Modern Control Engineering

270 Chapter 6 / Control Systems Analysis and Design by the Root-Locus Method H(s) G(s) R(s) C(s) +– Figure 6–1 Control system. a ...

Section 6–2 / Root-Locus Plots 271 The characteristic equation for this closed-loop system is obtained by setting the denominato ...

272 Chapter 6 / Control Systems Analysis and Design by the Root-Locus Method Test point Test point p 4 p 3 p 2 p 1 s s ...

Section 6–2 / Root-Locus Plots 273 R(s) K C(s) +– s(s+ 1) (s+ 2) Figure 6–3 Control system. EXAMPLE 6–1 Consider the negative fe ...

274 Chapter 6 / Control Systems Analysis and Design by the Root-Locus Method It can be seen that the angle condition is not sati ...

Section 6–2 / Root-Locus Plots 275 jv s v= 0 1 j 3 j 3 s + 1 – 3 v = 0 s + 1 + 3 v = 0 0 Figure 6–4 Three asymptotes. By sub ...

276 Chapter 6 / Control Systems Analysis and Design by the Root-Locus Method whereA(s)andB(s)do not contain K. Note that f(s)=0h ...

Section 6–2 / Root-Locus Plots 277 For the present example, the characteristic equation G(s)+1=0is given by or By setting dK/ds= ...

278 Chapter 6 / Control Systems Analysis and Design by the Root-Locus Method jv j 1 j 1 2 – 1 s+ 1 s+ 2 u 2 u 1 u 3 0 s s Figu ...

Section 6–2 / Root-Locus Plots 279 7.Determine a pair of dominant complex-conjugate closed-loop poles such that the damping rati ...

280 Chapter 6 / Control Systems Analysis and Design by the Root-Locus Method jv 0 s u9 2 s p 2 f 1 f9 1 z 1 p 1 u 2 u 1 Fi ...

Section 6–2 / Root-Locus Plots 281 jv j 1 j 1 4 – 3 – 2 0 1 s z= 0.7 line j 2 j 2 145 ° 1 Figure 6–10 Root-locus plot. Sin ...

282 Chapter 6 / Control Systems Analysis and Design by the Root-Locus Method The value of the gain Kat any point on root locus c ...

Section 6–2 / Root-Locus Plots 283 These two equations are the equations for the root loci for the present system. Notice that t ...

284 Chapter 6 / Control Systems Analysis and Design by the Root-Locus Method Note that the root loci are symmetrical about the r ...

Section 6–2 / Root-Locus Plots 285 If a test point is located very far from the origin, then by dividing the denominator by the ...

286 Chapter 6 / Control Systems Analysis and Design by the Root-Locus Method jv s Angle of departure u 2 f u 1 Figure 6–12 0 Con ...

Section 6–2 / Root-Locus Plots 287 of the root loci in this important region in the splane must be obtained with reasonable accu ...

288 Chapter 6 / Control Systems Analysis and Design by the Root-Locus Method R(s) C(s) (a) 1 s K (s+ 1) (s+ 2) R(s) C(s) (c) 1 s ...

Section 6–2 / Root-Locus Plots 289 Typical Pole–Zero Configurations and Corresponding Root Loci. In summa- rizing, we show sever ...