In Figure 1, Q1-Q4 are insulated-gate bipolar transistors (IGBTs), D1-D4 are fast-recovery free-wheeling diodes, L and C are inductor and capacitor of LC output filter, R is a load resistor, (a constant) is average voltage of DC

source, is inductor current, is load current, is load voltage and is modulation signal. The snubber circuits of SPA are omitted for simplicity, the design of which can be found in [10] .

Figure 2 illustrates the principle of generating the bipolar pulse width modulation (PWM) signals, where is the amplitude of the isosceles-triangle carrier, is the carrier period and also the switching period and the sampling period (the sampling frequency). According to the equivalent-area principle [11] [12] , the area of the curved-edge trapezoidal pulse ABCDE should be equal to the net area of the PWM pulses, i.e.. Because is very small, the area of ABCDE is close to the area of the rectangle A'C'DE (the shadowed area). and can be written as

where is the ordinate of the intersection point B, and are the turn-on time and turn-off time of Q1 and Q4. From, the following relationship is obtained:

where the relationship is considered. The prerequisite for accurate tracking is that the average output voltage between points a and b (seen in Figure 1) should be equal to, thus

where and. t_bon is called biased turn-on time (with an offset), and is called time-to-voltage transfer coefficient. It is easy to know that: 1) if, then; 2) if, then; 3) if, then. So, Equation (4) excellently describes the whole process of turn-on and turn-off, and is the correct open-loop model of SPA.

To realize the closed-loop control, the output of SPA should be fed back to affect, and there are 2 ways to do so: 1) let; 2) let is the current command signal, is the voltage command signal, is the output current deviation, is the output voltage deviation, is the current error amplification coefficient and is the voltage error amplification coefficient. Because can be assigned with either or, the SPA in Figure 1 can realize either current tracking or voltage tracking, though it is supplied by a voltage source. Given that most of the references are dedicated to discussing voltage tracking [13] [14] , and that the control method of current tracking can be transplanted to voltage tracking with slight modifications as discussed, this paper concentrates on discussing current tracking only.

in Equation (4) is the output of the controller and the output of SPA needs at least 1sampling period to be fed back to the controller, so the continuous model of SPA in frequency domain can be constructed as shown in Figure 3. The 3 blocks enclosed by the dash-dotted frame are the general model of SPA, which considers SPA, LC output filter and load resistor together:

, and are the Laplace transforms of, and, is the transfer function of the controller, and F is the proportional coefficient of feedback channel.

Although it is easy to write out the closed-loop transfer function according to Figure 3, it is difficult to design the controller due to the pure-delay term which leads the system to be a non-minimum phase system [15] . Further, although the system can be turned into a minimum phase system by expanding into a power series and taking a finite number of the fore terms, this would suffer a great loss of the system bandwidth. Therefore, it is wise to design the controller from another angle, i.e., in time domain.

For digital simulation in time domain, G(s) must be discretized in time domain.

The first step is to transform G(s) in s domain to G(z) in z domain by virtue of the relationship between Laplace transform and z transform:

where Z[・] denotes performing z transform on the expressions in the square brackets. To maintain the invariability of the system step response after z transform, a zero-order holder, i.e., , is introduced to Equation (6). G(z) is very complex if expressed with parameter symbols, so, instead, it is expressed in numerical type with detailed values of the parameters substituted into the expression and calculated (the values of the parameters are listed in Appendix A):

The second step is to perform inverse z transform on G(z) to get the difference equation:

where k is the integer index of the discrete time series,. It is assumed:.

The structure of single-neuron adaptive quasi-PID control method is presented in Figure 4, where, and are the 3 inputs of the single neuron:

And, and are the 3 connection weights:

The single neuron sums the 3 weighted inputs up by its adder component “Σ” to form a total input signal:

Substitute Equations (32) and (33) into Equation (34), it is seen that Equation (34) actually realizes the same calculation of Equation (31).

The 3 connection weights in Equation (34) should be normalized to maintain their relative magnitudes to promote the robustness of simulation and actual control. The normalization can be carried out by virtue of vector norms. There are 3 commonly used vector norms [16] : 1) 1-norm, the summation of the absolute values of the elements; 2) 2-norm, the square root of the quadratic sum of the elements; 3) ∞-norm, the maximum value of the absolute values of the elements. Comparisons show that 2-norm is of the greatest computational complexity, and simulations show that it does not give a better control effect than 1-norm. Although ∞-norm is of the lowest computational complexity, it always makes one of the 3 normalized connection weights equal to 1, causing the corresponding input to have the greatest impact on the control quantity and thus inducing oscillations on the output waveform during the first 1 or 2 power frequency periods. Therefore, 1-norm is the best choice, and the normalized type of Equation (34) based on 1-norm is

where is the connection weight vector, is the 1-norm of, and is defined to replace the coefficient of for short.

