Stabilizing and Destabilizing Control for a Piecewise-Linear Circuit

172 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I: FUNDAMENTAL THEORY AND APPLICATIONS, VOL. 45, NO. 2, FEBRUARY 1998 Stabilizing and Destabilizing Control for a Piecewise-Linear Circuit Tadashi Tsubone and Toshimichi Saito Abstract This paper considers a basic approach to generalize techniques of controlling chaos. First, we propose a novel manifold piecewiselinear (MPL) system as a control object. The novel MPL can generate both chaotic attractors and periodic attractors. Second, we propose a novel occasional proportional feedback (OPF) method and apply it to the MPL. The OPF can change a form of the return map of the MPL it can change chaos into a periodic attractor and change a periodic attractor into chaos. The OPF functions can be guaranteed theoretically. Third, we propose an implementation example of the OPF and confirm its function in the laboratory. Index Terms Chaos, hysteresis control, nonlinear circuit, PWL system. I. INTRODUCTION This paper discusses a basic approach to generalize techniques of controlling chaos [1] [5]. As is well known, Ott, Grebogi, and Yorke (OGY) have first introduced a technique that can stabilize a desired unstable periodic orbit (UPO) embedded in a chaotic attractor through perturbations in system parameters [1]. The OGY method uses a knowledge of linear approximation near the desired UPO. Hunt [5] has applied an occasional proportional feedback (OPF) method to stabilize a desired UPO from the diode resonator. The OPF does not require a knowledge of linear approximation and can be implemented easily; therefore, the OPF has been applied to stabilize laser systems [6], our heart [7], and so on. However, theoretical analysis of such control techniques is hard chaos generation is usually hard to be proven and the OPF function has not been guaranteed theoretically. We have considered a piecewise-linear OPF (PWLOPF) version [10] [12]. Saito and Mitsubori have proposed the PWLOPF in [10] and applied it to a three-dimensional (3-D) hysteresis chaos generator, and [11] has applied the PWLOPF to a manifold piecewise-linear (MPL) chaos generator [8] whose return map is exactly piecewise linear. In those papers, we have given some theoretical results for the chaos generation, UPO identification, and the OPF function. This paper takes a first step toward generalization of the PWLOPF. In Section II, we propose a novel MPL as a control object. The novel MPL can generate both chaotic attractors and periodic attractors (the old MPL cannot generate periodic attractors [8]). Using the mapping procedure, we clarify the system dynamics theoretically. In Section III, we propose a novel PWLOPF and apply it to the continuous-time system MPL. The novel PWLOPF can change a form of the return map it can change chaos into a periodic attractor and change a periodic attractor into chaos. These changes can be guaranteed theoretically. In Section IV, we propose an implementation example of the OPF and verify some of the theoretical results in the laboratory. Recently, [14] has discussed a trial to destabilize a limit cycle into chaos. The trial is basic to generalize techniques of controlling chaos; however, theoretical analysis and laboratory experiments are not sufficient. Manuscript received June 19, 1996; revised April 4, 1997. This paper was recommended by Associate Editor T. Endo. The authors are with the Department of Electrical and Electronics Engineering, Hosei University, Koganei, Tokyo 184, Japan. Publisher Item Identifier S 1057-7122(98)00973-8. Fig. 1. Switching of the novel MPL system ( >0, p>0). II. CHAOS GENERATOR We propose the following piecewise-linear system: x 0 2 _x + x = ph(x; _x; y); h(x; _x; y) = 1; for (x; _x; y) 2 D a 01; for (x; _x; y) 2 D b D a f(x; _x; y)jx Th;y =1g [f(x; _x; y)jx <Th;_x 0;y =1g D b f(x; _x; y)jx <Th;y= 01g [f(x; _x; y)jx Th; _x 0;y = 01g (1) where 1 denotes differentiation by a normalized time and where y 