Digital Power Control Enables System Identification

Nov 1, 2006 12:00 PM
By Brett Etter, Marketing Manager, and Ross Fosler, Senior Applications Engineer, Silicon Laboratori

System Identification

System identification is accomplished by injecting a known signal into the system at one point and reading data from another point. The data is analyzed to determine the response of the system between those two points.

A typical network analyzer perturbs the system with a sine wave and varies the frequency of that wave to get information about the system. Since the controller has visibility of the control data path, it is possible to have the controller inject an approximate sine wave or square wave, and then calculate the fourier transform for the input and output signal at the injected harmonic frequency. This allows the components used for control to become a part of the system identification. Thus, the frequency characteristic of the controller's internal ADC and PWM become part of the identification.

The fourier transform produces real and imaginary data enabling phase and gain to be extracted. The frequency response is the difference of gain in decibels and difference in phase between the input and output. The following is an expression for the discrete fourier transform at a given frequency point kFS/N:[1-2]

Another method of system identification is to use “white” noise to inject an impulse into the system.[3] By injecting an ideal delta function into a buck converter, the output of the system should be the typical impulse response of a second-order system.[4]

There are two fundamental mathematical properties that make white noise quite useful. First, the cross correlation of the input and the output of a system is mathematically equivalent to the autocorrelation of the input convolved with the transfer function of the system:[1-2]

Second, the autocorrelation of white noise over infinity is an ideal delta function making it possible to determine the impulse response of the system by injecting noise and correlating the input to the output:

The frequency response is just the fourier transform of the impulse response. Fig. 3 shows the actual response of a system using noise correlation compared with a simulation of the same system.

As shown in the previous equations, the data set was assumed to be infinite; this is not realistic. The data must be limited to some finite size. Controller memory is an important limiting factor, which is finite and usually quite small in all present digital power controllers. Another criteria for system identification is that the digital power controller must have some processing capability. Many existing digital power controllers are fixed-state-machine controlled and can support only limited functions. A state-controlled digital power controller would have to explicitly indicate support for more advanced data analysis functions to be able to perform system identification or be able to manage sizeable blocks of data that could be transmitted to an external computer and analyzed.

Since controllers have finite processing and memory, the data analysis results are limited. Fig. 4 shows the result for an injection of 2-bit data and sampling 6-bit data over a set of only 1024 points. Because the data is so tightly bounded in data word width and memory depth, the results are not nearly as ideal as with an infinite data set with infinite precision. When compared to Fig. 3, it is not as accurate or representative of the real system.

Fig. 3 already shows a clean plot using system identification (that is, analysis on finite data sets), so the results are far better than those shown in Fig. 4, even with small data sets. This can be achieved by averaging. Instead of injecting one random pattern of 1024 points, inject numerous random patterns each of 1024 points. Calculate the impulse response for each injected pattern and accumulate (average) the impulse response. The averaged impulse response yields a greatly improved frequency response. Typically, it only takes approximately 10 iterations depending on the injected signal amplitude to get good results. The plot shown in Fig. 3 is the result of several thousand iterations.

Figs. 3 and 4 only show the response of the power stage combined with the ADC and the PWM. For complete system identification, the power system designer must add the compensator response to the data plotted in Fig. 3. The compensator could be anything for a given topology; however, for most applications the power stage is a second-order, level-crossing-rate system. Thus, a typical compensator is usually a proportional-integral-derivative (PID) controller. A basic PID takes the following form in the z-domain:[5]

The frequency response of the PID is the fourier transform of the z-domain transfer function. The evaluation of this function is useful to up to half of the sampling frequency:

The addition of the calculated PID frequency response with the system identification data gives a complete picture of the loop response in Fig. 5.

Using digital control in power supplies offers significant benefits including better performance, higher efficiency and reliability, communications for telemetry monitoring and fault detection, and prevention. More importantly, it enables faster and more accurate development of the proper compensation and performance monitoring of the supply. System ID is just one extremely beneficial capability and is the basic building block that leads to systems that are self-tuning and self-optimizing.

References

1. Jackson, L.B. Signals, Systems and Transforms, Addison-Wesley, Reading, Mass., 1991.

2. Elliot, D.F., and Rao, K. R. Fast Transforms, Algorithms, Analyses, Applications, Academic Press, Orlando, Fla., 1982.

3. Miao, B., Zane, R., and Maksimovic, D. “System Identification of Power Converters with Digital Control Through Cross-Correlation Methods,” IEEE Transactions on Power Electronics, Vol. 20, No. 5, pp. 1093-1099, September 2005.

4. Erickson, R.W., and Maksimovic, D. Fundamentals of Power Electronics, Kluwer Academic Publishers, Boston, 2001.

5. Kuo, B.C. Digital Control Systems, Holt, Rinehart and Winston, 1980.

6. Luk, R.W.P., and Damper, R.I. Non-Parametric Linear Time-Invariant System Identification by Discrete Wavelet Transform, Elsevier, 2005.