zero and first order hold

Digital Control

Mo dule 1

Lecture 4

Module 1: Introduction to Digital Control Lecture Note 4

1

Data Data Reco Recons nstr truc ucti tion on

Most of the control systems have analog controlled processes which are inherently driven by analog analog inputs inputs.. Th Thus us the outputs outputs of a digita digitall contro controlle llerr should should first first be conve converte rted d into into analog analog signal signalss before before being applied applied to the system systems. s. Anothe Anotherr wa way y to look at the proble problem m is that that the high frequency components of  f (  f (t) should be removed before applying to analog devices. A low pass filter or a data reconstruction device is necessary to perform this operation. In contro controll system system,, hold hold operati operation on becomes becomes the most popular popular wa way y of recons reconstru tructi ction on due to its simplicity simplicity and low cost. Problem Problem of data reconstruct reconstruction ion can be formulate formulated d as:   “ Given a sequence of numbers,  f (0)  f (0),, f ( f (T ) T ), f (2 f (2T  T )), · · ·  , f ( f (kt) kt), · · · , a continuous time signal  f (  f (t), t ≥ 0, is to be reconstructed from the information contained in the sequence.” Data reconstruction process may be regarded as an extrapolation process since the continuous data signal has to be formed based on the information available at past sampling instants. Suppose the original signal f  signal  f ((t) between two consecutive sampling instants  kT   and (k (k + 1)T  1)T  is  k T  and to be estimated based on the values of  f  of  f ((t) at previous instants of  k of  kT  (k − 1)T  1)T ,, (k − 2)T  2)T ,, T ,, i.e., (k · · · 0. Power series expansion is a well known method of generating the desired approximation which yields  f (2) (kT ) kT ) + f  (kT )( (t − kT ) f k (t) = f ( f (kT ) kT ) + f  kT )(tt − kT ) kT ) + kT )2 + ..... 2! where, f k (t) = f ( 1) T  and f (t) for kT  ≤ t ≤ (k + 1)T  dn f ( f (t) for n  = 1, 2,... f (n)(kT ) kT ) = dtn t=kT  Since the only available information about f ( f (t) is its magnitude at the sampling instants, the derivatives of  f ( of  f ((kT ),  f (t) must be estimated from the values of  f  kT ), as 1  [f   [f ((kT ) 1)T )] )] f (1) (kT ) kT ) ∼ kT ) − f (( f ((k k − 1)T  = T  1 (1) Similarly, f (2) (kT )  [f   [f  (kT ) ((k − 1)T  1)T )] )] kT ) ∼ kT ) − f (1) ((k = T  1 where, f (1) ((k ((k − 1)T  1)T )) ∼  [f   [f (( ((k 1)T )) − f (( 2)T )] )] k − 1)T  f ((k k − 2)T  = T  (1)



I. Kar

1

Digital Control

1.1

Module 1

Lecture 4

Zero Order Hold

Higher the order of the derivatives to be estimated is, larger will be the number of delayed pulses required. Since time delay degrades the stability of a closed loop control system, using higher order derivatives of  f (t) for more accurate reconstruction often causes serious stability problem. Moreover a high order extrapolation requires complex circuitry and results in high cost.

For the above reasons, use of only the first term in the power series to approximate  f (t) during the time interval kT  ≤ t