宋　剛， 譚　川， 陳　果

(招商局重慶交通科研設(shè)計(jì)院有限公司橋梁工程結(jié)構(gòu)動(dòng)力學(xué)國(guó)家重點(diǎn)實(shí)驗(yàn)室，重慶 400067)

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基于輸出反饋的建筑結(jié)構(gòu)閉開(kāi)環(huán)次優(yōu)控制①

宋剛， 譚川， 陳果

(招商局重慶交通科研設(shè)計(jì)院有限公司橋梁工程結(jié)構(gòu)動(dòng)力學(xué)國(guó)家重點(diǎn)實(shí)驗(yàn)室，重慶 400067)

摘要：對(duì)傳統(tǒng)的結(jié)構(gòu)抗震閉開(kāi)環(huán)控制算法進(jìn)行改進(jìn)。基于地面運(yùn)動(dòng)自回歸模型，采用Kalman濾波利用可以量測(cè)到的地面加速度激勵(lì)對(duì)未來(lái)時(shí)段即將發(fā)生的地面加速度激勵(lì)進(jìn)行預(yù)估，并在微分方程的求解中引入精確高效的精細(xì)積分算法?？紤]到實(shí)際控制中量測(cè)全部狀態(tài)變量的困難，改進(jìn)算法僅需量測(cè)部分狀態(tài)變量。數(shù)值仿真表明，基于輸出反饋的閉開(kāi)環(huán)次優(yōu)控制策略能大大降低結(jié)構(gòu)的地震響應(yīng)。

關(guān)鍵詞：地震激勵(lì)； 輸出反饋； 閉開(kāi)環(huán)控制； Kalman濾波； 精細(xì)積分

0引言

近幾十年來(lái)控制在理論和實(shí)際工程中都獲得了很大進(jìn)展[1-3]，其中線性二次型調(diào)節(jié)器(LQR)在很多工程中得到了應(yīng)用[4-5]。在傳統(tǒng)的二次型調(diào)節(jié)器問(wèn)題中，目標(biāo)函數(shù)定義為由結(jié)構(gòu)狀態(tài)和控制力向量組成的二次表達(dá)式在一定時(shí)間區(qū)段上的積分，通過(guò)龐德里亞金極大值原理或貝爾曼動(dòng)態(tài)規(guī)劃等方法可推導(dǎo)出相應(yīng)的最優(yōu)控制器。然而，在公式推導(dǎo)過(guò)程中Riccati方程是通過(guò)忽略外激勵(lì)項(xiàng)而得出的。因此從這種意義上說(shuō)，傳統(tǒng)的二次型閉環(huán)最優(yōu)控制只是一種近似的最優(yōu)控制，在公式推導(dǎo)過(guò)程中保留外激勵(lì)項(xiàng)的閉開(kāi)環(huán)控制比閉環(huán)控制有一定的優(yōu)越性。但是傳統(tǒng)的閉開(kāi)環(huán)控制需要預(yù)先知道整個(gè)控制時(shí)間區(qū)段上的外激勵(lì)，這對(duì)結(jié)構(gòu)抗震等工程問(wèn)題來(lái)說(shuō)是無(wú)法實(shí)現(xiàn)的。地震激勵(lì)是隨機(jī)的，人們無(wú)法事先知道作用在結(jié)構(gòu)上的確切地震激勵(lì)。相應(yīng)地不需要事先知道作用在結(jié)構(gòu)上的確切外激勵(lì)的閉開(kāi)環(huán)次優(yōu)控制方法應(yīng)運(yùn)而生。文獻(xiàn)[6]基于地震自回歸模型，通過(guò)Kalman濾波預(yù)測(cè)一步或多步的地震動(dòng)輸入，采用Taylor級(jí)數(shù)進(jìn)行數(shù)值積分，提出一種次優(yōu)的結(jié)構(gòu)抗震閉開(kāi)環(huán)控制策略，并通過(guò)數(shù)值算例驗(yàn)證了該閉開(kāi)環(huán)次優(yōu)控制方法相對(duì)于其他控制方法的優(yōu)點(diǎn)。文獻(xiàn)[7]進(jìn)一步提出具有指定穩(wěn)定度的結(jié)構(gòu)閉開(kāi)環(huán)次優(yōu)控制算法。除此之外，為了避免Riccati方程的求解和確定未來(lái)時(shí)段的地震激勵(lì)，文獻(xiàn)[8]提出一種基于多點(diǎn)瞬態(tài)激勵(lì)的閉環(huán)控制。為了提高控制精度，文獻(xiàn)[9]在結(jié)構(gòu)瞬態(tài)閉環(huán)及瞬態(tài)閉開(kāi)環(huán)控制中采用了精細(xì)積分算法。

