#### Abstract

In this work we received the stationary set of the control system of the hysteresis deformation of the electro magneto elastic actuator in form the segment of the straight line. The aim of this work is to determine the condition of the absolute stability on the derivative for control system of the deformation of the electro magneto elastic actuator for nano biomedicine. We used the frequency methods for Lyapunov stable control system to calculate the absolute stability of the control system with electro magnetoelastic actuator. In result we determined the condition of the absolute stability on the derivative for the control system with the electro magneto elastic actuator for nano biomedicine.

**Keywords:** Condition Absolute Stability; Control System; Electro Magneto Elastic
Actuator; Piezo Actuator; Hysteresis and Butterfly Characteristics; Stationary Set;
Transfer Function

#### Introduction

In this work the condition of the absolute stability on the derivative for control system of the deformation of the electro magneto elastic actuator is calculated. The control systems with electro magneto elastic actuator on piezoelectric, electrostrictive and magnetostrictive effects solves problems of the precise matching in the nano biomedicine, the compensation of the temperature and gravitational deformations of the equipment, the wave front correction in the adaptive laser system [1-14]. The piezo actuator for nano biomedicine is used in the scanning tunneling microscope, the scanning force microscope, the atomic force microscope, in the gene manipulator [15-29]. The problems of using criteria absolute stability of the control system with electro magneto elastic actuator for nano biomedicine are discussed. The stationary set of the control system of the deformation of the electro magneto elastic actuator is received.

#### Condition Absolute Stability Control System

The electro magneto elastic actuator on piezoelectric, electrostrictive or magnetostrictive effects is used in the control systems for the micro and nano surgical repairs, the micro and nano robotics, the micro and nano manipulators and injectors for nano biomedicine [1-6]. The aim of this work is to calculate the condition of the absolute stability on the derivative for control system of the deformation of the electro magneto elastic actuator for nano biomedicine. The frequency methods for Lyapunov stable control system are used to determine the condition of the absolute stability of control system [2] with electro magneto elastic actuator for nano biomedicine.

We received the sufficient condition absolute stability control
system with the hysteresis nonlinearity of the electro magneto elastic
actuator using the Yankelovich absolute stability criterion with the
condition on the derivative. This criterion is the development of
the Popov absolute stability criterion [2]. For the Lyapunov stable
control system and the Yalubovich absolute stability criterion for
the systems with the single hysteresis nonlinearity provides the
simplest and pictorial representation of results of the investigation
of the stability of the strain control system with the electro magneto
elastic actuator for nano biomedicine. For description of the control
system we use the transfer function of the linear part of the system
*W*^{ij}(P) and the hysteresis function of the relative deformation *S _{j}* of
the electro magneto elastic actuator [16]. We have the hysteresis
characteristic of the electro magneto elastic actuator in the
following form

where S_{j} is the relative displacement of the cross section of the actuator along j axis, Ψ_{i} is the control parameter of the actuator along i axis. The hysteresis function S_{j} at each time instant t depends on the behavior of the function Ψ_{i}=E_{i} or Ψ_{i}=H_{i}, where E_{i} and H_{i} are the electric field strength and the magnetic field strength on the interval [0,t], the value of t, the initial value S_{j} (0), and the sign of the rate Ψ ̇_{i} of the field strength variation. Let us consider hysteresis type characteristic of the deformation of the electro magnetoelastic actuator for Nano biomedicine on Figure 1.

The set *S _{j}*(0) is the vertical segment [

*S*, -

^{0}_{j}*S*] bound by the points of intersection of the ordinate axis with the hysteresis loop at the maximum admissible field strength in the actuator. Therefore, we receive the stationary set for the deformation of the piezo actuator on Figure 1 for the stable linear part of the control system, where the equation for the straight-line L has the form

^{0}_{j}In static regime we have the value of the transfer function *W _{ij}*(0)
of the linear part of the control system for the deformation of the
piezo actuator. The set of points N’s for intersection of this straightline
L with the hysteresis characteristic represents the segment of
the straight line marked on Figure 1. At

