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Robust Synthesis for Master-Slave Synchronization of Lur'e Systems J.A.K. Suykens1, P.F. Curran2 and L.O. Chua3 1
Katholieke Universiteit Leuven, Dept. of Electr. Eng., ESAT-SISTA Kardinaal Mercierlaan 94, B-3001 Leuven (Heverlee), Belgium Tel: 32/16/32 11 11 Fax: 32/16/32 19 70 Email: [email protected] 145, Electronic and Electrical Eng. University College, Bel eld, Dublin 4, Ireland Tel: (353)-1-706 1846 Fax: (353)-1-283 0921 Email: [email protected] 2
Department of Electrical Engineering and Computer Science University of California at Berkeley, Berkeley, CA 94720, USA Tel: +1 (510) 642 3209 Fax: +1 (510) 643 8869 Email: [email protected] 3
( Running title: Robust synthesis for synchronization ) ( Corresponding author: Johan Suykens ) 1
Abstract In this paper a method for robust synthesis of full static state error feedback and dynamic output error feedback for master-slave synchronization of Lur'e systems is presented. Parameter mismatch between the systems is considered in the synchronization schemes. Sucient conditions for uniform synchronization with a bound on the synchronization error are derived based on a quadratic Lyapunov function. The matrix inequalities from the case without parameter mismatch between the Lur'e systems remain preserved, but an additional robustness criterion has to be taken into account. The robustness criterion is based on an uncertainty relation between the synchronization error bound and the parameter mismatch. The robust synthesis method is illustrated on Chua's circuit with the double scroll. One observes that it is possible to synchronize the master-slave systems up to a relatively small error bound even in the case of di erent qualitative behaviour between the master and the uncontrolled slave system, such as limit cycles and stable equilibria.
1 Introduction Master-slave synchronization schemes for Lur'e systems have been studied in [5, 6, 16] for the autonomous case with static state feedback or dynamic output feedback applied to the slave system. With respect to secure communications applications [8], the non-autonomous case has been studied in [17, 18], where an information carrying message signal is considered as an external input. In these works the synchronization problem has been approached from the viewpoint of control theory. The autonomous case has been studied with respect to absolute stability theory with sucient conditions for global asymptotic stability of the error system, either based on quadratic or Lur'e-Postnikov Lyapunov functions. For the non-autonomous case the synchronization scheme has been interpreted as a modelreference control scheme in standard plant form with exogenous input and regulated output, according to modern control theory [1, 11]. In this way additive channel noise has been taken into account in the design procedure. 2
However, these synchronization schemes assume that the master-slave Lur'e systems are identical. In this paper we study the in uence of parameter mismatch between the Lur'e systems. Previous work on robust synchronization has been reported in [21, 5, 19]. According to [21] the case of non-identical Lur'e systems requires the de nition of a synchronization error bound, because zero synchronization error cannot be achieved. In this paper we present matrix inequalities [2] for the synchronization schemes with static state feedback or dynamic output feedback. The matrix inequalities give sucient conditions for uniform synchronization with a certain error bound. The same form is preserved for the matrix inequalities as for the case of identical master-slave Lur'e systems. Uncertainty relations between the bound on the synchronization error and the parameter mismatch follow from the derived Theorems. The additive perturbation which is related to the parameter mismatch and is considered with respect to the nominal identical Lur'e systems, is assumed to be unstructured [11]. Finally, a robust synthesis method is presented which involves solving a nonlinear optimization problem. This optimization problem is based on the matrix inequality and the robustness criterion, where the latter follows from the uncertainty relation. The robust synthesis method is illustrated on Chua's circuit. From the simulation results one observes that a relatively large parameter mismatch can be allowed such that the systems remain synchronized with a relatively small synchronization error bound. In this sense it is possible to synchronize the slave system, which behaves as the double scroll in the uncontrolled case, to a master system which behaves either chaotically, periodically or has stable equilibrium points. These results have been obtained both for the case of full static state feedback and dynamic output feedback. This paper is organized as follows. In Section 2 we present the synchronization schemes with non-identical Lur'e systems for the case of static state error feedback and dynamic output error feedback. In Section 3 we describe the corresponding error systems. In Section 4 criteria for robust synchronization are derived. In Section 5 the robust synthesis method is presented, which is based on the derived matrix inequalities. Finally, in Section 6 the method is illustrated by applying it to Chua's circuit. 3
2 Synchronization Schemes In this Section we describe master-slave synchronization of Lur'e systems for two cases: full static state error feedback and dynamic output error feedback.
2.1 Full static state error feedback Consider the master-slave synchronization scheme with full static state error feedback [21, 5, 17] 8 > > M : x_ = A1x + B1 (C1x) > > > s > > > > > > > < > > > > > > > > > > > > :
Ss : z_ = A2 z + B2(C2z) + us
(1)
Cs : us = F (x ? z) with master system Ms, slave system Ss and controller Cs. The index s refers to the static feedback case. The master and slave system are Lur'e systems with state vectors x; z 2 Rn respectively and matrices Ai 2 Rnn , Bi 2 Rnnh , Ci 2 Rnh n (i = 1; 2). A Lur'e system is a linear dynamical system, feedback interconnected to a static nonlinearity () that satis es a sector condition [9, 20] (here it has been represented as a recurrent neural network with one hidden layer, activation function () and nh hidden units [15]). We assume that () : Rnh ! Rnh is a diagonal and continuous nonlinearity (but possibly nondi erentiable at a countable number of points) with i () belonging to sector [0; k], i.e. i ( )[i( ) ? k ] 0, 8 2 R for i = 1; :::; nh. The scheme aims at synchronizing the master system to the slave system by applying full static state error feedback to the slave system using the control signal us 2 Rn with feedback matrix F 2 Rnn . The Lur'e systems are assumed to be non-identical. The parameter mismatch between the systems, kA1 ? A2 k2; kB1 ? B2 k2; kC1 ? C2 k2 is assumed to be relatively `small'. The aim of this paper is precisely to design the feedback matrix F such that the scheme is robust with respect to this parameter mismatch.
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2.2 Dynamic output error feedback In addition to the state feedback scheme we will also study the dynamic output error feedback scheme, introduced in [16]:
Md :
8 > < > :
x_ = A1 x + B1 (C1x) p = Hx
Sd :
8 > < > :
z_ = A2 z + B2(C2 z) + Dud q = Hz
Cd :
8 > < > :
_ = E + G(p ? q) ud = M + N (p ? q)
(2)
with master system Md, slave system Sd and a controller Cd . The index d refers to the dynamic feedback case. For the master and slave system we consider the state vectors, system matrices and nonlinearity as described for (1). The output vectors of the master and slave system are p; q 2 Rl with l n. The slave system is controlled by means of the control vector ud 2 Rm through the matrix D 2 Rnm . The signal ud is the output of a linear dynamic output feedback controller. The input of this controller is the output error p ? q. The linear dynamic controller has state vector 2 Rnc and consists of the matrices E 2 Rnc nc , G 2 Rnc l, M 2 Rmnc , N 2 Rml . In [16] this scheme has been studied for identical Lur'e systems. As for the state feedback case we are interested here in designing the controller Cd such that a high robustness is obtained with respect to parameter mismatch between the Lur'e systems.
