ans4:adding zero

(A) THE ZERO S = –Z1 IS NOT PRESENT.
FOR DIFFERENT VALUES OF K, THE SYSTEM CAN HAVE TWO REAL POLES OR A PAIR OF COMPLEX
CONJUGATE POLES. THIS MEANS THAT WE CAN CHOOSE K FOR THE SYSTEM TO BE OVERDAMPED,
CRITICALLY DAMPED OR UNDERDAMPED.
(B) THE ZERO S = –Z1 IS LOCATED TO THE RIGHT OF BOTH POLES, S = – P2 AND S = –P1.
IN THIS CASE, THE SYSTEM CAN HAVE ONLY REAL POLES AND HENCE WE CAN ONLY FIND A VALUE
FOR K TO MAKE THE SYSTEM OVERDAMPED. THUS THE POLE–ZERO CONFIGURATION IS EVEN MORE
RESTRICTED THAN IN CASE (A). THEREFORE THIS MAY NOT BE A GOOD LOCATION FOR OUR ZERO,
SINCE THE TIME RESPONSE WILL BECOME SLOWER.
(C) THE ZERO S = –Z1 IS LOCATED BETWEEN S = –P2 AND S = –P1.
THIS CASE PROVIDES A ROOT LOCUS ON THE REAL AXIS. THE RESPONSES ARE THEREFORE LIMITED TO
OVERDAMPED RESPONSES. IT IS A SLIGHTLY BETTER LOCATION THAN (B), SINCE FASTER RESPONSES
ARE POSSIBLE DUE TO THE DOMINANT POLE (POLE NEAREST TO JAXIS) LYING FURTHER FROM THE J
AXIS THAN THE DOMINANT POLE IN (B).
(D) THE ZERO S = –Z1 IS LOCATED TO THE LEFT OF S = –P2.
THIS IS THE MOST INTERESTING CASE. NOTE THAT BY PLACING THE ZERO TO THE LEFT OF BOTH
POLES, THE VERTICAL BRANCHES OF CASE (A) ARE BENT BACKWARD AND ONE END APPROACHES THE
ZERO AND THE OTHER MOVES TO INFINITY ON THE REAL AXIS. WITH THIS CONFIGURATION, WE CAN
NOW CHANGE THE DAMPING RATIO AND THE NATURAL FREQUENCY (TO SOME EXTENT). THE
CLOSED-LOOP POLE LOCATIONS CAN LIE FURTHER TO THE LEFT THAN S = –P2, WHICH WILL PROVIDE
FASTER TIME RESPONSES. THIS STRUCTURE THEREFORE GIVES A MORE FLEXIBLE CONFIGURATION FOR
CONTROL DESIGN.
WE CAN SEE THAT THE RESULTING CLOSED-LOOP POLE POSITIONS ARE CONSIDERABLY INFLUENCED BY
THE POSITION OF THIS ZERO. SINCE THERE IS A RELATIONSHIP BETWEEN THE POSITION OF CLOSED-LOOP
POLES AND THE SYSTEM TIME DOMAIN PERFORMANCE, WE CAN THEREFORE MODIFY THE BEHAVIOUR OF
CLOSED-LOOP SYSTEM BY INTRODUCING APPROPRIATE ZEROS IN THE CONTROLLER.

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