A Comparison between Integer and Fractional Order PD^μ Controllers for Vibration Suppression

The fractional order PD controller transfer function is

HFO-PD(s)=kp(1+kdsμ),$$\begin{array}{} \displaystyle {{\rm{H}}_{{\rm{FO - PD}}}}(s) = {k_p}{\mkern 1mu} (1 + {k_d}{\mkern 1mu} {s^\mu }), \end{array}$$(5)

where μ is the fractional order of the differentiator belonging to the interval (0,2], k_p is the proportional gain, k_d is the derivative gain, and s is the Laplace variable. If μ = 1 or μ = 2, the controller is an integer order PD controller.

Replacing s with jω, we obtain the complex form of the fractional order controller:

HFO-PD(jω)=kp[1+kdωμ(cos(πμ2)+jsin(πμ2))].$$\begin{array}{} \displaystyle {{\rm{H}}_{{\rm{FO - PD}}}}(j\omega ) = {k_p}{\mkern 1mu} \left[ {1 + {k_d}{\mkern 1mu} {\omega ^\mu }\left( {\cos \left( {\frac{{\pi \mu }}{2}} \right) + j\sin \left( {\frac{{\pi \mu }}{2}} \right)} \right)} \right]. \end{array}$$(6)

Replacing Eq. (6) in (4) gives

|kp[1+kdωcgμ(cos(πμ2)+jsin(πμ2))]|=1|G(jωcg)|∠[1+kdωcgμ(cos(πμ2)+jsin(πμ2))]=−π+φm−∠G(jωcg)d(∠[1+kdωμ(cos(πμ2)+jsin(πμ2))])dω|ω=ωcg=−d(∠G(jω))dω|ω=ωcg.$$\begin{array}{} \displaystyle \begin{array}{*{20}{c}} {|{k_p}[1 + {k_d}\omega _{cg}^\mu (\cos (\frac{{\pi \mu }}{2}) + j\sin (\frac{{\pi \mu }}{2}))]|} \hfill & = \hfill & {\frac{1}{{|G(j{\omega _{cg}})|}}} \hfill \\ {\angle [1 + {k_d}\omega _{cg}^\mu (\cos (\frac{{\pi \mu }}{2}) + j\sin (\frac{{\pi \mu }}{2}))]} \hfill & = \hfill & { - \pi + {\varphi _m} - \angle G(j{\omega _{cg}})} \hfill \\ {\frac{{d(\angle [1 + {k_d}{\omega ^\mu }(\cos (\frac{{\pi \mu }}{2}) + j\sin (\frac{{\pi \mu }}{2}))])}}{{d\omega }}{|_{\omega = {\omega _{cg}}}}} \hfill & = \hfill & { - \frac{{d(\angle G(j\omega ))}}{{d\omega }}{|_{\omega = {\omega _{cg}}}}.} \hfill \\ \end{array} \end{array}$$(7)

Focusing on the phase equation from Eq. (4) and expanding it further leads to

kdωcgμsin(πμ2)1+kdωcgμcos(πμ2)=tan(−π+φm−∠G(jωcg)),$$\begin{array}{} \displaystyle \frac{{{k_d}{\mkern 1mu} \omega _{cg}^\mu {\mkern 1mu} \sin \left( {\frac{{\pi \mu }}{2}} \right)}}{{1 + {k_d}{\mkern 1mu} \omega _{cg}^\mu {\mkern 1mu} \cos \left( {\frac{{\pi \mu }}{2}} \right)}} = \tan ( - \pi + {\varphi _m} - \angle G(j{\omega _{cg}})), \end{array}$$(8)

while expanding the robustness condition leads to

μkdωcgμ−1sin(πμ2)1+2kdωcgμcos(πμ2)+kd2ωcg2μ=−d(∠G(jω))dω|ω=ωcg.$$\begin{array}{} \displaystyle \frac{{\mu {\mkern 1mu} {k_d}{\mkern 1mu} \omega _{cg}^{\mu - 1}\sin \left( {\frac{{\pi \mu }}{2}} \right)}}{{1 + 2{k_d}{\mkern 1mu} \omega _{cg}^\mu \cos \left( {\frac{{\pi \mu }}{2}} \right) + k_d^2\omega _{cg}^{2\mu }}} = - \frac{{{\rm{d}}(\angle G(j\omega ))}}{{{\rm{d}}\omega }}{|_{\omega = {\omega _{cg}}}}. \end{array}$$(9)

