Robust control simulation on following performance of high-speed railway in situ testing system

Abstract

Subgrade bears both the impacts of the high-speed train and the weight of superstructures. Its stability affects the line smoothness directly, but it is very hard to simulate its working condition, and the in situ testing method is inadequate. The article presents a railway roadbed in situ testing device, and it proposes an excitation hydraulic cylinder system to output static and dynamic pressure simultaneously in order to simulate the force of the train. The dynamic following performance is quite poor without control algorithm, so the robust control block is adopted to improve the following performance of the excitation system under different frequencies and waveforms. In addition, considering the effects of the external disturbances, simulation is carried out with a certain intensity of noise to test the effectiveness of the control scheme. The simulation results show that the robust control algorithm makes the excitation system achieving much better following performance.

Keywords

Following performance in situ testing method robust control excitation system

Introduction

Subgrade is a critical part of the high-speed rail system; it withstands not only the static pressure of the superstructures but also the dynamic pressure of the repeated changes with wheels’ movement.^1,2 Its stability directly affects the smoothness of the superstructures and concerns the life and property safety of passengers.

Generally, in order to solve the above problem, numerical simulation in situ testing and laboratory testing methods are adopted. There are so many control schemes such as conventional proportional -integral -derivative (PID),³ variants based on fuzzy PID^4–8 to overcome the linear characteristics of the conventional excitation system. Zheng et al.⁹ developed a valve-controlled cylinder excitation system and an adaptive model following control (AMFC) algorithm without consider external disturbances. Thus, considering the uncertainty factor in control plant, robust control schemes^10,11 have a widespread applications in industrial fields. Pradana et al.¹² addressed the stabilization of a model helicopter in a hover configuration subject to parametric uncertainty and external disturbances using robust H-infinity control theory. Akmeliawati et al.¹³ presented a development of robust controller to solve the major challenges associated with the deployment of autonomous small-scale helicopters in civilian unmanned aerial vehicle applications. Tijani et al.¹⁴ developed a multi-objective differential evolution (MODE)-based extended H-infinity controller for autonomous helicopter control and the article addressed the challenges involved in selection of weighting function parameters for H-infinity control synthesis to satisfy conflicting stability and time-domain objectives. Tijani and Budiyono¹⁵ presented a through procedure to design an extended H1 robust control of dynamic system especially to parametric uncertainty. At the same time, there are few methods for in situ testing railway roadbed stability except the application in the dynamic stability field test (DyStaFiT) system¹⁶ and the AMFC algorithm.^9,17 Both the DyStaFiT and the AMFC systems have some shortcomings. Their excitation waveform is simple and has many distortions, what is more, it has poor performance of anti-load variation capabilities.

For these problems above, the article develops an improved control structure and a valve control cylinder drive system and studies the static state characteristic and the dynamic pressure in both high frequency and different waveforms. First, the mathematical model of the excitation system is derived, and the exciting force and the waveform can be adjusted online. Simulation with different signals is carried out to verify the effectiveness of the robust controller. The following performance of the system with the mixed sensitivity H-infinity robust control can simulate the load to subgrade and contribute to high-speed rail research.

Mathematical model of the excitation system

The principle of the excitation system

Traditional mechanical method has a big vibration quality, bearing loss seriously, and low frequency, so it is difficult to meet the development of high-speed rail. Therefore, valve-controlled cylinder and pump control motor are two relatively advanced vibration modes. However, the pump control motor is essentially still based on the inertia vibration principle and it cannot effectively solve the problem above. A new valve–controlled hydraulic cylinder vibration mode is chosen to overcome the above shortcomings; the principle of the valve-controlled cylinder system is shown in Figure 1. The hydraulic cylinder consists of a static pressure cylinder in parallel with a dynamic pressure cylinder. The static force cylinder is controlled by static load, and the dynamic force cylinder is controlled by a pressure servo valve or flow servo valve. The valve-controlled cylinder model of vibration system can adjust vibration parameters online, largely improve the vibration frequency by improving the performance of the servo valve and changing the control strategy, and consequently, it can simulate various working conditions by superposition of two waveforms to imitate the dynamic load of the train. The parallel exciting servo hydraulic cylinder is the core of the system.

Figure 1.

Principle of the excitation testing system.

