2009 International Conference on Signals, Circuits and Systems
Master-Slave Synchronization for PWA Systems Olfa Boubaker National Institute of Applied Sciences and Technology, University of 7th November at Carthage, Tunis, Tunisia
[email protected]
Abstract— In this paper we give a new sufficient condition for master-slave synchronization of piecewise affine (PWA) systems. Using a Lyapunov approach and the so-called S-procedure, we show that the synchronization problem can be solved as an optimization problem subject to a set of linear matrix inequalities (LMIs) for which the slave state-feedback controller is designed efficiently. A mechanical system of two-degree-of-freedom (2DOF), generally used in chronizing control of wire bonders, is finally used to illustrate the efficiency of the proposed approach.
II.
Consider a partition of the state-space with polyhedral cells, noted by Λ q , q = 1, ! , N by a finite number d of hyperplanes Γl , l = 1, " , d . According to [11], each polyhedral cell is constructed as the intersection of a finite number rq of half-spaces. Thus, a polyhedral cell is definite by following polytopic description:
Keywords— Master-slave systems, synchronization, Lyapunov stability, switching functions, optimization problems.
I.
PROBLEM FORMULATION
{
}
Λ q = x H qT x + hq < 0
INTRODUCTION
Synchronization is one of basic nonlinear phenomenon that can occur when two or more nonlinear oscillators are coupled. Different kinds of synchronization exist. They are summarized in [1]. From the control point of view, the controlled synchronization is the most interesting. There exist two cases of controlled synchronization [2]: mutual synchronization see e.g. [3] and master-slave synchronization see e.g. [4, 5].
where H q ∈ ℜ
n× rq
and hq ∈ ℜ
rq ×1
(1)
, q ∈ Q = {1, ! , N } .
Consider, ∀ j ∈ Q , a PWA master system described by: x m = A j x m + Bu m + b j x m ∈ Λ j
(2)
and ∀ i ∈ Q , a PWA slave system described by: Recently, interesting works are obtained for a specific class of hybrid dynamical systems: piecewise affine systems (PWA). The PWA systems are switching systems for which the switching rule depends on the affine state vector. Many references treated the problem of stabilization of PWA systems see e.g. [6, 7, 8, 9] and references therein.
x s = Ai x s + Bu s + bi , x s ∈ Λi
where Ai , A j ∈ ℜ n×n , B ∈ ℜ n×m and bi , b j ∈ ℜ n . x m ∈ ℜn and x s ∈ ℜn are the states of the master and of the slave, respectively.
In this paper, we interest in master-slave controlled synchronization of continuous-time PWA systems. The master-slave synchronization problem will be formulated as a global stability problem of synchronization error which will be controlled via a state-feedback control law. The same problem was recently considered in [10] and in [11] but the solution given in this paper is very different since we use a unique Lyapunov function for the master-slave system to guarantee global stability and the so called S-procedure to reduce the conservatism of the classical Lyapunov approach.
u m is the input of the master system and u s is the control input for the slave system. The error between the dynamics of the PWA master system (2) and the PWA slave system (3) are described by: Δx = Ai Δx + BΔu + ΔAij x m + Δbij
where Δu = u s − u m , ΔAij = Ai − Aj , Δbij = bi − b j
This paper is structured as follows: the problem of master slave synchronization and the assumptions under which the problem is solved are formulated in section 2. The control law under a Lyapunov approach is designed in section 3. In section 4, a mechanical system of two-degree-of-freedom (2DOF), generally used in chronizing control of wire bonders, is finally used to illustrate the efficiency of the proposed approach. .
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(3)
(4)
and
i, j ∈ Q . Master slave problem: Design a control law u s for the slave system (3) based on the measured error (4) such that
xs ⎯ ⎯→ x m as t ⎯ ⎯→ ∞ (see fig.1).
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2009 International Conference on Signals, Circuits and Systems
The condition (11) satisfies the third Lyapunov condition (10) and furthermore makes V (Δx ) more negative. In the next it will be used to propose a design of a control law. Theorem 1. For a given decay α > 0 and for all i, j ∈ Q , If
matrices P ∈ ℜn×n and K ∈ ℜ m×n , column
there exist
vectors β 1,ij ∈ ℜ ri ×1 , β 2,ij ∈ ℜ
To solve the synchronisation problem let us adopt the following assumptions:
P = PT ≥ 0 ,
β1,ij ≤ 0 , β 2,ij ≤ 0 , β 3,ij ≤ 0 , β 4,ij ≤ 0
Assumption 1. The right-hand sides of (2) and (3) are Lipschitz, so the right-hand side of (4) is as well Lipchitz.
ªΔ1 « «∗ «∗ ¬
Assumption 2. The entire state vectors of both systems (master and slave) are measured.
PΔAij β 3,ij I ∗
º » »≤0 T T + 2 β1,ij hi + 2 β 2,ij h j »¼
PΔbij + H i β1,ij H i β1,ij + H j β 2,ij
β 4,ij
(12)
Δ1 = ( Ai + BK )T P + P( Ai + BK ) + αP
Assumption 3. A unique feedback control law incorporating a linear synchronization error (4) must be applied in closed loop such that:
are satisfied, then the master-slave synchronization error system (6) is globally asymptotically stable.
