The Stability of Linear Feedback Systems

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220 Chapter 5 The Stability of Linear Feedback Systems Then the Routh array is established as ~ 1 1 S; 1 1 s~ 0 0 ~ 1 O. Clearly, there arc roots on the shifted Imaginary axis and the rOOls can be obtained from the auxiliary equation, which is U(s.) - S; + 1 - (s. + j)(s. - j) _ (s+ 1 +j)(s+ 1 -j). (5.19) The shifting ofthe s·plane axis in order to ascertain the relative stability of a system is a very useful approach, particularly for higher-order systems with sev. eral pairs ofclosed·loop complex conjugate roots. III The Determination of Root Locations in the s-Plane The relative stability of a feedback control system is directly related to the loca· tion of the closed-loop roots of the characteristic equation in the s-plane. There· fore it is often necessary and easiest to simply determine the values of the roots of the characteristic equation. This approach has become particularly attractive today due to the availability of digital computer programs for determining the roots ofpolynomials. However, this approach may even be the most logical when using manual calculations ifthe order ofthe system is relatively low. For a thirdorder system or lower, it is usually simpler to utilize manual calculation methods. The determination of the roots ofa polynomial can be obtained by utilizing synlhelic dh>isioll which is based on the remainder theorem; that is, upon dividing the polynomial by a factor, the remainder is zero when the factor is a root of the polynomial. Synthetic division is commonly used to carry out the division p": cess. The relations for the roots of a polynomial as obtained in Eq. (5.6) are utilized to aid in the choice ofa first estimate of a root. • Example 5.7 Synthetic division Let us determine the roots ofthe polynomial q(s) - s' + 45' + 6s + 4. Establishing a table of synthetic division, we have 4 6 4 l=..! = trial rQot -I -3 -3 (5.20) 3 3 = remainder
5.4 The Determination of Root Locations in the s-Plane 221 root of 5 = - I. In this table, we multiply by the trial root and succesadd in each column. With a remainder ofone, we might try s = - 2, which in the form 4 6 4 1-2 -2 -4 -4 2 2 0 the remainder is zero, one root is equal to - 2 and the remaining roots obtained from the remaining polynomial (52 + 25 + 2) by using the qualOOt formula. search for a root ofthe polynomial can be aided considerably by utilizing of change of the polynomial at the estimated root in order to obtain a tllimate. The Newton-Raphson method is a rapid method utilizing synthetic to obtain the value of d~;) L" II is a first estimate of the root. The Newton-Raphson method is an iterach utilized in many digital computer root-solving programs. A new SUI ofthe root is based on the last estimate as [I8, 19] q(s.) SHl = s~ - q'(S~)' (5.21 )
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: 5.3 The Relative Stability of Feedb
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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