JP2007334843A - Optimum control method for system - Google Patents

Abstract

<P>PROBLEM TO BE SOLVED: To derive a control rule for a nonlinear optimum control problem in a finite evaluation section following an unpredictable time-varying target value. <P>SOLUTION: An approximate solution method for the nonlinear optimum control problem using Receding Horizon control is developed to derive a general technique applicable for real time control for a nonlinear system. Followability (in both responsiveness and convergency) is confirmed to be remarkably enhanced compared with general proportional control, as a result of application to a target side slip angle following control of a DYC in a turning limit area of an RR vehicle as a concrete example. A neutral steering characteristic can be attained even in the turning limit area. <P>COPYRIGHT: (C)2008,JPO&INPIT

Description

  The present invention mainly relates to a real-time optimal control method for a mechanical system. It can be applied to a wide range of systems, whether linear or non-linear.

Optimal control for tracking the state quantity of the system to the target value follows the time-varying target value in the finite evaluation interval in a nonlinear system, although there is a method proposed by Tsuchiya et al. When the system is linear (Non-Patent Document 1). It is difficult to analytically find the optimal control law.
For example, in recent years, DYC (Direct Yaw Moment Control) that actively controls the movement of a vehicle with a braking / driving force has become widespread. However, in conventional DYC research and practical systems, control for detecting the tire limit and inserting the tire operating state inside the limit range is widely performed (for example, Non-Patent Documents 2 and 3). reference). For the purpose of improving vehicle motion performance, research on control methods that maintain stability and controllability in the entire operating range of the tire in Fig. 2 is considered meaningful, but it is difficult to derive control laws. Therefore, except for Non-Patent Document 4, no research examples have been found so far.

Non-Patent Document 4 presents a general theory of a solution to this problem, followed by a rear-engine rear-wheel drive vehicle (RR vehicle), which incorporates DYC and devise controls to consider dynamics. It is stated that neutral steer in the entire drive range can be realized even in this state. The target value in this case is time-varying and the future value is unpredictable, and the tire slip becomes extremely strong when the side slip angle increases, and the evaluation interval is finite in general driving. It describes that the target value tracking is optimally controlled, and that the computation load of the control law is small and real-time control is possible. The present invention applies an exception of Article 30 of the Patent Law and applies for a part of the contents of Non-Patent Document 4.
Takeshi Tsuchiya, Tadashi Egami, Digital Predictive Control, (1992), 26-27, Industrial Books. Anton T. van Zanten, Rainer Erhardt, Georg Pfaff: VDC, The Vehicle Dynamics Control System of Bosch, SAE paper 950759 (1995) Masanori Yamamoto, Ken Takeshi, Yoshiki Fukada, Takuji Inagaki: Active Braking Force Control for Improving Vehicle Stability near the Limit, Automotive Technology Society Preliminary Proc. 953 (1995-5) Naoto Fukushima: Approximate Solution of Nonlinear Optimal Control Problem and Application to Large Side Slip Angle Tracking Control, Transactions of the Japan Society of Mechanical Engineers (C) Vol. 72, no. 713 (2006), 84-91.

  The challenge is to derive a control law for a nonlinear optimal control problem in a finite evaluation interval that follows an unpredictable time-varying target value.

  We propose a method for deriving a suboptimal control law by an approximate solution based on Receding Horizon control (for example, see Non-Patent Document 5). First, a general theory will be described, and then applied to the control that maintains the neutral steering characteristics during turning drive, assuming that the DYC is mounted on an RR vehicle as a vehicle motion control device. As a result, it is confirmed by simulation that it is possible to control the side slip angle of the vehicle in the turning limit region, which is extremely difficult to control, and to achieve almost the intended performance.

ここでｘ（τ，ｔ）∈Ｒ^ｎ，ｕ（τ，ｔ）∈Ｒ^ｒはτ軸上で定義された状態変数と制御入力である．ｆ

値で予見はできないものとする．式（２）を最小化する制御則を求めることが制御問題となる．The optimal control law by Receding Horizon control, in which the evaluation function to be minimized is defined by the finite evaluation interval T that moves along the time axis, is derived. Introduce virtual time τ (0 ≦ τ ≦ T) and describe nonlinear optimal control problem of target value tracking method. Thus, the initial values of the state variable and the control input at τ = 0 become the state variable x (t) and the control input u (t) on the real time axis.
Let Eq. (1) be the state equation for τ of the nonlinear system, and Eq. (2) be the evaluation function to be minimized.

Here, x (τ, t) εR ⁿ and u (τ, t) εR ^r are state variables and control inputs defined on the τ axis. f

The value cannot be predicted. Finding a control law that minimizes Equation (2) is a control problem.

