Sliding-mode-based robust controller design for one channel in thrust vector system with electromechanical actuators

Author’s Accepted Manuscript Sliding-mode-based Robust Controller Design for One Channel in Thrust Vector System with Electromechanical Actuators Nan Wang, Weiyang Lin, Jinyong Yu www.elsevier.com/locate/jfranklin

PII: DOI: Reference:

S0016-0032(16)30332-5 http://dx.doi.org/10.1016/j.jfranklin.2016.09.018 FI2729

To appear in: Journal of the Franklin Institute Received date: 16 March 2016 Revised date: 24 August 2016 Accepted date: 18 September 2016 Cite this article as: Nan Wang, Weiyang Lin and Jinyong Yu, Sliding-modebased Robust Controller Design for One Channel in Thrust Vector System with Electromechanical Actuators, Journal of the Franklin Institute, http://dx.doi.org/10.1016/j.jfranklin.2016.09.018 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

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Sliding-mode-based Robust Controller Design for One Channel in Thrust Vector System with Electromechanical Actuators Nan Wang, Weiyang Lin and Jinyong Yu

Abstract In this paper, a sliding-mode-based robust controller is proposed for the single channel thrust vector system (TVC) to suppress the disturbances and improve the tracking performance. Specifically, the dead-zone input-output relationship is analyzed to depict the mount gap in the mechanical shaft. The system mathematic representation including the mechanical and electrical sections, which suffers from the dead-zone nonlinearity, frictions and unstructured disturbance, is constructed. An adaptive-fuzzybased observer is developed to estimate and compensate the disturbances because the fuel combustion dynamic and frictions in TVC are inevitable but difficult to obtain the precise dynamic state. Based on the nominal model, a robust controller is designed via the sliding-mode variable structure approach, which is derived in the sense of Lyapunov stability theorem. Instead of the traditional hitting law in the sliding mode controller, the chattering problem due to the discontinuous switch law is addressed by a continuous function. In the end, various illustrative examples are provide to demonstrate the effectiveness of the designed method.

Nomenclature for single channel Vm

Equivalent viscous damping parameter in the actuator

Vn

Equivalent velocity related damping parameter in the sealing ring

Bn

Equivalent position related damping parameter in the sealing ring

H

Moment arm of the actuator about the sealing pivot

N. Wang, W. Lin and J. Yu are with the Research Institute of Intelligent Control and System, Harbin Institute of Technology, Harbin 150080, China. ([email protected], [email protected] and [email protected]). This work was partially supported by the National Natural Science Foundation of China (61673133).

2

Hm

Moment arm of the actuator rotor about the rolling screw

Jm

Equivalent inertia of the actuator

Jn

Equivalent inertia of the thrust nozzle

Kr

Stiffness coefficient of the mechanical shaft

N

Proportion factor of actuator rotation to mechanical shaft translation

Ng

Decelerating ratio of the gears

Lb

Pitch of the ball-screw

ηm

Efficiency of the transmission mechanism

xref (t)

Desired pitching or yawing displacement in the yawing or pitching channel

e(t)

Linear displacement error of nozzle in the yawing or pitching channel

Tl (t)

Required nozzle rotation torque in the yawing or pitching channel

u(t)

Control effort from rotation motor in the yawing or pitching channel

θn (t), θm (t)

Rotation angle of the thrust nozzle and the actuator, respectively

xn (t), xm (t)

Linear displacement of the thrust nozzle and the actuator, respectively I. I NTRODUCTION

