ASSIGNMENT:2 question 4

What would the effect of adding a zero to a control system? Consider the second-order system, G(s) = 1/((s+p1)*(s+p2)) , p1>0 , p2>0 The poles are s = –p1 and s = –p2 and the simple root locus plot for this system is shown in (a). When we add a zero at s = –z1 to the controller, the open-loop transfer function will be: G1(s)= K(s+z1)/((s+p1)*(s+p2)) , z1>0graph We can put the zero at three different positions with respect to the poles: 1. To the right of s = –p1 (b) 2. Between s = –p2 and s = –p1 (c) 3. To the left of s = –p2 (d) (a) The zero s = –z1 is not present. the system can have two real poles or a pair of complex conjugate poles ,may be overdamped, critically damped or underdamped. 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.System can have only real poles and system is 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 system is overdamped. 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 jw axis) lying further from the jw axis than the dominant pole in (b). d) The zero s = –z1 is located to the left of s = –p2. 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, the damping ratio and the natural frequency can be changed 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. Since there is a relationship between the position of closed-loop poles and the system time domain performance, the behavior of closed-loop system can be modified by introducing appropriate zeros in the controller.


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