ここで，ヤコビアンｆ_ｘ（ｔ），ｆ_ｕ（ｔ）とｆ_０（ｔ）は次式で定義される．

Next, the derivation method of the optimal control law is described. If the evaluation interval T is sufficiently small, Equation (1) is linearized as follows.

Here, Jacobian _{f x (t), f u} (t) and _f 0 (t) is defined by the following equation.

Formula 6

ここで．区間０≦τ≦Ｔにおけるｘ（ｔ），ｘ（Ｔ，ｔ），ｕ（ｔ），ｕ（Ｔ，ｔ）について考える．
まず，Ｒｅｃｅｄｉｎｇ Ｈｏｒｉｚｏｎ制御であるからｘ（ｔ）はτ＝０における状態量の初期値として与えられる．一般にｇが２次形式で与えられれば，最適性の原理（例えば、非特許文献６参照）より明らかに，ｕ（Ｔ，ｔ）＝０である．
従って，図１のイメージ図に示すように，白丸が未知，黒丸が既知という関係になり，Ｊ^＊を最小化する問題は白丸で示したｘ（Ｔ，ｔ）とｕ（ｔ）を求める２点境界値問題になる．ここで図１においてｕ（τ，ｔ）の変化を直線で近似すると，

と記述できる．この近似もＴが十分小さければ許容される．このような近似により２点境界値問題を初期値問題に転換できる．式（８）を式（３）に代入してこれをτについて解いてｘ^＊（τ，ｔ）を求め，これよりｘ^＊（Ｔ，ｔ）をｕ^＊（ｔ）の関数として求めることができる．従って式（７）のＪ^＊（ｔ）がｕ^＊（ｔ）の関数として定まる．Ｊ^＊（ｔ）を最小化する条件は次式から求まる．

ここで，式（７）のｘ（Ｔ，ｔ），ｕ（ｔ）をｘ^＊（Ｔ，ｔ），ｕ^＊（ｔ）でおきかえたものをあらためてＪ^＊（ｔ）としている．これより，準最適制御則ｕ^＊ _ｏｐｔ（ｔ）が得られる．ただし，制御装置は計測された状態量からヤコビアンｆ_ｘ（ｔ），ｆ_ｕ（ｔ）とｆ_０（ｔ）を算出する手段を有するものとする．本手法のような有限時間評価関数の最適化では，閉ループ系が有限時間発散系でないかぎり制御則は意味を持つ．従って得られる制御則は漸近安定の保証はないが，制御目的に合致した適切な評価関数を設定すれば，考えられる可能性の中のベストな制御則になっているということは保証される．
以上が一般論であり，次章への繋がりを考慮してｇの変数に目標値が含まれることを前提に述べてきたが，目標値を含まない形であってもここまでの議論は適用できる．f ₀ (t) = f (x (t), u (t)) − f _x (t) x (t) −f _u (t) u (t)
Similarly, if the evaluation interval T is sufficiently small, Equation (2) can be approximated as follows within this interval. In the following, approximates are identified with a * on the right shoulder.

here. Consider x (t), x (T, t), u (t), u (T, t) in the interval 0 ≦ τ ≦ T.
First, since it is Receding Horizon control, x (t) is given as an initial value of the state quantity at τ = 0. In general, if g is given in a quadratic form, u (T, t) = 0 clearly from the principle of optimality (for example, see Non-Patent Document 6).
Therefore, as shown in the image diagram of FIG. 1, there is a relationship that the white circle is unknown and the black circle is known, and the problem of minimizing J ^* is to obtain x (T, t) and u (t) indicated by white circles. It becomes a boundary value problem. Here, when the change of u (τ, t) in FIG. 1 is approximated by a straight line,

Can be described. This approximation is acceptable if T is small enough. This approximation can convert a two-point boundary value problem into an initial value problem. Substituting equation (8) into equation (3) and solving for τ finds x ^* (τ, t), from which x ^* (T, t) can be found as a function of u ^* (t). it can. Therefore, J ^* (t) in equation (7) is determined as a function of u ^* (t). The condition for minimizing J ^* (t) is obtained from the following equation.

Here, x ^* (T, t) and u (t) in the equation (7) are replaced with x ^* (T, t) and u ^* (t), and are rewritten as J ^* (t). From this, the suboptimal control law u ^* _opt (t) is obtained. However, the control device shall have a means for calculating the Jacobian _f x from the state amount measured _{(t), f u (t} ) and _f 0 (t). In the optimization of the finite-time evaluation function like this method, the control law is meaningful unless the closed-loop system is a finite-time divergent system. Therefore, the obtained control law is not guaranteed asymptotically stable, but if an appropriate evaluation function that matches the control objective is set, it is guaranteed that the control law is the best possible control law.
The above is general theory, and it has been described on the premise that the target value is included in the variable of g in consideration of the connection to the next chapter, but the discussion up to this point applies even if the target value is not included. it can.

