ANALOG FRACTIONAL ORDER CONTROLLER IN A TEMPERATURE CONTROL APPLICATION

Proceedings of the 2nd IFAC Workshop on Fractional Differentiation and its Applications Porto, Portugal, July 19-21, 2006

ANALOG FRACTIONAL ORDER CONTROLLER IN A TEMPERATURE CONTROL APPLICATION Gary W. Bohannan ,1



Wavelength Electronics, Bozeman, Montana, USA

Abstract: An analog fractional order PIα controller using a fractional order impedance device, a FractorTM (patent pending), is demonstrated in a temperature control application. The performance improvement over a standard PI controller was notable in reduction of overshoot and decreased time to stable temperature while retaining the long term stability and set point accuracy of the standard controller. The modi cation of the standard controller to a fractional order controller was as simple as replacing the integrator capacitor with a Fractor TM . Keywords: Fractional Order Control, fractance

1. INTRODUCTION There has been signi cant work focused on the fact that dynamic properties of materials, particularly electrical impedance, can often be described with high accuracy using non-integer order impedance descriptions.(Macdonald, 1987) The Warburg impedance, with the impedance varying as the square-root of frequency, was originally described at the end of the 19th century.(Warburg, 1899) The non-integer order impedance spectroscopic description has been tied to the fractional order calculus time domain dynamics by a number of researchers, see, e.g. (Westerlund and Ekstam, 1994). By reversing the point of view from the fractional calculus describing the dynamic behavior of a material to viewing the materials dynamics as carrying out the fractional calculus operation, we can create broadband analog fractional order operators with a high degree of accuracy. The output of a fractional order integrator can be summed with the output of a standard proportional ampli1

The author also holds an appointment as Affiliated Professor of Physics at Montana State University, Bozeman.

er to implement the PIα controller described by Podlubny and others. (Podlubny, 1999) The idea of using a non-integer order impedance to create a fractional order mathematical operator dates back to at least 1961. (Carlson and Halijak, 1961) Since thermal loads involve thermal di usion, a half-order controller represents a better match to the physics of the plant to be controlled. Earlier work (Petras and Vinagre, 2002) showed that a digital fractional order controller was e ective for temperature control and suggested that the test should be conducted with an analog circuit. In this paper, an analog fractional order controller, with   0.5, is demonstrated in a temperature control application.

2. THE FRACTORTM CIRCUIT ELEMENT A FractorTM is a two lead passive electronic circuit element similar to a resistor or capacitor, but exhibiting a non-integer order powerlaw impedance versus frequency. The term “fractance” has been used to describe fractional order impedance, consistent with the terms “resistance” and “capacitance”. A fractance has a nearly con-

Log |Z| [Ohms]

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(b) Fig. 1. Spectral response of the FractorTM used in this demonstration project; (a) the impedance magnitude and (b) impedance phase. The multiple lines show the variation over 26 impedance measurement scans. stant loss ratio (“tan”) over a large e ective bandwidth. The impedance behavior is accurately modelled with the power-law form K ZF rac (ω) = , (jω )α

(1)

where K is the impedance magnitude, in Ohms, at a calibration frequency ωC = 1/ , and  is the non-integer exponent, 0 <  < 1. The phase shift is related to the exponent by φ = 90o × . The fractance description in equation 1 also includes ideal resistance and capacitance in the limits as  → 0 and  → 1, respectively. Note that the impedance de nition of equation (1) uses only real, integer order units of Ohms, seconds, and Hertz. For actual circuit performance speci cation and design this provides for a direct link between the impedance spectroscopic

measurements and the ultimate device performance without trying to determine the meaning of such descriptions as Farads to a fractional power. (Westerlund and Ekstam, 1994) Figure 1 shows the frequency response for the FractorTM used in the temperature control demonstration. Note that the nearly constant phase covers over seven decades of frequency. Additional measurements have shown that the fractance behavior with   0.5 adheres to the milli-Hertz regime. Samples have exhibited stable fractance properties for over a year. FractorTM devices with phase shifts other than 45o have also been created. No digital implementation has yet achieved this e ective bandwidth. The phase ripple over frequency is no worse than that achieved through approximations. The prototype FractorTM elements are currently made by hand, but are still not much larger than typical through-hole ca-

