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applied sciences Article

A Modified Disturbance Observer Structure Based on Acceleration Measurement for Disturbance Suppression in Tracking Control System Jiuqiang Deng 1,2,3 , Wei Ren 1,2,3 , Hanwen Zhang 1,2,3 , Yong Luo 1,2,3 , Xi Zhou 1,2 and Yao Mao 1,2, * 1

2 3

*

Key Laboratory of Optical Engineering, Chinese Academy of Sciences, Chengdu 610209, China; [email protected] (J.D.); [email protected] (W.R.); [email protected] (H.Z.); [email protected] (Y.L.); [email protected] (X.Z.) Institute of Optics and Electronics, Chinese Academy of Sciences, Chengdu 610209, China Yanqihu Campus, University of Chinese Academy of Sciences, Beijing 100049, China Correspondence: [email protected]; Tel.: +86-135-4787-8788

Received: 1 August 2018; Accepted: 3 September 2018; Published: 6 September 2018







Abstract: The micro-electro-mechanical system (MEMS) accelerometer is widely adopted in many engineering control systems due to its extraordinary performance with high bandwidth, small size and low weight. However, massive drift caused by its insensitively at low frequency is the main factor which limits its performance. It leads to integral saturation when the feedforward method is used and hinders the improvement of disturbance suppression ability at low frequency, which is a significant factor for evaluating the closed-loop performance of a high-precision tracking system. To solve this problem, a modified disturbance observer structure and its corresponding new controller, which can improve disturbance suppression performance at low frequency by effectively rejecting more drift and weakening the occurrence possibility of integral saturation when drift exists, are proposed. Detailed analyses and a series of comparative experimental results verify that the proposed method can effectively enhance disturbance suppression performance at low frequency. Keywords: MEMS accelerometer; integral saturation; modified disturbance observer; disturbance suppression; drift rejection

1. Introduction The tracking control system (TCS) is widely used in long-distance communication, astronomical observation, target tracking and other scientific fields [1–5]. However, when applied to practical programs, the TCS suffers from some external interferences from atmosphere, ground or movable platforms such as vehicles, airplanes and ships [1,6]. In the TCS, gyroscopes and position detectors are generally used as sensors to implement a dual-loop feedback control (DFC) to stabilize the controlled platforms [7,8] Gyroscopes with a small delay are used to build a high-bandwidth inner loop and the tracking performance of the TCS mainly depends on the position information from position detectors such as a charge-coupled device (CCD). Control bandwidth and disturbance suppression ability are two significant factors for evaluating the closed-loop performance of the TCS. However, control bandwidth and disturbance suppression ability are limited by the mechanical resonance and the sensors’ performance such as the drift and time delay caused by the low sampling frequency [9,10]. In order to improve the performance of control bandwidth and disturbance suppression, the micro-electro-mechanical system (MEMS) accelerometer has been widely recommended for the TCS due to its high bandwidth, small size and low weight [11–15]. The MEMS accelerometer can compensate for the mechanical resonance in the acceleration feedback control loop, and is Appl. Sci. 2018, 8, 1571; doi:10.3390/app8091571

