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Analysis of control systems with complex nonlinearities and transportation lag
Analysis
of Control Systems
.jlr,nlinearities
and
With Complex
Transf.wrtatio?z Lag
bZJ K. W. HAN Chun-shan Institute and National Chio-tung University, Taiwan Visiting Research Associate, University of California, Berkeley Und G. J. THALER Department of Electrical Engineering U. S. Naval Postgraduate School, Monterey, SUMMARY: When a control system contains
California
a hysteretic nonlinearity, oriaidead time element
(transportation lag), or a distributed lag, the system may be described by a characteristic equation with complex coeficients which are junctions of dependent variables. The characteristic equation is partitioned and root locus methods are used to analyze stability, existence and characteristics of limit cycles, damping characteristics of stable systems, etc. Control systems with frequency dependent, hysteretic, nonlinearities and with adjustable parameters are analyzed using parameter plane and parameter space methods.
Formulation
of the Problem
The describing function of a hysteretic nonlinearity is a complex function of the amplitude of the driving signal (I), and may be symbolized as Gd = g - jb; the transfer function of dead time is e*r = exp [- (u + jw) T] = (~1 - j/3,; and the transfer function of distributed lag is exp [- (sT)ltZ] = a2 - jP,. In most feedback control problems the linear portion of the system may be described by an open-loop transfer function
(1)
G = N(s),‘D(s)
which is in cascade with the hysteretic element or the dead time so that the transfer function is of the form G loop
=
IT(s) /D (8)WT
@a)
CN(s)D(s)l(g - 3)
(24
or G loop =
from which the characteristic equation is of the form eaTI)
+
N(s)
= 0
GW
= 0
(3b)
or B(S) + (g - jb)N(s)
76
Systemswith Complex Sonlinearities where each of the coefficients of some terms are complex. The problem, then, is to analyze the effects of these complex coefficients on stability and dynamic performance. Although Eqs. 3a and 3b may be in a similar form, their characteristics are quite different. Since the complex coefficients in Eq. 3b are functions of a reference signal (I), which is a positive real number, the signs of the complex coefficients are independent of I; but in Eq. 3a the complex coefficients are transcendental functions of the Laplace operator s, and as the value of s changes from zero to infinity both the magnitudes and the signs of the complex coefficients change accordingly. Thus, a system with a transportation lag is more difhcult to analyze. If a system contains both hysteretic nonlinearity and transportation lag, there are few available methods for analysis. The main purposes of this paper are to use a general stability criterion to analyze the aforementioned systems, and to consider the effects of adjustable parameters and frequency dependent nonlinearities. Stability
Criterion
Consider an Nth order system with characteristic equation as F(s)
a, = 1,.
= 5 CLiSi= 0, i=a
a; = a!, + jP,
(4)
after the substitution s = jw, Eq. 4 becomes
F(s)
=
&(jw)i= 0
(3)
i--O
which can be partitioned into real and imaginary parts FR + FI = 0
(6)
where
(7a)
_jPd jwji+
FI = i,g. 1
I
1
.
5
2=1.3,5..
cxi(jw)i
= 0.
(71))
..
Both Eqs. 7a and 7b involve real coefficients only, and may be factored by any convenient method. The roots of Eq. 7a and 7b may be real, complex, or imaginary. However, converting Eqs. 7a and 7b to the s-variables:
any real roots of Eqs. 7a and 7b are imaginary roots of Eqs. 8s and 8b, because
Vol. 286, No. 1, July 1953
77
K. W. Han andG. J. Thder 8 = jw. In like manner, complex conjugate roots of Eqs. 7a and 7b become complex pairs of roots of Eqs. 8a and 8b with opposite signs for the real parts. Since Eqs. 8a and 8b are just partitions of Eq. 4, it follows that
which is a proper form for constructing root loci. From the general characteristics of root loci, conditions for stability are that: (a) all poles (pi) and zeros (zi) of Eq. 9 are imaginary except that the denominator may have zero root (if either partition is odd-order it should be chosen for the denominator), and (b) poles and zeros must occur in alternating sequences: (l-3)
***z-2
<
p-1
<
z-1 <
po <
21 <
pl <
Z2...
(10)
Note that in order to satisfy these conditions it is necessary that all roots of Eqs. 7a and 7b be real and related as defined in Eq. 10. In a later part of this paper, Eqs. 7a and 7b are called stubizityequath, and the curves which represent:the loci of the real roots of stability equations are called stub?.Zitycurves; (zi and pi) is used to representthe real roots of Eqs. 7a and 7b, respectively. Prediction
of Limit
Cycle
The complex coefficients provided by hysteresis,or dead time, or distributed lag are not constant numbers. However, assuming that the roots of the characteristic equation move slowly, Eq. 7 may be solved repetitively and the inequality of Eq. 10 checked at each step. If there exists a condition for which a pole is equal to a zero, then this condition defines an imaginary root of the characteristic equation and a limit cycle exists at the value of frequency (w) common to the pole zero, with amplitude as defined by the parameterthat causes the complex coefficients to change values. The proposed method has the following advantages: (a) It is relatively eazy to tid the amplitude and the frequency of a limit cycle, because the stability curves are the loci of the real roots of two stability equations; (b) More information can be obtained from a set of stability curves than that from two hodographs (4), or from the commonly used describing function method (5)) because for any value of I the values of the characteristicroots of a system can be predicted from the stability curves approximately (3). This can best be illustratedby the following example: Example
GI =
78
I.
Consider the system in Fig. 1, if the transfer functions are
0.8/(8 + I)(8 + 21,
G2 = 1,
Ga =
1/S,
Gd = 0.05
Journal of The F&
Institute
Systems will ComplexNonlinearities
p*0.05.N,q-lb d.0.025 Y&
FIG. 1. A block diiram of a nonlinear system.
then the characteristicequation is 8 + 3.S2+ (2 + 0.0+&N&+ 0.8Nr = 0
(11)
the stability equations are FR = -02 + 0.266~ = 0 FI = -cd + (2 + O.l6/7rl)w - 0.8b = 0.
(13)
For various values of I, the values of g and b can be found in (5,6) and the real roots of the stability equations can be calculated, or obtained with the root locus method (3) ; thus the stability curves can be plotted as in Fig. 2, which indicates that a stable limit cycle is at Qr.
FIG 2. Stability curves of a system
with two nonlinearities.
