飛行控制翻譯

1、A new nonlinear guidance logic, that has demonstrated superior performance in guiding unmanned air vehicles (UAVs) on curved trajectories, is presented. The logic approximates a proportional-derivative controller when following a straight line path, but the logic also contains an element of anticipa

2、tory control enabling tight tracking when following curved paths. The method uses inertial speed in the computation of commanded lateral acceleration and adds adaptive capability to the change of vehicle speed due to external disturbances, such as wind. Flight tests using two small UAVs showed that

3、each aircraft was controlled to within 1.6 meters RMS when following circular paths. The logic was ultimately used for air rendezvous of the two aircraft, bringing them in close proximity to within 12 meters of separation, with 1.4 meters RMS relative position errors一種新的非線性導(dǎo)航理論被提出了，它在無(wú)人飛行器在彎曲軌跡飛行的運(yùn)用

4、中表現(xiàn)出了優(yōu)良性能。當(dāng)直線飛行時(shí)，這個(gè)理論接近于一個(gè)比例微分控制器，但是，這個(gè)理論包含了一個(gè)預(yù)期控制的元素，使飛行器緊緊跟隨曲線路徑。該方法把慣性速度運(yùn)用于command橫向加速度的計(jì)算（command lateral acceleration）并且增加了針對(duì)由于外部干擾（例如風(fēng)）而引起飛行器的速度變化的自適應(yīng)能力。使用兩架小型無(wú)人飛行器的飛行試驗(yàn)表明：當(dāng)以圓形軌跡飛行時(shí)，飛行器將被控制在1.6米以?xún)?nèi)時(shí)。該理論最終運(yùn)用于飛行器的空中交會(huì)，使飛行器之間保持在接近12米的距離內(nèi)，允許1.4米的相對(duì)誤差Nomenclature 術(shù)語(yǔ)V Vehicle velocity 飛行速度 補(bǔ)償速度 L1 A

5、line defined from vehicle position to a reference point on a desired trajectory在預(yù)定軌跡中，定義前飛行位置到參考點(diǎn)的的直線（距離？）Angle created from V to the line L1 (clockwise direction is positive)速度方向V與我們的L1直線的方向的夾角（順時(shí)針?lè)较驗(yàn)檎゛scmd Acceleration command sideways i.e. perpendicular to vehicle velocity direction側(cè)向加速度，垂直于飛行器的

6、速度方向d Cross-track error交叉航跡誤差R Radius of circle or circular segment圓或則圓弧的半徑L Lyapunov function李亞普諾夫函數(shù)I. Introduction 1介紹Two approaches can be considered for the problem of trajectory tracking. One method separates the vehicle guidance and control problems into an outer guidance loop and an inner con

7、trol loop. The inner loop controls the vehicle to follow acceleration commands which are generated by the outer loop. Simple strategies,based on geometric and kinematic properties, are typically used in the outer guidance loop. The alternative method uses an integrated approach wherein the inner and

8、 outer loops are designed simultaneously. In this case, a number of modern control design techniques can be applied, such as receding horizon 1, differentialfatness 2, 3 and neural network based adaptive controls我們有兩套可以考慮的關(guān)于軌跡跟蹤問(wèn)題的方法，一種方法就是將飛行器引導(dǎo)和控制的問(wèn)題分為外引導(dǎo)回路和內(nèi)控制回路。內(nèi)引導(dǎo)回路控制飛行器跟隨由外控制回路產(chǎn)生的加速度?；趲缀魏瓦\(yùn)動(dòng)特性

9、的簡(jiǎn)單策略通常被運(yùn)用于外引導(dǎo)回路。另一種方法是應(yīng)用集成手段將內(nèi)外集成一體，其中內(nèi)外控制回路是同時(shí)設(shè)計(jì)的。在這種情況下，一些現(xiàn)代的控制設(shè)計(jì)技術(shù)可以被應(yīng)用，如：滾動(dòng)優(yōu)化，differential fatness，神經(jīng)網(wǎng)絡(luò)的自適應(yīng)控制。In most actual flight applications the separate inner and outer loop approach is more commonly taken because it is usually simpler and well-established design methods are available for

10、inner loop vehicle control. Linear controllers are commonly used for the outer loop guidance of an aircraft. Typically, proportional and derivative (PD) controllers are used on the cross-track error, which is the lateral deviation from a desired flight path. If the desired trajectory path is similar

