Performance Analysis of Learning-Based Disturbance Observer for Pulsed Superconducting Cavity Field Control

Abstract

The use of Disturbance Observer-based (DOB) control is widespread in stabilizing the electromagnetic field in superconducting radio frequency cavities to facilitate beam acceleration in particle accelerators. Repetitive disturbances such as beam loading and Lorentz force cavity detuning are compensated by DOB control, and their suppression is enhanced through the incorporation of a learning scheme into the conventional disturbance observer. This paper evaluated the performance of a learning-based disturbance observer for compensating beam loading and cavity detuning in pulsed superconducting radio frequency cavities and proposes modifications for better field stability. A superconducting cavity baseband model for $\pi$-mode was simulated in Matlab/Simulink with a trapezoidal beam pulse as the input disturbance and different cavity detuning values to analyze the controllers’ performance. The simulations were conducted for multiple observer filter bandwidths to evaluate the performance of the learning-based disturbance observer under plant model uncertainties and different detuning values. The results demonstrate that the learning-based disturbance observer yields faster convergence to the reference input and lower tracking errors during the flat top of pulse voltage in comparison to conventional disturbance observer control.

1. Introduction

Superconducting radio frequency (SRF) cavities are widely used in particle accelerators due to their lower wall losses and higher accelerating gradients as compared to normal conducting cavities [1]. In many large accelerator projects such as the international linear collider [2], the next-generation light source (e.g., Linac Coherent Light Source (LCLSII) [3], European X-Ray Free-Electron Laser Facility (E-XFEL) [4] and Subcritical Hybrid Intense Neutron Emitter (SHINE) [5]), China Spallation Neutron Source (CSNS-II) [6], China initiative Accelerator Driven System (CiADS) [7], compact energy recovery linac (cERL) [8], and Chinese Accelerator Driven System Front-end (CAFe) [9] SRF cavities are used. The SRF cavity is a vital part of modern accelerators and the stability of the accelerating field inside the cavities is very important for the desired parameters of charged particle beams. For this purpose, field detectors are used to measure the accelerating field in radio frequency (RF) cavities and then suitable feedforward and feedback control techniques are applied. A generalized cavity field control architecture is shown in Figure 1. The pickup antenna on the cavity picks up an RF signal which is then down-converted to an intermediate frequency (IF) by mixing it with a local oscillator signal (LO). The resulting IF signal is then processed by an IQ detector to separate the I and Q components. These components are subsequently filtered using an infinite impulse response (IIR) filter, before being compared to values in a set point table. Any errors generated by this comparison are then fed to the controller, which in this case is a proportional integrator (PI) controller and a disturbance observer-based (DOB) controller. The outputs of these controllers are the I and Q components, which are converted back into an IF signal by the IQ-Demodulator, and are then up-converted to an RF signal. This RF signal is then sent to the preamplifier to drive the high-power RF source (such as a klystron). The output of the klystron, via the waveguide and directional coupler, powers the RF cavity through the input power coupler, completing the feedback loop. In RF particle accelerators, high power RF sources such as klystrons and magnetrons are used to power accelerating cavities. The high power oscillating electric field inside the cavity is responsible for the acceleration of the charged particle beam introduced in the cavity. Since the magnitude of the field is time-varying, the beam must be introduced in the cavity at a particular phase to obtain the desired energy or acceleration. If the magnitude and phase of the RF field are not stable, then the quality of the beam will suffer, as different bunches of the charged particle beam will receive different amounts of energy. The stable operation of a superconducting accelerator requires the identification and control of sources of perturbation in the RF field inside the cavity. Superconducting cavities are narrow bandwidth cavities and the main sources of field error in superconducting cavities are Lorentz force detuning [10] and beam loading [11]. In SRF cavities, disturbances can be classified into two categories—predictable and unpredictable. Lorentz force detuning and beam loading are predictable disturbances that are repeated in each RF pulse, while microphonics are unpredictable and are generally not synchronized with the RF system [11,12]. We used a standard cavity model and the parameters were chosen according to a disturbance observer for a superconducting accelerator, the compact energy recovery linac [13]. In this paper, a trapezoidal beam was simulated as the disturbance to the cavity accelerating field [13,14] and the beam loading effect was compensated by the disturbance observer and learning-based disturbance observer control.

The remainder of the article is structured as follows. The existing research literature about DOB controllers and related work on cavity field control problems are discussed in Section 1.1. In Section 2, a system overview is given including the cavity model, DOB, and learning-based DOB control, and the proposed modified learning-based DOB for cavity field control is explained. The simulation results and performance comparison of DOB and modified learning-based DOB are presented in Section 3. The conclusion and future work are given in Section 4.

1.1. Literature Review

During the last two decades, due to rapid development in digital signal processing (DSP), many researchers have presented digital control algorithms for field control problems in particle accelerator cavities [15]. DOB is a powerful technique for estimating and compensating for external disturbances without utilizing additional sensors. It has been widely used in many applications including in quad-copters [16,17], lasers [18], UAVs [19], and robotics [20]. DOB problems can be formulated and solved using transfer functions, state space equations and input–output data sets used for modeling systems in the framework of data-driven approaches. In this paper, frequency-domain DOB and its iterative learning form will be explored for the pulsed superconducting cavity field control problem. An overview of the existing research and low-level RF control setup are discussed.

