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附件2、外文資料翻譯譯文
液壓驅(qū)動的無級變速器控制
4. 液壓約束
CVT的比率控制器（實際上）控制初級和次級壓力。幾個壓力的限制，必須考慮到該控制器：
1. 轉(zhuǎn)矩限制Pα≥Pα的扭矩，防止打滑的滑輪;
2. 較低壓力的約束Pα≥Pα，以保持兩個電路注滿油。在這里，相當任意的的PPlow=3[bar]選擇.為使有足夠的油流Qsa的附件電路和用于被動閥在該電路的一個適當?shù)牟僮魇潜匾腝sa大于最小流Qsa，最小。最小壓力Ps低4[bar]轉(zhuǎn) 證明是不夠的;
3. 上部壓力限制Pα≤Pαmax以防止損壞液壓管路汽缸和活塞.因此，Ppmax=25[bar]PS最大值=50[bar];
4. 液壓約束Pα≥Pα液壓，以保證主電路能快速放掉夠向漏和次級電路可以提供足夠的流動朝向初級電路.
壓力Pp，的扭矩和Ps扭矩限制1依賴于關(guān)鍵的夾緊力 Fcrit方程（5）.估計轉(zhuǎn)矩Tp是使用固定式發(fā)動機的的扭矩計算 地圖，變矩器特性和鎖止離合器模式，隨著慣性作用一起發(fā)動機輪和主齒輪箱軸.安全系數(shù)Ks=0.3相對于估計最大的主轉(zhuǎn)矩Tpmax已被引入到占上干擾估計轉(zhuǎn)矩Tp，例如沖擊負荷的車輪。然后帶輪的夾緊力（相等的兩個滑輪，而忽略了變速器效率）所需的扭矩傳遞變成了：
因此，所產(chǎn)生的壓力，可以很容易地使用公式推導（12）和（13）：
(26)
(27)
一模一樣的夾緊裝置已被以前使用參考。[ 3 ]試驗臺用于測量這款變速箱與測試車路。無滑移已經(jīng)實現(xiàn)，在任何這些實驗中這項工作的主要目標是改進比跟蹤行為，夾緊裝置維持不變。進一步的闡述制約4是基于質(zhì)量守恒的定律初級電路。首先，應當指出，對于本論述的泄漏流量Q p ，泄露漏和可壓縮長期可忽略不計相比。此外，它被再次提及，流量QSP與QPD永遠不能不等于零，則在同一時間。最后，可以選擇替換的比率變化rcvt率通過比移rcvt ，D所需的速率，這是由分層傳動系統(tǒng)控制器指定。如果 rcvt ，D <0，則油流出的主缸的給排水管，所以QPD > 0且QSP = 0 。約束4相對于主滑輪電路然后導致以下關(guān)系的壓力，hyd ：
(28)
其中APD，max是主閥的流路中由初級的最大開口氣缸組成。以類似的方式，對于次級帶輪電路壓力Ps的關(guān)系，HYD在約束4可以得出。這個約束是特別相關(guān)的，如果˙rcvt>0，也就是說，如果從流量QSP 次級到初級電路必須為正并且結(jié)果，QPD= 0。這就 液壓驅(qū)動的CVT397的結(jié)果：
(29)
Asp，max是主閥中的流道從次級最大開口到主電路。 對于CVT的比率控制器的設計是有利的配制，以約束夾緊力，而不是壓力方面。一個關(guān)聯(lián)的夾緊力Fα，β與壓力Pα，β和使用等式（12）和（13）這個結(jié)果的要求：
(30)
最小滑輪夾緊力：
(31)
5.控制設計
假定在本節(jié)中，在每個時間點t時，主speedωp （t）的比值rcvt （t）的初級壓力峰值（ t）和次級壓力Ps （t）的測量結(jié)果從已知的過濾和重建。此外，假定該無級變速器被安裝在一個車輛傳動系和所期望的CVT比rcvt ，D（t）和比值變化的所需速率˙ rcvt ，D（ t）由整體分層傳動系統(tǒng)控制器指定。這意味著，每個時間點的約束反力可被確定。本CVT控制器的主要目標是實現(xiàn)的快速和準確的跟蹤所需比例的軌跡。此外，控制器也應該是對干擾的控制性。一個重要的子目標是最大限度地提高效率。這是很合理的（和其他支持通過實驗， [3]） ，要實現(xiàn)這個子目標夾緊力Fp和Fs的必須為越小越好，考慮在方程的要求（30）考慮在內(nèi)。比例控制器的輸出是受方程（31）的約束。約束Fα ≥ Fα ，最小有效提高1帶輪的夾緊力的設定值，從而產(chǎn)生一個不良率的變化。這可以通過提高相對帶輪的夾緊來抵消力為好，使用基于模型的補償條款中的比例控制器。使用IDE的模式，即用式（ 10 ） ，表達式的比例變化迫使閃點，比例和Fs ，比（圖8 ）可以很容易地得出：
(32)
(33)
其中Fshift，d是期望的換檔力，基本上之間加權(quán)力差異兩個滑輪。如前所述，κ取決于τs這又取決于Fs的。這是一個隱式關(guān)系（FS，比例取決于FS），已解決由壓力計算κ測量?，F(xiàn)在將顯示在每一個時間，兩個夾緊力之一等于Fα，min，而其他確定的比值。用公式（30），（32）和（33）表示
圖8帶約束補償比例控制器
次級夾緊力Fp，d和Fs，d由下式給出：
（34）
（35）
實際上，該比例被控制在這樣一種方式，移動力FSHIFT變得等于Fshift，d。對于由此產(chǎn)生的換檔力擁有所以：
（36）
這適用于，只要夾緊力不會對它們的最大約束飽和 （Fα，_≤Fα，min）。在Fα，ratio≥Fα，max，F(xiàn)α，d =Fα，max，F(xiàn)shift= Fshift，d的情況。因此，該換檔速度是因為飽和執(zhí)行器有限。（根據(jù)完成控制器，F(xiàn)shift，d必須變速器的specified.As動態(tài) 到IDE的型號）都相當非線性，等效輸入u介紹，使用逆 該井模型Fshift，d的代表性：
（37）
當|ωp |與雙方的互補基本上是一個反饋線性化曲線。這將取消（已知的）非線性的變速器，max見，例如Slotine等。 [15]。另外，設定值前饋被引入，這將降低受控的相位滯后系統(tǒng)響應。由于模型不準確等因素（如上層鎖模力的限制），max差異˙rcvt和˙rcvt之間γ，max會發(fā)生d：
（38）
如果u代替γ以及獲得良好的跟蹤性能。線性反饋控制器 基于該（違背方程（10））中，有慣性的知識被選定進行u 參與，需要至少一個第二順序控制器。因此，使用的PID控制器。
比例控制是用于迅速減少錯誤，而集成所需的過程，以便跟蹤斜坡設定值與零誤差。某些微分作用證明有必要獲得更大的穩(wěn)定裕度（少振蕩響應）?？刂破鲗崿F(xiàn)如下：
（39）
其中Ke∈{0，1??}切換積分，并根據(jù)是一定條件下進一步解釋。控制器的微分作用只作用于所測CVT 信號，以避免在給定值的階梯式變化的過度控制響應。 此外，一個高頻極點已被添加到該過程的操作，以防止過度的頻率在高頻率?？刂破鲄?shù)P，I和D已被調(diào)諧手動。
在執(zhí)行器飽和的（因為最大的力約束）情況下， 閉環(huán)有效地打破（測量rcvt已經(jīng)不反應的變化,u）。這會導致性能下降，因為控制器的積分器的值繼續(xù)成長。這個所謂的積分器積分飽和是不可取的。有條件的抗飽和機制 已加入飽和期間限制積分器的值：
（40）
如果任飽和壓力（= ，max or= ，max），移動速度誤差γ必然變大?？狗e分飽和算法，確保穩(wěn)定，但跟蹤行為會 惡化。這是硬件限制其只能通過提高變換器來解決和液壓系統(tǒng)的硬件。有條件的抗飽和與一個標準的（線性）的優(yōu)點算法是線性方法需要調(diào)整的良好表現(xiàn)，而條件 辦法沒有。此外，有條件的算法的性能密切類似于一個良好的線性調(diào)整機制。
6.實驗結(jié)果
作為無級變速器已經(jīng)在測試車輛已經(jīng)實現(xiàn)，在車載實驗上的滾子長凳已經(jīng)進行調(diào)整和驗證新的比率控制器。為了防止非同步的油門和CVT比操作，油門踏板信號（見圖1）具有被用作輸入的驗證實驗。協(xié)調(diào)器將跟蹤 發(fā)動機的最大效率運行點。在巡航控制的一種半強迫降行動 背出了在一個單一的參考實驗已進行的?50公里每小時后跟一個踏板的速度。記錄的踏板角度（參見圖9）已被施加到所述協(xié)同控制器。 這種方法取消了有限的人力驅(qū)動的可重復性。
圖10的上圖顯示了從速度的測量計算出的CVT比反應 利用方程（1），描述的跟蹤誤差。因為這是一個相當苛刻實驗，跟蹤信號是足夠的?？梢缘玫礁玫母櫺阅?更光滑的設定點，但反應的特點將變得不那么明顯 為好。圖11示出了初級和次級帶輪壓力。最初的主峰在誤差信號（大約T =1.5秒）是飽和的二次壓力（下圖11地塊），由于泵的過流限制。一個更快的初步反應是必需的， 液壓硬件的適應是必要的。初始快速降擋后， 比再次降檔之前達到設定值（大約T =7次）。在轉(zhuǎn)移所有變動方向（T =1.3，T =1.6和t= 7.5s）發(fā)生相對少量的過沖， 這表明，該積分抗飽和算法表現(xiàn)良好。
