عنوان مقاله
کنترل معادله ریکاتی وابسته به حالت (SDRE)
فهرست مطالب
مقدمه
قوانین غیر خطی SDRE
موجودیت راه حل
تحلیل های ثبات
قابلیت ها و هنر طراحی SDRE
مسائلی برای بررسی
بخشی از مقاله
پایداری مجانبی جهانی
پایداری مجانبی جهانی سیستم حلقه بسته نشان میدهد که غیرممکن است که حالت های مبدأ را بدون توجه به موقعیت های اولیه تنظیم کنیم. و این مسلما یک خصوصیت مورد خواستار می باشد، هرچند که معمولا مشکل می باشد که آنرا انجام دهیم یا ثابت کنیم. بعلت طبیعت فرمولبندی LQR، تحت فرضیه 1، مبدأ سیستم کنترل شده SDRE از لحاظ موضعی پایدار است، یعنی اینکه، همه مقادیر ویژه ماتریس دینامیک حلقه بسته 6 قسمت های واقعی منفی در X=0 دارند.
تست پایداری برای برآورد ناحیه کشش
بعنوان یک راه چاره برای پایداری مجانبی جهانی بهتر است که قادر باشیم تا ناحیه کشش را برای پایداری مجانبی برآورد کنیم. اخیرا، مک کافری و بانکز، یک تست پایداری را برای تعیین اندازه ناحیه ای پیشنهاد کردند که در آن، پایداری مجانبی بزرگ-مقیاسی در الگوریتم SDRE وجود دارد. تست برآیند نیازمند ارزیابی نابرابری بین خط سیرهای سیستم دینامیکی هامیلتون، بدون نیاز به یافتن تابع ارزش می باشد.
کلمات کلیدی:
State-Dependent Riccati Equation (SDRE) Control: A SurveyTayfun Çimen** ROKETSAN Missiles Industries Inc., Ankara 06780Turkey (Tel: +90538-316-1356; e-mail: tcimen@roketsan.com.tr).Abstract: Since the mid-90’s, State-Dependent Riccati Equation (SDRE) strategies have emerged asgeneral design methods that provide a systematic and effective means of designing nonlinear controllers,observers, and filters. These methods overcome many of the difficulties and shortcomings of existingmethodologies, and deliver computationally simple algorithms that have been highly effective in a varietyof practical and meaningful applications. In a special session at the 17th IFAC Symposium on AutomaticControl in Aerospace 2007, theoreticians and practitioners in this area of research were brought together todiscuss and present SDRE-based design methodologies as well as review the supporting theory. It becameevident that the number of successful simulation, experimental and practical real-world applications ofSDRE control have outpaced the available theoretical results. This paper reviews the theory developed todate on SDRE nonlinear regulation for solving nonlinear optimal control problems, and discusses issuesthat are still open for investigation. Existence of solutions as well as stability and optimality propertiesassociated with SDRE controllers are the main contribution in the paper. The capabilities, design flexibilityand art of systematically carrying out an effective SDRE design are also emphasized.1. INTRODUCTIONDuring the 1950’s and 1960’s, aerospace engineeringapplications greatly stimulated the development of optimalcontrol theory, where the objective was to drive the systemstates in such a way that some defined cost function isminimized. This turned out to have very useful applicationsin the design of regulators (where some steady state is to bemaintained) and in tracking control strategies (where somepredetermined state trajectory is to be followed). Among suchapplications was the problem of optimal flight trajectories foraircraft and space vehicles. Linear optimal control theory, inparticular, has been very well documented and widelyapplied, where the plant that is controlled is assumed linearand the feedback controller is constrained to be linear withrespect to its input. In recent years, however, the availabilityof powerful low-cost microprocessors has spurred greatadvantages in the theory and applications of nonlinearcontrol. The competitive era of rapid technological changeand aerospace exploration now demands stringent accuracyand cost requirements in nonlinear control systems. This hasmotivated the rapid development of nonlinear control theoryfor application to challenging complex dynamical real-worldproblems, particularly those that bear major practicalsignificance in the aerospace, marine and defense industries.Despite recent advances, however, there remain manyunsolved problems, so much so that practitioners oftencomplain about the inapplicability of contemporary theories.For example, most of the techniques developed have verylimited applicability because of the strong conditionsimposed on the system. Control system designers continue tostrive for control algorithms that are systematic, simple, andyet optimize performance, providing tradeoffs betweencontrol effort and state errors.The State-Dependent Riccati Equation (SDRE) strategy iswell-known and has become very popular within the controlcommunity over the last decade, providing a very effectivealgorithm for synthesizing nonlinear feedback controls byallowing nonlinearities in the system states while additionallyoffering great design flexibility through state-dependentweighting matrices. This method, first proposed by Pearson(1962) and later expanded by Wernli & Cook (1975), wasindependently studied by Mracek & Cloutier (1998) andalluded to by Friedland (1996). The method entailsfactorization (that is, parameterization) of the nonlineardynamics into the state vector and the product of a matrixvaluedfunction that depends on the state itself. In doing so,the SDRE algorithm fully captures the nonlinearities of thesystem, bringing the nonlinear system to a (nonunique) linearstructure having state-dependent coefficient (SDC) matrices,and minimizing a nonlinear performance index having aquadratic-like structure. An algebraic Riccati equation (ARE)using the SDC matrices is then solved on-line to give thesuboptimum control law. The coefficients of this equationvary with the given point in state space. The algorithm thusinvolves solving, at a given point in state space, an algebraicstate-dependent Riccati equation, or SDRE. Thenonuniqueness of the parameterization creates extra degreesof freedom, which can be used to enhance controllerperformance. In Cloutier, D’Souza & Mracek (1996) andMracek & Cloutier (1998) it is shown that the SDREfeedback scheme for the infinite-time nonlinear optimalcontrol problem (with control terms that appear affine in thedynamics and quadratically in the cost) in the multivariablecase is locally asymptotically stable and locallyasymptotically optimal, and in the scalar case is optimal. It isalso shown in the general multivariable case that thePontryagin necessary conditions for optimality are satisfiedasymptotically by the algorithm.Proceedings of the 17th World CongressThe International Federation of Automatic ControlSeoul, Korea, July 6-11, 2008978-1-1234-7890-2/08/$20.00 © 2008 IFAC 3761 10.3182/20080706-5-KR-1001.3557