The single neuron takes through excitation function to generate the normalized control quantity:

where a linear proportional function with amplitude limitations is chosen as, and is the slope of the linear segment of. The choice of this ex- citation function lies on 2 considerations: (i) limiting the amplitude of the con- trol quantity is indispensable to prevent the control quantity from overreaching; (ii) has already been the required control quantity, further pro- cesses with complex excitation functions (such as the sigmoid function or the radial basis function) may not only deprive its physical meanings but induce unnecessary computational complexities, so it is better to choose a simple function to slightly adjust its amplitude. It should also be noted that the amplitude limitations of are set as ±5 instead of ±1 (±1 are the amplitude limitations defined in normalization theory). The reasons are: 1) avoiding pure decimal- fraction computations on fixed point microprocessor used in this paper, which may introduce large rounding errors to the calculated data; 2) slightly loosening the amplitude limitations to enhance the fault tolerance of the algorithm.

In Equation (8), the coefficients of and are far greater than those of and because the variables and their coefficients are all actual values. Seeing that Equations (35) and (36) are in normalized types, Equation (8) cannot be calculated with them if not normalized accordingly. So 2 steps are taken to normalize and: (i) according to Equations (4) and (36), , when, , so; (ii) the peak-to-peak values of the output current in this paper are designed as ±10A, so is normalized by being divided by 10A, and the result is the per-unit value. Based on the 2 steps above, Equation (8) is normalized as

The general learning rule [17] for connection weight adjustment is as follows:

where is the connection weight vector, is the incremental vector of, is the learning rate, is the input vector, d (a scalar quantity) is the expected output and is called the teacher signal, function is the learning signal and is a real constant.

Argument 1: If (e is the error between d and the actual output y, e and y are scalar quantities) and, then, which is perceptron learning rule based on least mean square standard. This learning rule includes d, so it is a supervised learning rule with teacher guidance, and theory [18] verifies that it is asymptotically stable. The expanded type is

where, in this paper, , , and. Simulations show that this learning rule possesses outstanding stability but lacks “independence” or “self-learning enthusiasm”. When illustrated on the output waveform, the phenomenon is that the steady-state errors of the output waveform are very small while the response speed is fairly slow.

Argument 2: If (is the control quantity and a scalar quantity) and, then, which is Hebb learning rule. This learning rule does not include d, so it is an unsupervised learning rule without teacher guidance, and theory [18] verifies that it is unstable under certain conditions. The expanded type is

where, in this paper,. Simulations show that this learning rule has strong “independence” and “self-learning ability”, and its learning speed is very fast. So the output waveform has a fairly high response speed. However, because of the lack of teacher guidance, the steady-state errors are relatively large.

To better illustrate the 2 arguments above, a periodic square waveform is chosen as an example. The reasons for the choice are: 1) for periodic waveform, comparisons can be made between different waveforms or among different segments of the same waveform; 2) for square waveform, it has rising and falling edges and smooth segments, so the steepness of the former can be used to compare the response speed while the smoothness of the latter can be used to compare the steady-state errors. The simulated output waveform using perceptron learning rule is presented in Figure 5(a), which shows that the rising and falling edges are not steep (i.e., the response speed is slow) but the smooth segments are very flat (i.e., the steady-state errors are very small). The simulated output waveform using Hebb learning rule is presented in Figure 5(b), which shows that the rising and falling edges are steeper than those in Figure 5(a) (i.e., the response speed is faster), but there exist oscillations and great overshoots in the smooth segments (i.e., the steady-state errors are large); the oscillations seem to grow larger, implying the likelihood to become unstable.

Given that the strong point of perceptron learning rule is the weak point of Hebb learning rule and vice versa, the authors creatively combine them together and propose the perceptron-Hebb learning rule:

The simulated output waveform using the new learning rule is presented in

Figure 5(c), which shows that the rising and falling edges are steeper than those in Figure 5(a) and the smooth segments are flatter than those in Figure 5(b), meaning that both the response speed and the steady-state errors are improved―the new learning rule inherits the strong points of the two but gets rid of their weak points to a large extent; moreover, the possible unstability of Hebb learning rule never exists.

