TUTORIAL ROOT LOCUS DESIGN

TUTORIAL ROOT LOCUS DESIGN - Root-Locus Design The...

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Root-Locus Design The root-locus can be used to determine the value of the loop gain , which results in a satisfactory closed-loop behavior. This is called the proportional compensator or proportional controller and provides gradual response to deviations from the set point. There are practical limits as to how large the gain can be made. In fact, very high gains lead to instabilities. If the root-locus plot is such that the desired performance cannot be achieved by the adjustment of the gain, then it is necessary to reshape the root-loci by adding the additional controller G to the open-loop transfer function. G must be chosen so that the root-locus will pass through the proper region of the -plane. In many cases, the speed of response and/or the damping of the uncompensated system must be increased in order to satisfy the specifications. This requires moving the dominant branches of the root locus to the left. K ( ) c s ( ) c s s The proportional controller has no sense of time, and its action is determined by the present value of the error. An appropriate controller must make corrections based on the past and future values. This can be accomplished by combining proportional with integral action or proportional with derivative action . One of the most common controllers available commercially is the controller. Different processes are suited to different combinations of proportional, integral, and derivative control. The control engineer's task is to adjust the three gain factors to arrive at an acceptable degree of error reduction simultaneously with acceptable dynamic response. The compensator transfer function is PI PD PID ( ) I c P K G s K K s s = + + D (1) For or controllers, the appropriate gain is set to zero. PD PI Other compensators, are lead, lag, and lead-lag compensators. A first-order compensator having a single zero and pole in its transfer function is 0 0 ( ) c s Z G s s P + = + (2) The pole and zero are located in the left half s-plane as shown in Figure 1. 0 0 p 0 z θ 0 p θ 1 s 1 s : × 0 z 0 p 0 z θ 0 p θ z × : (a) Phase-lead (b) Phase-lag Figure 1 Compensator phase angle contribution 7A.1
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For a given 1 1 s 1 j σ ω = + , the transfer function angle given by 0 ( c z p 0 ) θ θ θ = is positive if as shown in Figure 1 (a), and the compensator is known as the phase-lead controller . On the other hand if as shown in Figure 1 (b), the compensator angle 0 z p < 0 ( c z 0 0 0 z p > 0 ) p θ θ = θ is negative, and the compensator is known as the phase-lag controller In general, the open-loop transfer function is given by 1 2 1 2 ( )( ) ( ( ) ( ) ( )( ) ( m n K s z s z s z KG s H s s p s p s p + + + = + + + " " ) ) ) where is the number of finite zeros and is the number of finite poles of the loop transfer function. If , there are ( m n m n m > n zeros at infinity. The characteristic equation of the closed-loop transfer function is 1 ( ) ( ) KG s H s + = 0 Therefore 1 2 1 2 ( )( ) ( ) ( )( ) ( ) n m s p s p s p K s z s z s z + + + = − + + + " " From the above expression, it follows that for a point in the -plane to be on the root- locus, when 0 , it must satisfy the following two conditions.
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