算法 (例如 DGKF 算法[18] )求解充分
逼近最优解的次优解.此求解计算通过 Matlab
棒控制工具箱 (Robust control toolbox) 可容易实
.在文献 [15] 中有具体说明,这里不予详述.
2.3 加权函数的选取
H干扰观测器设计中,加权函数的选择
是关键的问题.尤其在 DOB Q滤波器设计中,
需要通过加权函数的适当选择实现阶次条件和构
造条件.2.2 节提出的步骤 4中出现的加权函数
WTD(s)(20) WSD(s)必须含有设计指标和
约束条件的信息,才能得到鲁棒性设计指标和给定
的阶次和构造条件的理想 Q滤波器.DOB 回路
的特殊构造,Q滤波器必须满足如下的阶次和构造
约束:
1) 为了保证 Q(s)P1(s)的实现,Q(s)一定
具有 k(P1(s)的相对阶次)以上的相对阶次.
2) 为了能够消除形为 d=pqtq+pq1tq1+
1期 尹正男等:保证闭环系统鲁棒稳定性的干扰观测器系统型设计方法 17
· · · +p0的干扰,Q(s)一定具有指标 q(内模阶)
有关的构造 ((21) 所显示).
3) 要满足阶次指标 n.
(9) 只表现对 Q滤波器的相对阶次约束.
了此约束之外,Q滤波器还要满足整体阶次条件和
内模阶次条件,:
Q(s)n,k,q (21)
其中, Ωn,k,q 为满足整体阶次 n相对阶次 k(n)
及内模阶次 q(m)的最小实现传递函数集合:
n,k,q ={G(s)|G(s) = M(s)
N(s)
N(s) =
n
X
i=0
aisi, M(s) =
m
X
j=0
bjsj,
an6= 0, bm6= 0, m =nk,
al=bl, l = 0,1,· · · , q }(22)
加权函数 WTD (s)WSD(s)选择应使(10)
设计结果满足式 (21).
最优 Q滤波器 Q(s)可以用加权混合灵敏度
问题 (10) 的解 e
K(s)得出:
Q(s) = L(s)(1 L(s))1
L(s) = e
P(s)e
K(s)(23)
e
K(s)的阶次不超过式 (10) 的广义受控对象的阶
,WTD(s)WSD(s)e
P(s)的阶次和.而且,
H范数优化解的性质,e
K(s)e
P(s)的所有
极点包含为零点,所以 L(s)的阶次等于 e
K(s)
阶次.由于 e
P(s)的相对阶次选择为 k,所以 L(s)
Q(s)的相对阶次都等于 k.鲁棒性设计指标和
阶次约nqWTD(s)WSD (s)的阶次及构
造决定.
2.3.1 W
W
WTD(s
s
s)的选取
加权函数 WTD(s)Q(s)的高频截止特性.
H控制理论可知,为了保证优化问题 (10) 的最
优解的存在性,W1
TD(s)的相对阶次应等于 e
P(s)
相对阶次.因此,保持相对阶次指标 k的最小阶次加
权函数为
W1
TD(s) = δ
MTD(s), MTD (s) =
k
X
i=0
cisi(24)
系数 δci(i= 0,1,· · · , k)必须满足式 (17).
了尽量减小设计的保守性,这些系数的选择应使
|W1
TD(jω)|在高频 (截止频率以上的频段)上尽量
E(ω). WTD(s)的选择过程中,除了阶次条件
k和鲁棒稳定条件 (17) 之外,其频率特性的成形也
对干扰观测器的干扰抑制性能有很重要的作.
决定反馈系统的保守性,因此在截止频率附近尽量
|W1
TD(jω)|靠近 E(ω), 具体选取方法在第 3.2
和第 4节中通过几个实例说明.
2.3.2 W
W
WSD(s
s
s)的选取
WTD(s)选择为多项式函数的情况下,Q(s)
的阶次 n和内模阶qWSD(s)决定.(10)
次优解的阶次等于 WSD(s)e
P(s)的阶次,n
=nW+k(标准问题的广义受控对象的阶次).