The control flow of single-neuron adaptive quasi-PID control method for simulation or actual control is summarized in Figure 6. It is shown that Equations (32), (41) and (36) are the 3 most important computational procedures of the flow chart, but they introduce only a small amount of floating additions and multiplications. These calculations are of relatively low computational complexities, meaning that the control method is very suitable for real-time control and for realization on low-end microprocessors.

From Equation (33), it is seen that is actually initialized by quasi-PID parameters, which, as mentioned above, would vary during operation. So is eventually initialized by values with small unpredictable errors. Therefore, it is necessary to analyze the impacts of the inaccuracy of the initial on system stability. Considering the great complexity of the calculation process, only the conclusive results are presented.

By performing z transforms on Equations (36) and (37) respectively, then solving the resultant simultaneous equations, the system function can be obtained. After rationalizations of both the numerator and the denominator polynomials of, the denominator polynomial becomes the characteristic polynomial that is in the following form:

where are the coefficients acquired from rationalizations. In terms of Jury criteria [19] , the constraint conditions for system stability are as follows:

1) The first criterion requires, and it is fulfilled straightforwardly because are all positive numbers;

2) The order of is 5, an odd number, so the second criterion requires, the calculation of which gives the in equation

3) The third criterion requires, the calculation of which gives the in equation

4) …

The punctuation “…” means the curtailment of the subsequent calculations. From calculations, it is found that as long as the choice of fulfills both Equation (43) and Equation (44), the curtailed in equations are fulfilled as well; what’s more, all the in equations have some margins to retain their inequalities, which not only gives the choice of certain freedom, but also makes the impacts of the small unpredictable errors caused by variations of quasi-PID parameters on the initializations of negligible. In short, the initializations of according to Equation (33) and the choice of according to in equations (43) and (44) can ensure the system stability.

A5A (RMS), 50 Hz sinusoidal waveform is chosen for the test. Here, in order to compare the actual performances of quasi-PID control method with the current command signal (with, the control method is actually the PI control method [20] ) and with the modified command current signal, the single neuron is temporarily thrown off. The results are presented in Figure 7, and it is clear that the simulated waveforms and the experimental ones are alike. Figure 7

shows that the output waveform with is fluctuating around the expected output waveform and is distorted in the vicinity of the peaks, which illustrates the existence and the detrimental effects of the inherent tracking error; however, the one with satisfactorily inhibits the fluctuations and distortions, which means the inherent tracking error is effectively counteracted. At the end of this subsection, it should be pointed out that the experimental waveforms in Figure 7(b) are obviously thicker than those in Figure 10(f), even the one with, which is the visible representation of the impacts of the circuit parameters drift and the random disturbances as mentioned in Section 4. These unpredictable errors can only be eliminated or compensated by the adaptability of the single neuron.

A ± 5 A (peak-to-peak value), 50 Hz square waveform (its characteristics were discussed in Subsection 4.2) is chosen to test the tracking speed (the response speed) and the tracking accuracy (the steady-state errors) of single-neuron adaptive quasi-PID control method. Normally, to current tracking, the load resistor R is of a few ohms, so 2 situations, R = 3 Ω and 10 Ω, are chosen for the test. The reason for the choice of this load difference (10 Ω − 3 Ω = 7 Ω) is that if the 2 values of R are fairly close, the results would be too close to distinguish. However, this choice gives rise to a problem. The maximum output power of the prototype machine in this paper is designed as 100 W. If the ±5 A square waveform is outputted to R = 3 Ω, the maximum output power is at least 75 W, and if outputted to R = 10 Ω, the maximum output power would be up to 250 W, which is unrealizable for the prototype machine. Thus, only the simulation results are presented (shown in Figure 8).

A comparison of Figure 8(a) and Figure 8(b) shows that the steepness of the rising and falling edges and the smoothness of the smooth segments are alike except the overshoots, so the tracking speed and the tracking accuracy are almost invariant for different values of R, which illustrates the good adaptability of single-neuron adaptive quasi-PID control method to different load resistors.

