Back recursive estimation of unknown frequency sinusoidal disturbance in superconducting RF cavities

ISA Transactions xxx (xxxx) xxx

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Research article

Back recursive estimation of unknown frequency sinusoidal disturbance in superconducting RF cavities ∗

Liping Zhang a , Xinyu Wen a,b , , Baoguang Wang b , Peng Zhao a , An Sun c a

Key Laboratory of Road Construction Technology and Equipment of MOE, Changan University, Xi’an, China School of Electronic Information Engineering, Taiyuan University of Science and Technology, Taiyuan, 030024, China c Institute of Energy Science, Nanjing University, Nanjing 210046, China b

article

info

Article history: Received 19 November 2017 Received in revised form 31 January 2020 Accepted 1 February 2020 Available online xxxx Keywords: Superconducting radio frequency (RF) cavities Back recursive observer Virtual disturbance Disturbance observer

a b s t r a c t A novel back recursive estimation (BRE) scheme is proposed for superconducting radio frequency (SRF) cavities. Microphonic,the main source of cavities detuning is modeled as unknown frequency sinusoidal disturbance. The disturbance property is excited by an auxiliary filter and the frequency information is estimated in observer framework. Furthermore, the sinusoidal disturbance is rearranged as a series of dynamics form using virtual disturbances. Back recursive signal is calculated according to the correlation between virtual disturbance and equivalent input disturbance. As a result, the asymptotic stability of estimation error can be obtained based on Lyapunov function, and robustness can be obtained if another external bounded disturbance exists. Simulations verify the effectiveness of the proposed method. © 2020 ISA. Published by Elsevier Ltd. All rights reserved.

1. Introduction Microphonics, the main error source of the cavities detuning in continuous wave (CW) mode, may cause the cavity resonance frequency to change at the vibration frequency [1]. As reported in [2], microphonics may be generated from heavy machinery, rotating machinery, fluid fluctuations, and ground motion, which will bring periodic signal into system and greatly affects the performance of the superconducting cavity. From the viewpoint of control, how to suppress detuning falls in anti-disturbance region. Traditional methods such as PI (proportional–integral) controller, high-gain feedback controller works well in rejecting microphonics at about frequency of 10 Hz, which deal with disturbance as a bounded signal. A recent promising strategy is to derive a compensator for microphonics if it can be equivalent to a input disturbance [3]. The adaptive and observer theory are foremost commonly microphonic compensation approaches. In [4], the microphonics, caused by mechanical vibrations, is modeled by an equivalent sinusoidal disturbance act on the same point as the control input. Phase locked loop (PLL) [5] is used to estimate the sinusoidal parameters directly and generate a compensation signal to cancel their effect. Based on the microphonics detuning information about spectral content, peak detuning values, and the driving terms, a cw-adapted fast detuning compensation ∗ Corresponding author at: School of Electronic Information Engineering, Taiyuan University of Science and Technology, Taiyuan, 030024, China. E-mail address: [email protected] (X. Wen).

algorithm is proposed in [6]. In [7], a microphonics estimation and prediction method is designed for superconducting accelerators, where the fast tuners is set up using the estimates of detuning during previous RF pulses. In [8], a disturbance observer based control (DOBC) is designed to compensate multiple disturbances, the disturbance signals in specified frequency ranges were reconstructed and were then removed in the feedforward model. From the analysis mentioned above we know that the microphonics compensation is effective strategy to suppress the effect of microphonics. But the present compensation techniques work when the periodic signal is of known frequency but unknown amplitude and phase. In general, the mechanical frequency concerned is usually unknown and varies in [60 Hz–200 Hz] region. The motivation for this paper is to develop a disturbance rejection method for CW mode, where the microphonics can be modeled by unknown frequency sinusoidal signals. From viewpoint of control theory, handles of such periodic signal have received much attentions from various of perspectives. In [9–13], the parameter identifying methods for sinusoidals are provided with detailed stability and robustness analysis. With above, a natural idea is estimating the input disturbance in combination with observer or adaptive theory. Another prevalent scheme aims to reformulate the input disturbance as a parametric form, and then the frequency information and equivalent disturbance state can be estimated in a overall framework [14–18]. While the aforementioned results provide efficient techniques, there are still some shortcomings that limit the application of disturbance

https://doi.org/10.1016/j.isatra.2020.02.001 0019-0578/© 2020 ISA. Published by Elsevier Ltd. All rights reserved.

