Yapıların optimal kontrolu
Optimal control of structures
- Tez No: 100686
- Danışmanlar: PROF.DR. MEHMET BAKİOĞLU
- Tez Türü: Doktora
- Konular: İnşaat Mühendisliği, Civil Engineering
- Anahtar Kelimeler: Aktif denetim, Binalar, Optimum denetim, Active control, Buildings, Optimum control
- Yıl: 1999
- Dil: Türkçe
- Üniversite: İstanbul Teknik Üniversitesi
- Enstitü: Fen Bilimleri Enstitüsü
- Ana Bilim Dalı: Belirtilmemiş.
- Bilim Dalı: Belirtilmemiş.
- Sayfa Sayısı: Belirtilmemiş.
Özet
YAPILARIN OPTIMAL KONTROLÜ ÖZET Yapıların aktif kontrolü ile ilgili yapılan çalışmaların büyük bir kısmı, optimalliğin gerekli koşullarından yararlanarak entegral formda tanımlanmış kuadratik amaç fonksiyonunun minimizasyonundan elde edilen ve lineer regülatör probleminin bir tür uygulaması olan klasik aktif kontrol algoritmalarına dayanmaktadır. Bu algoritmalardan sadece yaklaşık optimal olan kapalı-çevrim kontrolü yapılara uygulanabilmektedir. Kapalı-açık ve açık-çevrim kontrolleri ise depremin önceden bilinmesini gerektirdiklerinden yapı kontrolüne uygulanamamaktadırlar. Bu çalışmada yapı kontrolü için tanımlanan lineer regülatör probleminin analitik çözümü V. F. Krotov tarafından verilen optimalliğin yeterli koşullarından yararlanarak elde edilmiştir. Ayrıca kapalı-açık çevrim kontrol algoritmasının yaklaşık çözümü depremin yakın-gelecek ivme değerleri Kalman filtresi tekniği ile tahmin edilerek elde edilmiştir. Sayısal uygulama olarak sönümsüz üç ve sönümlü beş katlı iki binanın El Centro depremi altındaki davranışları incelenmiş ve depremin yakın gelecek ivme değerleri hassas olarak tahmin edilebildiği takdirde optimal çözüme yaklaşılabileceği gösterilmiştir. Daha sonra lineer regülatör problemi kararlılık mertebesi kavramı dahil edilerek çözülmüş ve kararlılık mertebesi ile tahmin edilen deprem ivme değerleri arasındaki ilişki sayısal olarak incelenmiştir. Klasik kontrol algoritmalarının mevcut eksikliğine ikinci bir alternatif olarak çok noktalı kuadratik ani amaç fonksiyonu önerilmiş ve Lagrange yöntemiyle elde edilen algoritma üç katlı sönümsüz yapıya uygulanarak sonuçlar karşılaştırılmıştır. Ayrıca, üzerine pasif ayarlı kütlesel sönümleyici yerleştirilmiş bir esnek deniz yapışma bir geminin yanaşma problemi Wilson- 0 yöntemiyle lineer ve lineer olmayan durumlar için sayısal olarak incelenmiş ve kütlesel sönümleyicinin sisteme olan etkisi enerji dağılımı hesaplanarak araştırılmıştır.
