Robust master-slave synchronization of chaos in a one-sided 1-DoF impact mechanical oscillator subject to parametric uncertainties and disturbances

Mechanism and Machine Theory 142 (2019) 103610

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Research Paper

Robust master-slave synchronization of chaos in a one-sided 1-DoF impact mechanical oscillator subject to parametric uncertainties and disturbances Hassène Gritli a,b,∗ a

Higher Institute of Information and Communication Technologies, Université de Carthage, 1164 Borj-Cedria, Tunis, Tunisia Laboratory of Robotics, Informatics and Complex Systems (RISC-Lab, LR16ES07), National Engineering School of Tunis, Université de Tunis El Manar, BP. 37, Le Belvédère, 1002 Tunis, Tunisia

b

a r t i c l e

i n f o

Article history: Received 26 June 2019 Revised 10 August 2019 Accepted 4 September 2019

Keywords: Impact mechanical oscillator Master-slave synchronization Chaos Robust stabilization Parametric uncertainties and disturbances LMI approach

a b s t r a c t This paper addresses the problem of robust master-slave synchronization of chaos in a 1DoF impact mechanical oscillator with a single rigid constraint. The master is considered to be with nominal parameters, whereas the slave impacting system is considered to be subject to polytopic parametric uncertainties and disturbances. We adopt a state-feedback controller for the robust stabilization of the master-slave synchronization error and the Linear Matrix Inequality (LMI) approach for the design of stability conditions. We use the S-procedure Lemma in order to only consider the regions within which the two impacting systems evolve. Thus, the stability conditions are formulated in terms of Bilinear Matrix Inequalities (BMIs). Via some technical Lemmas and congruence transformations, we transform these BMIs into LMIs, which are numerically traceable. An improved LMI-based optimization problem is proposed to solve the problem of high gains of the controller. Finally, the robustness of the proposed state-feedback feedback controller towards parametric uncertainties and disturbances is presented through simulation results showing then the master-slave synchronization of chaos in the one-sided 1-DoF impact mechanical oscillator. © 2019 Elsevier Ltd. All rights reserved.

1. Introduction Chaos has become very popular and has found its interest in almost all areas of research by exploiting the nonlinear dynamics of physical systems [1]. A chaotic system has an excessive sensitivity to initial conditions and its chaotic motion is locally unstable and globally bounded [2–5]. Some definitions of (or attempts to define) chaos can be found in [6,7]. Chaos can occur in many engineering fields such as nonlinear oscillators [3,8], non-smooth mechanical systems [9–11], legged walking robots [12,13], inverted pendulums with obstacles [14,15], impact mechanical oscillators [9,16–20] (see also the special issue in [21] and references inside) and other many dynamical systems [2–5], just to mention a few. In a large number of mechanical systems, there are some working conditions that lead to the appearance or generation of collisions between/of moving elements [21–24]. The operation principle of some mechanical systems such as pile drivers, impact dampers, inertial shakers, milling and forming machines, etc., is based mainly on the impact actions between ∗

Corresponding author. E-mail address: [email protected]

https://doi.org/10.1016/j.mechmachtheory.2019.103610 0094-114X/© 2019 Elsevier Ltd. All rights reserved.

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H. Gritli / Mechanism and Machine Theory 142 (2019) 103610

oscillatory elements [9,22]. Furthermore, the mechanical systems with clearances and by considering soft or hard rigid constraints are used to model impact dampers, inertial shakers, shock absorbers, drill-strings, impact print hammers, etc. [21– 23,25–29]. Even if the gap between interacting bodies is very small, it can lead to some undesirable behaviors and responses like the vibration, the fatigue phenomena, and the chaotic behavior [9,21,23]. Furthermore, impacts between moving elements may perturb operations in systems and hence the manufacturing process, where we expect to have a regular behavior. The degradation of the performance of impacting mechanical systems due to the presence of clearance between moving bodies has been recognized in mechanisms and machines theory [21,23,30]. In addition, the presence of impacts can lead to some dangerous effects such as severe forces, large acceleration impulses, wear, damage of interacting bodies, fracture of the mechanical devices, etc. [21,23,28,29,31,32]. These effects can lead then to the exhibition of quasi-periodic and chaotic behaviors. Thus, the main concern of mechanical and automation engineers is to improve the performance of these impacting systems and then ensuring their stability via control. However, the control of these systems is difficult to realize due to their impulsive and hybrid nature. Non-smooth dynamics of soft/hard impacting mechanical oscillators and their complex behaviors have received a significant amount of theoretical and experimental attentions to date. The impact mechanical systems are highly nonlinear and chaotic behaviors and bifurcations become the rule rather than an exception. Several mathematical and numerical works on these impacting systems have been carried out by analyzing and controlling chaos and bifurcations. The list of publication is too extensive. We refer our reader to see, e.g., [16–21,32–44,44–57]. Several other studies have also been carried out for multiple-degree-of-freedom impacting oscillators (see, e.g., [17,18,20,32,54,58] and references therein). However, almost these studies focused primarily on a 1-DOF or 2-DoF impact mechanical oscillator with one- or two-sided constraint. In this paper, we consider a periodically excited, 1-DoF, impact mechanical oscillator with only one-sided rigid constraint [9,17,59– 61]. This proposed 1-DoF impact oscillator is modeled by an impulsive hybrid non-autonomous linear dynamics, which is complex enough to exhibit various attractive phenomena and hence to control them [17]. The control subject of chaos has been realized using many different methods [2–5,62–65]. The objective behind the control of chaotic systems is to transform a chaotic motion into a periodic one. An important class of control approaches corresponds to synchronization [2,66–70]. Synchronization and control design of chaotic systems have been widely investigated in view of its potential applications in many fields; see, e.g., [71–74] and references therein. Because of its very complex nonlinear behavior, chaotic systems essentially resist synchronization. Therefore, the design of controllers for realizing synchronization is a significant problem. The problem of chaos synchronization lies in designing a master-slave coupling between two chaotic systems such that the state/trajectory of the slave system asymptotically tracks the state/trajectory of the master system. According to Fradkov and Evans [2], if the synchronization does not exist in the system without control, then the problem can regarded as that of finding the control law that ensures the master-slave synchronization in the closed loop. Synchronization was understood as the property by which two or several interconnected dynamical systems behave in a coordinated way, in another way it was understood as the asymptotic convergence of the states of the interconnected dynamical systems to each other. Therefore, controlled synchronization is the ability to induce synchronization via an external force, i.e. a control input. The idea is to develop an appropriate control law so that two interconnected systems, one called master and one called slave, are synchronized. In this concept, the slave system will try to track (synchronize with) the master system. In the field of impacting oscillators, several strategies to solve the problem of chaos control have been developed. We refer the readers to see, e.g., [17,75] and references therein. In [17,76], the Ott-Grebogi-Yorke (OGY) method was designed in order to control chaos in the 1-DoF impact mechanical oscillator subject to one-sided rigid constraint. In [17], an analysis of the nonlinear dynamic behavior of such impacting oscillator under the OGY controller was achieved. Thus, by varying some parameter (the excitation frequency) of the mechanical system, chaos and bifurcations appear. Uncertain parameters and external disturbances are very often encountered in almost all physical devices and then in practical control applications because, generally, it is difficult and may impossible to obtain exact mathematical models of these physical systems. The presence of uncertainties and disturbances may be due to environmental noises, vibrations, data errors, time-varying parameters, and so on. Furthermore, the presence of parametric uncertainties and disturbances may cause instability and bad performances on the controlled system. Therefore, many efforts and a tremendous of research works have been devoted to the robust stabilization of systems suffering from parametric uncertainties and disturbances. For a recent literature, we refer the reader to see, e.g., [77–80] and references inside. For the problem of chaos synchronization, the presence of uncertainties and disturbances may destroy the synchronization process. Therefore, it is very necessary to take into account the time-varying uncertainties of the system parameters and also the external disturbances for the robust master-slave synchronization problem, and especially of chaos, to counteract their influence via a control law. There are several approaches concerned with this robust synchronization of chaotic systems; see, e.g., [72–74,81,82] and references inside. In this paper, we are interested in the robust master-slave synchronization of chaos exhibited in the 1-DoF impact mechanical oscillator. To achieve this master-slave synchronization, our methodology lies first in adding an auxiliary control input for the slave impact mechanical oscillator in order to synchronize it with the master. Moreover, we consider that the slave oscillator is subject to parametric uncertainties and external disturbances. We will take the time-varying perturbation of the parameters of the one-sided 1-DoF impact mechanical oscillator and also we will take time-varying external disturbances. However, the master system is considered to be in the nominal case, that is without parametric uncertainties and

