MIT16_30F10_lec05_slides

# MIT16_30F10_lec05_slides - 16.30 Estimation and Control of...

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16.30 Estimation and Control of Aerospace Systems Topic 5 addendum: Signals and Systems Aeronautics and Astronautics Massachusetts Institute of Technology Fall 2010 (MIT) Topic 5 addendum: Signals, Systems Fall 2010 1 / 27

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Outline 1 2 3 4 5 Continuous- and discrete-time signals and systems Causality Time Invariance State-space models Linear Systems (MIT) Topic 5 addendum: Signals, Systems Fall 2010 2 / 27
Continuous- and discrete-time signals Continuous-time signal A (scalar) continuous-time signal is a function that associates to each time t R a real number y ( t ), i.e., y : t �→ y ( t ). Note: We will use the “standard” (round) parentheses to indicate continuous-time signals. Discrete-time signal A (scalar) discrete-time signal is a function that associates to each integer k Z a real number y [ k ], i.e., y : k �→ y [ k ]. Note: We will use the square parentheses to indicate discrete-time signals. y ( t ) y [ k ] t k (MIT) Topic 5 addendum: Signals, Systems Fall 2010 3 / 27

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Signals are vectors Multiplication by a scalar Let α R . The signal α y can be obtained as: ( α y )( t ) = α y ( t ) , and ( α y )[ k ] = α y [ k ] . Notice 0 y is always the “zero” signal, where 0( t ) = 0 for all t R , and 0[ k ] = 0 for all k Z , and 1 y = y . Addition of two signals Let u and v be two signals of the same kind (i.e., both in continuous or discrete time). The signal u + v is defined as: ( u + v )( t ) = u ( t ) + v ( t ) , and ( u + v )[ k ] = u [ k ] + v [ k ] . Notice that u u = u + ( 1) u = 0. (MIT) Topic 5 addendum: Signals, Systems Fall 2010 4 / 27
Systems Definition (system) A system is an operator that transforms an input signal u into a unique output signal y . u ( t ) y ( t ) t t u ( t ) y ( t ) System u [ k ] y [ k ] u [ k ] y ( t ) t k (MIT) Topic 5 addendum: Signals, Systems Fall 2010 5 / 27

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A classification Continuous-Time System: CT CT This is the kind of systems you studied in 16.06. Discrete-Time System: DT DT We will study this kind of systems in this class. Sampler: CT DT This class includes sensors, and A/D (Analog Digital) converters. Let us call a sampler with sampling time T a system such that y [ k ] = u ( kT ) . Hold: DT CT This class includes actuators, and D/A (Digital Analog) converters. A Zero-Order Hold (ZOH) with holding time T is such that y ( t ) = u �� t T �� . (MIT) Topic 5 addendum: Signals, Systems Fall 2010 6 / 27
Outline 1 2 3 4 5 Continuous- and discrete-time signals and systems Causality Time Invariance State-space models Linear Systems (MIT) Topic 5 addendum: Signals, Systems Fall 2010 7 / 27

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Static/Memoryless systems Definition (Memoryless system) A system is said to be memoryless (or static ) if, for any t 0 R (resp. k 0 Z ), the output at time t 0 (resp. at time k 0 ) depends only on the input at time t 0 (resp. at step k 0 ). This is the most basic kind of system. Essentially the output can be written as a “simple” function of the input, e.g., y ( t ) = f ( u ( t )) , y [ k ] = f ( u [ k ]) . Examples: A proportional compensator; A spring; An electrical circuit with resistors only.
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