The Stability of Linear Feedback Systems

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216 Chapter 5 The Stability of Linear Feedback Systems row preceding the row ofzeros is, in this case, obtained from the 5 1 -row. We that this row contains the coefficients ofthe even powers ofs and therefore i~eca.1I ~~wehave , thIs V(s) ~ 2s' + Ki' ~ 2s' + 8 ~ 2(s' + 4) ~ 2(s + j2)(s - j2). (5.13) [0 order to show that the auxiliary equation, U(s), is indeed a factor of the h acteristic equation, we divide q(s) by U(s) to obtain car. i's + 1 2s' + 8) s' + 2s' + 4s + 8 sJ + 45 2s' + 8 2s' + 8 Therefore, when K = 8, the factors of the characteristic equation arc q(s) ~ (s + 2)(s + j2Xs - j2). (5.14) 0'. • Example 5.4 Robot control Let us consider the control of a robot arm as shown in Fig. 5.4. It is predicted that there will be about 100,000 robots in service throughout the world by 1996 [6]. The robot shown in Fig. 5.4 is a six-legged micro robot system using highly flexible legs with high gain controllers that may become unstable and oscillale. Under this condition, we have the characteristic polynomial The Routh-Hurwitz array is q(s) = s' + l' + 4s' + 24s' + 3s + 63. s' l' s' s' s' Therefore the auxiliary equation is 1 4 3 21 o 1 24 63 -20 -60 0 63 0 o O. (5.15) V(s) = 21s' + 63 = 21(s' + 3) = 21(s + jV3)(s -jV3), (5. 16 ) which indicates that two roots are on the imaginary axis. In order to examine lhe remaining roots, we divide by the auxiliary equation to obtain ~ = Sl + s2 + S + 21. s'+3
5.2 The Routh-Hurwitz Stability Criterion 217 SA. A completely integrated, six-legged, micro robot system. The legged design maximum dexterity. Legs also provide a unique sensory system for tal interaction. It is equipped with a sensor network that includes 150 of 12 different types. The legs are instrumented so that it can determine the lay IaTain, the surface texture, hardness, and even color. The gyro-stabilized camera nder can be used for gathering data beyond the robot's immediatc reach. This perfonnance system is able to walk quickly, climb over obstacles, and perform • mOlions. Courtesy of IS Robotics Corporation. "nga Routh-Hurwitz array for this equation, we have s' 1 1 s2 I 21 Sl -20 0 f' 21 O. ~t.:abanges in sign in the first c:01umn indicate the presence of two roots in nd plane, and the system IS unstable. The roots in the right-hand plane - +1 ±jV6. • a.PIe S.S Disk-drive control disk-storag d . . t . e eVlces arc used with IOOay's computers [17]. The data head o ~lfferent psitions on th~ spinning disk and rapid, accurate response block dtagram of a disk-storage data·head positioning system is
Page 1 and 2: The Stability of Linear Feedback Sy
Page 3 and 4: 5.1 The Concept of Stability 209 St
Page 5 and 6: 5.2 The Routh·Hurwitz Stability Cr
Page 7 and 8: 5.2 The Routh-Hurwitz Stability Cri
Page 9: 5.2 The Routh·HuTWitz Stability Cr
Page 13 and 14: 5.3 The Relative Stability of Feedb
Page 15 and 16: 5.4 The Determination of Root Locat
Page 17 and 18: 5.5 Design Example: Tracked Vehicle
Page 19 and 20: , 5.6 Summary 225 2 " 06 o o so 70
Page 21 and 22: Problems 227 (.) Controlk, --------
Page 23 and 24: Problems 229 . the relative stabili
Page 25 and 26: Problems 231 A (c:edback control sy
Page 27 and 28: References 233 equation The equatio

The Stability of Linear Feedback Systems

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