次优解通过 γ最大化逼近最优解时,其阶次减 1,
n=nW+k1. 因此,为了实现阶次指标 n,
WTD(s)的阶次 nW应选为 nW=nk+ 1.
WSD(s) = MSD(s)
NSD(s)
NSD(s) =
nk+1
X
i=0
disi, MSD(s) =
nk+1
X
j=0
fjsj(25)
由式 (23) 可知,为了实现内模阶次条件 qal=bl
(l= 0,1,· · · , q), e
K(s)应具有 q+ 1 个值为零的极
,由式 (10) 可知,e
P(s)为稳定的情况下,e
K(s)
具有 q+ 1 个值为零的极点当且仅当 WSD(s)具有 q
+ 1 个值为零的极点,:
WSD(s) = 1
sq+1 ·MSD(s)
e
NSD(s)
e
NSD(s) =
nkq
X
j=0 e
djsj(26)
如果设计指标 q选择为 m(注意:Q(s)能达到的最
高的内模阶次为 m), e
NSD(s) = e
d0,因此
WSD(s) = e
d1
0
sq+1 ·MSD(s) (27)
考虑标准 H控制问题的提前条件,加权函数不应
该含有在虚轴上的极点.因此用充分小的正数 λ >
0把式 (26) 改为
WSD(s) = 1
(s+λ)q+1 ·MSD(s) (28)
因为式 (10) 含有优化参数 γ,e
d1
0可以视为 1.
MSD(s)的系数 fi(i= 0,1,· · · , n k+ 1) 要选择
使得 W1
SD (s)的频率响应在低频 (截止频率以下的
频段)上的部分与 Q(s)的理想频率响应一致.
以 上 考 察 的 内 容 表 明,为 了 实 现 Q滤 波 器
阶次及构造约束和最优鲁棒性设计,加权函数可
以用式 (24) (28) 构 成,总 之,H范数优化
问题的加函数 WTD (s)WSD (s)MTD(s)
18 自动化学报 38
MSD(s).系数 ci(i= 0,1,· · · , q)fj(j=
0, 1,· · · , n k+ 1) 应选择使得 W1
SD (s)在低频段
的特性和 W1
TD(s)在高频段的特性分别代表灵敏度
和补灵敏度频率响应,具体选取方法显示在第 3
和第 4.
3Q
Q
Q滤波器频率响应成形的其他设计指标
的实现
用直接的方法,能细密地形成 Q滤波器频率响
,是使II 型问题评价函数的特点之一.用此
,可以实现截止频率附近的高峰幅度等设计指.
DOB 局部系统的灵敏度函数 SDOB(s)和补灵敏度
函数 TDOB(s)的高峰幅度对系统性能有直接的影响.
如系统在冲击干扰作用下的响应等:同时,对低频周
期性干扰的抑制性能也有直接的作用.因此,将幅度
最大值也列入 DOB 设计性能指标有助于实现较佳
的干扰观测器[7].下面就采用 II 型评价函数实现该
指标进行讨论.
3.1 在混合灵敏度设计中对高峰幅度与频率特性之
间关系的分析
混合灵敏度问题的目的是,通过加权传递函数
的最小化,把灵敏度函数和补灵敏度函数尽量减小
在低频带宽和高频带宽.此时,可最小化的灵敏度函
数和补灵敏度函数在低频带宽和高频带宽的幅度下
限与截止频率附近的衰减程度 (即高峰幅度)有直接
关系.假设系统补灵敏度函数为
T(s) = 1
(σs)2+ 2ξ(σs)+1 (29)
4显示 T(s)和其灵敏度函数 S(s)=1T(s)
不同衰减率 ξ的频率响应.因为 S(s)T(s)的和
1, 所以这两个传递函数不可能同时在全频域任
意减.,使用加权混合灵敏度问题,S(s)
T(s)减小在低频和高频,但不能超过其下限.