The requirements of the excitation system are shown in Table 1. Output waveform can be controlled through computer. The dynamic valve can be switched from closed-loop force to closed-loop location control system. The excitation frequency is 1–40 Hz and adjustable. When it is closed-loop force control system, the output force is 200 ± 100 kN. When it is closed-loop displacement control system, the output displacement at 1 Hz is ±20 mm and at 40 Hz is ±0.5 mm. The electro-hydraulic servo valve is a complex component and plays an important role in the whole system. One servo valve loads static pressure on the static force cylinder and the other provides dynamic load on the dynamic force cylinder. The performance of the servo valve directly affects the whole system. In order to realize the requirements, the parameters of the two servo valve are shown in Table 2.

Table 1.

Parameters of the excitation system.

Name	Requirements
Excitation frequency	1–40 Hz
Output force (static, dynamic)	200 kN; ±100 kN
Excitation displacement	1 Hz:±20 mm; 40 Hz:±0.5 mm

Table 2.

Parameters of two servo valve.

Name	Dynamic valve	Static valve
Oil supply pressure	2–28 MPa	7–21 MPa
Pressure rated	21 MPa	21 MPa
Current rated	10 mA	10 mA
Linearity	<±7.5%	<±7.5%
Resolution	<1%	<2%
Internal leakage	100 L/min	>60 L/min
Amplitude frequency width, 3 dB	>50 Hz	>90 Hz
Phase frequency width $- 90^{\circ}$	>50 Hz	>90 Hz

Analysis and modeling of the excitation system

Due to the high natural frequency of the hydraulic load of the system, the load can be equivalent to a mass–spring system and the transfer function of the servo valve is considered as a second-order oscillation link. The electro-hydraulic dynamic servo excitation system designed in this article is a driving force control system. The transfer function is described as follows

G (s) = \frac{K_{sv}}{\frac{s^{2}}{{w_{sv}}^{2}} + \frac{2 ξ_{sv}}{w_{sv}} s + 1}

(1)

where $K_{sv}$ is the gain of the dynamic servo valve, $K_{sv} = K_{xv} K_{q}$ . $K_{xv}$ is the displacement of valve core, $K_{xv} = X_{v} / I_{n}$ . $K_{q}$ is the current gain, $K_{q} = Q_{om} / X_{v}$ . Joint three equations mentioned above, $K_{sv} = Q_{om} / I_{n}$ , while $Q_{om}$ is the no-load flow, $Q_{om} = q_{n} \sqrt{p_{s} / p_{n}}$ , $q_{n}$ is the rated flow, $q_{n} = 100 L / \min$ , $p_{s}$ is the oil pressure actual value, $p_{s} = 21 MPa$ , $p_{n}$ is the pressure drop, $p_{n} = 7 MPa$ . $I_{n}$ is the rated current, $I_{n} = 10 mA$ . According to the calculated parameters, $K_{sv} = 3 \times 10^{- 4} m^{3} / (s mA)$ . $w_{sv}$ is the natural frequency of the servo valve, $W_{sv} = 2 π f_{- 90^{\circ}} = 2 \times 3.14 \times 50 = 314 rad / s$ , and $ξ_{sv}$ is the integrated damping ratio. Generally, the value range of damping ratio is from 0.5 to 0.7; in the excitation system, the valve of damping ratio is 0.7. To sum up, the transfer function of dynamic valve control cylinder is

G (s) = \frac{3 \times 10^{- 4}}{\frac{s^{2}}{314^{2}} + \frac{2 \times 0.7}{314} s + 1}

(2)

The mathematics transfer function of the valve-controlled cylinder is

\frac{F_{g}}{Q} = \frac{\frac{A_{p}}{K_{ce}} (\frac{s^{2}}{ω_{m}^{2}} + 1)}{(\frac{s}{ω_{r}} + 1) (\frac{s^{2}}{ω_{0}^{2}} + \frac{2 ξ_{0}}{ω_{0}} s + 1)}

(3)

where $A_{p}$ is the area of dynamic valve piston, $K_{ce} = K_{c} + C_{tp}$ is the flow-pressure coefficient of the system, $ω_{m}$ is the natural frequency of the load, $ω_{r}$ is the corner frequency of the inertial link, $ω_{0}$ is the integrated natural frequency, and $ξ_{0}$ is the integrated damping ratio.

