## Abstract

The nonlinear behavior of bubble in the blood vessel restricts the applications of bubble technology in the cardiology. In this paper, a novel engineering solution is proposed to control the radius of the bubble and prevent bubble collapse. For this purpose, the nonlinear dynamics of the bubble in the blood vessel is introduced and then presented into the state-space form. The next part of this paper is devoted to designing a nonlinear control method, where the ultrasonic wave plays the role of control input and the output is the radius of the bubble in the blood vessel. In order to achieve faster, finite-time convergence and higher control precision, fast terminal sliding mode is proposed. The stability of the closed-loop system is ensured through the Lyapunov stability analysis. In order to tackle the chattering problem, the switching gain is adaptively tuned according to the fuzzy rule. In this method, the tracking error and its differential are considered as the inputs and the switching gain is considered as the output of the fuzzy logic system. Simulation results confirm the effectiveness of the proposed control method. Finally, the results of this research are of immediate interest for modern medical applications such as ultrasonic imaging, targeted drug delivery, and cancer treatment.

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## Appendix: Proof of Lemma 1

### Appendix: *Proof of Lemma 1*

When the system states reach nonlinear sliding surface, the following equation is hold

Equation (30) can be formulated as

Let \( \alpha = e^{{\left( {1 - \frac{q}{p}} \right)}} \), then Eq. (31) can be rewritten as

The solution of differential Eq. (32) is obtained as follows

Take the logarithm of Eq. (33) yields

With the help of \( \alpha \left( 0 \right) = \left( {e_{0} } \right)^{{\left( {1 - \frac{q}{p}} \right)}} \), the following result is obtained

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Badfar, E., Ardestani, M.A. Design of adaptive fuzzy gain scheduling fast terminal sliding mode to control the radius of bubble in the blood vessel with application in cardiology.
*Int. J. Dynam. Control* (2020). https://doi.org/10.1007/s40435-020-00641-y

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### Keywords

- Bubble dynamics
- Cavitation phenomenon
- Sliding mode controller
- Ultrasonic
- Chattering phenomenon