3 Error systems De ning the error signal as e = x?z, the aim of the synchronization schemes is to design the controllers Cs , Cd such that ke(t)k2 ! 0 as t ! 1. However, a zero error is only obtainable 5
in the case of identical master-slave systems. For the case of parameter mismatch between the systems an error bound must be considered instead. This will be further discussed in the following Section. Denoting the state feedback synchronization scheme as 8 > < > :
x_ = fs(x) z_ = gs(z; x)
(3)
with continuous nonlinear mappings fs() : Rn ! Rn , gs(; ) : Rn Rn ! Rn , one obtains the error system: (4) Es : e_ = fs(x) ? gs(z; x): Inspired by the proof of Theorem 14 in [21], we make the following decomposition for the error system: e_ = vs(x; z) + ws(x) (5) with
8 > > > vs(x; z) > > > > > > < > > > ws(x) > > > > > > :
= gs(x; x) ? gs(z; x) = (A2 ? F )e + B2 (C2e; z) = fs(x) ? gs(x; x) = Ax + B1(C1x) ? B2(C2x) where A = A1 ? A2, (C2e; z) = (C2e + C2z) ? (C2z). Denoting the dynamic synchronization scheme as 8 > > > > >
> > > > : _ = hd (; x; z ) with continuous nonlinear mappings fd () : Rn ! Rn , gd(; ; ) : hd (; ; ) : Rnc Rn Rn ! Rnc , one obtains the error system:
Ed : e_ = fd(x) ? gd (z; x; ): 6
(6) Rn
R n R nc ! R n , (7)
We make the following decomposition:
e_ = vd(x; z; ) + wd(x; ) with
(8)
8 > > > vd(x; z; ) > > > > > > < > > > wd (x; ) > > > > > > :
= gd(x; x; ) ? gd(z; x; ) = (A2 ? DNH )e + B2 (C2e; z) = fd(x) ? gd(x; x; ) = '(x) ? DM where A = A1 ? A2, (C2e; z) = (C2 e + C2z) ? (C2z) and '(x) = Ax + B1 (C1x) ? B2 (C2x).
4 Criteria for Robust Synchronization In order to derive criteria for synchronization of the schemes (1)-(2) with parameter mismatch between the systems, we rst have to introduce assumptions on the nonlinearity in the error system and on the norm of the state vector of the master system.
Assumption 1. The nonlinearity (C2e; z) belongs to sector [0; k]: (cT e; z) (cT e + cT z) ? (cT2i z) 0 i c2Ti e = i 2i c2Ti e k; 8e; z; i = 1; :::; nh (cT2i e 6= 0); (9) 2i
2i
where cT2i denotes the i-th row vector of C2 .
The following inequality holds then:
i(cT2i e; z) [i(cT2i e; z) ? kcT2i e] 0;
8e; z; i = 1; :::; nh:
(10)
It follows from the mean value theorem that for di erentiable () the sector condition [0; k] on () corresponds to [5] 0 d i(; z) k; 8; z; i = 1; :::; nh: (11) d 7
For ws() there exists a positive real constant s such that
kws(x)k2 < skxk2 ; 8x 2 Rn :
(12)
The same holds for the function '() with constant d.
Assumption 2. Master systems Ms and Md satisfy the condition that there exists a
positive real constant such that, for any initial condition x0 , there exists time T (x0 ) for which kx(t; x0 )k2 ; 8t T: (13)
This assumption is based on the work of [5]. From a practical point of view this is a reasonable assumption, in particular for chaotic Lur'e systems, because one is not interested in employing a master system that possesses unbounded trajectories. Because the master-slave systems are non-identical the synchronization error will not tend asymptotically to zero. Therefore the following de nition of synchronization with error bound is employed [21].
De nition 1. The synchronization schemes (1)-(2) uniformly synchronize with error bound if there exists a 0 > 0 and a T 0 such that if kx(0) ? z(0)k2 0 then kx(t) ? z(t)k2 for all t T . For the synchronization scheme with static feedback we consider the following positive de nite quadratic Lyapunov function (which is radially unbounded)
V (e) = eT Pe; P = P T > 0:
(14)
Theorem 1. Suppose Assumption 1 and 2 and that there exists a diagonal positive de nite matrix = diagfig 2 Rnh nh , a symmetric positive de nite matrix P 2 Rnn , a 8
matrix F 2 Rnn and a positive real constant such that the matrix inequality
Y
2 = Y T = 64
3
(A2 ? F )T P + P (A2 ? F ) + I PB2 + kC2T 7 5 = ; 8t: When c is determined such that E s = fe j eT Pe cg is the smallest ellipsoid containing the ball B1s = fe j eT e 2= 2g, the trajectory will enter the ellipsoid E s for every initial 9
condition e(0). Note that the ellipsoid E s, which is parametrized in terms of c, is directly related to the level set of the Lyapunov function V [20]. Hence for every initial condition, the limit set of the error system is nonempty, closed and bounded and belongs to E s [20]. Therefore the error system will uniformly synchronize with error bound pc2 where c2 is determined such that B2s is the smallest ball containing E s.
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Remarks We stress that the condition (15), being based on a quadratic Lyapunov function, is only sucient and possibly conservative. However, from absolute stability theory of Lur'e systems [9, 20] one knows that this matrix inequality is related to the circle criterion (by means of the KYP-Lemma) and in this sense is meaningful. The numerical algorithms which follow this Section have two tasks. Firstly to con rm that for chaotic Lur'e systems such as Chua's circuit a solution to the LMI exists and secondly to explain how one can nd one.
Convex optimization procedures for nding an outer approximation of a union of ellipsoids and an intersection of ellipsoids are discussed in [2](p.43). These methods can be used in order to construct B2s, E s given B1s and nding c, c2 . On the other (P ) hand it follows from Fig.1 that pc2 = (P )= with (P ) = max min (P ) the condition number of matrix P .
Note that for P = I the synchronization scheme (1) is uniformly synchronizing with error bound 2 s= . In this case the sets B1s, B2s and E s are identical. For the synchronization scheme with dynamic feedback we consider the following positive de nite quadratic Lyapunov function 2 P V ( ) = T P = [eT T ] 64 11
32 P12 7 6 54
P21 P22 10
e
3 7 5;
P = P T > 0:
(16)
with = [e; ].
Theorem 2. Suppose Assumption 1 and 2 and that there exists a diagonal positive de nite matrix = diagfig 2 Rnh nh , a symmetric positive de nite matrix P 2 R(n+nc )(n+nc ) ,
controller matrices E; G; M; N and positive real constants 1 ; 2 such that the matrix inequality
Z
2 66 Z11 + 1 I = Z T = 666 Z12T 4
Z13T
with
Z11 Z12 Z13 Z22 Z23 Z33
= = = = = =
Z12 Z22 + 2 I Z23T Z33
(17)
(A2 ? DNH )T P11 + P11(A2 ? DNH ) + H T GT P21 + P12GH (A2 ? DNH )T P12 + H T GT P22 ? P11 DM + P12E P11B2 + kC2T E T P22 + P22E ? M T DT P12 ? P21 DM P21B2 ?2
is satis ed. De ne PI = r
3 Z13 7 7 Z23 777 < 0 5
diagfIn=12; Inc =22g
with 1 =
1 2 1
+
1 2
r
12 21
+ 12 2 , 2 = 2
2
2 2
+
+ 222 where 1 = 2 d max (P11), 2 = 2 d max (P12 ). d , are de ned according to (12), (13) respectively. If there exist positive real constants c1 ; c2 ; c3 ; c4 such that (Fig.2) 1 2
12 1 2
2
B1d = fe j eT e 12g E1d = f j T PI 1g; B2d = f j T 22g E1d; B3d = fe j eT e c3g E3d = f j T PI c2g; B4d = f j T c4 g E3d and
E1d E2d = f j T P c1 g E3d
then the synchronization scheme (2) is uniformly synchronizing with error bound pc3 .