11、 to a straight line, then this simple strategy will provide reasonably good outer loop performance. However, when tasks require tight tracking of complex curved paths, linear feedback on the cross-track error may not provide satisfactory performance. The guidance logicpresented in this paper contain

12、s an anticipatory control element which overcomes the inherent limitation of feedback control in following curved paths.在大多數(shù)實(shí)際飛行應(yīng)用中，單獨(dú)的內(nèi)或外控制回路方法是比較常用的，因?yàn)檫@樣通常是比較簡(jiǎn)單的，而且對(duì)內(nèi)回路的飛行器的控制是一個(gè)非常有效的設(shè)計(jì)方法。線性控制器通常用于飛機(jī)的外環(huán)制導(dǎo)控制。通常情況下，比例以及它的衍生（比例微分）控制器被用于交叉誤差的跟蹤，交叉誤差就是偏離于期望飛行路徑的橫向偏差。如果期望的軌跡路徑近似一條直線，那么這個(gè)簡(jiǎn)單的策略將提供非常好的外環(huán)控制

13、性能。然而當(dāng)我們需要精確跟蹤復(fù)雜彎曲的路徑時(shí)，基于交叉軌跡跟蹤的線性反饋不能提供令人滿意的性能。在本文中提出的制導(dǎo)理論包含了一種預(yù)期控制的元素，它克服了在跟隨彎曲路徑的反饋控制的固有限制。There are several terminal phase guidance laws for short-range tactical missiles that can be used to do trajectory following by using an imaginary point moving along the desired flight path as a pseudo targ

14、et. Of these, proportional navigation generally provides the best performance, with less control effort, in constantvelocity intercepts, and it is widely accepted as the preferred method of guidance 57. The trajectory following guidance logic presented in this paper was motivated by this proportiona

15、l navigation method. An important element in the proportional navigation is the use of the change in the line-of-sight between a missile and a target. A similar feature is also found in the trajectory following guidance logic between a vehicle and a pseudo target on a desired path.An important di

16、74;erence between the two methods is that,unlike the proportional navigation, the speed of the pseudo target is not taken into account in the trajectory tracking guidance logic. A detailed discussion on the relationship of the trajectory following guidance logic to proportional navigation is provide

17、d in Section II-B.有幾種末端引導(dǎo)規(guī)律的短程戰(zhàn)術(shù)導(dǎo)彈，他被使用在軌跡跟蹤中，具體方式是通過(guò)使用一個(gè)假想的沿著期望的飛行路徑飛行的點(diǎn)作為偽目標(biāo)。其中，比例引導(dǎo)一般提供了最好的性能，在等速攔截中需要的控制力度更小，這種比例引導(dǎo)方法被廣泛地認(rèn)可為制導(dǎo)的首選方法。本文提出的軌跡跟蹤引導(dǎo)理論源于這個(gè)比例引導(dǎo)方法。在比例引導(dǎo)中一個(gè)關(guān)鍵元素就是導(dǎo)彈與目標(biāo)之間瞄準(zhǔn)線變化的使用。相似的特點(diǎn)也出現(xiàn)在飛行器與在期望路徑上的假想目標(biāo)之間的軌跡跟蹤理論中。兩種方法有一個(gè)非常大的不同之處，與比例引導(dǎo)不同的是，在軌跡追蹤引導(dǎo)理論中偽目標(biāo)的速度是不被考慮的。軌跡跟蹤引導(dǎo)理論與比例引導(dǎo)理論的關(guān)系我們將在第二部分

18、的B篇進(jìn)行詳細(xì)的討論。Section II introduces the guidance logic and describes related properties. While the guidance logic developed here is simple and easy to apply, it is shown to have a number of benefits over linear approaches for curved paths. First, it contains proportional and derivative controls on cros

19、s-track error. Second, it has an element of anticipation for the upcoming local desired flight path. This property enables tight tracking on curved flight trajectories. Third, it uses instantaneous vehicle speed in the algorithm. This kinematic factor adds an adaptive feature with respect to changes

20、 in vehicle inertial speed caused by external disturbances such as wind.第二部分介紹引導(dǎo)理論并敘述相關(guān)屬性。然而發(fā)展到現(xiàn)在，制導(dǎo)理論已經(jīng)可以簡(jiǎn)單容易的應(yīng)用了，它被證明有很多優(yōu)于對(duì)曲線路徑的線性控制方法。首先，它包含了在交叉跟蹤中的比例和微分控制。其次，它有一個(gè)預(yù)算的元素，預(yù)期將要來(lái)的本地期望飛行路徑。這個(gè)屬性可以使他緊緊跟隨曲線飛行軌跡。最后，它在算法中采用了瞬時(shí)的飛行速度。這個(gè)運(yùn)動(dòng)因素增加了自適應(yīng)功能，用于調(diào)節(jié)由于外部干擾（例如風(fēng)）引起的飛行器固有速度的變化The algorithm is easily implemen