Feng et al. presented a comprehensive analysis of the low-level radio frequency (LLRF) control, including the implementation of DOB control for pulsed superconducting LINAC [13]. Their study revealed that at an accelerating voltage of 10.5 MV and Lorentz force detuning (LFD) drift affected both the RF phase and magnitude, in contrast to 7.5 MV. The study also showed that the cavity voltage magnitude was less sensitive to detuning drift at resonance as compared to the cavity voltage phase. The PI-feedback control alone was not sufficient to reject the detuning drifts and compensate long beam pulse loading. In principle, beam loading can be compensated by high feedback gains but this approach increases the risk of an unstable system and increases the noise level of high-frequency components. RF trips occur because of high feedback gain so higher feedback gains are avoided.

In DOB control, the inverse of the nominal plant is required to estimate the disturbance, and often the plant inverse is not realizable. To tackle this problem, a filter “Q” is used to make the inverse realizable. Although DOB control can compensate disturbances that are either predictable or unpredictable, the Q filter’s bandwidth limits its performance [13]. Iterative learning control (ILC) can compensate predictable (repetitive) disturbances and it is not bandwidth-limited. To utilize the advantages of both DOB control and ILC control, Feng et al. [21] demonstrated a combination of both algorithms in a cavity simulator-based test bench, to compensate for predictable or repetitive disturbances (LFD and beam loading etc.) and unpredictable disturbances (microphonics etc.). In the ILC approach, error information from the last cycle is stored in the memory and the current cycle error is improved by the stored information. Many advanced ILC algorithms are in practice, depending upon the requirements of fast convergence rate or robustness. In [21], a plant-inversion-based algorithm was selected due to its fast convergence rate. The results show that, by using only proportional (P) control, the effect of beam loading was approximately −0.25% in the cavity voltage magnitude and the cavity voltage phase waveform had a tilt. The application of P + DOB control resulted in the compensation of these beam loading effects but, during the beam pulse transients, the effect in magnitude was not compensated. This was because the very fast rising and falling edge of the beam induces higher frequency components and the Q-filter in DOB control had a 3 KHz bandwidth. This beam loading effect was compensated by P+ILC control but the phase waveform still had a tilt, which may be due to microphonics or a 10% beam fluctuation in the simulator model. In the case of P + DOB + ILC control, both the cavity voltage magnitude and phase variation due to the beam loading effect were perfectly compensated. A zero-phase FIR filter was used with ILC control, which improved the performance of ILC control.

In another paper by Zheqiao Geng [22], a disturbance observer-based technique was simulated for both disturbance rejection and cavity modeling. Beam loading, Lorentz force detuning, microphonics and uncertainty in measurements were considered as disturbances, so the active disturbance rejection control (ADRC) was used as a robust control. ADRC and DOB are both control techniques used to improve tracking performance and disturbance rejection. In ADRC, an extended state observer is used to estimate the disturbance and in DOB, a nominal plant model is used for disturbance estimation so its performance is highly dependent on accurate plant modeling. ADRC is more robust to uncertainties while DOB is more simple and easy to implement as compared to ADRC. The cavity parameters are estimated in runtime, which could be used for the optimization of the control process. To make the ADRC effective, the observer was designed such that its poles are ten times faster than the closed loop system poles. ADRC provided promising results as an alternative to the existing PI control. Simulation results showed that ADRC resulted in better suppression of voltage errors as compared to proportional feedback control in both phase and magnitude during the flat portion of the pulse; the ADRC also provided a faster rise time as compared to proportional feedback control. For beam loading, compensation feed-forward control was considered. Initially, for the first pulse, the beam cavity parameter was not estimated so the the feed-forward signal was calculated by the desired beam current and, for the rest of the pulses, the feed-forward signal was refined by the beam voltage estimated by the disturbance observer. It was proposed that, for a better beam phase measurement, many pulse results should be averaged. In this paper, 100 pulses results were averaged for the calculation of magnitude and phase jitter. Zhang et al. proposed a modified active disturbance rejection control (ADRC), which exhibits a faster response, less overshoot, and lower tracking error. This modified ADRC could be investigated as a potential solution for field control problems [23].

In another paper, Dongbing Li et al. [24] used disturbance observer-based control for the suppression of low-frequency microphonics. The nominal plant model used was a first-order low pass filter, assuming resonance operation of the cavity. Therefore, the only parameter used in the plant model transfer function was the cavity half bandwidth, and the plant model was not estimated in runtime. Three types of filters with the same bandwidth were studied: first-order, second-order, and third-order. The third-order filter gave the best results at the cost of more computations. Therefore, the second effective filter for suppressing noise was selected as a compromise between noise suppression and computational complexity. A higher bandwidth filter showed better results in suppressing noise at low frequencies but may induce instabilities. The results clearly showed that PI + DOB control suppresses the microphonics noise, and the desired magnitude and phase stability were achieved.