Determining the controller parameters k_p, k_d and μ is done by solving the equations. The two parameters k_d and μ can be determined by solving the system of equations composed by Eqs. (8) and (9). An easy approach to system solving is the graphic method which consists of plotting k_d as functions of μ, based on Eqs. (8) and (9), and knowing that the solution is found at the intersection of these two plots.

After the derivative gain k_d and the fractional order of the differentiator μ are determined, k_p is easily computed using the following equation, derived from the magnitude condition in Eq. (7):

kp=1|G(jωcg)|11+2kdωcgμcos(πμ2)+kd2ωcg2μ.$$\begin{array}{} \displaystyle {k_p} = \frac{1}{{|G(j{\omega _{cg}})|}}\frac{1}{{\sqrt {1 + 2{k_d}{\mkern 1mu} \omega _{cg}^\mu {\mkern 1mu} \cos \left( {\frac{{\pi \mu }}{2}} \right) + k_d^2{\mkern 1mu} \omega _{cg}^{2\mu }} }}. \end{array}$$

The tuning of the integer order PD controller is done similarly to the fractional order one starting from its transfer function:

HPD(s)=kpTds+1TNs+1,$$\begin{array}{} \displaystyle {{\rm{H}}_{{\rm{PD}}}}(s) = {k_p}{\mkern 1mu} \frac{{{T_d}{\mkern 1mu} s + 1}}{{{T_N}{\mkern 1mu} s + 1}}, \end{array}$$(10)

where T_d is the derivative time constant and T_N is the filter time constant. For a derivative effect, it is imposed that T_N is much smaller than T_d. The integer order PD controller in Eq. (10) has exactly the same tuning parameters as the fractional order controller in Eq. (5). Thus, the same performance specifications at Eqs. (1)-(2) are used to tune the integer order PD controller. Using the phase condition in (4), it follows that:

∠(kpTdjωcg+1TNjωcg+1)=−π+φm−∠G(jωcg).$$\begin{array}{} \displaystyle \angle \left( {{k_p}{\mkern 1mu} \frac{{{T_d}j{\omega _{cg}} + 1}}{{{T_N}j{\omega _{cg}} + 1}}} \right) = - \pi + {\varphi _m} - \angle G(j{\omega _{cg}}). \end{array}$$(11)

Denoting the right-hand side of Eq. (11) as z = tan(−π + φ_m − ∠G(jω_cg)), the last equation can be expressed as:

TD(ωcg−zTNωcg2)−TNωcg+z=0.$$\begin{array}{} \displaystyle {T_D}({\omega _{cg}} - z{T_N}\omega _{cg}^2) - {T_N}{\omega _{cg}} + z = 0. \end{array}$$(12)

The robustness condition in Eq. (4) leads to

Td1+Td2ωcg2−TN1+TN2ωcg2=−∠G(jωcg)dω|ω=ωcg.$$\begin{array}{} \displaystyle \frac{{{T_d}}}{{1 + T_d^2\omega _{cg}^2}} - \frac{{{T_N}}}{{1 + T_N^2\omega _{cg}^2}} = - \frac{{\angle G(j{\omega _{cg}})}}{{{\rm{d}}\omega }}{|_{\omega = {\omega _{cg}}}}. \end{array}$$(13)

