﻿ MIMO仿射型极值搜索系统的输出反馈滑模控制
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MIMO仿射型极值搜索系统的输出反馈滑模控制

1. 北京航空航天大学 仪器科学与光电工程学院, 北京 100083;
2. 海军航空工程学院 研究生管理大队, 烟台 264001;
3. 海军航空工程学院 战略导弹工程系, 烟台 264001

Output-feedback sliding mode control for MIMO affine extremum seeking systems
ZUO Bin1 , ZHANG Lei2, LI Jing3
1. School of Instrumentation Science and Opto-electronics Engineering, Beijing University of Aeronautics and Astronautics, Beijing 100083, China;
2. Graduate Students' Brigade, Naval Aeronautical and Astronautical University, Yantai 264001, China;
3. Department of Strategic Missile Engineering, Naval Aeronautical and Astronautical University, Yantai 264001, China
Abstract: An output-feedback sliding mode control method is proposed for a class of multi-input multi-output (MIMO) affine nonlinear extremum seeking systems. Firstly, the original MIMO affine nonlinear extremum seeking system is decomposed into several single-input single-output (SISO) extremum seeking subsystems. Considering the subsystem's states are unmeasurable, the control method uses a simple ramp time function as the reference signal of the subsystem's output, constructs the sliding mode manifold by the output tracking error and the integral of the sign function of the tracking error, and designs the output-feedback extremum seeking control law with sliding mode. The stability analysis shows that the MIMO nonlinear seeking extremum system with the proposed control method is possible to achieve an arbitrarily small neighborhood of the desired optimal point under all initial conditions, and all the states in the closed-loop system remain uniformly bounded. Simulation results are presented to illustrate the effectiveness of the control method.
Key words: affine nonlinear systems     extremum seeking systems     output-feedback     sliding mode control     uniformly bounded

1 问题阐述

DΔ为关于极值点x*Δ邻域。显然,常数1(Δ)将随着Δ的变化而变化。为方便描述,1(Δ)简写为1

2 方法设计

3 控制系统的稳定性分析

 图 1 非线性极值搜索闭环控制系统框图 Fig. 1 Frame of nonlinear extremum seeking closed-loop control system

 图 2 函数S1(σ(t))和S2(σ(t))的曲线 Fig. 2 Curves of functions S1(σ(t)) and S2(σ(t))

S1(σ(t))和S2(σ(t))分别求取1阶微分,并代入式(8)和式(13),可得

2) 证明输出量在极大值*附近的振荡幅值是关于参数ε的无穷小量。

① 情况1:如果状态量i一直在邻域内进行振荡运动,则通过选取合适的2i,可以使得Δ任意小,从而满足|-*|=O(ε)。

② 情况2:如果状态量i的振荡运动会逃出邻域,然后再返回进入邻域,则就需要证明输出量在此以外的振荡范围也是关于参数ε的无穷小量。

① 如果t3>t2,则可将时间分为2个阶段:t∈[t1,t2)和t∈[t2,t3]。

t∈[t1,t2)时,切换函数σ(t)并不处于滑模面上,即<σ(t)<(k+1)ε,则|(t)|=|σ(t)-σ(t1)|=O(ε)。

t∈[t2,t3]时,切换函数σ(t)处于滑模面上,(t)=0。根据时刻t2t3的定义可知,在t∈[t2,t3]时,=kr+γ>0,则输出量(t)是单调递增运动,它将从滑模面上朝着极大值*方向运动,因此说明与(t1)距离最远的位置是(t2),而由于已证明得知|(t2)-(t1)|=O(ε),所以∀t∈[t2,t3],存在|(t)-(t1)|=O(ε)。

② 如果t2t3>t*,分析输出量t1运动到t3的情况,由于此时切换函数σ(t)不处于滑模面上,则对于∀t∈[t1,t3],输出量的运动情况可以类比于①中t∈[t1,t2)的情况,因而,可知此时|(t)-(t1)|=O(ε)。

4 仿真分析

 图 3 不存在干扰时状态量x1的仿真结果 Fig. 3 Simulation result of state x1 without interference
 图 4 不存在干扰时状态量x2的仿真结果 Fig. 4 Simulation result of state x2 without interference
 图 5 不存在干扰时输出量y1的仿真结果 Fig. 5 Simulation result of output y1 without interference
 图 6 不存在干扰时输出量y2的仿真结果 Fig. 6 Simulation result of output y2 without interference
 图 7 不存在干扰时控制输入量u1的仿真结果 Fig. 7 Simulation result of control input u1 without interference
 图 8 不存在干扰时控制输入量u2的仿真结果 Fig. 8 Simulation result of control input u2 without interference

 图 9 存在干扰时状态量x1的仿真结果 Fig. 9 Simulation result of state x1 with interference
 图 10 存在干扰时状态量x2的仿真结果 Fig. 10 Simulation result of state x2 with interference
 图 11 存在干扰时输出量y1的仿真结果 Fig. 11 Simulation result of output y1 with interference
 图 12 存在干扰时输出量y2的仿真结果 Fig. 12 Simulation result of output y2 with interference
 图 13 存在干扰时控制输入量u1的仿真结果 Fig. 13 Simulation result of control input u1 with interference
 图 14 存在干扰时控制输入量u2的仿真结果 Fig. 14 Simulation result of control input u2 with interference

5 结 论

1) 该控制方法可在任何初始条件下使系统的输出量全局收敛至其期望极值的任意小邻域内,且所有状态量均一致范数有界。

2) 该控制方法确保了切换函数在任何时刻都可以全局收敛至滑模面上,提升了控制方法的鲁棒性。

3) 该方法属于一种在线反馈控制方法,在许多状态量不易测量或者不可测量的MIMO极值搜索系统中有着广泛的应用前景。

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#### 文章信息

ZUO Bin, ZHANG Lei, LI Jing
MIMO仿射型极值搜索系统的输出反馈滑模控制
Output-feedback sliding mode control for MIMO affine extremum seeking systems

Journal of Beijing University of Aeronautics and Astronsutics, 2016, 42(4): 718-727.
http://dx.doi.org/10.13700/j.bh.1001-5965.2015.0297