然而，以上文獻(xiàn)[6-9]都假定結(jié)構(gòu)的全部狀態(tài)可以量測(cè)，這在實(shí)際控制中往往是難以做到的。鐘萬(wàn)勰近年來(lái)提出的精細(xì)積分法求解常微分方程精度之高[10-12]，是其他時(shí)域積分法無(wú)法比擬的。本文在文獻(xiàn)[6-9]的基礎(chǔ)上，提出基于輸出反饋的結(jié)構(gòu)抗震閉開(kāi)環(huán)次優(yōu)控制策略，對(duì)微分方程的求解采用精細(xì)積分法來(lái)代替文獻(xiàn)[6]和[7]中所采用的Taylor級(jí)數(shù)展開(kāi)法。

1問(wèn)題描述

以單維地震動(dòng)輸入下的n自由度線性結(jié)構(gòu)為例。假定地面均勻一致運(yùn)動(dòng)，結(jié)構(gòu)的運(yùn)動(dòng)方程可寫(xiě)為：

(1)

(2)

其中:

1.1基于狀態(tài)反饋的閉環(huán)控制

采用性能指標(biāo)

(3)

其中:tf表示地震激勵(lì)持續(xù)時(shí)間。引入哈密頓函數(shù)

(4)

由方程

(5)

(6)

(7)

(8)

假定

(9)

聯(lián)立式(5)～式(9)，可得

(10)

(11)

聯(lián)立式(7)和(9)，可得

(12)

由式(11)解得P(t)，代入式(12)，便得到t時(shí)刻作用器應(yīng)輸出的控制力。

1.2基于狀態(tài)反饋的閉開(kāi)環(huán)控制

(13)

聯(lián)立式(5)、(6)、(7)和(13)，可得

(14)

式(14)在任何時(shí)刻均成立。聯(lián)立式(8)和式(14)，可得

(15)

(16)

聯(lián)立式(7)和(13)，可得

(17)

2基于輸出反饋的閉開(kāi)環(huán)次優(yōu)控制

式(15)為微分Riccati方程，借助于鐘萬(wàn)勰近年來(lái)提出的精細(xì)算法[10-12]，微分Riccati方程可以得到簡(jiǎn)便地解決。然而在除臨近tf的時(shí)間段外，P(t)在大部分時(shí)間段都近似保持為常數(shù)矩陣，把式(15)簡(jiǎn)化為代數(shù)Riccati方程來(lái)處理，見(jiàn)下式：

(18)

式(16)的求解需要從終端時(shí)刻進(jìn)行后向積分，這需要事先知道整個(gè)控制時(shí)間區(qū)段作用在結(jié)構(gòu)上的地面加速度激勵(lì)。雖然地面加速度激勵(lì)可以在線量測(cè)，但卻無(wú)法事先知道。采用地面運(yùn)動(dòng)自回歸模型利用Kalman濾波技術(shù)對(duì)未來(lái)時(shí)段即將發(fā)生的地面加速度激勵(lì)進(jìn)行預(yù)估[6]。

對(duì)于每一時(shí)間步[tk-1,tk]，式(16)的解可表示為

(19)

其中:

(20)

(21)

記從初始時(shí)刻到終端時(shí)刻tf的積分步數(shù)為m，由q(tf)=0，可得

(22)

(23)

利用地面運(yùn)動(dòng)自回歸模型和Kalman濾波技術(shù)進(jìn)行地面加速度激勵(lì)l步預(yù)估，即在tk時(shí)刻利用過(guò)去時(shí)段(包括現(xiàn)在時(shí)刻)量測(cè)到的地面加速度激勵(lì)估計(jì)未來(lái)時(shí)段tk+1,tk+2,…,tk+l(1≤l≤m-k)時(shí)刻的地面加速度激勵(lì)。將式(23)截取前l(fā)項(xiàng)，可得

(24)

關(guān)于截?cái)嗨鶐?lái)的誤差的討論，可見(jiàn)文獻(xiàn)[6-7]。式(17)需要量測(cè)結(jié)構(gòu)全部狀態(tài)變量，對(duì)于高階系統(tǒng)，這在很多情況下是不可實(shí)現(xiàn)的。這里假定只有部分狀態(tài)變量可測(cè)，記為y(t)：

(25)

假定

(26)

定義目標(biāo)函數(shù)

(27)

(28)

將式(24)和(28)代入式(26)中，就可得到基于輸出反饋的結(jié)構(gòu)閉開(kāi)環(huán)次優(yōu)控制。相應(yīng)地忽略式(26)中的R-1BTq(t)，將式(28)代入式(26)，即得到基于輸出反饋的結(jié)構(gòu)閉環(huán)次優(yōu)控制。

