30079_27b.pdf

Frequency Response Plots

The frequency response of a fixed linear system is typically represented graphically, using one of

three types of frequency response plots. A polar plot is simply a plot of the vector H(jcS) in the

complex plane, where Re(o>) is the abscissa and Im(cu) is the ordinate. A logarithmic plot or Bode

diagram consists of two displays: (1) the magnitude ratio in decibels Mdb(o>) [where Mdb(w) = 20 log

M(o))] versus log w, and (2) the phase angle in degrees <£(a/) versus log a). Bode diagrams for

normalized first- and second-order systems are given in Fig. 27.23. Bode diagrams for higher-order

systems are obtained by adding these first- and second-order terms, appropriately scaled. A Nichols

diagram can be obtained by cross plotting the Bode magnitude and phase diagrams, eliminating

log a). Polar plots and Bode and Nichols diagrams for common transfer functions are given in

Table 27.8.

Frequency Response Performance Measures

Frequency response plots show that dynamic systems tend to behave like filters, "passing" or even

amplifying certain ranges of input frequencies, while blocking or attenuating other frequency ranges.

The range of frequencies for which the amplitude ratio is no less than 3 db of its maximum value

is called the bandwidth of the system. The bandwidth is defined by upper and lower cutoff frequencies

o)c, or by o> = 0 and an upper cutoff frequency if M(0) is the maximum amplitude ratio. Although

the choice of "down 3 db" used to define the cutoff frequencies is somewhat arbitrary, the bandwidth

is usually taken to be a measure of the range of frequencies for which a significant portion of the

input is felt in the system output. The bandwidth is also taken to be a measure of the system speed

of response, since attenuation of inputs in the higher-frequency ranges generally results from the

inability of the system to "follow" rapid changes in amplitude. Thus, a narrow bandwidth generally

indicates a sluggish system response.

Response to General Periodic Inputs

The Fourier series provides a means for representing a general periodic input as the sum of a constant

and terms containing sine and cosine. For this reason the Fourier series, together with the super-

position principle for linear systems, extends the results of frequency response analysis to the general

case of arbitrary periodic inputs. The Fourier series representation of a periodic function f(t) with

period 2T on the interval t* + 2T > t > t* is

jv N a° ^ i n/Trt i • n7rt\

/(O = -T + Zr I an cos — + bn sin — I

2, n=l \ i

i I

where

1 r+2^ nirt j

an = ~ J^ /(O cos — dt

bn = J'L f(f} sin T^dt

If f(t) is defined outside the specified interval by a periodic extension of period 27, and if f(t) and

its first derivative are piecewise continuous, then the series converges to /(O if f is a point of con-

tinuity, or to l/2 [f(t+) + /(*-)] if t is a point of discontinuity. Note that while the Fourier series in

general is infinite, the notion of bandwidth can be used to reduce the number of terms required for

a reasonable approximation.

27.6 STATE-VARIABLE METHODS

State-variable methods use the vector state and output equations introduced in Section 27.4 for

analysis of dynamic systems directly in the time domain. These methods have several advantages

over transform methods. First, state-variable methods are particularly advantageous for the study of

multivariable (multiple input/multiple output) systems. Second, state-variable methods are more nat-

urally extended for the study of linear time-varying and nonlinear systems. Finally, state-variable

methods are readily adapted to computer simulation studies.

27.6.1 Solution of the State Equation

Consider the vector equation of state for a fixed linear system:

x(t) = Ax(i) + Bu(t)

The solution to this system is

Fig. 27.23 Bode diagrams for normalized (a) first-order and (b) second-order systems.

x(t) = <l>(0*(0) + I $(f - r)Bu(r) dr

where the matrix <E>(0 is called the state-transition matrix. The state-transition matrix represents the

free response of the system and is defined by the matrix exponential series

Fig. 27.23 (Continued)

0(0 - eAt = I + At + ^-A2t2 + ... = 5) 1 A*r*

£=0 k\

where / is the identity matrix. The state transition matrix has the following useful properties:

0(0) - /

O-'(0 = O(-0

O*(0 = O(fo)

Oft + r2) = 0(^)0^)

Ofe - OOft - f0) = ^fe - O

0(0 - A0(0

The Laplace transform of the state equation is

sX(s) - Jt(0) = AX(s) + BU(s)

The solution to the fixed linear system therefore can be written as

XO = £-l[XW

= fi-^OWWO) + £Tl[<b(s)BU(s)]

where <&(s) is called the resolvent matrix and

0(0 = ^[OCs)] = ST^sI - A]'1

27.6.2 Eigenstructure

The internal structure of a system (and therefore its free response) is defined entirely by the system

matrix A. The concept of matrix eigenstructure, as defined by the eigenvalues and eigenvectors of

the system matrix, can provide a great deal of insight into the fundamental behavior of a system. In

particular, the system eigenvectors can be shown to define a special set of first-order subsystems

embedded within the system. These subsystems behave independently of one another, a fact that

greatly simplifies analysis.

System Eigenvalues and Eigenvectors

For a system with system matrix A, the system eigenvectors u,. and associated eigenvalues Az are

defined by the equation

Table 27.8 Transfer Function Plots for Representative Transfer Functions5

G(s)

Polar plot

Bode diagram

Srs + 1

O,+ l) (5r2 + l)

(Sr{+ 1) (ST2+l)(Sr3 + l)

K s

Table 27.8 (Continued)

Nichols diagram

Root locus

Comments

Stable; gain margin = oo

Elementary regulator; stable; gain

margin =00

Regulator with additional energy-storage

component; unstable, but can be made

stable by reducing gain

Ideal integrator; stable

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