Modeling and stabilization of eccentric gravity machinery

Abstract

In general, eccentric gravity machinery is a rotation mechanism with eccentric pendulum mechanism, which can be used to convert continuously kinetic energy generated by gravity energy to electric energy. However, a stable rotated velocity of the eccentric gravity machinery is difficult to be achieved only using gravity energy. In this article, a stable velocity control system applied to eccentric gravity machinery is proposed. The dynamic characteristic of eccentric gravity machinery is analyzed and its mathematical model is established, which is used to design the controller. A stable running velocity of the eccentric gravity machinery can be operated by the controlled servomotor. Due to disturbances being periodic, repetitive controller is installed to velocity control loop. The stability performance and control performance of the repetitive control system are discussed. The iterative algorithm of the repetitive control is executed by a digital signal processor TI TMS320C32 floating-point processor. Simulated and experimental results are reported to verify the performance of the proposed eccentric gravity machinery control system.

Keywords

Eccentric gravity machinery servomotor mathematical model repetitive controller

Introduction

Gravity power is a clean and natural energy from inherent characteristics of the earth. Therefore, gravity can be used to many industrial applications, such as puncher, gravity separation machine, and vibration machine. Nowadays, potential/kinetic energy can often be converted to electric power by eccentric gravity machinery (EGM). The converted efficiency of EGM depends on the stable velocity operation. However, without appropriate control system, large varying angular velocity of EGM can be easily generated. Therefore, in this article, a velocity control driven by servomotor is applied to EGM to improve the operated velocity stability. The disturbances of EGM produced from gravity/generator are periodic signals. Therefore, a repetitive controller is adopted to reduce the steady-state error.

In this article, a prototype of EGM is constructed, which is composed of eccentric pendulum mechanism, servomotor, and generator as shown in Figure 1. The 40-kg eccentric pendulum mechanism is composed of flywheel (see Figure 2(a)) and planetary gear set (as shown Figure 2(b)), which is used to generate potential/kinetic energy from the gravity effect. The 1.5-kW servomotor (see Figure 2(c)) connected to planet pinion of planetary gear set is used to control the rotation velocity of the eccentric pendulum mechanism. The motor torque is applied to drive the rotary shaft at a gear ratio of 1:20 and drive the generators by the sun gear with a gear ratio of 12:1. The generator including three 0.4 kW @1200 r/min alternators (as shown in Figure 2(d)) as a loading in EGM can produce electrical power converted by potential/kinetic energy from the motion of eccentric pendulum mechanism. In this case, rated velocity of the generator is 1200 r/min, and the rotated velocity of 100 r/min of the EGM is determined by the gear ratio 12:1. Therefore, the control target is that the EGM has to stable rotated by 100 r/min. The gravity center of flywheel is at top of the planetary gear to set 0°, while the gravity center of flywheel is at bottom of the planetary gear being 180°. As the flywheel being free rotated at 0°, within the first revolution, its velocity curve is measured by encoder as shown in Figure 3. As expected, the maximum velocity of 100 r/min occurs at about 180° and approaches to 1–3 r/min of the rotated velocity near 0°. Due to the large varying angular velocity of the flywheel, the efficiency of the electric energy power generated by EGM system is low. Therefore, a stable rotated velocity of the eccentric pendulum mechanism is required and can be done by the controlled servomotor. In practical, due to viscous friction, for free movement, the maximum rotated velocity of EGM within a time period will be decreased gradually to zero as shown in Figure 4. Therefore, extra energy to maintain stable rotated velocity of EGM system is required.

Figure 1.

Prototyped of EGM.

Figure 2.

Components of EGM: (a) flywheel, (b) planetary gear, (c) servomotor, and (d) generator mechanism.

Figure 3.

Measured velocity of the eccentric pendulum mechanism.

Figure 4.

Measured velocity of EGM decreasing gradually without control.

The dynamic performance of the servo-electromechanical systems is always influenced by uncertainties, for instance, a nonlinear disturbance or mechanical parameters. Among the system parameters such as eccentric inertia or damping coefficient are essential parameters in control design. In order to achieve a stable velocity output of the controlled plant and the specific transient specification, the accuracy and integrity of system modeling are required. The eccentric pendulum being rotated to generate centrifugal force to lead to difficulties in the velocity control, and the centrifugal force of eccentric pendulum produces a periodic torque.