The thrust vector control system (TVC) is one of the most significant power source in aircraft attitude control when a high manoeuvring acceleration and the low air density circumstance are required, such as missiles interception [1], satellite attitude adjusting [2], [3]. Traditionally, the TVC is driven by hydraulics actuators [4], [5]. Recently, owing to the high power, facilitated maintainability, reduced weight and volume, increasing interests on electromechanical actuators motivate the alternative to hydraulics ones and extensive research has been carried out in the Component Development Division of the Propulsion Laboratory at NASA Marshall Space Flight Center [6], [7] and the similar work in [8], [9]. Specifically, Schinstock et al. [6], [7] start the inspired work including the TVC model identification and parameters estimation approach. Li et al. adopt H∞ control and μ synthesis theory in [8] to tackle the large inertia tracking problem for the TVC. Following that, a mixed controller based on the proportion-integral-differential control (PID) and on-and-off control in [9] is designed for each channel actuator. However, some problems in the aforementioned results are ignored, such as the dead-zone character in the transmission shaft and the model perturbation. In addition, the friction model is simplified as a static map to the nozzle velocity. The disturbance forces, e.g., the counterforce

3

on the surface of the nozzle because of the fuel combustion dynamic effect and the air drag, are neglected in [8], [9]. Furthermore, in terms of the energy cost, the on-and-off control method is non-commercial and it may excite the resonance without the input shaping technology [11], [12]. Essentially, the TVC can be viewed as a two-channel-actuator-coordinated motion system, so the single-channel tracking controller design and two-channel decoupling method are two main issues in engineering. The two-channel decoupling method can be solved by the Euler transformation [9] and our work focuses on a robust controller design method to improve the system performance. In general, the disturbance forces and model perturbation can be integrated into a total disturbance. Following this idea, the adaptive robust control [14], [15], extended state observer [16], the disturbance observer-based approaches [17]–[19], H∞ control [20], [35] and slidingmode-based controller [21] will be effective methods for the disturbance compensation and robust controller design. Furthermore, the soft computing approaches, such as fuzzy logical [22]–[24], [31] and data-driven-based off-line model identification technology [25] are suitable for model estimations. On the other hand, the dead-zone is a significant type of input nonlinearity in the servo control science which can depict the mount gap in the mechanism. Due to the thermal expansion, this reserved mount gap is necessary but may be time varying and unmeasurable. Some control approaches are presented to confront this kind of input nonlinearity. To mention a few, the H∞ control and adaptive laws [27] are proven to be effective to deal with input nonlinearity for electro-hydraulic actuators in vibration suspension system. Moreover, the slidingmode-based controller is introduced for input nonlinearity in a general framework [28]. Whereas, the approaches need the prior knowledge of the bounds of the dead-zone width which may be difficult to obtain in practice. In conclusion, the tracking performance of the TVC may deteriorate due to various aforementioned disturbances. In this paper, the authors focus on a robust motion controller design method based on the sliding-mode variable structure control. Because the adaptive fuzzy logic has been proven to be effective for unknown function universal approximation [23] and insensitive to measurement noise, and the linguistic information from the engineers can be incorporated into the fuzzy rules, the disturbances are estimated by an adaptive-fuzzy-based observer and compensated in the feed-forward loop. It is worth mention that, for the air flow resistance, in the mechanism design process, one consideration is that the most components of the nozzle are

4

located inside the edge of rocket to ensure the least amount of the nozzle surface be exposed to air flow [13], so this kind of disturbance is ignored in this paper. Compared with the PID and on-and-off control methods in [9], the main features of the proposed controller include: 1) the control algorithms can be adjusted to get an expected performance like the PID controller, and are able to self-learn by means of the adaptive fuzzy inference for disturbances compensation; 2) a continuous function is utilized to alleviate the chattering phenomenon instead of the hitting law in conventional variable structure controller; 3) the designed controller will avoid the resonance and be more economical in power consuming compared with the on-and-off control. The outline of this paper is organized as following manners. In section II, the dynamic model of the TVC is formulated. Especially, the dead-zone input-output relationship is analyzed. Section III is dedicated to the main result about controller design method and the stability proof is provided in the sense of the Lyapunov stability. In section IV, several illustrative evaluations of the proposed controller are devoted to demonstrate the effectiveness and advantages. In the end, the conclusion is presented in section V. Throughout the paper, •ˆ denotes the estimation of •, the estimated error of • is •˜ = •ˆ − •. In addition, all the symbols and corresponding meanings used in this paper, which are not stated specially, are all listed in the table ‘Nomenclature for single channel’ before the introduction section. II. M ODEL

