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JBlg. 5.10    Component Bode-phase plots for Example 5.1.
   The combined magrutude and phase plots, which are the sum of component
plots, are shown in Fig. 5.11.
   MATLAB4 is a convenient tool for control system analysis and design. We
assume that the reader has access to this or a software with sinular capabilities.
Using MATLAB,4 the magnitude and phase plots of the given transfer function
are drawn as shown in Fig. 5.12. It is interesting to observe that the approximate
method that involves the concept of comer frequencies comes close to the more
accurate plots given by MATLAB.4
5.7   Stability of Closed-Loop Systems
         One ofthe most important requirements for a control systemis stability. A linear,
time-invariant system is said to be stable if a bounded input produces a bounded
output. In other words, for a stable system the output should reach a steady state.
1

20 /o91o
PERFORMANCE, STABILV fY, DYNAMICS, AND CONTROL
5.7.1  Routh-sStabilitY Criterion                            Lhe closed-/oop
   Routh's stabilitY criterion helps x;i~letermine whethe'  ple?cfwhthout actua/lY
 po/es arthpositive if ,eal or have positive rea/ parts if cornp
  solving the closed-loop chalracteristic equation.
.

\,
\

            \.

464          PERFORMANCE, STABILITY, DYNAMiCS, AND CONTROL
The procedure is as follows.
1) Express the characteristic polynomial in the following form:
aos" + ats l-l  + a2Sn-2 + - .. + an_is + an  - 0                  (5.115)
where the coefficients ao, ai,..., an are real quantities. We assume that an  7{: O
so that any zero root is removed.
    2) Examine the value of each coefficient. If any coefficient is zero or negative
when at least one other coefficient is positive, then Routh's criterion states that
there will be at least one root of the characteristic polynomial that is imaginary
or has a positive.real part. In such a case, the system is not'stable. Therefore, for
stability,vaII the coefficients must be positive or must have the same sign. This
forms the necessary condition for stability.
     3) To check whether the sufficiency condition is satisfied, form the Routh's array
as follows:
where
  bi -
ai a2 - aOa3
ai
Sn  00 a2 a4 a6
   b2 -
ai 04 - aoas
ai
ai a6  - aoai
ai
(5.116)
       ~ . .      (5.117)
       bla3 - alb2           bias -aIb3           bia7 - alb4
ci-  b  ' C2-  b  ' C3=  b  , ... (5.118)
and
di -
ci b2 - bi C2
CI
ci b3 - b} C3
CI
(5.119)
         (5.120)
This process is continued until nth row has been completed. The complete array of
coefficients is triangular. Note that the evaluation of b,, c,, and d,, etc., is continued
LINEAR SYSTEMS, THEORY, AND DESIGN: A BRIEF REVIEW       465
until the remaining ones ate zero.For example, for a fourth-degree polynornialin s,
as - a6 -.  .- .  - 0 so thatb3 - b4 =  ...  - O,c2 - C3 =  ...  = O,d2 - d3 - ...  = O,
e2-e3-. "- =O,and f2- f3= .*. -.0.
    Routh's stability criterion states that the number of roots of the characteristic
　
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