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Proof: Taking the time derivative of the Lyapunov function (16) and, using the inequalities (10), one obtains:
V_ = _T P + T P _ = [vd(x; z; ) + wd(x; )]T (P11e + P12) + _T (P21 e + P22 ) +(eT P11 + T P21)[vd (x; z; ) + wd(x; )] + (eT P12 + T P22 )_ [(A2 ? DNH )e + B2 ? DM + '(x)]T (P11 e + P12 ) + (E + GHe)T (P21 e + P22 ) +(eT P11 + T P21)[(A2 ? DNH )e + B2 ? DM + '(x)] + (eT P12 + T P22 )(E + GHe) ? Pi 2ii(i ? kcT2i e) T Z0 + 2 d max (P11)kek2 + 2 dmax (P12 )kk2 where = [e; ; ] and
3
2 Z11 6 6 Z0 = 666 Z12T 4
Z12 Z13 7 7 Z22 Z23 777 : 5 Z13T Z23T Z33 From (17) we obtain the upper bound T Z0 ? 1 eT e ? 2T and therefore V_ ? 1 kek22 ? 2kk22 + 2 d[max (P11 )kek2 + max (P12 )kk2] = kek2[? 1 kek2 + 2 d max (P11)] + kk2[? 2 kk2 + 2 d max (P12)] ?(p 1 kek2 ? 2p 1 1 )2 ? (p 2 kk2 ? 2p 2 2 )2 + 14 ( 121 + 222 ):
Hence V_ < 0 if 8t
ke(t)k2 > 1 k(t)k2 > 2 :
When c1 is selected such that E1d is the smallest ellipsoid containing the balls B1d and B2d , the trajectory of the error system will enter the ellipsoid E2d for every initial condition [e(0); (0)]. The ellipsoid E2d is directly related to the level set of the Lyapunov function. Hence the error system will uniformly synchronize with error bound pc3 where c3 is determined such that E2d is the smallest level set containing E1d. E3d is the smallest ellipsoid containing E2d, subject to the de nition of these sets.
2 12
5 Robust Synthesis The design of the controller Cs is based then on the Theorem 1, for which one derives the following uncertainty relation between the synchronization error bound and the parameter mismatch of the systems: =c ; (18) 0
where the positive real constant c0 is equal to 2max (P )(P )= . The interpretation of the uncertainty relation is twofold: supposing a larger parameter mismatch, which corresponds to an increasing value, suggests a larger synchronization error bound and vice-versa demanding a lower error bound requires a smaller parameter mismatch. An interpretation for follows by taking the 2-norm of the functions ws() and '(), which gives the conservative estimate = kAk2 + k kB1k2 kC1k2 + k kB2 k2 kC2k2: (19) Robust synthesis aims at minimizing the constant c0 in the uncertainty relation such that the matrix inequality (15) is satis ed. Since follows from the choice of the master system, this corresponds to the following design problem for the static feedback case: 8 > < > :
Y (F; P; ; ) < 0 (20) P = P T > 0; 0 and diagonal: For the dynamic feedback case, rather than considering the error bound as derived in Theorem 2, we will minimize the volume of the ball B1d. This is meaningful as long as the condition number of the matrix P remains relatively small. With respect to therball B1d the 2 2 uncertainty relation (18) has the positive real constant c0 equal to 1 = 2 11 + 21 121 + 12 2 and corresponds to d. The design for the dynamic feedback case can be done then as follows: 8 > > > Z (E; G; M; N; P; ; 1; 2) < 0 > > < max (P11 ) max (P12 ) such that > P = P T > 0; 0 and diagonal min E;G;M;N;P;; 1; 2 1 + 2 > > > > : (P ) 0 (21) min (P ) (P )= such that F;P;; max
13
where 0 is a user-de ned upper bound on the condition number of P . The problems (20) and (21) are non-convex optimization problems. Non-di erentiability might occur when the two largest eigenvalues of Y or Z coincide [14]. The constraint P > 0 can be eliminated by considering the parametrization P = QT Q. A similar idea applies to . An important observation is that the matrix inequalities, obtained for the case without parameter mismatch, are preserved (see [5, 6, 16]) and an additional robustness criterion can be taken into account in the design procedure.
6 Example: Chua's Circuit In this Section we illustrate the robust synthesis method on Chua's circuit. We take the following representation for Chua's circuit: 8 > > > x_ 1 > >
> > > > : x_ 3 = ?b x2
(22)
h(x1 ) = m1 x1 + 21 (m0 ? m1 ) (jx1 + cj ? jx1 ? cj)
(23)
with nonlinear characteristic
and parameters a = 9, b = 14:286, m0 = ?1=7, m1 = 2=7 in order to obtain the double scroll attractor [3, 4, 12]. The nonlinearity (x1 ) = 21 (jx1 + cj?jx1 ? cj) (linear characteristic with saturation) belongs to sector [0; 1]. Hence Chua's circuit can be interpreted as the Lur'e system x_ = Ax + B (Cx) where 2 66 A = 666 4
3
2
3
?a(m0 ? m1 ) 77 ?a m1 a 0 77 6 6 7 ; C = [1 0 0]: 7 0 1 ?1 1 777 ; B = 666 7 5 4 5 0 0 ?b 0
(24)
In the sequel, we consider the double scroll as the nominal system and we let it correspond to the slave system A2 ; B2; C2. 14
We illustrate rst the full static state feedback case. Sequential quadratic programming [7] has been applied in order to optimize (20) using Matlab's optimization toolbox (function constr). An additional constraint on the controller parameter vector kF k2 < 20 has been used. Instead of the inequality Y < 0 the constraint max (Y ) + 0:001 < 0 has been employed, where max () denotes the maximal eigenvalue of the symmetric matrix. As starting point for the iterative procedure, a random F matrix has been chosen according to a normal distribution with zero mean and variance 0.1 and Q = I , = 0:1 I . Simulation results for master-slave synchronization of the Chua's circuits are shown in Fig. 3-6, with perturbations on the element a11 of the A matrix of the master system. The simulation results show that it is possible to synchronize the master-slave systems with relatively small non-zero synchronization error bound, even in the case where the master system has di erent qualitative behaviour from the uncontrolled slave system (such as limit cycles and stable equilibrium points). In order to illustrate the dynamic output feedback case, suppose that we measure the rst state variables x1 and z1 only in order to synchronize the circuits and that we take a one-dimensional control signal in order to control the slave system. This corresponds to the choice H = [1 0 0], D = [1; 0; 0] (l = m = 1). We report the results here for a second order controller with nc = 2. A 2-norm constraint on the controller parameter vector [E (:); G(:); M (:); N (:)] (< 80) has been used for (21), where `:' denotes a columnwise scanning of a matrix. Instead of the inequality Z < 0 the constraint max (Z ) + 0:001 < 0 has been employed and 0 = 20 has been imposed. As starting point for the iterative procedure, a random controller parameter vector has been chosen according to a normal distribution with zero mean and variance 0.1. For the matrix Q a square random matrix was chosen according to the same distribution but with variance equal to 3. The matrix has been initialized as 100 I . Simulation results for master-slave synchronization of the Chua's circuits are shown in Fig. 7-8, for perturbations on the element a11 of the master system. As for the static state feedback case it is possible to synchronize the masterslave systems with relatively small non-zero synchronization error bound, even in the case 15
where the master system has di erent qualitative behaviour from the uncontrolled slave system. All simulation results have been obtained using a Runge-Kutta integration rule with adaptive step size (ode23 in Matlab) [13].