In [25], Zheqiao Geng considered the practical aspects of LLRF. The author discussed the use of the ILC algorithm for RF pulse flattening and its limitations. The ILC algorithm can be used to flatten small ripples in the pulse but not for large step variations. Therefore, it is activated after the filling time of the cavity in the middle portion of the pulse. The continuous operation of the ILC algorithm was not recommended as the error would accumulate from pulse to pulse.

In another paper by C.Y. Xu et al. [26], feed-forward control was simulated for the CiADS proton accelerator to compensate for beam loading. The study aimed to determine the threshold limit of beam parameter variation that could be compensated by feed-forward control to meet desired stability specifications. The parameters investigated included the beam current ripple frequency for different beam current magnitudes and the beam arrival time mismatch with feed-forward control activation.

In most of the discussed research, the conventional disturbance observer-based control was used for the compensation of cavity field disturbances. Disturbance observer control in combination with iterative learning control was used by Feng et al. [21], but both techniques were applied independently; iterative learning was not used to enhance the performance of the disturbance observer for improved estimation of disturbances and, furthermore, iterative learning control was applied from pulse to pulse, i.e., the error from the previous pulse was used to modify the control input in the current pulse. A learning-based disturbance observer was proposed by Zheng et al. [27] and was applied for UAV attitude control. The use of a learning-based disturbance observer (DOB) to improve cavity field stability in particle accelerators in the presence of beam loading and cavity detuning has not been previously explored, creating a research gap. Our study demonstrates that the application of a learning-based DOB to address this problem has the potential to significantly improve the stability of the cavity voltage.

2. Materials and Methods

The cavity’s model for $\pi$-mode is simulated in Simulink/MATLAB for the performance analysis of control algorithms including the PI controller, DOB controller and learning-based DOB controller. In this section, the mathematical model of the superconducting RF cavity and the conventional disturbance observer with cavity control loop is explained.

2.1. Cavity Model

Superconducting cavities can be a single cell or multiple cells, depending upon the accelerator design and applications. We used a standard cavity model for the simulations, available in the literature [28,29,30]. The parallel RLC circuit cavity model coupled to an RF source via an input coupler is the basis for analyzing multicell cavities and their properties [28]. The RF source and the beam are modeled as current sources. The input coupler is modeled by a $1:n$ transformer and the transmission line characteristic impedance is denoted by $Z_0$. The equivalent circuit by transferring the RF source to the cavity side is shown in Figure 2, where, the RF cavity is represented by a parallel combination of Resistance ‘R’, Capacitance ‘C’, and inductance ‘L’. The phasor ${\mathbf{i}}_{rf}$ is the drive current by RF power source, ${\mathbf{i}}_b$ represents the beam current. The phasor ${\mathbf{v}}_c$ is the resulting cavity voltage and the phasor ${\mathbf{i}}_c$ is the overall drive current given by

$${\mathbf{i}}_c={\mathbf{i}}_{rf}+{\mathbf{i}}_b.$$

(1)

The RF accelerator cavity is usually characterized by its loaded quality factor ($Q_L$), the cavity resonance frequency and the normalized shunt impedance (r/Q). The mathematical relations between RLC circuit elements and typical parameters of an accelerator cavity are given in

Table 1. Cavity circuit model parameters relations with typical cavity parameters.

Cavity Parameter	Symbol	Relation with Circuit Parameter
Resonance Frequency	$w_o$	$w_o=1/\sqrt{LC}$
Unloaded Quality Factor	$Q_o$	$Q_o=\frac{w_{o}W}{P_{cav}}=w_{o}RC=\frac{R}{L{w}_o}$
External Quality Factor	$Q_{ext}$	$Q_o=\frac{w_{o}W}{P_{ext}}=w_o{n}^2{Z}_{o}C$
Coupling Factor	$\beta$	$\beta =\frac{P_{ext}}{P_{cav}}=\frac{Q_o}{Q_{ext}}=\frac{R}{n^2{Z}_o}$
Loaded Quality Factor	$Q_L$	$Q_L=\frac{Q_O}{1+\beta }$
Normalized Shunt Impedance	$r/Q$	$R=\frac{1}{2}(r/Q)Q_o$
Loaded Resistance	$R_L$	$R_L=\frac{R}{1+\beta }=\frac{1}{2}(r/Q)Q_L$
Half Bandwidth	$w_{1/2}$	$w1/2=\frac{w_o}{2Q_L}$

, where:

• W = The electromagnetic energy stored in the cavity;
• $P_{cav}$ = The dissipated RF power in the cavity wall; $P_{ext}$ = The RF power reflected back to transmission line from the cavity.

The phasor Laplace transform can be used to analyze this model due to complex impedances of RLC circuit. The impedances of inductor and capacitor in the RLC model and their transformation from the s to the $\widehat{s}$ domain are presented in

Table 2. s and $\widehat{s}$ domain representation of impedance in cavity circuit model.

