4 - Lecture 4:Sampling and Reconstruction Sampling Data...

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1 Lecture 4:Sampling and Reconstruction • Sampling • Data Reconstruction (Hold) • Reading: Chapter 3 of the textbook A Typical Digital Control System In the above digital control system, sampling and data reconstruction of signals are needed to interface the digital computer with the physical world Continuous-time Signals Discrete-time Signals Sampling Data reconstruction discrete-time signal continuous-time signal digital signal Plant Computer A/D D/A y ( t ) Sensor T r ( t ) Data Hold continuous-time signal discrete-time signal continuous-time system discrete-time system Sampling Reconstruction

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2 A sampler obtains from a continuous-time signal a discrete-time signal by sampling every T seconds – Input is a continuous-time signal – Output is a discrete-time signal e ( kT ), k =0,1,… – Both signals have identical values at sampling moments – Not a lossless procedure in general – Not a time-invariant procedure e ( t ) t T e ( kT ) e ( t ), t ¸ 0 0 2 T 3 T 4 T T e ( t ) e ( kT ) Sampler Reconstruction: Zero-Order Hold Data reconstruction devices are needed to convert the discrete-time signal output by the digital controller into a continuous-time signal suitable for driving the physical plant Ideally both signals should agree at the sampling moment: 0, T , 2 T , … The simplest data reconstruction is zero-order hold Given a discrete time signal e ( kT ), k =0,1,… A continuous-time signal is produced The value of at any time t is equal to the value of e ( kT ) in the closest sampling moment preceding t : t e ( kT ) 0 T 2 T 3 T 4 T ¢¢¢ ¹ e ( t ), t ¸ 0 ¹ e ( t ) ¹ e ( t ) ¹ e ( t ) = e (0)[ u ( t ) ¡ u ( t ¡ T )] + e ( T )[ u ( t ¡ T ) ¡ u ( t ¡ 2 T )] +
3 Sampled-Data Control System Sampler Plant Digital controller Data hold e ( t ) y ( t ) e ( kT ) u ( kT ) u ( t ) Digital controller is designed together with sampler and data hold Does there exist a transfer function from e ( t ) to u ( t )? How to characterize the input-output relation from e ( t ) to u ( t ) based on the transfer function D ( z ) of the digital controller? Let us start from the simplest case: digital controller does nothing Sampler Data hold e ( t ) e ( kT ) ¹ e ( t ) sampler and hold Sampler and Zero-Order Hold Sampler and zero-order hold ¹ e ( t ) ¹ e ( t ) e ( t ) t 0 T 3 T 4 T 2 T In general, original signal e ( t ) can not be fully reconstructed: e ( t

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4 - Lecture 4:Sampling and Reconstruction Sampling Data...

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