Root Locus

The root locus method: famous curves, control designs and non-control applications Annraoi de Paor Department of Electronic and Electrical Engineering, National University of Ireland, Dublin, Ireland E-mail: [email protected] Abstract Famous named curves generated as root loci produce optimum and pseudo-optimum stability designs for realistic systems. Non-minimum-phase control involves a principle of topological certainty. A root locus with complex breakpoints is discussed. Cassini’s ovals in Brauer’s method of eigenvalue localisation are illustrated. A problem in dielectric theory, recast into an imaginary-parameter root locus, is solved via real-parameter theory. Continuation, translation and scaling are invoked. It is hoped to impart an appreciation of the versatility of root-locus-inspired thinking. Keywords eigenvalue localisation; named curves; optimum stability; root locus

Introduction In this paper, the root locus method is taken to be a set of theorems and techniques for visualising how the roots of the real-coefficient polynomial P(s)=N(s)+kM(s)

(1)

where N(s) and M(s) are monic polynomials of degrees n and m, respectively, with n≥m, move about the s plane as k varies over some defined range. The method was hinted at by Walter Evans1 in 1948, developed by him in 1950,2 and in 1954 given a definitive exposition which still informs modern textbooks.3 The status of the root locus method was summarised by O. I. Elgerd4 in 1967: ‘The Root Locus Method, introduced by Evans in 1948, won early popular acclaim and has since become our most important synthesis technique for single-input-single-output linear servomechanisms’. The years which have elapsed since this claim was made have enhanced rather than diminished the status of the root locus method, and real-parameter root locus plotting routines are now an essential constituent in all reasonably comprehensive control system design and analysis packages (e.g. MATLAB and Program CC). The Evans approach is purely geometrical, based on the polar form of the equation P(s)=0

(2)

which leads to two relations Arg N(s)−Arg M(s)=Arg(−k)=lp (angle condition: locus equation) (3) and |N(s)|/|M(s)|=|k| (magnitude condition: calibration equation) International Journal of Electrical Engineering Education 37/4

(4)

The root locus method: famous curves

345

In eqn (3), l is an integer, odd for k>0 and zero or even for k<0. The positive parameter locus (k>0) is much more common than that for negative parameter. Nonetheless the latter, via the principle of continuation, can be very valuable in design studies. Analytical study of the root locus method is well developed in the former Soviet sphere of influence (e.g. Teodorchik and Bendrikov5), but is largely neglected in Western Europe and the USA. This is unfortunate, since the Evans method, for all its power, obscures the fact that the locus equation is often simply computed, and only for n+m>6 requires factorisation of an equation more complicated than a quadratic.6 Only two Western textbooks with which the author is familiar make any attempt to introduce geometrical and graphical approaches on anything like an equal footing.7,8 The analytical approach is based on the Cartesian form of eqn (2). By letting s=s+ jv

(5)

expanding eqn (1) into real and imaginary parts (noting the restriction here to real k), equating both separately to zero in eqn (2) and eliminating k, the result is Re N Im M−Im N Re M=0

(6)

Equation (6) always contains v as a factor, showing that the real axis (v=0) is part of the locus, with different segments, readily separable by Evans’ rules, belonging to k positive and k negative. The remaining portion, Re N(Im M/v)−(Im N/v) Re M=0

(7)

describes the branches off or at isolated crossings of the real axis, the latter being the breakpoints of the Evans theory. This paper explores the equations and shapes of curves yielded by eqn (7) for various constellations of ‘poles’ (roots of N(s)) and ‘zeros’ (roots of M(s)). Configurations which seem at first to have little or no physical significance give rise to famous named curves9 which have simple constructions.10 These may, by invoking three basic principles of $

$ $

continuation (i.e. if k=k +dk, then P(s)=(N(s)+k M(s))+dkM(s)) 0 0 translation of axes scaling of axes

yield controller designs for realistic processes which possess the optimum stability property that the rightmost eigenvalue is as deep in the left half plane as possible. (This is a feature which the author has used with success over many years in many studies, of which Refs. 11, 12, 13 and 20 are just a selection.) Windmills on the mind: the Trisectrix In 198014 the author invented a novel aerodynamic theory of the optimum actuator disc, which is an idealised wind turbine having an infinite number of International Journal of Electrical Engineering Education 37/4

A. de Paor

346

infinitely then drag-free blades, and later extended the theory to a design tool for real wind turbines.15 Using s rather than g as the variable, an equation in the theory, P(s)=(s−1)3+k(s−1/3)

