By S. A. Marshall (auth.)
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70) wnt) ~ = 1, the system is said to be critically damped. 4. Note that cos rf> =~, the damping factor. 5 for values of the damping factor ~ between oand 1; as is evident, it is this damping factor that dominates the response whilst Wn merely affects the time scale. When ~ == 0 (no damping), the response is a sinusoid of natural frequency Wn, the undamped natural frequency. / xU) ! /' 0:;'\ 1// .... 2' . ~ .... o 7' l,y - .. ~ : \ ". :""- --~r ........ / ;'/ / / -- -.... '-', The Need for Some Mathematics 45 as the damped frequency, and the percentage overshoot decreases.
2 TRANSFER FUNCTIONS In chapter 2, mathematical models of various systems were developed by considering the physics and chemistry of these systems and writing down the relevant equations of motion. This led to models consisting of non-linear differential equations, which, it was stated, could be linearised. The argument advanced for linearisation was that most control-system design techniques are based on linear theory, and that systems under closed-loop control should experience only small perturbations about some reference value, and hence can be considered to behave in a linear fashion about that value.
These roots are called the pol~s of G(s); the roots of f(s), the numerator of G(s), are called the zeros of G(s); much more will be said about these parameters in chapter 6.