Literature suggests that the input signal for system identification should be a frequency-rich signal to ensure parameter estimation accuracy [2]. Such a signal can fully excite the system and cause a small, but measurable, dynamic disturbance in the system output. In theory, a white noise-like signal is ideal as the input signal for this purpose since it contain a wide variety of frequency
0 0.2 0.4 0.6 0.8 1 1.2 0 0.2 0.4 0.6 0.8 1 1.2 0 0.2 0.4 0.6 0.8 1 1.2 Step s ize For gett ing factor value EMSE EMSE RLS EMSE FAP
83
components and the variance is adjustable. It also has equal intensity across the frequency spectrum and has a mean value of zero. Using this signal as the input signal, system identification algorithms can readily estimate the parameters in a few iterations.
In SMPC system identification, the input signal is typically the duty cycle, which is calculated via the control algorithm and PWM unit in the DSP. True white noise cannot readily be injected into the control loop to generate the duty ratio. Therefore, in many practical system identification applications the input signal is usually an artificially-generated frequency-rich signal.
Much research has been carried out to generate an artificial random signal with similar properties to white noise in signal processing applications. The Pseudo Random Binary Sequence (PRBS) is such a signal which is commonly used in signal processing [2]. A major attraction of the PRBS is that it is very easy to implement on a DSP and requires little computational effort. Consequently, it lends itself well to system identification of digital controlled SMPCs.
The PRBS is a rectangular pulse sequence with a “random” modulation pattern, as shown in Fig.4.9. This sequence is generated by a fixed number of digital bits, shift registers, and an exclusive-or gate (XOR) in the feedback loop [69]. As an example, shown in Fig.4.9, a nine-bit PRBS signal is generated. At each iteration, the data bits are shifted right by one bit. The value which exits bit 9 is the PRBS output and is used as the excitation signal within the digital control loop. The XOR operator is applied to bits 5 and 9 and the output of this operation is used to refresh the value of bit 1 in the register following the shift.
In theory, the maximum length of the PRBS sequence before repetition is L =2m−1, where m is
the number of bits. The register bit number should be long enough to provide a sufficiently frequency rich signal for the intended application. A good way to determine the competence of the PRBS is to follow these steps: 1. perform a frequency response test, based on the nominal transfer function of the SMPC system. 2. From the frequency response of SMPC, establish the bandwidth of the system. 3. Select a PRBS sequence length which covers the frequency band and has enough data points for system identification (usually >200 data points). By following these basic steps, an 11-bits PRBS sequence is selected for this research. In a SMPC application, the duty ratio can only be changed at each switching interval. That means the input update frequency is limited by the switching frequency. Therefore, the maximum frequency component of PRBS perturbation signal
84
is intrinsically bound to the switching frequency, it cannot update and have an impact between normal duty ratio updates.
For completeness, Table 4.7 shows the maximum sequence length for different size PRBS. This table also shows which register bits should be XOR to ensure the maximum sequence. Due to the algorithmic generation of the PRBS signal, it cannot be considered as a pure random signal just like Gaussian white noise. A detailed test should be done before applying specific PRBS signal in practice.
In terms of the PRBS perturbation signal, this very much depends on the mathematical model, system sensitivity to disturbance, and the effectiveness of the control loop feedback. If a robust system control loop is applied, the PRBS amplitude can be selected as a higher value to increase the accuracy of parameters estimation, and vice versa.
85
Figure 4.12 PRBS bit generation method
Number of bits (m) m
L = 2 - 1 Bits in XOR operation
2 3 1 and 2 3 7 1 and 3 4 15 3 and 4 5 31 3 and 5 6 63 5 and 6 7 127 6 and 7 8 255 2, 3, 4 and 8 9 511 5 and 9 10 1023 7 and 10 11 2047 9 and 11 12 4095 4, 10, 11 and 12 13 8191 8, 11,12 and 13 14 16383 2, 12, 13 and 14 15 32767 14 and 15
86
To verify the impact of the PRBS signal with respect to the DC-DC buck converter operation and system identification results, simulation studies have been performed. The variation of duty ratio and output voltage is presented in Fig. 4.13. It can be observed that the duty ratio is perturbed when the PRBS sequence is injected into the control loop. This is also the same time that the system identification process is enabled. The output voltage begins to slightly fluctuate when the PRBS signal is injected. This level of disturbance may be greater in practice due to resonance and the effect of parasitic inductance or capacitance within the circuit. The amplitude of the injected PRBS signal must be kept small to minimise the impact on the output voltage in SMPC systems. It can be seen that there is a voltage oscillation after 0.3s where is the end of PRBS injection. It is caused by a feedback loop which contains a typical PID controller. Fig.4.12 shows the parameter estimation result without PRBS perturbation signal.
87
Figure 4.14 system identification result without PRBS, comparison between RLS and FAP,
=0.3
and =0.98
It can be seen in Fig.4.14 that, without the PRBS signal, both FAP and RLS cannot estimate the correct value for parameter 2 (red curve, see correct result in Fig.4.7). Also, the RLS output exhibits significant oscillations around 0.285s and ultimately does not converge to the correct value. In terms of convergence speed, it is much slower in the absence of the PRBS injection. This confirms that injecting the PRBS signal is essential in the system identification process and is necessary to establish an accurate transfer function model of the SMPC.
88
4.7 Chapter summary
In this chapter, a detailed study of the proposed Fast Affine Projection (FAP) algorithm is presented. A detailed mathematical analysis demonstrates that the proposed FAP algorithm is computationally simpler than conventional Recursive Least Squares (RLS) algorithm due to the special data regression model of SMPCs. Importantly, there is no division operation involved in FAP algorithm, which significantly reduces the computational cost compared to RLS. The mathematical derivation and simulation results also show that the FAP algorithm has a faster convergence speed compared with the RLS. Overall, the FAP algorithm is proven to be a viable candidate algorithm for system identification and adaptive control.
89