(5)
In (12) and in the following, ∗ denotes the symmetric bloc.
with K ∈ ℜm×n is some constant gain matrix to be designed.
Proof. See Appendix 1
Assumption 4. Since the master and the slave systems (2) and (3) are PWA systems, the closed loop system is again a PWA system with time varying input such that in closed loop, the error dynamics (4) under the assumption 3 are described by:
Δx = ( Ai + BK)Δx + ΔAij xm + Δbij
and constants β3,ij and
β 4,ij , such that the following inequalities:
Figure 1. Controlled master-slave system
Δu = KΔx
r j ×1
Remark. The sufficient condition (12) is a Bilinear Matrix Inequality (BMI). Until now, there are not yet easy algorithms devoted for BMIs’ resolution. So, the BMI must be reformulated in the following in LMIs to be efficiently solved by common software.
(6)
Theorem 2.
III.
CONTROLLER SYNTHESIS For a given decay α > 0 and for all i, j ∈ Q , if there exist
In order to synchronize the dynamics between the master and the slave PWA systems it is necessary to prove global stability of the error dynamics (6) via the control law (5). Impose then to the system (6) to have a unique Lyapunov function defined by: V (Δx ) = Δx T PΔx
matrices S ∈ ℜn×n , R ∈ ℜ m×n , diagonal matrices Eij ∈ ℜri ×ri and Fij ∈ ℜ
and some constants β ij and
ξ ij , such that
the following LMIs:
(7)
S = ST > 0 Eij < 0 , Fij < 0 , βij < 0 , ξ ij < 0
The master-slave synchronization error dynamics (6) is then global asymptotically stable if V (Δx ) satisfies the following conditions [13]: V (Δx ) = 0 if ∀Δx = 0 (8) V (Δx ) > 0 for ∀Δx ≠ 0 V (Δx ) < 0 for ∀Δx ≠ 0
r j ×r j
ª «ξij « «* « «* «¬
(9) (10)
Furthermore Let α > 0 be the desired decay rate for this Lyapunov function such that [13]: V (Δx ) ≤ −αV (Δx ) (11)
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ξij hi
T
1 Eij 2 *
Tº
ξij h j »
» » 0 0¢τ 1 ¢¢1, 0¢τ 2 ¢¢1 , (A.1) can be rewritten as:
I. Blekhman, A. Fradkov and H. Nijmeijer, “On self synchronisation and controlled synchronisation,” Systems & Control Letters, Vol. 31, 1997, pp. 299–305. H. Nijmeijer and A. Rodriguez-Angeles, Synchronization of mechanical systems., World Scientific Publishing Co. Pte. Ltd., Singapore, 2003. A. Rodriguez-Angeles and H. Nijmeijer, H, “Mutual synchronisation of robots via estimated state-feedback: a cooperative approach,” IEEE Trans. on Control Systems Techn., Vol. 12, 2004, pp. 542–554. P. Curran, J. Suykens and L. Chua, “Absolute stability theory and master-slave synchronisation,” International Journal of Bifurcation and Chaos, vol. 7, 1997., pp. 2891–2896. J. Suykens and J. Vandewalle, “Master-slave synchronisation using dynamic output feedback,” International Journal of Bifurcation and Chaos, Vol. 7, 1997, pp. 671–679. A. Bemporad & M. Morari, M., “Control of systems integrating logic, dynamics, and constraints,”. Automatica, Vol. 35, 1999, pp. 407-427. M. Johansson and A. Rantzer, “Computation of piecewise quadratic Lyapunov functions for hybrid systems,” IEEE Trans. Automatic Control, Vol. 43, 1998, pp. 555–559. O. Boubaker,“ Gain scheduling control: an LMI approach,” International Review of Electrical Engineering, Vol. 3, 2008, pp. 378-385.
T V (Δx ) + αV (Δx ) − τ 1x m x m − τ 2 ≤ V (Δx ) + αV (Δx ) ≤ 0
This can be also written as following: z T F0 z ≤ 0 where
[
(A.3)
]
z = Δx x m 1 ªΔ 1 « F0 = « ∗ «∗ ¬
PΔAij −τ 1I ∗
(A.2)
T
PΔbij º » 0 » −τ 2 » ¼
Δ 1 = ( Ai + BK ) P + P( Ai + BK ) + αP T
In F0 , ∗ denotes the symmetric bloc and I ∈ ℜn×n is the identity matrix.
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2009 International Conference on Signals, Circuits and Systems
If hq (k ) ≥ 0 then M q (k , k ) = 1
Relying on polytopic expression (1) of polyhedral cell, we can write for the slave and the master respectively ∀ i, j ∈ Q for
k =1,!, rq
If hq (k ) < 0 then M q (k , k ) = −1
all δ i ∈ ℜ ri ×1 , γ j ∈ ℜ r j ×1 , δ i ≥ 0 and γ j ≥ 0 , the following inequalities: (A.4) δ iT H iT x s + δ iT hi ≤ 0
γ Tj H Tj x m + γ Tj h j ≤ 0
k =1,!, rq
Using Schur complement [14], the BMI (12) is satisfied if the two following conditions are verified:
(A.5)
T
Assume that there exist some small positive constants such that: 0 < ε j