看著在T附近的初級壓力= 1.5秒，它可以被觀察到，這壓力峰值反復高于其設定值。此行為是由性能限制 主壓力控制器。所開發(fā)的控制器，保證只有一個當時帶輪壓力設定值上升高于其低限，并且只實現(xiàn)期望的比率。
圖9 踏板輸入的CVT動力總成 圖10 CVT速比響應和跟蹤誤差，輥板凳半降檔
圖11 初級和次級帶輪壓力，輥板凳半降檔 圖12 新控制器的滑輪壓力設定值減去低的限制
這被可視化在圖12中。更高的夾緊力導致更多的損失 無級變速器[10]，只要沒有宏觀滑移發(fā)生。主要原因是油泵電力需求 （大約與壓力呈線性），并在帶本身，這既增加而損失 增大夾緊壓力，通過測量[16]作為支撐。因此，該控制器具有用于提高CVT的效率的電位，相對于基于非模型控制器。
回顧圖10，第二（正）峰的下圖（之后的第一個負峰值由于執(zhí)行器飽和）代表的比例響應的超調(diào)，由于移動方向變化。這個量描述了控制器的跟蹤性能好，并且將被用于評估控制器的性能。超調(diào)在這里計算作為（正的）最大的比例誤差：最大（rcvt，d - rcvt）。另外，平均絕對誤差（1 / N）的| rcvt-d，rcvt|（在10秒的響應的N個數(shù)據(jù)點）將用于比較的結(jié)果。
同樣的實驗已經(jīng)執(zhí)行用于在控制器上幾個變化。對于每一個這些變型，所有約束都仍施加，但有些在補償方面 比控制器已被暫時關(guān)閉（在圖8中的垂直箭頭所示）。結(jié)果進行了比較的結(jié)果為總控制器和在圖13中被描繪。將要處理的情況下，有：
1.所有前饋和補償?shù)模偭浚?
2.沒有設定值前饋（斷），rcvt，d =0等式（37）。
3.沒有關(guān)鍵（無皮帶打滑）扭矩約束補償（T排版關(guān)），F(xiàn)torque=0。
4.無液壓約束補償（hydr樣圖關(guān)），F(xiàn)α，液壓=0。
5.無扭矩傳遞，也不液壓約束補償（T，hydr樣圖關(guān)），F(xiàn)torque=0，F(xiàn)α，液壓=0。
它是立即清除所有的替代品，與所有前饋總控制器和在上段所述（總量）補償器性能最佳，這意味著所有 控制器方面擁有盡可能降低跟蹤誤差了積極的貢獻。開關(guān)關(guān)閉或液壓約束補償項（hydr排版關(guān)閉）或轉(zhuǎn)矩傳遞 補償器（T補償關(guān)閉）不會嚴重降低質(zhì)量跟蹤。但是， 切換兩個補償關(guān)閉（T，hydr補償關(guān)閉）不會引入大的跟蹤誤差。這發(fā)生，因為這兩個約束的最大操作者取來計算補償 動作，并且如果一個約束補償器是零，最大運算器的輸出仍然
圖13幾個控制器的替代品沖和平均絕對誤差
會是非零的，由于第二個約束。兩個補償器關(guān)掉同時 有效地引入控制器輸出U一個“死區(qū)”，其結(jié)果是明顯的。與設定值前饋響應關(guān)閉（off'）中增加了錯誤的因產(chǎn)生反應的增加相位滯后?？偟牡玫降慕Y(jié)果開發(fā)的控制器顯示出更好的跟蹤行為（過沖和平均絕對誤差）和 較低的瞬時滑輪壓力（僅在比值的變化，如夾緊策略是相等的）與以前采用的控制器獲得，按文獻結(jié)果進行了比較。 這可能指示了可能改進新的CVT效率控制器如前所述。
圖14 在針尖的變化在測試賽道的實驗車輛 圖15CVT速比響應和跟蹤誤差，道路尖端移位
通過在主壓力控制器的局限性。這種現(xiàn)象降低了最大減檔的速度，并且是作為在t輕微凸點=6.2s和t=8.2有形之前。如所呈現(xiàn)的實驗的主要目標是展示一個新的比率控制器概念，實驗皮帶打滑時一直使用經(jīng)過驗證的鉗位裝置避免前面提到的。此外，網(wǎng)上基于模型的檢測算法被使用，驗證該。有兩種方法來檢測帶從測量數(shù)據(jù)滑落線（不直接在滑輪的皮帶的運行半徑來計算所謂的幾何測量比）的實驗后已被使用。第一，它已被證實，如果CVT的范圍幾何比例可能不超過（rlow≤rcvt≤rod）。其次，最大 轉(zhuǎn)移CVT的速度是有限的，由于有限的夾緊力和變速器的速度，看方程（10）。摩擦在推帶的過度（宏觀）滑移區(qū)域系數(shù)減小，滑差調(diào)速[8]。這將導致不穩(wěn)定的動態(tài)行為，因此速度滑將迅速增加，當AV-帶的扭矩容量是exceeded. As的比值從測量滑輪的速度，過快的比例變化（rcvt高值）可以指示皮帶打滑。每次測量的結(jié)果都經(jīng)過仔細審查，其結(jié)果并沒有顯示皮帶打滑影響的任何痕跡。
圖16 初級和次級帶輪的壓力，道路尖端移位
7.結(jié)論
一種新的比例控制器的金屬推帶式CVT用帶液壓夾緊系統(tǒng)被開發(fā)出來。在變速器和液壓系統(tǒng)動態(tài)模型補償?shù)幕A系統(tǒng)，設定值前饋和反饋線性化控制方面實施。反饋控制器是帶有條件抗飽和PID控制器保護?？偙戎悼刂破鞅ＷC，壓力設定點中的至少一個總是以最少的相對于他的約束而另一種是上述的最小凸電平，用于換檔。這種方法有可能用于提高CVT的效率。滾子工作臺和公路實驗具有內(nèi)置CVT表明足夠的跟蹤是一個車輛獲得的。從比例上設定的最大偏差由執(zhí)行機構(gòu)壓力飽和造成的。實驗與幾個控制器的變化具有速度向前的現(xiàn)象，所有已執(zhí)行的前饋和約束補償方面產(chǎn)生有益的最小化跟蹤誤差影響。提示轉(zhuǎn)移的實驗顯示對曲線性好執(zhí)行器飽和。
8
河北建筑工程學院
畢業(yè)設計（論文）外文資料翻譯
系別： 機械工程學院
專業(yè)： 機械設計制造及其自動化
班級： 機101
姓名： 侯躍龍
學號： 2010307107
外文出處： Vehicle System Dynamics
（用外文寫）
Vol.No.5，May 2006,387-406
附 件：1、外文原文；2、外文資料翻譯譯文。
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河 北 建 筑 工 程 學 院
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學 科 專 業(yè) 機械設計制造及其自動化
班 級 機101
姓 名 侯躍龍
指 導 教 師 王占英 任玉燦
輔 導 教 師
目錄
第1章 前言··········································································1
1.1 國內(nèi)外起重機發(fā)展情況·······················································1
1.2 橋式起重機定義及特點·······················································4
1.3 實習地點及實習內(nèi)容··························································4
第2章 總體設計···································································4
2.1 概述·············································································5
2.2 傳動方案的確定·······························································6
2.3 基本參數(shù)······································································10
第3章 起升機構(gòu)的設計計算···················································12
3.1 選擇鋼絲繩···································································12
3.2 滑輪和卷筒的計算···························································13
3.3 計算靜功率···································································15
3.4 選擇電動機···································································15
