Chapter 8 SENSITIVITY ANALYSIS OF INTERVAL SYSTEM BY ROOT LOCUS APPROACH

Transcription

1 Chapter 8 SENSITIVITY ANALYSIS OF INTERVAL SYSTEM BY ROOT LOCUS APPROACH

2 SENSITIVITY ANALYSIS OF INTERVAL SYSTEM BY ROOT LOCUS APPROACH 8.1 INTRODUCTION Practical systems are difficult to model, as this modeling is based on the assumption that, there is sufficient information available about the plant and with, the short term or unreliable data records available [28][29][30], In such systems the parameters of model are identified as a set of possible values and not one single value for a parameter. The systems with such parameters obtained as intervals instead of real numbers are termed as interval systems. It is desirable to know the effects of variations of the parameters of interval system on its overall performance. The pole zero approach is useful to define a measure of sensitivity in terms of the position of the roots of the characteristic equation as, the roots of the characteristic equation represent the dominant mode of the transient response [31]. Sensitivity analysis is a technique by which one can determine, with good approximation, whether or not the system will work within the specified tolerance when the parameters of its model vary between their limits [28][32]. Here the well known root locus technique is used for determination of sensitivity of interval systems to its parameter variation. Calculation of sensitivity using this technique can prove of importance in designing robust control for interval systems. 8.2 ROOT LOCUS TECHNIQUE Mathematical models of control systems differ from the real plant for many reasons. The major reasons being, system has incomplete knowledge as, it may not be physically existing, model known is too complex to consider and needs to be simplified with some approximations, and parameters of the system can change over a tolerance due to unknown, 93

3 unpredictable or phenomena difficult to understand. The model then available at the disposal of a control engineer consists of uncertainties. These uncertainties can be unstructured uncertainties where the system equations are not known or structured uncertainties where, the system equations are known but the parameters of these equations are not known. The parameters may be varying over closed intervals in these systems. These uncertainties effectively violate the optimality conditions leading to great material losses. Sensitivity analysis hence becomes a mandatory stage in designing a robust system. For examining effects of uncertain inputs within a model, various analytical and computational techniques as, Sum of Tolerances, Worst Case, Probabilistic Transformation of Variables, Method of Moments, Monte Carlo, Fuzzy Arithmetic and Modal Interval Arithmetic exist. These methods however, deal with the problem of obtaining maximum and minimum values of the model, defining the interval for parameter variations. This paper illustrates the use of the popular root locus technique extended for multiple parameter variation, for determining sensitivity of interval systems. As an inherent advantage of root locus technique, we can comment on stability of interval system over given intervals of parameter variations without any need of writing Kharitonov Polynomials [5] [7] and then applying conventional techniques like Routh- Hurwitz Criterion and Nyquist Envelop [33] Technique. Root locus technique uses characteristic equation of system for finding effect of variation of gain on the closed loop poles. Applying root locus technique to interval system needs determination of characteristic equation of the interval system from its open loop transfer function G(s). In the case of interval systems, the open loop transfer function being consisting intervals instead of real numbers, the characteristic equation has to be determined using rules in interval arithmetic [34] 94

4 8.3 PARAMETER VARIATION Root locus concept developed originally concerns with the determination of locations of roots of characteristic equation with change in system gain K from zero to infinity. However the same technique can be extended to investigate the effect on location of these roots with variation in any other parameter of the system [31]. Characteristic equation of a system can be written as, aks" + an a{s + a0 = Effect of variation of parameter ao on location of roots of above equation can be found using, ^ = ansn + an_xsn a,5 This equation being in l+g(s)h(s) = 0 form, the second term enables us to plot root locus for variation in parameter ao. Similarly to plot root locus for variation in parameter ai we can use equation, ^ = <*n-\s ~ o0 Let us see now, how these root loci help us finding sensitivity of interval system. 8.4 SENSITIVITY ANALYSIS In general the basic problem is to provide a quantitative measure of the deviation of a system function when the element that comprisesthis function varxefin some prescribed manner. This degree of variation or sensitivity is conventionally expressed as the ratio of percentage change in the function to percentage change in the parameter [32]. Thus, gt _dyly _ x dy x dx / x y dx 95

5 represents sensitivity of y to the variations in x (y being function of x). This idea can be used in studying the dependence of any system function on any of its parameters. As we have discussed earlier, root locus technique is useful to investigate variation of roots of characteristic equation of the system with, variation in any parameter. Roots of the characteristic equation being representing dominant mode of transient response of the system, study of variation of these roots is of vital importance in control system design [31][35]. Sensitivity of these roots to the system parameters is termed as root sensitivity. Plotting root locus for variations in different parameters, the change in location of roots can be determined to find root sensitivity. 8.5 SEQUENTIAL EXAMPLE Consider an interval system = [2,3]s2+[17.5,18.5]$ + [15,16] g4 K [2,3>3+[17,18]s2+[35,36> + [20.5,21.5] whose characteristic equation is $3 + [6.333,10.5]s2 + [17.5,27.25]$ + [11.833,18.75] = Let, [6.333,10.5] = a, [17.5,27.25] = (3, [11.833,18.75] = y. The characteristic equation of the interval system then can be written as, 53 +as2 + J3s + y = 0 Nominal values of the intervals are denoted as, a0,j30,y0. Their values be, 96

6 Re. a0 =8.4165, (30 = , y0 = Characteristic equation of interval system with nominal values of its parameters is written as, s s = 0 Roots of this characteristic equation are, rx = y } r2 = jl r3 = These are the dominant roots of the system; hence sensitivity of these roots is of interest. Variation of these roots with variation in a, P and y can be plotted (Fig. 8.1) using root locus technique [31] [34] when one parameter allowed varying and other two kept constant at some value in the interval. Root locus for variation in a can be plotted using equation, as s Root locus for variation in p can be plotted using equation, (3s s * Similarly root locus for variation in y can be plotted using equation, 1 + -j s s r =