Element	Symbol	Impedance in s Domain	Impedance in $\widehat{\mathit{s}}$ Domain
Resistance	R	R	R
Capacitance	C	$1/sC$	$1/(\widehat{s}+j{w}_c)C$
Inductance	L	$sL$	$(\widehat{s}+j{w}_c)C$

, where the carrier frequency is equal to the input radio frequency [30]. The cavity’s phasor transfer function ${\mathbf{G}}_c\left(\widehat{s}\right)$ is given by

$$\begin{array}{cc}\hfill {\mathbf{G}}_c\left(\widehat{s}\right)& =\frac{{\mathbf{V}}_c\left(\widehat{s}\right)}{{\mathbf{I}}_c\left(\widehat{s}\right)}\hfill \\ \hfill & =\frac{(\widehat{s}+j{w}_c)/C}{{(\widehat{s}+j{w}_c)}^2+(\widehat{s}+j{w}_c)/\left(R_{L}C\right)+1/\left(LC\right)},\hfill \end{array}$$

(2)

where ${\mathbf{I}}_c\left(\widehat{s}\right)$ and ${\mathbf{V}}_c\left(\widehat{s}\right)$ are the phasor Laplace transform of the ${\mathbf{i}}_c\left(input\right)$ and the ${\mathbf{v}}_c\left(output\right)$. Using the relations in Table 1, the transfer function with cavity parameters can be represented as

$${\mathbf{G}}_c\left(\widehat{s}\right)=\frac{2w_{1/2}{R}_L(\widehat{s}+j{w}_c)}{{(\widehat{s}+j{w}_c)}^2+2w_{1/2}{(\widehat{s}+j{w}_c)}^2+w_o^2}.$$

(3)

The two poles of ${\mathbf{G}}_c\left(\widehat{s}\right)$ are:

$${\widehat{s}}_{1,2}=-w_{1/2}-j{w}_c\pm j\sqrt{w_o^2-w_{1/2}^2}.$$

(4)

Since SRF cavities are narrow bandwidth cavities and the resonance frequency is much higher than the cavity half bandwidth ($w_{1/2}\ll w_o$), the poles can be written as

$${\widehat{s}}_1\approx -w_{1/2}+j(w_o-w_c)$$

(5)

$${\widehat{s}}_2\approx -w_{1/2}-j(w_o+w_c)$$

(6)

$$\Delta w=w_o-w_c.$$

(7)

Considering slow variations in the RF signal envelop and $\Delta w<<w_o$ [30], Equation (3) can be simplified as

$${\mathbf{G}}_c\left(\widehat{s}\right)=\frac{w_{1/2}{R}_L}{\widehat{s}+w_{1/2}-j\Delta w}.$$

(8)

The first order transfer function has an asymmetric frequency response due to the presence of a complex pole. In the time domain, the cavity’s transfer function corresponds to the following differential equation:

$${\dot{\mathbf{v}}}_c+(w_{1/2}-j\Delta w)v_c=w_{1/2}{\mathbf{R}}_L{\mathbf{i}}_c.$$

(9)

By representing input and output in I and Q components, the above model can be represented as a MIMO model in which I and Q components are coupled to each other, resulting in the state space representations, given as:

$$\begin{array}{c}\begin{array}{c}\hfill \left[\begin{array}{c}{\dot{v}}_{CI}\\ {\dot{v}}_{CQ}\end{array}\right]=\left[\begin{array}{cc}-w_{1/2}& -\Delta w\\ \Delta w& -w_{1/2}\end{array}\right]\left[\begin{array}{c}{v}_{CI}\\ v_{CQ}\end{array}\right]\\ \hfill +w_{1/2}{R}_L\left[\begin{array}{c}{i}_{CI}\\ i_{CQ},\end{array}\right]\end{array}\end{array}$$

(10)

where ${\mathbf{i}}_C=i_{CI}+j{i}_{CQ}$ and ${\mathbf{v}}_C=v_{CI}+j{v}_{CQ}$. The above state space model can be represented in following form, with $u_I$ and $u_Q$ as real and imaginary control inputs to the cavity plant transfer function.

$$\begin{array}{c}\begin{array}{c}\hfill \left[\begin{array}{c}{\dot{v}}_{CI}\\ {\dot{v}}_{CQ}\end{array}\right]=\left[\begin{array}{cc}-w_{1/2}& -\Delta w\\ \Delta w& -w_{1/2}\end{array}\right]\left[\begin{array}{c}{v}_{CI}\\ v_{CQ}\end{array}\right]\\ \hfill +\left[\begin{array}{cc}{w}_{1/2}& 0\\ 0& w_{1/2}\end{array}\right]\left[\begin{array}{c}{u}_I\\ u_Q.\end{array}\right]\end{array}\end{array}$$

(11)

The cavity electrical model transfer function is given in Equation (12), which is further split into two transfer functions to separate the real and imaginary parts, because MATLAB/Simulink does not handle the complex coefficient transfer function. The transfer function is multiplied and divided by the complex conjugate of its poles as given under:

$${\mathbf{G}}_c\left(\widehat{s}\right)=\frac{w_{1/2}{R}_L(\widehat{s}+w_{1/2}+j\Delta w)}{{\widehat{s}}^2+2w_{1/2}\widehat{s}+w_{1/2}^2+\Delta w^2}.$$