3.5 驗算電動機的發(fā)熱條件······················································15
3.6 減速機的初選································································16
3.7 校核減速機···································································16
3.8 制動器的選擇································································17
3.9 聯(lián)動器的選擇································································17
3.10 驗算起動時間·······························································18
3.11 浮動軸強度驗算····························································19
第4章 運行機構(gòu)的設計計算···················································21
4.1 確定機構(gòu)傳動方案···························································21
4.2 選擇車輪與軌道并驗算其強度··············································21
4.3 運行阻力計算································································23
4.4 選擇電動機···································································24
4.5 驗算電動機發(fā)熱條件························································25
4.6 選擇減速器···································································25
4.7 驗算運行機構(gòu)和實際所需功率··············································25
4.8 驗算起動時間································································26
4.9 驗算起動不打滑條件························································27
4.10 制動器的選擇·······························································27
4.11 選擇聯(lián)軸器··································································28
4.12 驗算低速浮動軸強度·······················································29
第5章 零部件的設計計算·····················································31
5.1 滑輪的尺寸計算與選擇······················································31
5.2吊鉤組的選擇·································································32
5.3 車輪軸的設計計算···························································35第6章 零部件的設計計算·····················································38
6.1 梁Ⅰ···········································································38
6.2 梁Ⅱ···········································································40
6.3 梁Ⅲ···········································································42
6.4 梁Ⅵ···········································································44
6.5 梁Ⅴ···········································································48
第7章 畢業(yè)設計小節(jié)····························································53
參考文獻············································································54
附：英文原文
英文譯文
畢業(yè)實習報告
Vehicle System Dynamics Vol. 44, No. 5, May 2006, 387406 Control of a hydraulically actuated continuously variable transmission MICHIEL PESGENS*, BAS VROEMEN, BART STOUTEN, FRANS VELDPAUS and MAARTEN STEINBUCH Drivetrain Innovations b.v., Horsten 1, 5612 AX, The Netherlands Technische Universiteit Eindhoven, PO Box 513, 5600 MB Eindhoven, The Netherlands Vehicular drivelines with hierarchical powertrain control require good component controller tracking, enabling the main controller to reach the desired goals. This paper focuses on the development of a transmission ratio controller for a hydraulically actuated metal push-belt continuously variable transmission (CVT), using models for the mechanical and the hydraulic part of the CVT. The controller consists of an anti-windup PID feedback part with linearizing weighting and a setpoint feedforward. Physical constraints on the system, especially with respect to the hydraulic pressures, are accounted for using a feedforward part to eliminate their undesired effects on the ratio. The total ratio controller guarantees that one clamping pressure setpoint is minimal, avoiding belt slip, while the other is raised above the minimum level to enable shifting. This approach has potential for