Vol.286,No. 1, July1968
FIQ. 3. A sketch of root loci for a nonlinear system with Z = 0.4.
79
K. W. Han
and G. J. Thakr
For eat_&value of I, a vertical line can be drawn on Fig. 2 and its intersections with the stability curves define a set of real roots of the stability equation; thus the values of the characteristic roots (for the specified value of I) can be predicted approximately (3). For I = 0.4, as indicated by the dotted line in Fig. 2, the root loci are sketched in Fig. 3, from which the characteristics of the system (for I = 0.4) can be predicted approximately (3). Control
Systems
and Frequency
with Adjustable Dependent
Parameters
Nonlinearities
When a system has multiple nonlinearities which are related by linear transfer functions, the describing functions become frequency dependent. The commonly used methods for analyzing such nonlinear systems are based on trial and error; i.e., draw families of curves for various values of frequencies (0) and magnitudes (I), and find a point with a proper set of values of w and I which places the system at its stability limit (4, 5). If a nonlinear system has adjustable parameters, and the purpose of analysis is to find the relations among the parameters and the limit cycles, the aforementioned methods are not suitable. In order to solve such problems a new method is introduced in this section. Consider a system with two adjustable parameters; since an intersection between the stability curves is a common solution of stability equations, for each value of I and various values of o a set of points forming a locus of limit cycle can be plotted in a parameter plane by solving the stability equations simultaneously. In other words, any point on the locus of limit cycle represents a set of values of the adjustable parameters which can make the system have a limit cycle with specified values of I and w. For ease in presentation, an example is given : Example II. In Fig. 4, if Gd is open and G1 = K,/ (s + 2) (s + 4)) Gz = Kz, Gs = l/s then the characteristic equation is s3 + 6s2 +
(8 + K&Vz)s
+ K1 (g -
jb)
= 0
(14)
where K1 and Kz are adjustable parameters, and the stability equat,ions are FR= -602+Klg=0 F, = -co3 + (8 + KlK,N2)o
(15)
-
K& = 0.
(16)
KzNzw)
(17)
From Eqs. 15 and 16 K1 = 6w*/g = (-w” + 80)/(b
-
which gives
Kz = (1/Nzw)IC(w2- 8>s/6~1 + bJ. 80
(18) Journal of The Franklin Institute
Systems
with Complex
A’onlinearifies
0.6.. 0.4.. 0.2.
4
FIG. 4(a).
I:::::
4 Nj ~rl,a
N2.E
I’- IIG3l
I%IG51
0
.
:::::
0.2
0.4
0.6
0.0
1.0
I’
(b>
(6)
FIG. 4(b). The g and b curves of a complex nonlinearity.
A block diagram of a nonlinear system.
For various values of I and w, the loci of limit cycle are plotted in a parameter plane as in Fig. 5; thus, for each set of specified values of K1 and & the amplitude (I) and frequency (w) of the limit cycle can be read out. The results given in Fig. 5 can be checked with the method used in the last section. For example, choosing a point Q1 in Fig. 5, for the corresponding values of K1, Kz and w the stability curves are plotted in Fig. 6, where a limit cycle with I equal to 0.5 is located at Q1. Another way of checking the results in Fig. 5 is to solve the stabi1it.y equations. For the selected point (Q1) in Fig. 5, the roots of the stability equations are z1 = 2.9, .z__~= -2.9 and p. = 2.64, pl = 2.9, p-l = -5.54. Thus the limit cycle is caused by the condition of z1 equal to pl. In order to test stability of a limit cycle, assume a small change in the value of I at the limit cycle, and solving the stability equations, the relations among the solutions indicate the characteristics of the limit cycle. For the limit cycle at Q1, results are given in Fig. 6, which indicate that Q1 is an unstable limit cycle.
. K2.. 0.06.
0.04 0.02.. L
0
50
FIG. 5. Limit-cycleloci
Vol. 286, No. 1, July 1968
loo
150
260
of a nonlinear system.
.
KI
FIG. 6. Stability
curves near a limit cycle.
81
K.
w. Han and G. J. Thab Note that, in Fig. 6, the value of w is assumed constant and equal to that at
the limit cycle. In order to check this assumption, a different method for testing stability of a limit cycle is used.
Rewrite Eq. 14 as G,-l
-I- GN = 0
(19)
where GL = Kd(sP + 6.9 + 8s)
(20)
GN=K&fl-i-g-jb
(21)
and
which are the modified linear and nonlinear parts of the characteristicequation, respectively. From Eq. 19 GLGN= -1
(22)
which is a proper form for constructing root loci. The conditions for having a characteristicroot are I GL-’I = I GN I
(23)
and /GN --
/GL-~ = +mr,
m = 1, 3, 5,
?? ??
*
w
thus each point on the limit-cycle-loci should meet these two conditions (4). Rewrite Eqs. 22 and 24 as GL(-GN)
= 1
(25)
and /-GN-
IGL_I=O
(26)
then for various values of I and (J, the loci of GL+ and -ON can be plotted in a polar plane. For the consideredexample, in the neighborhood of the limit cycle (Q), the results are given in Fig. 7, which indicates that for each value of I there is a value of wwhich can satisfy Eq. 23. This value of w is called “coincident frequency” in Ref. (4), and it has been proved that the system stability is decided by the difference between the phase angles of GL-~and -GN at the coincident frequency (4). (This phase angle is termed coincident phase angle in a later part of this paper.) For I equal to 0.4 and 0.6, the coincident frequencies are found, approximately, at 2.6 and 3.2, respectively, as indicated by the two arcs in Fig. 7. In Fig. 7, it is interestingto note that, for various values of I, a locus of coincident frequency can be plotted, and that in the neighborhood of a limit cycle
82
Journal of The Franklin Institute
Systems with Complex Nonlineardies this locus can only cross the locus of GA- 1 at the limit cycle, because any intersection between these two loci defines a limit cycle; thus a few points on the loci of GL-’ and -GN, near the limit cycle, are enough to define the stability characteristics of the liiit cycle, because only the signs of the coincident-phase-angles are required to test stability of a limit cycle. The few points in Fig. 7 are enough to indicate that the limit cycle at Q1 is unstable (4). Since one locus of the coincident frequencies is that of CL-‘, and the other one crosses the “locus of constant frequency” at the limit cycle as indicated in Fig. 7, the signs of the coincident phase angles are decided by the locus of GL-1 and that of the locus ofconstant frequency; thus the method of testing stability of a limit cycle by assuming constant frequency is justified. In short, two methods of testing stability of a limit cycle are available; i.e., (a) plot the stability curves, and (b) plot the loci of GL+ and -GN. The sta-
180.4
Imt
0
0.1
02
0.3
0.4
b
Ka ’
(4 4
FIG. 7. Loci of ff L-1 and -GN in a polar plane.
