《自动化仪表与过程控制》课程学习资料：The Root Locus Techniques

4. The root Locus Techniques 2022-2-11

2022-2-11 1 4. The Root Locus Techniques

4.1 Introduction A simple method for tracing the roots of the characteristic equation in the complex plane has been bound by r.W. Evans in 1948 and used extensively in control engineering. In this method. called the root locus method the roots of the characteristic equation are plotted for all values(usually from zero to infinity) of a particular system parameter(usually the gain)of our interest. The root locus method enables us to find the closed-loop poles from the open-loop poles and zeros as a parameter changes. Hence, by using the root locus method the designer can predict the effects on the location of the closed loop poles when varying the gain value or adding open-loop poles and/or open-loop zeros 2022-2-11

2022-2-11 2 4.1 Introduction A simple method for tracing the roots of the characteristic equation in the complex plane has been bound by R.W. Evans in 1948 and used extensively in control engineering. In this method, called the root locus method, the roots of the characteristic equation are plotted for all values (usually from zero to infinity) of a particular system parameter (usually the gain) of our interest. The root locus method enables us to find the closed-loop poles from the open-loop poles and zeros as a parameter changes. Hence, by using the root locus method the designer can predict the effects on the location of the closed￾loop poles when varying the gain value or adding open-loop poles and/or open-loop zeros

4.2 Root Locus Concept The root locus is the path of the roots of the characteristic equation traced out in the complex -plane as a system parameter is changed Consider a unity-feedback closed-loop control system shown below We want to trace the roots of the characteristic equation of the system as the gain varies. The transfer function is R(S K G T(s) 1+ KG(s (4.1) 2022-2-11 3

2022-2-11 3 4.2 Root Locus Concept ¨ The root locus is the path of the roots of the characteristic equation traced out in the complex -plane as a system parameter is changed. ¨ Consider a unity-feedback closed-loop control system shown below. ¨ We want to trace the roots of the characteristic equation of the system as the gain varies. The transfer function is (4.1)

KGis=K 1+j0 d(s) (4.2) This condition may be written into two separate conditions as follows The magnitude condition: kG()=km(-1 (4.3) The angle condition: ∠KG()=±180(2k+1) (4.4) where 毳=0,土1,土2, As an example consider a second-order system shown below 2022-2-11 4

2022-2-11 4 This condition may be written into two separate conditions as follows. The magnitude condition: The angle condition: where . As an example consider a second-order system shown below. (4.2) (4.3) (4.4)

1 5on±jnV1-52=-1八K-1for0≤≤1 (4.5) When the two roots are and notice that when the two roots are poles of open-loop transfer function. As increases we can trace the roots of the characteristic equation on the complex plane using the magnitude and angle condition, i.e KG(s) (46) S+2 ∠G(s)=∠G()=∠1 =±180°(2k+1 sS+ 2) (4.7) 2022-2-11 5

2022-2-11 5 When , the two roots are and . Notice that when , the two roots are poles of open-loop transfer function. As increases, we can trace the roots of the characteristic equation on the complex - plane using the magnitude and angle condition, i.e., 2 1 2 , 1 1 1 for 0 1 n n s s    j     j K         1 2 K KG s s s            1 180 2 1 2 KG s G s k s s           (4.5) (4.6) (4.7)

K 2 K Increasing K 6 K 2 K Increasing □= roots of the closed-lo 00 p K stem X= poles of th open-loop system K 2022-2-1 2 6

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et is a root of the characteristic equation Then for this root the angle condition can be written as K ∠s1-∠(s1+2)=-180 s(s+2 =51 s+2 2022-2-11 7

2022-2-11 7 Let is a root of the characteristic equation. Then for this root, the angle condition can be written as     1 1 1 2 180 2 s s K s s s s           

The gain condition may be used to find the required value of at the root as K KG()= =1(48) s(s+2 S or K=sN(+2)9 Another example for the root locus of a system for varying a parameter other than gain is introduced Consider a system shown in the figure below 2022-2-11 8

2022-2-11 8 The gain condition may be used to find the required value of at the root as       1 1 1 1 2 2 s s K K KG s s s s s       K  s1 s1  2 Another example for the root locus of a system for varying a parameter other than gain is introduced. Consider a system shown in the figure below. or (4.8) (4.9)

Another introduced. Consider a system shown in the figure below. example for the root locus of a system for varying a parameter other than gain Is jVK -八R G(5) R(→○ K s(s +a) s1+八K 2022-2-11 9

2022-2-11 9 Another introduced. Consider a system shown in the figure below. example for the root locus of a system for varying a parameter other than gain is

4.3 The root locus construction procedure for General system E(s) R()一 G(s) Y(s) H(S) For the general feedback control system shown above, the closed-loop transfer function is given by T()=-G(s) 1+GH(S) (4.10 2022-2-11 10

2022-2-11 10 4.3 The Root Locus Construction Procedure for General System For the general feedback control system shown above, the closed-loop transfer function is given by     1   G s T s GH s   (4.10)