(8)

had to be subjected to root locus analysis. Continuation is initially invoked by letting k=−3+dk

(9)

axes are then translated by making the substitution s=s∞+3

(10)

and finally, the plot is scaled by letting s◊=3Ts∞

(11)

This leaves the effective problem of exploring the roots of P(s)=(s+1/T )2s+k(s+8/(9T ))

(12)

where the notation has been returned to the form in eqn (1). This is the characteristic polynomial of the feedback system shown in Fig. 1, with G (s)=k /(s+1/T )2 p 1 G (s)=k (1+8/(9Ts)) (13) c 2 k=k k 1 2 We are now dealing with PI control of a double lag process. The positive parameter root locus of eqn (12), plotted by Program CC, is shown in Fig. 2. The system is asymptotically stable for all 0
(15)

This is the Trisectrix of Colin MacLaurin, invented by that famous Scottish mathematician in the early 18th century to facilitate trisection of an angle.9 D(s) + R(s)

+ –

GC(s)

+

GP(s)

Y(s)

Fig. 1 Basic feedback system. International Journal of Electrical Engineering Education 37/4

Imag s

The root locus method: famous curves

347

0

–1.5

–1

–.5

0

Real s

Fig. 2 Root locus of the T risectrix system.

1 step reference

Output

.8 .6 .4

step disturbance

.2 0 0

5

10

15

Time

Fig. 3 Responses of the T risectrix system.

We propose to call the system described by eqns (13) and (14) the Trisectrix system. PID control of a double integrator process: the Limac¸on of Pascal We consider the double integrator process G (s)=k /s2 p 1 under PID control:

(16)

G (s)=k (s+a)2/s (17) c 2 The double integrator is a significant test case for controller design, which runs ˚ stro¨ m and Wittenmark.15 The characteristic polyright through the book by A nomial is P(s)=s3+k(s+a)2

(18)

whose positive parameter root locus is plotted in full line in Fig. 4. The system is asymptotically stable for k>a/2

(19)

given k>0, a>0 as necessary conditions. Following the complex roots as they move into the left-half plane, we find that they meet as far to the left as possible International Journal of Electrical Engineering Education 37/4

A. de Paor

348

Imag s

2 1 0 –1 –2 –6

–5

–3

–4

–1

–2

0

1

Real s

Fig. 4 PID control of a double integrator process.

in a double root at s=−3a, for k=27a/4

(20)

With a slight softening of the principle enunciated before, we define eqn (20) to give a pseudo-optimum stability design. This is because, although the complex roots have fused into a real pair as far into the left half plane as possible, the third root is not as far to the left as possible, since it moves steadily towards s=a as k goes from 0 to 2. Time responses are given for k =1 in Fig. 5. 1 The analytical equation for the root locus of eqn (18) is the Limac¸ on of Etienne Pascal (discovered circa 1640 by the father of the more famous Blaise Pascal).17 In fairly standard notation it is (s2+v2+2as)2=a2(s2+v2)

(21)

Taking s as independent variable, it only requires repeated factorisation of a quadratic in v2 to plot the curve. The negative parameter locus, shown in dotted line in Fig. 4, fills in the minor lobe with which students of the Limac¸ on are familiar. Filtered PID control of a two-lag process: Strophoid and Circle We begin by considering the root locus of the polynomial

Output

P(s)=s4+k(s+a)2

(22) 1.2 1 .8 .6 .4 .2 0

step reference

step disturbance 0

1

2

3

4

5

6

7

8

Time

Fig. 5 Responses of the L imac¸on system: PID control of a double integrator International Journal of Electrical Engineering Education 37/4

The root locus method: famous curves

349

The analytical equation of this is readily factorised into the two curves (s+a)2+v2=a2

(circle)

(23)

v2=s2(s+a)/(a−s) (strophoid)

The strophoid is a 90° rotation of the famous Folium of Rene´ Descartes.9 The circle belongs to k<0 and the strophoid to k>0. Both pass through the origin. The leftmost breakpoint at s=−2a belongs to k=−16a2. Applying continuation with k =−16a2, which gives a real root at s=2a(1+√2), translating 0 the origin to that real root, scaling to bring the leftmost poles, now at s= −2a(2+√2), to the value s=−b, and finally reverting to the notation of eqn (1) leads to consideration of P(s)=(s+b)2(s+(2b√2)/(2+√2))s+k(s+(3+2√2)b/(2(2+√2)))2

