The Stability of Linear Feedback Systems

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228 Chapter 5 The Stability of Linear Feedback Systems PS.2. An antenna control system was analyzed in Problem 3.5 and it was determined in order to reduce the effect of wind disturbances, the gain of the magnetic amplifi thai should be as large as possible. (a) Determine the limiting value of gain for maintai ~t k, stab~e syster!l. (b) It is desired to ha.ve a ~ys~cm settling .time equal to two seconds. ~~ra a shIfted aXIs and the Routh-Hurwllz cnlenon, determme the value of gain thaI sat' 6"1 this requirement. Assume that the complex fOOls of the closed-loop system dominal 1S ts transient response. (Is this a valid approximation in Ihis case) e 1!'It PS.3. Arc welding is onc of the most important areas of application for industrial Tobo [6]. In most manufacturing welding situations, uncertainties in dimensions of the Pa ls geometry of the joint, and the welding process itself require the use of sensors for main. taining weld quality. Several systems use a vision system to measure the geometry Oft~ puddle of melted metal as shown in Fig. P5.3. This system uses a constant rate offeedifl& the wire to be melted. (a) Calculate the maximum value for K for the system that will rf1ult in an oscillatory response. (b) For ~ of the maximum value of K found in part (a), deter. mine the roots of the characteristic equation. (c) Estimate the overshoot of the system of part (b) when it is subjected to a step input. Desired diamet~r Controller Wire-melting process A. + Error --L current 1 ..I ,+2 (O.$s + t)(s + 1) - diameter """" Vision ~y~t~m M~asured diameter 1 0.005s + 1 Figure PS.3. Welder control. PS.4, A feedback control system is shown in Fig. P5.4. The process transfer function is G(s) ~ K(s + 30) , s(s + 10) and the feedback transfer function is H(s) :: I/(s + 20). (a) Determine the limiting v~ue ofgain K for a stable system. (b) For the gain that results in borderline stability, detennt~ the magnitude of the imaginary roots. (c) Reduce the gain to ~ the magnitude ofth~ ~ derline value and determine the relative stability ofthe system (1) by shifting the aXIS ' tht using the Routh-Hurwitz criterion and (2) by determining the root locations. ShoW roots are between - I and - 2. R(s) 1---..._ C(s) Figure PS.4. Feedback system.
Problems 229 . the relative stability ofthe systems with the following characteristic cqua ~Ii~ng the axis in the s-pla~e and using the Routh-Hurwitz criterion and (b) ~ ins the location of the rOOts In the s-plane: +Jr+4s+2- 0 +t.r + lOs! + 42s + 20:: 0 + t9r + 110$ + 200 - 0 unity-feedback control system is Sho;nfin Fi.&- P5.6. ,Dctc~minhe the rcla.tivchsta Ibe system with thc following !ranSlcr unctIOns by oeatmg t c roots In t c s- 65 + 335 )- ,'(,+ 9) Figure PS.6. Unity fecdback systcm. The linear model of a phase detector (phase-lock loop) can be represented by Fig. Tbe phase-lock systems arc designed to maintain zero difference in phase between carrier signal and a local voltage
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 and 20: , 5.6 Summary 225 2 " 06 o o so 70
Page 21: Problems 227 (.) Controlk, --------
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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