各輪の回転１自由度モデルは線形で特に新しさはないため，ここでは記述を省略する．各輪のタイヤ発生力Ｆ_ｘｉ，Ｆ_ｙｉは各輪のβ_ｉ，γ_ｉと輪荷重からブラシモデルを用いて計算される．各輪荷重は車両の前後，横加速度から計算される．
ＤＹＣによる性能向上ポテンシャルが高いＲＲ車の諸元を用いて具体的なＤＹＣ制御法を検討する．表１に使用したＲＲ（後置きエンジン後輪駆動）スポーツカーの諸元を示す．
ＤＹＣモデルは，図４のようなモータで直接左右輪に互いに逆相のトルクを発生させる左右駆動力配分方式のＤＹＣを想定する．各センサからの信号がコントローラに送られ，この出力信号でデフに組み込まれた油圧モータが駆動される．
モータトルクは左右駆動軸に互いに逆相で伝達され，エンジンによる駆動トルクに加算される．このモータトルクが，図４（ｂ）に示すように左右タイヤに互いに逆方向の制駆動力を発生させヨーモーメントを生じさせている．モータへのトルク指令値および出カトルクをＴ_ｍとすると，これによる左右輪の駆動力はそれぞれ−Ｔ_ｍ／ｒ，Ｔ_ｍ／ｒになる．これによる車両に加わる制御ヨーモーメントは，Ｍ_ｃ＝Ｔ_ｍＩ_ｔｒ／ｒとなる．ただしｒはタイヤ動半径である．Next, a method of applying the above general theory to DYC control and confirming it by simulation will be described. First, the simulation model for this purpose is explained first.
In order to evaluate the performance of DYC, it is necessary to express the longitudinal force in consideration of the wheel motion. For this purpose, a brush model (for example, see Non-Patent Document 7) is suitable. Fig. 2 shows an example of the calculation of the rear tire of this vehicle model.
The characteristics shown in Fig. 2 have a lateral slip angle β as a parameter and the lateral force is in the form of a bivalent function, but the lateral force and longitudinal force have β, slip ratio γ, and load as parameters. When is determined, it is uniquely determined. In any case, the condition of the tire in the turning limit area is near the circumference of Fig. 2, and it can be seen that it is extremely difficult to control from the viewpoint of vehicle motion control. For this reason, in DYC research aimed at driving safety and systems put into practical use, control to insert β and γ inside the limit range is widely performed (see Non-Patent Documents 2 and 3 above). For the purpose of improving vehicle motion performance, research on control methods that maintain stability and control in the entire operating range of the tire in Fig. 2 is considered meaningful, but there have been no examples of research so far. Absent.
The vehicle model is a model with three degrees of freedom as shown in Fig. 3, one degree of freedom for each wheel, and a total of seven degrees of freedom.

Since the one-degree-of-freedom model for each wheel is linear and not particularly new, its description is omitted here. The tire generation forces F _xi and F _yi of each wheel are calculated from the β _i and γ _i of each wheel and the wheel load using a brush model. Each wheel load is calculated from the longitudinal acceleration and lateral acceleration of the vehicle.
A specific DYC control method will be examined using the specifications of an RR vehicle with high potential for performance improvement by DYC. Table 1 shows the specifications of the RR (rear engine rear wheel drive) sports car used.
The DYC model is assumed to be a left / right driving force distribution type DYC in which the motor shown in FIG. The signal from each sensor is sent to the controller, and the hydraulic motor built in the differential is driven by this output signal.
The motor torque is transmitted to the left and right drive shafts in opposite phases and added to the drive torque from the engine. As shown in FIG. 4 (b), this motor torque generates braking / driving forces in opposite directions to the left and right tires to generate a yaw moment. When the torque command value to the motor and exits the force torque and T _m, respectively driving forces of the left and right wheels by which -T m _{/ r,} becomes T m / _r. The control yaw moment applied to the vehicle is M _c = T _m I _tr / r. Where r is the tire radius.

Table 1

ここで，Ｆ_ｆ＝Ｆ_ｙ１＋Ｆ_ｙ２，Ｆ_ｒ＝Ｆ_ｙ３＋Ｆ_ｙ４，Ｍ_ｃは各輪の制駆動力によるヨーモーメントである．
これを図示すると図５のように簡略化されたモデルになる．
さらに評価区間Ｔ内では．スリップ比γと各輪荷重もほぼ一定，横すべり角βは変数であるが変動幅は小さいとみなすことができるため，タイヤ横力特性を図６のように線形化することができる．スリップ比γと輪荷重が定まると一つの特性曲線が定まる．横すべり角βの変動幅は小さいと作動点は図中のＡ点近傍で変動するため，タイヤの横力特性は次のようにβに関して線形化表示できる．

