## Abstract

An impact oscillator is a non-smooth dynamical system with discontinuous state jumps whose dynamical behavior illustrates a variety of non-linear phenomena including a grazing bifurcation. This specific phenomenon is difficult to analyze because it coincides with an infinite stretching of the phase space in the neighborhood of the grazing orbit, resulting in the well-known problem of the square-root singularity of the Jacobian of the discrete-time map. A novel Takagi–Sugeno fuzzy model-based approach is presented in this paper to model a hard impacting system as a non-smooth dynamical system including discontinuous jumps. Employing non-smooth Lyapunov theory, the *structural stability* of the system is analyzed to predict the onset of the destabilizing chaotic behavior. The proposed stability results, formulated as a Linear Matrix Inequality (LMI) problem, demonstrate how the new method can detect the loss of stability just before the grazing bifurcation.

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## Notes

- 1.
For a definition of the degree of smoothness of a non-smooth system, refer to [11].

- 2.
Non-smooth or piecewise-smooth systems are the terms initially coined for dynamical equations with a discontinuous right-hand side. However, they constitute different classes of systems (than initially defined by Filippov [13]) including flows and maps. Here, we use the term non-smooth (dynamical) systems for non-smooth flows only as we limit our studies to this class. Refer to [11] for more detailed definitions and discussions.

- 3.
Normally a sliding region in Filippov-type systems can be attracting or repelling. However, in case of a dry-friction oscillator where the system flow is forward in time, a repelling sliding region is not realizable in its sliding dynamics.

- 4.
From this point on, we will use the terminology

*smooth TS fuzzy system*for describing TS fuzzy models with the well-known formalism of (5) and (6) capable of approximating smooth dynamical systems and the terminology*non-smooth TS fuzzy system*for describing TS fuzzy models capable of approximating non-smooth dynamical systems to an arbitrary accuracy as proposed by the general formalism (7). - 5.
cl. denotes the closure of a set.

- 6.
The system is stable if the multipliers of the Jacobian of the linearized discrete map lie inside the unit circle.

- 7.
The letter

*J*in this equation stands for the Jacobian of the stroboscopic map and should not be confused with the matrix*J*as presented in (11). - 8.
For a complete derivation of

*J*_{ ZDM }please refer to [11].

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## Appendix

### Appendix

### Proof of Theorem 1

It is already known that the discontinuity boundary *Σ* is the zero set of the smooth function *H*. If we assume the set \(\partial S^{+}_{1}=\{x:\frac{\partial H(x)}{\partial x_{1}}\}\) and the function \(\zeta \in \partial S^{+}_{1}\), which is defined as ζ:*x*↦*rx*, then the last condition of stability can be written in the sense of Lyapunov as

Following the procedure stated in Sect. 4, the above condition can be recast on LMI as given in Theorem 1.

Since the continuous fuzzy states (*x*,*m*
_{
i
}),*i*=1,2,…,*N* can become discontinuous without passing to another state space fuzzy region *Ω*
_{
q
} and without changing to the next discrete state *m*
_{
i
}, the switching manifold can be represented as the region *Λ*
_{
qq
} for such states. Moreover, due to state discontinuities, if *Λ*
_{
qr
}≠0, *Ω*
_{
q
} must not be a neighboring set to *Ω*
_{
r
}.

It is necessary to assume that for all the conditions in the theorem, there is a finite number of discontinuous states in finite time. This implies that the states defined by the function *χ* (Remark 2), cannot undergo consecutive discontinuous jump in an infinite manner. □

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Mehran, K., Zahawi, B. & Giaouris, D. Investigation of the near-grazing behavior in hard-impact oscillators using model-based TS fuzzy approach.
*Nonlinear Dyn* **69, **1293–1309 (2012). https://doi.org/10.1007/s11071-012-0348-8

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### Keywords

- Takagi–Sugeno fuzzy approach
- Non-smooth Lyapunov theory
- Linear Matrix Inequality
- Impact oscillator
- Grazing