By suppressing unwanted vibrations in real life applications, significant improvements can be made in increasing safety and reducing discomfort for the passengers of an airplane, spectators on a stadium or construction workers on a bridge. Vibrations can be mitigated passively by means of tuned mass dampers that increase the mass and stiffness of flexible structures [1], or actively by means of sensors and actuators.
Many active control algorithms have been developed and the efficiency of the closed loop system depends on the chosen algorithm. Direct proportional feedback [2], constant gain velocity feedback control [3], optimal control strategies [4, 5], fuzzy logic controllers [6–8], robust control [9], and neural networks [10] are just a few of the approaches embraced to mitigate vibrations.
In the case of smart beam vibration suppression, fractional order PID controllers were proposed [11,12] with good results, as well as fractional order PD controllers [13–15]. In the case of the fractional PID, its complexity represents a great disadvantage since there are five parameters that need to be tuned leading to solving a system of minimum five equations [16]. Alternative tuning methods are presented in [12] and [13] based on lowering the resonant peak on the frequency domain magnitude plot using optimization algorithms. The method proves useful in bringing a dramatic improvement in terms of settling time, but it does not tackle the robustness of the closed loop system.
In this study, two Proportional Derivative controllers were tuned using the same tuning method, making them eligible for an accurate comparison. Both controllers were tuned based on frequency domain specifications such as the gain crossover frequency, phase margin and robustness to gain variations. For the gain crossover frequency and the phase margin certain values are imposed, while the robustness is ensured by forcing the phase of the open loop system to be a straight line near the crossover frequency [17]. This type of tuning procedure has been discussed before, with numerous applications [16, 18–22]. However, its application to the problem of active smart beam vibration control is an original element of this manuscript.
The obtained controllers were tested on an experimental vibration stand built at the Technical University of Cluj-Napoca, Romania. The impulse responses of the closed-loop of the system were compared in terms of settling time and robustness.
The article is structured as follows: in the second section the tuning equations are detailed, in the third section the practical stand and the experimental results are presented, while the fourth section consists of conclusions.
In the study, two PD controllers were tuned with the purpose of suppressing unwanted vibrations. The parameters of both controllers were computed based on constraints imposed in the frequency domain: gain crossover frequency
Knowing that the value of the magnitude at the gain crossover frequency is 1 (0 dB), the gain crossover specification can be mathematically written as
The equation for the phase margin constraint is expressed using the phase equation for the open loop system as:
The last specification imposed in the calculus of the controllers is the robustness. This condition guarantees an almost constant overshoot when the gain varies. One way to ensure this requirement is by imposing a constant phase in the interval grasping the gain crossover frequency. A constant phase is characterized by a straight line on the phase plot. This means that the derivative of the phase will be 0 when
For a process described by the transfer function noted
Rewriting the frequency domain specifications (1)-(2) in the complex plane with respect to (3), the constraints can be re-written as
The tuning of the controllers is made by replacing
The fractional order PD controller transfer function is
where
Replacing
Replacing Eq. (6) in (4) gives
Focusing on the phase equation from Eq. (4) and expanding it further leads to
while expanding the robustness condition leads to
Determining the controller parameters
After the derivative gain
The tuning of the integer order PD controller is done similarly to the fractional order one starting from its transfer function:
where
Denoting the right-hand side of Eq. (11) as
The robustness condition in Eq. (4) leads to
Denoting
Then, based on Eq. (12) and (14), the derivative and filter time constants can be easily computed. Once these have been determined, using the magnitude condition in (4), the following result is obtained:
which can be used to determine the proportional gain
Since both controllers were tuned with similar methods based on imposed frequency domain specifications, in order to properly analyze and compare their real life behavior, both were tested on the same equipment under the same conditions.