21、ted, and flight test results showing excellent tracking performance are given in Section III. The proposed guidance logic was implemented in two unmanned air vehicles (UAVs) in the Parent Child Unmanned Air Vehicle (PCUAV) Project 8, 9 at MIT, under the sponsorship of Draper Laboratory.該算法易于實(shí)現(xiàn)，飛行測(cè)試中

22、所表現(xiàn)出來(lái)的優(yōu)良跟蹤性能將在第三部分講解。在麻省理工的贊助下，在Draper實(shí)驗(yàn)室中，該引導(dǎo)理論提案在兩個(gè)無(wú)人飛行器中應(yīng)用測(cè)試。II. The New Guidance LogicThe guidance logic presented in this paper selects a reference point on the desired trajectory, and generates a lateral acceleration command using the reference point.本文提出的引導(dǎo)理論在預(yù)期軌跡上選擇一個(gè)參考點(diǎn)，并且用這個(gè)參考點(diǎn)產(chǎn)生一個(gè)橫向加速度指令。

23、.The reference point is on the desired path at a distance (L1) forward of the Lateral Acceleration CommandThe lateral acceleration command is determined by （1）Two properties of the guidance equation are significant.1. The direction of the acceleration depends on the sign of the angle between the L1

24、line segment and the vehicle velocity vector. For example, if the selected reference point is to the right of the vehicle velocity vector, then the vehicle will be commanded to accelerate to the right, which is the case in Figure 1. In other words, the vehicle will tend to align its velocity directi

25、on with the direction of the L1 line segment加速度的方向取決于L1線段和飛行器速度矢量的夾角，例如，如果選定的參考點(diǎn)是在飛行器的速度矢量右邊，這個(gè)飛行器將被控制向右加速，這就是圖1所示情況，換句話說(shuō)，飛行器將會(huì)調(diào)整其速度方向，使其速度方向與L1直線方向?qū)R。2. At each point in time a circular path can be defined by the position of the reference point, the vehicle position, and tangential to the vehicle v

26、elocity vector; as indicated by the dotted line in Figure 1. The acceleration command generated by Eq. (1) is equal to the centripetal acceleration required to follow this instantaneous circular segment. This is readily shown by noting that每個(gè)瞬時(shí)點(diǎn)的圓是由參考點(diǎn)和飛行器的位置以及飛行器速度矢量的切向確定和定義的；如圖1虛線所示。由等式1所產(chǎn)生的加速度就是這

27、個(gè)瞬時(shí)圓的向心加速度。下面等式很容易證明，如下： （2）So Centripetal acceleration=sin =Hence the guidance logic will produce a lateral acceleration that is appropriate to follow a circle of any radius R.因此，引導(dǎo)理論將產(chǎn)生一個(gè)橫向加速度，這個(gè)加速度指向一個(gè)圓的半徑方向This section describes a discrete time simulation that was performed to gain further insig

28、hts about the performance of the nonlinear guidance law. First, consider Figure 2 showing the evolution of the guidance logic in one small time step increment. In this diagram, the reference point is to the right of the direction of the vehicle velocity. Therefore, at the next time step the velocity

29、 direction rotates clockwise due to the acceleration command.本節(jié)描述一個(gè)離散系統(tǒng)模擬，用于進(jìn)一步獲得的關(guān)于非線性引導(dǎo)定律在性能方面的表現(xiàn)。首先，圖2顯示了在一個(gè)很小的時(shí)間步長(zhǎng)增量下由引導(dǎo)理論控制而產(chǎn)生的變化。在這個(gè)圖中，參考點(diǎn)是在飛行器速度方向的右邊。因此，在加速度作用下，下一步的速度方向?qū)㈨槙r(shí)針旋轉(zhuǎn)。With this one time step increment in mind, Figure 3 shows the trajectory of the vehicle over several time steps, wher