Many literatures to focus on the corresponding control can be found. In De Wit and Praly,¹ rejecting oscillatory position-dependent disturbances with unknown frequency and unknown amplitude was proposed. These disturbances are produced by eccentricity in mechanical systems. In Kugi et al.,² an active compensation of the roll-eccentricity-induced periodic disturbances has been done for the strip exit thickness of hot and cold rolling mills. The factorization approach over the set of stable transfer functions was provided by an adaptive least-mean-squares algorithm. In Gongand and Cao,³ an active hybrid tilting-pad bearing with unsymmetrical flexible rotor and the hydraulic system was presented in this article. A dynamic model with the effect of gyroscopic moment was given. A proportional–integral (PI) controller and the finite difference method were used to improve the pressure distribution on each tilting pad surface. In Qaiser et al.,⁴ the stabilization and disturbance attenuation problems of the oscillating eccentric rotor were studied. With under-actuation property, a controller has been designed to de-spin the rotor and to reduce the translational oscillations. In Zhao et al.,⁵ the eccentric paddle mechanism with epicyclical gear is designed for amphibious search and rescue tasks. A gait planning method for conducting a legged crawl gait was proposed. Analysis of gait sequence and stability margin of the quadruped robot was then presented. In Das et al.,⁶ an active vibration control scheme for reducing transverse vibration of a rotor shaft due to unbalance was proposed. An electromagnetic exciters mounted on the stator at a plane was used for applying suitable force of actuation over an air gap to control transverse vibration. In Yang et al.,⁷ a rotor mass eccentricity compensation control strategy was presented to restrain the vibration of suspended rotor for bearing less induction motor. The suspension rotor dynamical model was deduced, and unbalanced vibration mechanism was analyzed. A compensator was designed to increase the radial force and improve the stiffness of the vibration signal.

Due to the periodic disturbance, a linear controller cannot be used to regulate the steady-state periodic error. Repetitive control has proven to be very effective for a system to reject periodic disturbance signal in practical applications. If the disturbance contains only single fundamental frequency, the repetitive control can be very effective to significantly reduce the steady-state error. Therefore, to stabilize the velocity output of the EGM with periodic disturbances, a repetitive controller is installed to existed velocity control loop in this article. Considering on improving required control performance, a dynamic model of the controlled system is required for the design of the controller parameters. Therefore, an appropriate identification method to determine mathematic model is needed. In this article, closed-loop identification derived from the concept of coprime factorization of the plant model is proposed.

In this article, EGM including eccentric pendulum mechanism, servomotor, and generator is designed. A dynamic characteristic is analyzed to establish the mathematical model of EGM. A stable rotated velocity of EGM can be operated by the controlled servomotor. Based on the periodic disturbances of EGM, a repetitive controller is installed to velocity control loop to reduce steady-state error. Simulated and experimental results are reported to verify the performance of the proposed EGM.

Modeling of EGM

To achieve a desired control performance, a dynamic model of EGM is required for controller design in that the appropriate identification of a mathematic model is the key point. As shown in Figure 5, system modeling is divided into three parts including of EGM system, generator system, and servomotor system. Note that the system parameters are clearly defined and are listed in Table 1. Observing Figure 1, the motor torque is applied to the ring gear, which drives the rotary shaft at a gear ratio of 1:20 and drives the generators by the sun gear with a gear ratio of 12:1. Angular position/velocity of the EGM is measured by optical encoder of 500 pulses per revolution installed in the right side of the rotation shaft.

Figure 5.

Proposed system modeling of EGM in Figure 1.

Table 1.

Definitions of the parameters in Figure 2.

Items	Description
$J_{m}$	Inertia of servomotor
$D_{m}$	Damping coefficient of servomotor
$θ_{m}$	Angular position of servomotor
$T_{m}$	Motor torque output
$D_{cb}$	Damping coefficient of mechanical coupling between servomotor and EGM
$K_{cb}$	Stiffness coefficient of mechanical coupling between servomotor and EGM
$J_{la}$	Equivalent inertia of the left side of EGM
$D_{la}$	Damping coefficient of the left side of EGM
$θ_{la}$	Angular position of left side of EGM
$K_{l}$	Stiffness coefficient of rotation shift
$J_{lb}$	Equivalent inertia of the right side of EGM
$D_{lb}$	Damping coefficient of the right side of EGM
$θ_{lb}$	Angular position of right side of EGM
$T_{d}$	Torque output of EGM
$J_{g}$	Inertia of generator
$D_{g}$	Damping coefficient of generator
$K_{cg}$	Stiffness coefficient of mechanical coupling between generator and EGM
$D_{cg}$	Damping coefficient of mechanical coupling between generator and EGM
$θ_{g}$	Angular position of generator
$T_{g}$	Torque of generator

EGM: eccentric gravity machinery.

