Dynamics of adaptive systems

zyxwvutsrqponmlkjihgfed zyxwvutsrq zyxwvutsrqponmlk zyxwvutsrqpon lt1.t TRANSACTIONS ON CIKCIIITS ANI) SYSTI!vIS. VOL. 37. NO. 4. APRIL 1990 547 Dynamics of Adaptive Systems D A H U R E R M A N A N D ELUMER Ahstract -In this paper we introduce a siniple adapthe control mechaniwi into nonlinear s>stem\ which are capable of complicated oscillaton \tates and chaotic dynamics. We show that it provides efficient regulation while displa! ing novel behavior. Sudden perturbations in the s!stem’\ parameters can degenerate into chaotic bursts with no precursors, When such bursts occur, the s!stem first re\erberates wildl! and then reco\ers in times that are in\er\ely proportional to the d f f n e s s of the control. W e a h exhibit a general control principle which provides a quantitatite relation between the nia\imum amplitude of a perturbation from which a s!\teni can recober, and the \peed at which it does \o. The existence of chaotic dynamics in systems which by their very nature are nonlinear poses serious problems for their control. The consequent unpredictability of their behavior implies that the observation of an error signal at a given time cannot be translated into an obvious procedure to reduce it to zero later on. Nevertheless. i t is well established that a number of natural and artificial systems manage to operate in regular and smooth fashion in spite of their intrinsic nonlinear:ities, a fact which raises the interesting question about the mechanisms underlying their stability. In many cases. equilibria are achieved through adaptive control mechanisms in which feedback signals are used to produce stable outputs withn a range of parameter values. In many process applications, it is often the case that stable outputs are required in spite of the presence of nonlinear effects that can be present in either the plant itself or its controls. While an extensive literature exists on the subject of linear controls (see, for example, [l]).little is known about their nonlinear counterparts [2]-[4]. In this context, Hubler and collaborators [ 5 ] , [6] have recently proposed a procedure for controlling chaotic systems which relies on statistical forecasting techniques whch are used to construct a model of the dynamical system and to then change its parameters. Since by their bery nature these techniques imply a delay between the detection of an error signal and the system’s prediction, they do not allow the immediate damping of errors after their sudden appearance. In this paper, we introduce a simple adaptive control mechanism into nonlinear systems whch are capable of complicated oscillatory states and chaotic dynamics. We show that besides providing efficient regulation, it displays novel behavior. In particular, we demonstrate how sudden perturbations in the system’s parameters can degenerate into chaotic bursts with no precursors. When such bursts occur, the system first reverberates wildly and then recovers in times that are inversely proportional to the stiffness of the control mechanism. Finally, we exhibit a general control principle whch provides a quantitative relation between the maximum amplitude of a perturbation from whch a system can recover, and the speed at which i t does so. Consider a general adaptive system with feedback of the type shown in Fig. 1. Its behavior is specified by the values of an zyxwvu I zyxwvu 1 1 zyxwv zyx Dynamical System / t Adaptive Algorithm dn Fig 1 Schematic diagram of an adapti\e feedback system. d , is the desired output. i n ,IS the o h s e n e d one and e , IS the error signal at t m e n. output variable y,, at discrete time intervals, and an error signal e,, which is equal to the difference between the actual ourput of the system at time 12 and a specified g o d output, d,. This error signal is in turn used to change the control parameters of the system so as to reduce the error to zero. These parameters might also be viewed as controlled inputs driving the system to its desired regime. Finally, the system might also be subject to important perturbations in its internal parameters or independant inputs, otherwise weakly fluctuating. Those perturbations typically occur in an asynchronous and unpredictable way. Such a picture can be used to describe a large variety of systems, ranging from a self-regulating biological unit [7] or complex social and economical organizations [ 81, to artificial adaptive structures such as adaptive recursive filters, adaptive controls [9] and recursive neural networks [lo]. We claim that for a large class of adaptive systems one can write their dynamics as zyxwvuts where F ( J , , k , ) is a general nonlinear function of class C, in the output variable J ; , k , the control parameter, and G ( e,, ,( d e , / d k , )) is a continuous function of the error e, and its derivative with respect to k , . Although discrete in time, these equations also describe (through their stroboscopic sampling) continuous dynamical systems, and are therefore not restricted to discrete controls. When the function G is expressed as the product of its variables, e,,+l.