﻿ 控制输入受限的准全控制利用率有限时间稳定控制
 控制与决策  2020, Vol. 35 Issue (5): 1039-1051 0

### 引用本文 [复制中英文]

[复制中文]
PU Ming, YUAN Jian-ying. Quasi-complete control utilization finite time stable control with input constraint[J]. Control and Decision, 2020, 35(5): 1039-1051. DOI: 10.13195/j.kzyjc.2018.1269.
[复制英文]

### 文章历史

Quasi-complete control utilization finite time stable control with input constraint
PU Ming , YUAN Jian-ying
Automation Engineering College, Chengdu University of Information Technology, Chengdu 610225, China
Abstract: For nonlinear systems with input constraint, it is proved that the convergence speed of state under the action of a finite time stable controller (FTSC) is faster than that under the action of a non-FTSC at any point in state space. Sufficient conditions in six cases are summarized to guarantee the conclusion and it is proved that these conditions are attainable with appropriate parameter selection. Then, the definition of a complete control utilization controller (CCUC) is proposed, and it is proved that the convergence speed under the action of the FTSC is faster than that under the action of fast FTSC with input constraint. A series of conclusions concerning the parameter optimization are proposed to improve the control performance. Numerical examples, simulations, figures and tables are used to illustrate the proposed theory and make the conclusions more understandable. The proposed conclusions are fit for the improvement of nearly all kinds of FTSC methods, such as terminal sliding mode control, adding one power integrator control and finite time stable backstepping control.
Keywords: input constraint    finite time stable control    control utilization ratio    nonlinear control systems    terminal sliding mode control    adding one power integrator control    finite time stable backstepping control
0 引言

1 问题陈述和预备知识

 (1)

 (2)

2 控制输入受限下FTSC必优于NFTSC

 (3)

t ≥ 0的时刻, 均要求控制器不超过允许的最大值.因此得到如下的第2个约束:

 (4)

 (5)

u1(x)中的可选参数有4个, 当k3(或k2)为负时, 虽然在约束(3)下也可满足控制系统的稳定性要求, 但不一定满足约束(4), 例如下例.

 图 1 例1控制器曲线

 (6)

k2 > 0, 根据约束(3)必有

 (7)

 (8)

 (9)

 (10)

 (11)

 (12)

 (13)

, 可得

 (14)

 (15)

l2 - 2 < l3 - 2, 所以由引理2有xl2 - 2 > xl3 - 2, ∀ x ∈ (0, x0).若

 (16)

0, 从而得到结论Θ (x) > 0, x ∈ (0, x0).

Θ (x)状态变化示意如图 2所示.

 图 2 情况4中Θ (x)曲线

 (17)

1) 当x = 1时.此时

2) 当x < 1时.由引理2可知xl2 > x, x0l2 < x0, 所以由式(17)有下式成立:

 (18)

Θ (x) > 0.

3) 当1 < x < x0时.式(17)的导数为

 (19)

= 0, 可解出

 (20)

 图 3 Θ(x)的3种可能曲线

 (21)

 (22)

x > 1(对应于曲线A), 则有

 (23)

 (24)

1) 当x = 1时, 有

 (25)

 (26)

2) 当x < 1或1 < x < x0时, 有

 (27)

f(0) = 1/x0l3.根据洛必达法则, 有

 (28)

 (29)

 (30)

 (31)

k3必满足式(27).式(31)右边是一常数, 较式(27)更便于k3的选择.

x0 = 1代入上式有g(1) = 0.再求上式关于x0的偏导, 得

3 准全控制利用率控制器设计

 (32)

 (33)

 图 4 控制利用率曲线

r2说明在输出状态x0和最大控制量umax给定的受限情况下, 不可能通过改变k1值的方法提高r2.这与控制输入不受限的情况是不同的.

 (34)

r1 = 61.33 %, 因此, r1r2提高了11.33 %.

2) 初始条件和l2k2k3取值不变, 而

r1 = 58.33 %.该方案下, 尽管r1r2提高了8.33 %, 但比方案1)中提高值要低.这是因为在(0, 1)段, u1xl3项的控制能力弱于xl2项的控制能力, 且l3越大其控制能力越弱.同理, 若l2l3不变, 而增大k3减小k2, 也会使得r1下降.

 (35)

2) 式(35)也可记为

3) 全控制利用率FTSC控制器为

 (36)

4) 令u11 (x) = - k21 xl2, k21 x0l2 = umax, u12 (x) = - k22 xl2 - k3 xl3, k3 > 0, k22 x0l2 + k3 x0l3 = umax.取比较函数为

 (37)

x0 < 1, 则

x0 > 1, 则,

4 仿真研究

 图 5 example 1 ~ example 6状态轨迹
 图 6 example 1 ~ example 6控制曲线

1) 根据表 1所得的控制器参数, 使得6种FTSC输出总是小于最大允许值umax, 表明了表 1中所得约束条件的正确性.