In this case of Figure 1, the inertia of rotation shift can be divided into two segments and assume the damping inside the shift being negligible. As shown in Figure 6, the torque $T_{d}$ is a periodic signal. Note that the maximum magnitude of $T_{d}$ is about 260 N m at 90° and 270°. The torque of $T_{g}$ is a function of rotation speed as shown in Figure 7, which can be determined by MATLAB Curve Fitting Toolbox with high-order polynomials, that is, $T_{g} = 6.959 \times 10^{- 9} {\overset{\cdot}{θ}}_{g}^{3} - 1.114 \times 10^{- 5} {\overset{\cdot}{θ}}_{g}^{2} + 6.288 \times 10^{- 3} {\overset{\cdot}{θ}}_{g} + 0.0558$ . The information of the disturbances in Figures 6 and 7 can be used to design the controller parameters in the following.

Figure 6.

Plot of $T_{d}$ being function of $θ_{lb}$ .

Figure 7.

Plot of $T_{g}$ being function of ${\overset{\cdot}{θ}}_{g}$ .

From Figure 5, the dynamic equations of the EGM system are obtained in the following.

For the servomotor system, the dynamic equation is given as

J_{m} {\overset{\cdot\cdot}{θ}}_{m} + D_{m} {\overset{\cdot}{θ}}_{m} + K_{cb} (θ_{m} - θ_{la}) + D_{cb} ({\overset{\cdot}{θ}}_{m} - {\overset{\cdot}{θ}}_{la}) = T_{m}

For the generator, the dynamic equation is given as

J_{g} {\overset{\cdot\cdot}{θ}}_{g} + D_{g} {\overset{\cdot}{θ}}_{g} + K_{cg} (θ_{g} - θ_{la}) + D_{cg} ({\overset{\cdot}{θ}}_{g} - {\overset{\cdot}{θ}}_{la}) = T_{g}

For right side of the eccentric pendulum mechanism, the dynamic equation is given as

J_{lb} {\overset{\cdot\cdot}{θ}}_{lb} + D_{lb} {\overset{\cdot}{θ}}_{lb} + K_{l} (θ_{lb} - θ_{la}) = T_{d}

For left side of the eccentric pendulum mechanism, the dynamic equation is given as

\begin{matrix} J_{la} {\overset{\cdot\cdot}{θ}}_{la} + D_{la} {\overset{\cdot}{θ}}_{la} + K_{l} (θ_{la} - θ_{lb}) + K_{cg} (θ_{la} - θ_{g}) \\ + D_{cg} ({\overset{\cdot}{θ}}_{la} - {\overset{\cdot}{θ}}_{g}) + K_{cb} (θ_{la} - θ_{m}) + D_{cb} ({\overset{\cdot}{θ}}_{la} - {\overset{\cdot}{θ}}_{m}) = 0 \end{matrix}

Laplace transformation of the above equations can be

\begin{array}{l} Φ_{m} = \frac{D_{c b} s + K_{c b}}{J_{m} s^{2} + D_{m} s + D_{c b} s + K_{c b}} Φ_{l a} + \frac{{\tilde{T}}_{m}}{J_{m} s^{2} + D_{m} s + D_{c b} s + K_{c b}} \\ Φ_{g} = \frac{D_{c g} s + K_{c g}}{J_{g} s^{2} + D_{g} s + D_{c g} s + K_{c g}} Φ_{l a} - \frac{{\tilde{T}}_{g}}{J_{g} s^{2} + D_{g} s + D_{c g} s + K_{c g}} \\ Φ_{l b} = \frac{K_{l}}{J_{l b} s^{2} + D_{l b} s + K_{l}} Φ_{l a} + \frac{{\tilde{T}}_{d}}{J_{l b} s^{2} + D_{l b} s + K_{l}} \\ Φ_{l a} = \frac{D_{c b} s + K_{c b}}{J_{l a} s^{2} + (D_{l a} + D_{c b} + D_{c g}) s + (K_{c b} + K_{l} + K_{c g})} Φ_{m} \\ + \frac{K_{l}}{J_{l a} s^{2} + (D_{l a} + D_{c b} + D_{c g}) s + (K_{c b} + K_{l} + K_{c g})} Φ_{l b} \\ + \frac{D_{c g} + K_{c g}}{J_{l a} s^{2} + (D_{l a} + D_{c b} + D_{c g}) s + (K_{c b} + K_{l} + K_{c g})} Φ_{g} \end{array}