Because response frequencies of the servo amplifier and the sensor are much higher than that of the system, they can be simplified as proportion link. After analyzed and simplified the elements of excitation system,¹⁸ the block diagram of the excitation system is derived as shown in Figure 2, where $K_{α}$ is the gain of the servo amplifier and $K_{fF}$ is the feedback coefficient of the force sensor. The computing steps and results in detail are presented in Appendix 1. The open-loop transfer function of the excitation system is as follows

\begin{matrix} G (s) H (s) = & \frac{41}{s} \times \frac{3 \times 10^{- 4}}{\frac{s^{2}}{314^{2}} + \frac{2 \times 0.7}{314} s + 1} \\ \times \frac{8.1 \times 10^{8} \times (\frac{s^{2}}{618^{2}} + 1)}{(\frac{s}{18.12} + 1) (\frac{s^{2}}{1202^{2}} + \frac{0.6}{1202} s + 1)} \end{matrix}

(4)

Figure 2.

Block diagram of the open-loop system.

Robust control schemes

In many industrial applications, conventional control schemes such as PID and fuzzy PID are widely used. Comparing with traditional control methods, the robust control schemes have its advantages which makes it more application in nonlinear control systems with uncertain and fuzzy factors, overcomes the defeats of classical control theory over-dependent on the accurate controlled mathematical model. We present robust control method for approximate linear servo control cylinder loop to improve its following performance. Its aim is to make the system output to approach to the reference model output. It can not only eliminate the effect of disturbances but also can be applied to the hydraulic system to realize the adaptive control.

The mixed sensitivity $H_{\infty}$ robust control is an important branch of the robust control theory. The $H_{\infty}$ robust control theory considered the effect of external disturbance and the uncertain factors of the model, which meet the requirement of actual complex working condition. This new method can not only be well automatically but also have intelligent to change its waveform as required online. To construct such a system, it does not need to know the accurate parameters of the excitation system, and its performance is almost insensitive to interrupts. Therefore, the following performance is assured. The practices have proved that it is relatively simple and effective.

Introduction of the mixed sensitivity H-infinity robust control

The standard $H_{\infty}$ optimal robust control problem is to find a stable controller, which minimizes the following equation (5)

‖ F ‖_{\infty} = sup \bar{σ} [F (s)] = sup \bar{σ} [F (jw)]

(5)

where $F (G, K)$ is the transfer function of the closed-loop system. In practical, all kinds of $H_{\infty}$ problem can be transformed into a standard $H_{\infty}$ control problem.

Assumed a linear time invariant system that shown in Figure 3, $w$ is $l$ dimensional external input signal, mainly includes the jamming signal and sensor noise. $z$ is $p$ dimensional general control output signal, generally includes following error and adjustment error. $u$ is $n$ dimensional control input signal. $y$ is $m$ dimensional observe output signal. Matrix $P (s)$ is the known generalized controlled objects. The transfer function matrix $K (s)$ is the robust controller that need to be solved. Assumed that the generalized matrix $P (s)$ and the robust controller matrix $K (s)$ are rational function matrix.

Figure 3.

Block diagram of a linear time invariant system.

The generalized object $P$ can be written as

P = [\begin{matrix} P_{11} & P_{12} \\ P_{21} & P_{22} \end{matrix}]

(6)

According to Figure 3 and equation (6)

z = P_{11} w + P_{12} u

(7)

y = P_{21} w + P_{22} u

(8)

u = Ky

(9)

If $(I - P_{22} K)$ exist real rational invertible matrix

z = (P_{11} + P_{12} K (I - P_{22} K)^{- 1} P_{21}) w

(10)

The transfer function matrix $T_{zw}$ between the external input signal and generalized control output signal is

T_{zw} = F_{l} (P, K) = P_{11} + P_{12} K (I - P_{22} K)^{- 1} P_{21}

(11)

Equation (11) is called the linear fractional transformation. The system which can use linear fractional transformation to represent its transfer function is able to convert to standard H-infinity control problem.

Theoretical background of the mixed sensitivity H-infinity control

The closed loop of the following control system is shown in Figure 4, $P_{0} (s)$ is the controlled object of the transfer function and $K (s)$ it the controller, $y$ is the output signal, $r$ is the external input signal, $d$ is the noise signal, and $e$ is the error signal.

Figure 4.

The closed loop of the following control system.