Please cite this article as: L. Zhang, X. Wen, B. Wang et al., Back recursive estimation of unknown frequency sinusoidal disturbance in superconducting RF cavities. ISA Transactions (2020), https://doi.org/10.1016/j.isatra.2020.02.001.

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where ω0 is the resonant angular frequency, and QL is the loaded quality factor. Then Eq. (2) could be expressed as follows [24]:

ω0 ˙ ω0 ˙ V + ω02 V = Rc I

(4)

V¨ + 2ω1/2 V˙ + ω02 V = 2ω1/2 Rc I˙

(5)

V¨ +

QL

QL

or

Fig. 1. Equivalent RLC circuit of the RF cavity.

compensation in practical systems. The present adaptive methods mostly involve large numbers of identifications, possibly for frequency, amplitude or disturbance state, etc. The numerous tasks add the computation complex and bring difficult in compensation analysis [14]. An important result of DOBC is proposed recently in [19–21] using two-layer observer. On the basis of auxiliary observer, only the estimate of frequency is necessary to generate compensation signal. The attractive property of this structure inspire us solve our problem in DOBC [22]. But the construction of auxiliary observer requires system state, which is unmeasurable signal in our RF cavity system. Our topic consider an output-based optimize compensation scheme with less design conservativeness, which estimate and reject the frequency offset caused by microphonics. The compensation method is realized by means of series of virtual disturbance using back recursive fashion. This paper provides detailed analysis of the correlation with respect to input disturbance and virtual disturbance, shows a significant improvement in offset compensation performance. The paper is organized as follows. Section 2 gives the measurement principle and control problem formulation, and introduce the estimate of disturbance parameter on the basis of virtual disturbance. In Section 3, the synthetical disturbance estimated value is obtained through back recursive observer. In Section 4, the designed observer is applied to the superconducting cavity system, simulations demonstrate the advantages of the proposed scheme. Section 5 gives conclusions and the future research work. 2. Frequency information estimation 2.1. Superconducting RF cavity model analysis Resonator at a certain operating frequency can be equivalent to RLC oscillation circuit [23], and this article introduce our disturbance rejection strategy based on this principle. A superconducting RF cavity can be equivalent to a parallel RLC circuit as shown in Fig. 1. Assuming that the cavity voltage is V and equivalent current is I, the differential equation for the circuit can be written as [24]:

C

dV dt

+

1 Rc

V+

1

∫ Vdt = I

L

(1)

Taking derivative with respect to time and dividing by C in both sides, then we can obtain: V¨ +

1 Rc C

V˙ +

1 LC

V =

1 C

I˙

(2)

The half of 3 dB bandwidth (Full width half height) is given by [25],

ω1/2 =

1 2Rc C

=

ω0 2QL

(3)

Here, Rc is the equivalent resistance of the cavity, L, and C are the equivalent conductance and capacitance respectively. For continuous wave (CW) incident power, at the steady state under the exciting signal with frequency ω, the equivalent current I(t) can be represented as I(ω, t) = I0 sin(ωt). Substituting I(ω, t) into Eq. (4), the cavity voltage is expressed as [1]: V (t) = V0 sin(ωt + ψ )

(6)

where R c I0 , V0 ≈ √ 1 + tan2 ψ

tanψ ≈ 2QL

∆ω ω

(7)

ψ is the cavity detuning angle, and ∆ω is the detuning frequency width. If the cavity operates at the resonant peak (the detuning angle is zero), the amplitude of cavity signal has the maximum value. Otherwise, the cavity operates in a detuning state, and a frequency control usually is required. Note that the detuning angle at different side of resonant peak has the opposite sign. It provides some information to judge the exact frequency tuning direction. The equations in (7) represents the basic expressions of cavity frequency tuning. By taking Laplace transform of Eq. (5), the impedance or transfer function of the cavity can be obtained as follows: ZL (s) =

