STABILITY OF INPUT-PERTURBATION EXTREMUM-SEEKING SYSTEMS J. W. WHITE,’ G. J. FOLEY, AND R. J. ALTPETER Department of Chemical Engineerinp., University of Wisconrin, Madison, W i s .
53706
A general stability analysis for input-perturbation extremum-seeking adaptive systems is presented.
The presence of interaction in the dual variable objective function intercouples the adaptive loops, resulting in a locus of critical loop gains with a unique critical frequency corresponding to each gain pair. The applicability of univariable techniques for determination of conditional stability in the absence of appreciable interaction or under canonical transformation is discussed. High quality band pass filtering is recommended for reduction of interloop interference, even though no decrease in loop intercoupling is obtained. The critical gains and system cycling frequencies are determined through implicit solution of the frequency characteristic equation of a linearized model. The computational procedure may b e applied to almost any specified optimizer or process configuration. The results of the hybrid simulation demonstrate the validity of this analysis for a particular example.
NPUT-perturbation extremum-seeking adaptive systems have
I received considerable attention in the literature since the
work of Draper and Li (1951). Although many studies of this variant of extremum adaptation have been performed (Box and Chanmugam, 1962; Deem, 1965; Douce and Ng, 1964; Eveleigh, 1961; Hoyer et al., 1967; Marson and May, 1963; Rajaraman, 1962; Tsien, 1954; Van der Grinten, 1962), little effort has been expended in quantitative consideration of system stability. This is particularly true for multivariable systems, where only qualitatative comments by a few authors (Douce and Ng, 1964; Eveleigh, 1961 ; Hammond and Duckenfield, 1963; Hoyer et a/., 1967; McGrath, 1960) are available. Eveleigh (1 961, 1963) presented a stability analysis of univariable (sinusoidal) input-perturbation extremum-seeking adaptive systems which employed carrier system principles and a modified describing function for derivation of an “equivalent low pass adaptive loop” containing no perturbation signal. The approximation of each dynamic element by its low pass equivalent is undesirable should the process contain transport lag, particularly if this model is to be employed for examination of optimizer transient response. Frey et al. (1966) developed a two-constant linearized model utilizing an over-all loop gain and the process transport lag to aid in study of optimizer stability and transient response. The range of application was restricted by the all-pass nature of the forward path model and the neglect of all process dynamics save transport lag. The two techniques just mentioned have been shown to yield acceptable agreement between predicted and experimental results under certain conditions. However, the present approach unifies and extends these treatments to produce a computational procedure which permits prediction of system cycling frequency and critical gain for almost any specified dual variable application. Stability analyses of several other adaptive systems are available (Donalson and Leondes, 1963; Dymock et al. 1965; Kuntsevich, 1965; Margolis and Leondes, 1959; Rajaraman and Wertz, 1963; Tarjan, 1962).
1
296
Present address, Rohm and Haas Co., Philadelphia, Pa. l&EC FUNDAMENTALS
Basic Optimizer
Sinusoidal perturbation adaptive optimization (SPAO) causes the adjustable process inputs to be manipulated until observed estimates of the first partial derivatives of the objective function with respect to these inputs vanish. This is both a necessary and sufficient condition for operation at a unique extremum as long as unimodality pertains. The objective function may be a steady-state process nonlinearity-the process statics. Dual variable SPAO finds application in processes where the process statics is a function of two manipulatable process inputs and where the presence of an unmeasurable and uncontrollable disturbance necessitates adjustment of both input variables in order to maintain operation at the extremum. The design of a basic dual variable SPA0 configuration requires specification of: Frequency, up,,amplitude, 6,,, and form of the perturbation signals. Band pass filter quality, Q,, and natural frequency, uF,, or high pass filter time constant, T r . Phase, e,, and form of the correlation signals. Feedback gains, k,,(integral) and kp,(proportional), for each loop.