FIG. 8. Limit-cycle-surface space.
/
in a parameter
bility criterion of the first method is given in Eq. (lo), and that of the second method is < 0 unstable; / - GN - /G&-l = 0 on stability limit; >0 stable. It is worthwhile to note that the considered system may have two limit cycles, because for large I the values of g and b approach unity and zero, respectively, and the roots of the stability equation approach zl, z-~ = f (K,/6)1’2 and PO = 0, pl, p-1 = ~t8 l12.Thus for large I, the necessary condition for stability (for having p. < z1 < pl) is K1 < 48. Since, for some systems, large amplitude signals may be expected, and a small limit cycle may be acceptable or desirable, the analysis may search for a stable limit cycle instead of an unstable one. In short, a convenient way of using the stability equations depends upon the purpose of analysis and design.
Vol. 286, No. 1,
July 1963
83
K. W. Han and G. J. Thaler High Order Nonlinear
Systems
with Multiple
Adjustable
Parameters
FOG a control system with more than two adjustable or variable parameters, with some of the parameters nonlinear, complex, and frequency dependent, commonly proposed methods are not suitable for analysis. In this section, a method of constructing a limit cycle surface in a three dimensional parameter space is presented; and methods of testing stability on both sides of this surface are proposed (7). If a subspace in a parameter space, which is bounded by a limit-cycle-surface, contains all the characteristic roots and all of them have negative real parts, then the system is stable in this subspace. If a system has a limit cycle for all combinations of parameters, then for each assumed value of I a limit cycle surface can be constructed. If a system can be proved stable between two limitcycle-surfaces, then the surface for large I defines the upper limit of the magnitude of the signal of the system, and that for the small I defines the stable limit cycle. Now the problem is to construct the limit-cycle surface and to test stability on both sides of the surface. The following example is presented as an illustration.
Example
ZZZ.
Consider the system in Fig. 4; if G1 = Gf = 6s = Gq = Gj =
K/(s3 + a3s2+ as + ad, KT, l/s, K,, s
then the characteristic equation is ~4 + aas34
(a~ + KK,N3)s2
+ (al + KKdV2)s
+ K (g
jb) = 0
(27)
and the stability equations are FE = w4 - (a2 + KK,iV3) w2 + Kg = 0 F1 = -aw3 + (al + KKTN2)w - K6 = 0.
(28) (2%
From Eqs. 28 and 29 K =
(w”
-
azw”)/(-g
+ KJV3w2) = (-a3w3 + alw)/(b
KTNw)
(39)
which gives K
T
_
N3(al N2(a2
-
a3w2) -
w2)
K
_ a
g(al
-
a3w2) N2u2(a2
+
b(~3 -
w2)
-
a,w)
(31)
By this manipulation, each pair of parameters are related by an equation which
84
Journal of The Franklin Institute
Systems with Complex Nonlinear&es
can be considered separately; thus, all the information in a three-dimensional parameter space can be represented by two parameter planes (7). Assume al = 60, az = 50, a3 = 12 and I = 0.4, for various values of frequencies (w) the results are plotted in Fig. 8, which represents a limit-cyclesurface in a three dimensional parameter space. It can be seen that for each value of I a limit-cycle-surface can be constructed. In order to illustrate that any point on the limit-cycle-surface in Fig. 8 represents a condition for the system to have a limit cycle with amplitude (I) equal to 0.4, and to test stability of the limit cycle, pick a point Q1 and for the corresponding values of K, KT, K, and w the stability curves are plotted in Fig. 9(a) ; likewise, the loci of GL-l and -GN are plotted in Fig. 9(b). The results in both figures indicate that Q1 is a stable limit cycle. By either method, the stability characteristics of the other points on the limit-cycle-surface can be tested; thus, the characteristics of the system on both sides of this surface can be defined. Consideration
of Parameter
Adjustment
From the results of the last two sections, the stability characteristics of a system are decided by the real roots of stability equations; thus, a method relating all the parameters of the system to the real roots of stability equations would be useful for analysis and design.
xw 16
, 110.35 waO.6
6 t
-0.3
-0. I
-0.2
C
Re --* u
(b)
-6
-16
FIG. 9(a).
Stability
Vol. 286, No. 1, July
1968
curves near a limit cycle.
FIG. 9(b).
Loci of GL-l and -G, plane.
in a polar
85
K.
W. Han
and
G. J. Thaler
since esch parameter has a range of variation, each real root of a stability equation has a range of variation. If the range of variation of a zero (zi) overlaps that of a pole (pi), then it is possible to have a limit cycle; on the other hand, if there is no overlap for any variations of the parameters, then the system is eitherab80lUtelySt&ii3or always unstable. Therefore, a direct approach for analysis is to find the range of variations of all poles (pi’s) and zeros (zi’s) as functions of the adjustable parameters. Consider a stability equation of the general form
cY”Fl(W,I) + @“FZ(W,I) + Fa(o,I)
= 0
(32)
where (Y”, 8” are adjustable parameters, and F,, R, F3 are functions of w and I. Let 9 = CY”FI(W, 0 +
F3(0,
(33)
I>
and [ = -B”Fz(u,
I)
(34)
then for various values of o and I, each part of the stability equation, as in Eqs. 33 and 34, can be plotted separately in the 7 V.S.w and E vs. o planes; thus the effects of each parameter can be considered separately and the real roots of the stability equations are defined by the intersectionsof the two families of curves; i.e., 7 = 5. The proposed method is useful for finding the real roots of high order stability equations with two or more adjustable parameters (3). For 3rd or 4th order systems with one or two adjustable parameters in each stability equation, a simplifiedmethod is presented along with the following example. Example IV. Consider the system in Example III, if al = 750, a2 = 275, a3 = 30 and K,, = 0, then the stability equations can be written as Kg/ (w’ - 27~~) = - 1
(35)
and Kb/[w” -
(25 + a’)~] = -1,
(Y’= KKTNs/~O
(36)
which are proper forms for constructing root loci. Since only real values of w are required, the relations between adjustable parameters and the real values of w can be representedby two sets of stability curves, as in Fig. 10, where the effect of each adjustable parameter can be realized directly (3). Using the stability criterion in section II, the following observations can be made: (a) The system always has a limit cycle, because Nz approaches infinity while I approaches zero, which makes pl equal to 22.