(de,+l/dk,), the adaptive equation (2) is identical to the LMS algorithm [9], widely used for the adaptation of linear systems. As mentioned before, we intend to investigate in this paper some general features of nonlinear systems undergoing adaptation. We will therefore use an adaptive mechanism which, besides providing efficient regulation, is the simplest + zyxwvuts Manuhcript receibed March 8. 1989: rsviscd Jul? 1 X . 19x9 This \\ark w a s wpported in part by the Office of Naial Research under Contract N00014-82069Y This paper was recommended by Associate Editor T. Matsurnoto. I3 A. Huhernan is with Xerox Palo Alto Research Center. Palo Alto. CA 94304 E Lurncr i \ with thc Department of Applied Physics. Stanford Uni\ersity. Stanlord. CA 94305 IEEE Log Number X933909. 0098-4094/90/0400-0.547$01.00 Cl990 IEEE 548 zyxwvutsrq zyxw zyxwvutsrqponmlkjih IEEF IRANSACIIONS ON CIRCIJITS A N D smitxs. VOL. 37, NO 4. APRIL 1990 ‘.Or- zyxwvuts zyxwvutsrqponmlk zyxwvutsrqponmlkj zyxwvutsrqp zyxwv 0.0 I ~~ 0.0 1 ~~~ 150 n ~~~ ~ ~~~~ ~ zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA I 300 Fig. 2 Time evolution of the outoput signal. r,. for a aystem with c = 0.05 F is a sigmoid function The ordinated values are normalized to the maximum possiblc value of the burst. At r1 = 150 a perturbation in the \ d u e of the control parameter increases it by a factor of 2 5 that allows a straightforward quantitative analysis. This turns out to be k,+l =k, - ( dzl] e.e,+l.sign __ We are mainly interested in the transients following any sudden change of the control parameter, and on time scales where the goal output d, does not change much. We can therefore set it to a constant value, ( y ) , so that (4) reads In what follows, we will concentrate on functions F that possess a parabolic maximum and verifying the condition sgn(dF(.v,, k , ) / d k , ) = l . Our choice is dictated by the fact that such functions are well studied and describe a wide universality class of dynamic systems [ l l ] , which range from nonlinear oscillators to distributed computational systems with no global controls [12]. With no controls, ( c = 0) (1) gives rise to a wide repertoire of dynamical behaviors. As the parameter k increases in value, one observes an infinite cascade of period doubling bifurcations leading to a chaotic regime characterized by extreme sensitivity to initial conditions and broad band power spectra. Within this chaotic regime there exist multiple periodic windows and intermittent phenomena [ll]. For non-zero values of the control stiffness c , (4a) provides a general adaptive control mechanism that in the absense of major disturbances keeps the evolution of the system arbitrarily close to the desired fixed point, ( y ) . The parameter c defines the stiffness of the adaptive control mechanism. Notice however, that for sudden disturbances of the parameter values these equations can lead to extremely complicated behavior. This is due to the fact that if k were to cross over to a regime that produces chaotic dynamics, the relaxation into equilibrium might proceed through a set of intermediate values which do not necessarily converge back to the original fixed point of (4a). A typical scenario is illustrated in Fig. 2, where we show the time evolution of the dynamical system represented by (1) and (4a) in the presence of a sudden change in the control parameter Fig 3. Time e\olution o f the adaptive \!stem for the same function a\ ~n Fig 2 hut ~ i t ch = 1 . At the time indicated b> the vertical arrom. the control parametcr undcrgoe\ ti 5uddc.n decrease h> :i factor of 2 5 k . The simulation is for a system whose dynamics is modelled as a sigmoid mapping, i.e. F( ~ , . k , , )= k,[ sgn( yn . u l ) sgn( yn, U>)] where u1 = 4, u2 = 8, and sgn(J.. U ) = e x p ( y n ) / ( l + e x p ( y U)). The stiffness coefficient is set to E = 0.05. The goal output is set to = 0.1, and the corresponding relaxed control parameter ( k ) is increased by a factor of 2.5 at 12 = 150. As can be seen, this disturbance induces a wild chaotic burst in a narrow time interval, after which the relaxation mechanism produces a quick damping of the error signal. T h s bursty regime, or relusution chaos, was originally observed in models of neurophysiological regulation [7]. Notice that if one were to attempt controlling this