(12)

The above equation can be written as

$${\mathbf{G}}_c\left(\widehat{s}\right)={\mathbf{G}}_R\left(\widehat{s}\right)+j{\mathbf{G}}_I\left(\widehat{s}\right),$$

(13)

where

$${\mathbf{G}}_R\left(\widehat{s}\right)=\frac{w_{1/2}{R}_L(\widehat{s}+w_{1/2})}{\widehat{s}+2w_{1/2}\widehat{s}+w_{1/2}^2+\Delta w^2}$$

(14)

$${\mathbf{G}}_I\left(\widehat{s}\right)=\frac{w_{1/2}\Delta w{R}_L}{\widehat{s}+2w_{1/2}\widehat{s}+w_{1/2}^2+\Delta w^2},$$

(15)

where ${\mathbf{G}}_R\left(\widehat{s}\right)$ and ${\mathbf{G}}_I\left(\widehat{s}\right)$ are real and imaginary parts of the cavity transfer function in Equation (12) and the coupling between these transfer functions is depicted in Figure 3. Here, $w_{1⁄2}$ and $R_L$ are assumed to be constant and the cavity detuning $\Delta w$ is variable due to Lorentz force and microphonics. If the cavity detuning $\Delta w$ is zero then IQ-loops will be decoupled.

2.2. DOB Control

The traditional DOB control block diagram with plant transfer function ${\mathbf{G}}_p\left(s\right)$, nominal plant transfer function ${\mathbf{G}}_n\left(s\right)$ and filter $\mathbf{Q}\left(s\right)$ is shown in Figure 4. In designing a conventional DOB, the inverse of the plant transfer function is required and the filter $\mathbf{Q}\left(s\right)$ is designed for the realizable plant inverse. Disturbance estimation may contain errors because the nominal plant transfer function ${\mathbf{G}}_n\left(s\right)$ used in DOB design could differ from the actual plant ${\mathbf{G}}_p\left(s\right)$. The bandwidth of filter $\mathbf{Q}\left(s\right)$ further limits the performance of DOB, as the disturbances with a higher frequency than the bandwidth of filter $\mathbf{Q}\left(s\right)$ will not be estimated and so would not be compensated [31].

2.3. Learning-Based DOB

Learning-based DOB was proposed by Zheng et al. [27]. The iterative learning control (ILC) is the motivation for learning-based DOB. It is particularly useful for repetitive disturbances. In our case, beamloading and Lorentz force detuning are repetitive disturbances. In learning-based DOB, the estimation of disturbance by conventional DOB ${\widehat{d}}^o$ is improved by the term ${\widehat{d}}_i^f$, which is computed by the historical data of the system. The learning-based DOB framework is presented in Figure 5. The disturbance estimate $\widehat{d}$ is given by

$$\widehat{d}={\widehat{d}}^f+{\widehat{d}}^o.$$

(16)

In the learning scheme, the recursive learning term computed from the previous iteration is added to the current iteration. The learning law for the iterative learning scheme is given as under

$${\widehat{d}}_{i+1}^f={\widehat{d}}_i^f+\mathbf{L}\left(z\right)e_i,$$

(17)

where $\mathbf{L}\left(z\right)$ is the learning function, ${\widehat{d}}_{i+1}^f$ is the learning term in thw current iteration, and ${\widehat{d}}_i^f$ and $e_i$ are the learning term and tracking error in the previous iteration, respectively. If ${\mathbf{G}}_r\left(z\right)=\frac{y}{r}$, ${\mathbf{G}}_f\left(z\right)=\frac{y}{{\widehat{d}}^f}$ and ${\mathbf{G}}_d\left(z\right)=\frac{y}{d}$, then the output y is given by

$$y={\mathbf{G}}_r\left(z\right)r+{\mathbf{G}}_f\left(z\right){\widehat{d}}^f+{\mathbf{G}}_d\left(z\right)d.$$

(18)

For the $i^{th}$ iteration output, $y_i$ would be

$$y_i={\mathbf{G}}_r\left(z\right)r_i+{\mathbf{G}}_f\left(z\right){\widehat{d}}_i^f+{\mathbf{G}}_d\left(z\right)d_i.$$

(19)

Similarly for ‘$i+1$’, iteration output $y_{i+1}$ would be

$$\begin{array}{cc}\hfill y_{i+1}& ={\mathbf{G}}_r\left(z\right)r_{i+1}+{\mathbf{G}}_f\left(z\right){\widehat{d}}_{i+1}^f+{\mathbf{G}}_d\left(z\right)d_{i+1}\hfill \\ \hfill & ={\mathbf{G}}_r\left(z\right)r_{i+1}+{\mathbf{G}}_f\left(z\right)[{\widehat{d}}_i^f+\mathbf{L}\left(z\right)(r_i-y_i)]\hfill \\ \hfill & +{\mathbf{G}}_d\left(z\right)d_{i+1}.\hfill \end{array}$$