(24)

The positive-parameter root locus of eqn (24) is shown in Fig. 6, for b=1. We are now considering filtered PID control of a two-lag process: G (s)=k /(s+b)2 p 1 (25) G (s)=k (s+(3+2√2)b/(2(2+√2)))2/(s(s+2b√2/(2+√2))) c 2 The intersections of the circle and strophoid give a quadruple eigenvalue which has the optimum stability property, and for which k =4b2/(k (2+√2)2) 2 1 Corresponding time responses are shown for k =1 in Fig. 7. 1

(26)

Optimum stability PI control of a non-minimum-phase system We now consider a process and controller

Imag s

G (s)=k (−s+b)/(s+a)2 p 1 G (s)=k (s+c)/s c 2 (a, b, c>0)

(27)

0

–1.5

–1

–.5

0

Real s

Fig. 6 Filtered PID control of a two-lag process

International Journal of Electrical Engineering Education 37/4

A. de Paor

350

1 step reference

Output

.8 .6 .4

step disturbance

.2 0 0

5

10

15

Time

Fig. 7 Responses of the Strophoid and Circle system: filtered PID control of a two-lag process.

The characteristic polynomial is P(s)=s(s+a)2−k(s−b)(s+c)

(28)

so that we are dealing with a negative parameter root locus. Given values of a and b, the existence of an optimum stability design is tackled by exploring all possible topologies for c>0. The author invokes here the principle of topological certainty, which follows from an approximation theorem for root locus topology discovered by him many years ago.18 This asserts that all topologies which do not violate Evans’ rules are certain to occur for some specified parameter values. In this case it turns out that if c is chosen so that for some value of k we get a triple root, then those values of k and c give an optimum stability design. To find the required values of c and k, we imagine that P(s) is a perfect cube, i.e. P(s)=s3+(2a−k)s2+(a2−k(c−b))s+kbc =s3+3ds2+3d2s+d3

(29)

Equation (29), after quite a bit of tedious algebra, yields the following equations for k=k k and c: 1 2 k3−(9b+6a)k2+(12a2+35ab+27b2)k−(8a3+9a2b)=0 (30) c=(2a−k)3/(27kb) For a, b>0, the pattern of signs in the first member of eqn (30) establishes that it has only one real positive root for k. As a simple example, taking a=b=1, there follows k=0.2377968, and c= 0.8523119. The corresponding root locus plot is in Fig. 8, and time responses for k =1 in Fig. 9. 1 PID control of a three-lag process with integrator: a singular breakpoint behaviour We now consider the characteristic polynomial P(s)=s2(s+a)3+k(s+b)2 International Journal of Electrical Engineering Education 37/4

(31)

The root locus method: famous curves

351

Imag s

1 0 –1 –1

1

0

3

2

4

Real s

Output

Fig. 8 Optimum stability PI control of a non-minimum-phase system.

1 .8 .6 .4 .2 0 –.2 –.4

step reference step disturbance

0

5

10

15

Time

Fig. 9 Responses for optimum stability control of a non-minimum-phase system.

which arises from G (s)=k /(s(s+a)3) p 1 G (s)=k (s+b)2/s c 2 The breakpoint equation,

(32)

N(s)(dM/ds)=M(s)(dN/ds)

(33)

after cancellation of common factors, simplifies to s2+(5b/3)(s+2a/5)=0

(34)

The positive b-parameter root locus for this is the circle centred on s=−2a/5, with radius 2a/5. The breakpoint at the origin simply reflects the fact that P(s) has a double root at the origin for the trivial case k=0, whereas the other real breakpoint at s=−4a/5 corresponds to b=0.96a. However, root locus analysis (or application of the Routh–Hurwitz test) shows that this value of b gives an unstable system for all k>0. Now a very singular behaviour, which the author has never before encountered, emerges: it turns out (initially from numerical experimentation but analysed below) that every other point on the circle, with the sole exception of that corresponding to b=0.36a

(35) International Journal of Electrical Engineering Education 37/4

A. de Paor

352

belongs to k complex. The condition b=0.36a itself gives complex breakpoints through which the locus is passing for k=k k =0.8a3 (36) 1 2 The corresponding root locus is shown in Fig. 10. It is seen that this is an optimum stability design. Corresponding time responses are shown for k =1 1 in Fig. 11. Because the breakpoint behaviour is so singular it seems to merit special analysis. By using eqn (34) repeatedly any polynomial in s of degree ≥2 can be reduced to one of first degree. Hence, in the equation P(s)=0, we can arrive at an expression of the form k=(cs+d)/(es+ f )