化するが評価区間Ｔ内では定数とみなすことができる．
続いて、ヤコビアンｆ_ｘ（ｔ），ｆ_ｕ（ｔ）およびｆ_０の算出法を述べる。線形近似したモデルのパラメータは実時間では刻々変化するが，評価区間０≦τ≦Ｔの間では，定数とみなしてｆ_ｘ（ｔ），ｆ_ｕ（ｔ）およびｆ_０を求めることができる．
計測される車両状態量と前後・横加速度から，まずＦ_ｆ ^０，Ｆ_ｒ ^０，Ｃ_ｆ，Ｃ_ｒを推定する．ヨ

速度から計算される．横速度Ｖ_ｙは計測もしくは他の計測データから推定されるものとする．以上のデータが制御演算装置に入力される．制御演算装置は前後輪ともブラシモデルを２ヶ有し，その１つには各輪荷重，各輪横すべり角，各輪スリップ比のデータを

Ｃ_ｒ＝Ｃ_３＋Ｃ_４として求める．Ｆ_ｆ＝Ｆ_ｙ１＋Ｆ_ｙ２，Ｆ_ｒ＝Ｆ_ｙ３＋Ｆ_ｙ４として，Ｆ_ｆ ^０，Ｆ_ｒ ^０は式（２２），（２３）から逆算する．
式（２０）のＭ_ｃは各輪の制駆動力によるヨーモーメントでありこの場合は制御入力になる．よって，Next, a vehicle model simplification method for deriving a control law will be described. While tire vehicle model shown in FIG. 2 and 3 is a complex non-linear, the smaller the evaluation interval T of Receding Horizon Control, vehicle speed _{V x} before and after in this interval variation than the other two state variables Since the rate is small, it can be regarded as a constant. For this reason, the model can be linearized and a suboptimal control law can be obtained by an approximate solution.

_{Here, F f = F y1 + F} y2, F r = F y3 + F y4, M c is the yaw moment by the braking driving force of each wheel.
This is a simplified model as shown in FIG.
Furthermore, in the evaluation section T. Since the slip ratio γ and the load on each wheel are also almost constant and the side slip angle β is a variable but the fluctuation range is small, the tire lateral force characteristics can be linearized as shown in FIG. When the slip ratio γ and wheel load are determined, one characteristic curve is determined. If the fluctuation range of the side slip angle β is small, the operating point fluctuates in the vicinity of point A in the figure. Therefore, the lateral force characteristics of the tire can be displayed linearly with respect to β as follows.

However, it can be regarded as a constant within the evaluation interval T.
Subsequently, Jacobian _f x _(t), describing the calculation method of the _f u (t) and _{f 0.} Although the parameters of the linearly approximated model change every moment in real time, f _x (t), f _u (t), and f ₀ can be obtained by considering them as constants during the evaluation interval 0 ≦ τ ≦ T.
First, F _f ⁰ , F _r ⁰ , C _f , and C _r are estimated from the measured vehicle state quantity and longitudinal / lateral acceleration. Yo

Calculated from speed. The lateral velocity V _y is estimated from measurement or other measurement data. The above data is input to the control arithmetic unit. The control arithmetic unit has two brush models for both the front and rear wheels, one of which is data on each wheel load, each wheel side slip angle, and each wheel slip ratio.

_{Calculate as} C _r = C ₃ + C ₄ . As F _f = F _y1 + F _y2 and F _r = F _y3 + F _y4 , F _f ⁰ and F _r ⁰ are calculated backward from equations (22) and (23).
M _c in this case a yaw moment by the braking driving force of each wheel is a control input of the formula (20). Therefore,

Formula 24

ここで，

非特許文献（８）では，エンジンからの駆動力を２Ｆ_ｄとすると，図７のような制御により，図８のように前後の横すべり角がほぼ同じ値になってニュートラルステアを維持した状態で静的にバランスすることを示した．比較参考のためＦＲ車の結果も表示している．
しかしながら，これを動的に実現するための制御法の検討は未着手であったため，本論文にて提案する手法を適用して評価を行う．M _c = u (t)

here,

In Non-Patent Document (8), assuming that the driving force from the engine is 2F _d , the front and rear side slip angles are almost the same as shown in FIG. It was shown to balance statically. The result of the FR vehicle is also displayed for comparison.
However, since the investigation of the control method to realize this dynamically has not been started, the method proposed in this paper is applied and evaluated.

