Volume 17 Number 1
February 2020
Article Contents
Hiroki Matsumori, Ming-Cong Deng and Yuichi Noge. An Operator-based Nonlinear Vibration Control System Using a Flexible Arm with Shape Memory Alloy. International Journal of Automation and Computing. doi: 10.1007/s11633-018-1149-4
Cite as: Hiroki Matsumori, Ming-Cong Deng and Yuichi Noge. An Operator-based Nonlinear Vibration Control System Using a Flexible Arm with Shape Memory Alloy. International Journal of Automation and Computing. doi: 10.1007/s11633-018-1149-4

An Operator-based Nonlinear Vibration Control System Using a Flexible Arm with Shape Memory Alloy

Author Biography:
  • Hiroki Matsumori received the B. Sc. degree in applied electrical and electronic engineering from Tokyo University of Agriculture and Technology, Japan in 2014. He is currently a master student in electrical engineering, the Graduate School of Engineering of Tokyo University of Agriculture and Technology, Japan. His research interest is nonlinear control system design. E-mail: ailliks0522@gmail.com (Corresponding author)ORCID iD: 0000-0001-8613-0743

    Ming-Cong Deng received the Ph. D. degree in systems science from Kumamoto University, Japan in 1997. From 1997 to 2000, he was at Kumamoto University, Japan as an assistant professor. From 2000 to 2001, he was at the University of Exeter, UK, and then spent one year at Communication Science Laboratories, Nippon Telegraph and Telephone Corporation (NTT), Japan. From 2002 to 2010, he worked at Okayama University, Japan where he was an assistant professor and then an associate professor. He is currently a professor of Tokyo University of Agriculture and Technology, Japan. His research interests include nonlinear system modelling, control and fault detection, strong stability-based control, and robust parallel compensation. E-mail: deng@cc.tuat.ac.jp

    Yuichi Noge received the M. Sc. and Ph. D. degrees in electrical and electronic systems engineering from Nagaoka University of Technology, Japan in 2010 and 2014, respectively. From 2014 to 2017, he was with Tokyo Metropolitan College of Industrial Technology, Japan as assistant professor. Since 2017, he has been with Tokyo University of Agriculture and Technology, Japan as assistant professor. His research interests include multilevel power converters, power factor correction techniques and high speed digital controllers. E-mail: noge@go.tuat.ac.jp

  • Received: 2018-03-07
  • Accepted: 2018-07-09
  • Published Online: 2018-10-17
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An Operator-based Nonlinear Vibration Control System Using a Flexible Arm with Shape Memory Alloy

Abstract: In the past, arms used in the fields of industry and robotics have been designed not to vibrate by increasing their mass and stiffness. The weight of arms has tended to be reduced to improve speed of operation, and decrease the cost of their production. Since the weight saving makes the arms lose their stiffness and therefore vibrate more easily, the vibration suppression control is needed for realizing the above purpose. Incidentally, the use of various smart materials in actuators has grown. In particular, a shape memory alloy (SMA) is applied widely and has several advantages: light weight, large displacement by temperature change, and large force to mass ratio. However, the SMA actuators possess hysteresis nonlinearity between their own temperature and displacement obtained by the temperature. The hysteretic behavior of the SMA actuators affects their control performance. In previous research, an operator-based control system including a hysteresis compensator has been proposed. The vibration of a flexible arm is dealt with as the controlled object; one end of the arm is clamped and the other end is free. The effectiveness of the hysteresis compensator has been confirmed by simulations and experiments. Nevertheless, the feedback signal of the previous designed system has increased exponentially. It is difficult to use the system in the long-term because of the phenomenon. Additionally, the SMA actuator generates and radiates heat because electric current passing through the SMA actuator provides heat, and strain on the SMA actuator is generated. With long-time use of the SMA actuator, the environmental temperature around the SMA actuator varies through radiation of the heat. There exists a risk that the ambient temperature change dealt with as disturbance affects the temperature and strain of the SMA actuator. In this research, a design method of the operator-based control system is proposed considering the long-term use of the system. In the method, the hysteresis characteristics of the SMA actuator and the temperature change around the actuator are considered. The effectiveness of the proposed method is verified by simulations and experiments.

Hiroki Matsumori, Ming-Cong Deng and Yuichi Noge. An Operator-based Nonlinear Vibration Control System Using a Flexible Arm with Shape Memory Alloy. International Journal of Automation and Computing. doi: 10.1007/s11633-018-1149-4
Citation: Hiroki Matsumori, Ming-Cong Deng and Yuichi Noge. An Operator-based Nonlinear Vibration Control System Using a Flexible Arm with Shape Memory Alloy. International Journal of Automation and Computing. doi: 10.1007/s11633-018-1149-4
    • Vibration suppression is an important problem in a lot of engineering fields. In the past, arms in industrial fields have been prevented from increasing their stiffness and mass to improve their damping effect. The importance of the vibration suppression has increased along with the need to decrease the weight of the arms for the purpose of realizing desired control performance, which means faster operation of the arms, reduced cost of their production, etc.