Denoting −∠G(jωcg)dω|ω=ωcg=x$\begin{array}{} \displaystyle - \frac{{\angle G(j{\omega _{cg}})}}{{{\rm{d}}\omega }}{|_{\omega = {\omega _{cg}}}} = x \end{array}$, the last equation in Eq. (13) leads to

Td2TNωcg2−Td(1+TN2ωcg2)+TN+x=0.$$\begin{array}{} \displaystyle T_d^2{T_N}\omega _{cg}^2 - {T_d}(1 + T_N^2\omega _{cg}^2) + {T_N} + x = 0. \end{array}$$(14)

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:

|kpTdjωcg+1TNjωcg+1|=1|G(jωcg)|,$$\begin{array}{} \displaystyle \left| {{k_p}{\mkern 1mu} \frac{{{T_d}j{\omega _{cg}} + 1}}{{{T_N}j{\omega _{cg}} + 1}}} \right| = \frac{1}{{|G(j{\omega _{cg}})|}}, \end{array}$$

which can be used to determine the proportional gain k_p:

kp=TN2ωcg2+1|G(jωcg)|⋅Td2ωcg2+1.$$\begin{array}{} \displaystyle {k_p} = \frac{{\sqrt {T_N^2\omega _{cg}^2 + 1} }}{{|G(j{\omega _{cg}})| \cdot \sqrt {T_d^2\omega _{cg}^2 + 1} }}. \end{array}$$

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.

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 CompactRIO^TM 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 ω_n was determined to be 95 rad/s. The second order model obtained is:

G(s)=1.245s2+1.14s+9025.$$\begin{array}{} \displaystyle G(s) = \frac{{1.245}}{{{s^2} + 1.14s + 9025}}. \end{array}$$(15)

The overlay of the impulse response of G(s) and the experimental data is shown in Fig. 3. For comparison purposes, 6^th and 10^th order models were also identified improving the fit, but also increasing the controller calculus. Thus, the second order approximation given in Eq. (15) is used for controller tuning.

Fig. 3

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 ω_cg = 250 rad/s and the phase margin φ_m = 65, as well as the robustness condition. These frequency domain specifications have been used in order to compute the parameters of both integer and fractional order controllers. Note that the same specifications are used in tuning.

Fig. 4

The controllers that were computed using the method explained in the previous section are:

HFO-PD(s)=465.9⋅(1+1.65s0.728)HPD(s)=9727.8⋅0.0177s+18.7029⋅10−4s+1.$$\begin{array}{} \displaystyle \begin{array}{*{20}{c}} {{{\rm{H}}_{{\rm{FO - PD}}}}(s){\rm{\;}}}&{ = 465.9 \cdot \left( {1 + 1.65{s^{0.728}}} \right)}\\ {{{\rm{H}}_{{\rm{PD}}}}(s)}&{ = 9727.8 \cdot \frac{{0.0177s + 1}}{{8.7029 \cdot {{10}^{ - 4}}s + 1}}.} \end{array} \end{array}$$(16)

Remark. Analyzing the structure of the tuned controllers, it can be observed that the integer order proportional-derivate controller is implemented using a low pass filter, while the fractional order is implemented using the ideal structure. The low pass filter ensures a proper transfer function representation of the integer order PD controller, as well as an equal number of tuning parameters in comparison to the fractional order PD controller, such that the same tuning procedure is applied. In this case, both controllers are expected to achieve similar closed loop performance results.

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.

Fig. 5

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, T_s = 0.001 s. The problems arise in implementing the fractional order transfer function. A novel digital implementation used to approximate fractional order transfer functions to integer order ones [23] consists in generating a function that maps the Laplace operator s to the discrete time operator z⁻¹, followed by the evaluation of the frequency response of the discrete time system. The impulse response of the discrete time fractional order system is computed based on its frequency response. The final step consists in finding a rational discrete time transfer function having the same impulse response as the original discrete time fractional order system.

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.

Fig. 6

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.

Fig. 7

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.

Fig. 8