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Essentials of Control Techniques and Theory_12

Tham khảo tài liệu 'essentials of control techniques and theory_12', kinh doanh - tiếp thị, quản trị kinh doanh phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | Design Methods for Computer Control 283 1 2w 2 w 1 2 _ T z 1 2 1 T z 1 t 4 z t- 4 _ 2t 2 z 2t- 2 4.1z - 3.9 _ 2.2z -1.8 when we substitute the value of 0.1 for T. A zero at z 0.951 and a pole at z 0.818 are added to the root locus diagram with a result illustrated in the plots shown in Figure 20.5. Stable control is now possible but the settling time is not short. How can we tell In the negative-real half of the t-plane the real part of t defines an exponential decay rate while the imaginary part defines a frequency. A complex pair of roots can be viewed as the roots of a quadratic t2 2zM0t 2 _ 0 where O 0 is the undamped natural frequency and z is the damping factor. Figure 20.5 Root locus with compensator. Screen grab of www.esscont.com 20 rootzcomp.htm 284 Essentials of Control Techniques and Theory The damping factor dictates the shape of the step response while the natural frequency determines its speed. A collection of responses for varying values of z are shown in Figure 20.6. Now constant values of 0 0 will correspond to points on circles around the origin of the r-plane. Constant values of damping factor will lie on lines through the origin at an angle cos-1Z to the negative real axis. This is shown in Figure 20.7. A value of z 0.7 gives a response of acceptable appearance. This is a very popular choice of damping factor apparently for no more complicated reason than that the poles lie on lines drawn at 45 through the r-plane origin The natural Figure 20.6 Responses for a variety of damping factors. 2 1.75 1.5 1.25 1 0.75 0.5 0.25 Figure 20.7 Lines of constant damping factor and of constant undamped natural frequency. Design Methods for Computer Control 285 Figure 20.8 z-plane loci for constant z and for constant ffl0. frequency the frequency at which the system would oscillate if the damping were removed is now represented by the distance from the origin. To translate these yardsticks into the discrete time world we must use the relationship z Th Damping