V (s) I(s)

=

2ω1/2 Rc s s2 + 2ω1/2 s + ω02

(8)

This is a typical representation form of the second-order control system. From description mentioned above we know that, if the cavity frequency is tuned equal to the RF frequency (it is called peak operating), the nominal accelerating cavity system looks like a low pass filter system in IQ representation. Fig. 2 shows the proposed RF Cavity architecture, where the PI controller is nominal feedback regulator. fref (t) is the reference signal. The pick-up signal from cavity was frequency downconverted to IF (Intermediate-Frequency) fcav (t), this process is omitted here. y˜ (t) is the error of fref (t) and nominal model output ym (t). The core of this paper is the compensation mechanism for microphonics called as back recursive observer constituting of a frequency function observer and a disturbance generator. A dynamic property between measurable signal and input disturbance is modeled through virtual disturbance and auxiliary filter. As a result, the disturbance generator yields the original disturbance in a polynomial form only about disturbance frequency estimates. For the preliminary study, we adopt a simplified motor model to verify the proposed method, which is presented as: Gm (s) =

1

λs + 1

(9)

where λ can be obtained by sweep-frequency technique. Thus the nominal model in Fig. 2 includes superconducting cavity and motor model, and the compensation signal will be applied at the same channel as PI controller. Similar to [4], the cavity detuning is seen as input disturbance: d(t) = Φ sin(ωt + ϕ )

(10)

which is a sinusoid with unknown amplitude Φ , frequency ω and phase ϕ . This article rewrite such type of disturbance as following

Please cite this article as: L. Zhang, X. Wen, B. Wang et al., Back recursive estimation of unknown frequency sinusoidal disturbance in superconducting RF cavities. ISA Transactions (2020), https://doi.org/10.1016/j.isatra.2020.02.001.

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Fig. 2. Block diagram of the proposed RF Cavity system architecture.

where ζ1,2 (t) are the inverse Laplace transform of ζ1,2 (s). From (11), (12) and (16) we get

exogenous system similar to [18]

{

.

w (t) = W w(t) d(t) = V¯ w (t)

(11)

where w (t) ∈ R , V¯ ∈ R1×2 , W ∈ R2×2 . For simplification of the following design procedure, the parameters W and V¯ are further

b2 w ˙ 1 (t) − w ˙ 2 (t)

ζ˙2 (t) −

b22 + ω2

2×1

described in detail as:

[ W =

0

−ϖ

1 0

]

, V¯ =

[

1

0

]

.

(12)

As we known, both the disturbance parameter and the equivalent state for (11) need to be estimated in most present work, which slow the estimation convergence speed. In our proposed DOBC structure, only one parameter needs to be estimated for reconstruction of d(t), which greatly improves the compensation efficiency. Remark 1. To illustrate the main idea, only back recursive observer with a expected accuracy is considered in this paper. Integrating a feedback controller (not limited in PI controller) to deal with multiple-uncertainties is our next research. Remark 2. In order to emphasize the main ideology, this paper does not take into account of some particular motor or other types of auctors. Without loss of generality, we suppose the motor has a satisfactory tracking performance in Fig. 2.

2ω1/2 Rc s (s2

+ 2ω1/2 s + ω02 )(λs + 1)

.

(13)

Define ζ1 (s) as a signal generated from the following Lyapunov stable low-pass filter F (s) =

b22 + ω2

)

(17)

ζ2 (t) = d2 (t) + σ2 (t),

(18)

where d(t) is virtual disturbance and d2 (t) =

b2 w1 (t) − w2 (t) b22 + ω2

.

(19)

It follows that the decay term

σ2 (t) = ζ2 (t) −

b2 w1 (t) − w2 (t) b2 + ω 2

,

(20)

and obeys

σ˙ 2 (t) = −b2 σ2 (t).