86
Journal of The Franklin Institute
Systems with Complex Nonlinearitiees
(b) The system is unstable at any of the following conditions: cz’> (16.6)2 Kb > 110, (Y = 0 Kg >
1.9 x 10’
(because pl > 22) (because Fr has complex roots) (because FR has complex roots)
assume that the permissible amplitude of the small limit cycle is 0.2, then from Fig. 4(b), for 0.2 < I < 00, the ranges of the nonlinear parameters can be defined as 0 < b < 0.32, 0.5 < g < 1, and 0 < Nz < 6.38. Thus, the ranges of the variations of real roots (pi’s and zk’s) for each selected value of K can be found. For K = 5000, the results are given in Fig. 10, which indicate that the condition for I less than 0.2, and with no other limit cycle, is 0 < CY’ < 240, which gives 0 < KT < 0.113.
PO PI (b)
FIG. 10. Stability curves for various values of adjustable parameters.
FIG. 11. Stability curves for a nonlinear system with a transportation lag.
Figure 10 also indicates that, for K T = 0, since ~1, z_~ are quite near pl, p+ the system may have lightly damped, complex, characteristic roots with frequencies between 3.25 and 5.1; likewise, for KT = 0.113, lightly damped characteristic roots with frequencies near 15 can be expected (3). Thus, the best selection of KT depends upon the desired characteristics of the system. For a system with multiple adjustable parameters and nonlinearities, it may not be easy to eliminate a limit cycle, or have a limit cycle with amplitude always less than a specified value; it is also diEcult to determine the effects of alI the adjustable parameters. The methods presented in this section provide a useful tool for analysis.
Vol. 286,No.
1, July
1968
87
K. W. Han and G. J. Thalm No&-r
Control
System
with Transportation
Lag
b mentioned in section I, the effect of a transportation lag is to make some of the coefficients in a characteristic equation depend upon s; thus the real roots of stability equations are functions of frequency (0) , and the stability criterion defined in Eq. 10 is no longer valid, except that the intersection points of the stability curves can be used to indicate where the characteristic roots will cross the imaginary axis of the s-plane; it is still true that, if there is no intersection among the stability curves, there is no limit cycle. Hence, in order to find the stability characteristics of a system, a judgement based upon the linear part of the system is necessary, since the linearized system may have an infinite number of characteristic roots. To illustrate: Example V. Consider the system in Fig. 1, if G1 = lO/(s + 2) (S + 4), Gz = e-*T, Ga = Kl/s, and G4 = Kz, then the characteristic equation is Sa+ 6s2 +
(8 +
10K&~)s
+ 1OKl 1 Nl 1 e--sTe--iO-O
(37)
where 6 = tan-’ b/g, Kl and Kz are adjustable parameters. Let 6 = KzNz and T = .I, then the stability equations are FR=
=0 -Owe+10K~~N~~cos(w+8) -I- (8 + 106)~ = 1OKl I Nl 1sin (6~ + 0) = 0.
F1 = -d
(38)
(30)
For 1 = 0.3 and Kz = 0.25, the stability curves are plotted in Fig. 11, where the frequencies (0) at which the root loci cross the imaginary axis of the s-plane are denoted by Qi)s. Point Q1 indicates that the system has a limit cycle with I = 0.3 if K1 = 0.5. If a locus of Q1, for all values of I, is plotted in a Q us. K1 plane, as in section IV, then it is a stability boundary, because Q1 is the first characteristic root which crosses the imaginary axis while KI and w change from zero to infinity. Conclusion
In this paper, various methods for analyzing nonlinear control systems with complex coefhcients in characteristic equations are presented. Methods of constructing limit-cycle-loci and limit-cycle surfaces are presented for high order systems with adjustable parameters and with multiple, frequency dependent, nonlinearities. Methods of plotting stability curves, and the loci of GL-l and - GN are proposed for testing stability of a limit cycle. Finally, nonlinear systems with a transportation lag are analyzed. Due to the limitations of the describing function theory, applications of the proposed methods are limited to those systems for which the linear parts are of low pass characteristics, but the amount of work required to find a limit cycle and test its stability can be greatly reduced.
88
Journal of The Franklin Institute
Systems with Complex Nonlinearities List of Symbols
s
T 0% Pi
97 b
Gi GL GN i
Laplace operator time of lag real and imaginary parts of complex coefficients real and imaginary parts of a describing function transfer function linear part of the characteristic equation nonlinear part of the characteristic equation integer frequency order of a system real and imaginary parts of characteristic equation real roots of FR and FI, respectively constants open-loop gain gains of feedback compensation signals parameters two parts of a stability equation magnitude of a reference (sinusoidal) signal intersection point of stability curves
References (1) M. L. Shooman, “Stability Analysis of Nonlinear Systems in the Parameter Plane,” IEEE Trans. on Automatic Control, Vol. AC-g, July, 1964. (2) D. D. Siijak, “Analysis and Synthesis of Feedback Control Systems in the Parameter Plane,” Parts I and II, IEEE Tram. Appl. and Id ,Nov., lQ64. (3) K. W. Han and G. J. Thaler, “High Order System Analysis and Design Using the Root Locus Method,” J. l%anklin Inst., Feb., 1966. (4) C. F. Chen and I. J. Haas, “An Extension of Oppelt’s Stability Criterion Based on the Method of Two Hodographs,” IEEE Trans. on Automatic Control, Vol. AC-14 Jan., 1965. (5) G. J. Thaier and R. G. Brown, “Analysis and Design of Feedback Control Systems,” New York, McGraw-Hill Book Co., 1960. (6) J. E. Gibson, “Nonlinear Automatic Control,” New York, McGraw-Hill Book Co., 1961. (7) K. W. Han and G. J. Thaler, “Analysis and Design of Control Systems Using a Parameter Space Method,” IEEE Trans. on Automatic Control July, 1966.