where $Φ_{m}$ , $Φ_{g}$ , $Φ_{la}$ , $Φ_{lb}$ , ${\tilde{T}}_{m}$ , ${\tilde{T}}_{g}$ , and ${\tilde{T}}_{d}$ are Laplace transformation of $θ_{m}$ , $θ_{g}$ , $θ_{la}$ , $θ_{lb}$ , $T_{m}$ , $T_{g}$ , and $T_{d}$ , respectively. According to the equations mentioned above, the block diagram of the controlled system can be obtained as Figure 8.

Figure 8.

Control block diagram of the controlled system in Figure 5.

The accuracy parameters in Figure 8 are difficult to be measured, such as the spring and damping coefficients of mechanical couplings. Assume that the characteristics of the mechanical couplings between servomotor, EGM, and planetary gear set are rigid. Due to the inertia of EGM larger than that of generator and planetary gear set, the inertias of generator, EGM, and planetary gear set can be integrated and are simplified by Figure 9, where $J_{l}$ and $D_{l}$ are the equivalent inertia and damping coefficient of simplified EGM, respectively; K and D are the equivalent stiffness and damping coefficients of the mechanical coupling between servomotor and simplified EGM.

Figure 9.

Simplified model of Figure 8.

Then, the dynamic equations of the system in Figure 9 is given by

\begin{array}{l} (J_{l} s + D_{l}) Ω_{g} + D (Ω_{g} - Ω_{m}) + K (Ω_{g} - Ω_{m}) / s \\ = {\tilde{T}}_{d} + {\tilde{T}}_{g} (J_{m} s + D_{m}) Ω_{m} + D (Ω_{m} - Ω_{g}) + K (Ω_{m} - Ω_{g}) / s = {\tilde{T}}_{m} \\ Ω_{g} = \frac{s}{(J_{l} s^{2} + D_{l} s + D s + K) (J_{m} s^{2} + D_{m} s + D s + K) - {(D s + K)}^{2}} {\tilde{T}}_{m} \\ + \frac{s (J_{m} s^{2} + D_{m} s + D s + K) ({\tilde{T}}_{d} + \tilde{T})}{(J_{l} s^{2} + D_{l} s + D s + K) (J_{m} s^{2} + D_{m} s + D s + K) - {(D s + K)}^{2}} \end{array}

where $Ω_{m}$ and $Ω_{g}$ are Laplace transformations of ${\overset{\cdot}{θ}}_{m}$ and ${\overset{\cdot}{θ}}_{g}$ , respectively. From the above derived results, the controlled system is a four-order mathematic form with input ${\tilde{T}}_{m}$ and output $Ω_{g}$ , where four poles and single zero can be found. According to the equations mentioned above, the block diagram of the simplified controlled system can be obtained as Figure 10.

Figure 10.

Control block diagram of the controlled system in Figure 8.

As indicated in Figure 10, with the output $T_{m}$ of the control rule, the control target is a stable control of ${\overset{\cdot}{θ}}_{g}$ under the effect of torque disturbances $T_{d}$ and $T_{g}$ . An accuracy mathematical model can be obtained by closed-loop identification, which is motivated by the insight into the dynamic interactions described with a linear representation. Consider closed-loop configuration as depicted in Figure 11, where G and C are defined as the controlled systems to be identified and a stabilizing controller, respectively. Assuming the input signal r can be measured and d is disturbance, based on Figure 11, the frequency responses of $y / r = P_{yr} (= CG / (1 + CG))$ and $u / r = P_{ur} (= C / (1 + CG))$ can be measured by dynamic spectrum analyzer. Then, an estimated G can be obtained as $G = P_{yr} P_{ur}^{- 1}$ , where C has no unstable zeros. This identification method based on the closed-loop data is derived from the concept of coprime factorization of the plant model.