H. Gritli / Mechanism and Machine Theory 142 (2019) 103610

3

Table 1 Nomenclature with SI-units. Symbol

Description

SI-unit

m k c Um w r d x x˙ u v

Mass of the mechanical oscillator Stiffness of the spring Viscous damping constant Amplitude of the excitation input Excitation frequency Restitution coefficient at the collision Distance between the mass m and the rigid constraint Position of the mass m Velocity of the mass m Excitation input Control input Disturbance input maximum amplitude of the disturbance δ

kg N/m N/(m/s) N rad/s – m m m/s N N N N

δ ρ

disturbances. Thus, the objective is to realize the robust master-slave synchronization of two chaotic impacting oscillators. The control law will enforce the slave impact oscillator to track the nominal chaotic master impact oscillator. The proposed strategy in this work lies in the stabilization of the synchronization error between the master state and the slave one. As a result, the problem of the master-slave synchronization of chaos will be reformulated as a stabilization problem of the master-slave synchronization error. We introduce a state-feedback control law to the slave system in order to stabilize the synchronization error. Two main keys are used in order to solve the master-slave synchronization problem of the two impact mechanical oscillators. The first key lies in the use of the S-procedure in order to reduce some conservatism by only taking into account the working region inside which the state of the two systems evolves [79,83]. In the case of the impact mechanical oscillator, the working region is that between impacts. We will use the Linear Matrix Inequality (LMI) approach [83,84] for the design of stability conditions of the controlled master-slave synchronization error. The problem of finding these stability conditions is formulated as an optimization problem defined in terms of Bilinear Matrix Inequalities (BMIs) [83,84]. The second key for achieving the master-slave synchronization lies in the employment of the Schur complement and the Matrix Inversion Lemma [79] and the use of some congruence transformations in order to transform these designed BMIs into LMIs. Thus, by solving these LMIs, we obtained the matrix gain of the control law. In the end of this paper, we show thus the effectiveness of the designed control law in the robust master-slave synchronization of chaos in the 1-DoF impact mechanical oscillator. Third different scenarios will be considered: the slave system in the nominal case, the slave system under parametric uncertainties, and the slave system under both uncertainties and disturbances. The remainder of this paper is outlined as follows. The impulsive hybrid non-autonomous linear dynamics of the 1-DoF impact mechanical oscillator is described in Section 2. Section 3 presents some preliminaries, such as the S-procedure, the Schur complement and the Matrix Inversion Lemma, and describe also the problem for the master-slave synchronization of chaos for the 1-DoF impact mechanical oscillator. In Section 4, we develop our approach for designing the state-feedback control law for the robust stabilization of the master-slave synchronization error. Transformation of the BMIs into LMIs is achieved also in this section. Section 5 is dedicated to simulation results. Finally, concluding remarks and some future works are drawn in Section 7. Notations The following notations will be used throughout this paper. Expression X T represents the transposed matrix of X . Rn and Rn×m denote a real n vector and a real n × m matrix. A real matrix Y is positive definite if the quadratic form xT Y x > 0 holds for all nonzero x ∈ Rn . In addition, in large matrix expressions, the symbol () replaces terms that are induced by symmetry. Moreover, X + Y + ( ) = X + Y + Y T . Furthermore, the null matrix with appropriate dimension is denoted by O , whereas the identity matrix is denoted by I . Nomenclature A full list of nomenclature with SI-Units used throughout this paper is presented in Table 1. 2. The one-sided 1-DoF impact mechanical oscillator and its impulsive dynamics 2.1. Description of the impact mechanical oscillator Fig. 1 shows an illustration of the 1-DoF impact mechanical oscillator subject to a single rigid mechanical stop [17]. This impact oscillator is composed of a mass m, a spring of stiffness k and a dashpot with a damping coefficient c. The left rigid stop is fixed at a distance d from the mass. The mass m is periodically excited via a sinusoidal input u(t ) = Um cos(wt ) where the parameters w and Um are the excitation frequency and the excitation amplitude, respectively. Thus, under this

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H. Gritli / Mechanism and Machine Theory 142 (2019) 103610

Fig. 1. The 1-DoF impact mechanical oscillator subject to a one-sided rigid constraint [17].

excitation input u, the mechanical system will oscillate along the horizontal axis x and will produce, if possible, impacts with the left rigid stop with a restitution coefficient r. 2.2. Impulsive hybrid dynamics of the impact oscillator The dynamics of the 1-DoF impact mechanical oscillator in Fig. 1 is composed of two phases: an oscillation phase and an



impact phase [17]. Let x and x˙ be the displacement and the velocity, respectively, of the mass m. Moreover, let z = x be the state vector. The impulsive hybrid dynamics of the one-sided 1-DoF impact mechanical oscillator is given by:

x˙

T

z˙ = Az + Bu(t )

as long as

z∈

(1a)

z + = Re z −

whenever

z∈

(1b)

where subscribes

+

and

−

denotes immediately just after and  just before  the impact  phase,  respectively. Moreover, the 0 1 0 1 0 different matrices in (1) are defined like so: A = , B = 1 and Re = . Furthermore, in (1), the set k 0 −r −m − mc m  describes the impact surface, whereas the set  represents the oscillation space. They are defined as follows:

   = z ∈ R2×1 , C T z > d

   = z ∈ R2×1 , C T z = d, C T z˙ < 0   T

(2a) (2b)

where C = 1 0 . Relying on the physical behavior of the impact mechanical oscillator, it is natural to consider that its movement to the right is also limited due to its physical technologies. Then, in the sequel of this paper, the set  will be defined as:

   = z ∈ R2×1 , d < C T z < σ

(3)

where σ > 0. 2.3. Behavior of the impacting oscillation It is well know that the one-sided 1-DoF impact mechanical oscillator generates complex nonlinear phenomena, such as periodic and chaotic behaviors with respect to the parameters r, d and w (see for example [9,17] and references therein). In this work, we choose m = 1 [kg], k = 1 [N/m], c = 0 [N/(m/s )], d = 0 [m], r = 0.8 and Um = 1 [N] as in [17]. In Fig. 2, we provide the bifurcation diagram with respect to the excitation frequency w [17]. This bifurcation diagram shows the velocity x˙ at each stroboscopic period T = 2wπ . The 1-DoF impacting mechanical oscillator displays periodic and chaotic behaviors, and also several types of bifurcation such as the period-doubling bifurcations. Fig. 2 reveals the chaotic attractor of the impact mechanical oscillator for the parameter w = 2.8 [rad/s]. 3. Preliminaries and problem statement 3.1. Preliminaries Definition 1 (Stability of uncertain polytopic LTI systems [83,84]). An uncertain polytopic Linear Time-Invariant (LTI) system can be described by the following representation:

x˙ = A(ν )x,

A(ν ) ∈ Co{A1 , . . . , AL }

(4)

H. Gritli / Mechanism and Machine Theory 142 (2019) 103610

5

Fig. 2. Dynamic behavior of the one-sided 1-DoF impacting mechanical oscillator. (a) shows a bifurcation diagram with respect to the variation of the parameter w. (b) displays the chaotic attractor generated for the parameter w = 2.8 [rad/s] (see [9,17]).

where A(ν ) is the state matrix with polytopic uncertainties ν . The system (4) is described by a convex hull of matrices (a matrix polytope) Ai , ∀i = 1, . . . , L, i.e. Ai is one of the vertices of the polytope. An alternative way of writing this is as follows [84]:

x˙ = A(ν )x,

A (ν ) =

L 

λi Ai

∀λi ≥ 0,

i=1

L 

λi = 1

(5)

i=1

A necessary and sufficient condition for the stability of the system (5) is:

ATi P + P Ai < 0,

∀i = 1 , . . . , L

(6)

where P = P T > 0. Lemma 1 (The Young Inequality [79]). Given constant matrices J and H with appropriate dimensions and any positive matrix M > 0, the following inequality holds:

J T H + HT J ≤ J T MJ + HT M−1 H

(7)

Lemma 2 (The Matrix Inversion Lemma [79]). Given invertible matrices X and Y such that X ∈ Rn×n and Y ∈ Rm×m . Moreover, given matrices U and V with appropriate dimensions: U ∈ Rn×m and V ∈ Rm×n . Then, the Matrix Inversion Lemma is described by the following mathematical relation:



−1 (X + U Y V )−1 = X −1 − X −1 U Y −1 + V X −1 U V X −1

(8)

Lemma 3 (The Schur complement Lemma [79,83]). Given matrices Q = QT , R = RT and S with appropriate dimensions, the following propositions are equivalent:



Q ST





S >0 R

(9a)

R>0 Q − S R−1 S T > 0

(9b)

Lemma 4 (The S-procedure Lemma [79,83,84]). Let F 0 , . . . , F p ∈ Rn×n be symmetric matrices. We consider the following condition on F 0 , . . . , F p :

ζTF 0ζ > 0

for all

ζ = 0

such that

ζ T F i ζ ≥ 0,

i = 1, . . . , p

(10)

If there exist scalar variables τ 1 ≥ 0, . . . , τ p ≥ 0, such that the following condition

F0 −

p 

τi F i > 0

i=1

is satisfied, then the inequality under constraints in (10) holds.

