The Stability of Linear Feedback Systems

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226 Chapler 5 The Stability of Linear Feedback Systems E5.2. A system has a characteristic equalion 53 + 9$1 + 265 + 24 "" O. (a) U . Routh criterion, show that the system is stable. (b) Using the Ncwton-Raphson Sing the find Ihe three roots. methOd, ES.3. Find the roots of the characteristic equation s· + 9.5$J + 30.5s 1 + 37$ + 1') - • 0 E:S.4. A control system has the structure shown in Fig. E5.4. Determine Ihe gain at w .. the system will become unstable. hleh Figure E5.4. Feedforward system. E5.5. A feedback system has a loop transfer function K GH(s) = ($ + IXs + 3){s + 6)' where K = 10. Find the roOIS of this system's characteristic equation. E5.6. For the feedback system of Exercise E5.5, find thc value of K when two rOOIS lie on the imaginary axis. Determine the value of the three roots. Answer: s = -10, ±j5.2 £5.7. A negative feedback system has a loop transfer function GH(s) = K(s + 2) s(s 2)· (a) Find the value ofthe gain when the roflhe closed-loop roots is equal to 0.707. (b) Find the value of the gain when the closed-loop system has two roots on the imaginary axis. E5.8. Designers have developed small, fast, venical-takeofffighter aircraft that are invisible to radar (Stealth aircraft). The aircraft concept shown in Fig. E5.8(a), on the next paIC uses quickly tumingjet nozzles to steer the airplane 119]. The control syslem for the heading or direction control is shown in Fig. E5.8(b). Determine the maximum gain of the system for stable operation. E5.9. A system has a characleristic equation Find the range of K for a stable system. sl + 3s2 + (K + I)s + 4 = 0 An.f~·er: K > l' E5.10. We all use our eycs and ears to achieve balance. Our orientation system allo.",S ~~ to sit or stand in a desired position even while in motion. This orientation system IS P Is marily run by the information received in the inner ear, where the semicircular can a
Problems 227 (.) Controlk, -------- (b) figun £5.8. Aircraft heading conlrol. ~C") ~ II,,',", IIliplar acceleration and the otoliths measure linear acceleration. But these acceler .....urements need to be supplemented by visual signals. Try the following cxpcri (I) Stand with one foot in front ofanother and with your hands resting on your hips dboW5 bowed outward. (b) Oose your eyes. Did you find that you experienced a !!'I"'''' oscillation that grew until you lost balance Is this orientation position sta - without the use of your eyes' Routh-Hurwitz criterion, dcterminc the stability of the following (b) s) + 4s 1 + 5s + 6 (d) s' + r + 2.fl + 4s + 8 (f) s' + s' + 2s" +s + 3 eaacs, determine the number of roots, ifany, in the right-hand plane. Also, when it ble, determine the range of K that results in a stable system.
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 and 10: 5.2 The Routh·HuTWitz Stability Cr
Page 11 and 12: 5.2 The Routh-Hurwitz Stability Cri
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: , 5.6 Summary 225 2 " 06 o o so 70
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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