A robust and fast control technology of AC power conditioning for high-speed micromachining

In this paper, a robust and fast control technology is used to AC power conditioning, thus increasing the performance of the high-speed micromachining. The robust and fast control technology is made up of a robust sliding function (RSF) and a computationally fast grey forecasting model (GFM). The RSF without singularity problem admits system state converged to zero within finite time so that the output-voltage with low harmonic distortion in AC power conditioning is obtained. Nevertheless, while a severe non-linear loading is applied to high-speed micromachining, the needless chattering around robust sliding function occurs. The chattering results in thermal breakdown and serious voltage distortion in AC power conditioning output, and the reliability and stability of the high-speed micromachining will be worsened. Therefore, the GFM with Fourier series is introduced as a computationally fast and algorithmically easy means of removing the chattering existing in RSF when the system uncertainty bound is overestimated. By using this presented control technology, the AC power conditioning provides a high-quality AC output-voltage with accurate steady state and fast transience under various loading conditions, thus obtaining the excellent reliability and stability of the high-speed micromachining. Experimental results are performed in support of the proposed control technology.


Introduction
The AC power conditioning has widely been applied in high-speed micromachining [1][2][3][4]. High performance AC power conditioning must supply the output AC voltage to the reference sinusoidal with low total harmonics distortion (THD) and fast transient response. To minimize THD, some control schemes have been proposed for AC power conditioning. PI control is frequently used in industry due to simple control structure and ease of design. But, PID control cannot give good control performance as the controlled plant is severe non-linear and uncertain [5], [6]. Deadbeat control can improve the shortcoming of the PI control, but it is highly dependent on the accuracy of the parameters [7]. The repetitive control and mu-synthesis approach can overcome system uncertainties. However, they have implementation difficult and algorithm complexity [8], [9]. Sliding mode control (SMC) is being given attention due to its robustness characteristics [10][11][12][13]. A number of SMC associated with AC power conditioning have been reported [14][15][16][17][18]. A fixed switching frequency sliding mode controlled AC power conditioning is presented in [14]. However, the control design uses a typical SMC and cumbersome analog implementation, and thus incurs distorted output voltage during steady-state operation with a non-linear load. In [15], the multiple-sliding-surface is suggested to improve the incomplete system dynamics of classic sliding surface. Though the system performance is improved, the proposed methodology has time-consuming operation in algorithms. The control scheme based on fixed-frequency SMC has also been applied to the design of grid-connected AC power conditioning. In this case, the resulting output voltage makes a concession between steady-state and transience [16]. Reference [17] employs an integral SMC law to achieve AC power conditioning. But, the system trajectory could not hit the desired sliding surface fast and accurately. The noticeable distortion exists in the output voltage waveform. A modified SMC with the elimination of the disturbances for AC power conditioning is developed by [18]; this technology has complicated hardware design and a chattering problem. As mentioned by [14][15][16][17][18], linear sliding surface is adopted. Its characteristic is that the system tracking error converges to zero asymptotically. A robust sliding function (RSF), which employs nonlinear sliding surface is developed instead of linear sliding surface. Compared with linear slidingsurface-based control, the RSF can drive the system tracking error to converge to zero in finite time and there is no singular problem [19][20][21][22][23]. From the point of view in practical high-speed micromachining application, if the load disturbance is a severe non-linear condition, the chattering around RSF occurs. The chattering leads to thermal breakdown and serious voltage distortion in AC power conditioning output, thus deteriorating the reliability and stability of the high-speed micromachining. A computationally fast grey forecasting model (GFM) with Fourier series is employed to describe and analyze the future trend of sequence numbers according to the past and nowadays data for dynamic system. The GFM improves the accuracy of the basic grey forecasting model through the Fourier series, and has been successfully applied in many areas of engineering [24][25][26][27][28]. Thus, a mathematically simple and accurately forecasted GFM is employed to eliminate the chattering while the system uncertainty bounds are overestimated. Combining RSF with GFM, the proposed control technology yields a closed-loop AC power conditioning with low total harmonic distortion and fast transience under different types of loading, thus increasing the performance of the high-speed micromachining. Experimental are shown to certify the performance of the proposed control technology. (1)

System description
and u is the control signal. Owing to the unpredictability of the load condition, the parameters 2 a is almost impossible to know exactly, however to have reasonable approximations, the following assumptions are made as , where the 2 a represents nominal parameters of the system, and 2 a is parameter uncertainties. The bound of the 2 a can be given by . If the switching frequency is much higher than the fundamental frequency of the AC output, pm K is regarded as a proportional gain of a pulse-width modulation (PWM) full-bridge converter and equals tr s v Vˆ; tr v is the amplitude of the triangular wave tr v in the PWM. Once u is determined, by comparing u with tr v , the PWM gating signals is produced and controls the power switches. Based on the state-space averaging and linearization technique, the product of u and pm K is equivalent to output voltage i v of the full-bridge converter. In equation (1), the output voltage o v is desired to be maintained as close as possible to a sinusoidal reference voltage re v . Generally, in which rms V and r f are the root-mean-square and frequency values of the desired sinusoidal reference voltage, respectively. Therefore, the must be maintained, the design problem of the AC power conditioning can be regarded as a path-following control problem.
Define the tracking errors as (2) From (1) and (2)   As can be seen from (3), the control signal u must be designed so that 1 e and 2 e can be converged to zero. This paper proposes a RSF by analyzing Lyapunov stability criterion and designing RSF parameters carefully. The RSF will drive the system tracking error to converge to zero within finite time and is without singular problem, thus the closed-loop stability of the RSF can be guaranteed. However the loading is not fixed, the RSF system easily has the chattering problem and may present a high THD, especially when the loading is a large step change or an uncertainty or even a severe non-linear condition. Therefore, the GFM is employed to eliminate the chattering for producing higherperformance AC output voltage and the control design is represented in the following. The design concept of this proposed control technology is to modify the classic SMC by introducing nonsingularity criterion and GFM, so as to resolve infinite-time convergence and chattering.