(11)

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H. Gritli / Mechanism and Machine Theory 142 (2019) 103610

3.2. Problem description Our main objective is to achieve the robust master-slave synchronization of chaos for the 1-DoF impact mechanical oscillator. Such goal will be realized by designing a control law, saying v(t), for the slave system in order to track asymptotically the master system. These two nonsmooth systems are completely chaotic for the nominal values of parameters. However, the presence of external disturbances, non-modeled terms and uncertainties in some system parameters is quite natural. In the next, we consider that only the slave impact oscillator presents such disturbances and parametric uncertainties. Furthermore, we consider that the uncertainties are polytopic (see Definition 1). Moreover, we assume that the system parameters m, k and c are uncertain. Remark 1. Notice that the two parameters r and d can be considered uncertain. However, they are not considered as uncertain parameters in this work. But, it will be considered for the problem of the global master-slave synchronization in the subsequent work of this paper when dealing with both the continuous dynamics (1a) of the oscillation phase, and the algebraic equation and the impact condition in (1b) of the impact phase. Let us consider the following change of variables: kˆ =



A=

0 −kˆ



k m,

cˆ =

1 −cˆ

c m

ˆ = and m

1 m.

 

B=

Thus, we will have:

0 ˆ m

(12)

ˆ will be considered as uncertain parameters and with In these matrices A and B, the three parameters kˆ , cˆ and m polytopic uncertainties:

kˆ min ≤ kˆ ≤ kˆ max

ˆ ≤m ˆ max ˆ min ≤ m m

cˆmin ≤ cˆ ≤ cˆmax

(13)

Consider a master one-sided 1-DoF impact oscillator described by the following impulsive hybrid dynamics:

z˙ m = Azm + Bu(t )

as long as

zm ∈ 

(14a)

− z+ m = Re z m

whenever

zm ∈ 

(14b)

The slave system under polytopic parametric uncertainties and subject to external disturbance δ (t) is defined by the following impulsive hybrid dynamics:

z˙ s = Ai zs + Bi u(t ) + Bi v(t ) + Bi δ (t )

as long as

zs ∈ 

(15a)

− z+ s = Re z s

whenever

zs ∈ 

(15b)

where v(t) is the control law and (Ai , Bi ) represents one of the vertices of the polytope (see Definition 1). In addition, it is assumed that δ (t)2 ≤ ρ 2 , with ρ > 0. In the dynamics (15), if v(t ) = 0, δ (t ) = 0 and the system parameters m, k and c are certain and thus correspond to their nominal values, then the slave system is chaotic. However, if we take into account the external disturbance δ (t) and the parametric uncertainties, the motion of the oscillator can change radically from a chaotic motion to a periodic one, or to another motion. Then, as we considered that the master system has not parametric uncertainties and is not subject to external disturbances, then the objective is to design the control law v for the slave system in order to realize the robust master-slave synchronization even if it is subject to external disturbances δ (t), or in the presence of parametric uncertainties. In order to realize the robust master-slave synchronization of the two systems, the objective is to design the control law v(t) canceling the error between the master state zm and the slave state zs . Thus, In order to achieve the master-slave synchronization, our approach lies in the stabilization of the synchronization error, e = zs − zm , between the slave and the master. Therefore, the master-slave synchronization problem will be treated and solved as an asymptotic stability problem of the master-slave synchronization error. It is easy to show that, during the oscillation phase of the master and slave systems for which their state vectors zm and zs belong to the region  described by the set (3), the synchronization error e belongs to the set :

  = e ∈ R2×1 , d − σ < C T e < σ − d

(16)

The synchronization error between the master and the slave is defined as follows:

e˙ = Ai e + A˜i zm + B˜i u(t ) + Bi v(t ) + Bi δ (t )



− z+ m = Re z m − z+ = R z e s s

whenever whenever

zm ∈  zs ∈ 

as long as

e∈

and

zm ∈ 

(17a)

(17b)

where A˜i = Ai − A and B˜i = Bi − B. The error dynamics (17a) reveals the state during the oscillating motion of the two systems, that is without impact with the rigid stop. However, the algebraic equations (17b) describes the behavior of each system at the impact phase.

H. Gritli / Mechanism and Machine Theory 142 (2019) 103610

7

In order to realize the stabilization of the system (17) via the control law v, we adopt the following state-feedback control law [79]:

 v = Ke 0

whenever elsewhere

e2 >

(18)

with K is a constant gain matrix to be designed, and is very small positive constant. Thus, under the control law (18), the continuous dynamics of the synchronization error (17a) is recast as:

e˙ = (Ai + Bi K )e + A˜i zm + B˜i u(t ) + Bi δ (t )

as long as

e∈

and

zm ∈ 

(19)

Relying on (16) and (18), the set in (18) will be defined as:

  = e ∈ R2×1 , d − σ < C T e < σ − d and e2 >

(20)

Moreover, as the master system is chaotic, then, its chaotic motion is globally bounded in the phase space. Then, we have: zm 2 < γ , with γ > 0. Then, in (19) the sequel, the set  will be redefined like so:

   = z ∈ R2×1 , d < C T z < σ , and z2 < γ

(21)

The closed-loop system (19) and the associated algebraic equations (17b) constitutes an impulsive hybrid dynamics. Our methodology to realize the master-slave synchronization is to achieve the asymptotic stabilization of only the continuous dynamics (19). Then, the master-slave synchronization of the two impacting mechanical oscillators and hence of chaos will be only achieved during the oscillation phase of the two systems. However, if the master oscillator or the slave one undergoes an impact with the rigid stop, then the controller will enforce the slave system to re-track the master. In order to stabilize the continuous system (19), we need then to design the gain matrix K of the control law v in (18). To achieve this goal, our methodology is to resort to the Lyapunov approach by defining a candidate Lyapunov function V (e ) = eT P e, with P = P T . Nevertheless, such classical Lyapunov approach requires that the error e belongs to the whole state space R2 . However, it is worth noting that the continuous dynamics (19) is not defined in the whole state space R2 . However, the two states e and zm belong to the useful working regions and , respectively. Accordingly, in order to reduce the conservatism of the stability conditions of the close-loop system (19), our strategy lies mainly on the use of the S-procedure Lemma in order to only consider these two regions and  in the design of stability conditions [79]. This problem will be treated in the next section. Remark 2. It is worth to mention that in [17,76], we have designed an OGY-based state-feedback control law in order to suppress chaos exhibited in the motion of the one-sided 1-DoF impact mechanical oscillator by stabilizing a desired oneperiodic hybrid limit cycle embedded within the chaotic attractor. Such controller was deigned by developing an explicit expression of the stroboscopic controlled hybrid Poincaré map. In this control process of chaos, the controller should be kept constant during the whole period T of the oscillation. In addition, in [17,76], we have not considered the problem of parametric uncertainties and external disturbances. Thus, the major drawback of the OGY-based control method is due to the long waiting time before the value of the controller is re-updated at the beginning of each period T [76]. Hence, such controller does not take into account the influence of perturbations, noises and parameters variation away from the Poincaré section. We showed in [17] how the dynamics of the impact oscillator under the OGY-based state-feedback controller exhibits a diversity of nonlinear phenomena when some parameter (the excitation frequency) varies. Therefore, the OGY-based state-feedback control law is not robust against parametric uncertainties and disturbances. In the present work, our contribution is to consider first the continuous-time impulsive dynamics of the impact oscillator instead of its associated Poincaré map. Moreover, we consider the presence of time-varying uncertainties in the parameters k, m and c of the mechanical system and also an external disturbing torque δ (t). Furthermore, as chaos is a source of instability, our objective is to achieve the robust master-slave synchronization of chaos in the impact mechanical oscillator. Thus, the slave system under parametric uncertainties and disturbances should track, via the controller v, the chaotic master system, which is adopted as a reference model.

4. Synthesis of the robust stability conditions 4.1. BMI stability conditions In this section, we aim at designing the matrix gain K of the state-feedback control law (18) stabilizing the masterslave synchronization error (19) relying on the S-procedure Lemma. We recall that, in our impact mechanical oscillator, we have d = 0. Then, without loss of generality, we expand the set  by taking a new value of d such that d < 0. Actually, such expansion is made because we need next d = 0 in the linearization of the BMI conditions. We state then the following theorem for the asymptotic stabilization of the system (19).