OF THE

S INGLE C HANNEL

OF THE

T HRUST V ECTOR S YSTEM

As shown in Fig 1, the TVC with electromechanical actuators consists of two channels according to the kinematics concepts, the yawing and pitching directions. In each channel, the electromechanical actuator accepts the motion commands from aircraft autopilot to achieve the angle pivot. In practice, engineers will fix the two channel actuators in vertical, and then the coupling motion for the thrust nozzle could be ignored. On the other hand, the parameter variation in the dynamic model for the TVC is omitted due to the small angle rotation, normally less than ten degrees. According to the statements above and related work in [6], [7], [9], the dynamic model for each channel electro-mechanism is subject to following conditions: (C1) There exist nonlinearities disturbances during the nozzle rotation. (C2) The motor rotation angle is detected through the incremental encoder.

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Sealing ring

Thrust nozzle

Actuators (including roll screw and rotor motor)

Fig. 1: The configuration diagram of TVC

(C3) The two channel actuators are installed orthogonally to remove the coupling movement. (C4) Two angular transducers are equipped for each channel to detect the nozzle angle displacement, respectively. (C5) The nozzle rotation angle is within the scale of ten degrees and there is no parameters vibration except the friction model in the contact surface between sealing ring and rolling balls.

Remark1: The model satisfies the practice situations that the rotation angle is limited, small and detectable in C4 and C5 [6]–[9]; therefore, the nozzle linear displacement can be calculated with the angle displacement with and moment arm H. The disturbances, nonlinearity and parameters vibration in C1 and C5 are considered to improve the TVC robustness and be more truthful. C2 is common in all motion control field. A. Model of the load with the fixed point motion According to the installation condition C3, the coupling movement between the two channel actuators is ignored. The authors focus on the controller design methodology for the single electromechanical actuator. Therefore, as shown in Fig 1, the pivot of the nozzle around sealing point can be decoupled into two independent directions, yawing and pitching. For tracking purpose, the desired angle command from the autopilot is separated into two channels for the nozzle. Also, because the two channel electro-mechanisms own the same

6

configuration, in the following, one of them is considered to continue the rest work. The desired linear position curve of the nozzle in single direction is defined as x ref (t) and the tracking error e∗ (t) is written as: e∗ (t) = xref (t) − xm (t)

(1)

Based on a nominal model of the nozzle and friction, the feedback signal employs the motor linear position xm (t) in [6], [7]. However, due to the model imprecision, the relationship between xm (t) and the nozzle’s linear position xn (t) is not linear time invariant any more. In this paper, the angular transducer in C4 is assumed to be used as the nozzle angle position measurement and the tracking error is formulated as: e(t) = xref (t) − xm (t) + (t)

(2)

(t) = k (xm (t) − xn (t))

(3)

Here, the (t) is the difference between the motor’s and nozzle’s linear displacements and k > 0 is the weight value which is tuned through simulations or experiments for each channel. B. System Modeling for Single Channel Actuator The installation gap in the mechanical connection will lead to a nonlinearity input during the force transformation process. For brevity, the nonlinearity input is considered as a dead-zone profile function D(·) and assumed that the dead-zone relationship between input v(t) and output vd (t) to be as eq. (4).

⎧ ⎪ v(t) − U+ (t) f or v(t) ≥ U+ (t) ⎪ ⎪ ⎪ ⎨ vd (t) = D [(v(t))] = 0 f or U− (t) < v(t) < U+ (t) ⎪ ⎪ ⎪ ⎪ ⎩ v(t) − U (t) f or v(t) ≤ U (t) − +

(4)

where the U+ (t), U− (t) are the dead-zone width. The key features of the dead-zone investigated in this paper are: (A1) the dead-zone edges are not available for measurement but their signs are known: U + (t) > 0, U− (t) < 0. (A2) the dead-zone parameters U+ (t), U− (t) are bounded and may be time varying due to the thermal expansion.