burst by setting the stiffness parameter to a higher value, the result can be paradoxical. as shown in Fig. 3 for e = l . In t h s case, rather than relaxing to the desired fixed point the system undergoes a transition into a persistent nonlinear oscillation. Thus, increased controllability can actually destabilize the system. The occurence of similar bursts. although never explicitly identified as relaxation chaos. have been reported in various artificial adaptive systems [13]-[15]. This leads us to believe that the model described above, despite its formal simplicity, contains most of the dynamical features of more complex systems. These results show that the goal of keeping the output of a nonlinear adaptive system at a fixed value is not always compatible with the wide dynamical repertoire it can exhibit. Rather, one has to compromise between the robustness of the system and the speed at which it recovers from perturbations. In order to establish this tradeoff quantitatively. we now compute the speed of recovery of the system as a function of the magnitude of the perturbation and the stiffness control value e . Let the fixed point of the dynamics be specified by ( ( . p ) , ( k ) ) . The stable basin, S,, around it is defined in such a way that given any initial value in that region. the system will relax in finite time to its fixed point. Consider the case where at an arbitrary time in the discrete dynamics, the parameter k is shifted from ( k ) to ( k ) - Sk and such that the point ( y o = k , = ( k ) - S k ) is in S,. In order to evaluate the time, N , it takes for the system to recover to its fixed point, we notice that right after the perturbation its state is determined by ~ 0 7 ) ( ~ 3 ) . yl= F( yo, k o ) k,=(k)-Sk -c(.v~-(J)) ( 5) (6) IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS. VOL. zyxwvutsrqponml zyxwvutsrqpo zyx zyxw 37, NO. 4, APRIL 1990 549 = 0.5, ( k ) = 2). The above considerations lead to the following result. First, notice that the fact that the stable basin S, is finite, implies that there is a maximum value of the perturbation, Sk,,, beyond which the system never recovers to its previous equilibrium value. Moreover, numerical experiments show that the maximum value which produces monotonic recovery depends inversely on the value of the stiffness control, i.e., Sk,, = g(c), with g(c) a decreasing function of E. We now conjecture that for every adaptive control in the universality class defined above, this relation will hold, with gt(c) inversely proportional to a linear power of c. This, together with (9) and (12) leads to the following control principle: y,,) and the dynamics occurs around the fixed point ( ( y ) A7 zyxwvutsrqponml zyxwvutsrqponm 250 0.0 - - .~ zyxwvutsrqpo 500 n Fig. 4. Recovery times for (1) and (4a) after a sudden change in the control parameter b y n o precursors. When such bursts occur, the system first reverber ( o n . k , ) = k , , v , ( l - y n ) . Curve (a) corresponds t o f = 0.05, (b) c = 0.1. (c) f = 0.2. so that if this process is continued for a number N of iterations, the finite precision with which the error is measured’ leads to the following equations: This equation can be rewritten as N = Sk/cS,y (9) T-’Sk,, = constant (13) independent of the control stiffness. In other words: there exists a quantitative relation between the maximum amplitude of a perturbation from which an adaptive system can recover, and the speed at which it does so. This relation is independent of both the control stiffness and of the detailed nature of the system, provided it belongs to the general class defined above. We have tested this principle by conducting numerical experiments for a number of different functions with parabolic maxima, including the sigmoid and logistic functions described above. To high precision, (13) is obeyed by a wide range of parameter values, which leads us to believe that it might be more general than for the class of functions for which it is formulated. These results lead to a number of conclusions. First, the control principle sets a sharp limitation on the speed of recovery for an adaptive control with given robustness. Second, the notion of recovery in our context has a stronger meaning that the one usually assigned to Lyapunov stability. Whereas the latter emphasizes recovery from infinitesimal perturbations, this study concentrates on recovery from large perturbations in a state space with multiple attractors. We close by pointing out that the simple adaptive control that we have proposed can be used in a number of processes where nonlinearities are unavoidable. It can be simply implemented for the control of nonlinear oscillators, and through the modulation of the intrinsic