2) 图 5中实际状态收敛时间与表 2中根据式(32)理论计算所得的收敛时间完全一致, 表明了该计算式正确.

3) FTSC的收敛时间总是小于NFTSC无穷大的收敛时间; FTSC的控制利用率r1总是高于NFTSC的50 %控制利用率.

4) 将情况1 ~情况3视为第1个对照组, 情况4 ~情况6视为第2个对照组, 随着l2的减小, 这两个对照组中控制利用率r1均上升, 收敛时间t1均下降.例如:情况3相对于情况1, r1提升了30.40 %, t1减小了85.33 %; 情况6相对于情况4, r1提升了88.12 %, t1减小了89.79 %.

5) 将情况1和情况4视为第1个对照组; 情况2和情况5视为第2个对照组; 情况3和情况6视为第3个对照组.每组中的两个l2l3的取值都相同, 但每组中后者的k3小于前者的k3.这3组均表明, 较小的k3对应着较大的控制利用率r1和较小的收敛时间t1.例如:情况4相对于情况1, r1提升了3.44 %, t1减小了8.24 %; 情况5相对于情况2, r1提升了20.00 %, t1减小了26.06 %; 情况6相对于情况3, r1提升了49.22 %, t1减小了36.16 %.

6) example 6中, l2 = 0.01, 在该准全控制利用率控制器作用下, r1=99.01 %, 接近于100 %, 而t1=0.404 1 s, 也接近于理论下限t1 = x0/umax = 4/10 = 0.4 s.同时, 控制器无抖振.

7) 对比情况6和情况3可知, 虽然两种情况下l2 = 0.01都为充分小的数, 但因为情况3中的k3 ≠ 0, 所以情况3中的r1远小于1.这说明准全控制利用率FTSC必须是传统的仅含终端吸引子项的控制器, 如u = -x0.01; 而不可以是含快速项的控制器, 如u = - x - x0.01.

8) 从图 6(a) ~ 图 6(c)看, 若控制器含有快速项, 则在状态靠近x0附近, 控制器曲线为一凹函数, 随着x减小迅速下降; 从图 6(d) ~ 图 6(f)看, 没有快速项的控制器在整个状态区间均为一凸函数, 所以下降较慢, 从而始终保持较大的控制量, 提高了控制利用率.