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Then, the dead-zone model eq. (4) is rewritten as following: vd (t) = D [(v(t))] = v(t) + Λ(t) ⎧ ⎪ − U+ (t) f or v(t) ≥ U+ (t) ⎪ ⎪ ⎪ ⎨ Λ(t) = − v(t) f or U− (t) < v(t) < U+ (t) ⎪ ⎪ ⎪ ⎪ ⎩ − U (t) f or v(t) ≤ U (t) − −

where

(5)

(6)

From assumption (A2), it can be concluded that Λ(t) is bounded and can be denoted as 0 ≤ |Λ(t)| ≤ τ (t)

(7)

where τ (t) is assigned as the maximum of the dead-zone width. The mathematical model of the single channel electric-drive TVC considered in this paper can be formulated as following: For rotation motor: u(t) = D[Tl (t)]N + Jm θ¨m (t) + Fm (t)

(8)

Fm (t) = Vm θ˙m (t)

(9)

N = Lb /(2πNg Hηm )

(10)

D [Tl (t)] = D [HKr (xm (t) − xn (t))] = Jn θ¨n (t) + Fn (t) + w(t)

(11)

Fn (t) = Vn θ˙n (t) + Bn θn (t)

(12)

Hm = Lb /(2πNg )

(13)

xm (t) = Hm θm (t)

(14)

xn (t) = Hθn (t)

(15)

For thrust nozzle:

Here, the control effort u(t) is the torque generated by the rotation motor and T l (t) is the required torque for the thrust nozzle rotation. Fm (t) and Fn (t) are friction torques in the motor and sealing ring, respectively. w(t) is the unstructured disturbance in the TVC. In this scenario, the servo driver will operate in the torque/current mode and with the current amplification, one can get the motor’s input command in the current unit; thus, this model formulation is reasonable. This

8

paper considers the position-depended Bn θn (t) and velocity-related friction Vn θ˙n (t), disturbance lump w(t) and a dead-zone relationship D [Tl (t)], which is a more accurate model compared with [6]–[9]. The disturbance lump w(t) consists of unstructured disturbance torques during the nozzle rotation. III. C ONTROLLER D ESIGN

FOR THE

TVC

This section is devoted to the positioning controller design method in accordance with the friction torque Fn (t) estimation and the controller design.

xref (t ), xm (t )

w(t ) D[Tl (t )]

Tl (t ) xm ( t )

xn ( t )

Actuator

e(t ), e(t )

+

s(t )

Load/Nozzle

u (t )

Controller

Jˆ1 , Jˆ2 , /ˆ

Fˆn (t )

Adaptive laws

Fuzzy inference

s(t ), T n (t ), T n (t )

Fig. 2: The control architecture diagram

A. Friction modeling The friction torque Fn (t) is the main obstruction in electro-mechanism for the nozzle rotation. Furthermore, depending on the experimental data to approximate the real one may be time consuming and conservative. An alternative is the adaptive-fuzzy-based inference to model F n (t). The indirect adaptive fuzzy observer employs the fuzzy logic to depict the unknown function in TVC and IF-THEN rules can be incorporated into the controller design process with intuitions and human experiences. Inspired by [23], the linear combination of the fuzzy basis function is able to approximate any continuous function. Moreover, due to the fuzzy logic robustness to the measurement noise, we adopt this idea to estimate the F n (t) on-linea and the Gaussian membership function is utilized for the fuzzy input variables. The Fn (t) is related to the nozzle angle position and velocity in eq. (12). When considering the tracking magnitude ranges are [−6◦ , 6◦ ] and [−6◦ /s, 6◦/s] respectively, the membership function of input θ = [θn , θ˙n ] is shown in Fig 3. The fuzzy inference IF-THEN rules are as following.