,S(s)T(s)可尽量减小的下限与其衰减率有
明显的关系.由图 4中的频率响应曲线可知,衰减率
越小,S(s)T(s)在低频和高频段的下限幅度就越
.这说明,除了截止频率附近,在其他高频和低频
,两个灵敏度函数之间的折中程度与其衰减率 ξ
有密切的关系.
在一般加权混合灵敏度问题中,通过加权函数
的调整,很容易达到需要的衰减率指标.但由式 (7)
可知,I型问题的评价函数中,其加权函数没有直
接反映此指标,其原因是在式 (7) , DOB
统的灵敏度函数和闭环系统的补灵敏度函数混在一
.然而,由式 (10) , II 型恰恰是一般加权混
合灵敏度问题的形式,因此完全符合此要求.下面考
察用加权函数实现衰减率和性能之间的矛盾折中的
实例.
4T(s)S(s)对不同衰减率的频率响应
Fig. 4 Frequency responses of S(s) and T(s) at
different damping rates
3.2 调整高峰幅度设计指标
在给定截止频率上幅度最大值指标时,通过加
权函数 WSD(s)WTD (s)的适当选择,实现类似的
补充设计指标.当然,最直接的方法是选择这些加权
函数使得满足截止频率附近的高峰幅度指标.把衰
减率 ξ < 1直接反映在加权函数中:
WSD =s2+ 2ξs + 1
(s+ 0.001)2
WTD =s2+ 2ξs + 1
1.5(30)
这样,按照 II 型问题的求解方法设计Q滤波
Q1=1.349s+ 0.7039
s3+ 1.202s2+ 1.349s+ 0.7039, ξ1= 0.2
Q2=1.458s+ 0.4608
s3+ 1.459s2+ 1.458s+ 0.4608, ξ2= 0.5
Q3=1.489s+ 0.3081
s3+ 1.906s2+ 1.489s+ 0.3081, ξ3= 0.7
(31)
5显示 SDOB (s)TDOB(s)频率响应.5
可知,(14) 的闭环系统鲁棒稳定性上限 E(ω)
图给定,对不同 ξ设计的三个结果都在 E(ω)曲线之
,说明满足闭环系统鲁棒稳定性.但其低频上的
干扰抑制性能都不同.ξ= 0.2,虽然截止频率
(1 rad/s) 上的幅度高一点,但低频带上的干扰抑制
性能比 ξ= 0.7的性能高约 4(5中显示两者相
12 dB 以上).
1期 尹正男等:保证闭环系统鲁棒稳定性的干扰观测器系统型设计方法 19
5SDOB(s)TDOB (s)对不同衰减率的频率响应
Fig. 5 Frequency responses of SDOB(s) and TDOB (s) at
different damping rates
当上限函数 E(ω) (如图 6 (a) 所示)以较小的斜
率与 0 dB 线交叉时,应注意调整加权函数的衰减率.
在此情况下,为满足鲁棒稳定性条件,高峰幅度的出
现频率只可在截止频率以.但此时的设计结果
如高峰幅度低而高频曲线靠近 E(ω)的设计结果.
了获得后者的设计结果,把加权函数选择为有实数
(ξ > 1) 的传递函数 (6 (a) W1
TD(s)
线):
WSD =1
(s+ 0.001)2
WTD =(s+ 0.3)(s+ 20)
50 (32)
按照求解算法得到的最优 Q滤波器为
Q(s) = 50s+ 47.11
s3+ 22.38s2+ 50s+ 47.11 (33)
6 (a) Q(s)的频率响应与 Q1(s), Q2(s)
Q3(s) .
棒稳定性充分条件,但其高峰幅度和频率带宽都不
.这种差异影响到低频上的干扰抑制性能.由图
6 (b) 可知,在四个设计结果中, 1 Q(s)的低频
性比其他三个好得多.
