What is the effect of adding a zero to a control system?
Consider second order closed loop transfer function of the form given by
C(s)/R(s) =ωn2/ s2 +2ζωns+ωn2
ζ=damping factor(or damping ratio)
ωn =undamped natural frequency
Let a zero at s=-z be added to this transfer function .Then we have
C(s)/R(s)= [(s+z)/z]ωn2 / s2 +2ζωns+ωn2 →→→→(1)
Note that multiplication term in numerator of above expression has been adjusted so that steady state gain C/R(0) of system is unity .This gives steady state value of output Css=1 When input is unit step.Thus the system will track the step input with zero steady error .
From equation we have
C(s)/R(s)= ωn2/ s2 +2ζωns+ωn2 +s/z{ωn2/ s2 +2ζωns+ωn2 }
Let Cz(t) be the response of the system with a zero at s=-z.Then from the above equation we have
Cz(t)=c(t) + 1/z ( dc(t)/dt) (a)
The effect of added derivative term may be seen by examining fig (1) where a case for a typical value of ζ(less than one )is considered.We dee from this figure that the effect of zero is to contribute a pronounced early peak to the system’s response wherby peak overshoot may increase appreciably.From equation(a) and fig (1)it is seen that smaller the value of z ie closer is the value of zero to the origin,the more pronounced is the peaking phenomenon.On account of this fact the zeros of real axis near the origin are generally avoided in design.However ,in asluggish system artful introduction of a zero at the proper position can improve the transient response.
We further observe from equation(a)that as z increases ie zero moves farther into left half of s plane ,its effect become less pronounced.For sufficiently large values of z,the effect of zero on transient response may become negligible.
For closed loop transfer function of eq(1) peak percentage overshoot for a unit step input can be read from log –log graphs of fig (1) as function of z/ζωn for various values of ζ<=1
Consider second order closed loop transfer function of the form given by
C(s)/R(s) =ωn2/ s2 +2ζωns+ωn2
ζ=damping factor(or damping ratio)
ωn =undamped natural frequency
Let a zero at s=-z be added to this transfer function .Then we have
C(s)/R(s)= [(s+z)/z]ωn2 / s2 +2ζωns+ωn2 →→→→(1)
Note that multiplication term in numerator of above expression has been adjusted so that steady state gain C/R(0) of system is unity .This gives steady state value of output Css=1 When input is unit step.Thus the system will track the step input with zero steady error .
From equation we have
C(s)/R(s)= ωn2/ s2 +2ζωns+ωn2 +s/z{ωn2/ s2 +2ζωns+ωn2 }
Let Cz(t) be the response of the system with a zero at s=-z.Then from the above equation we have
Cz(t)=c(t) + 1/z ( dc(t)/dt) (a)
The effect of added derivative term may be seen by examining fig (1) where a case for a typical value of ζ(less than one )is considered.We dee from this figure that the effect of zero is to contribute a pronounced early peak to the system’s response wherby peak overshoot may increase appreciably.From equation(a) and fig (1)it is seen that smaller the value of z ie closer is the value of zero to the origin,the more pronounced is the peaking phenomenon.On account of this fact the zeros of real axis near the origin are generally avoided in design.However ,in asluggish system artful introduction of a zero at the proper position can improve the transient response.
We further observe from equation(a)that as z increases ie zero moves farther into left half of s plane ,its effect become less pronounced.For sufficiently large values of z,the effect of zero on transient response may become negligible.
For closed loop transfer function of eq(1) peak percentage overshoot for a unit step input can be read from log –log graphs of fig (1) as function of z/ζωn for various values of ζ<=1
Reference: (1)Control Systems Engineering-I.J Nagarath,M.Gopal
(2)Control Systems Engineering-S.N Sivanandam
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