improving the efficiency of the CVT, compared to non-model based ratio controllers. Vehicle experiments show that adequate tracking is obtained together with good robustness against actuator saturation. The largest deviations from the ratio setpoint are caused by actuator pressure saturation. It is further revealed that all feedforward and compensator terms in the controller have a beneficial effect on minimizing the tracking error. Keywords: Continuously variable transmission; Feedforward compensation; Feedback linearization; Hydraulic actuators; Constraints 1. Introduction The application of a continuously variable transmission (CVT) instead of a stepped transmis- sion is not new. Already in the 50s Van Doorne introduced a rubber V-belt CVT for vehicular drivelines. Modern, electronically controlled CVTs make it possible for any vehicle speed to operate the combustion engine in a wide range of operating points, for instance in the fuel optimal point. For this reason, CVTs get increasingly important in hybrid vehicles, see for example 13. Accurate control of the CVT transmission ratio is essential to achieve the intended fuel economy and, moreover, ensure good driveability. The ratio setpoint is generated by the hierarchical (coordinated) controller of figure 1. This controller uses the accelerator pedal position as the input and generates setpoints for the local controllers of the throttle and of the CVT. *Corresponding author. Email: pesgensdtinnovations.nl Michiel Pesgens was previously affiliated with Technische Universiteit Eindhoven. Vehicle System Dynamics ISSN 0042-3114 print/ISSN 1744-5159 online 2006 Taylor F shift = F p (r cvt , prime s ) F s (10) An axial force difference F shift , weighted by the thrust ratio results in a ratio change, and is therefore called the shift force. The occurrence of p in the model (10) is plausible because an increasing shift force is needed for decreasing pulley speeds to obtain the same rate of ratio change. The reason is that less V-shaped blocks enter the pulleys per second when the pulley speed decreases. As a result the radial belt travel per revolution of the pulleys must increase and this requires a higher shift force. However, it is far from obvious that the rate of ratio change is proportional to both the shift force and the primary pulley speed. k r is a non-linear function of the ratio r cvt and has been obtained experimentally. Experimental data has been used to obtain a piecewise linear fit, which are depicted in figure 6. The estimation of k r has 392 M. Pesgens et al. Figure 5. Contour plot of (r cvt , prime s ). Figure 6. Fit of k r (r cvt ); greyed-out dots correspond to data with reduced accuracy. Hydraulically actuated CVT 393 Figure 7. Comparison of shifting speed, Ides model vs. measurement. been obtained using the inverse Ide model: k r (r cvt ) = r cvt | p |F shift (11) In the denominator F shift is present, the value of which can become (close to) zero. Obviously, the estimate is very sensitive for errors in F shift when its value is small. The dominant dis- turbances in F shift are caused by high-frequency pump generated pressure oscillations, which do not affect the ratio (due to the low-pass frequency behavior of unmodeled variator pulley inertias). The standard deviation of the pressure oscillations and other high-frequency distur- bances has been determined applying a high-pass Butterworth filter to the data of F shift .To avoid high-frequency disturbances in F shift blurring the estimate of k r , estimates for values of F shift smaller than at least three times the disturbances standard deviation have been ignored (these have been plotted as grey dots in figure 6), whereas the other points have been plotted as