由此例可知,在加权混合灵敏度问题中截止频
率附近的高峰幅度与低频灵敏度及高频补灵敏度有
直接的关系,可通过调整加权函数直接实现此折中
指标.
4实验
以硬盘驱动器跟踪控制为例,考察本文所提出
的鲁棒 DOB 设计方法及性能,并与已有的方法进行
比较和分析.控制目的是,在模型误差 (因磁头部弹
性而发生的高频振动)和干扰 (因磁盘高速旋转而发
生的周期性干扰)存在的情况下,使磁头以给定的精
确度跟踪对应的磁轨[12].受控对象如下:
P(s) = Kω2
n1
s2+ 2ζ1ωn1s+ω2
n1
·
ω2
n2
s2+ 2ζ2ωn2s+ω2
n2
·0.5TDs+ 1
0.5TDs+ 1 (34)
该系统的第一共振频率和衰减率分别为约
600 rad/s 0.3, 第二共振频率和衰减率为约
28 000 rad/s 0.12. (34)
标称模型构成反馈控制系统,设计 DOB 系统.
Pn(s) = 2
n1
s2+ 2ζ1ωn1s+ω2
n1
(35)
(a) 鲁棒稳定性分析
(a) Robust stability analysis
(b) 低频上的干扰抑制性能比较
(b) Comparison of disturbance attenuating
performances at low frequencies
6鲁棒稳定性分析及干扰抑制性能比较
Fig. 6 Robust stability analysis and comparison of
disturbance suppression performances
因此,(2) 的乘型不确定性为 ∆(s) = P(s)/
Pn(s)1. 为了开环系统 L(s)具有穿越频率
20 自动化学报 38
3 310 rad/s 位裕55,设计相位超前控制器
C(s) = 5 850(0.00084s+ 1)
0.00011s+ 1 (36)
考虑需要的抗干扰性和不确定性 ∆(s), I 型问题评
价函数 (7) 的加权函数被选择为
WSD1(s) = 0.5(s+ 2 000)
s+ 0.1
WT(s) = 1 647.41s(s+ 1.3×104)2
(s+ 106)(s+ 5 ×104)2(37)
通过图 8的曲线,可以验不确 ∆(s)其上
限函WT(s)之间的幅度关系.根据式 (28) 和式
(29), 可获得式 (14) E(ω), 然后,定义满足式 (17)
的稳定有理函数为加权函数 WTD(s) (见图 7),
择满足 |W1
SD (jω)|<|W1
SD1(jω)|,ωWSD (s):
WTD(s) = (s+ρ)(s2+ 2ξQωQs+ω2
Q)
1012
WSD(s) = 0.5(s+ 2 000)
s+ 0.001 (38)
其中,ρ= 100, ξQ= 2, ωQ= 15 000 rad/s. 此基
础上,按照 II 型问题的求解方法设计最优Q
波器 Q(s) (见图 7):
Q(s) =
9.999 ×1011
s3+ 6.35 ×104s2+ 4.4×108s+ 9.999 ×1011
(39)
7E(ω), W1
TD(s)Q(s)的频率响应
Fig. 7 Frequency responses of E(ω), W1
TD(s), and Q(s)
8采用式 (39) 的闭环系统鲁棒稳定性分析
Fig. 8 Robust stability analysis of closed-loop
system using (39)
由图 8,设计结果完全满足整体闭环系统
的鲁棒稳定性.在此情况下,进行低频上的干扰抑制
性能的分析.9显示了 1Q(s)的频率响应,
与文献 [12] 的结果 (图中 1QW(s)的曲线)相比
.由图中的曲线可知,2×103rad/s 以下的频
带上, 1 Q(s)的幅度比 1QW(s)6 dB 以上,
意味着,在同一个阶次和相对阶次的情况下,本文所
提出的设计方法提供了干扰抑制性能比先前结果高
4倍的最优干扰观测器.