pacitors, 2.5cm × 2.5cm × 0.6cm, it is feasible to implement fractional order control without the excessive space requirements of a network of integer-order analog elements, or the computing power requirements of a digital approximation. See (Tenriero Machado, 1997) for a description of some of these approximation techniques. Haba, et.al., demonstrated that it is possible to create fractional order impedances in the range of 100 kHz to 10 GHz by fabrication of a fractal structure on silicon. (Haba et al., 2005) Fractance behavior has now been demonstrated over the frequency range < 10 2 to > 109 Hz. As with any circuit element, the FractorTM has voltage and power dissipation limitations as well as some other restrictions on operating environment. These are being actively investigated. FractorsTM developed to-date do not contain lead or other hazardous substances currently covered under European RoHS standards.

VIN

VOUT

Fig. 2. Schematic for a fractional order integrator. ZF represents the FractorTM element. The schematic symbol for the FractorTM was designed to give the impression of a generalized Warburg impedance; a mixture of resistive and capacitive characteristics.

3. THE FRACTIONAL ORDER INTEGRATOR Analog fractional order operators can be designed using standard rules for operational ampli er circuits. The gain of an ampli er circuit, such as shown in gure 2, is represented in the frequency domain as the ratio of the feedback to input impedances. VOU T = G(ω) = VIN

ZF B (ω) . ZIN (ω)

K 1 . RI (jω )α

K 1 . RI (s )α

(4)

It is apparent that equation (4) has the form of the Laplace transform of a fractional order integrator of order . (Oldham and Spanier, 1974) When summed with the output of a proportional ampli er, the result is a PIα controller. Since the summing ampli er is also typically inverting, positive polarity is restored without further circuitry. While the impedance spectroscopy data is most useful in characterizing a FractorTM , the more convincing demonstration of the power-law nature of the fractional order integrator is shown in the time domain. Figure 3 shows the response of an analog fraction order integrator with   0.5 to a square wave input. The circuit of gure 2 was followed by a unity √ gain inverter to restore positive polarity. The t shape is clearly evident in the output.

4. THE THERMAL LOAD The thermal load (the test “plant”) for this demonstration was a thermoelectric cooler (TEC) mounted on a 14cm × 7.5cm × 0.5cm aluminum block. The system was driven to current limit to demonstrate the nonlinear e ect of “actuator saturation.” A 10 kΩ negative temperature coe cient (NTC) thermistor was used as the temperature sensor. The MPT Series controllers provide a constant 100A excitation current for probing the thermistor resistance. Since the impedance of the sensor varies nearly exponentially with temperature, there is an inherent non-linearity in the feedback gain. The Steinhart-Hart relation was used to convert resistance to temperature.

5. RESULTS (2)

In this instance the feedback impedance is given by equation (1) and the input impedance is a resistance, RI , resulting in an overall gain of G(ω) =

G(s) =

For the temperature control demonstration, a standard 5 Amp analog temperature controller (Wavelength Electronics MPT-5000) was modied by removing the 1F integrator capacitor and installing the FractorTM characterized by gure 1. No other changes were required.

ZFract

RI

Rewriting equation (3) in terms of the Laplace variable, s = jω,

(3)

A comparison of the e ectiveness of a conventional PI controller versus a PIα controller is shown in gures 4a and 4b. Note that the overshoot is virtually eliminated. Time to stable temperature is reduced by a factor of almost three. The integrator windup e ect due to the time spent in actuator saturation was virtually eliminated without the

Fractor with Square Wave Input

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Time [sec] Fig. 3. Time domain response of a half-order integrator using a FractorTM with properties similar to Input Output those shown in gure 1. Curve b 10/19/05

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(b) PIα control. Fig. 4. Comparison of results with standard PI (a) versus PIα control (b).

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extra complexity or expense of “anti-windup” circuitry.

ditionally, a number of projects for acoustic and image signal conditioning are been considered.

In addition to the simplicity in the arrangement, the system exhibited much less sensitivity to changes in other parameters.