      On the other hand, smart materials have received much attention and applied widely as new actuators in the past few decades. For example, piezoelectric, shape memory alloy (SMA), ionic polymer-metal composite (IPMC), and magnetorheological (MR) fluid are used widely to make actuators. There are particular advantages for SMA actuators[1, 2] such as large force of contraction and large displacement by their deformation, and light weight compared with other actuators. For instance, the practical produced force of the SMA wire (BMF100, TOKI Corporation) is 70 gf, and the SMA wire whose diameter is 0.1 mm and length is 1 m can generate the force of 0.4 kgf at maximum. To utilize the above advantages in applications to various fields, a lot of research using laminated SMAs[37], coiled SMAs[8, 9] and so on has been conducted. While SMA actuators possess the above advantages, there are undesirable characteristics such as hysteretic behavior and slow response speed. In particular, the hysteresis characteristics affect the performance of the actuators.

      An operator theory based method for nonlinear control has been proposed recently[1014], and that guarantees bounded-input bounded-output (BIBO) stability of the nonlinear feedback control system using the concept of robust right coprime factorization of the nonlinear system. More recently, a practical condition for the robust right coprime factorization has been proposed[15]. The robust stabilization of the nonlinear control systems can be obtained by using the method; for instance, nonlinear control for an aluminum plate thermal process[16], fault diagnosis for a nonlinear system[17], and nonlinear control for a travelling crane system[18]. It is expected that the operator theory can be applied to nonlinear control systems with hysteresis characteristics.

      In previous research[19], an operator-based control system including a hysteresis compensator has been proposed. The vibration of a flexible arm is dealt with as the controlled object; one end of the arm is fixed and the other end is free. The effectiveness of the hysteresis compensator has been confirmed by simulations and experiments. Nevertheless, the feedback signal of the previous designed system has increased exponentially, and that problem has made long-term use of the system difficult.

      Additionally, the SMA actuator radiates heat to the surroundings because the electric current passing through the SMA actuator provides heat, and then the strain on the SMA actuator is generated. With long-time use of the SMA actuator, the environmental temperature around the SMA actuator varies through radiation of the heat. There exists a risk that the ambient temperature change dealt with as disturbance affects the temperature and strain of the SMA actuator.

      This paper provides a vibration control scheme using a flexible arm with an SMA actuator. One end of the flexible arm is clamped and the other end is free. The control scheme is based on the operator theory, which can guarantee bounded-input bounded-output (BIBO) stability, and is designed for compensating the effect of the hysteresis and the change of the ambient temperature of the SMA wire to realize the desired damping performance and guaranteeing robust stability.

      Although there are various expression of hysteresis[20, 21], the Lipschitz operator-based PI hysteresis model is applied to express the hysteretic behavior of the SMA actuator and reduce the effect of hysteresis[2224]. The hysteresis model can be indicated in the sum of two integral terms, which are the invertible term and the bounded disturbance term. The disturbance exhibits the influence of the hysteresis and is compensated by the controller designed based on the operator theory. Also, the change of the temperature around the SMA actuator is defined as a thermal disturbance and is compensated by designing a thermal compensator based on the operator theory as with the design method of the hysteresis compensator. The effectiveness of the designed vibration control system as described above is verified by simulations and experiments.

    • In this section, the thermal model and PI hysteresis model of an SMA actuator, and the vibration model of a flexible arm are designed. In addition, objectives of this research are described.

    • The SMA actuators can be put large strain, specifically, 4–6%, and actuation force under a thermal input. The method for generation of the heat in this paper is the Joule effect by the controlled electric current passing through the SMA actuator. The SMA wire is well-known as a material which has slow response speed because of the conversions between the temperature and the input current, and between the strain and the temperature. To reduce the influence, the short SMA actuator is used to improve the rate of heat transfer. The model of the SMA actuator is utilized to design not only a stable controller on the operator theory but also the hysteresis compensator, and composed of a thermal model and a hysteresis model. In the beginning, the design method of the thermal model of the SMA actuator is explained.

    • The heat transfer equation for the SMA actuator is expressed in (1) by a lumped capacitance model using natural convection and electrical heating.

      $ \begin{split} & m{c_p}\frac{{{\rm d}(T - {T_a})}}{{{\rm d}t}} = {i^2}R - {h_c}{A_c}(T - {T_a})\\ & {T_a}(t) = {T_{a0}} + {T_e}(t) \end{split} $

      (1)

      where i and R show the electric current passing through the SMA wire and the resistance of the SMA wire, respectively. The resistance is calculated in real time and used to calculate the input power to the SMA wire. In addition, $A_c$ is the surface area, $c_p$ is the specific heat, m is the mass, $h_c$ is the heat convection coefficient, $T_a$ is the temperature around the SMA actuator, $T_{a0}$ is the initial temperature around the SMA actuator, and $T_e$ is the temperature difference between $T_a$ and $T_{a0}$, respectively.