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H. Gritli / Mechanism and Machine Theory 142 (2019) 103610

Theorem 1.  If, for a fixed constant , there exist some positive constants α i , β i , for all i = 1, . . . , 6, a positive-definite symmetric M1 O matrix M = , and matrices P = P T and K, such that the following matrix inequalities: M2 ( )

⎡

P (Ai + Bi K ) + ( )+

⎢ P J T MJ P + α3 I ⎢ ⎢ ( ) ⎢ ⎣ ( ) ⎡

P − β3 I

O

( )

β6 I ( )

⎣

( )

P A˜i

−(α1 − α2 )C

−α6 I

−(α4 − α5 )C

⎤

⎥ ⎥ ⎥<0 ⎥ − α3 + γ α6 + 2κ (α1 + α2 )+⎦ ( ) 2σ α4 − 2dα5 + D Ti M−1 D i ⎤ (β1 − β2 )C ⎦>0 (β4 − β5 )C

β3 − γ β6 − 2κ (β1 + β2 ) − 2σ β4 + 2dβ5

(22a)

(22b)

are feasible, then the system (19) under parametric uncertainties and disturbances is asymptotically stable. Hence, the master-slave synchronization of the 1-DoF impact mechanical oscillator is achieved with the control law (18). Proof. To demonstrate the results in Theorem 1, we consider first the candidate Lyapunov function V (e ) = eT P e, with P = P T . Then, the continuous dynamics (19) is asymptotically stable if there exists P such that:

V (e ) = eT P e > 0

such that

e∈

and

zm ∈ 

(23a)

V˙ (e ) = 2eT P e˙ < 0

such that

e∈

and

zm ∈ 

(23b)

Using the dynamics in (19), it follows that:

V˙ (e ) = 2eT P (Ai + Bi K )e + 2eT P A˜i zm + 2eT P B˜i u(t ) + 2eT P Bi δ (t )

(24)

Relying on Lemma 1 and as δ (t)2 ≤ ρ 2 , it yields: T

˜ 2eT P B˜i u(t ) ≤ eT P M1 P e + Um2 B˜i M−1 1 Bi

(25a)

2eT P Bi δ (t ) ≤ eT P M2 P e + ρ 2 BTi M−1 2 Bi

(25b)

with M1 = MT1 > 0 and M2 = MT2 > 0. Hence, using expression (24) and those in (25), it follows that: T

−1 2 T ˜ V˙ (e ) ≤ 2eT P (Ai + Bi K )e + 2eT P A˜i zm + eT P (M1 + M2 )P e + Um2 B˜i M−1 1 B i + ρ B i M2 B i

(26)

T

2B ˜i M−1 B˜i + ρ 2 BTi M−1 Bi . This quantity can be Posing U (e ) = 2eT P (Ai + Bi K )e + 2eT P A˜i zm + eT P (M1 + M2 )P e + Um 1 2 rewritten as follows:

⎡ ⎤T ⎡ e

U ( e ) = ⎣z m ⎦

 

1

P ( A i + B i K ) + (  ) + P J T MJ P

P A˜i

O

( ) ( )

O

O

( )

D Ti M−1 D i

⎣ 







⎤⎡ ⎤ e

⎦⎣zm ⎦

(27)

1

I M1 O Um B˜i , M= , and D i = , where (here and in the next) I is the identity matrix and O is a I M2 ρ Bi ( ) zero matrix with appropriate dimension. Furthermore, in (23a), the quantity V (e ) = eT P e can be rewritten like so: with J =

 T 

V (e ) =

e zm 1

P ( ) ( )

O O ( )

O O 0

  e zm 1

(28)

We recall that the Lyapunov function V (e ) in (23) is constrained with respect to the conditions e ∈ and zm ∈ , where and  are defined by the two sets (20) and (21), respectively. The constraints defining the set can be recast as:

 T 

W1 ( e ) =

e zm 1

 T  W2 ( e ) =

e zm 1

O ( ) ( )

O O ( )

C O −2κ

O ( ) ( )

O O ( )

−C O −2κ

  e zm 1

<0

(29a)

<0

(29b)

  e zm 1

H. Gritli / Mechanism and Machine Theory 142 (2019) 103610

 T  e zm 1

W3 ( e ) =

−I ( ) ( )

O O ( )

O O

9

  e zm 1

<0

(29c)

with κ = σ − d. Similarly, the constraints defining the set  can be rewritten like so:

 T  e zm 1

W4 ( e ) =

 T  e zm 1

W5 ( e ) =

 T  W6 ( e ) =

e zm 1

O ( ) ( )

O O ( )

O C −2σ

O ( ) ( )

O O ( )

O −C 2d

O ( ) ( )

O I ( )

O O −γ

  e zm 1

<0

(30a)

  e zm 1

<0

(30b)

<0

(30c)

  e zm 1

Hence, based on previous relations, the two constrained inequalities in (23) can be reformulated as follows:

V (e ) > 0

such that

Wi ( e ) < 0

∀i = 1 , . . . , 6

(31a)

U (e ) < 0

such that

Wi ( e ) < 0

∀i = 1 , . . . , 6

(31b)

Using the quadratic representation of all the quantities in the constraints in (31) and applying the S-procedure Lemma, the stability conditions (31a) and (31b) are equivalent to the existence of positive scalar variables β i and α i , ∀i = 1, . . . , 6, and a symmetric matrix M > 0 such that the two matrix inequalities in (22) hold.  Notice that the first matrix inequality (22b) is linear with respect to the variables P , β i , i = 1, . . . , 6. However, the second matrix inequality (22a) is defined as bilinear with respect to P , K, M, α i , i = 1, . . . , 6. Then, we emphasize that the stabilization problem of the master-slave synchronization error (19) is transformed into the solving problem of two BMIs (22b) and (22a). Here, the LMI (22b) can be considered as a BMI as we describe a complete set of stability conditions. To solve these BMIs, we should transform them into LMIs. This problem will be solved in the next section. 4.2. Linearization of the BMIs and synthesis of LMI conditions In this part, the objective is the linearization of the two BMIs in (22). Then, we will linearize first the second BMI (22a). The linearization procedure of the first BMI (22b) will be achieved with almost the same method for the BMI (22a). 4.2.1. Linearization of the BMI (22a) Posing S = P

−1

. Multiplying the BMI (22a by



S ( ) ( )

O I ( )

O O 1

 from the left and from the right leads to the following

matrix inequality:

⎡

(Ai S + Bi R ) + ( ) + J T MJ + α3 S 2 ⎢ ( ) ⎢ ⎣ ( )

A˜i −α6 I

( )

−(α1 − α2 )S C

⎤

⎥ ⎥<0 − α3 + γ α6 + 2κ (α1 + α2 )+⎦ 2σ α4 − 2dα5 + D Ti M−1 D i −(α4 − α5 )C

(32)

with R = KS . Relying on the Schur complement Lemma, and since M > 0, it is easy to demonstrate that the condition (32) is equivalent to:

⎡

(Ai S + Bi R ) + ( ) + J T MJ + α3 S 2 ⎢ ( ) ⎢ ⎣ ( ) ( )

A˜i

−(α1 − α2 )S C

O

−α6 I

−(α4 − α5 )C

O

( ) ( )

− α3 + γ α6 + 2κ (α1 + α2 ) + 2σ α4 − 2dα5

D Ti

( )

−M

⎤ ⎥ ⎥<0 ⎦ (33)

10

H. Gritli / Mechanism and Machine Theory 142 (2019) 103610

⎡

I ⎢O Furthermore, pre- and post- multiplying expression (33) by the matrix ⎣ O O tion:

⎡

(Ai S + Bi R ) + ( ) + J T MJ + α3 S 2 ⎢ ( ) ⎢ ⎣ ( ) ( )

O I O O

O O O I

⎤

O O⎥ ⎦, yields the following condiI O

⎤

A˜i

O

−(α1 − α2 )S C

−α6 I

O

−(α4 − α5 )C

( ) ( )

−M

Di

( )

− α3 + γ α6 + 2κ (α1 + α2 ) + 2σ α4 − 2dα5



( A i S + B i R ) + (  ) + J MJ ( ) ( ) ⎡ ⎤ 2κ 0 0 0 0 ⎢0 2κ 0 0 0⎥ ⎢ ⎥ Q−1 = ⎢ 0 0 2σ 0 0 ⎥, ⎣0 0 0 −2d 0⎦ T



⎥ ⎥<0 ⎦

 





(34)

S O G= O Z= O , O Di ⎡ ⎤ ⎡ ⎤ 1 μ1 0 0 0 0   ⎢1⎥ ⎢0 −S C SC O O O μ2 0 0 0⎥ ⎢ ⎥ ⎢ ⎥ L = ⎢1⎥, N = ⎢ 0 O O −C C O , 0 μ4 0 0 ⎥, ⎣ ⎦ ⎣ O O O O O 1 0 0 0 μ5 0⎦ 0 0 0 0 γ 1 0 0 0 0 μ6 μi = αi−1 for i = 1, . . . , 6, ξ1−1 = − α3 , and ϕ1 = ξ1−1 + 2κ (α1 + α2 ) + 2σ α4 − 2dα5 + γ α6 . This last quantity can be rewritten like: For

simplicity,

posing

X =

A˜i −α6 I ( )

O O , −M

Y=

ϕ1 = ξ1−1 + LT (QN )−1 L

(35)

Furthermore, inequality (34) can be reformulated as follows:





X + μ−1 YYT 3 ( )

Z + G N −1 L

<0

ϕ1

(36)

Applying the Schur complement Lemma to (36) yields:



−1

T ϕ1 Z + G N −1 L < 0

T −1 X + μ−1 L 3 Y Y − Z + GN

(37a)

ϕ1 < 0

(37b)

It is clear that, as d < 0, we have then Q > 0. Moreover, we have N > 0. Then, according to the inequality (37b), it is necessary that: ξ 1 < 0. Let consider the expression of ϕ 1 in (35). By applying the Matrix Inversion Lemma, we obtain the following relation:

ϕ1−1 = ξ1 − ξ12 LT H−1 L

(38)

H = ξ1 L L T + Q N

(39)

with

Then, using expression (38), we obtain:





T −1 X + μ−1 L 3 Y Y − Z + GN







T ξ1 − ξ12 LT H−1 L Z + G N −1 L < 0



(40)

T

Posing  = Z + G N −1 L ξ1 − ξ12 LT H−1 L Z + G N −1 L . By developing function (40), we obtain:

 = ξ1 Z Z T + ξ1 Z LT N −1 G T + ξ1 G N −1 LZ T + ξ1 G N −1 LLT N −1 G T − ξ12 Z LT H−1 LZ T −ξ12 Z LT H−1 LLT N −1 G T − ξ12 G N −1 LLT H−1 LZ T − ξ12 G N −1 LLT H−1 LLT N −1 G T

(41)

Moreover, using expression (39) of the matrix H, it easy to show that:

ξ1 G N −1 LLT N −1 G T = G N −1 HN −1 G T − G QN −1 G T

(42a)

ξ12 G N −1 LLT H−1 LLT N −1 G T = G N −1 HN −1 G T − 2G QN −1 G T + G QH−1 QG T

(42b)

ξ12 G N −1 LLT H−1 LZ T = ξ1 G N −1 LZ T − ξ1 G QH−1 LZ T

(42c)

Then, via the three expressions in (42), relation (41) is simplified as follows:

 = ξ1 Z Z T − ξ12 Z LT H−1 LZ T + ξ1 G QH−1 LZ T + ξ1 Z LT H−1 QG T + G QN −1 G T − G QH−1 QG T

(43)

H. Gritli / Mechanism and Machine Theory 142 (2019) 103610

11

It is easy to demonstrate that expression (43) can be rearranged as follows:





T  = ξ1 Z Z T − ξ1 Z LT − G Q H−1 ξ1 Z LT − G Q + G QN −1 G T

(44)

Let substitute the previous expression of the function  in the inequality (40). Thus, we obtain:





T ξ1 Z LT − G Q H−1 ξ1 Z LT − G Q − G QN −1 G T < 0

T T X + μ−1 3 Y Y − ξ1 Z Z +

Since N > 0, then G QN

−1

(45)

T

G > 0. Hence, the matrix inequality (45) is recast as:

T T X + μ−1 3 Y Y − ξ1 Z Z +





T ξ1 Z LT − G Q H−1 ξ1 Z LT − G Q < 0

(46)

Let us consider now the second condition (37b) where ϕ 1 is given by expression (35). As ξ 1 < 0 and QN > 0, then the Schur complement states that the condition (37b) is equivalent to:



ξ1−1 ( )

LT −QN



<0

(47)

By applying again the Schur Lemma to the matrix inequality (47), we obtain the following condition on the matrix H:

−QN − ξ1 LLT = −H < 0

(48)

Accordingly, as μ3 > 0 and H > 0, and based on the Schur complement Lemma, inequality (46) is equivalent to:



X − ξ1 Z Z T ( ) ( )

ξ1 Z L T − G Q

Y

−μ3 I ( )



O −H

<0

(49)

By substitution expression (39) of the matrix H in the matrix inequality (49) and using the Schur Lemma, we obtain the following condition:

⎡

X − ξ1 Z Z T ⎢ ( ) ⎣ ( ) ( )

ξ1 Z L T − G Q

Y

−μ3 I ( ) ( )

O −QN ( )

⎤

O O ⎥ <0 ξ1 L ⎦

(50)

ξ1

Thus, by substituting expressions of the quantities X , Y , Z , G , L and Q in the inequality (49), we obtain then the following matrix inequality:

⎡

 ⎢ ( ) ⎢ ⎢ ⎢ ( ) ⎢ ⎢ ( ) ⎢ ⎢ ( ) ⎢ ⎢ ( ) ⎢ ⎢ ⎢( ) ⎢ ⎢ ( ) ⎢ ⎣ ( ) ( )

A˜i

O

−α6 I

O

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

−M − ξ

T 1 Di Di

( ) ( ) ( ) ( ) ( ) ( ) ( )

S

1 SC 2κ

− 21κ S C

O

O

O

O

ξ1 D i

−μ3 I

( ) ( ) ( ) ( ) ( ) ( )

O

O

C

1 C 2d

O

ξ1 D i

ξ1 D i

ξ1 D i

ξ1 D i

O

O

O

O

O

O

2 −μ 2κ

O

O

O

( ) ( ) ( ) ( ) ( )

O

O

O

O

O

1 −μ 2κ

( ) ( ) ( ) ( )

O 1 2σ

− 2μσ4

μ5

( ) ( ) ( )

O

2d

− μγ6

( ) ( )

( )

O

⎤

O⎥ ⎥

⎥ ⎥ O⎥ ⎥ ξ1 ⎥ ⎥<0 ξ1 ⎥ ⎥ ξ1 ⎥ ⎥ ⎥ ξ1 ⎥ ⎥ ξ1 ⎦ ξ1 O⎥

(51)

with  = (Ai S + Bi R ) + ( ) + J MJ . μ Recall that μ3 = α3−1 and ξ1−1 = − α3 . Then, we have ξ1 = − 3 . Moreover, we have μ6 = α6−1 . Then, the matrix inequality (51) is not an LMI. To linearize it, we multiply it from left and right by the matrix diag(I , μ6 I , I , I , 1, 1, 1, 1, 1, 1 ). Thus, we obtain the following LMI: T

⎡

⎢ ( ) ⎢ ⎢ ( ) ⎢ ⎢ ( ) ⎢ ⎢ ⎢( ) ⎢ ⎢ ( ) ⎢ ⎢ ( ) ⎢ ⎢ ( ) ⎢ ⎣ ( ) ( )

μ6 A˜i −μ6 I ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

O

S

O

O

−M + μ 3 D i D Ti

( ) ( ) ( ) ( ) ( ) ( ) ( )

O −μ3 I

( ) ( ) ( ) ( ) ( ) ( )

1 SC 2κ

− 21κ S C

− μ 3 D i

O O

1 −μ 2κ

( ) ( ) ( ) ( ) ( )

O

O

− μ 3 D i

− μ 3 D i

− μ 3 D i

O

O

O

O

O

O

O

O

O

O

O

2 −μ 2κ

( ) ( ) ( ) ( )

O

− 2μσ4

( ) ( ) ( )

μ6

O

μ6 C 2σ μ3 − Di

2d

C

O

μ5 2d

( ) ( )

O O

O

− μγ6

( )

O

⎤

⎥ ⎥ O ⎥ ⎥ O ⎥ ⎥ ⎥ − μ 3 ⎥ ⎥<0 − μ 3 ⎥ ⎥ − μ 3 ⎥ ⎥ − μ 3 ⎥ ⎥ − μ3 ⎦ O

− μ 3

(52)

12

H. Gritli / Mechanism and Machine Theory 142 (2019) 103610

Accordingly, the BMI (22a) is transformed into the LMI (52). 4.2.2. Linearization of the BMI (22b) the linearization methodology of the BMI (22b) is almost similar to that of the BMI (22a). As S = P −1 , and by multiplying at left and at right the expression (22b) by the matrix diag(S , I , 1 ), we obtain thus the following condition:

⎡

S − β3 S 2

O

( ) ( )

β6 I ( )

⎣

⎤ (β1 − β2 )S C ⎦>0 (β4 − β5 )C

β3 − γ β6 − 2κ (β1 + β2 ) − 2σ β4 + 2dβ5

Posing ξ2−1 = β3 , ϕ2 = ξ2−1 − 2κ (β1 + β2 ) − 2σ β4 + 2dβ5 − γ β6 , diag(η1 , η2 , η4 , η5 , η6 ). Then, relying on the previous section, we obtain:

ϕ2 = ξ2−1 − LT (QN )−1 L



As in the previous section, let us pose: X =

S

( )





O SC ,G= −β6 I O

(53)

ηi = βi−1

with

−S C O

O C

O −C

for



i = 1, . . . , 6,

 

N =

(54)

O S , and Y = . Then, the O O

matrix inequality (53) can be recast as:



X − η3−1 Y Y T

G N −1 L



ϕ2

( )

>0

(55)

Based on the Schur complement Lemma, the matrix inequality (55) is equivalent to the following conditions:



T

X − η3−1 Y Y T − G N −1 Lϕ2−1 G N −1 L

>0

ϕ2 > 0

(56a) (56b)

By applying the Matrix Inversion Lemma on the function ϕ 2 in (54), we obtain:

ϕ2−1 = ξ2 − ξ22 LT H−1 L

(57)

H = ξ2 L L T − Q N

(58)

where

Substituting expression (57) in (56a) gives:

X − η3−1 Y Y T − ξ2 G N −1 LLT N −1 G T + ξ22 G N −1 LLT H−1 LLT N −1 G T > 0

(59)

Through expression of H in (58), we can show that:

ξ2 G N −1 LLT N −1 G T = G N −1 HN −1 G T + G QN −1 G T

(60a)

ξ12 G N −1 LLT H−1 LLT N −1 G T = G N −1 HN −1 G T + 2G QN −1 G T + G QH−1 QG T

(60b)

Let us substitute these two last expressions into (58). As a result, we obtain the following simplified condition:

X − η3−1 Y Y T + G QN −1 G T + G QH−1 QG T > 0

(61)

As N > 0, then from inequality (61), we obtain:

X − η3−1 Y Y T + G QH−1 QG T > 0

(62)

