Stability of control systems in time-domain

Stability of control systems in time-domain

1. Unit 5 Stability of control systems in time domain Dr. Ankita Malhotra SVKM's D J Sanghvi College of Engineering

2. Routh’s stability criterion Dr. Ankita Malhotra SVKM's D J Sanghvi College of Engineering Most linear closed-loop systems have closed-loop transfer functions of the form: where the a’s and b’s are constants and m< n.A simple criterion, known as Routh’s stability criterion, enables us to determine the number of closed-loop poles that lie in the right-half s plane without having to factor the denominator polynomial.

3. Dr. Ankita Malhotra SVKM's D J Sanghvi College of Engineering Routh’s Stability Criterion. Routh’s stability criterion tells us whether or not there are unstable roots in a polynomial equation without actually solving for them. This stability criterion applies to polynomials with only a finite number of terms. When the criterion is applied to a control system, information about absolute stability can be obtained directly from the coefficients of the characteristic equation. The procedure in Routh’s stability criterion is as follows: 1. Write the polynomial in s in the following form: where the coefficients are real quantities. We assume that an a n ≠ 0; that is, any zero root has been removed.

4. Dr. Ankita Malhotra SVKM's D J Sanghvi College of Engineering Routh’s Stability Criterion. Routh’s stability criterion tells us whether or not there are unstable roots in a polynomial equation without actually solving for them. This stability criterion applies to polynomials with only a finite number of terms. When the criterion is applied to a control system, information about absolute stability can be obtained directly from the coefficients of the characteristic equation.

5. Dr. Ankita Malhotra SVKM's D J Sanghvi College of Engineering The procedure in Routh’s stability criterion is as follows: 1. Write the polynomial in s in the following form: where the coefficients are real quantities. We assume that an a n ≠ 0; that is, any zero root has been removed. 2. If any of the coefficients are zero or negative in the presence of at least one positive coefficient, a root or roots exist that are imaginary or that have positive real parts. Therefore, in such a case, the system is not stable. If we are interested in only the absolute stability, there is no need to follow the procedure further. ---- 5.1

6. Note that all the coefficients must be positive. This is a necessary condition, as may be seen from the following argument: A polynomial in s having real coefficients can always be factored into linear and quadratic factors, such as ( s+a ) and (s2+bs+c), where a, b, and c are real. The linear factors yield real roots and the quadratic factors yield complex-conjugate roots of the polynomial. The factor (s2+bs+c) yields roots having negative real parts only if b and c are both positive. For all roots to have negative real parts, the constants a, b, c, and so on, in all factors must be positive The product of any number of linear and quadratic factors containing only positive coefficients always yields a polynomial with positive coefficients. It is important to note that the condition that all the coefficients be positive is not sufficient to assure stability. The necessary but not sufficient condition for stability is that the coefficients of Equation (5.1) all be present and all have a positive sign. (If all a’s are negative, they can be made positive by multiplying both sides of the equation by –1.) Dr. Ankita Malhotra SVKM's D J Sanghvi College of Engineering

7. 3. If all coefficients are positive, arrange the coefficients of the polynomial in rows and columns according to the following pattern: Dr. Ankita Malhotra SVKM's D J Sanghvi College of Engineering

8. The process of forming rows continues until we run out of elements. (The total number of rows is n+1.) The coefficients b1, b2, b3 , and so on, are evaluated as given below. The evaluation of the b’s is continued until the remaining ones are all zero. The same pattern of cross-multiplying the coefficients of the two previous rows is followed in evaluating the c’s , d’s , e’s , and so on. Dr. Ankita Malhotra SVKM's D J Sanghvi College of Engineering

9. Limitations of Routh -Hurwitz criterion Dr. Ankita Malhotra SVKM's D J Sanghvi College of Engineering

10. Special Cases of Routh -Hurwitz Criteria Dr. Ankita Malhotra SVKM's D J Sanghvi College of Engineering

11. The terms in the s3 row are all zero. (Note that such a case occurs only in an odd numbered row.) The auxiliary polynomial is then formed from the coefficients of the s4 row. Example of special case 1 : Example of special case 2 Dr. Ankita Malhotra SVKM's D J Sanghvi College of Engineering

12. The auxiliary polynomial P(s) is We see that there is one change in sign in the first column of the new array. Thus, the original equation has one root with a positive real part. Dr. Ankita Malhotra SVKM's D J Sanghvi College of Engineering

13. Examples Dr. Ankita Malhotra SVKM's D J Sanghvi College of Engineering

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18. Control system analysis and design by root locus method 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. The closed-loop poles are the roots of the characteristic equation. A 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. Dr. Ankita Malhotra SVKM's D J Sanghvi College of Engineering