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therefore utilized to enhance the stiffness of the controlled object. Tang implemented a dual-loop control by using the MEMS accelerometer and the CCD with a high bandwidth to improve control bandwidth [16]. In 2016, Tian proposed a multi-loop feedback control method (MFC) with the MEMS accelerometer, MEMS gyroscope and CCD, which successfully improved the disturbance suppression performance [17]. The overall disturbance suppression ability is the product of each single loop, hence the disturbance suppression ability of MFC is naturally better than that of DFC [18]. Moreover, the research about the disturbance suppression of the TCS is based on MFC for obtaining better performance. To improve the performance of the TCS, the feedforward method based on direct measurement is also introduced as it can suppress most of the theoretical external disturbances [19,20]. However, it involves additional cost for another sensor and specific environments with low measurable noise. As a result, the implementation feedforward method is limited in more complex conditions due to its disadvantages. In order to avoid the extra cost and application limitations, a basic idea to suppress disturbances emerges, which estimates the influences of external disturbances independently by using the disturbances observer and then eliminates the perturbation through the feedforward method. This feedforward method was first presented by Ohnishi in the 1980s and was named the disturbance observer (DOB) method [21–23]. It has been applied to several engineering processes, such as robot control, the hard disk, the magnetic hard drive servo system and permanent magnet synchronous motor control [24–29]. Beyond this, Deng recommended the disturbance observer for an acceleration feedback control loop with a MEMS accelerometer, which improved disturbance suppression ability only at intermediate frequency [30]. Disturbance suppression is still a large obstacle for the good tracking performance of the TCS. One problem that has not been handled by the afore-mentioned method is that there is considerable drift caused by the MEMS accelerometer due to its insensitivity at low frequency. As a result, the drift leads to integral saturation when feedforward methods are used to suppress disturbance [30]. As such, it is hard to improve disturbance suppression performance at low frequency with the MEMS accelerometer. To solve the above-mentioned problem, in this paper we propose the modified disturbance observer (MDOB) method to enhance disturbance suppression performance at low frequency in the TCS, which weakens the occurrence possibility of integral saturation when drift exists. We design the structure of the MDOB by changing the output node of the feedforward and present a corresponding new MDOB controller which is simple but effective. As a result, the drift rejection ability is effectively improved, making it possible to significantly improve the disturbance suppression ability at low frequency. The rest of the paper is organized as follows: Section 2 introduces the basic tracking control system and disturbance observer; in Section 3, a modified disturbance observer and its performance analyses are presented; Section 4 shows the verification experiments of this method; and lastly, concluding remarks are presented in Section 5. 2. Tracking Control System and Disturbance Observer A typical tracking control system is shown in Figure 1, which can be used to track the movement of stars and communicate between the earth and satellites. The CCD is utilized to detect the errors between the target position and pointing position. However, the disturbance under the pedestal affects the tracking accuracy. Moreover, the frame rate of the CCD is limited. In order to improve the disturbance suppression ability and tracking accuracy, a MFC and a disturbance observer are recommended in the TCS.

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Figure 1.typical AA typical tracking control system. Figure 1. A tracking control system. Figure 1. typical tracking control system.

The structure the multi-loop feedback control system is shown Figure 2 [13,17]. It is The structure multi-loop feedback control system is is shown in Figure 2 [13,17]. It is composed The structure of of thethe multi-loop feedback control system shown in in Figure 2 [13,17]. It is composed of a position feedback control (PFC) loop, a velocity feedback control (VFC) loop and of a position feedback controlcontrol (PFC) loop, feedback control (VFC) andloop an acceleration composed of a position feedback (PFC)a velocity loop, a velocity feedback controlloop (VFC) and an an acceleration feedback control (AFC) loop. There three kinds sensors—the position sensor, feedbackfeedback control (AFC) loop. There areThere three kinds of sensors—the position sensor, velocity sensor acceleration control (AFC) loop. areare three kinds of of sensors—the position sensor, velocity sensor and acceleration sensor—which are used toposition acquire position information, angular and sensor acceleration sensor—which are used toare acquire the information, angular velocity and velocity and acceleration sensor—which used to acquire thethe position information, angular velocity and angular acceleration, respectively. angular respectively. velocity and acceleration, angular acceleration, respectively. θd θd 2 s2 s

θ ref θ ref+ +

wref wref

Cp (sC ) p (s+)

- -

+

aref aref

Cv ( C s )v (+s )

- -

+

u u

Ca (C s )a ( s )

+

Ga (G s )a (+s )

- -

a a

+ +

w w

θ θ

1/ s1/ s

+

+ +

Acceleration Acceleration Sensor Sensor

1/ s1/ s +

ɑ driftɑ drift

Velocity Sensor Velocity Sensor Position Sensor Position Sensor

Figure The multi-loop feedback control system. Figure 2. 2. The multi-loop feedback control system. Figure 2. The multi-loop feedback control system.

