788 BASIC CONTROL SYSTEMS
M=C
R=K/τ
p+ 1 /τ1 +HK/τ
p+ 1 /τ=K/τp+1 +HK
τ(16.2.16)and the corresponding transient solution of the closed-loop system is of the form
cf(t)=Afe−[(^1 +H K)/τ]t (16.2.17)
wherecf(t) represents the transient response of the output variable with feedback. Comparing
Equations (16.2.15) and (16.2.17), it is clear that the time constant with feedback is smaller by
the factor 1/( 1 +HK), and hence the transient decays faster.
By treating the differential operatorpas the sinusoidal frequency variablejω, i.e.,p=jω=
j( 1 /τ ), it follows from Equation (16.2.14) that the bandwidth of the open-loop system spreads
over a range from zero to a frequency of 1/τrad/s. On the other hand, for the system with feedback,
Equation (16.2.16) reveals that the bandwidth spreads from zero to( 1 +H K)/τrad/s, showing
that the bandwidth has been augmented by increasing the upper frequency limit by 1+KH.Dynamic Response of Control Systems
The existence of transients (and associated oscillations) is a characteristic of systems that possess
energy-storage elements and that are subjected to disturbances. Usually the complete solution of
the differential equation provides maximum information about the system’s dynamic performance.
Consequently, whenever it is convenient, an attempt is made to establish this solution first.
Unfortunately, however, this is not easily accomplished for high-order systems. Hence we are
forced to seek out other easier and more direct methods, such as the frequency-response method
of analysis.
Much of linear control theory is based on the frequency-response formulation of the sys-
tem equations, and several quasi-graphical and algebraic techniques have been developed to
analyze and design linear control systems based on frequency-response methods. Although
frequency-response techniques are limited to relatively simple systems, and apply only to lin-
ear systems in the rigorous mathematical sense, they are still most useful in system design
and the stability analysis of practical systems and can give a great deal of information about
the relationships between system parameters (such as time constants and gains) and system
response.
Once the transfer function of Equation (16.2.3) is developed in terms of the complex frequency
variables, by lettings=jω, the frequency-response characteristic and the loop gainGH(jω) can
be determined. The Bode diagram, displaying the frequency response and root-locus techniques,
can be used to study the stability analysis of feedback control systems. The dc steady-state
response, which becomes one component of the step response of the control system, can also
be determined by allowingsto be zero in the transfer function. The step response, in turn, can
be used as a measure of the speed of response of the control system. Thus, the transfer function
obtained from the block diagram can be used to describe both the steady-state and the transient
response of a feedback control system.
The matrix formulations associated withstate-variabletechniques have largely replaced
the block-diagram formulations. Computer software for solving a great variety of state-equation
formulations is available on most computer systems today. However, in the state-variable for-
mulation, much of the physical reality of any system is lost, including the relationships between
system response and system parameters.