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20. The roots corresponding to a particular value of this parameter can then be located on the resulting graph. Note that the parameter is usually the gain, but any other variable of the open-loop transfer function may be used. Unless otherwise stated, we shall assume that the gain of the open-loop transfer function is the parameter to be varied through all values, from zero to infinity. 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 zeros. Therefore, it is desired that the designer have a good understanding of the method for generating the root loci of the closed-loop system. Dr. Ankita Malhotra SVKM's D J Sanghvi College of Engineering

21. ROOT- LOCUS PLOTS Angle 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 the denominator of transfer function equation equal to zero. That is, Control system Dr. Ankita Malhotra SVKM's D J Sanghvi College of Engineering

22. The values of s that fulfill both the angle and magnitude conditions are the roots of 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. Dr. Ankita Malhotra SVKM's D J Sanghvi College of Engineering

23. Let characteristic equation based on system gain K is given 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. Note that to begin sketching the root loci of a system by the root-locus method we must know the location of the poles and zeros of G(s)H(s). Remember that the angles of the complex quantities originating from the open-loop poles and open-loop zeros to the test point s are measured in the counterclockwise direction. where –p2 and –p3 are complex-conjugate poles, then the angle of G(s)H(s) is where angles are measured counterclockwise Dr. Ankita Malhotra SVKM's D J Sanghvi College of Engineering

24. The magnitude of G(s)H(s) for this system is Figure (a) and (b) Diagrams showing angle measurements from open-loop poles and open-loop zero to test point s. 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 Dr. Ankita Malhotra SVKM's D J Sanghvi College of Engineering

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27. Root –locus conclusion 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 system’s 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, we must modify the original root loci by inserting a suitable compensator . Once the effects on the root locus of the addition of poles and/or zeros are fully understood, we can readily determine the locations of the pole(s) and zero(s) of the compensator that will reshape the root locus as desired Dr. Ankita Malhotra SVKM's D J Sanghvi College of Engineering

28. Series Compensation and Parallel (or Feedback) Compensation . In compensating control systems, we see that the problem usually boils down to a suitable design of a series or parallel compensator. The choice between series compensation and parallel compensation depends on the nature of the signals in the system, the power levels at various points, available components, the designer’s experience, economic considerations, and so on. In general, series compensation may be simpler than parallel compensation; however, series compensation frequently requires additional amplifiers to increase the gain and/or to provide isolation. (To avoid power dissipation, the series compensator is inserted at the lowest energy point in the feed forward path.) Note that, in general, the number of components required in parallel compensation will be less than the number of components in series compensation, provided a suitable signal is available. A) Sereis compensation, b) parallel compensation Dr. Ankita Malhotra SVKM's D J Sanghvi College of Engineering

29. Effects of the Addition of Poles. The 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 system’s 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.) 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 feed forward 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 Dr. Ankita Malhotra SVKM's D J Sanghvi College of Engineering

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31. Lead Compensator In carrying out a control-system design, we place a compensator in series with the unalterable transfer function G(s) to obtain desirable behavior. The main problem then involves the judicious choice of the pole(s) and zero(s) of the compensator Gc (s) to have the dominant closed-loop poles at the desired location in the s plane so that the performance specifications will be met. Consider a design problem in which the original system either is unstable for all values of gain or is stable but has undesirable transient-response characteristics. In such a case, the reshaping of the root locus is necessary in the broad neighborhood of the jw axis and the origin in order that the dominant closed-loop poles be at desired locations in the complex plane. This problem may be solved by inserting an appropriate lead compensator in cascade with the feedforward transfer function Dr. Ankita Malhotra SVKM's D J Sanghvi College of Engineering

32. By drawing the root-locus plot of the uncompensated system (original system), ascertain whether or not the gain adjustment alone can yield the desired closed loop poles. If not, calculate the angle deficiency Փ .This angle must be contributed by the lead compensator if the new root locus is to pass through the desired locations for the dominant closed-loop poles. If static error constants are not specified, determine the location of the pole and zero of the lead compensator so that the lead compensator will contribute the necessary angle Փ . If no other requirements are imposed on the system, try to make the value of a as large as possible. Determine the value of Kc (gain of compensator)of the lead compensator from the magnitude condition. Dr. Ankita Malhotra SVKM's D J Sanghvi College of Engineering

33. Lag Compensation The problem of finding a suitable compensation network for the case where the system exhibits satisfactory transient-response characteristics but unsatisfactory steady-state characteristics. Compensation in this case essentially consists of increasing the open loop gain without appreciably changing the transient-response characteristics. This means that the root locus in the neighborhood of the dominant closed-loop poles should not be changed appreciably, but the open-loop gain should be increased as much as needed. This can be accomplished if a lag compensator is put in cascade with the given feed forward transfer function. To avoid an appreciable change in the root loci, the angle contribution of the lag network should be limited to a small amount, say less than 5°.To assure this, we place the pole and zero of the lag network relatively close together and near the origin of the s plane. Then the closed-loop poles of the compensated system will be shifted only slightly from their original locations. Hence, the transient-response characteristics will be changed only slightly. Dr. Ankita Malhotra SVKM's D J Sanghvi College of Engineering