JP2007233985A - Optimal control method of system - Google Patents

Abstract

<P>PROBLEM TO BE SOLVED: To clarify a process for obtaining a prospective optimal control rule to a nonlinear optimal control problem in which an optimal control rule is difficult to be analytically derived in the conventional theory. <P>SOLUTION: An evaluation function about input is defined as energy to be transmitted from a controller to a control target, and a function representing control performance is added to the evaluation function to be an evaluation function for the whole. Further, an expression of perfect power income and expenditure of the control target is added as a collateral condition to be a functional. Since the functional is a linear expression about a control variable in this way, the optimal control rule can be analytically derived without solving a differential equation from a condition that minimizes the functional, regardless of linearity or nonlinearity of the control target. Consequently, an optimal control rule can be obtained even about a nonlinear control problem having energy regenerative control and non-horonomic constraint in which the optimal control rule is analytically difficult to be obtained in the conventional practice. <P>COPYRIGHT: (C)2007,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, nonlinear, holonomic or nonholonomic.

  Pontryagin introduced adjoint variables and rewritten the optimal control problem in Hamiltonian form and found the maximum principle, which had a great influence on the subsequent development of optimal control theory. However, the situation in which simultaneous differential equations for adjoint variables and state variables have to be solved remains the same, and it is not easy to derive the optimal control law for nonlinear systems analytically. Therefore, various numerical search methods and inference methods have been devised. However, when the control law is a numerical solution, even if the control system designer tries to improve the control law obtained, the correspondence with the parameters of the control target system is not known and must be trial and error. To get out of this situation, an analytical method with good prospects is necessary.

Problems to be solved by the invention

Against this background, the present invention presents a method for analytically deriving control laws for a relatively wide range of optimal control problems, mainly including nonlinear systems of mechanical dynamics systems, using a method different from the conventional one. In order to easily derive a control law that asymptotically stabilizes the system by the concept of passivity focusing on the energy balance of the controlled object (see Non-Patent Document 1, for example), the present invention also focuses on the energy balance of the controlled object. The evaluation function related to the input is the energy transmitted from the control device to the control target, and the function that expresses the control performance is added to this to make the overall evaluation function. In addition, as an incidental condition, an equation for the complete energy balance of the controlled object is added to make a functional. In this way, the functional is linear with respect to the control variable, so the optimal control law can be derived analytically without solving the differential equation from the condition that minimizes the functional regardless of whether the controlled object is linear or non-linear.
The condition that minimizes this functional is a necessary condition for optimal control, and is a weak condition that includes undetermined constants. However, a useful control law can be obtained by making good use of the structural properties of the controlled object. It is possible. The usefulness is shown in a later example.

  Formulate the optimal control problem. First, the characteristic representation of the controlled object is described. In the control theory, the characteristics of the controlled object are described by the equation of motion. Here, the equation of balance of the total input / output power of the controlled object is used. Specifically, the equation of motion for each degree of freedom of the system is displayed as a vector. It is multiplied by the velocity vector. The input power includes not only the control input but also the disturbance input, and the controlled object is not limited to passive elements, so if there is an energy source inside and this affects the motion, it is treated as a disturbance input.

Formula 1

ここで，ｄ，ｅ，ｑ，ｕ，ｖ，ｚ ∈Ｒ^ｎ ，Ｍ ∈Ｒ^ｎ×ｎ は正定対称な慣性マトリクス，ｎは制御対象の自由度である．ｑは一般化座標，ｕは制御入力，ｖは力入力の外乱，ｚは変位入力の外乱である．ｄはコリオリ力や遠心力やダンピング力など，ｅはポテンシャル力である．
制御対象が非線形であっても制御装置を合理的に設計すれば，式（１）のようにｕを直接Ｍに作用させることができる．本論文では，このような合理的に設計された機械力学系システムを想定して最適制御則を導く．
上記のシステムに対し，次の評価関数を考える．

Here, d, e, q, u, v, z ∈ R ⁿ and M ∈ R ^{n × n} are positive definite inertia matrices, and n is the degree of freedom of the controlled object. q is a generalized coordinate, u is a control input, v is a force input disturbance, and z is a displacement input disturbance. d is Coriolis force, centrifugal force and damping force, and e is potential force.
Even if the object to be controlled is non-linear, if the controller is rationally designed, u can be directly applied to M as shown in Equation (1). In this paper, the optimal control law is derived assuming such a reasonably designed mechanical system.
Consider the following evaluation function for the above system.

Formula 2

置が御対象に加えるパワーである．ｒは重み係数である．
式（２）を最小化する制御ｕ（ｔ）を求めることが最適制御の課題である．
有本卓，新版ロボットの力学と制御，（２００２），ＰＰ．６８−６９．朝倉書店．

This is the power that the device will apply to the object. r is a weighting factor.
Finding the control u (t) that minimizes Equation (2) is the problem of optimal control.
Takashi Arimoto, Dynamics and Control of New Version Robot, (2002), PP. 68-69. Asakura Shoten.

Means for solving the problem

  In the conventional optimal control theory, an adjoint variable is introduced and a differential equation is derived and the original state equation is coupled to solve it. In this paper, a method completely different from this method is shown. The following scalar function L is defined to find the necessary conditions for optimal control.