［ｒａｄ／Ｎ］とした．Ｆ_ｄはドライバが与える駆動力であるから，この問題の目標値はドライバから刻々与えられることになり予見はできない．
また，車両がニュートラルを維持し，外側に膨れたり（アンダステア）内側に回り込んだり（オーバステア）しないようにする目的から，初期のヨーレートを維持するように目標ヨーレートを設定した．

ここで，車速８０Ｋｍ／ｈ，旋回半径６０ｍの初期条件より，Ｋ_ａ＝０．３７ ｒａｄ／ｓｅｃとした．
式（２）の被積分関数ｇを次式で与える．今回のような制御対象の自由度に対して独立した制御の数が少ない場合は，次式のようにｇにｕを入れなくても評価関数は凸性を持つ．

式（７）のＲｅｃｅｄｉｎｇ Ｈｏｒｉｚｏｎ制御形式の評価関数は次式になる．

制御問題は，時刻ｔにおける状態ｘ（ｔ）を初期値として，式（２９）の評価関数を最小にする準最適制御則ｕ^＊ _ｏｐｔ（ｔ）を導けということになる．
ｕ^＊ _ｏｐｔ（ｔ）を求めるため，２．２節の手順に従いまずｘ^＊（Ｔ，ｔ）を求める．式（３）の一般解は次式で与えられる．

制御では計測されたｘ（ｔ）がこの値になる．
式（８）を式（３０）のｕ（τ′，ｔ）のに代入しｘ^＊（Ｔ，ｔ）を求める．

式（３１）を式（２９）に代入し，最適条件式（９）を適用しその解をｕ^＊ _ｏｐｔ（ｔ）とすれば，

ここで，

Ｔ＝０．４ｓｅｃとした．
今回のように評価関数がｕを陽に含まないケースでも，最適性の原理からｕ（Ｔ，ｔ）＝０が成立するのかという疑問が残るため，この点を明らかにする．仮に本論文において，式（２９）の評価関数にＴＲｕ^２／２を加えて２次形式とした場合は，式（３２）の分母はＳ_１ ^２＋Ｓ_２ ^２＋ＲとなるだけでありＲ→０の極限値を考えることにより，Ｒ＝０の場合においても最適性の原理からｕ（Ｔ，ｔ）＝０の成立が確認できる．
以上で準最適制御則が定まった．次章に述べるシミュレーションにより，制御効果を評価する．比較のため，次式で表される比例制御則ｕ_０（ｔ）についてもシミュレーションを行った．

ここで，Ｋ_１，Ｋ_２は式（３２）における対応するゲインを考慮し，かつ制御入力の大きさが同程度になるように，Ｋ_１＝２９．０，Ｋ_２＝−１５４．０とした．
今回は，目標値を式（２６），（２７）のように旋回状態に限定して定めたが，本手法の制御則導出のプロセスから明らかなように制御則は式（３２）のままで目標値を任意設定できる．従ってより広い走行条件における目標性能に合わせて目標値を設定すれば適用域を拡大することができる．
大塚敏之，非線形最適フィードバック制御のための実時間最適化手法，計測と制御，３６−１１（１９９７），７７６−７８１． 高橋安人，システムと制御（下），（１９７８），４５６，岩波書店． 安部正人，自動車の運動と制御，３０−３６，山海堂． 福島直人，車両の超旋回限界域におけるＤＹＣ制御法に関する研究，（第２報，後置エンジン後輪駆動車への適用），日本機械学会論文集（Ｃ偏），Ｖｏｌ．７１，Ｎｏ．７０５（２００５），１７１−１７８． Next, the formulation of the DYC control problem and the method for deriving the suboptimal control law are described. Driving force equivalent to 4-speed full throttle (propulsion shaft torque 400 Nm, driving force 2) with a rise time of 1.0 sec during steady turning (steering angle fixed, vehicle speed 80 Km / h, turning radius 60 m) near the limit lateral force of the tire. (3KN / 1 wheel) is added to accelerate, and after a period of 1.0 sec, the throttle is returned to the original position, and then a constant driving force is maintained.
A target value is given for each state quantity. The target value of the side slip angle of the vehicle is set so as to realize the relationship between the side slip angle and the driving force shown in FIG. It is quite difficult to control following such a large target slip angle, and it is expected to be difficult to achieve with simple proportional control.
the target value of x ₂ was given by the following equation.

[Rad / N]. Since F _d is the driving force given by the driver, the target value of this problem is given from the driver every moment and cannot be predicted.
In addition, the target yaw rate was set to maintain the initial yaw rate in order to keep the vehicle neutral and not to swell outward (understeer) or inward (oversteer).

Here, K _a = 0.37 rad / sec based on the initial conditions of a vehicle speed of 80 Km / h and a turning radius of 60 m.
The integrand g of equation (2) is given by If the number of independent controls for the degree of freedom of the control target is small, the evaluation function has convexity even if u is not included in g as in the following equation.

The evaluation function of the Receding Horizon control form of Equation (7) is as follows.