(21)

Similarly, we can find ζ1 (t) will track a sinusoidal too with frequency ω. We first estimate the disturbance information ϖ = ω2 using ζ1 (t), by introducing the auxiliary filter

{

ξ1 (s) =

ζ

1 (s) s+g 1 s (s) s+g 1

(22)

ζ

where g is a selectable positive scalar. By virtue of (16) and (22),

We are now in position to introduce virtual disturbances to derive a equivalent input disturbance form. From Fig. 2 we see the transfer function from d to y˜ ZL (s)Gm (s) =

b2 w1 (t) − w2 (t)

where wi (t) (i = 1, 2) is the ith element of w (t). Eq. (17) indicates that ζ2 (t) will converge to a sinusoidal asymptotically with frequency ω. For simplicity, denoting

ξ2 (s) = 2.2. Virtual disturbance

= −b2 (ζ2 (t) −

(s2 + 2ω1/2 s + ω02 )(λs + 1) 2ω1/2 Rc s(s + b1 )(s + b2 )

(14)

driven by y˜ (s), where the constant positive scalars b1,2 are selectable. To demonstrate the relation between d(s) and ζ1 (s), we introduce ζ2 (s) satisfying

ζ2 (s) d(s) , ζ2 (s) = ζ1 (s) = s + b1 s + b2

(15)

(23)

where ξ1,2 (t) are the inverse Laplace transforms of ξ1,2 (s). Theorem 1. If there exists dynamics ξ1 (t), ξ2 (t) satisfying (23) then the virtual disturbance d2 (t) can be rearranged as d2 (t) = ξ1 (t)(b1 g − ϖ ) + ξ2 (t)(b1 + g) + θ T δ (t)

(24)

where

θ=

[

b1 g − ϖ b1 + g

]

,L =

[

0 1

] (25)

and ϖ = ω2 , and δ (t) obeys

In consideration of (14) and (15), we have

ζ˙1 (t) = −b1 ζ1 (t) + ζ2 (t), ζ˙2 (t) = −b2 ζ2 (t) + d(t)

ξ˙1 (t) = ξ2 (t) ξ˙2 (t) = −g(−g ξ1 (t) + ζ1 (t)) + ζ˙1 (t) = g 2 ξ1 (t) − g ζ1 (t) + ζ2 (t) − b1 ζ1 (t) = g 2 ξ1 (t) − (g + b1 )(ξ2 (t) + g ξ1 (t)) + ζ2 (t) = −b1 g ξ1 (t) − (b1 + g)ξ2 (t) + ζ2 (t)

(16)

δ˙ (t) = Gδ (t) − Lσ2 (t).

(26)

Please cite this article as: L. Zhang, X. Wen, B. Wang et al., Back recursive estimation of unknown frequency sinusoidal disturbance in superconducting RF cavities. ISA Transactions (2020), https://doi.org/10.1016/j.isatra.2020.02.001.

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= b2 d2 (t) + b2 σ2 (t) + d˙ 2 (t) + b2 σ˙ 2 (t)

where

[ G=

0 −b1 g

]

1

−b1 − g

(27)

Proof. According to Theorem 1 in [21] we known that for a give vector ξ˘ (t) ∈ R2×1 satisfies

ξ˙˘ (t) = Gξ˘ (t) + Ld2 (t),

(28)

the sinusoidal disturbance d2 (t) can be denoted as d2 (t) = ξ˘1 (t)(b1 g − ϖ ) + ξ˘2 (t)(b1 + g) + θ T δ˘ (t)

Thus d2 (t) = ξ˘ (t)T θ + θ T δ˘ (t)

[

ξ1 (t)

ξ2 (t)

]T

(31)

(32)

(33)

Theorem 1 yields the virtual disturbances d2 (t) presentation in a parametric form related to ϖ . In consideration of (21) and (26), the signal σ2 (t), δ (t) all converge to zero. As such, ξ1,2 (t) are inherently stable, and will converge to a sinusoidal with frequency ω. Correspondingly, consider the designed observer

⎧ ˆ (t) = z(t) + p(t) ⎨ ϖ ˙z (t) = −αξ12 (t)ϖ ˆ (t) + αξ22 (t) ⎩ p(t) = −αξ1 (t)ξ2 (t)

