Digital Power Control Enables System Identification

With digital supplies, the number of points that can be monitored and analyzed far surpasses the capabilities of analog including viewing values and parameters not available in analog supplies.

Typically, power-supply designers can monitor the output voltage, output current and error amp voltage. Sometimes, sense current also can be gauged. To know the actual transfer characteristics of the control loop, often referred to as system identification, the loop must be measured with a network analyzer or similar capable piece of external equipment. However, a network analyzer adds significant equipment and labor costs to the typical design cycle.

With the implementation of a digital controller, the designer has complete visibility into the entire control loop to measure frequency response in real time without the use of a network analyzer. The frequency response is communicated to a host system, such as a typical PC.

Digital Control

Silicon Laboratories' Si8250 digital power controller has a unique architecture that enables a digital control loop and a host of other capabilities. Fig. 1 shows the architecture of the Si8250, which includes a high-speed control processor and a system management processor.

The high-speed control processor runs at four discrete frequencies between 1.25 MHz and 10 MHz and is usually operated at 10x to 25x the switching frequency of the supply. For example, a 50-kHz PFC, 200-kHz full-bridge, 400-kHz POL, and 750-kHz to 1-MHz POL would use the 1.25-MHz, 2.5-MHz, 5-MHz and 10-MHz settings, respectively. This control processor includes a specialized DSP filter with which all of the loop compensation is accomplished.

The system management processor runs independently of the control processor and performs a variety of tasks. It is based on a Silicon Laboratories 8051 MCU core with 32 kbytes of Flash memory and a host of typical microcontroller-like peripherals. Additionally, it includes a 200-kHz auto-sequencing analog-to-digital converter (ADC).

One set of tasks it performs includes external tasks such as monitoring a fan tachometer. Based on the internal supply temperature, it can modulate the fan speed using one of the PCA channels to regulate the temperature to a programmed level. With a UART and an I2C interface, it can manage all communications with the host system or outside world through PMBus, Z-1 or any other future power-supply communications protocol.

Its second major task is to monitor and control the control processor itself. For example, during a load transient, there is a transient detector that generates an interrupt to the system management processor, which can then modify the coefficients in the DSP filter. This helps resolve the transient much faster than normal. With this nonlinear control capability, the control loop can be modified in real time so that when a transient occurs, the control loop can be changed to have a very high bandwidth for resolving the transient. Once it is over, the loop can be returned to a low bandwidth for improved noise rejection.

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A simplified block diagram of the Si8250 (Fig. 2) will be used as the basis for demonstrating system identification without the use of a network analyzer. The voltage feedback is immediately converted to digital through an ADC connected to a compensation processing engine, such as a proportional, integral and differential (PID), before passing to the pulse width modulator (PWM). Each of these circuit components of the controller is usually either part of or interfaced to a microcontroller.

The microcontroller acts as a management element in the system. The low-bandwidth signals such as input voltage and current used for monitoring are connected to the microcontroller. There are numerous signals and, with all of these, there are multiple access points for peeking at and injecting data or signals.

System identification tests the system to monitor the real-time response of the control loop. However, not all digital signals are useful for system identification, which is primarily a function of power control and not power management. Therefore, the signals associated with digital control are more desirable than the low-bandwidth signals for digital power management. Ideal points are before and after the compensator as shown in Fig. 2.

One option is to inject data into the compensator and read back from the ADC, which offers visibility into the entire loop. Injecting the signal into the system after the compensator enables the digital controller to analyze the power train without including any problems or signal modification by the compensator.

In an analog system, injecting into the compensator is problematic because of the integral operation that occurs in the system. The integrator tends to drive the output to saturation due to noise and small offsets within the amplifiers. However, this is not normally the case in a digitally compensated system because it is completely deterministic.

The only requirement to keep the system from drifting and saturating is maintaining integral balance on the injected signal. The sum of the positive error must equal the sum of the negative error data injected; in a digital system, integral balance is completely controllable.

Another option is to inject into the PWM and sample at the output of the compensator, which enables insight into the entire loop. However, unlike the digital compensator, the analog power stage is not absolutely deterministic. Even if integral balance is maintained, small offsets due to noise could potentially yield slightly higher or lower ADC results; therefore, the integrator could drift away from the quiescent operating point and eventually saturate.

The best option for a power-supply designer is to inject data into the PWM and then sample data from the ADC output. This leaves the compensation out of the analysis; however, a digital compensator is deterministic, thus the compensator frequency response is known and can be calculated. Once calculated, the data is added to the captured system identification data to get the complete loop response.

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.

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