9

1

0.9

0.8

Membership function degree

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0 -0.1

-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

0.1

x

Fig. 3: Membership functions for input sets

R(j) : IF θn is Al1 AND θ˙n is Al2 , THEN Fˆn (t) is B l1 l2 . where Al1 and Al2 are fuzzy subsets for input variables θ and θ˙n , B l1 l2 is for output variables. l1 , l2 ∈ 1, 2, ..., M. For lim [e1 (t)] = lim [Fn (t) − Fˆn (t)] = 0, we employ the singleton fuzzification, the Gaussian t→∞

t→∞

membership functions, the center-of-gravity defuzzification and the product inference in the fuzzy system, the Fˆn (t) can be written as: M 

Fˆn (t) =

M 

l1 =1 l2 =1 M 

2 

ρl1 l2 (

i=1

M 

(

2 

l1 =1 l2 =1 i=1

l (θ)) i i

μA

μA

i

(16)

li (θ))

where ρl1 l2 is the value when the fuzzy membership μAi li (θ) rises to its maximum. Then, the linear form for eq. (16) is represented as: Fˆn (t) = ρT ξ(θ)

(17)

in which ρ = (ρl1 l2 ) is a parameter vector and ξ(θ) = (ξ l1 l2 (θ)) is a regressive vector with 2 

ξ l1 l2 (θ) =

M 

i=1 M 

l (θ) i i

μA (

2 

l1 =1 l2 =1 i=1

l (θ)) i i

(18)

μA

In this section, the fuzzy approximation for F n (t) in the linear formulation is established as eq. (17). In the following, the self-learning method for ρ by the adaptive method will be deduced as well as the fuzzy estimation residual error e1 (t).

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B. Design of the control law With sliding-mode variable s(t), s(t) = e(t) ˙ + ke(t)

(19)

where k is a positive scalar. Actually, the s(t) is an attractor; if s(t) converges to zero, so does the tracking error e(t) exponentially. Theorem 1: Considering the single channel electromechanical actuator depicted in section II, a robust controller eq. (20) and the adaptive laws eqs. (24) to (27) exist such that the system is stable and the position tracking error e(t) converges to zero;

ue (t) = [(1 − k )Hm a1 ]

−1



u(t) = ue (t) + ur (t) ˆ + k HJ n −1 Fˆn (t) f (θm ) + f (θ˙m ) + f (θn ) + f˙1 + f2 Λ

(20)  (21)

ur (t) = [(1 − k )Hm a1 ]−1 kb s(t)

(22)

γ1 s−1 (t) + f2 γˆ2 s−1 (t) kb = η + (k HJn−1 )ˆ

(23)

with f1 = x˙ ref (t) + ke(t) −1 − k HJn−1 f2 = (1 − k )Hm NJm

f (θm ) = θm (t)[a2 (1 − k )Hm − k Ha5 ] f (θ˙m ) = θ˙m (t)[(1 − k )Hm a4 ] f (θn ) = θn (t)[(k − 1)Hm a3 + k Ha6 ] and a1 = 1/Jm , a2 = HHmKr N/Jm , a3 = H 2 Kr N/Jm , a4 = Vm /Jm , a5 = HHmKr /Jn , a6 = Kr H 2 /Jn . The adaptive laws are ρˆ˙ = λ1 k HJ n −1 ξ(θ)s(t)

(24)

ˆ˙ = λ2 f2 s(t) Λ

(25)

γˆ˙1 = λ3 k HJ n −1 s(t)

(26)

γˆ˙2 = λ4 f2 s(t)

(27)

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where k, k , η, λ1 , λ2 , λ3 , λ4 are tunable and positive constants. Proof: The candidate Lyapunov function is chosen as V (t) = V1 (s(t)) + V2 (t) V1 (s(t)) = 12 s2 (t) V2 (t) =