*E*(0) and

_{i0}=E_{0}*S*(0) we have the stationary solution to the control system with hysteresis. The stationary set N of the system is the marked segment of straight-line

_{j0}=S_{j}*L*in Figure 1 with the set of pairs (

*E*). Each point of intersection of the hysteresis nonlinearity with the partial loops and the straight-line

_{i0}, S_{j0}*L*corresponds to one equilibrium position with the coordinates (

*E*). Therefore, for the stationary set the equation has the following form

_{i0}, S_{j0}Let us consider butterfly type characteristic of the deformation of the electro magneto elastic actuator for nano biomedicine. For the actuator with the electrostrictive effect the deformation characteristic on butterfly wings is observed for unipolar change of the electric field strength on Figure 2.

The particular cycle on one wing of butterfly is the hysteresis
loop. For butterfly type characteristic deformation of actuator
in the control system the coordinate origin is moved to new zero
with top dash on Figure 2. For hysteresis loop on butterfly type
characteristic we have stationary set *N* of the system marked
segment of straight-line *L* in Figure 2. For the magnetostrictive
actuator the deformation characteristic has the butterfly type. We
have the continuous function *S _{j}(E_{i})* of the hysteresis loop of the
piezo actuator with the quantities of the derivative.

where the quantities of the derivative and are calculated using
the hysteresis characteristic on Figure 1 for the maximum admissible
electric field strength in the piezo actuator. The quantities 𝒱_{1ij}=0
and 𝒱_{2ij}=𝒱_{ij} are the minimum and the maximum values of the
tangent of the inclination angle of the tangent line to the hysteresis
nonlinearity of the piezoactuator. Thus, we obtain

𝒱_{33}: 𝒱_{31}: 𝒱_{35}= 𝒹_{33}: 𝒹_{31}: 𝒹_{15}

where the ratios of the tangents of the inclination angle of the
tangent line to the hysteresis nonlinearity of the piezo actuator for
longitudinal, transverse and shift piezo effects are proportional
to the ratios of the piezo modules. We have the expression for
the sufficient absolute stability condition of the system with the
hysteresis nonlinearity of the electro magneto elastic actuator
using the Yakubovich absolute stability criterion with the condition
on the derivative. The Yalubovich criterion is the development of
the Popov absolute stability criterion [2]. For the Lyapunov stabile
control system the Yalubovich absolute stability criterion for the
system with the single hysteresis nonlinearity provides the simplest
and pictorial representation of results of the investigation of the
stability control system. The sufficient absolute stability condition
of the control system of the deformation of the electro magneto
elastic actuator at 𝒱_{1ij} =0 and 𝒱_{2ij}=𝒱_{ij} have the form

Where v_{ij}W_{ij}(jω) is the amplitude-phase characteristic of the
open-loop system and in brackets j is the imaginary unity and ω is
the frequency. The amplitude-phase characteristic of the open-loop
system v_{ij}W_{ij}(jω) on Figure 3 should be situated to the right of the
straight line

Re v_{ij}W_{ij}(jω)=-1

for all values of ω≥0.

On the plane of the logarithmic amplitude frequency characteristic and the phase frequency characteristic for the absolute stability criterion the control system with electro magneto elastic actuator we have

We have the corrected logarithmic amplitude frequency characteristic below the boundary curve in the form

For the piezo actuator from PZT we received the value of the maximum tangent of the inclination angle of the tangent line to hysteresis nonlinearity about 1nm/V for longitudinal piezo effect and about 0.6nm/V for transverse piezo effect. We used the frequency methods for Lyapunov stable control system to calculate the condition the absolute stability of the control system with electro magneto elastic actuator for nano biomedicine.

#### Conclusion

We received the stationary set of the control system of the deformation of the electro magneto elastic actuator. The stationary set of the system is the segment of straight line for hysteresis or butterfly characteristic deformation of the actuator. We determined condition of the absolute stability on the derivative for the control system with the electro magneto elastic actuator for nano biomedicine.

#### Acknowledgement

None.

#### Conflict of Interest

No conflict of interest.

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