 图 7 AOPIC与改进AOPIC的控制器、滑模面及状态曲线

5 结论

 [1] Zhai J Y, Ai W Q, Fei S M. Global output feedback stabilisation for a class of uncertain nonlinear systems[J]. IET Control Theory and Applications, 2013, 7(2): 305-313. DOI:10.1049/iet-cta.2011.0505 [2] Li S H, Wu C S, Sun Z X. Design and implementation of clutch control for automative transmission using terminal sliding mode control and uncertain observer[J]. IEEE Transactions on Vehicular Technology, 2016, 65(4): 1890-1898. DOI:10.1109/TVT.2015.2433178 [3] Xu S D, Chen C C, Wu Z L. Study of nonsingular fast terminal sliding mode fault-tolerant control[J]. IEEE Transactions on Industrial Electronics, 2015, 62(6): 3906-3913. [4] 赵明元, 魏明英, 何秋茹. 基于有限时间稳定和backstepping的直接力/气动力复合控制方法[J]. 宇航学报, 2010, 31(9): 2157-2164. (Zhao M Y, Wei M Y, He Q R. Research on method of lateral jet and aerodynamic fins compound control based on finite time stability and backstepping approach[J]. Journal of Astronautics, 2010, 31(9): 2157-2164. DOI:10.3873/j.issn.1000-1328.2010.09.015) [5] Xie X J, Zhang X H, Zhang K M. Finite time state feedback stablilisation of stochastic high-order nolinear feedforward systems[J]. International Journal of Control, 2016, 89(7): 1332-1341. DOI:10.1080/00207179.2015.1129439 [6] Tian B L, Yin L P, Wang H. Finite time reentry attitude control based on adaptive multivariable disturbance compensation[J]. IEEE Transactions on Industrial Electronics, 2015, 62(9): 5889-5898. DOI:10.1109/TIE.2015.2442224 [7] Zhang R M, Wang L, Zhou Y J. On-line RNN compensated second order nonsingular terminal sliding mode control for hypersonic vehicle[J]. International Journal of Intelligent Computing and Cybernetics, 2012, 5(2): 186-205. DOI:10.1108/17563781211231534 [8] Lu P L, Gan C, Liu X D. Finite time distributed cooperative attitude control for multiple spacecraft with actuator saturation[J]. IET Control Theory and Applications, 2014, 8(18): 2186-2198. DOI:10.1049/iet-cta.2014.0147 [9] Li S H, Zhou M M, Yu X H. Design and implementation of terminal sliding mode control method for PWSM speed regulation system[J]. IEEE Transactions on Industrial Information, 2012, 9(4): 1879-1891. DOI:10.1109/tii.2012.2226896 [10] 蒲明.近空间飞行器鲁棒自适应滑模控制[D].南京: 南京航空航天大学自动化学院, 2012. (Pu M.Robust adaptive sliding mode control for near space vehicle[D].Nanjing: College of Automation Engineering, Nanjing University of Aeronatutics and Astronuatics, 2012.) [11] Barbot J P, Boutat D, Busawon K. Utility of high-order sliding mode differentiators for dynamical left inversion problems[J]. IET Control Theory and Applications, 2015, 9(4): 538-544. DOI:10.1049/iet-cta.2014.0124 [12] Chen S Y, Lin F J. Robust nonsingular terminal sliding mode control for nonlinear magnetic bearing system[J]. IEEE Transactions on Control Systems Technology, 2011, 19(3): 636-643. DOI:10.1109/TCST.2010.2050484 [13] Tao C W, Taur J S, Chan M L. Adaptive fuzzy terminal sliding mode controller for linear systems with mismatched time-varying uncertainties[J]. IEEE Transactions on Systems, Man, and Cybernetics, Part B:Cybernetics, 2004, 34(1): 255-262. DOI:10.1109/TSMCB.2003.811127 [14] 张健.不确定非线性系统全局自适应收敛控制设计[D].济南: 山东大学控制科学与工程学院, 2012. (Zhang J.Global adaptive stability control design for several classess of uncertain nonlinear systems[D].Jinan: School of Control Science and Eingineering, Shandong University, 2012.) [15] Sun Z Y, Zhang X H, Xie X J. Global continuous output-feedback stabilization for a class of high-order nonlinear systems with multiple time delays[J]. Journal of the Franklin Institute-Engineering and Applied Mathematics, 2014, 35(18): 4334-4356. [16] Xiao F, Wang L, Chen T. Finite time consensus in networks of integrator-like dynamic agents with directional link failure[J]. IEEE Transactions on Automatic Control, 2014, 59(3): 756-762. DOI:10.1109/TAC.2013.2274705 [17] Zhao Y, Duan Z S, Wen G H. Finite-time consensus for second-order multi-agent systems with saturated control protocols[J]. IET Control Theory and Applications, 2015, 9(3): 312-319. DOI:10.1049/iet-cta.2014.0061 [18] He X Y, Wang Q Y, Yu W W. Finite-time containment control for second-order multi agent systems under directed topology[J]. IEEE Transactions on Circuits and Systems, 2014, 61(8): 619-623. DOI:10.1109/TCSII.2014.2327473 [19] 蒲明, 吴庆宪, 姜长生, 等. 新型快速Terminal滑模及其在近空间飞行器上的应用[J]. 航空学报, 2011, 32(7): 1283-1291. (Pu M, Wu Q X, Jiang C S, et al. Novel fast terminal sliding mode and its application to near space vehicle[J]. Acta Aeronautica et Astronautics Sinica, 2011, 32(7): 1283-1291.) [20] 王连祥, 方德植, 张鸣镛, 等. 数学手册[M]. 北京: 高等教育出版社, 2011: 21-22. (Wang L X, Fang D Z, Zhang M Y, et al. Mathematics manual[M]. Beijing: Higher Education Press, 2011: 21-22.) [21] 薛定宇, 陈阳泉. 控制数学问题的Matlab求解[M]. 北京: 清华大学出版社, 2007: 157-174. (Xue D Y, Chen Y Q. Matlab solutions to mathematical problems in control[M]. Beijing: Tsinghua University Press, 2007: 157-174.) [22] Levant A, Li S H, Yu X H. Accuracy of some popular non-homogeneous 2-sliding modes[J]. IEEE Transactions on Automatic Control, 2013, 58(10): 2615-2619. DOI:10.1109/TAC.2013.2256674 [23] Huang X Q, Lin W, Yang B. Global finite-time stabilization of a class of uncertain nonlinear systems[J]. Automatica, 2005, 41(5): 881-888. DOI:10.1016/j.automatica.2004.11.036