      For convenience of calculation, the electric power $i^2 R$ are defined as follows:

      $ {u_d}(t) = {i^2}(t)R. $

      (2)

      From (2), (1) is transformed as follows:

      $ \begin{aligned} T(t) = & {T_h}({u_d})(t)=\\ & \frac{1}{{m{c_p}}}\int_0^t {{{\rm e}^{( - \gamma (t - \tau ))}}} {u_d}(\tau ){\rm d}\tau + {T_a}(t) \end{aligned} $

      with

      $ \gamma = \frac{{{h_c}{A_c}}}{{m{c_p}}}. $

      (3)
    • In this research, the Lipschitz operator-based PI model is utilized for describing the hysteresis. The PI hysteresis model is a subclass of Preisach model and is composed of a weighted superposition of elementary hysteresis operators, which are play hysteresis operators and stop hysteresis operators. In this research, the play hysteresis operator is fitted for the expression of the hysteresis behavior of the SMA actuator.

      $ {F_h}[u](t) = \left\{ {\begin{aligned} & {u(t) + h},\quad {{\rm{if}}\;u(t) \ge {F_h}[u]({t_i}) - h}\\ & {{F_h}[u]({t_i})}, \quad\!\! {{\rm{if}}\; - h < u(t) - {F_h}[u]({t_i}) < h}\\ & {u(t) - h},\quad {{\rm{if}}\;u(t) \le {F_h}[u]({t_i}) - h} \end{aligned}} \right. $

      (4)

      where

      $ {F_h}[u](0) = \max (u(0) - h, \min (u(0) + h, {q_{ - 1}})) $

      (5)

      where $0 = t_0 < t_1 < \cdots < t_N = t_E$ is a partition of $[0, t_E]$ such that the function u is monotone on each of the sub-intervals $[t_i, t_{i+1}]$. h is the threshold value and satisfies $h > 0$, and $F_h$ is deformed by h. $q_{-1}$ is an initial value.

    • The PI model of the play hysteresis operator is shown as follows:

      $ \begin{split} {u^*}(t) = & \int_{{h_0}}^H p (h){F_h}[T](t){\rm d}h=\\ & {D_{PI}}(T)(t) + {\Delta _{PI}}(T)(t) \end{split} $

      (6)

      where

      $ \begin{split} & {D_{PI}}(T)(t) = K \times T(t), \;K = \int_{{h_0}}^H p (h){\rm d}h\\ & {\Delta _{PI}}(T)(t) = - \int_{{h_0}}^{{h_x}} {{S_n}} hp(h){\rm d}h + \int_{{h_x}}^H p (h){F_h}[T]({t_i}){\rm d}h \end{split} $

      $ {S_n} = \left\{ {\begin{aligned} & {\;1, \;\;} & {{\rm{if}}\;T(t) - {F_h}[T]({t_i}) \ge 0}\\ & { - 1, \;\;} & {{\rm{if}}\;T(t) - {F_h}[T]({t_i}) < 0} \end{aligned}} \right.\quad\quad\quad\quad\quad\;\, $

      (7)

      and where, on $[{h_0}, \;H]$, $h_x$ is the maximum number which satisfies the condition, $h \ge |T(t) - F_h[T](t)|$, when $h \in [{h_0}, \;{h_x}]$. $p(h)$ is an unknown continuous density function, which satisfies the following condition:

      $ p(h) > 0, \;\;\;\;\int_{{h_0}}^\infty h \times p(h){\rm d}h < \infty . $

      (8)

      When H exists and $h > H$, it is assumed that the density function $p(h)$ becomes zero. Suppose that the density function $p(h) = a \times {{\rm e}^{b{{(h - 1)}^2}}}$. The weight parameters a and b are decided by the experiments. The reader is referred to [19] for further details of the experiments. The relationship between the bending moment $M_a(t)$ which is the input of the plant and the strain of the SMA actuator is linear. The bending moment $M_a(t)$ is defined as follows based on Mohr′s theorem.

      $ {M_a}(t) = {M_{a0}} \times {P_I}(u)(t) $

      (9)

      where $P_I(u)(t)$ is the output of the PI hysteresis operator, and

      $ {M_{a0}} = \frac{{3EI}}{{l_1^2}}. $

      (10)

      E, I and $l_1$ denote the Young′s modulus, the area moment of inertia, and the attachment position of the SMA actuator. Specifically, the SMA actuator is located from the base of the flexible arm to the position which is $l_1$ from the base of the arm.