As mentioned in Section 4, there are many types of system parameters drift, so, for brevity, the drift of the load resistor R at different temperatures is chosen as a test example, where it is assumed that R varies from 3 Ω to 5 Ω with temperature increase. In practice, this variation is actually very slow, but for convenience the variation of R is further assumed to be abrupt because the fast variation can encompass the slow variation as its special case. A5A (RMS), 50 Hz sinusoidal waveform is again chosen for the test instead of the square waveform, because as to a square waveform, choosing the abrupt variation point at the rising or falling edge would seem to be too special while choosing at the smooth segment would lack representativeness. It is unsafe to abruptly vary the load resistor by a switch or a relay on-line on the prototype machine, and the switch may introduce side effect to the circuit, so again, only the simulation results are presented (shown in Figure 9).

Figure 9(a) shows that R varies abruptly from 3 Ω to 5 Ω at 0.042 s, and the induced disturbance on the output waveform is nearly undetectable. For clear presentation, the dynamic tracking error e_i is presented in Figure 9(b) to illustrate the disturbance. Figure 9(b) indirectly illustrates the rapid adjustment of single-neuron adaptive quasi-PID control method towards the abrupt disturbance.

Different types of output waveforms contain different frequency components, the content and duration of which, in practice, may be unpredictable. Although the predesigned sampling frequency determined by the hardware fixes the theoretically maximum bandwidth of the open-loop system, the controller may lead to a great loss of the predesigned bandwidth of the close-loop system. From many simulations and experiments, the authors found that most of the control methods (such as those mentioned in Section 1) are more suitable for generating and amplifying sinusoidal waveforms or specified waveforms only,

since the frequency components of these waveforms are predetermined and the parameters of the controller can be directly adjusted towards these frequency components to acquire a relatively high and stable tracking accuracy; however, for waveforms with unpredictable frequency components, the tracking accuracy of these control methods may decline uncontrollably if there exist some frequency components not preconsidered during the design process of the controller due to the poor adaptability of these control methods. Thus, in this subsection, the adaptability of single-neuron adaptive quasi-PID control method to different types of waveforms, with and without unpredictable frequency components, is tested, and the simulation and experimental results are presented in Figure 10.

Figures 10(a)-(f) show that the simulation waveforms and the experimental waveforms match with each other closely, and Figures 10(g)-(j) show that the experimental fault current waveform and the experimental fault voltage waveform match the corresponding ones recorded by digital fault recorder (DFR) satisfactorily. The waveforms in Figures 10(a)-(f) contain invariable (deterministic) frequency components that can be obtained by Fourier series expansion, and the waveforms in Figures 10(g)-(j) contain variable (unpredictable) frequency components, the unpredictability of which is generated by the random characteristics of the faults in electrical power systems. These figures illustrate excellent adaptabilities of single-neuron adaptive quasi-PID control method to waveforms with deterministic and unpredictable frequency components from qualitative angle. A new comparison of Figure 10(f) and Figure 7(b), which is correspondent to the one in Subsection 5.4, shows that the sinusoidal waveform in Figure 10(f) is slimmer and smoother than that in Figure 7(b), demonstrating that single-neuron adaptive quasi-PID control method adaptively inhibits the circuit parameters drift and the random disturbances.

However, merely assessing the accuracy of the output waveforms from a qualitative angle, i.e., from the subjective impression, is very superficial, especially when the waveform is too complex to discriminate its subtle discrepancies. Therefore, a quantitative criterion for accuracy assessment is constructed, which is able to assess the accuracy of the waveforms by making point-to- point comparisons between the actual output waveform and the expected one and then give a score. This quantitative criterion is mean square error, where the notation signifies the mathematical expectation computation. For discrete time series, is estimated by a statistical quantity:

where N is the length of the time series. As an example, Equation (45) is performed on the experimental fault current waveform in Figure 10(h), which is a waveform with unpredictable frequency components, and the result is. For comparison, the same fault current waveform is generated by quasi-PID control method (without the single neuron), and the result is; by PI control method,; by repetitive control method,; by deadbeat control method,; by sliding mode control method,. It is clear that with the introduction of the quasi-D term, quasi-PID control method gains a relatively high accuracy by compensating the inherent tracing error of PI control method, which, to some extent, is already comparable to the accuracies of deadbeat control method and sliding mode control method. And the improvement owing to the single neuron is more impressive, and this high accuracy has made the applications of SPAs in some of the high-accuracy fields possible. For example, the regenerated fault current and fault voltage in Figure 10(h) and Figure 10(j) can be used for protective relay test or other similar tests [21] , the accuracy requirement of which is generally prescribed as ≤1.5%; this means that by virtue of single-neuron adaptive quasi- PID control method SPAs can also be used in protective relay test equipment so as to make the equipment small in volume and weight but high in performance.