(37)

We now select any point s=g+ jh on the circle described by eqn (34) and pose the problem: under what condition will it give k real? Substituting this value of s into eqn (37), the required condition is readily obtained as cf =ed

(38)

Now, c, f, e and d are all functions of a and b and working through the tedious

Imag s

2 1 0 –1 –2 –4

–3

–2

–1

0

1

2

3

4

Real s

Fig. 10 Optimum stability PID control of a three-lag process with integrator.

Fig. 11 Responses of three-lag process with integrator under optimum stability PID control. International Journal of Electrical Engineering Education 37/4

The root locus method: famous curves

353

algebra eventually yields b3−(28/25)ab2+(1031/625)a2b−(236/625)a3=0 which factorises to (b−0.36a)(b−0.96a)(b−a)=0

(39)

in accordance with previous findings.

Eigenvalue localisation: Gershgorin circles and Cassini ovals In state space analysis of systems in vector-matrix form, dx/dt=Ax+Bu

(40)

y=Cx

it is sometimes convenient to get a preliminary idea of the locations of the eigenvalues of A by invoking the familiar theorem of Gershgorin.19 If R is the i sum of the moduli of the off-diagonal elements from the ith row (or column) of A, then the eigenvalues lie on or within the union of the circles |s−a |=R (i=1, 2, ..., n) (41) ii i The circle described by eqn (41) is the off-real-axis portion of the positiveparameter root locus of the polynomial P(s)=(s−a −R )2+k(s−a ) (42) ii i ii Since Program CC, for example, allows root locus plots to be superimposed, it may be conveniently used to apply Gershgorin’s theorem. A generalisation of Gershgorin’s theorem, due to A. Brauer,19 is that the eigenvalues of A lie on or within the union of 1/2n(n−1) Cassini ovals |s−a | |s−a |=R R ii jj i j

(i=1, 2, ..., n−1; j=i+k, k=1, 2, ..., n−i) (43)

The main problem in applying Brauer’s theorem, as explained to the author by mathematician friends, is the sheer tedium of drawing the Cassini ovals. However, root locus theory provides a way forward. The author has discovered that the typical oval described by eqn (43) is the negative-parameter root locus of the polynomial P(s)=(s2−(a +a )s+a a +R R )2+k(s−a )(s−a ) (44) ii jj ii jj i j ii jj Cassini ovals for a =1, a =−1 and the three values of R R , 2, 1, and 0.3 ii jj i j are shown in Fig. 12. The intermediate value R R =1 gives the Lemniscate of i j Bernoulli; R R greater than this gives a single oval, and R R less than this i j i j gives two disjoint closed curves. International Journal of Electrical Engineering Education 37/4

A. de Paor

354

RiRj=2

Imag s

1

RiRj=1

0

RiRj=.3

–1 –3

–2

–1

0

1

2

3

Real s

Fig. 12 Cassini ovals as root loci.

A problem in dielectric theory As final illustration of the power of root-locus-inspired thinking, we consider a problem in the behaviour of complex permittivity, posed to the author by his friend Professor B. K. P. Scaife of Trinity College, Dublin. In developing a theory of complex permittivity in liquid dielectrics, Professor Scaife came up with the equation whose notation is here modified: (s−1)(2s+1)/s=A/(1+ jaT )

(45)

The problem is whether, for 0
(46)

By studying the imaginary parameter root locus, Arg N(s)−Arg M(s)=−p/2

(47)

it is possible to deduce quite readily that the model is reasonable. However, imaginary parameter root locus plotting is not available as an option in any package with which the author is familiar. To study the problem via real parameter locus theory, eqn (47) is recast as N(s)=−jaT M(s), which leads to N2(s)+a2T 2M2(s)=0

(48)

The right half plane portion of the root locus of eqn (48), drawn for k=a2T 2, is shown for A=20 by the full curve of Fig. 13. The lower half is the portion of physical interest. The broken line shows the circle generated from the relevant International Journal of Electrical Engineering Education 37/4

The root locus method: famous curves

355 complex permittivity locus

Imag s

5

0 circle

–5 0

5

15

–10

20

Real s

Fig. 13

A problem in dielectric theory.

poles and zeros in eqn (46): P∞(s)=(s−(A+1)/4−1 √((A+1)2+8)))2+k(s−1)2 4 It is seen that the full curve does indeed lie outside the circle.