(34)

(35)

if virtual disturbance d2 (t) can be obtained, the remaining work is to reconstruct d(t) based on (16) and (18), which is detailed in the next section. 3. Disturbance generator and stability In the most existing methods, it is necessary to construct the equivalent state of d(t), this procedure is removed in our design. As shown in Fig. 2, disturbance generator provides an alternative form of the equivalent input disturbance based on virtual disturbance using BRE fashion. From (16), (18) and (21) we can see d(t) = b2 ζ2 (t) + ζ˙2 (t)

(38)

As so far, we have deduced the alternative forms of d(t), which is only depend on unknown parameter ϖ . Then, we applied dˆ 2 (t) into (38) to reconstruct the compensation signal, which is detailed in Theorem 2.

−2α Λ = ⎣ α θ¯ α

⎡

α θ¯ T T γ2 (G P + PG) −γ2 LT P T

⎤ α −γ2 PL ⎦ < 0, −2b2 γ1

− (b1 + g)ξ1 (t)ϖ ˆ (t) + b1 g ξ2 (t) − ξ2 (t)ϖ ˆ (t)

(40)

ˆ converges to zero as t → +∞. Then the error dynamic d(t) − d(t) Proof. Denoting disturbance estimation error

˜ = d(t) − d(t) ˆ = −b2 ξ1 (t)ϖ d(t) ˜ (t) − (b1 + g)ξ1 (t)ϖ ˜ (t) − ξ2 (t)ϖ ˜ (t) + θ T Gδ (t) − b22 σ2 (t) + θ T δ (t).

(41)

From (34) and (35) we can get

˙ˆ (t) = αξ 2 (t)ϖ ϖ ˜ (t) − αξ1 (t)σ2 (t) − αξ1 (t)θ T δ (t), 1 then the estimation error

˙˜ (t) = −αξ 2 (t)ϖ ϖ ˜ (t) + αξ1 (t)σ2 (t) + αξ1 (t)θ T δ (t). 1

(42)

Define the following Lyapunov function V (t) = ϖ ˜ T (t)ϖ ˜ (t) + γ1 σ2T (t)σ2 (t) + γ2 δ T (t)P δ (t),

where α > 0 is a given constant value. According to Eq. (24) the estimation forms of virtual disturbance dˆ 2 (t) can be given by dˆ 2 (t) = ξ1 (t)(b1 g − ϖ ˆ ) + ξ2 (t)(b1 + g)

+ θ T Gδ (t) − b22 σ2 (t)

ˆ = b1 b2 g ξ1 (t) − b2 ξ1 (t)ϖ d(t) ˆ (t) + b1 b2 ξ2 (t) + b2 g ξ2 (t)

In the following, we are ready to track the unknown constant scalar ϖ by constructing a nonlinear observer. Combining with (18), (23) and (24) we have

ξ˙1 (t) = ξ2 (t) ξ˙2 (t) = −ϖ ξ1 (t) + θ T δ (t) + σ2 (t)

Then the unexpected input d(t) can be reformulated as

where θ¯ is upper bound of θ , then reconstruct input disturbance d(t) as

ξ˙˘ (t) − ξ˙ (t) = G(ξ˘ (t) − ξ1 (t)) − Lσ2 (t) ξ˘˙ (t) − ξ˙ (t) + δ˙˘ (t) = G(ξ˘ (t) − ξ1 (t) + δ˘ (t)) − Lσ2 (t).

Combining the above and equations of (25) and (32), it can be seen that (24) upholds by inserting δ (t) = ξ˘ (t) − ξ1 (t) + δ˘ (t) into (31). □

{

(37)

(39)

According to (18), (23), (28) and (30) it follows that

{

= θ T W ξ (t) + θ T Gδ (t) − b2 σ2 (t)

Theorem 2. Applying the ϖ estimation law (35), for some γ1,2 > 0 and α > 0, if there exist P > 0 satisfying

where

ξ (t) =

d˙ 2 (t) = θ T W ξ (t) + θ T δ˙ (t) + σ˙ 2 (t)

d(t) = b2 ζ2 (t) + ζ˙2 (t) = b2 d2 (t) + θ T W ξ (t)

(30)

= (ξ˘ (t) − ξ (t) + ξ (t))T θ + θ T δ˘ (t) = ξ T (t)θ + θ T δ˘ (t) + θ T (ξ˘ (t) − ξ (t))

The remaining task of back recursive observer is using estimators dˆ 2 (t) in replace of d(t). From (23) and (24) we can derive the derivative of d2 (t) directly as

(29)

where δ˘ (t) satisfies

δ˙˘ (t) = Gδ˘ (t).