30、e the vehicle initially starts from a location far away from the desired path, and eventually converges to the desired path. Given a certain length L1 as shown in Figure 3, it can be inferred that帶著時(shí)間步長(zhǎng)增量的概念，圖3顯示了一些時(shí)間步長(zhǎng)后的飛行軌跡，其中飛行器最初從一個(gè)遠(yuǎn)離期望路徑的位置開(kāi)始，并最終收斂到期望路徑。給定某一個(gè)長(zhǎng)度L1如圖3所示，那么可以推斷： The direction of L

31、1 makes a large angle with the desired path, when the vehicle is far away from the desired path.當(dāng)飛行器遠(yuǎn)離目標(biāo)時(shí)，L1方向?qū)?huì)與期望路徑之間有一個(gè)很大的夾角。 The direction of L1 makes a small angle with the desired path, when the vehicle is close to the desired path.當(dāng)飛行器接近目標(biāo)時(shí)，L1的方向?qū)?huì)與期望路徑的夾角變小Therefore, if the vehicle is far aw

32、ay from the desired path, then the guidance logic tends to rotate the vehicle so that its velocity direction approaches the desired path at a large angle. On the other hand, if the vehicle is close to the desired path, then the guidance logic rotates the vehicle so its velocity direction approaches

33、the desired path at a small angle.因此，如果飛行器是遠(yuǎn)離期望路徑，引導(dǎo)控制理論將旋轉(zhuǎn)飛行器使其速度方向與期望路徑的夾角是一個(gè)大角度。另一方面，如果飛行器接近期望路徑，那么引導(dǎo)控制理論將旋轉(zhuǎn)飛行器使其速度方向與期望路徑的夾角是一個(gè)小的夾角。Consider the reference point as a target and the aircraft as a missile. Then, an interesting similarity is found in relation to proportional navigation missile guid

34、ance. The formula in Eq. (1) for the lateral acceleration command in the trajectory following guidance logic can be shown to be equivalent to the formula把參考點(diǎn)視作目標(biāo)，飛行器視作導(dǎo)彈。有趣的是，我們發(fā)現(xiàn)導(dǎo)彈中使用的比例引導(dǎo)法。在軌跡追蹤引導(dǎo)理論中的橫向加速度的等式1的形式可以證明是與這個(gè)公式是等價(jià)的。for the acceleration command perpendicular to the line-of-sight in the

35、proportional navigation with a navigation constant of N0=2, under the assumption that the reference point is stationary in the computation of the line-of-sight rate and the closing velocity. This equivalence can be shown using Figure 4.對(duì)于垂直于在比例引導(dǎo)中的瞄準(zhǔn)線的加速度的引導(dǎo)常數(shù)是2.，假設(shè)在瞄準(zhǔn)線的rate和接近速度的計(jì)算中參考點(diǎn)是靜止的，這個(gè)等式如圖4所

36、示，首先注意到，在這個(gè)飛行器的橫向加速度和這個(gè)垂直于LOS的加速度之間的不同Case 1: Following a Straight-line and Selection of L1Figure 5 defines the notation used in the linearization. L1 is the distance from the vehicle to the referencepoint, d is the cross-track error, and V is a vehicle nominal speed. Assuming is small in magnitude圖

37、5定義了一個(gè)在線性化中的標(biāo)記，L1是飛行器和參考點(diǎn)之間的距離，d是交叉跟蹤誤差，V是飛行器的理論速度，是一個(gè)很小的量Hence, linearization of the nonlinear guidance logic yields a PD (proportional and derivative) controller for the cross-track error. Also, the ratio of the vehicle speed V and the separation distance L1 is an important factor in determining t

38、he gains of the proportional and derivative controllers. For instance, a small value forL1 would lead to a high control gain and the ratio L1=V determines the time constant of the PD controller.The separation distance can be chosen by performing a stability analysis with the linear plant model and t

39、he derived linear controller. The plant model should include the vehicle dynamics with inner-loop bank angle controller (if bank angle is used to generate lateral acceleration for aircraft) and any sensor dynamics in the associated loop transmission function.因此，非線性引導(dǎo)理論的線性化然后做成一個(gè)PD控制方法來(lái)消除橫向軌跡誤差。在確定比例