Figure 11.

Closed-loop system identification.

In the closed-loop system identification of this illustrated example, the rotation velocity of the generator $y = {\overset{\cdot}{θ}}_{g}$ and the motor torque input $u = T_{m}$ are given. A proportional–integral–derivative controller (PID controller) $C (s) = 32 + (0.1 / s) + 213 s$ is determined to stabilize the closed-loop control system. A swept sine excitation signal r can be generated by dynamic spectrum analyzer. The frequency responses of $P_{yr} = y (= {\overset{\cdot}{θ}}_{g}) / r$ and $P_{ur} = u (= T_{m}) / r$ can be measured simultaneously and to calculate the frequency response of G. In this case, four times measured results of the closed-loop identification are given as shown in Figure 12. Based on the analysis of Figure 12, an appropriate result can be determined by curve fitting, that is

G (s) = \frac{s}{365.45 s^{4} + 19.12 s^{3} + 0.256 s^{2} + 0.0015 s + 0.00001}

Figure 12.

Closed-loop identification of the illustrated example.

Repetitive controller design to stabilize velocity of EGM

Owing to the result of the system modeling, a velocity controller is designed to achieve stable rotation of EGM. The torques caused by eccentric pendulum and the generator are periodic disturbances; therefore, a repetitive controller is installed to existed velocity control loop as shown in Figure 13. For a controlled plant of $G (s)$ and a pre-determined controller of $C (s)$ , an unity feedback system $G_{p} (s) / 1 + G_{p} (s)$ is internally stable where $G_{p} (s) = C (s) G (s)$ . The repetitive controller of $K_{q} (s)$ and $K_{b} (s)$ can be designed by Yao and colleagues,^8,9 where $K_{q} (s) = 1 / ((s^{2} / ω_{n}^{2}) + (2 ζ s / ω_{n}) + 1)$ and $K_{b} (s) = e^{τ_{b} s}$ are given. Based on the design rule^8,9 of $K_{q} (s)$ and $K_{b} (s)$ , the magnitude of the sensitivity function in Figure 13 can be reduced at the harmonic frequencies of the reference signal within the designed frequency range $ω_{n}$ of $K_{q} (s)$ .

Figure 13.

Velocity control loop with repetitive controller.

Consider the repetitive control system of Figure 13, where the sensitivity function is $S_{γ} (s) = (1 - K_{q} e^{- s τ_{d}}) / (1 - K_{q} e^{- s τ_{d}} + G_{p} (1 - K_{q} e^{- s τ_{d}} + K_{b} K_{q} e^{- s τ_{d}}))$ . Then, the tracking error is given by $E (s) = S_{γ} (s) [R (s) - T_{l} (s)]$ or equivalently $E (s) = E_{0} ((1 - K_{q} e^{- s τ_{d}}) / (1 - {\hat{S}}_{0} K_{q} e^{- s τ_{d}}))$ , where $E_{0} = ((R (s) - T_{l} (s)) / (1 + G_{p} (s)))$ , $S_{0} = (1 / (1 + G_{p} (s)))$ , and ${\hat{S}}_{0} = 1 - ((K_{b} (s) G_{p} (s)) / (1 + G_{p} (s)))$ . Note that $E_{0} (s)$ denotes the tracking error of Figure 13 without repetitive controller. Based on the analysis results of Yao and colleagues,^8,9 if the main harmonics of r and $T_{l}$ are all within the designed bandwidth, the steady-state tracking error will be attenuated significantly. Assume that the regeneration spectrum^8,9 satisfies $| K_{q} {\hat{S}}_{0} (j ω) | < 1$ for all $ω$ , that is, the stability condition is satisfied. The tracking error with repetitive controller can be represented as

