Introduction to Control Systems and Mathematical Review

Chapter 4_Part3Root Locus MethodPrepared by Nurul Muthmainnah, Faculty of Mechanical UiTMModern Control EngineeringFifth Edition

Katsuhiko Ogata

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Course Outcome

Understand the meaning/concept of root locus. CO1Sketch the root locus for transfer function given with 8 steps. CO1

Prepared by NMMN2Introduction The basic characteristic of the transient response of a closed-loop system is closely related to the location of the closed-loop poles. If the system has a variable loop gain, then the location of the closed-loop poles depends on the value of the loop gain chosen.

It is important, therefore, that the designer know how the closed-loop poles move in the s plane as the loop gain is varied. From the design viewpoint, in some systems simple gain adjustment may move the closed-loop poles to desired locations.

Then the design problem may become the selection of an appropriate gain value. If the gain adjustment alone does not yield a desired result, addition of a compensator to the system will become necessary.Prepared by NMMN3The closed-loop poles are the roots of the characteristic equation. Finding the roots of the characteristic equation of degree higher than 3 is laborious and will need computer solution. MATLAB provides a simple solution to this problem.

However, just finding the roots of the characteristic equation may be of limited value, because as the gain of the open-loop transfer function varies, the characteristic equation changes and the computations must be repeated.Prepared by NMMN4A simple method for finding the roots of the characteristic equation has been developed by W. R. Evans and used extensively in control engineering. This method, called the root-locus method, is one in which the roots of the characteristic equation are plotted for all values of a system parameter.

The roots corresponding to a particular value of this parameter can then be located on the resulting graph. By using the root-locus method the designer can predict the effects on the location of the closed-loop poles of varying the gain value or adding open-loop poles and/or open-loop zerosPrepared by NMMN5In designing a linear control system, we find that the root-locus method proves to be quite useful, since it indicates the manner in which the open-loop poles and zeros should be modified so that the response meets system performance specifications.This method is particularly suited to obtaining approximate results very quickly.Prepared by NMMN6ROOT-LOCUS PLOTSAngle and Magnitude Conditions.

Consider the negative feedback system shown in Figure. The closed-loop transfer function is

The characteristic equation for this closed-loop system is obtained by setting thedenominator of the right-hand side of equation of transfer function equal to zero. That is,1 + G(s)H(s) = 0 or G(s)H(s) = -1 - Equation 1Prepared by NMMN7Here we assume that G(s)H(s) is a ratio of polynomials in s. Since G(s)H(s) is a complex quantity, Equation 1 can be split into two equations by equating the angles and magnitudes of both sides, respectively, to obtain the following:

The values of s that fulfill both the angle and magnitude conditions are the roots of the characteristic equation, or the closed-loop poles. A locus of the points in the complex plane satisfying the angle condition alone is the root locus. The roots of the characteristic equation (the closed-loop poles) corresponding to a given value of the gain can be determined from the magnitude condition.Prepared by NMMN8In many cases, G(s)H(s) involves a gain parameter K, and the characteristic equation may be written as

Then the root loci for the system are the loci of the closed-loop poles as the gain K is varied from zero to infinity.Sketching the root loci of a system by the root-locus methodMust know the location of the poles and zeros of G(s)H(s)The angles are measured in the counterclockwise directionPrepared by NMMN9Example 1

where p2 and p3 are complex-conjugate poles, then the angle of G(s)H(s) is

where 1 , 1 , 2 , 3 , and 4 are measured counterclockwise as shown in Figure (a) and (b).The magnitude of G(s)H(s) for this system is

where A1 , A2 , A3 , A4 , and B1 are the magnitudes of the complex quantities s+p1 , s+p2, s+p3, s+p4 , and s+z1 , respectively, as shown in Figure (a).

Note that, because the open-loop complex-conjugate poles and complex-conjugate zeros, if any, are always located symmetrically about the real axis, the root loci are always symmetrical with respect to this axis.

Therefore, we only need to construct the upper half of the root loci and draw the mirror image of the upper half in the lower-half s plane.