When there is no external disturbance ( $d = 0$ ), the transfer function (the complementary sensitivity transfer function) is obtained as follows

T_{ry} = \frac{P_{0} (s) K (s)}{1 + P_{0} (s) K (s)}

(12)

Now consider the changes of $P_{0} (s)$

\begin{matrix} Δ T_{ry} & = \frac{[P_{0} (s) + Δ P (s)] K (s)}{1 + [P_{0} (s) + Δ P (s)] K (s)} - \frac{P_{0} (s) K (s)}{1 + P_{0} (s) K (s)} \\ = \frac{Δ P (s) K (s)}{[1 + [P_{0} (s) + Δ P (s)] K (s)] [1 + P_{0} (s) K (s)]} \\ = \frac{Δ P (s)}{1 + [P_{0} (s) + Δ P (s)] K (s)} \cdot \frac{T_{ry} (s)}{P (s)} \end{matrix}

(13)

The formula (13) is rearranged as follows

\frac{Δ T_{ry} / T_{ry} (s)}{Δ P (s) / P_{0} (s)} = \frac{1}{1 + [P_{0} (s) + Δ P (s)] K (s)}

(14)

When $Δ P (s) \to 0$ , the sensitivity $S$

S = \frac{1}{1 + P_{0} (s) K (s)}

(15)

When there is no external disturbance ( $d = 0$ ), formula (14) is the closed-loop transfer function from the input signal $r$ to the error signal $e$ . Assumed that there is no input signal $r$ ( $r = 0$ ), formula (15) is the closed-loop transfer function from the external disturbance $d$ to the output signal. Therefore, the sensitivity function can reflect the influence of between external disturbance and the output, and the influence between external input and the error signal. The formula (12) is the complementary sensitivity function; the relationship between the sensitivity function and the complementary sensitivity function is

T (s) + S (s) = I

(16)

The calculation above considered the following performance, capacity of resisting disturbance, and the robust stability of the excitation system, but the improvement of the three kinds of performance at the same time is a contradictory problem; therefore, the mixed sensitivity optimization problem arises. H-infinity mixed sensitivity control has been widely used, and it can be simplified into a standard H-infinity control problem.

The mixed sensitivity $H_{\infty}$ control optimization design is shown in Figure 5, which $P_{0} (s)$ , $K (s)$ represent controlled object and controller, $y$ is the output signal, $r$ is the external input signal, $d$ is the noise signal, $e$ is the error signal. $W_{1} (s)$ , $W_{2} (s)$ , and $W_{3} (s)$ are the system performance weighting function, controller output weighted function and the system robust weighted function, respectively. The mixed sensitivity $H_{\infty}$ can be transformed into standard $H_{\infty}$ control as shown in Figure 6. The closed loop of the system can be written in the following form

{\begin{matrix} z_{1} \\ z_{2} \\ z_{3} \\ y_{2} \end{matrix}} = [\begin{matrix} W_{1} & - W_{1} P_{0} \\ 0 & W_{2} \\ 0 & W_{3} P_{0} \\ I & - P_{0} \end{matrix}] {\begin{matrix} r \\ r_{1} \end{matrix}}

(17)

Figure 5.

Schematic of mixed sensitivity $H_{\infty}$ design.

Figure 6.

The transformed standard $H_{\infty}$ control.

Rewrite equation (17) contains two input variables and two output variables

{\begin{matrix} z \\ y_{2} \end{matrix}} = [\begin{matrix} G_{11} & G_{12} \\ G_{21} & G_{22} \end{matrix}] {\begin{matrix} r \\ r_{1} \end{matrix}}

(18)

The closed-loop matrix $F (s)$ is

F (s) = [G_{11} (s) + G_{12} (s) K (s) (I - G_{22} (s) K (s))^{- 1} G_{21} (s)]

(19)

Expanded equation (19) and the standard $H_{\infty}$ controller is designed to minimize the following function

‖ F (s) ‖_{\infty} = ‖ \begin{matrix} W_{1} S \\ W_{2} R \\ W_{3} T \end{matrix} ‖_{\infty} < γ

(20)

Three weighting functions play a critical role in ensuring good performance of the designed $H_{\infty}$ controller. The system performance weighting function $W_{1} (s)$ must satisfy the equation $\bar{σ} [S (j ω)] < | W_{1}^{- 1} (j ω) |$ to guarantee inner stabilization of the system. The system robust weighted function $W_{3} (s)$ must satisfy the equation $\bar{σ} [T (j ω)] < | W_{3}^{- 1} (j ω) |$ to guarantee inner stabilization of the system. The controller output weighted function $W_{2}$ can chose a suitable constant after the other two weighting function parameters set.

