Stability
241
jytj C2 , for t 2 0; 1, where C2 is a nite constant. If for all possible bounded
inputs the corresponding outputs of the system are also bounded, then the system is
said to be BIBO stable.
Finally, it is mentioned that asymptotic stability i
The Root Locus Method
2 1
;
nm
0; 1; . . . ; jn mj 1
The root locus for K
following angles:
2
;
nm
277
7:3-12
0 approaches asymptotically the straight lines having the
0; 1; . . . ; jn mj 1
7:3-13
Proof
The characteristic equation (7.3-10) can be
Stability
265
4p12 1
p11 3p12 2p22 0
2p12 6p22 1
The above equations give the following matrix
!
51
44
P 1 1
4
4
If we apply the Sylvesters criterion (Sec. 2.12), it follows that the matrix P is positive
denite. Therefore, the sytem is asymptotically stab
Stability
Figure 6.3
253
Block diagram of the automatic depth control system of a submarine.
approximated by a second-order transfer function. The depth of the submarine is
measured by a depth sensor with transfer function Fd s. It is remarked that, as th
The Root Locus Method
Figure 7.8
289
The root locus of the supersonic airplane closed-loop system.
Example 7.3.4
Consider the closed-loop control system which controls the thickness of metal sheets,
shown in Figure 1.10 of Chap. 1. The system is approxima
Frequency Domain Analysis
Figure 8.44
349
The weighting function W u lncoth juj=2.
relation G j ! 0 F j ! 908, the phase margin is about 908. To obtain a more
desired (smaller) value for the phase margin, it sufces that the gradient of the
logarithmic amp
Frequency Domain Analysis
Figure 8.32
337
The block diagram of the automatic thickness control system.
the closed-loop system is stable. The controller transfer function Gc s is specied as
follows:
(a) Gc s K , i.e., the controller is a gain amplier
(b) G
Frequency Domain Analysis
325
itate the construction of the Nyquist diagram of H s we divide the Nyquist path N
into four segments and construct the Nyquist diagram for each of the corresponding
four segments of N as follows.
a. The segment for s 2 j 0 ;
Frequency Domain Analysis
313
Figure 8.8
(a) Amplitude M and (b) output yt curves of two different systems which show
the relation between Tr and bandwidth BW.
Figure 8.9 (a) The amplitude M and (b) the output yt of four different systems, which
show the