(20)

From Equation (19) we can write

$${\mathbf{G}}_f{\widehat{d}}^f=-{\mathbf{G}}_r\left(z\right)r_i-{\mathbf{G}}_d\left(z\right)d_{i+1}+y_i.$$

(21)

Substituting the value in Equation (20), we get

$$\begin{array}{c}\hfill y_{i+1}=[1-{\mathbf{G}}_f\left(z\right)\mathbf{L}\left(z\right)]y_i+{\mathbf{G}}_r\left(z\right)(r_{i+1}-r_i)\\ \hfill +{\mathbf{G}}_d\left(z\right)(d_{i+1}-d_i)+{\mathbf{G}}_f\left(z\right)\mathbf{L}\left(z\right)r_i.\end{array}$$

(22)

Now, the tracking error is given by

$$e_{i+1}=r_{i+1}-y_{i+1}$$

(23)

$$\begin{array}{c}\hfill e_{i+1}=[1-{\mathbf{G}}_f\left(z\right)\mathbf{L}\left(z\right)](r_i-y_i)-[1-{\mathbf{G}}_r\left(z\right)](r_{i+1}-r_i)\\ \hfill -{\mathbf{G}}_d\left(z\right)(d_{i+1}-d_i).\end{array}$$

(24)

Assuming r and d are consistent over iterations then $r_{i+1}=r_i$ and $d_{i+1}=d_i$, resulting in

$$\begin{array}{cc}\hfill e_{i+1}& =[1-{\mathbf{G}}_f\left(z\right)\mathbf{L}\left(z\right)](r_i-y_i)\hfill \\ \hfill & =[1-{\mathbf{G}}_f\left(z\right)\mathbf{L}\left(z\right)]e_i.\hfill \end{array}$$

(25)

Learning convergence is achieved by iteratively refining disturbance estimation. The learning function is designed in such a way that the tracking errors will be reduced with time. This can be described as an infinity-norm minimization problem:

$$\parallel 1-{\mathbf{G}}_f{\left(z\right)\mathbf{L}\left(z\right)\parallel }_{\infty }<1.$$

(26)

2.4. Modified Learning-Based DOB for Field Control Problem

The iterative learning Equation (17), when applied to a field control problem in a superconducting accelerator cavity model with a PI controller, resulted in an unstable system. Therefore, the learning law is modified by introducing recursive coefficient a, and the modified learning equation is given as

$${\widehat{d}}_{i+1}^f=a{\widehat{d}}_i^f+\mathbf{L}\left(z\right)e_i.$$

(27)

In the simulation when we choose the recursive coefficient ‘a = 1’, the cavity voltage grows exponentially after the start of the iterative algorithm, clearly showing the unstable system. With the set parameters of the PI gain, sampling rate, and cavity dynamics, the value of the recursive coefficient ‘a = 0.9’ resulted in a stable cavity voltage. We incorporated learning-based DOB and DOB with a PI controller, shown in the block diagram given in Figure 5. Secondly, we activated the learning scheme after the transient peak of cavity voltage to avoid excessive overshoot in cavity voltage response, and gradually increased the recursive coefficient ‘a’ to its final value ‘0.9’ for a better transient response. By using the modified learning Equation (27), Equation (22) will be changed to

$$\begin{array}{cc}\hfill y_{i+1}& =[a-{\mathbf{G}}_f\left(z\right)\mathbf{L}\left(z\right)]y_i+{\mathbf{G}}_r\left(z\right)(r_{i+1}-a{r}_i)\hfill \\ \hfill & +{\mathbf{G}}_d\left(z\right)(d_{i+1}-a{d}_i)+{\mathbf{G}}_f\left(z\right)\mathbf{L}\left(z\right)r_i;\hfill \end{array}$$

(28)

with the above updated equation, the error in Equation (25) will change to

$$\begin{array}{cc}\hfill e_{i+1}& =[a-{\mathbf{G}}_f\left(z\right)\mathbf{L}\left(z\right)]e_i+[1-{\mathbf{G}}_r\left(z\right)](r_{i+1}-a{r}_i)\hfill \\ \hfill & -{\mathbf{G}}_d\left(z\right)(d_{i+1}-a{d}_i).\hfill \end{array}$$

(29)

For consistent values of r and d over iteration as assumed for Equation (25), for a = 0.9, we can write

$$\begin{array}{cc}\hfill e_{i+1}& =[0.9-{\mathbf{G}}_f\left(z\right)\mathbf{L}\left(z\right)]e_i+[1-{\mathbf{G}}_r\left(z\right)]\left(0.1r_i\right)\hfill \\ \hfill & -{\mathbf{G}}_d\left(z\right)\left(0.1d_i\right).\hfill \end{array}$$

(30)