1 2 ρ˜ 2λ1

+

1 ˜2 Λ 2λ2

it is easy to know that V (t) is positive. Then, if its first order derivative is negative definite, the s(t), e(t) → 0 |t→∞ . Then, with the dynamic model in part B, section II, the time derivative of s(t) can be written as: s(t) ˙ = (k − 1)Hm a1 u(t) + f (θm ) + f (θn ) + f (θ˙m ) + Λf2 + k HJn−1 Fn (t) + k HJn−1 w(t) + f˙1 Considering the control laws eqs. (20) to (23), the time derivative of the Lyapunov candidate function is represented as:   1 1 ˜ ˆ˙ −1 ˆ −1 ˙ ˆ V (t) = s(t) −k HJn (Fn − Fn ) − f2 (Λ − Λ) + k HJn w(t) − kb s(t) + ρ˜ρˆ˙ + Λ Λ λ1 λ2 Substituting eq. (17) and the adaptive laws eqs. (24) and (25), we obtain ρT ξ(θ) + −k HJn−1 s(t)˜ ˜+ −f2 s(t)Λ

˙ 1 ˜ˆ ΛΛ λ2

1 ρ˜ρˆ˙ λ1

=0

=0

V˙ (t) = k HJn−1 w(t)s(t) + k HJn−1 e1 (t)s(t) + f2 e2 (t)s(t) − kb s2 (t) ≤ −ηs2 (t) ˆ is the dead-zone width estimation residual error. with e2 (t) = Λ − Λ In addition, for alleviating the chattering in control effort u(t), the −ηs 2 (t) is employed instead of zero in the proof of the negative definite V˙ (t). In theory, if kb ≥ η + k HJn−1 w(t)s−1(t) + k HJn−1 e1 s−1 (t) + f2 e2 s−1 (t), the V˙ (t) will be negative. Here, the adaptive laws eqs. (26) and (27) are employed to estimate the values which guarantee the system performance. Therefore, s(t) ∈ L2 ; on the other hand, from the definition of s(t), it implies that s(t) ∈ L ∞ and s(t) ˙ ∈ L∞ . It follows from the Barbalats lemma that s(t) converges to zero asymptotically. Also, in accordance to the relationship between s(t) and e(t), one can conclude that the tracking error e(t) will converge to zero too.

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The proof is completed. Remark 2: In Theorem 1, six positive scalars, k, η, λ1 , λ2 , λ3 , λ4 , which determine the tracking accuracy and convergence speed. In addition, the fuzzy-based friction Fˆn (t) is valid with the adaptive laws even if Fˆn (t) is not accuracy enough, which will be seen in the illustrative examples. Moreover, the kb is for ensuring the robustness, the cancelation of unstructured disturbance w(t) and estimated residual errors e1 (t), e2 (t). Remark 3: The chattering phenomenon in the sliding-mode control is a serious drawback in application due to the limited bandwidth in the electric driver, energy cost, wear, etc [32], [33]. In this paper, instead of the sign function in the conventional method, a continuous control function in kb is proposed to attenuate the chattering phenomenon. More importantly, in eq. (23), instead of the positive constants γ 1 and γ2 , adaptive laws are employed to prevent the controller eq. (22) staying non-zero even if s(t) converts to zero which generates steady-state error. On the other hand, more work about advanced controller [10], [30] and vision-detection-based feedback [26], [29], [34] will be carried out in the future. IV. I LLUSTRATIVE