The control problem is to derive a sub-optimal control law u ^* _opt (t) that minimizes the evaluation function of Equation (29) with the state x (t) at time t as an initial value.
To find u ^* _opt (t), first find x ^* (T, t) according to the procedure in section 2.2. The general solution of equation (3) is given by

In control, the measured x (t) is this value.
Substituting Equation (8) into u (τ ′, t) in Equation (30), finds x ^* (T, t).

Substituting equation (31) into equation (29), applying optimal condition equation (9), and taking the solution as u ^* _opt (t),

here,

T = 0.4 sec.
Even in the case where the evaluation function does not explicitly include u as in this case, the question remains whether u (T, t) = 0 holds from the principle of optimality. In If this paper, the case of a quadratic evaluation function by adding ^TRu 2/2 of the formula (29), the denominator of formula (32) is only the _{^{S 1 2 + S 2 2 +}} R R → 0 From the principle of optimality, it can be confirmed that u (T, t) = 0 holds even when R = 0.
The sub-optimal control law has now been determined. The control effect is evaluated by the simulation described in the next section. For comparison, a simulation was also performed for the proportional control law u ₀ (t) expressed by the following equation.

Here, K ₁ and K ₂ are K ₁ = 29.0 and K ₂ = −154.0 so that the corresponding gains in the equation (32) are considered and the magnitudes of the control inputs are approximately the same. did.
This time, the target value is limited to the turning state as shown in equations (26) and (27), but the control law remains as in equation (32) as is clear from the control law derivation process of this method. Target value can be set arbitrarily. Therefore, if the target value is set according to the target performance under wider driving conditions, the applicable range can be expanded.
Toshiyuki Otsuka, Real-time Optimization Method for Nonlinear Optimal Feedback Control, Measurement and Control, 36-11 (1997), 776-781. Yasuhito Takahashi, System and Control (below), (1978), 456, Iwanami Shoten. Masato Abe, Movement and control of automobiles, 30-36, Sankaido. Naoto Fukushima, research on DYC control method in the super turning limit region of vehicles, (2nd report, application to rear engine rear wheel drive vehicle), Japan Society of Mechanical Engineers papers (C bias), Vol. 71, no. 705 (2005), 171-178.

The invention's effect

ここでβ_ｉは各輪の横すべり角である．αが正の値なら前輪の横すべり角が大きくアンダステア，αが負なら後輪の横すべり角が大きくオーバステアになる．
図１０は各条件でαの変化の様子を調べたものである．制御なしは駆動力が増した０．５＜ｔ＜２．５ｓｅｃの間でアンダが増しその後の復元性も悪い．準最適制御はαの値が最も小さく抑えられておりニュートラルステアに近い状態が維持されている．比例制御も最初のうちはαの値が小さく抑えられているが制御の遅れがあるため図１２に示すようにヨーレートにオーバシュートが生じこれが急激なアンダステアとなって現れる．
図１１は次式で定義されるタイヤ発生力指数の比較を示す．

Ｃ_ｃは旋回におけるタイヤの有効活用度合いを表し，この値が大きいほど装着したタイヤのポテンシャルを引き出していると見ることができる．この評価をみても準最適制御が最も優れていることがわかる．
図１２は，目標ヨーレートに対する追従性を示す．制御なしは目標横すべり角に追従しないためヨーレート変化は小さい．逆に目標横すべり角に追従するように制御するとどうしてもヨーレート変化が生じるが，準最適制御は比例制御に比べヨーレートの変動幅が小さいことがわかる．
図１３は横すべり角についての追従性を比較したものである．目標値はドライバが意図した駆動力に応じて図の細い実線のように変化する．当然のことながら，制御なしは全く追従しない準最適制御は比例制御に比べ遅れが少なくオーバーシュートもなく収束が早いことがわかる．
図１４は．準最適制御則と比例制御則の指令値をＤＹＣのモータトルクに換算して示している．これより比例制御に比べ準最適制御は遅れが改善されていることが分かる．指令値の大きさも０．５ｋＮｍ以下であり過大すぎず現実的なレベルに収まっている．
以上により，準最適制御が比例制御に比べて優れていることが明らかになった．式（３２），（３３）の比較から推測できるように，第一の要因は比例制御では定数であった制御ゲインが，準最適制御では図１５に示すように車両の状況に応じて変化することによる．制御ゲインが低下する０．５＜ｔ＜３．５ｓｅｃの間は横すべり角が増大して車両の復元性が低下しており，安定性を確保する必要性からゲインが低下しているものと考えられる．第二の要因は，準最適制御では目標値と比較する量をＶ_１（ｔ），Ｖ_２（ｔ）としており，式（３２）の補足式から明らかなようにこれらは制御が無い場合のＴｓｅｃ先の状態量であることである．図１６に示すように、状態量ｘ_１（ｔ），ｘ_２（ｔ）と比較すれば，この予測的制御により遅れが補償されていることがわかる．
一般に旋回中は後内輪荷重が減るため，ＤＹＣによって後内輪に過大な制御力を加えると空転が生じこれが性能面での拘束になっているが，ＲＲ車で大横すべり角制御を行えば後内輪のこの問題は生じない．これを確認したのが図１７に示す後輪のスリップ比の結果である．旋回駆動時の後内輪のスリップ比γ_３は最大０．３になるもののほとんどの時間帯で０．１以下に収まっていることがわかる．ここでスリップ比γとは各輪が取り付けられている部位での車体前後速度に対する各輪のスリップ速度の比である．
以上により，本手法による車両の大横すべり角制御が可能であることが確認できた．A simulation based on the above driving conditions is performed. The driving force is increased during 0.5 <t <1.5 sec and returned during 1.5 <t <2.5 sec. Using MATLAB / SIMULLINK, a simulation was performed for 4 sec with a fixed step of 10 ⁻³ sec.
The simulation results are shown in Figs.
Figure 9 shows the vehicle trajectory. Without control, understeer becomes noticeable and the trajectory bulges out by applying a driving force equivalent to a 4-speed full throttle. On the other hand, the proportional control u _n (t) has an effect of reducing understeer, but still swells outward. It can be seen that in the sub-optimal control u ^* _opt (t), the turning radius is substantially maintained at 60 m and the neutral steer is maintained.
The following index α was introduced to see the change in the steering characteristics.