Design of the mixed sensitivity H-infinity control

Electro-hydraulic servo vibration system is shown in Figure 2; it can be seen that the controlled object is a five-order system. Because the order of the robust controller is equal to the order of the controlled object plus the weighted function order, so the order of the robust controller is 7. There are two methods to reduce the robust controller order: (1) reduce the order of the controlled object and (2) reduce the order of the robust controller. Here we chose the first method. The response frequency of the servo valve is much higher than the response frequency of the hydraulic cylinder, so we simplified the servo valve link as proportional component to reduce the order of the $H_{\infty}$ controller. Figure 7 compares the original system transfer function and simplified transfer function under step single, the red dashed curve is the response of the original model, and the black curve is the response of the simplified model. It is shown in Figure 7 that (a) it the comparison between the original system transfer function and simplified transfer function without control algorithm, while (b) added control algorithm. Two curves in Figure 7(b) fit well verify that the feasibility of the simplification under control algorithm.

Figure 7.

The contrast curve of the before and after simplification.

The controlled object after simplification

G (s) = \frac{7512 s^{2} + 2.869 \times 10^{9}}{s^{3} + 78.22 s^{2} + 1.446 \times 10^{6} s + 2.618 \times 10^{7}}

(21)

Three weighting functions are selected based on the following objectives:

$W_{1}$ is the weighting function of the sensitivity function S and minimize the largest singular values $\bar{σ} (S (j ω))$ at low frequency for good performance (e.g. tracking, disturbance rejection). To meet the requirement of system stability, the weighting function is constrained by the following equation. A PI structure $W_{1} (s) = k_{1} / (s / ω_{1} + 1)$ is used for weight function $W_{1}$ to add integrator for good tracking and disturbance rejection at low frequency, and the parameter $ω_{1}$ is the bandwidth at the low frequency and the parameter $k_{1}$ is the attenuation coefficient for the low-frequency disturbance

\bar{σ} [S (j ω)] \leq | W_{1}^{- 1} (j ω) |

(22)

$W_{3}$ is the weighting function of the complementary sensitivity function T, and minimize the largest singular values $\bar{σ} (T (j ω))$ at high frequency to deal with un-modeled dynamics. To meet the requirement of system stability, the weighting function is constrained by the following equation. A PI structure $W_{3} (s) = k_{3} s / (s / ω_{3} + 1)$ is used for weight function $W_{3}$ , and the parameter $ω_{3}$ is the bandwidth at the high frequency and the parameter $k_{3}$ is the attenuation coefficient for the high-frequency disturbance

\bar{σ} [T (j ω)] \leq | W_{3}^{- 1} (j ω) |

(23)

$W_{2}$ is selected as a constant gain and the choice of the weighting functions’ structure ( $W_{2}$ ) is motivated by the desire to keep the controller order low as much as possible.

Three weighting functions are selected based on the experience and trial-and-error approach

W_{1} = \frac{5}{s + 0.001}

(24)

W_{2} = 20

(25)

W_{3} = \frac{5 s}{s + 100}

(26)

Using the robust toolkit function, the mixed sensitivity $H_{\infty}$ controller is derived as

K (s) = \frac{12.33 s^{4} + 2197 s^{3} + 1.792 \times 10^{7} s^{2} + 2.105 \times 10^{9} s + 3.227 \times 10^{10}}{s^{5} + 2386 s^{4} + 2.309 \times 10^{6} s^{3} + 1.456 \times 10^{9} s^{2} + 3.348 \times 10^{11} s + 3.348 \times 10^{8}}

(27)

As shown in Figures 8 and 9, the sensitivity and complementary sensitivities responses stay below $W_{1}^{- 1}$ and $W_{3}^{- 1}$ for all frequencies. According to equations (22) and (23) and the robustness analysis,¹³ the designed controller is considered to have met the requirements of stability.

Figure 8.

Weighting function $W_{1}$ .

Figure 9.

Weighting function $W_{3}$ .

Simulation

Here, simulation model is developed using equations of the excitation system given in section “Mathematical model of the excitation system” in order to verify the effectiveness of the mixed sensitivity H-infinity control. To show the outstanding tracking performance of the mixed sensitivity H-infinity control, simulation is carried out with sinusoidal signals, square signals, and sawtooth signals and the frequencies are 10 and 30 Hz, respectively. Meanwhile, the same signals with a certain intensity of white noise are used in the simulation model.

Simulation without the mixed sensitivity H-infinity control

The signals response curves without the mixed sensitivity H-infinity control are shown in Figure 10, where the red dashed curve is the input signal and the black curve is the signals response. It can be seen that when the system is uncorrected, the signals response waveform distorts seriously and lags much, and it distorts more severely when the frequency increases. Therefore, the system cannot be used directly and should be appropriately corrected to improve the response performance.