Since, in the absence of disturbance, the system is tracking well, the reference input in the steady state, ${\mathbf{G}}_r$, is approximately equal to one; therefore, $[1-{\mathbf{G}}_r\left(z\right)]\left(0.1r_i\right)$ would not contribute much to the error in each iteration and its effect is also reduced by the ${\mathbf{G}}_d\left(z\right)\left(0.1d_i\right)$ term, which is verified by the results presented in Section 3. In recursive error Equation (30), the dominant term is $[0.9-{\mathbf{G}}_f\left(z\right)\mathbf{L}\left(z\right)]$. The smaller the magnitude of this term, the faster the convergence of the learning scheme, as the tracking error in each iteration will be reduced. In Figure 6, a bode magnitude diagram of $[0.9-{\mathbf{G}}_f\left(z\right)\mathbf{L}\left(z\right)]$ is plotted for different values of L = 1:0.25:5. It is evident that below 20 KHz, the convergence of the learning scheme would be much better, so the low-frequency disturbances will be compensated much faster than the higher frequency disturbances. Beam loading is a phenomenon that occurs when a beam of charged particles, such as electrons or protons, is accelerated in a particle accelerator. As the beam travels through the accelerator, it can interact with the structure of the accelerator, causing some of the energy from the beam to be transferred to the accelerator structure. In the case of beam loading, the higher frequency components are only present at the beginning and end of the beam pulse. This can be mitigated through feed-forward control or by reducing the steepness of the rising and falling edges of the beam pulse.

3. Results and Discussion

The cavity baseband model for $\pi$-mode was simulated in MATLAB/Simulink with PI, DOB and learning-based DOB controllers. PI gains $K_p=80$ and $K_I=1.3\times {10}^5$ were used [13]. For disturbance observer-based (DOB) control, we assumed the nominal plant model with the mismatch in cavity half bandwidth and a $-20$ Hz offset in cavity detuning, 1 $\mathsf{\mu }$s loop delay and 50 Hz cavity half bandwidth. For better transient results, the PI and DOB gains were gradually increased during the rise time of the pulse. A trapezoidal beam pulse was introduced during the flat portion of the voltage pulse [14], having a 66 $\mathsf{\mu }$A peak current [13] as shown in Figure 7. A simplified rectangular beam pulse profile was used by Chengye et al. in a proposed modified iterative learning algorithm for cavity field control problem, to implement the algorithm inside a Field Programmable Gate Array (FPGA) [32].

For the simulation, we used a low-pass second order $\mathbf{Q}\left(s\right)$ filter given by

$$\mathbf{Q}\left(s\right)=\frac{1}{{(\tau s+1)}^2},$$

(31)

where $\tau$ defines the filter bandwidth. In the case of the sharp rising and falling edge of the beam pulse, a higher bandwidth filter is required in the disturbance observer, otherwise higher frequency components in the disturbance would not be estimated or compensated. If the sampling time is $T_s$, then the actual system’s time delay $T_d$ is modeled as digital delays equal to the floor value of ratio of $T_d$ and $T_s$, given as

$$Delay=z^{-k},wherek=\lfloor \frac{T_d}{T_s}\rfloor .$$

(32)

In Figure 8, the cavity voltage magnitude for DOB + PI and learning-based DOB + PI controllers are plotted. The reference tracking convergence time and steady-state tracking error are much better in the case of learning-based DOB as compared to DOB control. Due to faster convergence, flat top duration is increased and a wider beam pulse load can be managed. The cavity voltage phase is plotted in Figure 9, showing better tracking of the reference phase in the case of learning-based DOB + PI as compared to a conventional DOB + PI controller. The control inputs, highlighting the flat top or beam loading duration, for the case of LDOB + PI and DOB + PI control, are plotted in Figure 10; here, the value of the recursive coefficient “$a=0.9$” was used for the simulation. In

Table 3. Performance of conventional DOB and learning-based DOB in presence of beam loading and different cavity detuning.

Detuning ${\mathrm{w}}_{\mathrm{d}}$	PI + DOB		PI + LDOB
	$\mathrm{Q}=8\mathrm{k}\mathrm{H}\mathrm{z}$		$\mathrm{Q}=8\mathrm{k}\mathrm{H}\mathrm{z}$		$\mathrm{Q}=16\mathrm{k}\mathrm{H}\mathrm{z}$		$\mathrm{Q}=32\mathrm{k}\mathrm{H}\mathrm{z}$
	$\Delta \mathrm{A}/\mathrm{A}(\%\mathrm{rms})$	$\Delta \Phi$	$\Delta \mathrm{A}/\mathrm{A}(\%\mathrm{rms})$	$\Delta \Phi$	$\Delta \mathrm{A}/\mathrm{A}(\%\mathrm{rms})$	$\Delta \Phi$	$\Delta \mathrm{A}/\mathrm{A}(\%\mathrm{rms})$	$\Delta \Phi$
0 Hz	$0.1502$	0	$0.0488$	0	$0.0195$	0	$0.0082$	0
$-1$ Hz	$0.1484$	$5.2\times {10}^{-4}$	$0.0480$	$4.96\times {10}^{-4}$	$0.0188$	$4.95\times {10}^{-4}$	$0.0088$	$4.94\times {10}^{-4}$
$-5$ Hz	$0.1400$	$0.0026$	$0.0447$	$0.0025$	$0.0178$	$0.0025$	$0.0143$	$0.0025$
$-10$ Hz	$0.1275$	$0.0052$	$0.0426$	$0.0050$	$0.0225$	$0.0050$	$0.0251$	$0.0049$
$-15$ Hz	$0.1132$	$0.0078$	$0.0448$	$0.0075$	$0.0334$	$0.0074$	$0.0389$	$0.0074$
$-20$ Hz	$0.0970$	$0.0104$	$0.0526$	$0.0099$	$0.0483$	$0.0090$	$0.0550$	$0.0099$
$-25$ Hz	$0.0780$	$0.0130$	$0.0658$	$0.0124$	$0.0662$	$0.0124$	$0.0730$	$0.0124$