The schematic of the closed loop system is depicted in Fig. 1. Since the main purpose of the controller is to cancel the oscillations of the beam, the reference position will always be kept at 0. The measured structural displacement is given to the controller, which computes the control signal
The experimental setup has been entirely developed at the Technical University of Cluj-Napoca, Romania. The most important part of the stand is the smart beam which is 240 mm long, 40 mm broad, and 3 mm narrow. The beam is fixed on one end, while the other is allowed to vibrate freely. The beam is “smart” because its location is permanently known with the help of two Honeywell SS495A Miniature Ratiometric Linear (MRL) Hall Effect sensor. The experimental setup can be seen in Fig. 2.
On the surface of the beam there are two magnets that were hot pressed into its surface. The position of the smart beam is controlled by controlling the current/voltage through the coils which controls the magnetic flux, attracting the magnets, hence the beam. The coils can only attract the beam without offering the possibility to reject it. Only Coil 1 is used for vibration attenuation, while Coil 2 is used to give measurable impulse type disturbances. The reason that Coil 1 is used for control is the fact that it is closer to the fixed end of the beam, making it easier to stop the beam’s movements.
The National Instruments modules NI 9215 and NI 9263 are used to read data from the sensors and to control the magnetic flux in the coils. The control algorithm was implemented using LabVIEWTM and the real time CompactRIOTM 9014 controller. The sample rate is 1 ms.
The model was experimentally identified as a second order transfer function based on data acquired by giving an impulse on Coil 1. A swept sine having frequencies in the range [5,60] Hz has been applied to the beam. Based on the measured data, the resonant frequency of the beam has been determined to be at 15 Hz. Transforming 15 Hz to rad/s, the natural frequency
The overlay of the impulse response of
The Bode diagram of the uncompensated structure is illustrated in Fig. 4. Both controllers were designed based on frequency domain constraints. The imposed gain crossover frequency was
The controllers that were computed using the method explained in the previous section are:
The frequency domain response of the open loop with both controllers is plotted in Fig. 5. As can be seen, both the gain crossover and the phase margin specifications are honored. The main difference between the controllers is the robustness specification. When the tuning was performed, the phase line including the phase margin was imposed to be a flat line. The fractional order controller makes the phase flat on a longer interval than the integer order one, leading, in this particular situation, to a more robust system to gain variations. This will be further analyzed in the impulse response of the closed loop system and by performing a simple practical robustness test.
In order to implement the controllers on the practical vibration unit, the digital forms of the two controller transfer functions in Eq. (16) have to be obtained first. Since the sensors read the position of the beam every 1 ms, the sampling time of the discretization has to be kept the same,
The practical results with the fractional order controller can be seen in Fig. 6. In the left plot, the impulse response of the uncompensated structure is presented. As it can be seen, the settling time is greater than 6 seconds. In the center plot, the impulse response of the closed loop system with the HFO-PD controller for two consecutive tests is plotted, while in the right plot a zoomed impulse response is shown for a more detailed analysis. The closed loop settling time is less than 200 ms.
The results obtained with the integer order controller can be seen in Fig. 7. The center and the zoomed impulse response plots (right) illustrate a greater settling time than in the case of the fractional controller. The number of oscillations until the vibration is completely cancelled is also greater. Comparing the zoomed impulse responses for both controllers, it is clear that the fractional order controller is superior to the integer order one.
In order to practically compare the robustness of the two controllers a simple test was performed. Near the free end of the beam, a weight of 2 grams was attached bending and changing the initial position of the beam and indirectly the gain of the identified second order transfer function. The beam was then subjected to the same impulse excitation and the results can be seen in Fig. 8. Once more, the fractional order controller has an outstanding performance.
In this study, a fractional order and an integer order controller were tuned for vibration attenuation in a smart beam using the same method and the same frequency domain constraints: gain crossover frequency, phase margin and robustness. The robustness specification was imposed indirectly by imposing a flat phase around the crossover frequency. The experimental results, obtained by applying an impulse disturbance to the free end of the smart beam, demonstrate that the fractional order controller is more robust than the integer order one. Also, the fractional order controller mitigates vibrations in less time, making it the better choice to solve vibration attenuation problems.