Vol. 286, No. 1, July 1968
89
of Control Systems
.jlr,nlinearities
and
With Complex
Transf.wrtatio?z Lag
bZJ K. W. HAN Chun-shan Institute and National Chio-tung University, Taiwan Visiting Research Associate, University of California, Berkeley Und G. J. THALER Department of Electrical Engineering U. S. Naval Postgraduate School, Monterey, SUMMARY: When a control system contains
California
a hysteretic nonlinearity, oriaidead time element
(transportation lag), or a distributed lag, the system may be described by a characteristic equation with complex coeficients which are junctions of dependent variables. The characteristic equation is partitioned and root locus methods are used to analyze stability, existence and characteristics of limit cycles, damping characteristics of stable systems, etc. Control systems with frequency dependent, hysteretic, nonlinearities and with adjustable parameters are analyzed using parameter plane and parameter space methods.
Formulation
of the Problem
The describing function of a hysteretic nonlinearity is a complex function of the amplitude of the driving signal (I), and may be symbolized as Gd = g - jb; the transfer function of dead time is e*r = exp [- (u + jw) T] = (~1 - j/3,; and the transfer function of distributed lag is exp [- (sT)ltZ] = a2 - jP,. In most feedback control problems the linear portion of the system may be described by an open-loop transfer function
(1)
G = N(s),‘D(s)
which is in cascade with the hysteretic element or the dead time so that the transfer function is of the form G loop
=
IT(s) /D (8)WT
@a)
CN(s)D(s)l(g - 3)
(24
or G loop =
from which the characteristic equation is of the form eaTI)
+
N(s)
= 0
GW
= 0
(3b)
or B(S) + (g - jb)N(s)
76
Systemswith Complex Sonlinearities where each of the coefficients of some terms are complex. The problem, then, is to analyze the effects of these complex coefficients on stability and dynamic performance. Although Eqs. 3a and 3b may be in a similar form, their characteristics are quite different. Since the complex coefficients in Eq. 3b are functions of a reference signal (I), which is a positive real number, the signs of the complex coefficients are independent of I; but in Eq. 3a the complex coefficients are transcendental functions of the Laplace operator s, and as the value of s changes from zero to infinity both the magnitudes and the signs of the complex coefficients change accordingly. Thus, a system with a transportation lag is more difhcult to analyze. If a system contains both hysteretic nonlinearity and transportation lag, there are few available methods for analysis. The main purposes of this paper are to use a general stability criterion to analyze the aforementioned systems, and to consider the effects of adjustable parameters and frequency dependent nonlinearities. Stability
Criterion
Consider an Nth order system with characteristic equation as F(s)
a, = 1,.
= 5 CLiSi= 0, i=a
a; = a!, + jP,
(4)
after the substitution s = jw, Eq. 4 becomes
F(s)
=
&(jw)i= 0
(3)
i--O
which can be partitioned into real and imaginary parts FR + FI = 0
(6)
where
(7a)
_jPd jwji+
FI = i,g. 1
I
1
.
5
2=1.3,5..
cxi(jw)i
= 0.
(71))
..
Both Eqs. 7a and 7b involve real coefficients only, and may be factored by any convenient method. The roots of Eq. 7a and 7b may be real, complex, or imaginary. However, converting Eqs. 7a and 7b to the s-variables:
any real roots of Eqs. 7a and 7b are imaginary roots of Eqs. 8s and 8b, because
Vol. 286, No. 1, July 1953
77
K. W. Han andG. J. Thder 8 = jw. In like manner, complex conjugate roots of Eqs. 7a and 7b become complex pairs of roots of Eqs. 8a and 8b with opposite signs for the real parts. Since Eqs. 8a and 8b are just partitions of Eq. 4, it follows that
which is a proper form for constructing root loci. From the general characteristics of root loci, conditions for stability are that: (a) all poles (pi) and zeros (zi) of Eq. 9 are imaginary except that the denominator may have zero root (if either partition is odd-order it should be chosen for the denominator), and (b) poles and zeros must occur in alternating sequences: (l-3)
***z-2
<
p-1
<
z-1 <
po <
21 <
pl <
Z2...
(10)
Note that in order to satisfy these conditions it is necessary that all roots of Eqs. 7a and 7b be real and related as defined in Eq. 10. In a later part of this paper, Eqs. 7a and 7b are called stubizityequath, and the curves which represent:the loci of the real roots of stability equations are called stub?.Zitycurves; (zi and pi) is used to representthe real roots of Eqs. 7a and 7b, respectively. Prediction
of Limit
Cycle
The complex coefficients provided by hysteresis,or dead time, or distributed lag are not constant numbers. However, assuming that the roots of the characteristic equation move slowly, Eq. 7 may be solved repetitively and the inequality of Eq. 10 checked at each step. If there exists a condition for which a pole is equal to a zero, then this condition defines an imaginary root of the characteristic equation and a limit cycle exists at the value of frequency (w) common to the pole zero, with amplitude as defined by the parameterthat causes the complex coefficients to change values. The proposed method has the following advantages: (a) It is relatively eazy to tid the amplitude and the frequency of a limit cycle, because the stability curves are the loci of the real roots of two stability equations; (b) More information can be obtained from a set of stability curves than that from two hodographs (4), or from the commonly used describing function method (5)) because for any value of I the values of the characteristicroots of a system can be predicted from the stability curves approximately (3). This can best be illustratedby the following example: Example
GI =
78
I.
Consider the system in Fig. 1, if the transfer functions are
0.8/(8 + I)(8 + 21,
G2 = 1,
Ga =
1/S,
Gd = 0.05
Journal of The F&
Institute
Systems will ComplexNonlinearities
p*0.05.N,q-lb d.0.025 Y&
FIG. 1. A block diiram of a nonlinear system.
then the characteristicequation is 8 + 3.S2+ (2 + 0.0+&N&+ 0.8Nr = 0
(11)
the stability equations are FR = -02 + 0.266~ = 0 FI = -cd + (2 + O.l6/7rl)w - 0.8b = 0.
(13)
For various values of I, the values of g and b can be found in (5,6) and the real roots of the stability equations can be calculated, or obtained with the root locus method (3) ; thus the stability curves can be plotted as in Fig. 2, which indicates that a stable limit cycle is at Qr.
FIG 2. Stability curves of a system
with two nonlinearities.