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(a) and (b) Diagrams showing angle measurements from open-loop poles and open-loop zero to test point s.Prepared by NMMN11General Rules for Constructing Root LociFirst, obtain the characteristic equation 1 + G(s)H(s) = 0

Then rearrange this equation so that the parameter of interest appears as the multiplying factor in the form

In the present discussions, we assume that the parameter of interest is the gain K, whereK>0. (If K0. It is seen that G(s) has a pair of complex-conjugate poles at

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Step 1:Determine the root loci on the real axis.

Step 2: Determine the angle of departure from the complex-conjugate open-loop poles

Step 3: Determine the break-in point

Step 4: Sketch a root-locus plot, based on the information obtained in the foregoing steps.Prepared by NMMN24ROOT-LOCUS APPROACH TO CONTROL-SYSTEMS DESIGNi) Preliminary Design Consideration

In building a control system, the proper modification of the plant dynamics may be a simple way to meet the performance specifications. This, however, may not be possible in many practical situations because the plant may be fixed and not modifiable. Then adjust parameters other than those in the fixed plant.

In practice, the root-locus plot of a system may indicate that the desired performance cannot be achieved just by the adjustment of gain (or some other adjustable parameter). In fact, in some cases, the system may not be stable for all values of gain (or other adjustable parameter).Then it is necessary to reshape the root loci to meet the performance specifications.

The design problems, therefore, become those of improving system performance by insertion of a compensator. Compensation of a control system is reduced to the design of a filter whose characteristics tend to compensate for the undesirable and unalterable characteristics of the plant.ROOT-LOCUS APPROACH TO CONTROL-SYSTEMS DESIGNii) Design by Root-Locus Method

The design by the root-locus method is based on reshaping the root locus of the system by adding poles and zeros to the systems open-loop transfer function and forcing the root loci to pass through desired closed-loop poles in the s plane.In designing a control system, if other than a gain adjustment (or other parameter adjustment) is required, must modify the original root loci by inserting a suitable compensator.In essence, in the design by the root locus method, the root loci of the system are reshaped through the use of a compensator so that a pair of dominant closed-loop poles can be placed at the desired location.ROOT-LOCUS APPROACH TO CONTROL-SYSTEMS DESIGNiv) Commonly Used Compensators

If a sinusoidal input is applied to the input of a network, and the steady-state output (which is also sinusoidal) has a phase lead, then the network is called a lead network. (The amount of phase lead angle is a function of the input frequency.) If the steady-state output has a phase lag, then the network is called a lag network. In a laglead network, both phase lag and phase lead occur in the output but in different frequency regions; phase lag occurs in the low-frequency region and phase lead occurs in the high-frequency region.A compensator having a characteristic of a lead network, lag network, or laglead network is called a lead compensator, lag compensator, or laglead compensator.

ROOT-LOCUS APPROACH TO CONTROL-SYSTEMS DESIGNiii) Series Compensation and Parallel (or Feedback) Compensation

(a) Series compensation(b) parallel or feedback compensation.ROOT-LOCUS APPROACH TO CONTROL-SYSTEMS DESIGNEffects of the Addition of PolesThe addition of a pole to the open-loop transfer function has the effect of pulling the root locus to the right, tending to lower the systems relative stability and to slow down the settling of the response. (Remember that the addition of integral control adds a pole at the origin, thus making the system less stable.)

(a) Root-locus plot of a single-pole system; (b) root-locus plot of a two-pole system; (c) root-locus plot of a three-pole system.Effects of the Addition of Zeros

The addition of a zero to the open-loop transfer function has the effect of pulling the root locus to the left, tending to make the system more stable and to speed up the settling of the response.

(Physically, the addition of a zero in the feedforward transfer function means the addition of derivative control to the system. The effect of such control is to introduce a degree of anticipation into the system and speed up the transient response.)

(a) Root-locus plot of a three-pole system; (b), (c), and (d) root-locus plots showing effects of addition of a zero to the three-pole system.Figure (a) show that is stable for small gain but unstable for large gain. Figures (b), (c), and (d) show root-locus plots for the system when a zero is added to the open-loop transfer function.Notice that when a zero is added to the system, it becomes stable for all values of gain.

END CHAPTER 4