Our immediate concern now is the condition (56b). Consider expression (54). By applying the Schur Lemma on the condition (56b) and as ξ 2 > 0 and QN > 0, the following matrix inequality is then deduced:



ξ2−1 ( )

LT QN



>0

(63)

Reapplying once again the Schur complement Lemma on the condition (63). We obtain then the following condition:

QN − ξ2 LLT > 0

(64)

Hence, relying on expression (58), we have H < 0. Therefore, the Schur complement Lemma states that the condition (62) is equivalent to the following matrix inequality:

⎡

X

Y

GQ O

( )

η3 I ( ) ( )

⎢ ( ) ⎢ ⎣ ( )

QN

( )

O

⎤

⎥ ⎥>0 ξ2 L⎦ O

ξ2

(65)

H. Gritli / Mechanism and Machine Theory 142 (2019) 103610

Therefore, we obtain an LMI:

⎡

S

⎢ ( ) ⎢ ⎢ ( ) ⎢ ⎢ ( ) ⎢ ⎢ ⎢ ( ) ⎢ ⎢ ( ) ⎢ ⎢ ( ) ⎢ ⎣ ( ) ( )

O

S

β6 I ( ) ( ) ( ) ( ) ( ) ( ) ( )

1 SC 2κ

− 21κ S C

O

O

O

η3 I ( ) ( ) ( ) ( ) ( ) ( )

O

O

2κ

η1

O

O

O

1 C 2d

O

O

O

O

O

O

O

O

2κ

O

O

O

O

O

1 2σ

η2

( ) ( ) ( ) ( ) ( )

C

η4 2σ

( ) ( ) ( ) ( )

− 2ηd5

( ) ( ) ( )

( ) ( )

O η6 γ

( )

O

13

⎤

O⎥

⎥

O⎥ ⎥

ξ2 ⎥ ⎥ ⎥ ξ2 ⎥ > 0 ⎥ ξ2 ⎥ ⎥ ξ2 ⎥ ⎥ ξ2 ⎦

(66)

ξ2 η

We posed previously η3 = β3−1 , η6 = β6−1 and ξ2−1 = β3 . Then, ξ2 = 3 . Left and right multiplying the matrix inequality (66) by the matrix diag(I , η6 I , I , 1, 1, 1, 1 ) yield the following LMI:

⎡

S

⎢ ( ) ⎢ ⎢ ( ) ⎢ ⎢ ( ) ⎢ ⎢ ⎢ ( ) ⎢ ⎢ ( ) ⎢ ⎢ ( ) ⎢ ⎣ ( ) ( )

O

S

η6 I ( ) ( ) ( ) ( ) ( ) ( ) ( )

O

η3 I ( ) ( ) ( ) ( ) ( ) ( )

1 SC 2κ

O

− 21κ S C O

O

O

2κ

O

η1

O

η6

2σ

η2 2κ

( ) ( ) ( ) ( ) ( )

O

C

( ) ( ) ( )

C

O

O

O

O

O

O

O

O

O

O

O

O

η4 2σ

( )

η6

2d

O

− 2ηd5

( ) ( ) ( )

( ) ( )

O

⎤

O⎥

⎥

O⎥ ⎥

η3 ⎥

⎥ η3 ⎥

⎥>0 η3 ⎥ ⎥

⎥ η3 ⎥

⎥ η3 ⎦

η3

O η6 γ

( )

(67)

Theorem 2. The dynamics (19) of the master-slave synchronization error under polytopic parametric uncertainties and subject to external perturbations is robustly stabilizable by means of the control law (18), if, for some positive parameter fixed a priori  M1 O to be small enough, there exist a symmetric matrix S , a matrix R, a positive-definite symmetric matrix M = , M2 ( ) scalars μj and ηj , j = 1, . . . , 6, such that the following LMIs

⎡

⎢ ( ) ⎢ ⎢ ( ) ⎢ ⎢ ( ) ⎢ ⎢ ⎢( ) ⎢ ⎢ ( ) ⎢ ⎢ ( ) ⎢ ⎢ ( ) ⎢ ⎣ ( ) ( ) ⎡ S

⎢ ( ) ⎢ ⎢ ( ) ⎢ ⎢ ( ) ⎢ ⎢ ⎢ ( ) ⎢ ⎢ ( ) ⎢ ⎢ ( ) ⎢ ⎣ ( ) ( )

μ6 A˜i −μ6 I ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

O

S

O

O

−M + μ 3 D i D Ti

O

S

η6 I ( ) ( ) ( ) ( ) ( ) ( ) ( )

O

η3 I ( ) ( ) ( ) ( ) ( ) ( )

1 SC 2κ

O

−μ3 I

− 21κ S C O

O

2κ

( ) ( ) ( ) ( ) ( )

− μ 3 D i

− μ 3 D i

− μ 3 D i

− μ 3 D i

− μ 3 D i

1 −μ 2κ

O

O

O

O

O

O

O

O

O

2d

O

O

( ) ( ) ( ) ( ) ( ) ( )

O η1

− 21κ S C

O

O

( ) ( ) ( ) ( ) ( ) ( ) ( )

1 SC 2κ

( ) ( ) ( ) ( ) η6

2σ

C

η6

O

C

O O

O

O

O

O

O

O

O

2κ

O

O

O

O

O

η2

( ) ( ) ( ) ( )

η4 2σ

( ) ( ) ( )

− 2ηd5

( ) ( )

are feasible. In (68a),  = (Ai S + Bi R ) + ( ) + J T MJ .

2σ

O η6 γ

( )

C

O

− 2μσ4

( ) ( ) ( ) ( )

O

2d

μ6

O

2 −μ 2κ

( )

O

O

( ) ( ) ( )

O

O

μ6 2d

C

O

μ5

( ) ( )

O O O

− μγ6

( )

O

⎤

⎥ ⎥ ⎥ ⎥ O ⎥ ⎥ ⎥ − μ 3 ⎥ μ3 ⎥ < 0 − ⎥ ⎥ − μ 3 ⎥ ⎥ − μ 3 ⎥ ⎥ − μ3 ⎦ O

O

(68a)

− μ 3

⎤

O⎥

⎥

O⎥ ⎥

η3 ⎥

⎥ η3 ⎥

⎥>0 η3 ⎥ ⎥

⎥ η3 ⎥

⎥ η3 ⎦

η3

(68b)

14

H. Gritli / Mechanism and Machine Theory 142 (2019) 103610

The matrix gain K of the control law (18) is defined and computed via the following relation:

K = RS −1

(69)

5. Numerical studies and enhanced LMIs 5.1. Numerical studies for Theorem2 In previous sections, we designed a state-feedback controller for the robust stabilization of the master-slave synchronization error (19) subject to polytopic parametric uncertainties and external disturbances. In the present section, we provide some numerical studies in order to show the validity and effectiveness of the proposed methodology for the design of the matrix gain K of the controller (18). Thus, the main objective is to demonstrate the feasibility of the developed LMI conditions (68a) and (68b). The feedback gain K is computed via expression (69), where S and R are solutions of the LMIs (68a) and (68b). Recall that in Theorem 2, we should fix a priori the constant to be as minimum as possible. Moreover, in the sequel of this paper, we take the following fixed constants: σ = 3, γ = 5 and d = −0.01. Table 2 shows some numerical values of the matrix gain K for some values of the parameters and ρ . We emphasize that as decreases and for a fixed ρ , the size of the gain K increases. Moreover, for a fixed and by increasing ρ , the size of the gain K becomes more large. In addition, we note that for all proposed values of the two parameters and ρ , the size of K is large. This big size in the gain matrix K will provoke a considerable increase in the control effort. Then, from the results in Table 2, and due to some technological restriction of the actuators, all the computed feedback gains K cannot be implemented (in practice) for the robust stabilization of the one-sided 1-DoF impact mechanical oscillator. For a practical application, we should have a feedback gain K with a reasonable size, saying about 102 and 103 . 5.2. Enhanced LMIs and results 5.2.1. Improved LMIs Our immediate concern is the “big” size of the gain matrix K. To make the outcomes of the problem with the two LMIs (68a) and (68b) realizable in a practical application, we should reduce and limit this size. According to the expression (62), we can limit the size of the matrix gain K by limiting the size of the two matrices R and S −1 . Let assume that:

RT R <  R I ,

R > 0

S −1 < S I ,

(70a)

R > 0

(70b)

These two conditions (70a) and (70b) can be converted, respectively, into the following LMIs:

 

R I R S I ( )

 ( ) I

I S

>0

(71a)

 >0

(71b)

In order to limit the size of the gain matrix K, the two scalars R and S should be minimized. Thus, according to these new conditions (71a) and (71b), the LMI problem in Theorem 2 becomes a minimization problem of R and S and with respect to four LMIs, as follows. Theorem 3. If, for some positive parameter  fixed a priori  to be small enough, there exist a symmetric matrix S , a matrix R, M1 O a positive-definite symmetric matrix M = , where M1 and M2 are with appropriate dimensions, and scalars μj , M2 ( )

Table 2 Numerical values of the gain matrix K for different values of the two parameters and ρ . KT