Interaction on Open Loop
An open loop analysis is a prerequisite for investigation of optimizer stability. The dual variable system depicted in Figure 1 is studied here. This may be reduced to a univariable system through operation of only one loop (White, 1968). If each process input, x i , is sinusoidally perturbed ( w p l + w p 2 ) about a point off extremum, xi,-i.e., if xi = .xi0 bpi sin wpit-the output of the linear time-invariant dynamic element preceding the process statics (GiD1), for each loop, may be expressed as
+
there are sufficient dynamic elements between the correlator output and the process statics to perform such averaging, the output of the ith correlator may be approximated by
I
9ic = 6 P l ( 2 - < ) 6 p 2 ( t - l ) ~ ~ ~ ( * - j . j - sin(wpit l)
/
+ +> w c if the phase and attenuation contributions of the dynamics following the VOL. 7
NO. 2
MAY 1968
299
Table 1. Description 1st-order dynamics before statics
4
Attenuation and Phase Lag Contributions Attenuation 111
+ (wpiT)'l[l+ ( ~ c T ) z-'" ll
+ (W~~T)']-'/~
Statics before 1st-order dynamics
[l
Transportation lag
1
High pass filter ( H P F )
[ w p i T ~ I [ 1 -I-
Band pass filter (BPF)o
Q P + ( ~ Q W ~ / W , ) --liZ ~I
Valid only when
WF
tan-' 1
+%T (WpiT)'
WeT d
tan-'
(WpiT~)~l-~''
+
WcTF
1 (wP~TF)' 2Q% tan-' - WP
(natural frequency of filter) equals u p .
statics, with the exception of transport lag, are to be adequately estimated by the formulas in Table I. T h e optimizer was assumed to be cycling about the extremum with the process inputs X degrees out of phase and with different amplitudes of oscillation, 6,1 and 6,2. T h e phase disparity, A, and the cycling ratio, ~ 3 ~ 2 / 6 , l ,may be obtained from equations resulting from combination of Equation 17 with Equation 18. Algebraic manipulation then yields
If the gains, KlZ and K 2 1 , which contain the interaction are small with respect t o the over-all loop gains, K1l and K22, then Equation 21 reduces, by equating amplitudes and phase shifts, t o the univariable stability equations @duct) =
-
Kit = u c t
P
2
(22) (23)
If the gains, K12 and K21, are appreciable, the same procedure yields
Implicit solution of Equations 24 and 25 yields four of the five variables-critical frequency, two integrator gains (cf. one Equation 20), phase disparity, and cycling ratio-when of them is fixed within its permissible range. Thus the interacting dual variable optimizer has one degree of freedom. A locus of critical integrator gains exists and a distinct cycling frequency, phase disparity, and cycling ratio correspond to each point on the locus. T h e key critical parameters are most easily estimated by iterative solution based upon the characteristic equation (Equation 14) rather than on Equations 24 and 25. T h e remaining two unknown variables (cycling ratio and phase disparity) may then be obtained from the above equations. Canonical Transformation
T h e performance of the dual variable adaptive optimizer is influenced by the nature of the process statics, since interaction there couples the loops and decreases the efficiency of the optimizer. T h e interaction may be minimized through re300
Phase Lag tan-' w,T
I&EC F U N D A M E N T A L S
definition of the system inputs in terms of canonical variables and implementation of extremum adaptation in these new variables (Deem, 1965; Foley, 1968; McGrath, 1960); T h e canonical variables, u t , are defined such that the canonical coordinates are parallel to the principal axes of the elliptical contours. First the major and minor axes of the contours are estimated. Then 4, the angle through which the major axis must be rotated, until it is parallel to the abscissa, is measured. The transformation for the dual variable system is then
The contours are circularized by a scaling transformation which makes use of the ratio of the minor axis length, d , to the major axis length, f . Therefore
where u j is the final canonical variable; Since the canonical variables are linear functions of the process variables, a linear combination of sine waves of frequencies, wpl and w p 2 , is applied to each of the process variables to produce a pure sine wave in each of the canonical variables. T h a t is, if the canonical variables are perturbed sinusoidally, the process variable perturbations are given by