7 Conclusion By means of Chua's circuit we illustrated that a relatively large parameter mismatch between Lur'e systems can be allowed in order to maintain synchronization with a relatively small synchronization error bound, even in the case where the master system behaves qualitatively di erent from the uncontrolled slave system. These results have been obtained by deriving a robustness criterion in addition to matrix inequalities, based upon a quadratic Lyapunov function. The matrix inequalities basically take the same form as for the case of identical Lur'e systems. The robustness criterion is based on an uncertainty relation between the synchronization error bound and the parameter mismatch. The parameter mismatch has been interpreted in terms of unstructured perturbations on the system matrices of the Lur'e systems. Both static state error feedback and dynamic output error feedback have been studied. The proposed synthesis method o ers a straightforward design procedure by means of solving a nonlinear optimization problem. It completes previous work on control theoretic interpretations of synchronization schemes.
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Acknowledgment This research work was carried out at the University of California at Berkeley, in the framework of the Belgian Programme on Interuniversity Poles of Attraction, initiated by the Belgian State, Prime Minister's Oce for Science, Technology and Culture (IUAP-17) and in the framework of a Concerted Action Project MIPS (Modelbased Information Processing Systems) of the Flemish Community. The work is supported in part by the Oce of Naval Research under grant N00014-96-1-0753 and the Fulbright Fellowship Program. We thank the reviewers for constructive comments.
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References [1] Boyd S. & Barratt C., Linear controller design, limits of performance, Prentice-Hall, 1991. [2] Boyd S., El Ghaoui L., Feron E. & Balakrishnan V., Linear matrix inequalities in system and control theory, SIAM (Studies in Applied Mathematics), Vol.15, 1994. [3] Chua L.O., Komuro M. & Matsumoto T., \The Double Scroll Family," IEEE Trans. Circuits and Systems-I, 33(11), 1072-1118, 1986. [4] Chua L.O., \Chua's circuit 10 years later," Int. J. Circuit Theory and Applications, 22, 279-305, 1994. [5] Curran P.F. & Chua L.O., \Absolute stability theory and the synchronization problem," Int. J. Bifurcation and Chaos, Vol.7, No.6, pp.1375-1382, 1997. [6] Curran P.F., Suykens J.A.K. & Chua L.O., \Absolute Stability Theory and MasterSlave Synchronization," Int. J. Bifurcation and Chaos, Vol.7, No.12, 1997. [7] Fletcher R., Practical methods of optimization, Chichester and New York: John Wiley and Sons, 1987. [8] Hasler M., \Synchronization principles and applications," Circuits & Systems: Tutorials IEEE-ISCAS '94, 314-326, 1994. [9] Khalil H.K., Nonlinear Systems, New York: Macmillan Publishing Company, 1992. [10] LaSalle J.P. & Lefschetz S., Stability by Liapunov's direct method with applications, New York: Academic Press, 1961. [11] Maciejowski J.M., Multivariable feedback design, Addison-Wesley, 1989. [12] Madan R.N. (Guest Editor), Chua's Circuit: A Paradigm for Chaos, Signapore: World Scienti c Publishing Co. Pte. Ltd, 1993. 18
[13] Parker T.S. & Chua L.O., Practical numerical algorithms for chaotic systems, New York: Springer-Verlag, 1989. [14] Polak E. & Wardi Y., \Nondi erentiable optimization algorithm for designing control systems having singular value inequalities," Automatica, 18(3), 267-283, 1982. [15] Suykens J.A.K., Vandewalle J.P.L. & De Moor B.L.R., Arti cial Neural Networks for Modelling and Control of Non-Linear systems, Boston: Kluwer Academic Publishers, 1996. [16] Suykens J.A.K., Curran P.F. & Chua L.O., \Master-slave synchronization using dynamic output feedback," International Journal Bifurcation and Chaos, Vol.7, No.3, pp.671-679, 1997. [17] Suykens J.A.K., Vandewalle J. & Chua L.O., \Nonlinear H1 Synchronization of Chaotic Lur'e Systems," International Journal of Bifurcation and Chaos, Vol.7, No.6, pp. 1323-1335, 1997. [18] Suykens J.A.K., Curran P.F., Yang T., Vandewalle J. & Chua L.O., \Nonlinear H1 synchronization of Lur'e systems: dynamic output feedback case," IEEE Transactions on Circuits and Systems-I, Vol.44 No.11, pp.1089-1092, Nov. 1997. [19] Suykens J.A.K., Curran P.F., Vandewalle J. & Chua L.O., \Robust nonlinear H1 synchronization of chaotic Lur'e systems," IEEE Transactions on Circuits and SystemsI, Special Issue on Chaos Synchronization, Control and Applications, Vol.44, No.10, pp.891-904, Oct. 1997. [20] Vidyasagar M., Nonlinear Systems Analysis, Prentice-Hall, 1993. [21] Wu C.W. & Chua L.O., \A uni ed framework for synchronization and control of dynamical systems," Int. J. Bifurcation and Chaos, 4(4), 979-989, 1994.
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Captions of Figures Fig. 1. Illustration of the balls B1s, B2s and the ellipsoid E s with respect to Theorem 1. Fig. 2. Illustration of the balls B1d , B2d , B3d , B4d and the ellipsoids E1d, E2d, E3d with respect to Theorem 2. A simpli ed case is shown for n = 1, nc = 1 such that the balls correspond to line segments. Fig. 3. Master-slave synchronization of Chua's circuits using static state feedback. This gure shows the simulation results of the robust synthesis method in case there is no parameter mismatch. (Top) master system; (Middle) slave system; (Bottom) ke(t)k2 on logarithmic scale. The synchronization error is asymptotically converging to zero. Fig. 4. Static state feedback (continued). In this case there is parameter mismatch between the master and slave Chua's circuit. The slave system is considered to be the nominal system and behaves as the double scroll in case of a zero control input. A perturbation is considered on the A matrix of the master system (a11 = 1:2). The behaviour of master and slave system is shown on (Top) and (Middle) respectively. (Bottom) ke(t)k2 on logarithmic scale. The Chua's circuits synchronize with non-zero error synchronization bound. Fig. 5. Static state feedback (continued). Similar to Fig. 2, but with perturbation a11 = 1:8. The master system shows limit cycle behaviour, while the uncontrolled slave system behaves as the double scroll. Fig. 6. Static state feedback (continued). Similar to Fig. 2, but with perturbation a11 = 1:95. The master system possesses stable equilibria, while the uncontrolled slave system behaves as the double scroll. 20
Fig. 7. Master-slave synchronization of Chua's circuits using dynamic output feedback. A SISO second order linear dynamic controller is considered. A perturbation of a11 = 1:5 is considered on the A matrix of the master system. (Top-left) master system; (Top-right) slave system; (Bottom-left) ke(t)k2 on logarithmic scale. The Chua's circuits synchronize with non-zero error synchronization bound. (Bottom-right) k(t)k2 on logarithmic scale. Fig. 8. Dynamic output feedback (continued). Similar to Fig. 5, but with perturbation a11 = 2. The master system possesses stable equilibria, while the uncontrolled slave system behaves as the double scroll.