Here, β _i is the side slip angle of each wheel. If α is positive, the side slip angle of the front wheel is large and understeer, and if α is negative, the side slip angle of the rear wheel is large and oversteer.
Figure 10 shows the changes in α under each condition. Without control, under increases for 0.5 <t <2.5 seconds when the driving force increases, and the restorability after that increases. In sub-optimal control, the value of α is minimized, and a state close to neutral steer is maintained. In the proportional control, the value of α is initially kept small, but there is a delay in the control, so an overshoot occurs in the yaw rate as shown in FIG. 12, and this appears as a sudden understeer.
Fig. 11 shows a comparison of tire force index defined by the following equation.

C _c represents the degree of effective use of the tire in turning, and it can be seen that the larger the value, the more potential the tire is mounted. This evaluation shows that suboptimal control is the best.
Fig. 12 shows the followability to the target yaw rate. Without control, the yaw rate change is small because the target side slip angle is not followed. On the contrary, yaw rate changes inevitably occur when control is performed so as to follow the target side slip angle, but it can be seen that sub-optimal control has a smaller fluctuation range of yaw rate than proportional control.
Figure 13 shows a comparison of the trackability of the side slip angle. The target value changes like the thin solid line in the figure according to the driving force intended by the driver. Naturally, it can be seen that suboptimal control, which does not follow at all without control, has less delay than proportional control and converges quickly without overshoot.
FIG. The command values of the sub-optimal control law and proportional control law are converted into DYC motor torque. This shows that the delay of sub-optimal control is improved compared to proportional control. The size of the command value is 0.5kNm or less, which is not too large and is within a realistic level.
From the above, it became clear that suboptimal control is superior to proportional control. As can be inferred from the comparison of the equations (32) and (33), the first factor is that the control gain, which is a constant in the proportional control, changes according to the vehicle situation as shown in FIG. 15 in the sub-optimal control. It depends. During the period of 0.5 <t <3.5 seconds when the control gain decreases, the side slip angle increases and the vehicle's resilience decreases, and it is considered that the gain has decreased due to the need to ensure stability. It is possible. The second factor is that V ₁ (t) and V ₂ (t) are compared with the target values in the sub-optimal control, and as is apparent from the supplementary equation of Equation (32), these are the cases where there is no control. The state quantity is Tsec ahead. As shown in FIG. 16, when compared with the state quantities x ₁ (t) and x ₂ (t), it can be seen that the delay is compensated by this predictive control.
In general, the load on the rear inner ring decreases during turning, and if excessive control force is applied to the rear inner ring by DYC, slipping occurs and this imposes a constraint on performance. This problem does not occur. This was confirmed by the result of the slip ratio of the rear wheel shown in FIG. An inner ring of the slip ratio gamma ₃ after the time of turn driving it can be seen that falls to 0.1 or less most of the time zone but maximized 0.3. Here, the slip ratio γ is the ratio of the slip speed of each wheel to the vehicle body longitudinal speed at the part where each wheel is attached.
From the above, it was confirmed that the large slip angle control of the vehicle by this method is possible.