40、和微分控制器的增益中，改飛行器的速度變化率和L1的距離是一個(gè)非常關(guān)鍵的因素。例如，一個(gè)很小的L1的值將會(huì)將會(huì)產(chǎn)生一個(gè)很高的控制增益，L1/V的變化率決定著PD控制器的時(shí)間常數(shù)我們可以通過(guò)線性控制模型和衍生線性控制器來(lái)進(jìn)行穩(wěn)定性分析，然后來(lái)選擇間隔距離。裝置模型應(yīng)該包括動(dòng)態(tài)飛行器的內(nèi)環(huán)坡度控制（如果傾斜角用于給飛行器產(chǎn)生橫線加速度）和相關(guān)環(huán)路傳送功能的任何動(dòng)力學(xué)傳感。Eq. (4) indicates that an approximate linear model, for the case of following a straight line, is a simple second or

41、der system that always has a damping ratio of 0.707 and its natural frequency is determined by the ratio of the vehicle speed and the length for the reference point selection.等式4指示一個(gè)近似的追蹤直線的線性模型，是一個(gè)簡(jiǎn)單的二階系統(tǒng)，它的阻尼比是0.707，并且它的自然震蕩頻率是由飛行速度和參考點(diǎn)選擇的長(zhǎng)度的比率所決定。Case 2: Following a Perturbed Non-Straight Line追蹤一

42、個(gè)擾動(dòng)且非直線的路徑The tracking capability on a curved path is demonstrated in this section by performing a linear analysis for a case with non-straight desired trajectory as shown in Figure 6. In this case the desired path is a perturbed curved line from a nominal straight line. In Figure 6, d is the latera

43、l position of the current vehicle location and indicates the position of the reference point. Assuming the magnitude of the angles, and are small本節(jié)中，我們將通過(guò)對(duì)非直線軌跡的線性分析來(lái)證明曲線追蹤能力，如圖6所示。期望路徑是由一條繞著標(biāo)稱(chēng)直線的擾動(dòng)曲線。d是飛行器當(dāng)前位置的橫向位置，指示參考點(diǎn)的橫向位置（相對(duì)于標(biāo)稱(chēng)直線）。假設(shè)角度和都很小。Eq. (6) represents a second order low-pass linear syste

44、m with a unity steady state gain from the reference point input to the vehicle position. The damping ratio（） is 0.707 and the undamped natural frequency( ) is determined by . The input of the transfer function in Eq. (6) is the lateral position of the referencepoint, not the position of the desired

45、path at current vehicle location (i.e., d?ref:pt: not d?). The use of a reference point in front enables phase recovery around the bandwidth frequency . For example, consider a sinusoidal trajectory command written as where A is a small path amplitude, Lp is a length-scale of the sinusoid, and x is

46、distance along the path. Assuming x = V t then if Lp =, the commanded trajectory expressed by Eq. (7) will excite the system at the bandwidth frequency (recall ). For a well-damped second order system asin Eq. (6) the phase lag at this frequency is 90 degrees. But this phase loss is from position in

47、put of the reference point , not from d*, in Figure 6. Recalling that the reference point is at L1 = Lp=4:4 (about a quarter of period) distance away, there will be about 90 degrees of phase lead in over d*. Therefore,these two effects will cancel each other, and the phase difference between the veh

48、icle position and the desired path at current vehicle location (i.e. between d and d*) will be significantly reduced.等式6代表一個(gè)二階低通線性系統(tǒng)，并且從參考點(diǎn)輸入到飛行器位置的一個(gè)穩(wěn)態(tài)增益。阻尼比是0.707,無(wú)阻尼自然震蕩頻率是由決定。等式6中傳遞函數(shù)的輸入是參考點(diǎn)的橫向位置（），特別注意不是當(dāng)前飛行器位置所期望的參考點(diǎn)位置。前面一個(gè)參考點(diǎn)的使用會(huì)使得帶寬頻率周?chē)南辔换謴?fù)。例如：式中，A是一個(gè)小的路徑幅度，Lp is a length-scale of the sinus

49、oid(Lp是正弦上的一段尺寸)，x是沿著路徑的距離，假設(shè)x=V*t 且Lp= 。由等式7所表示的方程將會(huì)刺激系統(tǒng)的帶寬頻率。對(duì)于一個(gè)阻尼較好的二階系統(tǒng)（等式6）,。在這個(gè)頻率上的相位滯后是90度。但是這個(gè)相位的損失是由于參考點(diǎn)（）的位置輸入引起的（而不是）。Recalling這個(gè)參考點(diǎn)是在遠(yuǎn)處（大約1/4周期）。將會(huì)超前90度的相位。這兩種效應(yīng)將會(huì)相互抵消。飛行器的當(dāng)前位置和飛行器應(yīng)該所在預(yù)定軌跡上的位置之間的相位差將會(huì)減少(i.e. between d and d*) 。本系統(tǒng)的伯德圖，以L1/Lp為橫軸的函數(shù)，如圖7所示，圖清楚地展示了，由于期望所產(chǎn)生在系統(tǒng)帶寬附近的相位改善（當(dāng)L1/L