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Katholieke Universiteit Leuven, Dept. of Electr. Eng., ESAT-SISTA Kardinaal Mercierlaan 94, B-3001 Leuven (Heverlee), Belgium Tel: 32/16/32 11 11 Fax: 32/16/32 19 70 Email: [email protected] 145, Electronic and Electrical Eng. University College, Bel eld, Dublin 4, Ireland Tel: (353)-1-706 1846 Fax: (353)-1-283 0921 Email: [email protected] 2
Department of Electrical Engineering and Computer Science University of California at Berkeley, Berkeley, CA 94720, USA Tel: +1 (510) 642 3209 Fax: +1 (510) 643 8869 Email: [email protected] 3
( Running title: Robust synthesis for synchronization ) ( Corresponding author: Johan Suykens ) 1
Abstract In this paper a method for robust synthesis of full static state error feedback and dynamic output error feedback for master-slave synchronization of Lur'e systems is presented. Parameter mismatch between the systems is considered in the synchronization schemes. Sucient conditions for uniform synchronization with a bound on the synchronization error are derived based on a quadratic Lyapunov function. The matrix inequalities from the case without parameter mismatch between the Lur'e systems remain preserved, but an additional robustness criterion has to be taken into account. The robustness criterion is based on an uncertainty relation between the synchronization error bound and the parameter mismatch. The robust synthesis method is illustrated on Chua's circuit with the double scroll. One observes that it is possible to synchronize the master-slave systems up to a relatively small error bound even in the case of di erent qualitative behaviour between the master and the uncontrolled slave system, such as limit cycles and stable equilibria.
1 Introduction Master-slave synchronization schemes for Lur'e systems have been studied in [5, 6, 16] for the autonomous case with static state feedback or dynamic output feedback applied to the slave system. With respect to secure communications applications [8], the non-autonomous case has been studied in [17, 18], where an information carrying message signal is considered as an external input. In these works the synchronization problem has been approached from the viewpoint of control theory. The autonomous case has been studied with respect to absolute stability theory with sucient conditions for global asymptotic stability of the error system, either based on quadratic or Lur'e-Postnikov Lyapunov functions. For the non-autonomous case the synchronization scheme has been interpreted as a modelreference control scheme in standard plant form with exogenous input and regulated output, according to modern control theory [1, 11]. In this way additive channel noise has been taken into account in the design procedure. 2
However, these synchronization schemes assume that the master-slave Lur'e systems are identical. In this paper we study the in uence of parameter mismatch between the Lur'e systems. Previous work on robust synchronization has been reported in [21, 5, 19]. According to [21] the case of non-identical Lur'e systems requires the de nition of a synchronization error bound, because zero synchronization error cannot be achieved. In this paper we present matrix inequalities [2] for the synchronization schemes with static state feedback or dynamic output feedback. The matrix inequalities give sucient conditions for uniform synchronization with a certain error bound. The same form is preserved for the matrix inequalities as for the case of identical master-slave Lur'e systems. Uncertainty relations between the bound on the synchronization error and the parameter mismatch follow from the derived Theorems. The additive perturbation which is related to the parameter mismatch and is considered with respect to the nominal identical Lur'e systems, is assumed to be unstructured [11]. Finally, a robust synthesis method is presented which involves solving a nonlinear optimization problem. This optimization problem is based on the matrix inequality and the robustness criterion, where the latter follows from the uncertainty relation. The robust synthesis method is illustrated on Chua's circuit. From the simulation results one observes that a relatively large parameter mismatch can be allowed such that the systems remain synchronized with a relatively small synchronization error bound. In this sense it is possible to synchronize the slave system, which behaves as the double scroll in the uncontrolled case, to a master system which behaves either chaotically, periodically or has stable equilibrium points. These results have been obtained both for the case of full static state feedback and dynamic output feedback. This paper is organized as follows. In Section 2 we present the synchronization schemes with non-identical Lur'e systems for the case of static state error feedback and dynamic output error feedback. In Section 3 we describe the corresponding error systems. In Section 4 criteria for robust synchronization are derived. In Section 5 the robust synthesis method is presented, which is based on the derived matrix inequalities. Finally, in Section 6 the method is illustrated by applying it to Chua's circuit. 3
2 Synchronization Schemes In this Section we describe master-slave synchronization of Lur'e systems for two cases: full static state error feedback and dynamic output error feedback.
2.1 Full static state error feedback Consider the master-slave synchronization scheme with full static state error feedback [21, 5, 17] 8 > > M : x_ = A1x + B1 (C1x) > > > s > > > > > > > < > > > > > > > > > > > > :
Ss : z_ = A2 z + B2(C2z) + us
(1)
Cs : us = F (x ? z) with master system Ms, slave system Ss and controller Cs. The index s refers to the static feedback case. The master and slave system are Lur'e systems with state vectors x; z 2 Rn respectively and matrices Ai 2 Rnn , Bi 2 Rnnh , Ci 2 Rnh n (i = 1; 2). A Lur'e system is a linear dynamical system, feedback interconnected to a static nonlinearity () that satis es a sector condition [9, 20] (here it has been represented as a recurrent neural network with one hidden layer, activation function () and nh hidden units [15]). We assume that () : Rnh ! Rnh is a diagonal and continuous nonlinearity (but possibly nondi erentiable at a countable number of points) with i () belonging to sector [0; k], i.e. i ( )[i( ) ? k ] 0, 8 2 R for i = 1; :::; nh. The scheme aims at synchronizing the master system to the slave system by applying full static state error feedback to the slave system using the control signal us 2 Rn with feedback matrix F 2 Rnn . The Lur'e systems are assumed to be non-identical. The parameter mismatch between the systems, kA1 ? A2 k2; kB1 ? B2 k2; kC1 ? C2 k2 is assumed to be relatively `small'. The aim of this paper is precisely to design the feedback matrix F such that the scheme is robust with respect to this parameter mismatch.
4
2.2 Dynamic output error feedback In addition to the state feedback scheme we will also study the dynamic output error feedback scheme, introduced in [16]:
Md :
8 > < > :
x_ = A1 x + B1 (C1x) p = Hx
Sd :
8 > < > :
z_ = A2 z + B2(C2 z) + Dud q = Hz
Cd :
8 > < > :
_ = E + G(p ? q) ud = M + N (p ? q)
(2)
with master system Md, slave system Sd and a controller Cd . The index d refers to the dynamic feedback case. For the master and slave system we consider the state vectors, system matrices and nonlinearity as described for (1). The output vectors of the master and slave system are p; q 2 Rl with l n. The slave system is controlled by means of the control vector ud 2 Rm through the matrix D 2 Rnm . The signal ud is the output of a linear dynamic output feedback controller. The input of this controller is the output error p ? q. The linear dynamic controller has state vector 2 Rnc and consists of the matrices E 2 Rnc nc , G 2 Rnc l, M 2 Rmnc , N 2 Rml . In [16] this scheme has been studied for identical Lur'e systems. As for the state feedback case we are interested here in designing the controller Cd such that a high robustness is obtained with respect to parameter mismatch between the Lur'e systems.