Vol.286,No. 1, July1968
FIQ. 3. A sketch of root loci for a nonlinear system with Z = 0.4.
79
K. W. Han
and G. J. Thakr
For eat_&value of I, a vertical line can be drawn on Fig. 2 and its intersections with the stability curves define a set of real roots of the stability equation; thus the values of the characteristic roots (for the specified value of I) can be predicted approximately (3). For I = 0.4, as indicated by the dotted line in Fig. 2, the root loci are sketched in Fig. 3, from which the characteristics of the system (for I = 0.4) can be predicted approximately (3). Control
Systems
and Frequency
with Adjustable Dependent
Parameters
Nonlinearities
When a system has multiple nonlinearities which are related by linear transfer functions, the describing functions become frequency dependent. The commonly used methods for analyzing such nonlinear systems are based on trial and error; i.e., draw families of curves for various values of frequencies (0) and magnitudes (I), and find a point with a proper set of values of w and I which places the system at its stability limit (4, 5). If a nonlinear system has adjustable parameters, and the purpose of analysis is to find the relations among the parameters and the limit cycles, the aforementioned methods are not suitable. In order to solve such problems a new method is introduced in this section. Consider a system with two adjustable parameters; since an intersection between the stability curves is a common solution of stability equations, for each value of I and various values of o a set of points forming a locus of limit cycle can be plotted in a parameter plane by solving the stability equations simultaneously. In other words, any point on the locus of limit cycle represents a set of values of the adjustable parameters which can make the system have a limit cycle with specified values of I and w. For ease in presentation, an example is given : Example II. In Fig. 4, if Gd is open and G1 = K,/ (s + 2) (s + 4)) Gz = Kz, Gs = l/s then the characteristic equation is s3 + 6s2 +
(8 + K&Vz)s
+ K1 (g -
jb)
= 0
(14)
where K1 and Kz are adjustable parameters, and the stability equat,ions are FR= -602+Klg=0 F, = -co3 + (8 + KlK,N2)o
(15)
-
K& = 0.
(16)
KzNzw)
(17)
From Eqs. 15 and 16 K1 = 6w*/g = (-w” + 80)/(b
-
which gives
Kz = (1/Nzw)IC(w2- 8>s/6~1 + bJ. 80
(18) Journal of The Franklin Institute
Systems
with Complex
A’onlinearifies
0.6.. 0.4.. 0.2.
4
FIG. 4(a).
I:::::
4 Nj ~rl,a
N2.E
I’- IIG3l
I%IG51
0
.
:::::
0.2
0.4
0.6
0.0
1.0
I’
(b>
(6)
FIG. 4(b). The g and b curves of a complex nonlinearity.
A block diagram of a nonlinear system.
For various values of I and w, the loci of limit cycle are plotted in a parameter plane as in Fig. 5; thus, for each set of specified values of K1 and & the amplitude (I) and frequency (w) of the limit cycle can be read out. The results given in Fig. 5 can be checked with the method used in the last section. For example, choosing a point Q1 in Fig. 5, for the corresponding values of K1, Kz and w the stability curves are plotted in Fig. 6, where a limit cycle with I equal to 0.5 is located at Q1. Another way of checking the results in Fig. 5 is to solve the stabi1it.y equations. For the selected point (Q1) in Fig. 5, the roots of the stability equations are z1 = 2.9, .z__~= -2.9 and p. = 2.64, pl = 2.9, p-l = -5.54. Thus the limit cycle is caused by the condition of z1 equal to pl. In order to test stability of a limit cycle, assume a small change in the value of I at the limit cycle, and solving the stability equations, the relations among the solutions indicate the characteristics of the limit cycle. For the limit cycle at Q1, results are given in Fig. 6, which indicate that Q1 is an unstable limit cycle.
. K2.. 0.06.
0.04 0.02.. L
0
50
FIG. 5. Limit-cycleloci
Vol. 286, No. 1, July 1968
loo
150
260
of a nonlinear system.
.
KI
FIG. 6. Stability
curves near a limit cycle.
81
K.
w. Han and G. J. Thab Note that, in Fig. 6, the value of w is assumed constant and equal to that at
the limit cycle. In order to check this assumption, a different method for testing stability of a limit cycle is used.
Rewrite Eq. 14 as G,-l
-I- GN = 0
(19)
where GL = Kd(sP + 6.9 + 8s)
(20)
GN=K&fl-i-g-jb
(21)
and
which are the modified linear and nonlinear parts of the characteristicequation, respectively. From Eq. 19 GLGN= -1
(22)
which is a proper form for constructing root loci. The conditions for having a characteristicroot are I GL-’I = I GN I
(23)
and /GN --
/GL-~ = +mr,
m = 1, 3, 5,
?? ??
*
w
thus each point on the limit-cycle-loci should meet these two conditions (4). Rewrite Eqs. 22 and 24 as GL(-GN)
= 1
(25)
and /-GN-
IGL_I=O
(26)
then for various values of I and (J, the loci of GL+ and -ON can be plotted in a polar plane. For the consideredexample, in the neighborhood of the limit cycle (Q), the results are given in Fig. 7, which indicates that for each value of I there is a value of wwhich can satisfy Eq. 23. This value of w is called “coincident frequency” in Ref. (4), and it has been proved that the system stability is decided by the difference between the phase angles of GL-~and -GN at the coincident frequency (4). (This phase angle is termed coincident phase angle in a later part of this paper.) For I equal to 0.4 and 0.6, the coincident frequencies are found, approximately, at 2.6 and 3.2, respectively, as indicated by the two arcs in Fig. 7. In Fig. 7, it is interestingto note that, for various values of I, a locus of coincident frequency can be plotted, and that in the neighborhood of a limit cycle
82
Journal of The Franklin Institute
Systems with Complex Nonlineardies this locus can only cross the locus of GA- 1 at the limit cycle, because any intersection between these two loci defines a limit cycle; thus a few points on the loci of GL-’ and -GN, near the limit cycle, are enough to define the stability characteristics of the liiit cycle, because only the signs of the coincident-phase-angles are required to test stability of a limit cycle. The few points in Fig. 7 are enough to indicate that the limit cycle at Q1 is unstable (4). Since one locus of the coincident frequencies is that of CL-‘, and the other one crosses the “locus of constant frequency” at the limit cycle as indicated in Fig. 7, the signs of the coincident phase angles are decided by the locus of GL-1 and that of the locus ofconstant frequency; thus the method of testing stability of a limit cycle by assuming constant frequency is justified. In short, two methods of testing stability of a limit cycle are available; i.e., (a) plot the stability curves, and (b) plot the loci of GL+ and -GN. The sta-
180.4
Imt
0
0.1
02
0.3
0.4
b
Ka ’
(4 4
FIG. 7. Loci of ff L-1 and -GN in a polar plane.
