Modern Control Engineering

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Section 5–4 / Higher-Order Systems 181

C(s). If there is a closed-loop zero close to a closed-loop pole, then the residue at this

pole is small and the coefficient of the transient-response term corresponding to this pole

becomes small. A pair of closely located poles and zeros will effectively cancel each

other. If a pole is located very far from the origin, the residue at this pole may be small.

The transients corresponding to such a remote pole are small and last a short time. Terms

in the expanded form of C(s)having very small residues contribute little to the transient

response, and these terms may be neglected. If this is done, the higher-order system may

be approximated by a lower-order one. (Such an approximation often enables us to es-

timate the response characteristics of a higher-order system from those of a simplified

one.)

Next, consider the case where the poles of C(s)consist of real poles and pairs of

complex-conjugate poles. A pair of complex-conjugate poles yields a second-order term

ins. Since the factored form of the higher-order characteristic equation consists of first-

and second-order terms, Equation (5–33) can be rewritten

where we assumed all closed-loop poles are distinct. [If the closed-loop poles involve

multiple poles,C(s)must have multiple-pole terms.] From this last equation, we see that

the response of a higher-order system is composed of a number of terms involving the

simple functions found in the responses of first- and second-order systems. The unit-

step response c(t), the inverse Laplace transform of C(s), is then

fort 0 (5–34)

Thus the response curve of a stable higher-order system is the sum of a number of

exponential curves and damped sinusoidal curves.

If all closed-loop poles lie in the left-half splane, then the exponential terms and

the damped exponential terms in Equation (5–34) will approach zero as time tincreases.

The steady-state output is then c(q)=a.

Let us assume that the system considered is a stable one. Then the closed-loop poles

that are located far from the jvaxis have large negative real parts. The exponential

terms that correspond to these poles decay very rapidly to zero. (Note that the hori-

zontal distance from a closed-loop pole to the jvaxis determines the settling time of tran-

sients due to that pole. The smaller the distance is, the longer the settling time.)

Remember that the type of transient response is determined by the closed-loop

poles, while the shape of the transient response is primarily determined by the closed-

loop zeros. As we have seen earlier, the poles of the input R(s)yield the steady-state

response terms in the solution, while the poles of C(s)/R(s)enter into the exponential

transient-response terms and/or damped sinusoidal transient-response terms. The zeros

ofC(s)/R(s)do not affect the exponents in the exponential terms, but they do affect the

magnitudes and signs of the residues.

+ a

r

k= 1

ck e-zk^ vk^ tsinvk 21 - z^2 kt,

c(t)=a+ a

q

j= 1

aj e-pj^ t+ a

r

k= 1

bk e-zk^ vk^ tcosvk 21 - z^2 kt

C(s)=

a

s

+ a

q

j= 1

aj

s+pj

+ a

r

k= 1

bkAs+zk vkB+ck vk 21 - z^2 k

s^2 + 2 zk vk s+v^2 k

(q+2r=n)