3 Error systems De ning the error signal as e = x?z, the aim of the synchronization schemes is to design the controllers Cs , Cd such that ke(t)k2 ! 0 as t ! 1. However, a zero error is only obtainable 5
in the case of identical master-slave systems. For the case of parameter mismatch between the systems an error bound must be considered instead. This will be further discussed in the following Section. Denoting the state feedback synchronization scheme as 8 > < > :
x_ = fs(x) z_ = gs(z; x)
(3)
with continuous nonlinear mappings fs() : Rn ! Rn , gs(; ) : Rn Rn ! Rn , one obtains the error system: (4) Es : e_ = fs(x) ? gs(z; x): Inspired by the proof of Theorem 14 in [21], we make the following decomposition for the error system: e_ = vs(x; z) + ws(x) (5) with
8 > > > vs(x; z) > > > > > > < > > > ws(x) > > > > > > :
= gs(x; x) ? gs(z; x) = (A2 ? F )e + B2 (C2e; z) = fs(x) ? gs(x; x) = Ax + B1(C1x) ? B2(C2x) where A = A1 ? A2, (C2e; z) = (C2e + C2z) ? (C2z). Denoting the dynamic synchronization scheme as 8 > > > > >
> > > > : _ = hd (; x; z ) with continuous nonlinear mappings fd () : Rn ! Rn , gd(; ; ) : hd (; ; ) : Rnc Rn Rn ! Rnc , one obtains the error system:
Ed : e_ = fd(x) ? gd (z; x; ): 6
(6) Rn
R n R nc ! R n , (7)
We make the following decomposition:
e_ = vd(x; z; ) + wd(x; ) with
(8)
8 > > > vd(x; z; ) > > > > > > < > > > wd (x; ) > > > > > > :
= gd(x; x; ) ? gd(z; x; ) = (A2 ? DNH )e + B2 (C2e; z) = fd(x) ? gd(x; x; ) = '(x) ? DM where A = A1 ? A2, (C2e; z) = (C2 e + C2z) ? (C2z) and '(x) = Ax + B1 (C1x) ? B2 (C2x).
4 Criteria for Robust Synchronization In order to derive criteria for synchronization of the schemes (1)-(2) with parameter mismatch between the systems, we rst have to introduce assumptions on the nonlinearity in the error system and on the norm of the state vector of the master system.
Assumption 1. The nonlinearity (C2e; z) belongs to sector [0; k]: (cT e; z) (cT e + cT z) ? (cT2i z) 0 i c2Ti e = i 2i c2Ti e k; 8e; z; i = 1; :::; nh (cT2i e 6= 0); (9) 2i
2i
where cT2i denotes the i-th row vector of C2 .
The following inequality holds then:
i(cT2i e; z) [i(cT2i e; z) ? kcT2i e] 0;
8e; z; i = 1; :::; nh:
(10)
It follows from the mean value theorem that for di erentiable () the sector condition [0; k] on () corresponds to [5] 0 d i(; z) k; 8; z; i = 1; :::; nh: (11) d 7
For ws() there exists a positive real constant s such that
kws(x)k2 < skxk2 ; 8x 2 Rn :
(12)
The same holds for the function '() with constant d.
Assumption 2. Master systems Ms and Md satisfy the condition that there exists a
positive real constant such that, for any initial condition x0 , there exists time T (x0 ) for which kx(t; x0 )k2 ; 8t T: (13)
This assumption is based on the work of [5]. From a practical point of view this is a reasonable assumption, in particular for chaotic Lur'e systems, because one is not interested in employing a master system that possesses unbounded trajectories. Because the master-slave systems are non-identical the synchronization error will not tend asymptotically to zero. Therefore the following de nition of synchronization with error bound is employed [21].
De nition 1. The synchronization schemes (1)-(2) uniformly synchronize with error bound if there exists a 0 > 0 and a T 0 such that if kx(0) ? z(0)k2 0 then kx(t) ? z(t)k2 for all t T . For the synchronization scheme with static feedback we consider the following positive de nite quadratic Lyapunov function (which is radially unbounded)
V (e) = eT Pe; P = P T > 0:
(14)
Theorem 1. Suppose Assumption 1 and 2 and that there exists a diagonal positive de nite matrix = diagfig 2 Rnh nh , a symmetric positive de nite matrix P 2 Rnn , a 8
matrix F 2 Rnn and a positive real constant such that the matrix inequality
Y
2 = Y T = 64
3
(A2 ? F )T P + P (A2 ? F ) + I PB2 + kC2T 7 5 = ; 8t: When c is determined such that E s = fe j eT Pe cg is the smallest ellipsoid containing the ball B1s = fe j eT e 2= 2g, the trajectory will enter the ellipsoid E s for every initial 9
condition e(0). Note that the ellipsoid E s, which is parametrized in terms of c, is directly related to the level set of the Lyapunov function V [20]. Hence for every initial condition, the limit set of the error system is nonempty, closed and bounded and belongs to E s [20]. Therefore the error system will uniformly synchronize with error bound pc2 where c2 is determined such that B2s is the smallest ball containing E s.
2
Remarks We stress that the condition (15), being based on a quadratic Lyapunov function, is only sucient and possibly conservative. However, from absolute stability theory of Lur'e systems [9, 20] one knows that this matrix inequality is related to the circle criterion (by means of the KYP-Lemma) and in this sense is meaningful. The numerical algorithms which follow this Section have two tasks. Firstly to con rm that for chaotic Lur'e systems such as Chua's circuit a solution to the LMI exists and secondly to explain how one can nd one.
Convex optimization procedures for nding an outer approximation of a union of ellipsoids and an intersection of ellipsoids are discussed in [2](p.43). These methods can be used in order to construct B2s, E s given B1s and nding c, c2 . On the other (P ) hand it follows from Fig.1 that pc2 = (P )= with (P ) = max min (P ) the condition number of matrix P .
Note that for P = I the synchronization scheme (1) is uniformly synchronizing with error bound 2 s= . In this case the sets B1s, B2s and E s are identical. For the synchronization scheme with dynamic feedback we consider the following positive de nite quadratic Lyapunov function 2 P V ( ) = T P = [eT T ] 64 11
32 P12 7 6 54
P21 P22 10
e
3 7 5;
P = P T > 0:
(16)
with = [e; ].
Theorem 2. Suppose Assumption 1 and 2 and that there exists a diagonal positive de nite matrix = diagfig 2 Rnh nh , a symmetric positive de nite matrix P 2 R(n+nc )(n+nc ) ,
controller matrices E; G; M; N and positive real constants 1 ; 2 such that the matrix inequality
Z
2 66 Z11 + 1 I = Z T = 666 Z12T 4
Z13T
with
Z11 Z12 Z13 Z22 Z23 Z33
= = = = = =
Z12 Z22 + 2 I Z23T Z33
(17)
(A2 ? DNH )T P11 + P11(A2 ? DNH ) + H T GT P21 + P12GH (A2 ? DNH )T P12 + H T GT P22 ? P11 DM + P12E P11B2 + kC2T E T P22 + P22E ? M T DT P12 ? P21 DM P21B2 ?2
is satis ed. De ne PI = r
3 Z13 7 7 Z23 777 < 0 5
diagfIn=12; Inc =22g
with 1 =
1 2 1
+
1 2
r
12 21
+ 12 2 , 2 = 2
2
2 2
+
+ 222 where 1 = 2 d max (P11), 2 = 2 d max (P12 ). d , are de ned according to (12), (13) respectively. If there exist positive real constants c1 ; c2 ; c3 ; c4 such that (Fig.2) 1 2
12 1 2
2
B1d = fe j eT e 12g E1d = f j T PI 1g; B2d = f j T 22g E1d; B3d = fe j eT e c3g E3d = f j T PI c2g; B4d = f j T c4 g E3d and
E1d E2d = f j T P c1 g E3d
then the synchronization scheme (2) is uniformly synchronizing with error bound pc3 .