black dots. The white line is the resulting fit of this data. The few points with negative value for k r have been identified as local errors in the map of . To validate the quality of Ides model, the shifting speed r cvt , recorded during a road exper- iment, is compared with the same signal predicted using the model. Model inputs are the hydraulic pulley pressures (p p , p s ) and pulley speeds ( p , s ) together with the estimated primary pulley torque ( T p ). The result is depicted in figure 7. The model describes the shifting speed well, but for some upshifts it predicts too large values. This happens only for high CVT ratios, i.e. r cvt 1.2, where the data of is unreliable due to extrapolation (see figure 5). 3. The hydraulic system The hydraulic part of the CVT (see figure 3) consists of a roller vane pump directly connected to the engine shaft, two solenoid valves and a pressure cylinder on each of the moving pulley 394 M. Pesgens et al. sheaves. The volume between the pump and the two valves including the secondary pulley cylinder is referred to as the secondary circuit, the volume directly connected to and including the primary pulley cylinder is the primary circuit. Excessive flow in the secondary circuit bleeds off toward the accessories, whereas the primary circuit can blow off toward the drain. All pressures are gage pressures, defined relative to the atmospheric pressure. The drain is at atmospheric pressure. The clamping forces F p and F s are realized mainly by the hydraulic cylinders on the move- able sheaves and depend on the pressures p p and p s . As the cylinders are an integral part of the pulleys, they rotate with an often very high speed, so centrifugal effects have to be taken into account and the pressure in the cylinders will not be homogeneous. Therefore, the clamping forces will also depend on the pulley speeds p and s . Furthermore, a preloaded linear elastic spring with stiffness k spr is attached to the moveable secondary sheave. This spring has to guarantee a minimal clamping force when the hydraulic system fails. Together this results in the following relations for the clamping forces: F p = A p p p + c p 2 p (12) F s = A s p s + c s 2 s k spr s s + F 0 (13) where c p and c s are constants, whereas F 0 is the spring force when the secondary moveable sheave is at position s s = 0. Furthermore, A p and A s are the pressurized piston surfaces. In the hydraulic system of figure 3, the primary pressure is smaller than the secondary pressure if there is an oil flow from the secondary to the primary circuit. Therefore, to guarantee that in any case the primary clamping force can be up to twice as large as the secondary clamping force, the primary piston surface A p is approximately twice as large as the secondary surface A s . It is assumed that the primary and the secondary circuit are always filled with oil of constant temperature and a constant air fraction of 1%. The volume of circuit ( = p, s) is given by: V = V ,min + A s (14) V ,min is the volume if s = 0 and A is the pressurized piston surface. The law of mass conservation, applied to the primary circuit, combined with equation (14), results in: oil V p p p = Q sp Q pd Q p,leak Q p,V (15) Q sp is the oil flow from the secondary to the primary circuit, Q pd is the oil flow from the primary circuit to the drain, Q p,leak is the (relatively small) oil flow leaking through narrow gaps from the primary circuit and Q p,V is the oil flow due to a change in the primary pulley cylinder volume. Furthermore, oil is the compressibility of oil. The oil flow Q sp is given by: Q sp = c f A sp (x p ) radicalBigg 2 |p s p p |sign(p s p p ) (16) where c f is a constant flow coefficient and is the oil density. A sp , the equivalent valve opening area for this flow path, depends on the primary valve stem position x p . Flow Q pd follows from: Q pd = c f A pd (x p ) radicalBigg 2 p p (17) Here, A pd is the equivalent opening area of the primary valve for the flow from primary circuit to the drain. The construction of the valve implies that A sp (x p ) A pd (x p ) = 0 for all possible x p . Hydraulically actuated CVT 395 Flow Q p,leak is assumed to be laminar with leak flow coefficient c pl , so: Q p,leak = c pl p p (18) The flow due to a change of the primary pulley cylinder volume is described by: Q p,V = A p s p (19) with s p given by equation (4). Application of the law of mass conservation to the secondary circuit yields oil V s p s = Q pump Q sp Q sa Q s,leak Q s,V (20) The flow Q pump , generated by the roller vane pump, depends on the angular speed e of the engine shaft, on the pump mode m (m = SS for single sided and m = DS for double sided mode), and the pressure p s at the pump outlet, so Q pump = Q pump ( e ,p s ,m). Q sa is the flow from the secondary circuit to the accessories and Q s,leak is the leakage from the secondary circuit. Flow Q sa is modeled as: Q sa = c f A sa (x s ) radicalBigg 2 |p s p a |sign(p s p a ) (21) where A sa , the equivalent valve opening of the secondary valve, depends on the valve stem position x s . The laminar leakage flow Q s,leak is given by (with flow coefficient c sl ): Q s,leak = c sl p s (22) The flow due to a change of the secondary pulley cylinder volume is: Q s,V = A s s s (23) with s s according to equation (3). The accessory circuit contains several passive valves. In practice, the secondary pressure p s will always be larger than the accessory pressure p a , i.e. no backflow occurs. The relation between p a and p s is approximately linear, so p a = c a0 + c a1 p s (24) with constants c a0 0 and c a1 (0, 1). Now that a complete model of the pushbelt CVT and its hydraulics is available, the controller and its operational constraints can be derived. 4. The constraints The CVT ratio controller (in fact) controls the primary and secondary pressures. Several pressure constraints have to be taken into account by this controller: 1. the torque constraints p p ,torque to prevent slip on the pulleys; 2. the lower pressure constraints p p ,low to keep both circuits filled with oil. Here, fairly arbitrary, p p,low = 3 bar is chosen. To enable a sufficient oil flow Q sa to the accessory circuit, and for a proper operation of the passive valves in this circuit it is necessary that 396 M. Pesgens et al. Q sa is greater than a minimum flow Q sa,min . A minimum pressure p s,low of 4 bar turns out to be sufficient; 3. the upper pressure constraints p p ,max , to prevent damage to the hydraulic lines, cylinders and pistons. Hence, p p,max = 25 bar, p s,max = 50 bar; 4. the hydraulic constraints p p ,hyd to guarantee that the primary circuit can bleed off fast enough toward the drain and that the secondary circuit can supply sufficient flow toward the primary circuit. The pressures p p,torque and p s,torque in constraint 1 depend on the critical clamping force F crit , equation (5). The estimated torque T p is calculated using the stationary engine torque map, torque converter characteristics and lock-up clutch mode, together with inertia effects of the engine, flywheel and primary gearbox shaft. A safety factor k s = 0.3 with respect to the estimated maximal primary torque T p,max has been introduced to account for disturbances on the estimated torque T p , such as shock loads at the wheels. Then the pulley clamping force (equal for both pulleys, neglecting the variator efficiency) needed for torque transmission becomes: F torque = cos() (| T p |+k s T p,max ) 2 R p (25) Consequently, the resulting pressures can be easily derived using equations (12) and (13): p p,torque = 1 A p parenleftBig F torque c p 2 p parenrightBig (26) p s,torque = 1 A s parenleftbig F torque c s 2 s k spr s s F 0 parenrightbig (27) Exactly the same clamping strategy has been previously used by ref. 3 during test stand efficiency measurements of this gearbox and test vehicle road tests. No slip has been reported during any of those experiments. As the main goal of this work is to an improved ratio tracking behavior, the clamping strategy has remained unchanged. A further elaboration of constraints 4 is based on the law of mass conservation for the primary circuit. First of all, it is noted that for this elaboration the leakage flow Q p,leak and the compressibility term oil V p p p may be neglected because they are small compared to the other terms. Furthermore, it is mentioned again that the flows Q sp and Q pd can never be unequal to zero at the same time. Finally, it is chosen to replace the rate of ratio change r cvt by the desired rate of ratio shift r cvt,d , that is specified by the hierarchical driveline controller. If r cvt,d 0 and Q sp = 0. Constraint 4 with respect to the primary pulley circuit then results in the following relation for the pressure p p,hyd : p p,hyd = oil 2 parenleftbigg A p p max(0, r cvt,d ) c f A pd,max parenrightbigg 2 (28) where A pd,max is the maximum opening of the primary valve in the flow path from the primary cylinder to the drain. In a similar way, a relation for the secondary pulley circuit pressure p s,hyd in constraint 4 can be derived. This constraint is especially relevant if r cvt 0, i.e. if the flow Q sp from the secondary to the primary circuit has to be positive and, as a consequence, Q pd = 0. This then Hydraulically actuated CVT 397 results in: p s,hyd = p p,d + oil 2 parenleftbigg A p p max(0, r cvt,d ) c f A sp,max parenrightbigg 2 (29) where A sp,max is the maximum opening of the primary valve in the flow path from the secondary to the primary circuit. For the design of the CVT ratio controller it is advantageous to reformulate to constraints in terms of clamping forces instead of pressures. Associating a clamping force F , with the pressure p , and using equations (12) and (13) this results in the requirement: F ,min F F ,max (30) with minimum pulley clamping forces: F ,min = max(F ,low ,F ,torque ,F ,hyd ) (31) 5. Control design It is assumed in this section that at each point of time t, the primary speed p (t), the ratio r cvt (t), the primary pressure p p (t) and the secondary pressure p s (t) are known from measurements, filtering and/or reconstruction. Furthermore, it is assumed that the CVT is mounted in a vehicular driveline and that the desired CVT ratio r cvt,d (t) and the desired rate of ratio change r cvt,d (t) are specified by the overall hierarchical driveline controller. This implies, for instance, that at each point of time the constraint forces can be determined. The main goal of the local CVT controller is to achieve fast and accurate tracking of the desired ratio trajectory. Furthermore, the controller should also be robust for disturbances. An important subgoal is to maximize the efficiency. It is quite plausible (and otherwise supported by experiments, 3) that to realize this sub-goal the clamping forces F p and F s have to be as small as possible, taking the requirements in equation (30) into account. The output of the ratio controller is subject to the constraints of equation (31). The constraints F F ,min can effectively raise the clamping force setpoint of one pulley, resulting in an undesirable ratio change. This can be counteracted by raising the opposite pulleys clamping force as