Özet (Çeviri)
OPTIMAL CONTROL OF STRUCTURES SUMMARY In recent decades, many advances have been made in modern control theory and their applications to engineering systems. Since the pioneer works of J. T. P. Yao based on the control theory, considerable progress has been made in the applications of the structural control. The optimal control problem for the structures which is given by T. T. Soong and the other authors can be solved by using the classical method (The method of calculus of variations), Pontryagin's maximum principle or the method of dynamic programming of R. Bellman. A great number of the researches presented previously are based on the classical active control algorithms, which are the applications of the regulator problem. These algorithms which are based on the minimization of a quadratic performance index lead to the Riccati type differential equations. Optimal regulator is derived generally based on the necessary conditions of optimality. Depending on whether the algorithm depends on the state of the system or directly on the forcing excitation it may be classified as closed-loop, open-loop or closed-open loop. Among these, only classical closed-loop control is applicable to structural control problems. However, since the Riccati equation is obtained by ignoring the earthquake excitation term classical closed-loop control is approximately optimal and does not satisfy the optimality condition. On the other hand, even though the classical closed-open loop and open- loop control algorithms are superior to the closed-loop control, since they require the whole knowledge of the earthquake acceleration history they are not applicable to earthquake-excited structures. Therefore, it is almost impossible to find the optimal control exactly for the structures under earthquake and wind forces and in terms of application in control of structures, none of these methods can be considered as optimal control. In an attempt to resolve these problems, in Sections 2-4 of this thesis analytical solution of the linear regulator problem for structural control is derived based on the sufficient conditions of optimality given by V. F. Krotov and then the solution of the closed-open loop control is carried out based on the numerical solution with the prediction of near-future earthquake excitation using the Kalman filtering technique. The summary of the work done in Sections 2-4 is given below. Introducing a 2N-dimensional state vector, Z(t), as followsthe equation of motion of a shear-beam lumped mass linear building structure under the one-dimensional earthquake ground acceleration and the control can be described in state-space form as in which X(Y)=( x\, X2,-, x“ )T is the «-dimensional response vector denoting the relative displacement of the each story unit with respect to the ground; M is the (nxn)-dimensional diagonal constant mass matrix with diagonal elements mt = mass of / th story (z-1, 2,..., n); C and K are (nxn)-dimensional viscous damping and stiffness matrices, respectively; V=(1,...,1)T is the «-dimensional vector; L is the (nxr)-dimensional location matrix of r controllers (V and L indicate the locations of the earthquake excitation terms and the controllers in the matrix equation, respectively); \J(t) is the r-dimensional active control force vector and scalar function fit) is the one-dimensional earthquake acceleration. In obtaining the optimal control law; the classical quadratic performance measure is minimized under the constraints imposed by Eq. 2, where [0, t{\ time interval is the control time and defined to be longer than that of the external excitation duration; Q is a (2nx2n) dimensional positive semidefinite symmetric weighting matrix and R is a (rxr) dimensional positive definite symmetric weighting matrix. Numerical values for the elements of Q and R matrices are assigned according to the relative importance of the state variables and the control forces in the minimization procedure in order to adjust the power requirements in the actuators. The sufficient conditions, which are used for the solution of the optimal control problem given by the Eqs. 2 and 4 are defined by using a definite function !f(/, Z) which is a generalization of Bellman function. For a given function n(t, Z)e Q(cr), a=[0, ?i]xR2”, the following functions can be defined: where XllIf (Z (0,U (/.)) is an admissible process and if there exists at least one function such that then we say that the process (Z (i),XJ (t)) satisfies the extended maximum principle. The function f{t,Z) in the definition of extended maximum principle is selected as where 2n-dimensional column vector q(t) and (2nx2n)-dimensional symmetric matrix P(/) are considered as unknowns. Since our performance index is quadratic, the function f{t,Z) is selected in this form. Using the given function ¥(t,Z) and the sufficient conditions given by Eqs. 7 and 8, the optimal control force is obtained as where P(t) and q(f) are the solutions of the following equations: and Upon substituting the vector function v* (t, Z) given by Eq. 10 into Eq. 2 in place of U, the equation of the structure under optimal control is obtained as Backward integration of Eq. 12 requires the complete knowledge of the earthquake excitation in the whole control interval. Since it is impossible to know the earthquake Xlllexcitation a priori the closed-open loop control can not be applied to the structural control. So, forward integration of Eq. 12, instead of backward, is