9(39) QW(s)之间低频干扰抑制性能的比较
Fig. 9 Comparison of disturbance attenuating
performances in low frequencies between (39) and QW(s)
下面再考察用 ρ= 2 000, ξQ= 0.4 (<1
3.2 节的第一个类), ωQ= 2 000 rad/s 的设计.
,加权函数为
WTD(s) =
s3+ 3.6×103s2+ 7.2×106s+ 8 ×109
4.5×1010
(40)
此函数频率带宽比上面的加权函数还窄 (见图 10),
1期 尹正男等:保证闭环系统鲁棒稳定性的干扰观测器系统型设计方法 21
而且,因衰减率小而高峰幅度,属于第 3.2 节的第
一个类型.采用式 (32), 设计 Q滤波器的结果为
Q(s) =
4.5×1010
s3+ 7.2×103s2+ 2.5×107s+ 4.5×1010
(41)
由图 11 ,其也满足闭环系统鲁棒稳定性.
12 显示此结果的干扰抑制性能曲线,,
2 000 rad/s 的频率以下, 1 Q(s)的幅1
QW(s)还小,但比式 (39) 的结果 (3.2 节的第二
类型)还大.说明,3.2 节所提出的 ξ > 1时的
加权函数选择方法 ((30)) 适合于此控制问题,
此可看出,本文给出的干扰观测器设计方法不仅提
供系统性的鲁棒最优化求解算法,而且能提供更多
的设计自由度,以满足更多的重要设计指标.
10 低频干扰抑制性能的比较
Fig. 10 Comparison of disturbance attenuating
performances at low frequencies
11 采用式 (41) 的闭环系统鲁棒稳定性分析
Fig. 11 Robust stability analysis of closed-loop
system using (41)
12 (41) QW(s)之间低频干扰抑制性能的比较
Fig. 12 Comparison of disturbance attenuating
performances at low frequencies between (41) and QW(s)
5结论
本文研究了兼顾闭环系统鲁棒稳定性的干扰观
测器系统性设计方法,提出了满足整体闭环系统
棒稳定性的一种充分条件,把它融入标准 H
观测器设计时对加权函数选取过程中.就把能
虑到整体反馈系统鲁棒稳定性和提供系统性求解算
法的设计方法结合起来,给出了保证闭环鲁棒稳定
性的系统性设计方.提出了反映鲁棒性能和阶
条件及结构约束信息的频率加权函数的选取方法.
此外,提供加权函数选定的自由,以反映任意给定
Q滤波器截止频率上的幅度设计指标,有助于
实现高峰幅度与低频干扰抑制性能之间的最佳折中,
给出的设计实验结果验证了所提方法的有效性和性
.
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尹正男 上海交通大学自动化系博士研
究生.1998 年在朝鲜金日成综合大学
自动化系获硕士学位.主要研究方向为
机器人运动控制和鲁棒控制.本文通
作者. E-mail: yjn2914@yahoo.cn
(YIN Zheng-Nan Ph. D. candi-
date in the Department of Automation,
Shanghai Jiao Tong University. He re-
ceived his bachelor and master degrees from Kim II Sung
University, D. P. R. Korea in 1998. His research interest
covers robot motion control and robust control. Corre-
sponding author of this paper.)
苏剑波 上海交通大学自动化系教授.
主要研究方向为机器人控制,多传感
信息融合,机器学习与人机交互.
E-mail: jbsu@sjtu.edu.cn
(SU Jian-Bo Professor in the De-
partment of Automation, Shanghai Jiao
Tong University. His research interest
covers robot control, multi-sensor fu-
sion, machine learning, and human-computer interaction.)
高秀行 朝鲜金策工业综合大学自动化
系副教授.主要研究方向为机器人控制,
智能控制及其应用.
E-mail: ksh616@126.com
(GAO Xiu-Xing Associate profes-
sor in the Department of Automa-
tion, Kim Chaek Industrial University,
D. P. R. of Korea. His research interest
covers robot control, intelligent control and its application.)
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