Inthis this figure, denotes theacceleration acceleration controlled plant, C )(s)denotes denotes theacceleration acceleration a (s) figure, GG a(s) denotes controlled plant, In In this figure, Ga(s) denotes thethe acceleration controlled plant, thethe acceleration Ca (C s )a ( sadenotes controller, C ( s ) denotes the velocity controller, C ( s ) denotes the position controller, θ the v ) denotes the velocity controller, C p( s ) denotes the position controller, θd denotes controller, denotes controller, the velocity controller, C p ( s )p denotes the position controller, θ d denotes C v (C s )v ( sdenotes d disturbance angle, and θre f denotes the given target position. disturbance angle, and denotes given target position. thethe disturbance angle, thethe given target position. θ ref θ refdenotes In addition toand the CCD, the MEMS accelerometer has also been widely used in target tracking In addition to the CCD, the MEMS accelerometer has been widely used target tracking In addition to the CCD, the MEMS accelerometer has also been widely used in in target tracking systems in recent years due to its excellent performance.also Due to its high bandwidth and specific systems recent years excellent Due high bandwidth and systems in in recent years due to to its its excellent performance. to to its its high bandwidth and dynamic characters, thedue transfer function ofperformance. the MEMSDue accelerometer equals constant 1.specific Asspecific a result, dynamic characters, transfer function MEMS equals constant 1. As a result, dynamic characters, thethe transfer function of of the MEMS accelerometer equals constant 1. As a result, the acceleration transfer function of AFC isthe depicted inaccelerometer Equation (1). the acceleration transfer function of AFC is depicted in Equation Error! Reference source not found. the acceleration transfer function of AFC is depicted in Equation Error! Reference source not found. Ca Ga 1 Ca Ga . . a= a + s2 θ d − a (1) 1 + Ca Ga re f 1 + Ca Ga 1 + Ca Ga dri f t Ca Ga Caa Ga CG 1 1 2 s 2θ −Ca G (1) a =a = a a arefa+ref + s θ d −d a adrift (1)up C G C G Caa Gadrift + + + 1 1 1 The MEMS accelerometer can also be utilized to measure the frequency characteristic to 1 + Ca Gaa a 1 + Ca Gaa a 1 + Ca G e 1 kHz, which means high-precision nominal plant Ga ≈ Ga can be acquired through system The MEMS accelerometer also utilized measure frequency characteristic up The MEMS accelerometer cancan also be be utilized to to measure thethe frequency characteristic upfrequency to to 1 1 identification [11,12,17]. However, there is massive drift caused by the insensitivity at low  kHz, which means high-precision nominal plant can be acquired through system  G ≈ G kHz,of which means high-precision be acquired Ga ≈ G a a the MEMS accelerometer whennominal angular plant acceleration is acan detected. The driftthrough makes it system difficult to identification [11,12,17]. However, there is massive drift caused by the insensitivity at low frequency identification [11,12,17]. However, there is massive drift caused by the insensitivity at frequency improve the disturbance suppression ability at low frequency with the feedforward low method. The harm the MEMS accelerometer when angular acceleration is detected. The drift makes it difficult of of the MEMS accelerometer angular is detected. The drift makes it with difficult to to from the drift is especiallywhen obvious whenacceleration the DOB method is introduced into the TCS the MEMS improve disturbance suppression ability low frequency with feedforward method. The improve thethe disturbance suppression at at low frequency with thethe feedforward method. The accelerometer as shown in Figure 3.ability harm from the drift is especially obvious when the DOB method is introduced into the TCS with harm from the drift is especially obvious when the DOB method is introduced into the TCS with thethe MEMS accelerometer shown in Figure MEMS accelerometer as as shown in Figure 3. 3.

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θd s2 aref

Ca ( s )

+

-

u

a

+

Ga ( s )

+

+

+

-

+

ɑ drift

MEMS Accelerometers

G a ( s )

-

+

C f (s)

Figure 3. structure A structure of disturbance observer acceleration feedback control (AFC). Figure 3. A of disturbance observer in in acceleration feedback control (AFC).

The acceleration transfer function in Figure 3 is depicted in Equation (2). It is clear that the drift The acceleration transfer function in Figure 3 is depicted in Equation rejection ability when the DOB is recommended is even worse than that of the AFC according to the Error! Reference source not found.. It is clear that the drift rejection ability when the DOB is comparison of Equations (1) and (2). When a DOB controller is designed, there are multiple integration recommended is even worse than that of the AFC according to the comparison of Equations elements which are used to offset the multiple differential elements in G . As a result, considerable Error! Reference source not found. and Error! Reference source not found..a When a DOB controller drift and the multiple integration of the DOB controller finally lead to integral saturation when the is designed, there are multiple integration elements which are used to offset the multiple differential MEMS accelerometer works at low frequency [30]. elements in G a . As a result, considerable drift and the multiple integration of the DOB controller

finally lead to integral saturation when the MEMS at low frequency [30]. eaaccelerometer 1−G Cf 2 Ga Cworks Ca Ga f + Ca Ga a= are f + s θd − adri f t , (2)  CCa Ga 1+ 1C +a G Caa Ga 1 −1G+ Ga C f + CaCGaaGa a f 2 a= aref + s θd − adrift , (2) 1 + Ca Ga 1 + Ca Ga 1 + Ca Ga where C f denotes the DOB controller. ea −1 . As the nominal Equation (2), it is clear that the ideal DOB controller is C f _ideal = G where Through C f denotes the DOB controller. ea of the controlled object is a strictly proper transfer function, the ideal DOB controller is plant G Through Equation Error! Reference source not found., it is clear that the ideal DOB controller system, which can be handled a low-pass filter  is a non-causal . As the nominal plant of thewith controlled object is a[29,30]. strictly proper transfer function, G C f _ id ea l = G a − 1