3算例

一個(gè)三層無(wú)阻尼結(jié)構(gòu)受水平方向地面加速度激勵(lì)，相鄰層間安裝主動(dòng)作用器，如圖1所示[6]。各層質(zhì)量均為48×103kg，層間剛度分別為k1=46 420 kN/m,k2=41 780 kN/m,k3=23 210 kN/m。地面加速度激勵(lì)采用El Centro地震波南北分量，地震加速度峰值為3.417 m/s2，時(shí)程記錄間距為0.02 s。假定結(jié)構(gòu)速度可測(cè)，即Cy=[0I] 。對(duì)未來(lái)時(shí)段地面加速度激勵(lì)進(jìn)行3步預(yù)估，采用基于輸出反饋的閉開(kāi)環(huán)控制策略對(duì)結(jié)構(gòu)進(jìn)行控制。取加權(quán)矩陣為

(29)

其中:μ為可調(diào)參數(shù)。不同的μ值對(duì)應(yīng)不同的控制力和控制效果，經(jīng)試算，取μ=1.0×108。

圖1　結(jié)構(gòu)模型Fig.1　Structural model

這里僅給出閉開(kāi)環(huán)控制和無(wú)控時(shí)結(jié)構(gòu)峰值響應(yīng)對(duì)比，見(jiàn)表1。關(guān)于閉開(kāi)環(huán)控制和閉環(huán)控制控制效果的詳細(xì)比較，可見(jiàn)文獻(xiàn)[6-7]。從表1可以看出，施加控制力后，結(jié)構(gòu)頂層的位移、速度、加速度峰值響應(yīng)和2～3層層間相對(duì)位移峰值響應(yīng)都大大降低。無(wú)控時(shí)，結(jié)構(gòu)頂層位移峰值為12.7 cm，施加控制力后結(jié)構(gòu)頂層位移峰值降為1.1 cm。

表 1　控制前后結(jié)構(gòu)響應(yīng)對(duì)比

4結(jié)論

針對(duì)實(shí)際控制中量測(cè)全部狀態(tài)的困難，提出一種基于輸出反饋的結(jié)構(gòu)閉開(kāi)環(huán)次優(yōu)控制算法，并對(duì)一個(gè)地震激勵(lì)下的三層無(wú)阻尼結(jié)構(gòu)進(jìn)行數(shù)值仿真。結(jié)果表明，輸出反饋閉開(kāi)環(huán)次優(yōu)控制可以很好地實(shí)現(xiàn)控制效果。

參考文獻(xiàn)(References)

[1]Housner G W,Bergman L A,Caughey T K,et al.Structural Control:Past,Present and Future[J].Journal of Engineering Mechanics (ASCE),1997,123(9):897-958.

[2]李慧,王亞楠,杜永峰.近場(chǎng)脈沖型地震動(dòng)作用下TMD-基礎(chǔ)隔震混合控制結(jié)構(gòu)的減震效果分析[J].地震工程學(xué)報(bào),2013,35(2):208-212.

LI Hui,WANG Ya-nan,Du Yongfeng.An Effectiveness Analysis on Hybrid Control System of Tuned Mass Damper-base Isolation under Near-fault Pulse-like Ground Motion[J].China Earthquake Engineering Journal,2013,35(2):208-212.(in Chinese)

[3]賈斌,羅曉群,張其林,等.粘滯阻尼器對(duì)空間結(jié)構(gòu)的振動(dòng)控制效應(yīng)[J].地震工程學(xué)報(bào),2014,36(1):39-46.

JIA Bin,LUO Xiao-qun,ZHANG Qi-lin,et al.Vibration Control Effect of Viscous Damper on Space Structure[J].China Earthquake Engineering Journal,2014,36(1):39-46.(in Chinese)

[4]Chandak A S,Patil A J.Robust LQR Control Design of Gyroscope[J].International Journal of Advanced Computer Research,2013,3(9):7-12.

[5]Kim Y M,You K P,You J Y.Active Control of Along-wind Response of a Tall Building with AMD Using LQR Controller[J].Applied Mechanics and Materials,2014,491: 1063-1067.

[6]Bakioglu M,Aldemir U.A new Numerical Algorithm for Sub-optimal Control of Earthquake Excited Linear Structures[J].International Journal for Numerical Methods in Engineering,2001,50(12):2601-2616.

[7]Aldemir U,Bakioglu M.Active Structural Control Based on the Prediction and Degree of Stability[J].Journal of Sound and Vibration,2001,247(4):561-576.

[8]Akhiev S S,Aldemir U,Bakioglu M.Multipoint Instantaneous Optimal Control of Structures[J].Computers and Structures,2002,80(11):909-917.

[9]李金橋,于建華.基于精細(xì)積分的結(jié)構(gòu)主動(dòng)最優(yōu)控制算法[J].西南交通大學(xué)學(xué)報(bào), 2004,39(1):77-81.