Much work remains to be done to bound thermal, aging and other e ects in the FractorTM devices. Additional, assembly and packaging options for shrinking the size of the devices are being investigated. Undoubtedly, di erent circumstances will dictate the impedance magnitude, phase, and frequency band requirements, inevitably leading to a catalog of fractance devices with a range of sizes and shapes. The requirements for additional FractorTM properties will evolve, guiding additional work into the electrochemistry of the devices.

6. ONGOING RESEARCH AND FUTURE PROSPECTS By allowing for digital control of device selection, it is possible to consider an auto-tuning mechanism for selection of gain magnitude and phase. See, e.g. (Chen et al., 2004). Additionally, hybrid analog/digital computational processors are demonstrating the potential to increase computational throughput over digital only systems by many orders of magnitude while dramatically reducing power requirements. (Cowen et al., 2005) Offloading fractional order operators to an analog co-processor o ers even greater savings in computational overhead while guaranteeing stability and gaining immunity from aliasing and roundo errors. Other circuit con gurations besides that shown in gure 2 are possible, such as a fractional order derivative operator, Dβ , by rearrangement of the input and feedback impedances. Exponent “arithmetic” is also possible by incorporating FractorsTM of di erent exponent orders. Numerical modelling to predict the behavior of these circuits, particularly when non-linear switched parameter conditions are included, stretches the current state-of-the-art in solving fractional order di erential equations. See, e.g. (El-Mesiry et al., 2004). Given the very deep memory of the FractorTM , it becomes all the more critical to bound the memory disturbance e ect outlined by Hartley and Lorenzo. (Hartley and Lorenzo, 2002) Notably, the memory of the rst step transition did not negatively a ect the performance at the second temperature step. With devices exhibiting such nearly exact single exponent behavior it will now be possible to arrange experiments to test the physical meaning of the initial conditions in the fractional calculus. While many mathematical de nitions have been o ered, nature should determine which of these should be accepted for physical system modelling. (Podlubny, 1999) More complex control demonstration projects are being considered, including control of flexible robotic arms to give them more “natural” or human-like motion such as would be required in prosthetic devices or other human augmentation systems. These projects have given impetus to better understand the process of characterizing plants for e ective fractional order control. Ad-

7. CONCLUSION The FractorTM will soon add a very simple device to the catalog of electronic elements available to the system designer, allowing for intuitive and straightforward design of fractional order controllers. Circuit con gurations that would be impractical or impossible to implement with conventional devices should start to become commonplace.

ACKNOWLEDGEMENT The author is indebted to Dr. S. Hurst of Montana State University, Bozeman, Montana, for development of the FractorTM devices.

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Haba, T. C., G. Ablart, T. Camps and F. Olivie (2005). Influence of the electrical parameters on the input impedance of a fractal structure realised on silicon. Chaos, Solitons and Fractals 24, 479–490. Hartley, T. T. and C. F. Lorenzo (2002). Dynamics and control of initialized fractional-order systems. Nonlinear Dynamics 29, 201–233. Macdonald, J. Ross (1987). Impedance Spectroscopy. John Wiley & Sons. New York. Oldham, K. and J. Spanier (1974). The Fractional Calculus. Academic Press. New York. Petras, I. and B. M. Vinagre (2002). Practical application of dgital fractional-order controller to temperature control. Acta Montanistica Slovaca 7, 131–137. Podlubny, I. (1999). Fractional Di erential Equations: An Introduction to Fractional Derivatives, Fractional Di erential Equations, to Methods of their Solution and some of their Applications. Vol. 198 of Mathematics in Science and Engineering. Academic Press. San Diego, CA. Tenriero Machado, J. A. (1997). Analysis and design of fractional-order digital control systems. SAMS 27, 107–122. Warburg, E. (1899). Uber das Verhalten sogenannter unpolarisierbarer Electroden gegen Wechselstrom. Ann. Phys. Chem. 67, 493– 499. Westerlund, S. and L. Ekstam (1994). Capacitor theory. IEEE Trans. Dielectrics and Electrical Insulation 1(5), 826–839.