Equation 28 is valid also for any periodic perturbation wave form. Figure 3 illustrates the required change in the basic dual variable block diagram (Figure 1) for extremum adaptation in the canonical variables. An important feature to be noted is the requirement of identical phase shift for the similar components of XI and XZthat is, the up' frequency component of XI must enter the process statics in phase with the w p l frequency component of XP. This is accomplished by delaying (or advancing) the input perturbations with the phase shift elements [$1Dl(wpl) ' $ 2 ~ 1 ( W p l ) ] and ['$2~~(Wp2) - ' $ I D I ( W ~ ~ )Without ]. this provision, additional correlating components would be required to extract the desired slope information. One restriction here is that the response function shape must be invariant. Disturbances may translate the position of the optimum, but may not modify the shape without necessitating redefinition of the canonical variables. T h e dual variable linearized model equations (cf. Equations 11 and 12 may be rewritten in terms of the canonical variables as
4 -dt
= -K*[ui(t
- .*)
- u**(t
-
Ti)]
Table II. SPAO Data for Univariable Application Perturbation signal 2.5 Amplitude, volts 1.5 Frequency, c.p.s. Process 1/(0.25 s 1)(0.50 s 1) Dynamics, G D I ( ~ ) 9 = 100 - O . ~ ( X - 50)' Statics exp(-2.0 s) Dynamics, G D S ( S ) Q = 20 Band pass filter W F = w,; Correlator signal e = 12250 Phase
+
PHASE
+
ELEMENTS
Table Ill. Experimental and Predicted Results for Univariable Application
Predicted (Digital Calculation)
Obscrued (Hybrid Simulatzon)
0,2647 0.2647
0.266 0,263
K l c , sec.-l G.~,rad./sec.
U
Figure 3. System modification operation in canonical variables
required
Experimental Verification
for
where
Equation 29 is the univariable linearized model equation. Thus each canonical loop cycles at a critical frequency obtained from iterative solution of Equation 22 even though the actual process inputs, x i , contain a mixture of the two critical frequencies. Canonical variables are recommended (Foley, 1968) as a modification to dual variable SPAO whenever the quadratic coefficients are such that
b i i :> 0.4 b 2 o - b ~ I ~ and
or equivalently whenever the contours are such that (1) the major axis is more than 10' off either coordinate axis, and (2) the ratio of major axis length to minor axis length is greater than 5/4. T h e scaling transformation (Equation 27) is worth-while even when contour rotation is not necessary and is recommended whenever condition 2 is encountered.
Table IV.
The assertions of the previous sections are best substantiated through discussion of the results obtained from examples of square wave perturbation adaptive optimization of a linear time-invariant plant with quadratic statics. Consider the univariable application summarized in Table 11. T h e specified configuration was simulated on a hybrid computer in order to obtain an exact (digital) representation of the process transport lag, GD2, as Pad6 approximates possess undesirable transient characteristics. T h e critical values of over-all loop gain, K,,, and system cycling frequency, w,, determined from hybrid simulation and by numerical iteration, are summarized in Table 111. T h e sustained cycling behavior of the conditionally stable SPAO loop is presented in Figure 4. As may be observed, the transport lag, G D 2 , was introduced after, rather than before, the band pass filter, GF. Although such a situation would not be encountered in practice, it is a valid simulational procedure (GD2 and GF are commutative) which conserves intermachine conversion (ADC, DAC) channels. T h e high quality factor of the band pass filter (Q = 20) was employed primarily for greater signal clarity through attenuation of useless components usually present a t the input to the correlator. I n actual practice a value of 1 to 5 generally represents a satisfactory compromise be tween resolution and speed of response. The effect of a second loop may be examined through consideration of the dual variable application summarized in Table IV. T h e process dynamics were selected to ensure
SPAO Data for Dual Variable Application
Loop I Perturbation signal Amplitude, volts Frequency, C.P.S. Process Dynamics, GL)I Statics Dynamics, G L ~ Z Band pass filter Correlator signal Phase
Loop I I
2.5 1.5
4.0 2.0