EXAMPLES

The validation examples of the proposed controller are presented in this section. The related model parameters are cited from [9] for comparability, shown in TABLE II. The relative results are numerically calculated in MATLAB/SIMULINK package. Firstly, the step response simulation of the proposed controller is provided. According to the dynamic performance indexes for the single channel actuator in [9], the setting time is less than 0.2s, the steady-state error is smaller than 1% and the overshoot should be limited under 1%. The designed parameters are k = 15, k = 1.98, η = 200, λ1 = 2500, λ3 = 50000; here, λ2 = 0, λ4 = 0 and Bn = 0 because in [9], there is no dead-zone or disturbance. The point-to-point tracking result is shown in Fig 4. Compared with the step response in [9], it is clear that the control current is within the range of 0 ∼ 20A if the motor torque constant equals to 0.075 N·m·s−1 as that in the same reference. The result demonstrates that the proposed controller can be adjusted for the desired performance indexes as the PID controller and a smaller control current value is needed. Following that, the continuous tracking mode is carried out with another group of designed parameters. The position related friction torque B n = 5 N·m·rad−1 and other parameters are

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TABLE II: Model parameters Parameter(unit)

Value

Vm (N ·m·A−1 )

1.0×10−3

Jm (kg·m2 )

1.78×105

Ng

3

ηm

8.5×10−1

Lb (m)

6.0×10−3

H(m)

3.447×10−1

Kr (N ·m−1 )

2.0×108

Vn (N ·m·rad−1 ·s)

55

Jn (kg·m2 )

9.2

1.2

1 Desired angle trajectory Actual angle trajectory

Position /degree

1

Control input

0.8

0.8 0.6

0.6

0.4 0.2

0.4

-0.2 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Time /s

Angle error

Error /degree

1

Control effert /N.m

0 0.2

0

-0.2

0.5

-0.4

0

-0.6

-0.5

-0.8

-1 -1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Time /s

(a) Step response of the engine

0.9

1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Time /s

(b) Control effort u(t) in step response

Fig. 4: Point-to-point tracking mode in one channel the same as these in TABLE II. Also, the projection algorithm [15] is adopted to limit the Fˆn (t) inside a constraint set. With the designed parameters k = 10, k  = 1.1, η = 40, λ1 = 300000, λ3 = 10000 without dead-zone or disturbance. The relative results are shown in Fig 5, Fig 6 and Fig 7. As seen in Fig 5, for a continuous trajectory, the proposed controller can keep

14

0.04

6

Desired trajectory Actual trajectory

Desired angle trajectory Actual angle trajectory

4

Position /degree

Position /m

0.02

0

2 0 -2

-0.02 -4 -6

-0.04 0

2

4

6

8

10

12

14

16

18

20

0

2

4

6

8

Time /s

8

10

12

14

16

18

20

Time /s

×10-4

0.1 Error

Angle error

0.05

4

Error /degree

Following error /m

6

2 0 -2

0

-0.05

-4 -0.1

-6 0

2

4

6

8

10

12

14

16

18

0

20

2

4

6

8

10

12

14

16

18

20

Time /s

Time /s

(a) Linear displacement of the engine

(b) Angle displacement of the engine

Fig. 5: Continuous tracking mode in one channel 10

4 Estimated friction Actual friction

γ ˆ1

8 3 6 2

1 2

γˆ1

Fuzzy adaptive friction /N.m

4

0

0

-2 -1 -4 -2 -6 -3 -8

-10

-4 0

2

4

6

8

10

12

14

16

18

20

0

2

Time /s

4

6

8

10

12

14

16

18

20

Time /s

(a) Friction approximation result Fˆn (t)

(b) Friction approximation error compensation γˆ1

Fig. 6: Friction in the engine the tracking accuracy and dynamic performance. For the adaptive-fuzzy-based friction torque estimation result in Fig 6, the Fˆn (t) can approximate the real one Fn (t) with relative small errors. In Fig 7, we can conclude that the control effort u(t) is smooth without the chatting phenomenon.