2f01; 1g is a dummy variable in order to describe the behavior of h. Letting EX a f(x; _x; y)jx <Th;_x =0;y =1g (respectively, EX b f(x; _x; y)jx Th; _x = 0;y = 01g) be an exit of D a (respectively, D b ), h is switched from 1 to 01 if the solution on D a hits EX a and is switched from 01 to 1 if the solution on D b hits EX b. Fig. 1 illustrates the switching behavior. Two parameters and p control a damping and equilibrium points, respectively. For simplicity, we focus on the following parameter range: 01 <<1 p 2 R Th =0: This system is a developed version of a MPL system proposed in [8], where we have considered neither negative nor parameter p. 1057 7122/98$10.00 1998 IEEE

IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I: FUNDAMENTAL THEORY AND APPLICATIONS, VOL. 45, NO. 2, FEBRUARY 1998 173 Fig. 2. Return maps and corresponding trajectories. The chaotic attractor (=! =0:11, p =1:0). The two-periodic attractor (=! = 00:1, p = 00:4). If is negative, the novel MPL can exhibit a periodic attractor; it plays an important role in the control procedure. Equation (1) has complex characteristic roots 6 j!;! p 1 0 2, and the solution on D a is given by x 0 p = e ((x(0) 0 p)cos! + 1 (_x(0) 0 (x(0) 0 p)) sin!): (2)! It rotates around the equilibrium point (p; 0; 1), as is shown in Fig. 1. The solution on D b is symmetric to (2). Note that the switching occurs only on the x-axis on which x takes an extremum. In order to derive the return map, let I a f(x; _x; y)jx 0; _x = 0;y =1gD a;i b f(x; _x; y)jx <0; _x =0;y = 01g D b and let I I a [ I b. Noting that a trajectory started from I at =0must return to itself at = =!, we can define one-dimensional (1-D) return map F from I to itself. Letting any point in I be represented by its x-coordinate, x 0 be a starting point in I, and letting x 1 be a return point in I, the return map can be described by f+(x0); for x0 0 x 1 = F (x 0 )= f0(x 0 ); for x 0 < 0 f +(x 0) 0A(x 0 0 p) +p f0(x 0 ) 0A(x 0 + p) 0 p (3) where A exp((=!)) > 0. As is shown in Fig. 2, the return map is exactly piecewise linear. Now the system dynamics is simplified into the iteration x n+1 = F (x n) which is characterized by two parameters and p. We introduce some basic definitions for the return map. Definition 1: A point 2 I is said to be a k-periodic point if F k () = and F l () 6= for 0 < l < k, where F k denotes the k-fold composition of F.Ak-periodic point is said to be stable (respectively, unstable) if jdf k ()j < 1 (respectively, jdf k ()j > 1), where DF k denotes a derivative of F k. A sequence of a k-periodic point ff (); 111;F k ()g is said to be a periodic sequence with period k. We refer to a stable periodic sequence as a periodic attractor. Hereafter, we abbreviate an unstable (respectively, stable) periodic point by UPP (respectively, SPP), and abbreviate an unstable (respectively, stable) periodic sequence by UPS (respectively, SPS). Definition 2: A subset I s in I is said to be a stable interval if F (I s) I s: Definition 3: Let I s be a stable interval of F. If there exists some positive integer n such that jdf n j > 1 on I s, F is ergodic and has a positive Lyapunov exponent [13]. In this case, we say that F exhibits chaos. Then we have Theorem: 1) If (A; p) 2f(A; p)ja >1;p < 0g[f(A; p)ja >2g; then x n diverges to either 1 or 01 except for jx 0 j = p.

174 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I: FUNDAMENTAL THEORY AND APPLICATIONS, VOL. 45, NO. 2, FEBRUARY 1998 Fig. 3. Novel PWLOPF. OPF method for the MPL. Dotted trajectories start from the edges of W. The controlled return map. Fig. 4. Changing chaos into a periodic attractor (=! =0:11, p =1:0, q =2:0, =0:4, K =0:1). The fourfold composition of F c on W. The periodic attractor changed from the chaotic attractor of Fig. 2. 2) If 1 <A< 2 and p> 0, then F exhibits chaos as shown in Fig. 2. 