E (s) = E_{0} {1 + [{\hat{S}}_{0} - 1] K_{q} e^{- s τ_{d}} + [{\hat{S}}_{0} - 1] {\hat{S}}_{0} K_{q}^{2} e^{- s 2 τ_{d}} + [{\hat{S}}_{0} - 1] {\hat{S}}_{0}^{2} K_{q}^{3} e^{- s 3 τ_{d}} + \dots}

or equivalently, in the time domain

e (t) = e_{0} (t) + \sum_{k = 1}^{\infty} L^{- 1} {E_{0} [{\hat{S}}_{0} - 1] {\hat{S}}_{0}^{k - 1} K_{q}^{k} e^{- sk τ_{d}}}

which can give the tracking error $e (t) = e_{0} (t) = L^{- 1} [E_{0} (s)]$ for $t \in [0, τ_{d}]$ . Note that $L^{- 1} [•]$ denotes the inverse Laplace transform operator. Obviously, the repetitive controller does not affect the tracking response in the first time period, that is, $t \in [0, τ_{d}]$ . Therefore, the steady-state tracking error being after the second time period of the input can be improved under the designed parameters $K_{q} (s)$ and $K_{b} (s)$ .

From Figure 13, the iterative algorithm of the repetitive controller can be constructed with sampling time $τ_{s}$ , where the periodic signal period $τ_{d} = N_{d} τ_{s}$ can be given. Note that an integer number $N_{d}$ is valid for a constant period $τ_{d}$ . The z-transform of the repetitive control system in Figure 13 is considered as Figure 14, where the designed parameters of ${\hat{K}}_{q} (z)$ and ${\hat{K}}_{b} (z)$ can be, respectively, given as

{\hat{K}}_{q} (z) = \frac{1}{\sqrt{1 - ζ^{2}}} \frac{z e^{- ζ ω_{n} τ_{s}} \sin (τ_{s} ω_{n} \sqrt{1 - ζ^{2}})}{z^{2} - 2 z e^{- ζ ω_{n} τ_{s}} \cos (τ_{s} ω_{n} \sqrt{1 - ζ^{2}}) + e^{- 2 ζ ω_{n} τ_{s}}}

and ${\hat{K}}_{b} (z) = z^{N_{b}}$ , where $N_{b} = τ_{b} / τ_{s}$ .

Figure 14.

Discrete-time repetitive control scheme.

The sensitivity function of Figure 14 is obtained as

\begin{matrix} \frac{E (z)}{R (z)} = \\ \frac{1 - {\hat{K}}_{q} (z) z^{- N_{d}}}{1 - {\hat{K}}_{q} (z) z^{- N_{d}} + {\hat{G}}_{p} (z) (1 - {\hat{K}}_{q} (z) z^{- N_{d}} + {\hat{K}}_{q} (z) z^{- N_{d} + N_{b}})} \\ = \frac{1 - {\hat{K}}_{q} (z) z^{- N_{d}}}{1 + {\hat{G}}_{p} (z)} \frac{1}{1 + (1 - \frac{z^{- N_{b}} {\hat{G}}_{p} (z)}{1 + {\hat{G}}_{p} (z)}) {\hat{K}}_{q} (z) z^{- N_{d}}} \end{matrix}

Clearly, the stability of the control system can be analyzed by consequence of the stability of all cascaded parts¹⁰ including $1 - {\hat{K}}_{q} (z) z^{- N_{d}}$ , $1 / (1 + {\hat{G}}_{p} (z))$ , and $1 / (1 + (1 - ((z^{- N_{b}} {\hat{G}}_{p} (z)) / (1 + {\hat{G}}_{p} (z)))) {\hat{K}}_{q} (z) z^{- N_{d}})$ . The third item plays an important role for the stability of the control system, where we can have

\frac{1}{1 + (1 - \frac{z^{- N_{b}} {\hat{G}}_{p} (z)}{1 + {\hat{G}}_{p} (z)}) {\hat{K}}_{q} (z) z^{- N_{d}}} = \frac{1}{1 + (1 - \hat{T} (z) z^{- N_{b}}) {\hat{K}}_{q} (z) z^{- N_{d}}}

By the small gain theorem, its stability can be assured for

‖ (1 - \hat{T} (z) z^{- N_{b}}) {\hat{K}}_{q} (z) ‖_{\infty} < ‖ (1 - \hat{T} (z) z^{- N_{b}}) ‖_{\infty} ‖ {\hat{K}}_{q} (z) ‖_{\infty} < ‖ {\hat{K}}_{q} (z) ‖_{\infty} < 1

with the corresponding open-loop transfer function internal stability, where $‖ • ‖_{\infty}$ denotes the $H_{\infty} - norm$ .