calculated by using the few steps ahead predictions of the earthquake excitation. This prediction is called near-future prediction. Eqs. 11, 12 and Eq. 13 can be put into a similar form of the first-order matrix differential equation, which is given below: When D is constant, the expression of S* can be given by the following simple form: k I Here, Gj+i is given by As seen in Eq. 17, since i>j, Sj values at any pointy are dependent on the c, terms that are completely beyond the point j and include the unknown excitation terms. These terms will be predicted by using their known values in the past. Besides that, if the norm of the matrix G (j-i<0), (2-norm), decreases sharply with the increasing i, the prediction of the first few excitation terms will be sufficient to obtain solutions close enough to optimal solution. The excitation terms in the more distant future do not effect the results significantly. Eq. 12 can be solved approximately using this property. Kalman filtering technique is used for the prediction of several time steps ahead earthquake excitation. The effectiveness of the prediction algorithm is confirmed by comparing the predicted and observed time histories of NS component of the 1940 El Centro earthquake. For one step ahead prediction, the observed and the predicted acceleration values follow each other closely which indicates the efficiency of the method. But, it is an expected result that the prediction sensitivity decreases with the time step increase. Therefore, the prediction algorithm can predict the seismic excitations in the near future with high accuracy, although it lacks accuracy in predicting them in the more distant future. As is well known, AR model orders for earthquake ground motion are generally selected between two and five. Thus, the earthquake ground motion is modeled as a second order AR process. Namely, only two acceleration values in the past are used for the prediction of future values. XIVIn Section 5, to demonstrate the effectiveness of the algorithm, firstly undamped vibration of a three-story building and then damped vibration of a five-story building subject to seismic forcing is investigated. Numerical results of the proposed algorithm for the closed-open loop control are compared with the closed-open loop control in which the earthquake excitation is known a priori and the closed-loop control. To show the efficiency of the approximate solution, the relative displacements, interstory displacements, maximum control forces and the performance measures are compared. In the first example, it is shown that the results of the approximate solutions of the closed-open loop control are better than the results of the closed-loop control. It is also shown that the approximate solution for the closed-open loop control gives better results than those of the instantaneous optimal control for the investigated problem. In the second example, similar results are obtained and it is shown that as we predict the more distant future accurately we can obtain results close to those of the closed-open loop in which the earthquake is assumed to be known. In section 6, the solution of the linear regulator problem is investigated by taking into account the concept of stability. Using the transformations the linear regulator problem given by (2) and (4) are obtained as where A=A+a I and a is the stability order. Using the sufficient conditions of optimality, the problem given by (20) and (21) is solved and the optimal closed-open loop control force is found as where and P are the solutions of the following equations;where A= A+2a I. Using these results, undamped behavior of the three-story structure is investigated for different values of a and the relation between a and the number of the predicted earthquake excitation values is discussed. In Section 7, to overcome the prementioned deficiency of the classical active control algorithms a new multipoint quadratic time dependent performance index that has the form is proposed as a second alternative. In above expression, Qk's can be treated as the penalty coefficients of Z(t+k& ). For sake of simplicity, necessary derivations will be made by considering three-point instantaneous functional instead of m points. Three- point instantaneous functional can be expressed as follows: The method of Lagrange multipliers is used to find the active control for the proposed performance index. First, an active control algorithm including near-future states Z{t+q ) and Z(M-28 ) and then some approximations of this algorithm are found in the form of closed-loop. The effectiveness of the proposed algorithm is demonstrated numerically in the case of real-time control of a three-story structure subject to earthquake excitation. For the case of earthquake excited structure, displacement and the control forces are compared with those of the structure under the instantaneous optimal control and uncontrolled structure for different cases. It is also shown that the instantaneous optimal control algorithm is a special case of the proposed algorithm. In section 8, a passive tuned mass damper is placed on an offshore structure and the berthing problem of a ship to this flexible marine structure is investigated. Tuning the damper vibration frequency to the vibration frequencies of the system, the effect of the damper on the system behavior is studied and the energy distribution is calculated. XVI
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