a

the3.ideal DOB controller is aObserver non-causal system, which can be handled with a low-pass filter [29,30]. Modified Disturbance The reason for the occurrence 3. Modified Disturbance Observer of integral saturation is that there is too much drift that cannot be effectively rejected by the DOB when the MEMS accelerometer detects movements at low frequency. The reason for the integral occurrence of integral saturation is that is too much drift that be The key to avoiding saturation is that more drift of there the accelerometer needs tocannot be rejected. effectively rejected by the DOB when the MEMS accelerometer detects movements at low frequency. Therefore, we propose a creative MDOB structure as shown in Figure 4, where we change the node The to avoiding integral the saturation thatofmore of the Caccelerometer needs to be rejected. of key feedforward by moving output is node DOBdrift controller f from the outer loop of the nominal Therefore, weinner propose a creative MDOB structure as to shown in Figure where we change the node plant to its loop. Moreover, in order to adapt this change, we4,then propose a corresponding of new feedforward by moving the output node of DOB controller from the outer loop of the nominalthe C f simple but effective way to design the MDOB controller, which enormously simplifies plant to its inner loop. in orderand to adapt to thisimproves change, we then propose a corresponding design process of theMoreover, MDOB controller effectively both drift rejection and disturbance new simple but effective way to design the MDOB controller, which enormously simplifies the design suppression performance. process of the MDOB controller and effectively improves both drift rejection and disturbance suppression performance.

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θd s2 aref

+

Ca ( s )

-

uc

+

u

a

+

Ga ( s )

+

+

-

+

ɑ drift MEMS Accelerometers

+

C f ( s)

G a ( s )

-

Figure Figure 4. 4. Control Control structure structure of of modified modified disturbance disturbance observer observer (MDOB) (MDOB) in in AFC. AFC.

The mathematical analyses in Figure 4 are as follows: The mathematical analyses in Figure 4 are as follows: a a==θθddss2 2++ uGaa

(3) (3)

e)C ucuGcaG u =u u=c u−c −( a( a++aadri driftf t−− a )fC f

(4) (4)

uc = uc (=are (afref−−aa − − aadrift dri )f C t )aCa

(5) (5)

After the MDOB: MDOB: After substitution, substitution, itit is is easy easy to to obtain obtain the the following following transfer transfer function function of of the the AFC AFC with with the  ea+CG eaf C f (1 GG GaaG GaaG C Ca GaC(a1G+a G +aC aC f )a C f ) fC aC f + 11 a adri f t are faref++ ss22θθdd −− e e    GeCa C) f )drift 1+ G1a+CG + C G ( 1 + G C ) 1 + G C + C G ( 1 + G C ) 1 + G C + C G ( 1 + 1 + Ga aCf f + C 1 + GaCaf +f CaGa a (1a + G a aG a a (1 + G a a Cf f ) f a C fa + aCa Ga (1a+ G f aC f ) a f

a = a=

(6) (6)

e a , Equation (6) can also be simplified as Equation (7). As shown in Equation (7), G Due ≈G aa ≈ Due to to GG a , Equation Error! Reference source not found. can also be simplified as Equation the disturbance transfer function and drift transfer function can be described as Equations (8) and (9), Error! Reference source not found.. As shown in Equation Error! Reference source not found., the respectively. Compared with the acceleration feedback closed loop without the DOB or MDOB shown in disturbance transfer function and drift transfer function can be described as Equations Equation (1), Tdri f t is totally changed by the recommended MDOB. This means that the drift rejection Error! Reference source not found. and Error! Reference source not found., respectively. Compared ability can be greatly improved by designing the MDOB controller. Beyond this, the tracking performance with the acceleration feedback closed loop without the DOB or MDOB shown in Equation is not influenced by introducing the MDOB into the AFC, as shown in Equations (1) and (7). Error! Reference source not found., Tdrift is totally changed by the recommended MDOB. This means