(36)

where P , γ1,2 > 0. Then from (21), (26) and (42) V˙ (t) = −2α∥ξ1 (t)ϖ ˜ (t)∥2 + 2αδ T (t)θξ1 (t)ϖ ˜ (t)

+ 2ασ2 (t)ξ1 (t)ϖ ˜ (t) + γ2 δ T (t)(GT P + PG)δ (t) T − 2γ2 δ (t)PLσ2 (t) − 2b2 γ1 ∥σ2 (t)∥2 [ ]T [ ] ξ1 (t)ϖ ˜ (t) ξ1 (t)ϖ ˜ (t) ¯ δ (t) δ (t) = Λ . σ2 (t) σ2 (t) where

−2α ¯ = ⎣ αθ Λ α

⎡

αθ T T γ2 (G P + PG) −γ2 LT P T

⎤ α −γ2 PL ⎦ < 0 −2b2 γ1

¯ < 0 can be In consideration that θ¯ is upper bound of θ , Λ guaranteed by (39). The above analysis means ϖ ˜ (t), σ2 (t) and δ (t) are bounded, and we can test that V¨ (t) is bounded too. Based on Barbalat lemma, ϖ ˜ (t), δ1 (t), σ2 (t) → 0 as t → +∞. ˜ Remembering (41), we follow that the error dynamic d(t) can converge to zero, which indicates Theorem 2 is uphold. □

Please cite this article as: L. Zhang, X. Wen, B. Wang et al., Back recursive estimation of unknown frequency sinusoidal disturbance in superconducting RF cavities. ISA Transactions (2020), https://doi.org/10.1016/j.isatra.2020.02.001.

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5

Fig. 3. Equivalent procedure of back recursive observer design.

Table 1 Resonant cavity’s parameters and its measurement data. Parameters

Symbol

Value

Cavity’s resonant frequency

f0

805 MHz

Cavity’s intrinsic quality factor

Q0

1.0E+10

Cavity’s loaded quality factor

QL

7.3E+5

Cavity’s intrinsic shunt impedance

R Q

279 

Servo motor inertia time constant

λ

0.01

Taking into account the compensation performance with (41) and (42), b1,2 , g , α can be chosen as desired to guarantee convergence speed and robustness according to (21), (25) and (27). In comparison with multiple estimates in present algorithms, our algorithm involves only one unknown parameter. With the above comprehensive analysis, procedure of back recursive observer shown in Fig. 3 mainly includes three aspects: (1) Obtain the measurable ζ1 (t) using low-pass filter F (s) according to (14). Represent the virtual disturbance in a uncertain parametric form of ξ (t) based on auxiliary observer (14). (2) Design a frequency factor observer to estimate disturbance key information ϖ (t). Calculate the correlation between d(t), ξ (t) and d2 (t) in back recursive fashion. (3) Apply ϖ ˆ (t) into (35), the virtual disturbance estimates dˆ 2 (t) can be obtained, along with which the equivalent disturbance d(t) is further reconstructed by (41). Remark 3. In the Eqs. (21), (26), (41) and (42), we give the relationship between parameters α, b1 , b2 , g and convergence ˜ performance of ϖ ˜ (t) and d(t). Furthermore, if we have a priori information on the range of the disturbance frequency ω(t), a reasonable selection of α and G can help avoid large initial values of ϖ ˜ , δ (t) and σ2 (t), thus achieving a faster convergence property. 4. Simulation In this section, a superconducting RF cavity designed for Spallation Neutron (SNS) Source is taken for simulation. According to the data above, we can make the following calculation:

ω0 = 2π f0 = 5.0579641723 × 10 rad/s ω0 1 ω1/2 = = = 3464.4 rad/s 9

Rc =

2Rc C R/Q Q0

2QL

≈ 27.9 n

Table 1 lists our measurement data and the intrinsic parameters of the resonator. Next we replace the above parameters into (8) to obtain the transfer function model of the resonator ZL (s) as 1.93311 × 10−4 s s2 + 6928.8s + 2.55830015682704240873 × 1019 We first take sinusoidal disturbance d(t) with ω = 80 for the simulation, and the design parameters are chosen as b1,2 =

Fig. 4. Estimation performance using proposed observer (d = 2000sin(80t)).

1, g = 20, α = 500. Fig. 4 shows the estimation performance using proposed observer, the estimation error of ϖ and original disturbance d(t) converges to zero with little error after 8 s. In the next, we adopt b1,2 = 1, g = 20, α = 5000 to test the convergence speed for higher frequency sinusoidal 2000sin(150t). It is noticed that we can track disturbance d(t) with good convergence performance (Fig. 5). Let us compare adaptive algorithm [18] with our proposed method using the same example. The results presented in Fig. 6 indicate that the high-frequency disturbance may seriously damage the adaptive estimation quality. To test the proposed observer robustness property, we introduce another noise ∆d(t) = 200ϵ (t), ϵ (t) is a white noise into system. Although the robustness analysis is not given in our paper for sake of applications, this property should not be neglected. The design parameters may be set large enough to improve convergence speed and robustness, we adopt b1,2 = 1, g = 50, α = 10 000 for simulation. Fig. 7 demonstrates that the ω2 can be well tracked with little static error less than 400, and thus the input disturbance estimation error comes to less than 50 after 5 s. As we can conclude, computer simulation illustrates that robustness properties keep safe with respect to unexpected noise ∆d(t), which confirms the conclusion. 5. Conclusion This paper gives a microphonics compensation approach for a class of superconducting RF cavities system. With the back recursive observer, the equivalent disturbance can be reconstructed through formulation of virtual disturbances. Simulations show the effectiveness of proposed algorithm from the perspective of convergence and robustness. The further research is to provide a comprehensive DOBC framework for nonlinear uncertainties, and develop an improved back recursive method for non-minimum phase systems with multi-frequencies sinusoidal. Declaration of competing interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Please cite this article as: L. Zhang, X. Wen, B. Wang et al., Back recursive estimation of unknown frequency sinusoidal disturbance in superconducting RF cavities. ISA Transactions (2020), https://doi.org/10.1016/j.isatra.2020.02.001.

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Acknowledgments The project was supported by the National Science Foundation of China under Grants (61203049), Shanxi Province Natural Science Fund (201801D121132), the key laboratory of expressway construction machinery of shanxi province (310825171131, 300102258205, 300102259513), Collaborative Innovation Center of Internet+3D Printing in Shanxi Province (CiCi3DP) 20122048. National natural science foundation of China (51405027), and open research foundation of state key lab of digital manufacturing equipment and technology in huazhong university of science and technology (dmetkf2015015, ProvinceCiCi3DP), Shanxi Province Innovation Program for Postgraduates (2019SY489). References

Fig. 5. Estimation performance using proposed observer (d = 2000sin(150t)).

Fig. 6. Estimation performance using adaptive algorithm (d = 2000sin(150t)).

Fig. 7. Estimation performance for multi-disturbances (d = 2000sin(150t) +∆d(t)).

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Please cite this article as: L. Zhang, X. Wen, B. Wang et al., Back recursive estimation of unknown frequency sinusoidal disturbance in superconducting RF cavities. ISA Transactions (2020), https://doi.org/10.1016/j.isatra.2020.02.001.

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Please cite this article as: L. Zhang, X. Wen, B. Wang et al., Back recursive estimation of unknown frequency sinusoidal disturbance in superconducting RF cavities. ISA Transactions (2020), https://doi.org/10.1016/j.isatra.2020.02.001.