well, using model-based compensator terms in the ratio controller. Using Ides model, i.e. using equation (10), expressions for the ratio change forces F p,ratio and F s,ratio (figure 8) can be easily derived: F p,ratio = F shift,d + F s,min (32) F s,ratio = F shift,d + F p,min (33) where F shift,d is the desired shifting force, basically a weighted force difference between both pulleys. As explained earlier, depends on prime s , which in turn depends on F s . This is an implicit relation (F s,ratio depends on F s ), which has been tackled by calculating from pressure measurements. It will now be shown that at each time, one of the two clamping forces is equal to F ,min , whereas the other determines the ratio. Using equations (30), (32) and (33) the desired primary 398 M. Pesgens et al. Figure 8. Ratio controller with constraints compensation and secondary clamping forces F p,d and F s,d are given by: F p,d = F p,ratio F s,d = F s,min bracerightBigg if F shift,d + F s,min F p,min (34) F p,d = F p,min F s,d = F s,ratio bracerightBigg if F shift,d + F s,min F p,min F p,min F s,ratio = F shift,d if F shift,d + F s,min F p,max F s,ratio F s,max (40) If either pressure saturates (p p = p p,max or p s = p s,max ), the shifting speed error inevitably becomes large. The anti-windup algorithm ensures stability, but the tracking behavior will deteriorate. This is a hardware limitation which can only be tackled by enhancing the variator and hydraulics hardware. The advantage of a conditional anti-windup vs. a standard (linear) algorithm is that the linear approach requires tuning for good performance, whereas the con- ditional approach does not. Furthermore, the performance of the conditional algorithm closely resembles that of a well-tuned linear mechanism. 6. Experimental results As the CVT is already implemented in a test vehicle, in-vehicle experiments on a roller bench have been performed to tune and validate the new ratio controller. To prevent a non- synchronized operation of throttle and CVT ratio, the accelerator pedal signal (see figure 1) has been used as the input for the validation experiments. The coordinated controller will track the maximum engine efficiency operating points. A semi kick-down action at a cruise-controlled speed of 50 km/h followed by a pedal back out has been performed in a single reference exper- iment. The recorded pedal angle (see figure 9) has been applied to the coordinated controller. This approach cancels the limited human drivers repeatability. The upper plot of figure 10 shows the CVT ratio response calculated from speed measure- ments using equation (1), the plot depicts the tracking error. As this is a quite demanding experiment, the tracking is still adequate. Much better tracking performance can be obtained with more smooth setpoints, but the characteristics of the responses will become less distinct as well. Figure 11 shows the primary and secondary pulley pressures. The initial main peak in the error signal (around t = 1.5 s) is caused by saturation of the secondary pressure (lower plot of figure 11), due to a pump flow limitation. If a faster initial response were required, adaptation of the hydraulics hardware would be necessary. After the initial fast downshift, the ratio reaches its setpoint (around t = 7 s) before downshifting again. All changes in shifting 400 M. Pesgens et al. Figure 9. Pedal input for the CVT powertrain. direction (t = 1.3, t = 1.6 and t = 7.5 s) occur with a relatively small amount of overshoot, which shows that the integrator anti-windup algorithm performs well. Looking at the primary pressure in the vicinity of t = 1.5 s, it can be observed that this pressure peaks repeatedly above its setpoint. This behavior is caused