BEST MODE FOR CARRYING OUT THE INVENTION

Figure 4 shows the situation where DYC is installed in an RR vehicle with high performance improvement potential. Signals from the steering angle sensor 1, the yaw rate sensor 4, the G sensor 5 that detects the longitudinal acceleration of the vehicle body and the wheel speed sensors 6, and the throttle angle signal 3 from the engine controller 2 enter the DYC controller 7. A hydraulic unit 8 is arranged in the differential case. The hydraulic unit includes a hydraulic motor 9, a pump 10, and a control valve 11, and torque generated by the hydraulic motor is transmitted to the left and right wheels through the differential gear 12 as opposite phase torques. As a matter of course, these torques and the driving torque from the engine are added together and the final torque added is the tire driving force.
The hydraulic motor torque transmitted to the left and right drive shafts in opposite phases generates a braking / driving force in opposite directions on the left and right tires as shown in FIG. Assuming that the torque command value and output torque to the motor are T _m , the driving forces of the left and right wheels are −T _m / r and T _m / r, respectively. The control yaw moment applied to the vehicle is M _c = T _m I _tr / r. Where r is the tire radius.

  Since this control method has an excellent effect when controlling a nonlinear system in real time, it is highly likely that it will be widely used not only in vehicle motion control but also in robotics.

Conceptual diagram of this control method Tire characteristics using brush model Vehicle model diagram DYC system configuration diagram Simple vehicle model diagram Illustration of linearization of tire characteristics Control signal explanatory diagram of Non-Patent Document 8 Side slip angle characteristic diagram of Non-Patent Document 8 Simulation results (vehicle trajectory comparison) Simulation results (Comparison of changes in steering characteristics) Simulation results (comparison of tire utilization) Simulation results (yaw rate comparison) Simulation result (side slip angle comparison) Simulation results (control signal comparison) Simulation result (control gain) Simulation result (Comparison of state quantity and predicted quantity) Simulation results (rear wheel slip ratio)

Explanation of symbols

1. 1. Steering angle sensor Engine controller 3. 3. Throttle angle signal 4. Yaw rate sensor G sensor 6. 6. Wheel speed sensor DYC controller8. Hydraulic unit 9. Hydraulic motor 10. Pump 11. Control valve 12. Differential gear

Claims (2)

An optimal control method for a system characterized in that the control law obtained in the following steps is included in part.
The nonlinear control system and the evaluation function to be minimized are described as a Receding Horizon control problem where the current time is t and the virtual time is τ, the state equation of the nonlinear system is expressed by equation (101), and the evaluation function to be minimized is expressed by equation ( 102).

Where x (τ, t) εR ⁿ and u (τ, t) εR ^r are state variables and control inputs defined on the virtual time τ axis.

The target value, T, is defined as the evaluation interval.
Equation (101) is linearized as follows.

Here, Jacobian _{f x (t), f u} (t) and _f 0 (t) is defined by the following equation.

[Formula 106] f ₀ (t) = f (x (t), u (t)) − f _x (t) x (t) −f _u (t) u (t)
Similarly, equation (102) is approximated as follows. In the following, approximates are identified with a * on the right shoulder.

x (t) is given as an initial value of the state quantity at τ = 0. Here, the change of u (τ, t) is approximated by the following straight line.

By substituting equation (108) into equation (103) and solving for τ, x ^* (τ, t) is obtained, and from this, x ^* (T, t) is obtained as a function of u ^* (t). J ^* (t) in equation (107) is obtained as a function of u ^* (t). The suboptimal control law that minimizes J ^* (t) is obtained from the following equation.

Here, x ^* (T, t) and u (t) in the equation (107) are replaced with x ^* (T, t) and u ^* (t), and are rewritten as J ^* (t). The suboptimal control law u ^* _opt (t) is obtained from the partial differential result of Equation (109) by algebraic calculation. Here, the control device comprises means for calculating the measured state quantity from the Jacobian _{f x (t), f u} (t) and _f 0 (t). 2. The vehicle control method according to claim 1, wherein the optimal control law that partially forms a part of the vehicle is an expression (115) obtained in the following steps.
The nonlinear control system is a vehicle with left and right wheel driving force control devices, and the state equation of the vehicle is linearized as follows.

here,

x ₁ is the yaw rate, _{x 2} is a side slip velocity. M is the vehicle mass, I _z is the yaw moment of inertia,

Is the ratio of the change in lateral force to the minute change in the side slip angle at each operating point of the front and rear wheel tires, δ is the front wheel steering angle, and F _f ⁰ and F _r ⁰ are linear approximations of the characteristics at the operating points of the front and rear wheel tires, respectively. This is the value of the lateral force at zero side slip angle, and the correspondence with equation (103) is as follows.

Let the evaluation function be equation (111).

The evaluation function of the Receding Horizon control form of Equation (107) is as follows.

The general solution of equation (103) is given by

In control, the measured x (t) is this value.
Substituting equation (108) into u (τ ′, t) in equation (113), find x ^* (T, t).

Substituting equation (114) into equation (112) and applying optimal condition equation (109), the solution u ^* _opt (t) is obtained as follows.

It is.

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