that the drift rejection ability by designing G the controller. Beyond ea C MDOB 1 Ca Ga can be greatly improved f a= anot s2 θthe adrithe re f + f t AFC, as shown(7) d − MDOB into this, the tracking performance is influenced by introducing in ea C f ) ea C f 1 + Ca Ga (1 + Ca Ga )(1 + G 1+G Equations Error! Reference source not found. and Error! Reference source not found.. 1 s2 θ G C (8) Ca GTa dis (s) = 1 easC2θf d) − d a f adrift aref + (1 + Ca Ga )(1 + G (7)   1 + Ca Ga (1 + Ca Ga )(1 + Ga C f ) 1 + Ga C f e Ga C f Tdri f t (s) = a (9) 1 ea C f dri f ts 2θ d Tdis ( s ) = 1+G (8)  (1 + Ca Ga )(1 + Ga C f ) Focusing on the changes in the disturbance transfer function and drift transfer function with G a Cthat the MDOB, we should design a MDOB controller is beneficial for both drift rejection and f Tdrift ( s) = adrift (9)  disturbance suppression. A new type of MDOB 1controller C f , which is a causal system and might + Ga C f be a very suitable one, is depicted in Equation (10). It is clear that there is only one parameter K that on the changes in the transfer and drift transfer function with the needsFocusing to be determined, meaning thedisturbance design process will function be enormously simplified. MDOB, we should design a MDOB controller that is beneficial for both drift rejection and disturbance After the design, the overall drift rejection ability with the MDOB is shown in Equation (11). It is 1 suppression. A new of MDOB which is a causal system and might be MDOB a very an inertial element andtype its inertia time controller constant is C f. ,The disturbance transfer function with the

a=

K

controller is shown in Equation (12), and theReference disturbance suppression ability low frequency can be suitable one, is depicted in Equation Error! source not found.. It isatclear that there is only enhanced from constant. one parameter that needs to be determined, meaning the design process will be enormously K K e −1 simplified. Cf = G (10) a s

Cf =

G a C f Tdrift ( s ) = adrift = 1 + G C

Appl. Sci. 2018, 8, 1571

where

Ga−1

K  −1 Ga s

a

f

(10)

K s a = 1 a drift 1 K drift 1+ s +1 s K

(11) 6 of 12

K ea C f G 11 s s Tdri f t (s) = a = a ft = 1 a 2ft dri f t 1 K dri 1 2 K 1 dri e =s ⋅s+ Tdis ( s ) = 1 + Ga C f s 1θ+ s θd , K d (1 + Ca Ga ) 1 s + 1 (1 + Ca Ga )(1 + G a C f ) K K1 s 2 1 1 Tdis (s) = s2 θ d = · s θ , ea C f )  (1 + Ca Ga ) K1 s + 1 d (1 +ofCnominal G a Ga )(1 +plant denotes the inverse Ga .

(11) (12) (12)

−1 denotes thethe bandwidth the AFC loop eageneral, ea .is designed as up to 200 Hz, hence the cut-off whereInG inverse ofof nominal plant G frequency of its drift rejection is also about 200 Hz. According toHz, thehence comparison of frequency Equations In general, the bandwidth of the AFC loop is designed as up to 200 the cut-off Error! Reference found. andAccording Error! Reference source notof found., the drift rejection of its drift rejectionsource is also not about 200 Hz. to the comparison Equations (1) and (2), theability drift with the DOB is even worse when the DOB is introduced. In other words, the cut-off frequency of the rejection ability with the DOB is even worse when the DOB is introduced. In other words, the cut-off drift rejection with the DOB is larger than 200 Hz. By contrast, the cut-off frequency of the drift frequency of the drift rejection with the DOB is larger than 200 Hz. By contrast, the cut-off frequency K constant rejection the MDOB is much smaller for constant , which means thatmeans it can bring better of the driftwith rejection with the MDOB is much smaller for K, which that itmuch can bring drift rejection ability as shown in Figure 5. much better drift rejection ability as shown in Figure 5. Overall Drift Rejection AFC AFC + DOB AFC + MDOB

5 0

Magnitude/dB

-5 -10 -15 -20 -25 -30 0

1

10

2

10

10

3

10

Frequency/Hz

Figure Figure5.5.The Thecomparison comparisonof ofoverall overalldrift driftrejection. rejection.

Under structure as shown in Figure 2, the 2, disturbance suppression ability ofability the position Underthe theMFC MFC structure as shown in Figure the disturbance suppression of the loop is depicted as Equation (14), and the disturbance suppression ability of the position loop with the position loop is depicted as Equation Error! Reference source not found., and the disturbance MDOB is shown in Equation (15). Through the comparison of Equations (14) and (15), the disturbance suppression ability of the position loop with the MDOB is shown in Equation suppression ability of the positionnot loop is changed by introducing MDOB to theof MFC,Equations and the Error! Reference source found.. the thecomparison Through e improvement can besource represented as 1/(and 1 + GError! 1. Error! Reference not found. Reference source not found., the disturbance a C f )