    • In this subsection, the dynamics model of a flexible arm is introduced. One end of the arm is clamped and the other end is free. It is assumed that the flexible arm is initially straight along the horizontal axis, and the effect of the gravity and rotatory inertia of the flexible arm cross-sections are ignored. Using Euler-Bernoulli beam theory, the dynamics of the arm can be expressed as follows:

      $ \begin{split} & \rho S\frac{{{\partial ^2}y(x, t)}}{{\partial {t^2}}} + \frac{{{\partial ^2}}}{{\partial {x^2}}}\left[{EI\left( {1 + {C_m}\frac{\partial }{{\partial t}}} \right)\frac{{{\partial ^2}y(x, t)}}{{\partial {x^2}}}} \right]=\\ & \quad\quad \frac{{{\partial ^2}}}{{\partial {x^2}}}M(x, t) \end{split} $

      (11)

      where $\rho$ is the density of the arm, S is the cross-section area of the arm, E is the Young′s modulus of the arm, I is the moment of inertia of area, C is the damping modulus, and M is the moment working on the flexible arm, respectively.

      In this research, the moment is generated by the SMA actuator. Assume that the flexural displacement of the flexible arm $y(x, t)$ is exhibited in (12).

      $ y(x, t) = \sum\limits_{m = 1}^\infty {{\omega _m}} (x){f_m}(t) $

      (12)

      where $\omega_m (x)$ and $f_m (t)$ are mode shape functions and the generalized coordinates. In (11), because the vibration of the arm is not affected by the SMA actuator in the case of free vibration, the bending moment $M = 0$. That is

      $ \quad \quad \rho S\frac{{{\partial ^2}y(x, t)}}{{\partial {t^2}}} + \frac{{{\partial ^2}}}{{\partial {x^2}}}\left[{EI\left( {1 + {C_m}\frac{\partial }{{\partial t}}} \right)\frac{{{\partial ^2}y(x, t)}}{{\partial {x^2}}}} \right] = 0. $

      (13)

      The following differential equation is derived from (12) and (13).

      $ \frac{{{{\rm d}^2}f(t)}}{{{\rm d}{t^2}}} + {k^2}C\frac{{{\rm d}f(t)}}{{{\rm d}t}} + {k^2}f(t) = 0 $

      (14)

      $ \frac{{{{\rm d}^4}\omega (x)}}{{{\rm d}{x^4}}} - {\lambda ^4}\omega (x) = 0 \quad\quad\quad\quad\quad $

      (15)

      $ {\lambda ^4} = \frac{{{k^2}\rho S}}{{EI}}.\quad\quad\quad\quad\quad\quad\quad\quad\quad\, $

      Each of the solutions of (15) indicates an m-th natural vibration mode function $\omega_m (x)$ and is shown as follows:

      $ \begin{split} {\omega _m}(x) = & {B_m}[(\sinh {\lambda _m}l + \sin {\lambda _m}l)(\cosh {\lambda _m}x-\cos {\lambda _m}x)-\\ & (\cosh {\lambda _m}l + \cos {\lambda _m}l)(\sinh {\lambda _m}x-\sin {\lambda _m}x)] \end{split} $

      (16)

      where $B_m$ is an arbitrary constant, and $\omega$ satisfies

      $ \int_0^l {{\omega _m}} (x){\omega _n}(x){\rm d}x = \left\{\!\!\! {\begin{array}{*{20}{l}} {0, } & {(m \ne n)}\\ {{\psi _m}, } & {(m = n).} \end{array}} \right. $

      (17)

      $\lambda_m$ can be obtained from the following condition.

      $ 1 + \cos {\lambda _m}l\cosh {\lambda _m}l = 0. $

      (18)

      Fig. 1 illustrates the configuration of the arm with the SMA and that the SMA is furnished with the base of the arm. x, y, l, and $l_1$ represent the distance along the flexible arm, the flexural displacement of the flexible arm, the length of the flexible arm, and the length of the SMA. From (11) and Fig. 1, the dynamics of the arm with the SMA can be described as follows:

      Figure 1.  Configuration of the flexible arm with SMA

      $ \begin{split} & \rho S\frac{{{\partial ^2}y(x, t)}}{{\partial {t^2}}} + \frac{{{\partial ^2}}}{{\partial {x^2}}}\left[{EI\left( {1 + {C_m}\frac{\partial }{{\partial t}}} \right)\frac{{{\partial ^2}y(x, t)}}{{\partial {x^2}}}} \right]=\\ & \quad\quad\frac{{{\partial ^2}}}{{\partial {x^2}}}\left[{{M_a}\delta (x-{l_1})} \right]. \end{split} $

      (19)

      Then,

      $ \begin{split} & y(x, t)=\\ &\quad\quad \sum\limits_{m = 1}^\infty {\left[{{A_m}\int_0^t {{{\rm e}^{-\frac{{{\alpha _m}}}{2}(t-\tau )}}} \times \sin \frac{{{\beta _m}}}{2}(t-\tau ) \times {M_a}(\tau ){\rm d}\tau } \right]} \end{split} $

      (20)

      where

      $ \begin{split} & {A_m} = \frac{{2{\omega _m}(x)}}{{\rho S{\psi _m}{\beta _m}}}{{\omega ''}_m}({l_1}) \\ & {\alpha _m} = k_m^2{C_m}\\ & {\beta _m} = \sqrt {4k_m^2 - k_m^4C_m^2}\\ & {k_m} = \sqrt {\frac{{{\lambda ^4}EI}}{{\rho S}}} . \end{split} $

    • The vibration of the flexible arm with the SMA actuator is dealt with as a controlled object in this research. The objectives of this research are as follows.