ρ=0

=2

−9.9563 103 × −4.2463

=1

104 ×

= 0.1

105 ×

= 0.01

106 ×









ρ=1



104 ×

−2.1734 −0.8409

105 ×

−3.4671 −1.3147

106 ×





−1.4769 104 × −0.5758

−2.3235 −0.9209















ρ=5





−2.1423 104 × −0.6163



−3.6402 −1.3299

105 ×

−3.5398 −1.2301

106 ×

−2.1218 −0.7721

107 ×











ρ = 10  

−1.0958 −0.2538

104 ×

−1.6752 −0.3550

105 ×







−3.6590 −0.7437



−6.0707 104 × −1.2216

 106 ×

ρ = 20  105 ×



−7.7882 −1.5486

105 ×



−8.2353 −1.3666

106 ×

−7.8180 −1.2878

107 ×





−1.3742 −0.1071













−2.3917 −0.2270

−1.1709 −0.0986 −4.7168 −0.1610

H. Gritli / Mechanism and Machine Theory 142 (2019) 103610

15

Table 3 Gain matrix K obtained by solving the optimization problem in Theorem 3 for the same values of and ρ used in Table 2. KT

ρ=0 

=2 

=1 

= 0.1

−53.6339 −108.6439

103 ×





103 ×



−2.7594 −3.0663

 103 ×

 

−0.9721 −1.2335







−5.1362 −5.6229

103 ×

−754.6257 −967.5815

103 ×

 

−2.0564 −1.9021

ρ = 20  103 ×





−1.6897 −1.9921

103 ×

−8.4586 −8.8731

104 ×

 103 ×



−1.0661 −1.2926

 103 ×

−3.7047 −4.1918

 104 ×

ρ = 10 



−469.4121 −631.2280



−184.1540 −293.1321



−480.1618 −661.0553

ρ=5 



−119.4892 −199.3597



−85.8935 −159.1364



= 0.01

ρ=1 





(!)





−2.4972 −2.8467



−3.9702 −4.4244



−2.3311 −2.0439

(!)

ηj , for all j = 1, . . . , 6, R and S , such that the following minimization problem subject to LMIs holds. minimize

R + S

(72a)

subject to

LMIs (68a ), (68b), (71a ) and (71b)

(72b)

Hence, the system (19) of the master-slave synchronization error under parametric uncertainties and external perturbations is robustly stabilizable by implementing the state-feedback control law introduced in (18), where the gain matrix K is computed via expression (69). 5.2.2. Numerical results Table 3 illustrates numerical results of the matrix gain K for the same values of the two parameters and ρ used in Table 2. Obviously, the gain matrices in Table 3 have a much smaller size than those illustrated in Table 2. Note that the symbol (!) introduced in Table 3 means that the optimization problem in Theorem 3 is unfeasible. It is worth to note that for the case = 0.1 and in the absence of external disturbance, i.e. ρ = 0, we obtained a reasonable gain matrix K. Furthermore, according to numerical results in Table 3, we can conclude that by decreasing ρ and increasing , the size of K decreases. Remark 3. In this work, we have considered polytopic uncertainties in the parameters of the one-sided 1-DoF impact mechanical oscillator as a chaotic system and which is forced via the excitation input u. Thus, via the control law v, we achieved the robust master-slave synchronization of chaos in the impact oscillator. Actually, it is difficult to find in the literature some works that can be applied directly for the case of the impact mechanical oscillator and then using the disturbed system (19) with polytopic uncertainties. We found three references; namely [71,82,85], that dealt with the robust master-slave synchronization of chaotic systems. First, in [71], authors considered norm-bounded parametric uncertainties in only the state matrix A. Moreover, they added the term BKe + α as the control input. Here, α ∈ Rn×1 is a nonlinear input; see expression (9) in [71]. It depends mainly on only the master state and the synchronization error e. Then, for the impact oscillator, we have n = 2 and hence we have a vector α. However, for the impact oscillator, we have Bi v as the term containing the control input v. Then, by comparing the previous terms, and according to the method adopted in [71], we should have Bi Ke + α where in the present ˆ , with α ˆ is a scalar term. Then, the problem lies in finding the expression of this function α ˆ . Hence, we have case α = Bi α ˆ that should be determined. In fact, according to the expression of α in [71], the linear gain K and the nonlinear function α we can not use it because we have several matrices Bi instead of only one. It is possible to use the design method of the controller adopted in [71] by considering the norm-bounded parametric uncertainties in the state matrix A and also the input matrix B. This will be considered as a future work. In [82], authors considered also the problem of norm-bounded parametric uncertainties in the state matrix A (and in the gain matrix K). Moreover, they used the same uncertainties’ state matrix A(t ) in the master system and the slave one. Hence, such parameter uncertainty was eliminated in the dynamics of the synchronization error e and then the problem becomes without norm-bounded uncertainties. In addition, authors in [82] introduced the same external input signal h(t) and two different external disturbances d1 (t) and d2 (t) in the master and the slave, respectively. As a result, the external input signal h(t) was eliminated in the error dynamical system. However, in our case of the impact oscillator, in the master system, the external input signal is h1 (t ) = Bu(t ), where u(t ) = Um cos(wt ) is the excitation input (and that was considered 2 ), and in the slave system, the second external input signal is h (t ) = B u (t ). also as an unknown input such that u(t )2 ≤ Um 2 i Then, in the dynamics of the synchronization error, we have the term B˜i u(t ) as the external input signal, where B˜i = Bi − B. Furthermore, in [82], authors used two disturbances d1 (t) and d2 (t) that are vectors (with the same dimension). This choice has simplified the development of the LMI stability condition by using an assumption e ≥ r, where r > 0 (as our condition e2 > imposed in expression of the state-feedback control law v in (18)). However, in our case, we have d (t ) = Bi δ (t ) as the disturbance term. For this general case and using the constraint e2 > , and via the S-procedure Lemma, we developed the LMI stability conditions.

16

H. Gritli / Mechanism and Machine Theory 142 (2019) 103610

Although the developed method introduced in [82] is almost similar to that adopted by us, but we can not transpose it directly for the impact oscillator. It requires a complete development of the LMI conditions by considering the case of norm-bounded uncertainties in the state and input matrices. Moreover, in [82] and also [71], the LMI condition conditions were designed for a free system, that is without constraints imposed on its state vector. In addition, authors in [85] considered also nonlinear systems with Lipschitz nonlinearity and subject to norm-bounded parametric uncertainties presented in the input and state matrices. Furthermore, as in [71,82], the system was without state constraints. It is possible to develop LMI conditions for the 1-DoF impact oscillator by considering the dynamics (19) and normbounded parametric uncertainties. Nevertheless, a complete development should be realized and added in the paper. This problem will be considered as a future direction of this work. 6. Simulation results for the robust master-slave synchronization of chaos In this section, we present some simulation results showing the effectiveness of the designed control law (18) for the master-slave synchronization of chaos in the impacting motion of the one-sided 1-DoF mechanical oscillator. The feedback gain K of such controller is computed via expression (69) after solving the minimization problem subject to LMIs in Theorem 3. In order to take into consideration the control input limitations for a practical application and then in order to avoid a high gain of the control input v(t), we introduce a saturation function as follows:

v=

⎧ ⎨vmax

if

v > vmax

Ke

if

−vmax ≤ v ≤ vmax

−vmax

if

v < −vmax

⎩

(73)

In the sequel, we choose vmax = 5. To show the efficiency of the designed controller in the master-slave synchronization of chaos, we will consider three different scenarios. In the first scenario, we will consider a nominal slave impact oscillator with nominal parameter (i.e. without uncertainties) and without disturbances. In the second scenario, we will only consider uncertain parameters injected in the dynamic model of the slave system. In the third scenario, we will consider the case of both external disturbances and uncertainties. Recall that the dynamics of the master oscillator is considered to be nominal, without uncertainties and disturbances. 6.1. First scenario: The nominal case In this first scenario, the dynamics of the master-slave synchronization error (24) is without uncertainties and disturbances δ (t), and then ρ = 0. Thus, in the system (19), we will have Ai = A and Bi = B, and hence A˜i = O and B˜i = O . In this nominal case, we choose = 10−4 . Then, the resolution   of the LMI-based minimization problem in Theorem 3 gives the following feedback gain K = −1.2350 − 1.9721 . Obviously, the size of this matrix gain is very small compared to those provided in Table 2 and Table 3. Then, using this matrix gain K, we obtain the results in Fig. 3. Fig. 3 shows the temporal evolution of the position of the master impacting oscillator and the position of the slave oscillator. Fig. 3 reveals the velocities of the two systems. Fig. 3 depicts the chaotic attractor of the master and the trajectory of the slave. Fig. 3 shows the control input v(t) applied to the slave impact oscillator to track the chaotic master oscillator. These simulation results show clearly the achievement of the master-slave synchronization of the 1-DoF impact mechanical oscillator and then the control of chaos. In addition, we emphasize that the control effort is small where the maximal value is about 3.5. Furthermore, when the synchronization is achieved, the control law is zero. It is worth noting that, physically speaking, the control input v is an auxiliary active input employed for the impact oscillator. As seen in the dynamics (15) of the slave impact oscillator, the controller v was added to the sinusoidal excitation signal u(t), which is generated via an actuator as, e.g., the DC motor. We can realize this control law v(t) using another DC motor that can be mounted at the same axis of that of the other motor providing the excitation input u(t). Moreover, such control signal v(t) can be provided using the same motor generating the signal u(t). In this case, which is much simpler in practice than the first one, the force that will applied to the 1-DoF impact mechanical oscillator via a single DC motor is the quantity (u(t ) + v(t ) ). 6.2. Second scenario: The uncertainty case In this second scenario, we will only consider parametric uncertainties in the dynamics of the master-slave synchronization error (24). The disturbance δ (t) is kept to be zero, then we take ρ = 0. We choose also = 1. After resolving the optimization problem in Theorem 3, we obtain then the following matrix gain K = [−85.8935 − 159.1364] (see Table 3). ˆ , kˆ and cˆ as three time-varying random functions verIn the simulation, we take the values of uncertain parameters m ˆ ˆ − 0.9, m ˆ max = m ˆ + 5, kˆ ifying conditions (13), with m =m = kˆ − 0.5, kˆ max = kˆ + 2, cˆ = cˆ − 0, cˆmax = cˆ + 0.5. Recall that cˆ =

c m

min

= 0. Thus, we have:

ˆ min ) + m ˆ (t ) = Rand1 (t )(m ˆ max − m ˆ min m

min

min

(74a)

H. Gritli / Mechanism and Machine Theory 142 (2019) 103610

17

Fig. 3. Simulation results of the synchronization of the slave impact mechanical oscillator with the master in the nominal case.

kˆ (t ) = Rand2 (t )(kˆ max − kˆ min ) + kˆ min

(74b)

cˆ(t ) = Rand3 (t )(cˆmax − cˆmin ) + cˆmin

(74c)

where Randi (t), i = 1, 2, 3, is the random function (of Matlab Software) that generates uniformly distributed random values between 0 and 1. The objective is to see if the designed state-feedback controller v is able to compensate these parametric uncertainties in (74). Simulation results of the master-slave synchronization are illustrated in Fig. 4. Fig. 4a–c show clearly that the slave system is synchronization with the master one after about 5 [s]. Hence, it is obvious that the designed state-feedback controller is able to ensure the robust master-slave synchronization of the impact mechanical oscillator despite the presence of time-varying parametric uncertainties. Moreover, it worth to note from the results in Fig. 4 that the behavior of the slave system appears smooth. However, only some fluctuations appear in the control signal v (Fig. 4d). The instantaneous pics raised in the controller are generated at the impact either of the master system or the slave one with the rigid constraint. The width of these peaks is very small. 6.3. Third scenario: The disturbance and uncertainty case In this third scenario, we inject the external disturbing torque δ (t) in the slave impact mechanical oscillator and show the ability of this system to synchronize with the chaotic master system. We take = 0.001 and ρ = 2. Thus, we obtain the

18

H. Gritli / Mechanism and Machine Theory 142 (2019) 103610

Fig. 4. Simulation results showing the master-slave synchronization of chaos in the 1-DoF impact mechanical oscillator subject to polytopic parametric uncertainties. (a) (resp. (b)) shows the position (resp. the velocity) of the two systems, the master and the slave, (c) reveals the master attractor and the slave trajectory in the state space. (d) depicts the control input v.

feedback gain K = [−262.6034 − 222.8331]. In the simulation, we take a randomly time-varying disturbing torque δ (t) as follows:

δ (t ) = ρ (2Rand(t ) − 1 )

(75)

The simulation results are given by Fig. 5. In Fig. 5a–c, we obtained almost the same results as in the previous scenario. However, the control signal in Fig. 5d fluctuates around zero, between ± 1. Obviously, the controller is found to compensate the effect of the randomly time-varying external disturbing torque δ (t) and then ensuring the robust master-slave synchronization of chaos in the impact oscillator. Remark 4. This work dealt with the robust master-slave synchronization of the one-sided 1-DoF impact mechanical oscillator. Then, a slave oscillator should track the state of the master system via the control input v. Actually, we can think about two possibilities concerning the practical application of the proposed control approach. The first possibility lies in using two real physical systems of the impact mechanical oscillator. Thus, via sensors implemented in the two systems to measure the two state vectors zm and zs and by applying an external disturbance torque on the slave system, the objective is then that the disturbed slave system follows the master. In order to consider parametric uncertainties, we can suppose that the two systems have not the same values of the inertial parameters, i.e. m, k and c. Thus, the slave impact oscillator should behave, by means of the robust state-feedback control law v, as the master impact oscillator even in the presence of the uncertainties and the disturbances.

H. Gritli / Mechanism and Machine Theory 142 (2019) 103610

19

Fig. 5. Simulation results revealing the robust master-slave synchronization of the impacting oscillator and its ability to track the chaotic master system despite the presence of parametric uncertainties and the external disturbance δ (t). (a) shows the position x of the two systems, (b) shows their velocity x˙ , (c) reveal the two attractors, and (d) displays the applied control law v.

The second possibility for a practical application of the state-feedback controller in the robust master-slave synchronization is to consider that the master impact oscillator as a reference model that will be implemented within a numerical calculator. Thus, we use the 4th-order Runge-Kutta method to solve the differential equation of the master system (as we have already done for the numerical simulation of the two systems in the present paper). Therefore, using only sensors for the slave impact oscillator and the numerical model for the master, we compute the state-feedback control law v and we apply it to the real physical slave system. Actually, this second case of practical application will be our objective in the future for experimental results using the 1-DoF inverted pendulum subject to two asymmetric vertical barriers. Remark 5. The one-sided (or even the two-sided) 1-DoF impact mechanical oscillator is a classical model introduced for the study of discontinuous dynamics, and has been investigated by several authors in the literature; see, e.g., [17] and references inside. In addition, it has been used as a classical testbed to develop new control strategies for systems with impulse effects. Based on this model, we can extend the developed approach of the robust master-slave synchronization to the case of 2-DoF and 3-DoF planar biped robots, namely the compass-gait biped robot [86] and the torso-driven biped robot [87], which are the two biped models on which we are actively working. As in our recent paper [17] realized on the one-sided 1-DoF impact mechanical oscillator under the OGY-based state-feedback controller, we investigated the dynamics of these two planar biped robots under the OGY-control method, as, e.g., in [13,88–91], and then we showed the exhibition of attractive and interesting nonlinear phenomena: several types of local bifurcations and chaos. Hence, as we have achieved in the present work the master-slave synchronization of the 1-DoF impact mechanical oscillator, as an impulsive hybrid system, our objective will be

20

H. Gritli / Mechanism and Machine Theory 142 (2019) 103610

the application of this control approach for the two previous planar biped robots, where their dynamics is more complex than that of the impact oscillator. In addition, the 1-DoF impact oscillator and also other basic simple models (see Remark 2 in [17]) are a quite fertile research subject in different fields; see, e.g., [23] and references therein. These simple models have immediate applications to more complex mechanical systems such as gears [92], printer head [93], and models of rocking blocks and bumpers [27,29,32], just to mention a few. Thus, we can consider these complex applications to achieve the robust master-slave synchronization. Some other more complex impact mechanical oscillators are the vibro-impact systems with dry friction and on a conveyor belt [24,36–38,57,58]. Moreover, the multiple-degree-of-freedom impact mechanical oscillators are also more complicated than the one-sided 1-DoF impact oscillator and especially when dealing with multiple soft or rigid constraints. All these complex systems can be considered in the future. In addition, as noted previously, the objective is to extend the present work for the biped robots. 7. Conclusion and future works In this paper, we dealt with the master-slave synchronization of chaos exhibited in the dynamics of a 1-DoF impact mechanical oscillator with a single rigid constraint. The slave system was considered to be subject to time-varying parametric uncertainties and also external disturbances. The synchronization problem was then converted into the robust stabilization of the master-slave synchronization error by designed a state-feedback controller. We adopted the LMI approach in the design of such robust controller. In addition, as the motion of the impact oscillator is limited by the rigid constraint, then we used the S-procedure Lemma in order to synthesis less restrictive stability conditions that were only satisfied in the working region. As a result, the stabilization problem of the master-slave synchronization error is represented as a BMI problem. In order to obtain LMI stability conditions allowing the computation of the feedback gain, we used the Schur complement and the Matrix Inversion Lemma. Moreover, we presented enhanced results by introducing additional LMIs permitting the reduction of the size of the gain matrix. Finally, by considering three cases: a nominal case, an uncertainty case and a disturbance case, the simulation results show the effectiveness of the designed state-feedback controller in the robust master-slave synchronization of the impact mechanical oscillator and hence of chaos. In this work, we only considered the motion between impact and then the motion during the oscillation phase in the design of the state-feedback controller. Our future concern is to take into consideration the impact dynamics of the oscillator in the synthesis of the LMI stability conditions. Moreover, as a future work, the objective is to design an observerbased controller or a static output feedback controller for the one-sided 1-DoF impact mechanical oscillator to achieve the master-slave synchronization of chaos and other desired periodic and non-periodic motions. Furthermore, the objective is to extend the present work to other multiple-degree-of-freedom impact mechanical oscillators and with several rigid or soft constraints. 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