15

0.04

1 Sliding mode

Control input

0.8

0.03 0.6

0.02 0.4

Control effert /N.m

Sliding mode

0.01

0

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(a) Sliding mode function s(t)

(b) Input effort u(t)

Fig. 7: Chattering free result Finally, another example is proposed to verify the robustness when considering the dead-zone relationship D [Tl (t)] and the disturbance w(t) in the electromechanical actuator. It is assumed that the dead-zone in the mechanism shaft is −15 ∼ 20 N·m and the disturbance is represented as w(t) = 10sin(t) N·m between 5s and 10s which may mimic the time varying unstructured disturbance. Additionally, according to the values in TABLE II, f 2 and k HJn−1 in eqs. (21) and (23) are both sufficiently small. As a result, the dead-zone and disturbance compensation in control laws is integrated into eq. (26) and, for simplicity, the designed and model parameters are the same as ones in continuous tracking mode. The corresponding results are Fig 8, Fig 9 and Fig 10. From the results, we can conclude that the designed controller can guarantee the robustness as well as the tracking accuracy. V. C ONCLUSIONS In this paper, the electromechanical actuator dynamic model suffering from the friction torque, the dead-zone relationship and the unstructured disturbance is established and analyzed for the TVC system in spacecrafts. The key problem of the TVC motion control is decoupled into a single channel tracking issue. The authors propose a robust controller which is suitable for both the point-to-point (Fig 4) and continuous (Fig 5, Fig 8) tracking patterns, and the stability proof

16

0.04

6

Desired trajectory Actual trajectory

Desired angle trajectory Actual angle trajectory

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×10-4 Error

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(b) Angle displacement of the engine

Fig. 8: Continuous tracking mode in one channel including D [T l (t)] and w(t)

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Estimated friction Actual friction

8

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Fuzzy adaptive friction /N.m

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(a) Friction approximation result Fˆn (t)

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(b) Robustness compensation γ ˆ1

Fig. 9: Friction in the engine including D [T l (t)] and w(t)

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Control input

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0.8

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Sliding mode

0.01

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(a) Sliding mode function s(t)

(b) Input effort

Fig. 10: Chattering free result including D [Tl (t)] and w(t) is provided in the sense of Lyapunov method. In the meanwhile, the chattering problem in the sliding-mode-based control is confronted by a continuous function (Fig 7, Fig 10). With the designed controller, the results of various illustrative examples have been presented to show the effectiveness and advantages of the designed controller for the TVC system. R EFERENCES [1] F. Yeh, “Adaptive sliding mode guidance law design for missiles with thrust vector control and divert control system”, IET Control Theory and Applications, 6 (4) (2012) 552-559. [2] B. Xiao, Q. Hu and P. Shi, “Attitude stabilization of spacecrafts under actuator saturation and partial loss of control effectiveness”, IEEE Transactions on Control Systems Technology, 21 (6) (2013) 2251-2263. [3] K. Lu, Y. Xia, Z. Zhu and M. Basin, “Sliding mode attitude tracking of rigid spacecraft with disturbances”, Journal of the Franklin Institute, 349 (2) (2012) 413-440. [4] J. Yao, Z. Jiao, D. Ma and L. Yan, “High-accuracy tracking control of hydraulic rotary actuators with modeling uncertainties”, IEEE Transactions on Mechatronics, 19 (2) (2014) 633-641. [5] J. Yao, Z. Jiao and D. Ma, “A practical nonlinear adaptive control of hydraulic servomechanisms with periodic-like disturbances”, IEEE Transactions on Mechatronics, 20 (6) (2015) 2752-2760. [6] D. Schinstock, D. Scott and T. Haskew, “Transient force reduction in electromechanical actuators for thrust vector control”, Journal of Propulsion and Power, 17 (1) (2001) 65-72. [7] D. Schinstock, D. Scott and T. Haskew, “Model and estimation for electromechanical thrust vector control of rocket engines”, Journal of Propulsion and Power, 14 (4) (1998) 440-446. [8] H. Lu, Y. Li and C. Zhu, “Robust synthesized control of electromechanical acturator for thrust vector system in spacecraft”, Computers and Mathematics with Applications, 64 (2012) 699-708.

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