      Firstly, the reconstruction of the operators of the previous control system is considered such that the outputs of the operators do not increase in an exponential manner. Accordingly, the hysteresis compensator is also redesigned for the hysteretic behavior of the SMA actuator.

      Secondly, the other compensator which compensates the effect of the ambient temperature of the actuator is proposed and designed. The compensator is called a “thermal compensator” in this paper.

      Finally, the effectiveness of the operators and compensators which are designed based on the operator theory is verified by simulations and experiments.

    • In this section, a design of the nonlinear system with hysteresis nonlinearity and thermal disturbance is proposed based on the operator theory. The stability of the plant is guaranteed by an operator-based robust right coprime factorization approach. The plant can be guaranteed to be robustly stable on this procedure. A bending moment generated by the strain of the SMA with hysteresis is considered as an input of the plant, and the difference between the temperature around the SMA at a certain time and the room temperature is regarded as a temperature disturbance. However, the operator theory can guarantee the only stability between input and output space. Therefore, sufficient vibration suppression performance cannot be obtained. To improve performance, the damping compensator is designed at the next step of the guarantee of the plant. Concurrently, the non-invertible part of hysteresis model and the temperature disturbance are also compensated in the damping compensator.

    • In this subsection, the control design scheme is indicated based on the operator theory. The operator theory can express a behavior of a control object in time domain. The framework of the considered system without compensators for the hysteresis and thermal disturbance is indicated in Fig. 2. U, $U^*$, W and Y are the input space, the output space of PI hysteresis model, quasi-state space, and the output space of the plant, respectively. N, $\Delta N$, and D are defined as follows:

      Figure 2.  Framework of proposed control system without compensators

      $ N:W \to Y, \;\Delta N:W \to Y, \;{\rm{and}}\;D:W \to {U^*}. $

      $r^* \in U$ is the reference input, $e \in U$ is the error signal between $r^*$ and b, $b \in U$ is the output of the controller A, $u_d \in U$ is the input power to the SMA actuator, $T \in U$ is the temperature of the actuator, and $M_a \in U^*$ is the bending moment generated by the strain of the actuator, respectively.

      The model of the plant is expressed in (20). In this paper, a first order mode, and second or more order mode are considered as the nominal vibration mode and uncertainties, respectively. Hence, the plant with uncertainties is represented as follows:

      $ \begin{split} & [P + \Delta P]({M_a})(t)=\\ & \quad\quad (1 + \Delta ){A_1}\int_0^t {{{\rm e}^{ - \frac{{{\alpha _1}}}{2}(t - \tau )}}} \times \sin \frac{{{\beta _1}}}{2}(t - \tau ) \times {M_a}(\tau ){\rm d}\tau \end{split} $

      (21)

      where $\Delta$ shows uncertainties of the plant, and $M_a$ is a bending moment by the SMA. Next, the plant $P + \Delta P$ is factorized into (22) and (23).

      $ \begin{split} & \quad [P + \Delta P]({M_a})(t) = N{D^{ - 1}}({M_a})(t)\\ & \quad N(w)(t) = {A_1}\int_0^t {{{\rm e}^{ - \frac{{{\alpha _1}}}{2}(t - \tau )}}} \sin \frac{{{\beta _1}}}{2}(t - \tau ) \times w(\tau ){\rm d}\tau \end{split} $

      (22)

      $ D(w)(t) = w(t).\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad $

      (23)

      These equations are called “right factorization”.

    • Generally, it is difficult to observe a bending moment. Hence, the hysteresis of the SMA actuator is regarded as one part of the plant. Namely, $D_{PI}$ in the PI model given in (6) and $D^{-1}$ of the plant applied right factorization are considered as a new invertible factor $\tilde{D}^{-1} = D^{-1} D_{PI} T_h$, and let $\Delta_{PI}$ be the bounded disturbance.

      Controllers A and B are designed to satisfy Bezout identity (24).

      $ AN + \tilde B\tilde D = \tilde M\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\;\; $

      (24)

      $ A({y_a})(t) = {K_1}{y_a}(t)\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\, $

      (25)

      $ \tilde B({u_d})(t) = (I - {K_2}A)N{{\tilde D}^{ - 1}}({u_d})(t)\quad\quad\quad\quad\quad\quad\, $

      (26)

      $ \tilde M(w)(t) = {A_1}(1 - {K_1}{K_2} + {K_1})N(w)(t) \quad\quad\quad\quad\, $

      (27)

      $ {{\tilde M}^{ - 1}}({r^*})(t) = \frac{2}{{{A_1}{\beta _1}(1 - {K_1}{K_2} + {K_1})}}{N^{ - 1}}({r^*})(t). $

      (28)

      Equations (27) and (28) indicate operator $\tilde{M}$ is a unimodular operator, and $K_1$ and $K_2$ are arbitrary constants which satisfy the following condition:

      $ {K_1}, {K_2} \in (0, 1).\quad\quad\quad\quad\quad\quad\quad\quad $

      (29)

      Moreover, the robust stability condition is shown by the following inequality:

      $ {\left\| {[A(N + \Delta N)-AN]{{\tilde M}^{ - 1}}} \right\|_{Lip}} < 1. $

      (30)

      If the designed system satisfies (30), the proposed control system can be robustly stable.