11
Proof: Taking the time derivative of the Lyapunov function (16) and, using the inequalities (10), one obtains:
V_ = _T P + T P _ = [vd(x; z; ) + wd(x; )]T (P11e + P12) + _T (P21 e + P22 ) +(eT P11 + T P21)[vd (x; z; ) + wd(x; )] + (eT P12 + T P22 )_ [(A2 ? DNH )e + B2 ? DM + '(x)]T (P11 e + P12 ) + (E + GHe)T (P21 e + P22 ) +(eT P11 + T P21)[(A2 ? DNH )e + B2 ? DM + '(x)] + (eT P12 + T P22 )(E + GHe) ? Pi 2ii(i ? kcT2i e) T Z0 + 2 d max (P11)kek2 + 2 dmax (P12 )kk2 where = [e; ; ] and
3
2 Z11 6 6 Z0 = 666 Z12T 4
Z12 Z13 7 7 Z22 Z23 777 : 5 Z13T Z23T Z33 From (17) we obtain the upper bound T Z0 ? 1 eT e ? 2T and therefore V_ ? 1 kek22 ? 2kk22 + 2 d[max (P11 )kek2 + max (P12 )kk2] = kek2[? 1 kek2 + 2 d max (P11)] + kk2[? 2 kk2 + 2 d max (P12)] ?(p 1 kek2 ? 2p 1 1 )2 ? (p 2 kk2 ? 2p 2 2 )2 + 14 ( 121 + 222 ):
Hence V_ < 0 if 8t
ke(t)k2 > 1 k(t)k2 > 2 :
When c1 is selected such that E1d is the smallest ellipsoid containing the balls B1d and B2d , the trajectory of the error system will enter the ellipsoid E2d for every initial condition [e(0); (0)]. The ellipsoid E2d is directly related to the level set of the Lyapunov function. Hence the error system will uniformly synchronize with error bound pc3 where c3 is determined such that E2d is the smallest level set containing E1d. E3d is the smallest ellipsoid containing E2d, subject to the de nition of these sets.
2 12
5 Robust Synthesis The design of the controller Cs is based then on the Theorem 1, for which one derives the following uncertainty relation between the synchronization error bound and the parameter mismatch of the systems: =c ; (18) 0
where the positive real constant c0 is equal to 2max (P )(P )= . The interpretation of the uncertainty relation is twofold: supposing a larger parameter mismatch, which corresponds to an increasing value, suggests a larger synchronization error bound and vice-versa demanding a lower error bound requires a smaller parameter mismatch. An interpretation for follows by taking the 2-norm of the functions ws() and '(), which gives the conservative estimate = kAk2 + k kB1k2 kC1k2 + k kB2 k2 kC2k2: (19) Robust synthesis aims at minimizing the constant c0 in the uncertainty relation such that the matrix inequality (15) is satis ed. Since follows from the choice of the master system, this corresponds to the following design problem for the static feedback case: 8 > < > :
Y (F; P; ; ) < 0 (20) P = P T > 0; 0 and diagonal: For the dynamic feedback case, rather than considering the error bound as derived in Theorem 2, we will minimize the volume of the ball B1d. This is meaningful as long as the condition number of the matrix P remains relatively small. With respect to therball B1d the 2 2 uncertainty relation (18) has the positive real constant c0 equal to 1 = 2 11 + 21 121 + 12 2 and corresponds to d. The design for the dynamic feedback case can be done then as follows: 8 > > > Z (E; G; M; N; P; ; 1; 2) < 0 > > < max (P11 ) max (P12 ) such that > P = P T > 0; 0 and diagonal min E;G;M;N;P;; 1; 2 1 + 2 > > > > : (P ) 0 (21) min (P ) (P )= such that F;P;; max
13
where 0 is a user-de ned upper bound on the condition number of P . The problems (20) and (21) are non-convex optimization problems. Non-di erentiability might occur when the two largest eigenvalues of Y or Z coincide [14]. The constraint P > 0 can be eliminated by considering the parametrization P = QT Q. A similar idea applies to . An important observation is that the matrix inequalities, obtained for the case without parameter mismatch, are preserved (see [5, 6, 16]) and an additional robustness criterion can be taken into account in the design procedure.
6 Example: Chua's Circuit In this Section we illustrate the robust synthesis method on Chua's circuit. We take the following representation for Chua's circuit: 8 > > > x_ 1 > >
> > > > : x_ 3 = ?b x2
(22)
h(x1 ) = m1 x1 + 21 (m0 ? m1 ) (jx1 + cj ? jx1 ? cj)
(23)
with nonlinear characteristic
and parameters a = 9, b = 14:286, m0 = ?1=7, m1 = 2=7 in order to obtain the double scroll attractor [3, 4, 12]. The nonlinearity (x1 ) = 21 (jx1 + cj?jx1 ? cj) (linear characteristic with saturation) belongs to sector [0; 1]. Hence Chua's circuit can be interpreted as the Lur'e system x_ = Ax + B (Cx) where 2 66 A = 666 4
3
2
3
?a(m0 ? m1 ) 77 ?a m1 a 0 77 6 6 7 ; C = [1 0 0]: 7 0 1 ?1 1 777 ; B = 666 7 5 4 5 0 0 ?b 0
(24)
In the sequel, we consider the double scroll as the nominal system and we let it correspond to the slave system A2 ; B2; C2. 14
We illustrate rst the full static state feedback case. Sequential quadratic programming [7] has been applied in order to optimize (20) using Matlab's optimization toolbox (function constr). An additional constraint on the controller parameter vector kF k2 < 20 has been used. Instead of the inequality Y < 0 the constraint max (Y ) + 0:001 < 0 has been employed, where max () denotes the maximal eigenvalue of the symmetric matrix. As starting point for the iterative procedure, a random F matrix has been chosen according to a normal distribution with zero mean and variance 0.1 and Q = I , = 0:1 I . Simulation results for master-slave synchronization of the Chua's circuits are shown in Fig. 3-6, with perturbations on the element a11 of the A matrix of the master system. The simulation results show that it is possible to synchronize the master-slave systems with relatively small non-zero synchronization error bound, even in the case where the master system has di erent qualitative behaviour from the uncontrolled slave system (such as limit cycles and stable equilibrium points). In order to illustrate the dynamic output feedback case, suppose that we measure the rst state variables x1 and z1 only in order to synchronize the circuits and that we take a one-dimensional control signal in order to control the slave system. This corresponds to the choice H = [1 0 0], D = [1; 0; 0] (l = m = 1). We report the results here for a second order controller with nc = 2. A 2-norm constraint on the controller parameter vector [E (:); G(:); M (:); N (:)] (< 80) has been used for (21), where `:' denotes a columnwise scanning of a matrix. Instead of the inequality Z < 0 the constraint max (Z ) + 0:001 < 0 has been employed and 0 = 20 has been imposed. As starting point for the iterative procedure, a random controller parameter vector has been chosen according to a normal distribution with zero mean and variance 0.1. For the matrix Q a square random matrix was chosen according to the same distribution but with variance equal to 3. The matrix has been initialized as 100 I . Simulation results for master-slave synchronization of the Chua's circuits are shown in Fig. 7-8, for perturbations on the element a11 of the master system. As for the static state feedback case it is possible to synchronize the masterslave systems with relatively small non-zero synchronization error bound, even in the case 15
where the master system has di erent qualitative behaviour from the uncontrolled slave system. All simulation results have been obtained using a Runge-Kutta integration rule with adaptive step size (ode23 in Matlab) [13].