(49)

Conclusion Some unusual and, it is hoped, useful perspectives on the root locus method have been given by using famous named curves, along with invocation of the principles of continuation, translation of axes and scaling, to deduce optimum and pseudo-optimum stability designs of controllers for processes of varied structures. Differentiation of topologies with the help of the breakpoint equation has also been illustrated in the problem of PID control of a three-lag-plusintegrator process. The author believes that his method of analysis in this situation is original, and he hopes that it may help to shed light on the occurrence of complex breakpoints in general in real-parameter loci. Finally, it has been shown that a non-real-parameter root locus problem may, in a special case, be converted to a real-parameter one, and it is hoped that this feature also may be the kernel of future developments. Two of the problems treated — those from windmill theory and dielectrics — arose outside the domain of control theory, and it is hoped that they may help to illustrate the largely untapped versatility of the root locus method. It certainly deserves to be recognised as a general problem-solving tool whose scope far transcends the domain in which it was originally proposed.

Acknowledgements The author wishes to acknowledge that a shortened version of this paper was presented, under the title ‘50 years of root locus: some new thoughts’, at the 2nd IFAC workshop on New Trends in Design of Control Systems, at Smolenice, Slovakia, September 7–10, 1997. He also wishes to thank his friends Toma´ s Ward and Brian Cogan for very helpful discussions. International Journal of Electrical Engineering Education 37/4

A. de Paor

356

References 1 2 3 4 5

6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

W. R. Evans, ‘Graphical analysis of control systems’, T rans. AIEE, 67 (1948), 547–551. W. R. Evans, ‘Control system synthesis by the root locus method’, T rans. AIEE, 69 (1950), 66–69. W. R. Evans, Control System Dynamics (McGraw-Hill, New York, 1954). O. I. Elgerd, Control Systems T heory (McGraw-Hill, New York, 1967). K. F. Teodorchik and F. A. Bendrikov, ‘The method of plotting root paths of linear systems and for qualitative determination of the path type’, in T heory of Continuous L inear Control Systems (Butterworth, London, 1963), pp. 7–10. M. Harley and H. M. Power, ‘Analytical equations and geometrical constructions for some common root loci’, Int. J. Elec. Enging. Educ., 11 (1973), 45–59. S. H. Lehnigk, Stability T heorems for L inear Motions (Prentice Hall, Englewood Cliffs, 1966). H. M. Power and R. J. Simpson, Introduction to Dynamics and Control (McGraw-Hill, Maidenhead, 1978). RCA (=R.C. Archibald), ‘Curves, special’, Encyclopaedia Britannica, 6 (1957), 887–899. K. Rektorys (ed.), Survey of Applicable Mathematics (translated from the Czech), (Iliffe, London, 1964). H. M. Power, ‘Analytical root locus design study of servomechanisms with inertial damping’, Int. J. Elec. Enging. Educ., 9 (1971), 411–424. H. M. Power, ‘Analytical design of servomechanisms with spring-coupled inertial. dampers’, Int. J. Control, 14 (1971), 497–511. A. M. de Paor, ‘A modified Smith predictor and controller for unstable processes with time delay’, Int. J. Control, 41 (1985), 1025–1036. A. M. de Paor, ‘Teoiric u´ r don teascghnı´omhro´ ir agus don mhuileann gaoithe’ (in Irish=A new theory for the actuator disc and the windmill ), T echnology Ireland, 11 (1980), 35–38. A. M. de Paor, ‘Aerodynamic design of optimum wind turbines’, Appl. Energy, 12 (1982), 221–228. ˚ stro¨ m and B. Wittenmark, Computer-controlled Systems: T heory and Application, K. J. A 2nd edn (Addison Wesley, Reading, 1990). C. B. Boyer, A History of Mathematics, 2nd edn, revised by U. C. Merzbach (John Wiley, New York, 1991). H. M. Power, ‘Approximation theory for establishing root locus topology’, Electron. L ett., 5 (1969), 97–99. R. A. Horn and C. R. Johnson, Matrix Analysis (Cambridge University Press, Cambridge, 1985). A. M. de Paor, ‘Concepts of optimum stability for linear feedback systems’, Int. J. Elec. Enging. Educ, 36 (1999), 46–64.

International Journal of Electrical Engineering Education 37/4