    • The stability of the proposed system is guaranteed by the design of the controllers A and B. However, there is a risk that sufficient vibration suppression performance cannot be obtained. In this subsection, the damping compensator is designed based on the operator theory. On the procedure of the design of the damping compensator, the hysteresis compensator $N_c$ and the thermal compensator $T_c$ are designed to compensate the hysteresis characteristics of the SMA actuator and the thermal disturbance $T_e$, respectively.

      From Fig. 2, the plant output y without uncertainties is shown as follows:

      $ y(t) \!=\! N{\tilde M^{ - 1}}({r^*} \!+\! \tilde BD_{PI}^{ - 1}T_h^{ - 1}{\tilde \Delta _{PI}} \!+\! \tilde BT_h^{ - 1}({T_e})(t)) $

      (31)

      where $\tilde BD_{PI}^{ - 1}T_h^{ - 1}{\tilde \Delta _{PI}}$ and $\tilde BT_h^{-1}({T_e})(t)$ are the terms into which ${\tilde \Delta _{PI}}$ and $T_e$ are converted equivalently. Their terms are added to the addition point of $r^*$ and b. $N_c$ and $T_c$ are designed to negate the disturbance terms $\tilde BD_{PI}^{-1}T_h^{-1}{\tilde \Delta _{PI}}$ and $\tilde BT_h^{-1}({T_e})(t)$ in (31). That is, the damping compensator shown below is designed to satisfy (31).

      $ C(r(t), z) = (1 - {K_2} + \frac{1}{{{K_1}}})A(r)(t) - {z_1} - {z_2} $

      (32)

      where ${z_1} = \tilde BT_h^{-1}D_{PI}^{-1}{\tilde \Delta _{PI}}$ and $z_1$ is adjusted using the error signal e. Using $T_e$, $z_2(= T_c)$ is shown as follows:

      $ {T_c}({T_e})(t) = \tilde BT_h^{ - 1}({T_e})(t).\quad\quad\quad\quad\quad\quad\quad\quad $

      (33)

      The disturbance part of the PI model $\Delta_{PI}$, which has nonlinearity and is used to design the damping compensator, is calculated based on the designed model and the input power to SMA actuator $u_d$.

      Finally, the proposed control system with the above compensators is shown in Fig. 3.

      Figure 3.  Proposed control system with compensator based on operator theory

    • In this subsection, the effectiveness of the proposed method using the operator theory is verified by numerical simulations. To begin with, parameters and conditions utilized in the simulations are illustrated.

    • The numerical simulations are conducted by using the parameters of the SMA actuator described in Table 1 and the flexible arm shown in Table 2. The parameters are decided based on the actual experimental devices. The displacement of the end of the flexible arm ($x = l$) is measured in this verification. The lower limit of the input power is 0 W. The upper limit of the input power is set as 1 W to prevent the damage to the SMA actuator by the excessive heat caused by the electric power and the ambient environment. The flexible arm is vibrated by adding electric power to the SMA actuator for 10 s before the vibration control begins, and the vibration is controlled for 15 s. In other words, the simulation time is 25 s. The input power $u_d$ for vibrating the arm is indicated as follows:

      Explanation Symbol (Unit) Value
      Length $ {{l_a}}$ (mm) 100
      Diameter $ {d}$ (mm) 0.1
      Resistance $ {R}$ (Ω) 13.5
      Heat convection coefficient $ {{h_c}}$ (W/m2°C) 689
      Surface area $ {{A_c}}$ (mm2) 3.14×10–5
      Weight $ {m}$ (kg) 5×10–6
      Specific heat $ {{c_p}}$ (J/kg°C) 7 349

      Table 1.  Parameters of SMA wire

      Explanation Symbol (Unit) Value
      Density of arm $ {\rho}$ (kg/m3) 2 700
      Cross-section area of arm $ {S}$ (m2) 10×10–6
      Young′s modulus of arm $ {E}$ (N/m2) 6.9×1010
      Moment of inertia of area $ {I}$ (m4) 1.7×10–12
      Damping modulus of first order mode $ {C_1}$ (–) 0.001 5
      Position of SMA $ {l_1}$ (m) 0.1
      Length $ {l}$ (m) 0.8

      Table 2.  Parameters of flexible arm

      $ {u_d}(t) = 0.24\sin \frac{{{\beta _1}}}{2}t + 0.40{\rm{W}}. $

      (34)

      The vibration modes considered in the simulations are chosen as $m = 5$. Namely, the nominal plant P is the first order mode of the vibration of the flexible arm, and the uncertainties $\Delta P$ are regarded as the second to fifth order modes. Parameters such as the constants used in the designed controller, and the other parameters are explained in Table 3.