3) If A<1and p>0, then F has two stable one-periodic points p and 0p. x n converges to one of them. 4) If A < 1 and p < 0, then F has one stable two-periodic sequence f0p(1+a=10a);p(1+a=10a)g that corresponds to a two-periodic attractor, as shown in Fig. 2. Proof: Since proof of (1), (2), and (4) is possible almost directly from graphs in Fig. 2, we show only for (2). If 1 <A< 2 and p> 0, then F satisfies jdf j = A> 1 and F (I s ) =I s, where I s (0p(1 + A), p(1 + A)). Letting d = 0p(1 + A)=1 0 A) and letting J (0d; d), I s J is satisfied and x n enters eventually into I s for all x 0 in J. Referring to Definition 3, F exhibits chaos. III. DESTABILIZING AND STABILIZING CONTROL We propose a control method for the MPL. First, let a trajectory intersect I at =0and let x sample be the intersection. Noting that x sample corresponds to the state x n of the return map, the control method for 0 <<(=!) is described as follows: x 0 2 _x + x =(p +1p)h(x; _x; y) 1p = KM (x sample 0 q); for x sample 2 W 0; otherwise K M = A 0 K (4) A +1 where W = f(x; _x; y)jq 0 <x<q+ ; _x =0gI and where 0 < q. Fig. 3 illustrates an example of the control procedure. That is, the control value 1p = K M (x sample 0 q) is added to the parameter p for 0 < <(=!) if a trajectory is trapped into the window W at =0. Noting that the 1p uses sampled state x sample in W, the 1p is constant for 0 <<(=!). Hence, the controlled trajectory is on D a for 0 < <(=!) and the state at =(=!) is given by x! 0 p = 0A(x sample 0 p) + (1 + A)K M (xsample 0 q): (5)

IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I: FUNDAMENTAL THEORY AND APPLICATIONS, VOL. 45, NO. 2, FEBRUARY 1998 175 Fig. 5. attractor changed from the periodic attractor of Fig. 2. Changing a periodic attractor into chaos (=! = 00:1, p = 00:4, q =2:84, K = p 2). The controlled return map. The chaotic Fig. 6. An implementation example. MPL system. Control system. For > =!, the same procedure is repeated the sampled state x sample on I is renewed at = n(=!) and the control method (4) is valid for n(=!) << (n + 1)(=!), where n is a positive integer. For the controlled system (4), the return map is to be x n+1 = F c (x n ) 0K(x n 0 q) +F (q); for x n 2 W F (x n); otherwise (6) where F (W ) I is assumed. Fig. 3 shows an example of the controlled map. 1 Adjusting K and q can change the form of the return map, and the controlled system (4) may exhibit various interesting 1 The conventional PWLOPF [10] corresponds to the special case of (6): is a unique intersection of a desired K =0 and q = x u, where x u uninterruptible power system (UPS) and W. Since chaotic orbit is ergodic, x n 2 W must be satisfied for some n. In this case, we obtain F(x n )= F(x u) for x n 2 W. Even if small noise exists, repeating this procedure can stabilize the desired UPP.

176 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I: FUNDAMENTAL THEORY AND APPLICATIONS, VOL. 45, NO. 2, FEBRUARY 1998 (d) (e) (c) (f) Fig. 7. Laboratory data for the OPF function (R =10:0k, : Rs =5:1k, : : : : : : R 1 =10:0k, R2 =24:5k, r1 =1:0k, c1 = c2 =0:068F). : The chaotic attractor (r 2 =3:2k). Horizontal scale: 2[v/div]; vertical scale: 1[v/div]. The periodic attractor changed from the chaotic attractor of. Horizontal scale: 2[v/div]; vertical scale: 1[v/div]. (c) Time-domain waveform of. Horizontal scale: 1[msec/div]; vertical scale: 1[v/div]. (d) The periodic : attractor (r 2 =4:0k). Horizontal scale: 2[v/div]; vertical scale: 1[v/div]. (e) The chaotic attractor changed from the periodic attractor of (d). Horizontal scale: 1[v/div]; vertical scale: 0.5[v/div]. (f) Time-domain waveform of (e). Horizontal scale: 1[msec/div]; vertical scale: 0.5[v/div]. phenomena. However, general discussion is extremely hard. In this paper, we consider the following two cases. A. Changing Chaos into a Periodic Attractor For 1 <A<2 and 0 <p, F exhibits chaos and we can change the chaos into a periodic attractor. It is guaranteed by Lemma 1: Fc has a unique stable m +1periodic point in W if F m+1 (q) 2 W; F m+1 is continuous on W; and ja m Kj < (7) where Fc m+1 (q) = F m+1 (q) q 0 and where is the minimum value of q + 0 q 0 and q 0 0 (q 0 ).

IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I: FUNDAMENTAL THEORY AND APPLICATIONS, VOL. 45, NO. 2, FEBRUARY 1998 177 Proof: As is shown in Fig. 4, (7) guarantees that Fc m+1 is continuous, monotone, and contracting on W ; hence, Fc m+1 has a unique stable fixed point in W: Fig. 4 shows an example of periodic attractors changed from chaotic attractors. B. Changing a Periodic Attractor into Chaos We assume that F exhibits a periodic attractor and that W includes at least one SPP. We can change the periodic attractor into chaos. It is guaranteed by Lemma 2: Fc generates chaos if W =[0; 2q]; Fc(0) < 2q; F 3 c (0) <Fc(0); and AK > 1: Proof: As is shown in Fig. 5, let I 1 [0;Fc(0)], I 2 [F 2 c (0); 0), and let I3 I1 [ I2. Noting Fc(xn) =0K(xn 0 q) + F (q) on I 1, Fc(0) < 2q and F 3 c (0) <Fc(0) guarantee Fc(I 3 ) I 3. Noting Fc(I 2 ) 2 I 1, AK > 1 guarantees DF 2 c > 1 on I 3 : Fig. 5 shows an example of chaotic attractors changed from periodic attractors. Note that generalization of this lemma is not easy, as suggested in [9]. IV. IMPLEMENTATION Fig. 6 shows an implementation example of the MPL, where the switching terminals (+) and (0) of S 1 correspond to h = 1and h = 01, respectively. The state >0 (respectively, <0) can be realized if the switch S2 connects to (A) (respectively, (B)). In this figure, v 1 and v 2 are proportional to x and _x, respectively. Using the following transformations, the circuit equation of Fig. 6 can be transformed into (1): =t x = v 1 jej = 1 21 = R r2c2r2 p = R 1 Rs E jej 2 R = : (8) r1r2c1c2r1 we are trying to develop the control method to be suitable for general chaotic circuits and to realize simple control circuits on chip. ACKNOWLEDGMENT The authors wish to thank the following investigators for their helpful comments: H. Torikai, Prof. T. Ushio, Prof. G. Chen, Dr. K. Mitsubori and Dr. K. Jin no. REFERENCES [1] E. Ott, C. Grebogi, and J. A. Yorke, Controlling chaos, Phys. Rev. Lett., vol. 64, no. 11, pp. 1196 1199, 1990. [2] T. Shinbrot, C. Grebogi, and E. Ott, Using chaos to target stationary states of flows, Phys. Lett. A, vol. 64, pp. 349 354, 1992. [3] K. Pyragas and A. Tamasevicius, Experimental control of chaos by delayed selfcontrolling feedback, Phys. Lett. A, vol. 180, pp. 99 102, 1993. [4] G. Chen and X. Dong, From chaos to order-perspectives and methodologies in controlling chaotic nonlinear dynamical system, Int. J. Bifurcation Chaos, vol. 3, pp. 1363 1409, 1993. [5] E. R. Hunt, Stabilizing high-period orbits in a chaotic system: The diode resonator, Phys. Rev. Lett., vol. 67, no. 15, pp. 1953 1955, 1991. [6] R. Roy and T. W. Murphy, Jr., T. D. Masier, Z. Gills, and E. R. Hunt, Dynamic control of a chaotic laser: Experimental stabilization of a globally coupled system, Phys. Rev. Lett., vol. 67, no. 15, pp. 1953 1955, 1991. [7] A. Garfinkel, M. L. Spano, W. L. Ditto, and J. N. Weiss, Controlling cardiac chaos, Science vol. 257, pp. 1230 1235, Aug. 1992. [8] H. Fujita and T. Saito, Continuous chaos represented by a nonlinear differential equation with manifold piecewise linear characteristics, in Proc. Theoretishe Elektrotechnik, Ilmenau, Germany, 1981, pp. 11 14. [9] T. Saito, A chaos generator based on a quasiharmonic oscillator, IEEE Trans. Circuits Syst., vol. CAS-32, pp. 320 331, Apr. 1985. [10] T. Saito and K. Mitsubori, Control of chaos from a piecewise linear hysteresis circuit, IEEE Tran. Circuits Syst. I, vol. 41, pp. 168 172, Mar. 1995. [11] K. Mitsubori and T. Saito, Control of piecewise linear chaos by occasional proportional feedback, Nonlinear Dynamics: New Theoretical and Applied Result, J. Awejcewics, Ed. Berlin, Germany: Akademie Verlag, 1995, pp. 361 375. [12] T. Tsubone, K. Mitsubori and T. Saito, Stabilizing high-period orbits in chaos and generation of islands: a piecewise linear approach, in Proc. IEEE/ISCAS III, Atlanta, GA, May 1996, pp. 257 260. [13] T. Y. Li and J. A. Yorke, Ergodic transformation from an interval into itself, Trans. Amer. Math. Soc., vol. 235, pp. 183 192, 1978. [14] T. Kousaka, T. Ueta and H. Kawakami, Destabilizing control of stable orbits, in Proc. Int. Symp. Nonlinear Theory Appl., Las Vegas, NV, Dec. 1995, pp. 997 1000. The dependent switch S 1 can be realized by using some comparators and one flip-flop. Fig. 6 shows an implementation example of the OPF, where vs, vq, v, and 1E correspond to x sample, q,, and 1p, respectively (vs is a sampled value of v 1 when v 2 =0). When S 1 connects to terminal (+), the circuit operates as follows. The sample-and-hold circuit samples vs when v 2 is zero and holds it by the time when v2 becomes zero again. If vs 2 WI [vq 0 v;vq + v], we add the control signal 0KM (vs 0 vq) to 0E by the time when v 2 becomes zero again. The system repeats this manner. The symmetric version of the OPF is also possible. Fig. 7 shows the laboratory verification of Figs. 4 and 5. V. CONCLUSION We have considered a novel control method of chaotic circuits. It can change chaos into a periodic attractor and change a periodic attractor into chaos. Using the novel MPL as a control object, the control function is confirmed in both theory and experiments. Now