7 Conclusion By means of Chua's circuit we illustrated that a relatively large parameter mismatch between Lur'e systems can be allowed in order to maintain synchronization with a relatively small synchronization error bound, even in the case where the master system behaves qualitatively di erent from the uncontrolled slave system. These results have been obtained by deriving a robustness criterion in addition to matrix inequalities, based upon a quadratic Lyapunov function. The matrix inequalities basically take the same form as for the case of identical Lur'e systems. The robustness criterion is based on an uncertainty relation between the synchronization error bound and the parameter mismatch. The parameter mismatch has been interpreted in terms of unstructured perturbations on the system matrices of the Lur'e systems. Both static state error feedback and dynamic output error feedback have been studied. The proposed synthesis method o ers a straightforward design procedure by means of solving a nonlinear optimization problem. It completes previous work on control theoretic interpretations of synchronization schemes.
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Acknowledgment This research work was carried out at the University of California at Berkeley, in the framework of the Belgian Programme on Interuniversity Poles of Attraction, initiated by the Belgian State, Prime Minister's Oce for Science, Technology and Culture (IUAP-17) and in the framework of a Concerted Action Project MIPS (Modelbased Information Processing Systems) of the Flemish Community. The work is supported in part by the Oce of Naval Research under grant N00014-96-1-0753 and the Fulbright Fellowship Program. We thank the reviewers for constructive comments.
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References [1] Boyd S. & Barratt C., Linear controller design, limits of performance, Prentice-Hall, 1991. [2] Boyd S., El Ghaoui L., Feron E. & Balakrishnan V., Linear matrix inequalities in system and control theory, SIAM (Studies in Applied Mathematics), Vol.15, 1994. [3] Chua L.O., Komuro M. & Matsumoto T., \The Double Scroll Family," IEEE Trans. Circuits and Systems-I, 33(11), 1072-1118, 1986. [4] Chua L.O., \Chua's circuit 10 years later," Int. J. Circuit Theory and Applications, 22, 279-305, 1994. [5] Curran P.F. & Chua L.O., \Absolute stability theory and the synchronization problem," Int. J. Bifurcation and Chaos, Vol.7, No.6, pp.1375-1382, 1997. [6] Curran P.F., Suykens J.A.K. & Chua L.O., \Absolute Stability Theory and MasterSlave Synchronization," Int. J. Bifurcation and Chaos, Vol.7, No.12, 1997. [7] Fletcher R., Practical methods of optimization, Chichester and New York: John Wiley and Sons, 1987. [8] Hasler M., \Synchronization principles and applications," Circuits & Systems: Tutorials IEEE-ISCAS '94, 314-326, 1994. [9] Khalil H.K., Nonlinear Systems, New York: Macmillan Publishing Company, 1992. [10] LaSalle J.P. & Lefschetz S., Stability by Liapunov's direct method with applications, New York: Academic Press, 1961. [11] Maciejowski J.M., Multivariable feedback design, Addison-Wesley, 1989. [12] Madan R.N. (Guest Editor), Chua's Circuit: A Paradigm for Chaos, Signapore: World Scienti c Publishing Co. Pte. Ltd, 1993. 18
[13] Parker T.S. & Chua L.O., Practical numerical algorithms for chaotic systems, New York: Springer-Verlag, 1989. [14] Polak E. & Wardi Y., \Nondi erentiable optimization algorithm for designing control systems having singular value inequalities," Automatica, 18(3), 267-283, 1982. [15] Suykens J.A.K., Vandewalle J.P.L. & De Moor B.L.R., Arti cial Neural Networks for Modelling and Control of Non-Linear systems, Boston: Kluwer Academic Publishers, 1996. [16] Suykens J.A.K., Curran P.F. & Chua L.O., \Master-slave synchronization using dynamic output feedback," International Journal Bifurcation and Chaos, Vol.7, No.3, pp.671-679, 1997. [17] Suykens J.A.K., Vandewalle J. & Chua L.O., \Nonlinear H1 Synchronization of Chaotic Lur'e Systems," International Journal of Bifurcation and Chaos, Vol.7, No.6, pp. 1323-1335, 1997. [18] Suykens J.A.K., Curran P.F., Yang T., Vandewalle J. & Chua L.O., \Nonlinear H1 synchronization of Lur'e systems: dynamic output feedback case," IEEE Transactions on Circuits and Systems-I, Vol.44 No.11, pp.1089-1092, Nov. 1997. [19] Suykens J.A.K., Curran P.F., Vandewalle J. & Chua L.O., \Robust nonlinear H1 synchronization of chaotic Lur'e systems," IEEE Transactions on Circuits and SystemsI, Special Issue on Chaos Synchronization, Control and Applications, Vol.44, No.10, pp.891-904, Oct. 1997. [20] Vidyasagar M., Nonlinear Systems Analysis, Prentice-Hall, 1993. [21] Wu C.W. & Chua L.O., \A uni ed framework for synchronization and control of dynamical systems," Int. J. Bifurcation and Chaos, 4(4), 979-989, 1994.
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Captions of Figures Fig. 1. Illustration of the balls B1s, B2s and the ellipsoid E s with respect to Theorem 1. Fig. 2. Illustration of the balls B1d , B2d , B3d , B4d and the ellipsoids E1d, E2d, E3d with respect to Theorem 2. A simpli ed case is shown for n = 1, nc = 1 such that the balls correspond to line segments. Fig. 3. Master-slave synchronization of Chua's circuits using static state feedback. This gure shows the simulation results of the robust synthesis method in case there is no parameter mismatch. (Top) master system; (Middle) slave system; (Bottom) ke(t)k2 on logarithmic scale. The synchronization error is asymptotically converging to zero. Fig. 4. Static state feedback (continued). In this case there is parameter mismatch between the master and slave Chua's circuit. The slave system is considered to be the nominal system and behaves as the double scroll in case of a zero control input. A perturbation is considered on the A matrix of the master system (a11 = 1:2). The behaviour of master and slave system is shown on (Top) and (Middle) respectively. (Bottom) ke(t)k2 on logarithmic scale. The Chua's circuits synchronize with non-zero error synchronization bound. Fig. 5. Static state feedback (continued). Similar to Fig. 2, but with perturbation a11 = 1:8. The master system shows limit cycle behaviour, while the uncontrolled slave system behaves as the double scroll. Fig. 6. Static state feedback (continued). Similar to Fig. 2, but with perturbation a11 = 1:95. The master system possesses stable equilibria, while the uncontrolled slave system behaves as the double scroll. 20
Fig. 7. Master-slave synchronization of Chua's circuits using dynamic output feedback. A SISO second order linear dynamic controller is considered. A perturbation of a11 = 1:5 is considered on the A matrix of the master system. (Top-left) master system; (Top-right) slave system; (Bottom-left) ke(t)k2 on logarithmic scale. The Chua's circuits synchronize with non-zero error synchronization bound. (Bottom-right) k(t)k2 on logarithmic scale. Fig. 8. Dynamic output feedback (continued). Similar to Fig. 5, but with perturbation a11 = 2. The master system possesses stable equilibria, while the uncontrolled slave system behaves as the double scroll.
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