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# Dynamic Systems and Contr ol Lecture Dynamical Systems Readings DD V Chapter Emilio razzoli Aeronautics and str onautics Massachusetts Institute of echnology eb rua ry E PDF document - DocSlides

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6.241 Dynamic Systems and Contr ol Lecture 6: Dynamical Systems Readings: DD V, Chapter Emilio razzoli Aeronautics and str onautics Massachusetts Institute of echnology eb rua ry 23, 2011 E. razzoli (MIT) Lecture 6: Dynamical Systems eb 23, 2011 10

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Signals Signals: maps from set to set Time axis top ological semigroup in ractice and combinations ereof, such Signal space vecto space, ypically fo some ﬁxed Discrete-time signals ps from (o to Continuous-time signals ps from (o to ypically constraints imp osed on maps to qualify as continu ous- time signals: Piecewise-continuit Lo cal (squa re) integrabilit DT and CT signals can given the structure of vecto spaces in the obvious (i.e., time-wise addition and scala multiplication of signal values). It is ossible to mix DT and CT signals (e.g., to describ digi sensing of physical ro cesses, zero-o rder holds, etc.). Semigroup: group without identit and/o nve se. E. razzoli (MIT) Lecture 6: Dynamical Systems eb 23, 2011 10

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Outline Behavio ral Mo dels Input-Output Mo dels E. razzoli (MIT) Lecture 6: Dynamical Systems eb 23, 2011 10

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Behavio ral Mo dels 1/2 system can deﬁned as set of constraints on signa ls Behavio ral mo del of dynamical syst em Given time axis and signa space ehavio ra mo del of system is subset of all ossible signals system is linea if its ehavio ral mo del is vecto space, i.e., if system is time-inva riant if its ehavio ral mo del is closed ith resp ect to time shift. an signal deﬁne the time-shift op erato as )( system is time-inva riant if fo any E. razzoli (MIT) Lecture 6: Dynamical Systems eb 23, 2011 10

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Behavio ral Mo dels 2/2 system is memo ryless if, fo any and ny the signal deﬁned as if if is also in In other rds, system is memo ryless if ossible futures re indep endent of the past. system is strictly memo ryless if there exists function True False such that )) True In other rds, system is strictly memo ryless if the constraints imp os ed on the signals re purely algeb raic, oint-wise in time (e.g., no derivatives integrals, etc.). Note: ny notion of regula it imp osed on the signals (as whole), such piecewise continu it integrabilit etc. requires system not to strictly memo ryless. (CT systems alw ys have some kind of memo ry .) E. razzoli (MIT) Lecture 6: Dynamical Systems eb 23, 2011 10

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Example: Memo ryless vs Strictly Memo ryless systems Consider ehavio ra mo del such that if and only if is piecew ise constant, i.e., if there exists ﬁnite pa rtition of into sets over which is constant. This system is memo ss, but is not strictly memo ryless. E. razzoli (MIT) Lecture 6: Dynamical Systems eb 23, 2011 10

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Outline Behavio ral Mo dels Input-Output Mo dels E. razzoli (MIT) Lecture 6: Dynamical Systems eb 23, 2011 10

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Input-Output mo dels 1/2 Behavio ral mo dels treat all comp onents of signals constrained the system equally without any i erences in their role interp retation. In many appl ic tions, it is use fu to mak distinction et een some of the comp onents of th signals (called input and the others (called the output ). An input-output del is map from set of input signa ls in in and et of output gnals out out In ehavio ral te rms, an input-output mo del is the set Su ypically will consider deterministic input-output, i.e., systems that asso ciate unique output signal to each input signal, the time axis is combinations thereof. convenience, will often assume in out E. razzoli (MIT) Lecture 6: Dynamical Systems eb 23, 2011 10

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Input-Output mo dels 2/2 Prop erties of ehavio ral mo dels map easily to input-output mo dels. An input-output ystem is linea if, fo all input signals Su Su An input-output system is time-inva riant if it commutes with the time -shift op erato r, i.e., if Su An input-output ystem is memo ryless (o static if there exists function in out such that, fo all Su )( )) E. razzoli (MIT) Lecture 6: Dynamical Systems eb 23, 2011 10

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Causalit An input-output ystem is causal if, fo any the output at time dep ends only on the values of the input on ]. In other rds, deﬁne the truncation op erato as )( fo fo Then an inp ut-output system is causal if SP An input-output ystem is strictly causal if, fo any the output at time dep ends only on the values of the input on ). E. razzoli (MIT) Lecture 6: Dynamical Systems eb 23, 2011 10 10

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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 .