      Explanation Symbol (Unit) Value
      Sampling time – (s) 5×10–3
      Upper limit of input power – (W) 1
      Lower limit of input power – (W) 0.025
      Initial ambient temperature of SMA wire $ {T_{a0}}$ (°C) 23
      Arbitrary parameter $ {K_1}$ (–) 0.5
      $ {K_2}$ (–) 0.5

      Table 3.  Parameters of flexible arm

      Also, in this research, although vibration modes which are higher than the first mode are considered as the uncertainties, the vibration modes from the second mode to the fifth mode are utilized in the simulations because the vibration of the flexible arm possesses infinite vibration modes, which are not able to be expressed on the simulator. The uncertainties of the plant are given as follows:

      $ \Delta (t) = \sum\limits_{m = 2}^5 {{A_m}} \int_0^t {{{\rm e}^{ - \frac{{{\alpha _m}}}{2}(t - \tau )}}} \sin \frac{{{\beta _m}}}{2}(t - \tau ) \times {M_a}(\tau ){\rm d}\tau $

      (35)

      with

      $ \begin{aligned} & \left(\; {\begin{aligned} {{A_2}}\\ {{A_3}}\\ {{A_4}}\\ {{A_5}} \end{aligned}} \;\right) = \left(\; {\begin{aligned} {0.027\;3}\\ {0.011\;5}\\ {0.005\;0}\\ {0.002\;0} \end{aligned}} \;\right), \;\;\;\;\left(\; {\begin{aligned} {{\alpha _2}}\\ {{\alpha _3}}\\ {{\alpha _4}}\\ {{\alpha _5}} \end{aligned}} \;\right) = \left(\; {\begin{aligned} {43.628}\\ {99.355}\\ {2.589\;1}\\ {3.537\;6} \end{aligned}} \;\right) \\ & \left(\; {\begin{aligned} {{\beta _2}}\\ {{\beta _3}}\\ {{\beta _4}}\\ {{\beta _5}} \end{aligned}}\; \right) = \left(\; {\begin{aligned} {90.628}\\ {263.54}\\ {551.90}\\ {912.34} \end{aligned}} \;\right). \end{aligned} $

    • Figs. 47 exhibit the displacement of the flexible arm and the input electric power to the SMA actuator. $N_c$ and $T_c$ denote the hysteresis compensator and the thermal compensator, respectively. The results that the vibration is suppressed with the hysteresis compensator faster than without that from Figs. 4 and 5 are obtained.

      Figure 4.  Displacement of the arm without $N_c$

      Figure 5.  Displacement of the arm with $N_c$

      Figure 6.  Input power without $N_c$

      Figure 7.  Input power with $N_c$

      Next, assuming that the temperature around the SMA wire increases linearly as shown in (36) and Fig. 8.

      Figure 8.  Ambient temperature of the SMA wire

      $ {T_a}(t) = {T_{a0}} + 0.4t. $

      (36)

      Assuming that $T_{a0} = 25$. With the assumption, the above hysteresis compensator is applied. Figs. 912 show the displacement of the arm and the input power with the hysteresis compensator and the thermal compensator. The input power is adjusted by the thermal compensator following an increase in the ambient temperature, and the effect of the thermal disturbance is eliminated. It is confirmed that the effect of the thermal disturbance is eliminated by the thermal compensator.

      Figure 9.  Displacement of the arm without $T_c$

      Figure 10.  Displacement of the arm with $T_c$

      Figure 11.  Input power without $T_c$

      Figure 12.  Input power with $T_c$

      Finally, the result of robust stability analysis by (30) is shown in Fig. 13. From Fig. 13, it is found that the designed control system is robustly stable relative to the assumed uncertainties.

      Figure 13.  Robust stability analysis of simulation results

    • In this subsection, the effectiveness of the proposed control system based on the operator theory is verified by the following experiments.

    • Two kinds of experiments, the cases of the hysteresis compensator and the thermal compensator, are conducted. The parameters used in the controller and compensators are the same as the above simulations. The environmental temperature is 19 °C. The ambient temperature of the SMA actuator is measured by using an integrated circuit (IC) temperature sensor (LM35, Texas Instruments). Also, the ambient temperature of the SMA actuator is decided every second by taking averages of the temperature for 1 s. The decided temperature is utilized as an input of the thermal compensator. The space near the SMA actuator is warmed up by heating a nichrome wire instead of generating heat by the SMA wire. When conducting the experiments, the input power to the nichrome wire is the same on each of the experiments. Depending on reference input power, the control input current was supplied to the SMA actuator via the D/A converter and voltage-current converter. The upper limit of the input power is set as 1 W. The lower limit of that is decided as 0.025 W because if the lower limit is 0 W, the current becomes 0 A, and the resistance of the SMA actuator cannot be calculated from its voltage and current. The sampling frequency is 200 Hz. Each of the input powers for vibrating the arm is indicated as follows:

      $ \begin{split} & {u_d}(t) = 0.15\sin \frac{{{\beta _1}}}{2}t + 0.40\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \\ & ({\rm{with}}\;{\rm{hysteresis}}\;{\rm{compensator}})\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \end{split} $

      (37)

      $ \begin{split} &{u_d}(t) = 0.13\sin \frac{{{\beta _1}}}{2}t + 0.30\\ & ({\rm{with}}\;{\rm{hysteresis}}\;{\rm{and}}\;{\rm{thermal}}\;{\rm{compensator}}). \end{split} $

      (38)

      Since the date when each of the experiments have been conducted is different, the input power for vibrating the arm is also different. The evaluation indices of the verifications of two compensators; the hysteresis compensator and the thermal compensator, are the time that the amplitude of the vibration is suppressed to 5% of the maximum amplitude and the mean absolute error (MAE), respectively. The time is called $T_{sup}$ in this paper for convenience. The MAE in each of the experimental results is computed from the following equation:

      $ {\rm{MAE}} = \frac{1}{N}\sum\limits_{n = 1}^N {\left| {{y_a}(n) - r(n)} \right|} $

      (39)

      where N, $y_a$ and r denote the number of samples, the displacement of the arm, and the reference input. The displacements for the last 10 s before the end of execution time; from 15 s to 25 s of the execution time of the experiments, is evaluated by the MAE.

    • Figs. 1423 express the displacement of the arm, the input power, and the temperature around the SMA wire with and without the hysteresis compensator. The time $T_{sup}$ of the cases with and without the hysteresis compensator are shown in Table 4. Table 4 explains that the suppression of the vibration with the hysteresis compensator is 4.0 s faster than without that, and the effectiveness of the hysteresis compensator is confirmed.

      Explanation $ {T_{sup}}$ (Unit)
      With hysteresis compensator 6.3 s
      Without hysteresis compensator 10.3 s

      Table 4.  $T_{sup}$ of the experimental results

      Figure 14.  Displacement of the arm without $N_c$

      Figure 16.  Input power without $N_c$

      Figure 17.  Input power with $N_c$

      Figure 15.  Displacement of the arm with $N_c$

      Figure 22.  Input power without $T_c$

      Figure 23.  Input power with $T_c$

      Secondly, the results in the case with and without the thermal compensator are shown in Figs. 18 and 23. The rise of approximately 5 °C is confirmed in both of Figs. 18 and 19. Fig. 20 denotes the excessive displacement is caused by an increase in the ambient temperature although the vibration of the arm is suppressed. On the other hand, it is confirmed from Fig. 21 that the excessive displacement is compensated by the thermal compensator. Slight vibration of the arm from 17 s to 25 s in the experimental time is also confirmed. It is speculated that the cause of the remaining vibration is the output of the thermal compensator. Obviously, the input of the thermal compensator is the temperature $T_e$ which is the difference between the present temperature around the actuator and the initial temperature. However, $T_e$ in this research is the average at 1 s intervals and is step-wise. The output of the compensator also becomes stepwise along with the input of that.

      Figure 18.  Temperature around the SMA wire w/o $T_c$

      Figure 19.  Temperature around the SMA wire $\dfrac{w}{T_c}$

      Figure 20.  Displacement of the arm without $T_c$

      Figure 21.  Displacement of the arm with $T_c$

      The evaluation results from using the MAE are represented in Table 5. Not only Figs. 20 and 21 but also Table 5 demonstrate that the effectiveness of the designed thermal compensator is verified.

      Explanation MAE $ {(15 \le t \le 25)}$
      With thermal compensator 0.159 3
      Without thermal compensator 0.948 6

      Table 5.  Calculation of MAE

    • In this paper, a design of the thermal compensator and re-design of the hysteresis compensator are conducted, and the control design method for the vibration suppression of the flexible arm with the SMA actuator is proposed. The thermal compensator was designed to remove the effect of an increase on environmental temperature regarded as a disturbance. As with the thermal compensator, the hysteresis compensator was redesigned to remove the hysteretic effect that the SMA actuator possesses.

      First, the vibration of the arm was modelled using Euler-Bernoulli beam theory, and the thermal model and the hysteresis model of the SMA actuator were constructed considering the ambient temperature of the SMA actuator. Second, the operator-based nonlinear vibration control system including the above designed compensators was designed using the above models.

      Finally, simulations and experiments for verification of the effectiveness of the hysteresis compensator and thermal compensator were conducted. From Figs. 14, 15 and Table 4, the vibration in the case with the hysteresis compensator is suppressed earlier than without one. In addition, from Fig. 20, deviation from the target value is confirmed. However, Fig. 21 and Table 5 show that the deviation is compensated by the thermal compensator. As a result, the effectiveness of the proposed system with the operator-based damping compensator with the hysteresis compensator and the thermal compensator is confirmed by the simulations and experiments.

Reference (24)

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