, a comparison of the performance of DOB and learning-based DOB is presented. The learning function was tuned to the value of $4.4$ for $-10$ Hz cavity detuning; for other cavity detuning, it needs to be optimized, which is why, for $L=4.4$ at a detuning of $-10$ Hz, learning-based DOB produces the best results and the variation in cavity voltage magnitude during the flat top is reduced to 33% of the voltage variation in the case of a conventional disturbance observer. The overall performance in the magnitude and phase stability of learning-based DOB + PI control is much better than with conventional DOB + PI control. From the results presented, it is evident that learning-based DOB provides a better magnitude and phase stability in the superconducting cavity field control problem as compared to conventional DOB.

Qiu et al. [13] present the performance of a conventional disturbance observer (DOB) with proportional control for beam loading compensation, which is found to be better than using proportional (P) only or proportional integral (PI) only control. Proportional control without integral action results in a steady-state error, which can be improved by using feed-forward control. Furthermore, the flat top portion is smaller in proportional plus DOB control compared to PI plus DOB control, making it unsuitable for longer beam pulse operations. We conducted simulations for PI plus DOB and PI plus learner-based DOB and observed improvements in magnitude and phase stability, as presented in Table 3.

Michizono et al. [21] tested iterative learning control (ILC) plus DOB control in the RF cavity simulator test bench. However, in ILC, data from the previous pulse were used to improve the current pulse, and the performance of the disturbance observer was not improved during the current pulse operation. In learning-based DOB, the error is reduced by iterative learning during the current pulse. In the future, a combination of learning-based disturbance observer and iterative learning control could be tested for further improvement of cavity voltage stability from pulse to pulse.

For different values of recursive coefficient ‘a’, the learning- based DOB was simulated and the cavity voltage was plotted as shown in Figure 11. It is apparent that, when the recursive coefficient ‘a’ is increased, the tracking error is reduced compared to when ‘a’ has smaller values, during the flat top of the pulse. The percentage overshoot and voltage magnitude stability during flat top are given in

Table 4. Effect of recursive coefficient a on overshoot and magnitude stability.

a	%Overshoot	$\mathbf{\Delta }$A/A (% rms)
0.9	3.88	0.0447
0.8	2.47	0.0849
0.7	2.17	0.1067
0.6	2.05	0.1161
0.5	1.97	0.1213
0.4	1.93	0.1245
0.3	1.90	0.1267
0.2	1.88	0.1283
0.1	1.86	0.1295

. The percentage overshoot during transient response is lower at lower values of the recursive coefficient ‘a’ in the learning scheme. However, $\Delta \mathbf{A}/\mathbf{A}$ (%rms) is degraded, so there is a tradeoff between the transient overshoot voltage that the cavity can withstand and the performance of the learning scheme. One way to avoid excess cavity voltage during the transient is to make the recursive coefficient variable and switch it on after the transient peak, gradually increasing to its final value. In Figure 12, the cavity voltage magnitude is plotted, and the recursive coefficient ‘a’ is switched on at 1.5 msec. Gradually, its value is increased to 0.9 at 2 msec. By doing so, the percentage overshoot during the transient is decreased by a considerable amount, and $\Delta \mathbf{A}/\mathbf{A}$ (%rms) during beam time is also acceptable compared to DOB control.

4. Conclusions

This paper presented a performance analysis of a learning-based disturbance observer controller for compensating beam loading and cavity detuning in pulsed superconducting radio frequency cavities. The cavity standard $\pi$-mode model, with a PI controller and both the observers, i.e., conventional DOB and learning-based DOB, were simulated in MATLAB/Simulink. A modified learning-based DOB was proposed to address stability issues and transient additional overshoot problems with the existing learning-based DOB. Simulation results demonstrated that the proposed learning-based DOB outperformed the conventional DOB controller in terms of disturbance estimation and cancellation, resulting in a $33\%$ reduction in cavity voltage magnitude variation during the flat top. Additionally, the cavity voltage phase variation was slightly improved with the use of a learning-based DOB. The learning function may need to be optimized for better results at different cavity detuning values. To compensate for additional overshoot in the transient response, the recursive coefficient in the learning scheme was varied and the learning scheme was switched after the transient peak time, gradually increasing the recursive coefficient to its final value. The effect of measurement noise and other disturbances, such as power supply ripples and microphonics, and higher-order adaptive disturbance observer filters, will be studied in future research.