International Journal of Bifurcation and Chaos, Vol. 8, No. 12 (1998) 2433–2438c World Scientific Publishing CompanyCONTROL OF CHAOS USINGSAMPLED-DATA FEEDBACK CONTROLTAO YANG and LEON O. CHUAElectronics Research Laboratory andDepartment of Electrical Engineering and Computer Sciences,University of California at Berkeley,Berkeley, CA 94720, USAReceived June 14, 1998; Revised August 5, 1998In this paper we present a theory for control of chaotic systems using sampled data. The outputof the chaotic system is sampled at a given sampling rate and the sampled output is used by afeedback subsystem to construct a control signal, which is held constant by a holding subsystem.Hence, during each control iteration, the control input remains unchanged. Theoretical resultson the asymptotic stability of the resulting controlled chaotic systems are presented. Numericalexperimental results via Chua’s circuit are used to verify the theoretical results.1. IntroductionThe control of chaos by sampled data has beenstudied and observed in experiments [Yang &Chua, 1997a, 1997b; Panas et al., 1998; Dedieu& Ogorzalek, 1994]. Previous results used thesampled data to change the state variables ofthe chaotic system “impulsively”. In this paper,we redesign the controller such that the controlsignal, which is constructed from the sampling se-quence of the output of the chaotic system, is fedinto the chaotic system as a control input. In thissampled-data feedback control scheme, the statevariables of the chaotic system are subject to con-tinuous changes instead of “impulsive” changes.Unlike most of the previous results where thecontrol input is constructed by the continuous ob-servations of the output of the chaotic system, thecontroller presented in this paper uses the samplesof the output of the chaotic system to constructcontrol signals.The main motivation for controlling chaosusing sampled data is to exploit well-developed dig-ital control techniques. In a digital controller, theoutput of the chaotic system is sampled and thesampled data is used to construct the appropriatecontrol signals. Assuming that a finite time dura-tion is needed by a digital processor to calculate thecontrol signals, then the sampling frequency is lim-ited by this time duration. On the other hand, afast sampling device is usually more expensive thana slow one. It is important therefore to develop atheory to predict the performance of the controlledchaotic system with a given sampling rate.The authors of [Dedieu & Ogorzalek, 1994] hadpresented some experimental results for controllingchaotic systems to referenced trajectories by usingonly sampled values. Although it is widely believedthat the control of continuous chaotic systems byusing digital controllers is possible, so far, there ex-ists no theoretical results to guarantee the asymp-totic stability of such controlled chaotic systems.In this paper, we present theoretical results whichguarantee the asymptotic stability of sampled-datafeedback control of chaotic systems.The organization of this paper is as follows. InSec. 2, the structure and theory of the sampled-datafeedback controller are presented. In Sec. 3, thesampled-data feedback control of a typical chaotic24332434 T. Yang & L. O. Chuasystem (Chua’s circuit) is given. In Sec. 4, someconcluding remarks are given.2. Control of Chaos UsingSampled DataIn this section we present the structure and thetheory of a sampled-data feedback controller.2.1. Structure of control systemThe proposed structure of a chaotic control sys-tem with sampled data is shown in Fig. 1. Thestate variables of the chaotic system are observed(measured) by transducers and the result is usedto construct the output signal y(t)=Dx(t) whereD is a constant matrix to be defined below. Theoutput y(t) is then sampled by the sampling blockto obtain y(k)=Dx(k) at discrete moments k∆,where k =0,1,2,..., and ∆ is the sampling du-ration. Then Dx(k) is used by the controller tocalculate the control signal u(k). During the timeinterval [k∆, (k + 1)∆), the output of the holdingblock is u(k), which is fed back into the chaotic sys-tem as a fixed control input during the entire timeslot [k∆, (k + 1)∆).2.2. Theory of stabilityConsider a chaotic dynamic system modeled by thestate equation˙x = f (x)(1)where x ∈<nis the state variable, f : <n7→ <nisa nonlinear function and f(0)=0.The controlled chaotic system is defined by(˙x(t)=f(x(t)) + Bu(k),t∈[k∆,(k+1)∆)u(k+1)=Cu(k)+Dx(k),k=0,1,2,...(2)where u ∈<m,B∈<n×<m, C ∈<m×<m,D ∈<m×<n,andt∈<+;x(k) is the sampledvalue of x(t)att=k∆; ∆ is the sampling dura-tion. Observe that since f (0)=0,xu=00isan equilibrium point of the system in Eq. (2). Theasymptotic stability of this equilibrium point canbe determined from that of the trivial solution ofthe associated linearized system:(˙x = Ax + Bu(k),t∈[k∆,(k+1)∆)u(k+1)=Cu(k)+Dx(k),k=0,1,2,...(3)where A ∈<n×<nis given byA∆=∂f(0)∂x. (4)Definition 1. An n × n matrix Γ is said Schurstable if, and only if, all eigenvalues of Γ lie in theunit disc centered at the origin.Fig. 1. Structure of a proposed sampled-data chaotic control system.Control of Chaos Using Sampled-Data Feedback Control 2435Lemma 1. Suppose f ∈ C1[<n, <n], then theequilibrium point (x, u)=(0,0)of the controlledchaotic system (2) is uniformly asymptoticallystable if the associated matrixH∆=eA∆ Z∆0A∆−τdτ!BDC(5)is Schur stable.Proof. The proof follows directly from Theorem 5.1of [Ye et al., 1998], which assumes that ∆ = 1.The generalization for arbitrary ∆, hence, followseasily.Lemma 1 is not practical because it involvesintegration of matrices. The following theoremgives a more explicit criterion for the stability ofthe controlled chaotic system.Theorem 1. Suppose f ∈ C1[<n, <n] and A isnonsingular, then the equilibrium point (x, u)=(0,0)of the controlled chaotic system (2) is uni-formly asymptotically stable if the spectral radiusρ(T ) of the matrixT∆= eA∆A−1eA∆B − A−1BDC!(6)is less than unity; i.e. ρ(T ) < 1.Proof. Observe that Z∆0A∆−τdτ!B = −Z∆0A∆−τd(∆ − τ )!B= Z∆0Atdt!B=A−1eAt|∆0B= A−1(eA∆− I)B (7)where t =∆−τ and I ∈<n×<nis the identitymatrix. Substituting (7) into (5), obtain T = H.Since the spectral radius ρ(T) < 1, we know that Tis Schur stable, and the theorem therefore followsfrom Lemma 1.3. Control of Chua’s Circuitwith