In this illustrated example, the period $τ_{d} (= 0.6 s)$ of the disturbance is measured from this case of Figures 6 and 7. Based on Figure 13, the controlled plant of $G_{p} (s) = C (s) \hat{G} (s) = (213 s^{3} + 32 s^{2} + 0.1 s) / (365.45 s^{4} + 19.12 s^{3} + 0.256 s^{2} + 0.0015 s + 0.00001)$ can be obtained. For the design of ${\hat{K}}_{q} (z)$ and ${\hat{K}}_{b} (z)$ , the parameters of $ζ = \sqrt{2} / 2$ , $τ_{b} = 0.011 s$ , and $ω_{n} = 20 rad / s = 3.2 Hz$ can be determined. Under the required EGM velocity output of 100 r/min, the simulated error responses of the control system with/without the repetitive controller are shown in Figure 15. It can be seen that the tracking error of the proposed repetitive control is smaller, and its error decays rapidly within the first two cycles of the reference signal. In the illustrated example, the magnitude plot of the sensitivity function of Figure 13 is shown in Figure 16. The magnitude of the sensitivity function is decreased, even close to zero, within the first two harmonics, that is, $k ω_{0} < ω_{n} (= 3.2 Hz)$ , $ω_{0} = 1 / τ_{d} = (1 / 0.6) Hz$ , $k = 1, 2$ . From the comparisons of the error responses with/without repetitive controller in Figure 15, the results show that the magnitude of the error can be reduced significantly after first time period (i.e. 0.6 s).

Figure 15.

Error responses of the control system with (solid line)/without (dashed line) repetitive controller.

Figure 16.

Magnitude plot of sensitivity function of Figure 13.

Experimental results

The iterative algorithm of the repetitive control is executed by a digital signal processor TI TMS320C32 floating-point processor, where the sampling period was set to $τ_{s} = 0.001 s$ . The iterative algorithm is obtained as shown in Figure 17, where the repetitive controller of ${\hat{K}}_{q} (z) = (\sqrt{2} e^{- 0.01 \sqrt{2}} \sin (0.02 \sqrt{2}) z) (z^{2} - 2 z e^{- 0.01 \sqrt{2}} \cos (0.02 \sqrt{2}) + e^{- 0.02 \sqrt{2}})$ and ${\hat{K}}_{b} (z) = z^{11}$ can be determined.

Figure 17.

Iterative algorithm of repetitive control rule.

The control rule can be rewritten as

u (k + 1) = u (k) + e (k) + y_{q} (k - N)

where $y_{q} (k) = K_{q} (z) u_{q} (k)$ , $u_{q} (k) = e (k) + y_{q} (k - N - N_{b})$ , $e (k) = r (k) - y (k)$ , and $N = N_{d} - N_{b}$ .

In experimental results, a steady-state velocity outputs of EGM with repetitive controller (solid line), with PID controller (dashed line), and without control (dash-dot line) are shown in Figure 18. From Figure 18, the rotation velocity of EGM can be driven to required velocity of 100 r/min with PID velocity controller; however, large velocity variation is still produced. With the repetitive controller, a stable velocity of 100 r/min can be done.

Figure 18.

Velocity outputs of EGM with repetitive controller (solid line), only with PID controller (dashed line), and without control (dash-dot line).

Conclusion

In this article, an EGM with 12-kW generator is constructed and composed of eccentric pendulum mechanism and servomotor, which can be applied to convert continuously kinetic energy generated by gravity energy to electric energy. A stable velocity control system with repetitive controller applied to EGM is required for a stable power output. Therefore, the dynamic characteristic of EGM is analyzed, and its mathematical model is established. Moreover, the torques of EGM and generator are measured, and their characteristics are described. Due to periodic characteristics of disturbance, a repetitive controller is designed and is installed to velocity control loop. The stability performance and control performance of the repetitive control system are discussed. The iterative algorithm of the repetitive control is executed by a digital signal processor TI TMS320C32 floating-point processor. Simulated and experimental results are reported to verify the performance of the proposed EGM. Under the required EGM velocity output of 100 r/min, its error decays rapidly within the first two cycles of the reference signal.

Footnotes

Handling Editor: Yong Chen

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Wu-Sung Yao

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