50、p1/4.4=0.23）。如果Lp是在期望路徑最高頻成分的波長(zhǎng)，要想準(zhǔn)確跟蹤期望路徑，那么L1的選擇就必須少于Lp/4.4。例3，曲線路徑追蹤圖8為例，展示了圓弧路徑的追蹤。以下分析，我們都假定和是一個(gè)很小的量，但并不是足夠的小?？梢宰⒁獾剑慕鞘桥c圓弧段有關(guān)的。飛行器的當(dāng)前位置由r=R+d和指定的。表示速度方向，是由速度方向和預(yù)定圓弧的切向所形成的夾角。D. Comparison of the New Guidance Logic with the Traditional Linear MethodD.新的引導(dǎo)理論與傳統(tǒng)的線性方法比較In the previous section, it w

51、as shown that the nonlinear guidance logic approximates a linear PD controller,on cross-track error, in following a straight line. This section will compare, by simulations, the performance of the nonlinear guidance logic and the associated linear controller, for various cases of trajectories and wi

52、nd conditions.In the simulation analysis presented below, 25 m/s of nominal vehicle speed and the separation distance(L1)of 150 m were used for the associated linear controller given by Eq. (3).在上一節(jié)中，我們看到，非線性制導(dǎo)理論在對(duì)直線追蹤中的軌道誤差的控制近似為一個(gè)一個(gè)線性PD控制法。本節(jié)將通過(guò)仿真討論在不同軌跡和風(fēng)力條件下非線性控制理論和線性控制法的性能。在如下仿真分析中，飛行器的速度是25m/s

53、，相隔距離是150m(L1),使用相關(guān)的線性方法來(lái)進(jìn)行控制。Comparison 1 - Straight Trajectory FollowingFirst, the two methods were applied for tracking a straight line. Figure 9 shows the simulation setup with an initial cross-track error of 10 meters and the associated results using the two methods. The simulation results indi

54、cate that the performances of the two methods are roughly the same in following a straight trajectory.比較1:直線路徑追蹤第一，兩種方法都被運(yùn)用于直線路徑追蹤。仿真設(shè)置起始軌跡誤差為10米，兩種控制方法的結(jié)果如圖9所示，仿真結(jié)果表明，在直線追蹤中兩種控制方法是幾乎一致的。Comparison 2 - Curved Trajectory FollowingNext, the two methods were applied to tracking a curved line. Figure 10

55、 shows the simulation setup, the desired curved flight trajectory, and the associated simulation results. The aircraft is initially at level flight heading due north. The trajectory plot (a) in Figure 10 is the case where the linear controller was used. ThePD controller resulted in a steady state er

56、ror of about 40 meters. The steady state error can be explained by noting that the system is type 2. There are two pure integrators in the associated loop transmission with a plant model and the PD controller. The two integrators are from the kinematics of the plant model - fromacceleration input to

57、 position output. The steady state error occurs because the position reference command for cross-track is imposed in a parabolic fashion, when the desired path is a circle.比較2:曲線路徑追蹤：接著，兩種方法被應(yīng)用于曲線追蹤。圖10仿真設(shè)置了期望的飛行曲線路徑，顯示了相關(guān)仿真結(jié)果。飛行器最初飛行方向是水平正北。圖10中的路徑圖a是線性控制器的控制結(jié)果圖。PD控制方法將會(huì)產(chǎn)生40米的穩(wěn)態(tài)誤差。這個(gè)系統(tǒng)是一個(gè)2型系統(tǒng)，我們可以使用2型系統(tǒng)來(lái)解釋這個(gè)誤差。在這個(gè)plant模型和PD控制器相關(guān)環(huán)路的傳遞函數(shù)有兩個(gè)1/s。這兩個(gè)純積分plant模型的運(yùn)動(dòng)學(xué)中來(lái)自加速度輸入和位置輸出。當(dāng)期望路徑是一個(gè)圓的時(shí)候，穩(wěn)態(tài)誤差的產(chǎn)生是由于對(duì)于交叉誤差的參考點(diǎn)是拋物線類(lèi)型的。In order to eliminate the steady state error, an integrato