payoff values given to resources, it can also be used to stabilize computational systems with no global controls. Although we have only reported here its success to control fixed points, it has been applied to control limit cycles in both discrete and continuous dynamics [18]. Last but not least, it provides a simple model with which to study issues of self-regulation and control in many other systems. zyxwvutsrq where If we further restrict ourself to nonlinear mappings F(y,, k , ) that factor out as k , P ( y , ) , the actual otuput following the perturbation, y l , can be written as (y)(1- ( S k / ( k ) ) ) . Assuming that the relaxation proceeds exponentially at a rate ~ / T , S J Jis then approximated by the expression If the recovery time N is much bigger than 7 T, (10) simply reads as Sk REFERENCES or equivalently, Sk By comparing (9) and (12), we conclude that under our assumptions, the control stiffness c is proportional to the relaxation rate of the controlled dynamics. This relation is illustrated in Fig. 4, where the relaxation of the output variable following a change in the control parameter by a factor of 0.05 has been plotted for increasing values of the stiffness c = 0.05, 0.1, and 0.2. The nonlinear mapping is the logistic function F ( y n ,k , ) = k , y , , ( l ‘Notice that the notion of recovery is well defined only for systems whose behavior is measured with a finite precision. An infinite precision in the measurement of a n error signal. for example. would imply the impossibility of deciding when it becomes zero. For a study of such effects, see [16] and [17]. zyxw See, for example, B. D 0. Anderson er U / . , Sruhr/ig A n d i m s of Adapricv Systems: Pusrir~rrva i d A i w a g i i i g A m ~ l w s . Cambridge. MA: M I T Press. 1986. I. M. Y. Mareels and R. R. Bitmead, “Bifurcation effects in robust adaptive control,” I E E E Trails. Circuits . Y u r . . vol 35. pp. 835-841, July 1988. “ O n the dynamics of an equation arising in adaptive control,” in Proc. 25rh I E E E Conf. oil Decision und Control, Athens, Greece. pp. 1161-1166, 1986. F. M . A. Salam and S . Bai. “Complicated dynamics of a prototype continuous-time adaptive control system.” I E E E Trails Circuits Svsr.. vol. 35. pp. 842-849. July 1988. A. Hubler and E. Luscher. “Resonant stimulation and control o f nonlinear oscillators.” T U M preprint L R 3895. 1988. A. Hubler. “Adaptive control of chaotic systems.” T U M preprint LR 3X95.1988. R. King, J. D. Barchas. and B. A. Huberman. “Theoretical psychopathology: An application of dynamical svstcms theory to human behavior.” in Swer,qeru s o/ rhe Rruitr. (E. Basar. H. Flohr. and H. Haken, Eds.) New York: Springer-Verlag. pp 352-364. 1983 _. zyxwvutsrqponml zyxwvutsrqponml zyxwvutsrqponmlkj zyxwvutsrqponmlkji zyxwvutsrq R L Su inth. Or.,quni:utmnu/ Si~strvnsjot Z l r i n u p n c n t l k \ i q t i i u y , Pki,intn,q ( i d 1 ~ 1 1 p l ~ ~ n i ~ ~ ~ tColumbus. r ~ t r t i ~ ~ i O H . ( i n d . 1974 R . Widrow and S. D Stearns, , 4 d u p r 1 1Siguul ~~ Pr.mc\\inq En_elcnood Cliffs. N J : Prentice-Hall, 19x5 L. B. Almeida. “Backpropagation in pcrceptronh u i t h feedhxk.“ S A T 0 AS1 Series, vol F41. Ncural Computcrs. R Eckmiller .ind C‘h Ma\burg. Springer-Verlag. 1988 H. <i Schustcr. Dererniiiiisric Chuos \‘CH Weinhelm. 19XX B. A. Hubcrman and T Hogg, “The Hehavior of Computational Ecologies,” in Tltr f:i~olog~i f Contpurutron. B. A Hubcrman. Ed North-Holland p p 77-116, 19x8 B D 0. Anderson. “Adaptive Sbstems. Lack of Persistench of Excitution and Bursting Phenomena.” 4titoniui1iu. vol. 71. no. 3. pp 247-25X. 1985. 0. Macchi and M. Jaidane-Saidane, tability of Adaptive Rccur\i\e Filters.” I E E E Intenrurroriol Conferenc‘ n A c o w t i < . ~S, p e e d i . ntid Si,qnu/ P r o < e s s ~ n gvol . 3. pp. 1503-1505. 19Xt W . A . Sethares. C R. Johnson, Jr. and C. E Rohrs. Bursting in Telephony Loops with Adaptive H!brids.” I tio~iulCon/erence on A c o u s t i i ~ ~Speeih. . o w / S i g r i u l Pro~e\.\irt,y. \ol. 3. pp 1612-1615. 1988 B. A. Huberman and W. Wolff. “Finite precision and transient heha\ior,” P h w . Rev. A . vol. 3 2 . pp. 3768-3771. 1985 W. Wolff and B. A. Huberman. “Transients and as>mptotics in granular phase space,” Z . P h w B. vol. 63. pp. 3‘17-405, 19x6 R. Ramaswamy. private communication. 