241 Dynamic Systems and Contr ol Lecture 6 Dynamical Systems Readings DD V Chapter Emilio razzoli Aeronautics and str onautics Massachusetts Institute of echnology eb rua ry 23 2011 E razzoli MIT Lecture 6 Dyn ID: 23804

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6.241 Dynamic Systems and Contr ol Lecture 6: Dynamical Systems Readings: DD V, Chapter Emilio razzoli Aeronautics and str onautics Massachusetts Institute of echnology eb rua ry 23, 2011 E. razzoli (MIT) Lecture 6: Dynamical Systems eb 23, 2011 10

Page 2

Signals Signals: maps from set to set Time axis top ological semigroup in ractice and combinations ereof, such Signal space vecto space, ypically fo some ﬁxed Discrete-time signals ps from (o to Continuous-time signals ps from (o to ypically constraints imp osed on maps to qualify as continu ous- time signals: Piecewise-continuit Lo cal (squa re) integrabilit DT and CT signals can given the structure of vecto spaces in the obvious (i.e., time-wise addition and scala multiplication of signal values). It is ossible to mix DT and CT signals (e.g., to describ digi sensing of physical ro cesses, zero-o rder holds, etc.). Semigroup: group without identit and/o nve se. E. razzoli (MIT) Lecture 6: Dynamical Systems eb 23, 2011 10

Page 3

Outline Behavio ral Mo dels Input-Output Mo dels E. razzoli (MIT) Lecture 6: Dynamical Systems eb 23, 2011 10

Page 4

Behavio ral Mo dels 1/2 system can deﬁned as set of constraints on signa ls Behavio ral mo del of dynamical syst em Given time axis and signa space ehavio ra mo del of system is subset of all ossible signals system is linea if its ehavio ral mo del is vecto space, i.e., if system is time-inva riant if its ehavio ral mo del is closed ith resp ect to time shift. an signal deﬁne the time-shift op erato as )( system is time-inva riant if fo any E. razzoli (MIT) Lecture 6: Dynamical Systems eb 23, 2011 10

Page 5

Behavio ral Mo dels 2/2 system is memo ryless if, fo any and ny the signal deﬁned as if if is also in In other rds, system is memo ryless if ossible futures re indep endent of the past. system is strictly memo ryless if there exists function True False such that )) True In other rds, system is strictly memo ryless if the constraints imp os ed on the signals re purely algeb raic, oint-wise in time (e.g., no derivatives integrals, etc.). Note: ny notion of regula it imp osed on the signals (as whole), such piecewise continu it integrabilit etc. requires system not to strictly memo ryless. (CT systems alw ys have some kind of memo ry .) E. razzoli (MIT) Lecture 6: Dynamical Systems eb 23, 2011 10

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Example: Memo ryless vs Strictly Memo ryless systems Consider ehavio ra mo del such that if and only if is piecew ise constant, i.e., if there exists ﬁnite pa rtition of into sets over which is constant. This system is memo ss, but is not strictly memo ryless. E. razzoli (MIT) Lecture 6: Dynamical Systems eb 23, 2011 10

Page 7

Outline Behavio ral Mo dels Input-Output Mo dels E. razzoli (MIT) Lecture 6: Dynamical Systems eb 23, 2011 10

Page 8

Input-Output mo dels 1/2 Behavio ral mo dels treat all comp onents of signals constrained the system equally without any i erences in their role interp retation. In many appl ic tions, it is use fu to mak distinction et een some of the comp onents of th signals (called input and the others (called the output ). An input-output del is map from set of input signa ls in in and et of output gnals out out In ehavio ral te rms, an input-output mo del is the set Su ypically will consider deterministic input-output, i.e., systems that asso ciate unique output signal to each input signal, the time axis is combinations thereof. convenience, will often assume in out E. razzoli (MIT) Lecture 6: Dynamical Systems eb 23, 2011 10

Page 9

Input-Output mo dels 2/2 Prop erties of ehavio ral mo dels map easily to input-output mo dels. An input-output ystem is linea if, fo all input signals Su Su An input-output system is time-inva riant if it commutes with the time -shift op erato r, i.e., if Su An input-output ystem is memo ryless (o static if there exists function in out such that, fo all Su )( )) E. razzoli (MIT) Lecture 6: Dynamical Systems eb 23, 2011 10

Page 10

Causalit An input-output ystem is causal if, fo any the output at time dep ends only on the values of the input on ]. In other rds, deﬁne the truncation op erato as )( fo fo Then an inp ut-output system is causal if SP An input-output ystem is strictly causal if, fo any the output at time dep ends only on the values of the input on ). E. razzoli (MIT) Lecture 6: Dynamical Systems eb 23, 2011 10 10

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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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