طراحي كنترلر فعال بهينه ي سازه ها با استفاده از الگوريتم تكامل تفاضلي

Design of optimal controller of structures: Differential evolution algorithm

از مهمترين اهداف روش هاي كنترل بهينه ي سازه ها، دستيابي به حد مطلوب ارتعاشات با صرف انرژي كنترلي حداقل است. در دهه هاي اخير الگوريتم هاي كنترلي بسياري پيشنهاد شده اند. بخش عمده اي از اين روش ها از طريق بهينه كردن يك شاخص عملكرد مرتبه ي دوم اقدام به محاسبه ي نيروهاي كنترل مي نمايند. وجود فرضيات ساده كننده در فرمول بندي، محدوديت روش هاي بهينه سازي مورد استفاده در بهينه سازي شاخص عملكرد و همچنين عدم ملاحظه ي اثر تحريك خارجي در محاسبه ي نيروهاي كنترل، عملاً حصول جواب بهينه براي مساله ي كنترل را دور از دسترس مي سازد. در اين پژوهش الگوريتم كنترلي ارائه مي شود كه با بهره گيري از روش بهينه سازي فراكاوشي تكامل تفاضلي به عنوان يك روش نوين و توانمند كه تاكنون بطور گسترده اي در كنترل سازه ها مورد استفاده قرار نگرفته، شاخص عملكرد سنتي را بهينه مي نمايد. عناصر ماتريس بهره ي كنترل توسط روش تكامل تفاضلي به نحوي در فضاي مساله جستجو مي شوند تا منجر به كمينه شدن شاخص عملكرد گردند. روند تكراري الگوريتم پيشنهادي و عدم نياز به حل معادله ي ديفرانسيل ريكاتي، ملاحظه ي اثر تحريكات زلزله را در محاسبه ي نيروهاي كنترلي ممكن مي سازد. نتايج حاصل از روش مذكور در سازه هاي نمونه تحت تاثير ركوردهاي زلزله ي مختلف بيانگر كاهش پاسخ ها و نيروهاي كنترلي مورد نياز نسبت به تنظيم كننده ي خطي مرتبه ي دوم LQR است. همچنين، ماكزيمم نيروي كنترل و ماكزيمم پاسخ تغييرمكان و نه شاخص بنچ مارك كه در سازه هاي كنترل شده سنجيده مي شود نيز محاسبه و با روش LQR مقايسه شده اند. كاهش اين شاخص ها در حالت اعمال روش پيشنهادي نسبت به روش LQR نيز بيانگر كارايي روش ارائه شده در كنترل بهينه ي ارتعاشات سازه ها مي باشد.

One of the most important goals of optimal control of structures is achieving the desired reduction in responses using minimal control forces. Regarding many researches conducted in the field of active control, several control algorithms have been presented over the past few decades. Most of these researches calculate the required control forces by optimizing a second-order performance index. There exist simplifying assumptions in formulation of these classic algorithms and constraints in mathematical optimization techniques that have been used in optimizing the performance index. For example, because of unknown nature of earthquakes, the LQR classic controller cannot consider the external forces -as earthquake excitation- in calculation of control signal. This may make difficulties in finding the optimal solution for optimization problem. Metaheuristic optimization methods, such as differential evolution, are modern algorithms and because of their special capabilities in finding global optima are powerful tools that can be used in solving complex problems. Despite of many advantages, these methods has not been used extensively for solving civil engineering problems, especially in the field of active control of structures. In this paper active control of structures is considered as an optimization problem and a controller is proposed. The controller uses the differential evolution metaheuristic algorithm for finding gain matrix elements of active control problem. The gain matrix elements are globally searched by differential evolution algorithm to minimize the LQR performance index. The proposed method is repetitive and does not need to solve the Riccati differential equation. Therefore, it is possible to consider the effect of external excitation in finding the gain matrix and calculation of control signal. The controller is applied on sample 2DOF and 10DOF structures. Responses of these structures under several excitations from the historical earthquake records are obtained by MATLAB programming. In addition to the performance index, the maximum control force, maximum displacement and 9 benchmark indexes -previously measured in controlled structures- are calculated in this study. These indexes represent the reduction of controlled maximum and average responses of structure in comparison with uncontrolled responses. In order to evaluate the effectiveness of the proposed controller, these 9 performance indexes are calculated for 2DOF and 10DOF examples against 7 historical earthquakes and are compared for proposed and LQR controller. The simulation results indicate that the proposed method is effective in keeping the controlled responses of structures in desired range. This is also efficient in reducing the vibrations of structures with lower need to control the amount of energy in comparison with LQR algorithm. Because of the great capabilities of DE algorithm in searching large spaces and due to the iterative nature of controller, it considers the effects of external forces in control process. Numerical simulation shows that performance of the presented control algorithm is better than the LQR controller in finding the optimal displacements and control forces. Therefore, metaheuristic algorithms such as differential evolution can be used in active control of structures to achieve more efficient results in comparison with classic controllers.