1YXY therefore be realized. This simplification removes the need for checking positivity of ( I I 1) entries of the Maria and Fahmy’s table [1] for all lzll= 1. Instead only one point, say, z1=1 needs to be tested. The dividing operation in the proposed table [3], however. may cause difficulty- in computing its entries, especially when 1 7 , 2 4. uhcre t i 2 is the degree of z2 in the 2-D polynomial F( z , , z 2 ) . Clearly the major difficulty in applying table form test to a 2-D filter with orders l i l and 1 7 2 is the computation of its and ?, (or entries. This is because they are now polynomials of ’) instead of numbers. Recently. in [ h ] a modified array (table) form has been suggested for a polynomial ~ zyxwvutsrqponm zyxwv zyxwvutsrqponm zyxwvutsrqponmlk zyxwvutsrqp 2-D Filter Stability Tests Using Polynomial Array XIHENG HU A N D T S. NG A bsrracr --In this paper a stability table for :, n F( s.z ) 1 = x);J U/( (1) I-0 where coefficients U, (s),J = 0,1,. . ’ , 17 are first-order polynomials of the unknown real parameter x. It has been shown that entries of the i th row of this array are polynomials of x of order 2‘ and the term polynomial array was used to describe such a table. In addition a systematic procedure for computing the coefficients of polynomial entries of the array has also been given. In this paper the polynomial array will be extended to the general case, i.e.. for F ( z , z ) Lvith orders nl and n 2 , where nl, t i 2 are orders of u/(.x) and 2. respectively. The polynomial array is then applied to testing the stability of 2-D polynomial F ( z , , z 2 ) (i.e. the denominator of 2-D filters). This is accomplished by forming 2n2 F(.x,,z,) = where U / ( z) are real polynomials of order )zl, called pol~nomialarra). is presented, and a systematic procedure for constructing the entries of the polynomial array is given. The polynomial array is then applied to test stability of 2-D filters. This is accomplished by first forming a pol~nomial F(x,. z 2 ) with real coefficients from the 2-D filter characteristic pol!nomial F(2,. z 2 ) , followed by constructing the polynomial array and then applying appropriate stability test. This procedure is proved to he equivalent to testing F ( z l , z 2 ) directly. A nuinher of examples is given to illustrate the proposed methodology. INTRODUCTION ~ ( z , . : ~ ) ~ ( : , ~ z= , ) 1u,(.x,):~on IZ,I=I J-0 where Z, is the complex :onjugate of 2 , . and x1 = (2, + 2,)/2. The coefficients of z 2 in F(.x,. z 2 ) .i.e. u , ( s , ) are realpolynomials of I, with degree 1 7 , . The polynomial array fcr F ( x , , z 2 ) is constructed and a stability test is then applied to F ( x l , z 2 ) . This procedure will be shown to bc equivalent to testing the stability of F ( z , . z , ) directly. Note that the proposed method requires tables for polynomials of order 2n, as compared with tables for polynomials of order n 2 in the previous methods [l].[3]. However, this disadvantage is compensated for by the following advantages of the proposed method. The problem of root distribution cf a polynomial with respect to the unit circle in the z-plane has continued to receive much attention. Recently, table form test has been extended to stability 1) The construction of the polynomial array is systematic and analysis for two- or multi-dimensional filters [1]-[3]. Maria and Fahmy [l]proposed a table form to test the multi-dimensional can easily be performed by a computer program regardless of the condition of Huang [4] by checking the positivity of entries of the values of 1 1 , and t i 2 . In addition. stability testing is simple and first column in the table for all 1, 1 = 1 or lxll < 1, where s1= ( z1 straightforward and can also be performed by a computer. 2) The table form test for real polynomials is well developed. + Z1)/2. The recent work by Jury [3:1 suggested a modified table for the same purpose. The appropriate entries of this table are There are results which engineers are familiar with and they can readily be applied to the proposed method. By contrast, results identical to the principal minors of the Hermitian Schur-Cohn matrix. Some simplification of the stability condition [5] can for table form test with complex coefficients have received much less attention. They are scattered in various publications and are not widely known by engineers. zyxwvut Manuscript received February 16, 1989: revised October 1. 19x9 This uork was supported by the University of Wollongong Re\earch and Postgraduate Studies Committee and by the Australian Research Grants Scheme. This paper was recommended by Associate Editor D M Goodman X. Hu was with the Universit) of Wollongong. Wollongong. Australia He i \ now with the School of Electncal Engineering. Universit! of S>dne\. N . S W 2006. Australia. T. S. N g is with the Department of Electrical and Computer Engineering. Universitv of Wollongong. Wollongong. Austrdia IEEE Log Numher 8933907. POLYNOMIAL ARRAY In this paper. 1- and 2-D ;transforms are defined in terms of negative powers of z for convenience, i.e.. m z[{ h ( n 7 ) } ] 0098-4094/90/0400-0550$01 .OO C1 990 IEEE View publication stats = /1( m=-m n7)frn (2 4