241 Dynamic Systems and Contr ol Lecture 25 Synthesis Emilio razzoli Aeronautics and str onautics Massachusetts Institute of echnology Ma 11 2011 E razzoli MIT Lecture 25 Synthesis Ma 11 2011 12 brPage 2br ID: 24772 Download Pdf

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241 Dynamic Systems and Contr ol Lecture 25 Synthesis Emilio razzoli Aeronautics and str onautics Massachusetts Institute of echnology Ma 11 2011 E razzoli MIT Lecture 25 Synthesis Ma 11 2011 12 brPage 2br

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6.241 Dynamic Systems and Contr ol Lecture 25: Synthesis Emilio razzoli Aeronautics and str onautics Massachusetts Institute of echnology Ma 11, 2011 E. razzoli (MIT) Lecture 25: Synthesis Ma 11, 2011 12

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Standa rd setup Consider the follo wing system, fo Ax (0) zw zu yw yu where is an exogenous disturbance input (also reference, noise, etc.) is control inp ut, computed th controller is the erfo rmance output This is “virtual output used only fo design. is the measured output This is what is available to controller It is desired to synthesize controller (itself

dynamical system), with input and output such that the closed lo op is stabilized, and the erfo rmance output is minimized, given class of disturbance inputs. In pa rticula r, will lo ok at controller synthesis with and criteria. E. razzoli (MIT) Lecture 25: Synthesis Ma 11, 2011 12

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Optimal synthesis? In rinciple, ould lik to ﬁnd controller such that minimizes the energy gain of the closed-lo op sys tem, i.e., that minimizes zw sup =0 Ho ever, the optimal controller(s) re such that max zw is constant over all frequencies, the resp onse es not roll o at high frequencies,

and the controller is not strictly rop er. (The optimal controller is not unique.) In addition, co mpu ng an optimal controller is numerically challenging. E. razzoli (MIT) Lecture 25: Synthesis Ma 11, 2011 12

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Sub-optimal synthesis etter app roach in ractice is to pursue sub-optimal design, i.e., given 0, ﬁnd controller such that zw if one exists. In other rds, assume that the controller and the disturba nc re pla ying zero-sum game, in which the cost is what is the smallest such that the controller can win the game (i.e., achieve negative cost)? E. razzoli (MIT)

Lecture 25: Synthesis Ma 11, 2011 12

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App ro ximately optimal synthesis The optimal erfo rmance can app ro ximated rbitra rily ell, bisection metho d, main ining lo er nd upp oun ds Init to, e.g., 0, the no rm of the optimal design. Let the optimal controller. Let 2. Check whether controller exists such that zw If es, set and set to the controller just designed. Otherwise, set Rep eat from step until Return E. razzoli (MIT) Lecture 25: Synthesis Ma 11, 2011 12

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Simpliﬁed setup simplicit consider the case in hi zu 0, i.e., the cost if of the rm Qx Ru dt 0,

i.e., ro cess noise and senso noise re unco rrelated. yw zu yw zu yw E. razzoli (MIT) Lecture 25: Synthesis Ma 11, 2011 12

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ull info rmation case (intuition) Assume that the state and the disturba nc re available fo measurement, i.e., Assume that the optimal control is of the fo rm and th at the optimal disturbance is of the fo rm The evolution of the system is completely determined the initial condition In pa rticula r, deﬁning the energy of the erfo rmance output is computed as dt The en ergy of the disturbance computed as dt E. razzoli (MIT) Lecture 25: Synthesis Ma

11, 2011 12

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ull info rmation case (intuition) Hence the cost of the game is where is the observabilit Gramian of the pair ), with rom the rop erties of the observabilit Gramian, it must the case that Assuming that there exist such that and and expanding, get E. razzoli (MIT) Lecture 25: Synthesis Ma 11, 2011 12

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Guess fo the structure of he sub optimal cont roller ossible solution ould e: This is Riccati equation, but notice that the quadratic term is not necessa rily sign deﬁnite. Simila on de ations hold fo the “observer Riccati equation The observer

gain ould Note the inversion of the matrix E. razzoli (MIT) Lecture 25: Synthesis Ma 11, 2011 12

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Sub optimal controller Assuming the follo wing technical conditions hold: stabili zable, detectable The matrices must have full ro rank. controller such that exists if and only if The foll wing Riccati equation has stabi lizing solution 0: The foll wing Riccati equation has stabi lizing solution 0: The matrix is ositive eﬁnite. E. razzoli (MIT) Lecture 25: Synthesis Ma 11, 2011 10 12

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Current resea rch Distributed control systems Net rk ed control systems

(quantization, bandwidth li itation etc.) Computational metho ds Nonlinea ystems/robustness (ISS, IQCs, olynomial systems, SoS, etc .) Hyb rid/switched systems System ID/Mo del reduction Robust/Adaptive control E. razzoli (MIT) Lecture 25: Synthesis Ma 11, 2011 11 12

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Other classes 6.231 Dynamic Programming and Sto chastic Control 6.242 Advanced Linea Control Systems 6.243 Dynamics of Nonlinea ys tems 6.245 Multiva riable Control Systems 6.256 Algeb aic ec hn iques and Semideﬁnite Optimiz tion 6.246-7 Advanced opics in Control 2.152 Nonlinea Control System Design 10.552

Advanced Systems Engineering (R. Braatz on LMIs fo optimal/robust control) 16.322 Sto chastic Es timation and Control 16.323 Principles of Optimal Control 16.333 Aircraft bilit and Control E. razzoli (MIT) Lecture 25: Synthesis Ma 11, 2011 12 12

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MIT OpenCourseWare http://ocw.mit.edu .241J / 16.338J Dynamic Systems and Control Spring 20 11 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms .

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