7 Although these equations give root locus for variation in parameter from zero to infinity, variation of parameter only over the given intervals is of our interest. As it can be observed from Fig. 8.2, these root loci when superimposed on each other intersect at points where roots of the characteristic equation are located, which is obvious. 8.6 CALCULATION OF SENSITIVITY Let us now find sensitivity of roots of the characteristic equation to, variations in a, P and y. To start with, we will calculate root sensitivity to +1% change in a, P and y from their nominal values. Change in a by +1% : a0 = , A«= 0.01 x cr0 = New value of a,a+f -a0+aa = With this change, the characteristic equation becomes, s3 + 8,500665s s = Roots of this equation are, rx = j r2 = y r3 = These are the new locations of the roots of characteristic equation, s + ctqs + /?o s + y g = 0 Comparing ri, r2, r3 in eqn. 8.7 and eqn 8.12, we can find change in root locations as shown in Fig

8 The change in roots are A r, = y' A r2 = ~y A r3 = Hence sensitivities of these roots of eqn. 8.6 with +1% change in a from cto can be found as, rx Ar, _ y a + 1 Aa/a 0.01 = J23.06 = 23.34Z Fig. 8.2 Root loci for a, p and y superimposed r2 &r2 _ 0, a + 1 A a/a 0.01 = y'23.06 = 23.34Z ^3 Ar3 = a + 1 A a/a 0.01 = 1.15Z0 Similarly we can calculate sensitivity of the roots to -1% variation in a from oto and variation in (3 and y from Po and yo respectively. 99

9 8.7 SIMULTANEOUS VARIATION OF PARAMETERS We have discussed so far, how to determine sensitivity of roots for variation of only one parameter at a time when other parameters are kept constant. Now we will calculate sensitivity for simultaneous variation in more than one parameter. For this purpose we have to follow a simple algorithm. Start: Write characteristic equation of given interval system, and find nominal values of parameters. Fig. 8.3 Change in root location due to change in a, p and y by +1% Step 1: Consider change in any one of the parameters and keep all others constant. This change causes the characteristic equation to change and hence change in root locations. Step 2: Consider change in the second parameter with first parameter kept at value as changed in step one and the other parameters at their original value. Characteristic equation changes with this change giving new locations of roots. Sensitivity calculated by comparing the new roots with the original roots is for simultaneous variation of first two parameters. Roots compared with those in step one give, sensitivity of roots in step one to variation in second parameter only. Step 3: Consider change in third parameter with first two parameters kept at new value and the remaining at their original value. Roots of the new characteristic equation obtained by changing third parameter give sensitivity 100

10 of original roots to simultaneous change in all the three parameters. Roots when compared with those found in step two give sensitivity of roots in step two to change in the third root only. This process can be continued for as many steps as the number of simultaneously variable parameters appear in characteristic equation. 8.8 SEQUENTIAL EXAMPLE Considering same interval system whose characteristic equation is given as in eqn. 8.6, we will find root sensitivity for simultaneous variation of parameters a, p and y. Step 1: Consider change in a by +1% from its nominal value do, with P and y kept at their nominal values. The effect of variation of only a by +1% gives sensitivities as calculated earlier. Step 2: Now let us vary p by +1% from its nominal value keeping a at its new value and y at nominal value. Aj3 = New characteristic equation is, s s s = Roots of this equation are, rt = y' r2 = y r3 = These roots when compared those of eqn. 8.6 give root sensitivity to simultaneous variation of a and p by +1 %. Sensitivities are calculated as, 101

11 5 1 = S.135Z51.538' a/ Vl = 8.135Z -5\.538 and, a/3 + l S h = 1.72Z1800 ajff + 1 The roots when compared with the roots found in Step 1 give the sensitivities of roots in Step 1 to +1% variation in p. S r' = 16.75Z S ^ = 16.75Z S T 3 =2.87Z180 >0 + 1 These sensitivities shall be compared with sensitivities of roots of characteristic equation with nominal values of the parameters. Step 3: Now we will consider change in the third parameter y by +1% when a and p maintained at their new values as in Step 2. New characteristic obtain can be written as,.s S = Roots of this equation are, r, = y' r2 = j r3 = Comparison of roots above with roots as in eqn. 8.7 give sensitivity of former roots to simultaneous variation in a, P, and y by +1% from their nominal values. 102

12 = Z65.2' a/3y + \ S Tl = Z ^ = 0.14Z00 af3y +1 Comparing roots found above with those in Step 1 we can find sensitivity of roots in step 1 to simultaneous variation in p and y by +1% from their nominal values. Comparison with roots in Step 2 gives sensitivities of roots in Step 2 for +1% change in y only. Sensitivities of roots of eqn. 8.6 to -1% variation in all the three parameters varied simultaneously are calculated as, S r' = 4.14Z a/3y-1 S ^ =4.14Z af3y-1 S r3 =0.14Z180 afiy CONCLUSION Root locus technique being simple to understand and easy for calculations, the technique can be readily applied to interval systems. Sensitivity of roots at breakaway points is maximum compared to that at any other location. Calculation of sensitivity for interval systems for variation in any (combination) of the parameters by any amount is possible. A comment can be made on stability of interval system without writing Kharitonov polynomials and then applying the conventional Routh-Hurwitz or Nyquist criteria. Sensitivity analysis of interval system enables design of robust controller for the system for better performance. 103