Control technology design
For the error dynamics (3), the robust sliding function is defined as where 0 and 1 , 0 , 2 1 k k , e u denotes the equivalent control with non-singularity, and s u displays the sliding control for compensating the perturbation influences. Thus, the system will be driven to the sliding mode 0 and converged within finite time, and the perturbation Proof: Let us use the following Lyapunov candidate function: 1 is reached, then the states of the system (3) will converge to zero within finite time. However, FCSMC has chattering in AC power conditioning system design. This is because of the changeable load, so once the loading is a severe uncertain condition, the system (3) will not provide accurate tracking performance. Thus, the control signal ) (t u (5) is modified by the addition of the Fourier modified grey control ( gfm u ), which eliminates chattering in AC power conditioning system. The modeling steps of the GFM are described below.
Step 1: Input the original sample data sequence Letting the original data sequence be denoted as where ) 0 ( x stands for the set of n original sample data.
Step 2: Accumulated generating operation (AGO) By taking the AGO on ) 0 ( x , the following firstorder AGO sequence is expressed as Step 3: Grey model Based on the accumulated data sequence, x , a firstorder ordinary differential grey model, GM(1,1) is formed as where a and b stand for model coefficients, and need to be decided.
By employing MEAN generating operation to The following matrices can be presumed as We can solve the estimated parameters via means of least square method below.
To solve the grey differential equation, substituting (15) into (12), and the forecast output is computed as where '^' denotes forecasted value, and ) 1 ( ) 1 ( k x symbols the approximation solution of differential equation in (16).
Step 4: Inverse accumulated generating operation (IAGO) By the use of the IAGO, the data sequence ) ( ) 0 ( k x can be estimated as Let n k ,..., 2 , 1 , the forecasted value yields

ICPMMT 2017
To improve the accuracy of forecasting models, the Fourier series is used in modifying the residuals in grey forecasting model GM(1,1) so that a Fourier modified grey model (FGM(1,1)) can be obtained.
Step 5: Get the residual series from GM (1,1) Based on the forecasted series, a residual series is defined as where Step 6: Defining Fourier modified grey model (FGM(1,1)) The Fourier series can approximate the residual series as (20) where n k ,..., 3 , 2 , 1 n T and 1 ) 2 Therefore, the residual series is restated as PC r (21) where .
The parameters Step 7: Correct original forecast series The original forecast series of FGM ( f x ) can be corrected as Therefore, the control law of (5) is rewritten as where the added compensation component is Fourier modified grey control, gfm u that can eliminate the chattering. (26) where ) ( k represents for the forecasted value of ) (k , K is a constant, and symbols the system boundary.

Experimental results
To evaluate the performance of the proposed control technology, the results of the proposed control technology are compared with the results of the classic SMC. The system parameters are listed in Table 1.  Figure 2 and Fig. 3 show the output voltage and the load current of the AC power conditioning with the proposed control technology and the classic SMC, respectively, under full resistive load. Their outputvoltages are close to sinusoidal waveforms. In order to verify the control technology under transient circumstances, step load change with linear resistive load is explored. Figure 4 shows the waveform obtained using the proposed control technology under step load change from no load to full load. Note that the transient behaviour is satisfactory, i.e., the output voltage dip is small and the recovery time is very speedy. On the contrary, the waveform obtained using the classic SMC, displayed in Fig. 5 has a significant voltage dip and a slow recovery time at the firing angle. Under rectifier load shown in Fig. 6, the output-voltage waveform with the proposed control technology is almost sinusoidal (%THD=1.72%), but that with the classic SMC shown in Fig. 7 has a high %THD of 10.51%. Also, Fig. 8 plots the error convergence time and it clearly demonstrates that the proposed control technology does give for reaching 0 ) ( k in finite time and therefore reduces the distortion of the waveform.

Conclusions
This paper describes a high-performance AC power conditioning controlled high-speed micromachining by associating RSF with GFM. Classic SMC is intrinsically robust against internal parameter variations and external disturbances, but it will undergo infinite system-state convergence time. The RSF guarantees finite systemstate convergence time and is singularity-free. But, while the system uncertainty bounds are overestimated, the chattering may occur. A computationally fast and algorithmically easy GFM is employed to resolve the chattering problem. Experimental results display that THD, transient response and chattering elimination results from an AC power conditioning under the presented system exceed the results achieved under the SMC system with various loads.