FIG. 8. Limit-cycle-surface space.
/
in a parameter
bility criterion of the first method is given in Eq. (lo), and that of the second method is < 0 unstable; / - GN - /G&-l = 0 on stability limit; >0 stable. It is worthwhile to note that the considered system may have two limit cycles, because for large I the values of g and b approach unity and zero, respectively, and the roots of the stability equation approach zl, z-~ = f (K,/6)1’2 and PO = 0, pl, p-1 = ~t8 l12.Thus for large I, the necessary condition for stability (for having p. < z1 < pl) is K1 < 48. Since, for some systems, large amplitude signals may be expected, and a small limit cycle may be acceptable or desirable, the analysis may search for a stable limit cycle instead of an unstable one. In short, a convenient way of using the stability equations depends upon the purpose of analysis and design.
Vol. 286, No. 1,
July 1963
83
K. W. Han and G. J. Thaler High Order Nonlinear
Systems
with Multiple
Adjustable
Parameters
FOG a control system with more than two adjustable or variable parameters, with some of the parameters nonlinear, complex, and frequency dependent, commonly proposed methods are not suitable for analysis. In this section, a method of constructing a limit cycle surface in a three dimensional parameter space is presented; and methods of testing stability on both sides of this surface are proposed (7). If a subspace in a parameter space, which is bounded by a limit-cycle-surface, contains all the characteristic roots and all of them have negative real parts, then the system is stable in this subspace. If a system has a limit cycle for all combinations of parameters, then for each assumed value of I a limit cycle surface can be constructed. If a system can be proved stable between two limitcycle-surfaces, then the surface for large I defines the upper limit of the magnitude of the signal of the system, and that for the small I defines the stable limit cycle. Now the problem is to construct the limit-cycle surface and to test stability on both sides of the surface. The following example is presented as an illustration.
Example
ZZZ.
Consider the system in Fig. 4; if G1 = Gf = 6s = Gq = Gj =
K/(s3 + a3s2+ as + ad, KT, l/s, K,, s
then the characteristic equation is ~4 + aas34
(a~ + KK,N3)s2
+ (al + KKdV2)s
+ K (g
jb) = 0
(27)
and the stability equations are FE = w4 - (a2 + KK,iV3) w2 + Kg = 0 F1 = -aw3 + (al + KKTN2)w - K6 = 0.
(28) (2%
From Eqs. 28 and 29 K =
(w”
-
azw”)/(-g
+ KJV3w2) = (-a3w3 + alw)/(b
KTNw)
(39)
which gives K
T
_
N3(al N2(a2
-
a3w2) -
w2)
K
_ a
g(al
-
a3w2) N2u2(a2
+
b(~3 -
w2)
-
a,w)
(31)
By this manipulation, each pair of parameters are related by an equation which
84
Journal of The Franklin Institute
Systems with Complex Nonlinear&es
can be considered separately; thus, all the information in a three-dimensional parameter space can be represented by two parameter planes (7). Assume al = 60, az = 50, a3 = 12 and I = 0.4, for various values of frequencies (w) the results are plotted in Fig. 8, which represents a limit-cyclesurface in a three dimensional parameter space. It can be seen that for each value of I a limit-cycle-surface can be constructed. In order to illustrate that any point on the limit-cycle-surface in Fig. 8 represents a condition for the system to have a limit cycle with amplitude (I) equal to 0.4, and to test stability of the limit cycle, pick a point Q1 and for the corresponding values of K, KT, K, and w the stability curves are plotted in Fig. 9(a) ; likewise, the loci of GL-l and -GN are plotted in Fig. 9(b). The results in both figures indicate that Q1 is a stable limit cycle. By either method, the stability characteristics of the other points on the limit-cycle-surface can be tested; thus, the characteristics of the system on both sides of this surface can be defined. Consideration
of Parameter
Adjustment
From the results of the last two sections, the stability characteristics of a system are decided by the real roots of stability equations; thus, a method relating all the parameters of the system to the real roots of stability equations would be useful for analysis and design.
xw 16
, 110.35 waO.6
6 t
-0.3
-0. I
-0.2
C
Re --* u
(b)
-6
-16
FIG. 9(a).
Stability
Vol. 286, No. 1, July
1968
curves near a limit cycle.
FIG. 9(b).
Loci of GL-l and -G, plane.
in a polar
85
K.
W. Han
and
G. J. Thaler
since esch parameter has a range of variation, each real root of a stability equation has a range of variation. If the range of variation of a zero (zi) overlaps that of a pole (pi), then it is possible to have a limit cycle; on the other hand, if there is no overlap for any variations of the parameters, then the system is eitherab80lUtelySt&ii3or always unstable. Therefore, a direct approach for analysis is to find the range of variations of all poles (pi’s) and zeros (zi’s) as functions of the adjustable parameters. Consider a stability equation of the general form
cY”Fl(W,I) + @“FZ(W,I) + Fa(o,I)
= 0
(32)
where (Y”, 8” are adjustable parameters, and F,, R, F3 are functions of w and I. Let 9 = CY”FI(W, 0 +
F3(0,
(33)
I>
and [ = -B”Fz(u,
I)
(34)
then for various values of o and I, each part of the stability equation, as in Eqs. 33 and 34, can be plotted separately in the 7 V.S.w and E vs. o planes; thus the effects of each parameter can be considered separately and the real roots of the stability equations are defined by the intersectionsof the two families of curves; i.e., 7 = 5. The proposed method is useful for finding the real roots of high order stability equations with two or more adjustable parameters (3). For 3rd or 4th order systems with one or two adjustable parameters in each stability equation, a simplifiedmethod is presented along with the following example. Example IV. Consider the system in Example III, if al = 750, a2 = 275, a3 = 30 and K,, = 0, then the stability equations can be written as Kg/ (w’ - 27~~) = - 1
(35)
and Kb/[w” -
(25 + a’)~] = -1,
(Y’= KKTNs/~O
(36)
which are proper forms for constructing root loci. Since only real values of w are required, the relations between adjustable parameters and the real values of w can be representedby two sets of stability curves, as in Fig. 10, where the effect of each adjustable parameter can be realized directly (3). Using the stability criterion in section II, the following observations can be made: (a) The system always has a limit cycle, because Nz approaches infinity while I approaches zero, which makes pl equal to 22.
