Masaharu Kuroda

In this paper, we introduce fractional-order calculus into linear quadratic regulator (LQR) control, which is a typical modern control, and discuss control of a magnetic levitation system using a state observer described by a fractional-order differential equation. In the previous paper, LQR control using a fractional-order state observer with output feedback was applied to a linear control object. It is necessary to verify if the control is also effective for the nonlinear model. For an unstable target, an integrator is required to be designed to trace the target value. In this report, we propose an output-feedback-type servo LQR control system with a fractional-order state observer for a voltage-controlled suspention-type magnetic levitation system with strong nonlinearity.

This research deals with a vibration suppression problem occurring in a viscoelastic damper system modeled with a fractional-order differential equation using a fractional-order linear quadratic regulator (LQR) as a typical modern control. In the previous report, the fractional-order LQR was configured with a state feedback style. However, ordinary sensors cannot measure a fractional-order state in practice, because a fractional-order state is composed of displacement, velocity and acceleration. Therefore, a state observer is required to estimate all of the states including fractional-order states. This report aims at realizing the fractional-order LQR with an output feedback style by constructing a state observer in order to estimate fractional-order states.

This study aims at applying linear quadratic regulator (LQR) theory to control a vibratory system modelled by a fractional-order differential equation. First, an iteration-based method for solving the algebraic Riccati equation is proposed to obtain a fractional-order LQR. Next, Newmark-<tt>𝛽</tt>method is modified to be able to simulate a fractional-order differential equation. Finally, as an example, a vibration control problem is considered for a generalized Voigt model. Numerical calculations demonstrate that the Fractional-order LQR control can suppress vibrations occurring in the vibratory system with viscoelastic damping.

<p>This study aims at developing a design method of a fractional-order PD controller and clarifying merits of the fractional-order PD controller. A fractional-order PD controller contains the non-integer differential order as a design parameter in addition to the proportional gain and the derivative gain. Therefore, the fractional-order PD controller can satisfy one more design specification than a normal integer-order PD controller can. In this report, besides the requirements on the gain crossover frequency and the phase margin, the condition on robustness is added. Based on those relationships, the proportional gain, the derivative gain and the non-integer differential order of the fractional-order PD controller are obtained with a graph-aided design method. Finally, the proposed design procedure of the fractional-order PD controller is applied to a vibration suppression problem of a flexible cantilever beam. Comparing with a usual PD controller, some control effects of the fractional-order PD controller are investigated by numerical simulation. Especially, the characteristics of the proportional gain, the derivative gain and the non-integer differential order of the fractionalorder PD controller are obtained as the gain crossover frequency is varied. It is confirmed that the fractional-order PD controller can be designed in the gain crossover frequency range where the normal PD controller can not satisfy the design specifications.</p>

In aircrafts and space structures, there are many structural components that can be considered as flexible beams. Vibrations can be harmful to the performance of such structures. In this study, fractional calculus was applied to the vibration suppression of a flexible beam, and the controller parameters were designed. The tuning of the controller parameters has been performed in some previous studies using a trial-and-error approach. However, this tuning process is time-consuming, and there is a risk of destabilization. As an alternative tuning method, a function that implements an optimization algorithm (e.g., the &ldquo;fmincon&rdquo; function in MATLAB) has been used in some studies. However, this complicated algorithm can place a heavy load on the computer. Therefore, in this study, a fractional-order controller was designed as an alternative non-trial-and-error approach that is simpler than previously implemented optimization algorithms. This paper proposes a fractional-order controller design method for the vibration suppression of a flexible structure. First, the process of designing a controller to meet the specifications of the Nyquist diagram is explained. An improvement to the design method that allows the attenuation level of a resonant peak to be treated as a specification is then discussed. Numerical simulations performed in this study demonstrate that the proposed fractional-order controller is more robust against spillover instability than the integer-order controller.

<p>Achieving active wave control is both an old and a new problem. A vibration suppression problem for a thin cantilevered beam is presented as an example for discussion. Results clarified that the active wave controller includes √s and √s terms. Those terms are realized as a 1/2-order derivative and 3/2-order derivative using fractional calculus. The active wave controller is realized as connected using fractional calculus, which is shown to be an important step for analysis. Results show that the control effect is extremely high when both a shear-force actuator and a bending-moment actuator are applied. However, the control effect is degraded considerably when only the bending-moment actuator is used because it is not called perfect active-wave-control anymore. As a subject for future work, the realization of the perfect active wave control supplemented with a shear-force actuator can be pointed out. A noncontact-type shear-force actuator using an electromagnet and so on will be required.</p>

This study considers a measurement method of fractional order derivative responses for the purpose of the vibration control of a mechanical structure. There are some proposed methods to implement fractional order differentiation, but most of them have technical difficulties. The simple method proposed in this paper enables the fractional order derivative responses to be measured by a combination of displacement signals and velocity signals at some locations on the structure. This method is based on the use of the expanded equations of motion involving fractional derivative. The modal matrix of the expanded system relate the integer order state vector and the fractional order state vector. In the case study, we design a measurement system with conventional displacement sensors and velocity sensors for the fractional derivative response of the vibrating cantilever beam. The first simulation results illustrate effectiveness of the proposed method. The second simulation results, considering the use of the fewer conventional sensors, show performance deterioration of the measurement system. However, the system performance is improved by adjusting the positions of the sensors. Each of the sensor positions is evaluated and the optimal position is obtained.