Figure 10.

System responses without the mixed sensitivity H-infinity control in different frequencies: (a) sinusoidal response, 10 Hz; (b) sinusoidal response, 30 Hz; (c) square response, 10 Hz; (d) square response, 30 Hz; (e) sawtooth response, 10 Hz; and (f) sawtooth response, 30 Hz.

Simulation with the mixed sensitivity H-infinity control

The signals response curves with the mixed sensitivity H-infinity control are shown in Figure 11. It is clear that with the mixed sensitivity H-infinity control, the system is of good following performance.

Figure 11.

System responses with the mixed sensitivity H-infinity control under different frequencies: (a) sinusoidal response, 10 Hz; (b) sinusoidal response, 30 Hz; (c) square response, 10 Hz; (d) square response, 30 Hz; (e) sawtooth response, 10 Hz; and (f) sawtooth response, 30 Hz.

Simulation under a certain intensity of white noise

It is well known that rail irregularity is the main external excitation to the subgrade, and its effect to the subgrade can be considered as a random process. To simulate its effect, a certain intensity of white noise (20 dB) is added to the model to verify the effectiveness of the mixed sensitivity H-infinity control. Comparison of system responses with respect to different control input single and frequency as shown in Figures 12 and 13. The rmses (root mean square error) of system response without the mixed sensitivity H-infinity control are 0.5336, 0.8311, 0.8853, 3.6323, 4.7375, 2.3851 on sinusoidal response (10 Hz), square response (10 Hz), sawtooth response (10 Hz), sinusoidal response (30 Hz), square response (30 Hz), sawtooth response (30 Hz), respectively. The rmses (root mean square error) of system response with the mixed sensitivity H-infinity control are 0.4759, 0.7812, 0.5122, 0.7304, 1.0672, 0.6354 on sinusoidal response (10 Hz), square response (10 Hz), sawtooth response (10 Hz), sinusoidal response (30 Hz), square response (30 Hz), sawtooth response (30 Hz), respectively. Compared Figures 11 with 13, it can be seen that the system with the mixed sensitivity H-infinity control still maintains a good following performance with external disturbance.

Figure 12.

System responses without the mixed sensitivity H-infinity control in different frequencies with a certain intensity of white noise: (a) sinusoidal response, 10 Hz; (b) sinusoidal response, 30 Hz; (c) square response, 10 Hz; (d) square response, 30 Hz; (e) sawtooth response, 10 Hz; and (f) sawtooth response, 30 Hz.

Figure 13.

System responses with the mixed sensitivity H-infinity control in different frequencies with a certain intensity of white noise: (a) sinusoidal response, 10 Hz; (b) sinusoidal response, 30 Hz; (c) square response, 10 Hz; (d) square response, 30 Hz; (e) sawtooth response, 10 Hz; and (f) sawtooth response, 30 Hz.

Conclusion

In the article, we develop an improved control structure and a valve control cylinder drive system for the high-speed railway in situ testing system and study the static state characteristic and the dynamic pressure in both high frequency and different waveforms. To meet the requirement of the improvement of the system’s frequency response and realization of the high-frequency vibration, the article presents a mixed sensitivity H-infinity robust control to improve the system’s frequency response to simulate the load to subgrade. Several conclusions are as follows:

The dynamic response of the uncorrected system of the high-speed railway in situ testing system is very slow and lags much, so it can hardly be used directly.

Controlled by the mixed sensitivity H-infinity control, the following performance of the hydraulic servo excitation system improved a lot.

The system responses with the mixed sensitivity H-infinity control are insensitive to interruption and they can meet the requirements basically.

The weight functions are selected based on the experience and trial-and-error approach; optimization algorithm may be used in the future research to address the selection of weighting functions.

Footnotes

Appendix 1

The requirements of the excitation system are shown in Table 1, according to the table, when at 1 Hz, the amplitude of the system is about 20 mm, when reaches 40 Hz, the amplitude is about 0.5 mm. Usually the piston stroke is designed more than twice of the maximum amplitude, so we take the servo cylinder piston stroke $S$ to 50 mm.

Handling Editor: Anand Thite

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The authors would like to offer their gratitude to the National Science Foundation of China for their financial support (the grant nos 51027002, 51475338, 51405350 and 51705377).

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