86
Journal of The Franklin Institute
Systems with Complex Nonlinearitiees
(b) The system is unstable at any of the following conditions: cz’> (16.6)2 Kb > 110, (Y = 0 Kg >
1.9 x 10’
(because pl > 22) (because Fr has complex roots) (because FR has complex roots)
assume that the permissible amplitude of the small limit cycle is 0.2, then from Fig. 4(b), for 0.2 < I < 00, the ranges of the nonlinear parameters can be defined as 0 < b < 0.32, 0.5 < g < 1, and 0 < Nz < 6.38. Thus, the ranges of the variations of real roots (pi’s and zk’s) for each selected value of K can be found. For K = 5000, the results are given in Fig. 10, which indicate that the condition for I less than 0.2, and with no other limit cycle, is 0 < CY’ < 240, which gives 0 < KT < 0.113.
PO PI (b)
FIG. 10. Stability curves for various values of adjustable parameters.
FIG. 11. Stability curves for a nonlinear system with a transportation lag.
Figure 10 also indicates that, for K T = 0, since ~1, z_~ are quite near pl, p+ the system may have lightly damped, complex, characteristic roots with frequencies between 3.25 and 5.1; likewise, for KT = 0.113, lightly damped characteristic roots with frequencies near 15 can be expected (3). Thus, the best selection of KT depends upon the desired characteristics of the system. For a system with multiple adjustable parameters and nonlinearities, it may not be easy to eliminate a limit cycle, or have a limit cycle with amplitude always less than a specified value; it is also diEcult to determine the effects of alI the adjustable parameters. The methods presented in this section provide a useful tool for analysis.
Vol. 286,No.
1, July
1968
87
K. W. Han and G. J. Thalm No&-r
Control
System
with Transportation
Lag
b mentioned in section I, the effect of a transportation lag is to make some of the coefficients in a characteristic equation depend upon s; thus the real roots of stability equations are functions of frequency (0) , and the stability criterion defined in Eq. 10 is no longer valid, except that the intersection points of the stability curves can be used to indicate where the characteristic roots will cross the imaginary axis of the s-plane; it is still true that, if there is no intersection among the stability curves, there is no limit cycle. Hence, in order to find the stability characteristics of a system, a judgement based upon the linear part of the system is necessary, since the linearized system may have an infinite number of characteristic roots. To illustrate: Example V. Consider the system in Fig. 1, if G1 = lO/(s + 2) (S + 4), Gz = e-*T, Ga = Kl/s, and G4 = Kz, then the characteristic equation is Sa+ 6s2 +
(8 +
10K&~)s
+ 1OKl 1 Nl 1 e--sTe--iO-O
(37)
where 6 = tan-’ b/g, Kl and Kz are adjustable parameters. Let 6 = KzNz and T = .I, then the stability equations are FR=
=0 -Owe+10K~~N~~cos(w+8) -I- (8 + 106)~ = 1OKl I Nl 1sin (6~ + 0) = 0.
F1 = -d
(38)
(30)
For 1 = 0.3 and Kz = 0.25, the stability curves are plotted in Fig. 11, where the frequencies (0) at which the root loci cross the imaginary axis of the s-plane are denoted by Qi)s. Point Q1 indicates that the system has a limit cycle with I = 0.3 if K1 = 0.5. If a locus of Q1, for all values of I, is plotted in a Q us. K1 plane, as in section IV, then it is a stability boundary, because Q1 is the first characteristic root which crosses the imaginary axis while KI and w change from zero to infinity. Conclusion
In this paper, various methods for analyzing nonlinear control systems with complex coefhcients in characteristic equations are presented. Methods of constructing limit-cycle-loci and limit-cycle surfaces are presented for high order systems with adjustable parameters and with multiple, frequency dependent, nonlinearities. Methods of plotting stability curves, and the loci of GL-l and - GN are proposed for testing stability of a limit cycle. Finally, nonlinear systems with a transportation lag are analyzed. Due to the limitations of the describing function theory, applications of the proposed methods are limited to those systems for which the linear parts are of low pass characteristics, but the amount of work required to find a limit cycle and test its stability can be greatly reduced.
88
Journal of The Franklin Institute
Systems with Complex Nonlinearities List of Symbols
s
T 0% Pi
97 b
Gi GL GN i
Laplace operator time of lag real and imaginary parts of complex coefficients real and imaginary parts of a describing function transfer function linear part of the characteristic equation nonlinear part of the characteristic equation integer frequency order of a system real and imaginary parts of characteristic equation real roots of FR and FI, respectively constants open-loop gain gains of feedback compensation signals parameters two parts of a stability equation magnitude of a reference (sinusoidal) signal intersection point of stability curves
References (1) M. L. Shooman, “Stability Analysis of Nonlinear Systems in the Parameter Plane,” IEEE Trans. on Automatic Control, Vol. AC-g, July, 1964. (2) D. D. Siijak, “Analysis and Synthesis of Feedback Control Systems in the Parameter Plane,” Parts I and II, IEEE Tram. Appl. and Id ,Nov., lQ64. (3) K. W. Han and G. J. Thaler, “High Order System Analysis and Design Using the Root Locus Method,” J. l%anklin Inst., Feb., 1966. (4) C. F. Chen and I. J. Haas, “An Extension of Oppelt’s Stability Criterion Based on the Method of Two Hodographs,” IEEE Trans. on Automatic Control, Vol. AC-14 Jan., 1965. (5) G. J. Thaier and R. G. Brown, “Analysis and Design of Feedback Control Systems,” New York, McGraw-Hill Book Co., 1960. (6) J. E. Gibson, “Nonlinear Automatic Control,” New York, McGraw-Hill Book Co., 1961. (7) K. W. Han and G. J. Thaler, “Analysis and Design of Control Systems Using a Parameter Space Method,” IEEE Trans. on Automatic Control July, 1966.
Vol. 286, No. 1, July 1968
89