The history of fractional calculus, in which the order of differentiation and integration are expanded to non-integer values, is actually almost as long as that of classical integer-order calculus. In describing an equation of motion, fractional calculus can express larger region of dynamics than integer-order calculus. For this reason, fractional calculus has been applied in various scientific and engineering fields, for example, rheology, electrochemistry, and general transport problems. However, fractional calculus has been used only in a theory, it has been rarely applied to a real engineering target. In this study, as an application of fractional calculus in the control engineering field, a vibration suppression problem is considered based on feedback control using the fractional derivative response of the mechanical structure. Specifically, a vibration control problem of a thin-cantilevered beam is examined to explain the method of constructing a control system. Compared with conventional PD feedback control, which is widely used in industry, fractional PD feedback control is investigated experimentally. A vibration control problem of a flexible cantilevered beam using the fractional PD (Pf-D) feedback is presented as an example.

The history of fractional calculus, in which the orders of differentiation and integration are expanded to non-integer values, is in fact almost as long as that of classical integer-order calculus. For this reason, fractional calculus has been applied in a variety of scientific and engineering fields, for example, rheology, electrochemistry, and general transport problems. In this research, as an application of fractional calculus in the control engineering field, a vibration suppression problem is considered based on feedback control using the fractional derivative response of the mechanical structure. Existing methods to realize a fractional derivative response of a dynamical structure are divided into two major categories. In the analogue approximation method, the signal from a conventional sensor is converted into the fractional derivative response using a special analogue electronic device assembled for post-processing. On the other hand, the digital approximation method converts conventional sensor output into a fractional derivative response using a digital signal processor in which a program based on a discretized fractional calculus definition is loaded. We examine the latter method in this report. In this paper we impose a limitation on the number of the sensors in use, specifically to one, in order to realize a fractional derivative response sensor. Specifically, a vibration control problem of a thin cantilevered beam is taken up to explain how to construct a control system.

We report a development of a micro viscosity sensor, named η-MEMS, which is manufactured by processing technology of micro electro mechanical systems. The small package of the sensor-holder made from stainless steel has a size of a USE memory. The viscosity can be easily measured by dipping the sensor holder into the sample liquid directly. The basic shape of the viscosity sensor is double spiral. In the present study, we evaluated the performance of the USB memory sized holder and new signal processing method. In the new system, the viscosity of the liquid is obtained by detecting the amplitude of the vibration of the sensing spiral. The performance of the holder and the system were evaluated. In experiment with DC bridge amplifier, electromagnetic noise disturbed the appropriate signal processing. On the other hand, AC bridge amplifier decreased the noise and detected the reasonable signal. Finally, The theoretical relationship between the amplitude and viscosity is verified.

This paper reports a MENS based viscosity sennsor suitable for in-process measurement in industrial uses. The sensor is based on the vibrating viscometer. However, the vibrating body is unique dual spiral geometry. The viscosity of the liquid is obtained by curve fitting of the frequency response curve of the spirals. In the present study, we propose an improved chip holder having an electromagnet coil to drive a permanent magnet implanted in the vibrating body of the sensor. The frequency response curve obtained in the experiment shows a good agreement with the theoretical curve.

This paper proposes the measurement system of cubic nonlinear elasticity of tissues. The system consists of a vibrator, a measuring object and a control circuit. We apply a nonlinear feedback proportional to velocity and displacement squared of the vibrator to actuate the vibrator for inducing van der Pol type self-excited oscillation and control the amplitude of the oscillation by changing the nonlinear feedback gain. When the vibrator touches the measuring object, the frequency shift of the vibrator occurs depending on elasticity of the object. When the object has nonlinear elasticity, it causes a natural frequency shift related to the amplitude of the vibrator. Therefore, by the change of the response amplitude based on the variation of the nonlinear feedback gain, we estimate the nonlinear elasticity of the object. We theoretically examine the validity of the above proposed measurement method for the cubic nonlinear elasticity of the object and experimentally confirm the performance for the cubic nonlinear elasticity of the system with magnetic restoring force.

This study aims at improving the precision of measurements with a frequency modulation (FM) mode atomic force microscope (AFM). While an observation with the FM-AFM is carried out, a vibrating microcantilever probe scans over the sample surface. The measurement accuracy depends on how exactly a change in the eigen-frequency of the microcantilever is obtained. The amplitude suppression enables not only to make the oscillation frequency agree exactly with the eigen-frequency, but also to realize a non-contact observation for easily-deformable or easily-damageable specimens. However, it is difficult to cope with both keeping the microcantilever vibrating firmly and making the amplitude of its vibration as small as possible. To overcome this difficulty, a nonlinear feedback is applied to control the amplitude of the microcantilever probe used in the AFM. As a result, it is achieved that the AFM microcantilever can oscillate with a considerably reduced amplitude in the steady state. The key to the success of this nonlinear feedback control method is that the cantilever vibration can be categorized into the van der Pol-type self-excited oscillation. A prototype AFM was built to demonstrate experimentally the advantage of this amplitude control method. To date, the vibration amplitude less than one nanometer has been achieved in the steady state.

When you observe biological samples in liquid by using an Atomic Force Microscopy (AFM), it is difficult to estimate the eigen frequency of microcantilever from the frequency response curve based on external excitation, because of low Quality factor of frequency response curve due to the high viscosity. For the measurement of biological samples which can be deformed on weak impact, it is important subject to oscillate cantilever with a low-amplitude without contact of the tip of the cantilever to the sample. In order to overcome these difficulties, we have proposed a control method by nonlinear feedback and realized the dynamics of van der Pol oscillator in the microcantilever probe; van der Pol oscillator has the response frequency corresponding to the natural frequency and produces a limit cycle due to the nonlinearity and its magnitude determes the steady state amplitude of the oscillator. By applying the linear and nonlinear viscous damping forces based on the feedback of the deflection of the microcantilever, we experimentally establish a van der Pol-type self-excited microcantilever probe. High-gain nonlinear feedback carries out a low response amplitude with 1.5nm. It is confirmed that the response frequency is approximately equal to the natural frequency of the microcantilever probe.