LI Jin-qiao,YU Jian-hua.Algorithm of Optimal Active Structural Control Based on Precise Integration[J].Journal of Southwest Jiao Tong University,2004,39(1):77-81.(in Chinese)

[10]鐘萬(wàn)勰.線性二次最優(yōu)控制的精細(xì)積分法[J].自動(dòng)化學(xué)報(bào),2001,27(2):166-173.

ZHONG Wan-xie.The Precision Integration of LQ Control Problems[J].Acta Automatica Sinica,2001,27(2):166-173.(in Chinese)

[11]鐘萬(wàn)勰.卡爾曼-布西濾波的精細(xì)積分[J].大連理工大學(xué)學(xué)報(bào),1999,39(2):504-513.

ZHONG Wan-xie.Precise Integration of Kalman-Bucy Filtering Problems[J].Journal of Dalian University of Technology,1999,39(2):504-513.(in Chinese)

[12]ZHONG Wan-xie.On Precise Integration Method[J].Journal of Computational and Applied Mathematics,2004,163(1):59-78.

Closed/Open-loop Sub-optimal Control of Structures Based

on Output Feedbacks

SONG Gang, TAN Chuan, CHEN Guo

(StateKeyLaboratoryofBridgeEngineeringStructuralDynamics,ChinaMerchantsChongqingCommunications

Research&DesignInstituteCo.Ltd.,Chongqing400067,China)

Abstract：Most recent studies have been based on the application of linear quadratic regulator control to earthquake-excited structures. In linear quadratic regulator control problems, the objective function is defined as the integral of a quadratic expression in the control interval with respect to structural states and control vectors, and the optimal regulator can be derived using Pontryagin’s maximum principle or Bellman’s method of dynamic programming. In traditional linear quadratic regulator control problems, the Riccati equation is obtained without considering the earthquake excitation term. To optimize control and satisfy the optimality condition, in this study, we propose a new closed/open-loop control strategy for structures under earthquake excitation. We derive an analytical solution to a linear regulator problem for structural control without neglecting unknown disturbances. The optimal regulator depends on both the state and disturbances. The solution for this closed/open-loop control requires the knowledge of the earthquake in the control interval, which is approximated based on the real-time prediction of near-future earthquake excitation using the Kalman filtering technique. Earthquake excitation is modeled as an autoregressive process. The prediction algorithm can predict seismic excitation in the near future with high accuracy, although it lacks prediction accuracy for more distant future events. Considering the measurement difficulty of all state variables, especially for some high-order systems, the proposed control strategy only requires the measurement of a partial state. In the calculation of a state transition matrix, which is required to solve a differential equation, large rounding errors may occur when the time-step size is excessively small. To overcome this limitation, we introduce a precise integration algorithm to solve the differential equation. This algorithm is always numerically stable and yields very high precision solutions for numerical integration problems. To demonstrate the effectiveness of the proposed control strategy, we investigated the undamped vibration of a three-story building subjected to horizontal seismic forces. We assumed that the columns of the building are massless and that the mass of the structure is concentrated at floor levels. We implemented control using actuators exerting forces on each story. We also assumed that floor velocities can be measured in real time by sensors installed in every story unit. We used the NS component of the 1940 El Centro earthquake ground acceleration record as the excitation source and performed calculations for its entire duration. We modeled the columns of the building as linear elastic springs and assumed the response mitigation effect of the actuators to be sufficient for the building to behave in a linear elastic manner during earthquake excitation. We did not consider the soil-structure interaction or the dynamic characteristics of the actuators. We investigated the controlled and uncontrolled behavior of the three-story undamped building and compared the relative displacement, velocity, acceleration, and inter-story displacement responses. Our numerical simulation results show that the proposed closed/open-loop sub-optimal output feedback control strategy can significantly reduce structural earthquake responses.

Key words：earthquake excitation; output feedback; close/open-loop control; Kalman filter; precise integration

DOI:10.3969/j.issn.1000-0844.2015.04.0933

中圖分類(lèi)號(hào):TU352.1

文獻(xiàn)標(biāo)志碼：A

文章編號(hào):1000-0844(2015)04-0933-05

作者簡(jiǎn)介：宋剛(1980-)，男，河南鎮(zhèn)平人，博士，副研究員，主要從事結(jié)構(gòu)振動(dòng)控制及安全監(jiān)測(cè)研究。E-mail: gangsong2008@163.com。

基金項(xiàng)目：交通運(yùn)輸部應(yīng)用基礎(chǔ)研究項(xiàng)目(2013319740080, 2014319740160)；交通運(yùn)輸部信息化技術(shù)研究項(xiàng)目(2013364740600)；重慶市科技人才培養(yǎng)計(jì)劃項(xiàng)目(cstc2013kjrc-qnrc30001)

收稿日期：①2014-08-20