Formula 3

ここで，κは未定定数である．右辺の｛｝内は，式（１）の左辺と同じで制御対象の全パワー収支であるからエネルギ保存則を満たし常にゼロである．従って式（３）で表されるＬの積分を最小化する条件は，式（２）も最小化する．
従って関数Ｌにｑを変数とする変分原理を適用した次式はｕが最適制御であるための必要条件を与える．Ｌはｕに関して１次式であるから，∂Ｌ／∂ｕは意味がなく次式に制御に関するすべての情報が含まれる．

Where κ is an undetermined constant. The inside of {} on the right side is the same as the left side of Equation (1) and is the total power balance of the controlled object, so it satisfies the energy conservation law and is always zero. Therefore, the condition for minimizing the integral of L expressed by equation (3) also minimizes equation (2).
Therefore, the following equation applying the variational principle with q as a variable to the function L gives a necessary condition for u to be optimal control. Since L is a linear expression with respect to u, ∂L / ∂u has no meaning and the following expression contains all information related to control.

Formula 4

Ｌにはｑの２階の導関数が含まれるため一般的なオイラーの方程式に第３項が追加されている（例えば非特許文献２参照）．
式（４）を積分し積分定数をゼロとすると次式になる．

Since L includes the second derivative of q, the third term is added to the general Euler equation (see, for example, Non-Patent Document 2).
If equation (4) is integrated and the integration constant is zero, the following equation is obtained.

Formula 5

式（３）を式（５）に代入し左辺第１〜３項の順に記すと次のようになる．

Substituting equation (3) into equation (5) and writing in the order of the first to third terms on the left side, it becomes as follows.

Equation 6

上式から制御則が次のように求まる．

From the above equation, the control law is obtained as follows.

Equation 7

このように，本手法はＪを最小化するｑとｕの関係を直接導くことができるため，従来のように２点境界値問題を最適性の原理を用いて解くプロセスが不要になる．κは未定定数であるが，κ＝０の時はｕが制御の評価関数のパラメータのみで定まり，κ＝∞の時はｕが制御対象のパラメータのみで定まることからｕが最適であるためにはκはゼロでない有限値でなければならないことは明らかである．
上式の第１行は外力ｖと慣性のｑ依存性に対する制御，第２行はコリオリ力や遠心力やダンピング力に対する制御，第３行はポテンシャル力とそのｑ依存性および外力ｚに対する制御，第４行は評価関数を低減させる制御でありそれぞれ意味が明確である．式（７）には未実行の微積分項が含まれているが，全ての外力と状態量の検出あるいは推定が可能とすればこれらの実時間での実行は可能である．式（５）において積分定数をゼロとしたが，上記の結果より積分定数は制御則に一定のバイアスを与えることになるためゼロとすることが妥当であることがわかる．式（７）中の積分も同様である．
式（７）ではアクチュエータの数と系の自由度が同じであることを想定しているが，アクチュエータの数が少ない場合には次のような処置が必要になる．例えばアクチュエータが二つの独立な慣性Ｍ_ｉ，Ｍ_ｉ＋１の間に置かれるような場合は，制御ベクトルｕにその拘束条件を含めておき最適制御則は二つの制御則を重みを付けて加算したものにする．

In this way, since this method can directly derive the relationship between q and u that minimizes J, the conventional process of solving the two-point boundary value problem using the principle of optimality becomes unnecessary. κ is an undetermined constant. However, when κ = 0, u is determined only by the parameter of the control evaluation function, and when κ = ∞, u is determined only by the parameter to be controlled. It is clear that κ must be a non-zero finite value.
The first line of the above equation is the control for the external force v and the q dependence of the inertia, the second line is the control for the Coriolis force, the centrifugal force and the damping force, the third line is the control of the potential force and its q dependency and the external force z, The fourth line is a control that reduces the evaluation function, and the meaning of each is clear. Equation (7) includes unexecuted calculus terms, but these can be executed in real time if all external forces and state quantities can be detected or estimated. In equation (5), the integration constant is set to zero. From the above results, it can be seen that it is appropriate to set the integration constant to zero because it gives a constant bias to the control law. The integration in equation (7) is similar.
Equation (7) assumes that the number of actuators and the degree of freedom of the system are the same, but the following measures are required when the number of actuators is small. For example, when the actuator is placed between two independent inertias M _i and M _{i + 1} , the constraint condition is included in the control vector u, and the optimal control law is the sum of the two control laws with weights added. .

Equation 8

ここでｕ_ｉ，ｕ_ｉ＋１はそれぞれが独立な制御として導いた場合の最適制御則であり，ρ_ｉ，ρ_ｉ＋１は重み係数である．これらの重み係数や未定定数κの決定法については，次の具体例で示す．
Ｌ．Ｅ．エルスゴルツ，瀬川富士訳，科学者・技術者のための変分法，（１９７２），ｐｐ．３４−３７．ブレイン図書出版．

Here, u _i and u _{i + 1} are optimum control laws when they are derived as independent controls, and ρ _i and ρ _{i + 1} are weighting factors. The following specific examples show how to determine these weighting factors and undetermined constant κ.
L. E. Els Golz, Segawa Fuji, Variation Method for Scientists and Engineers, (1972), pp. 34-37. Brain book publication.

The invention's effect

  Here, this method is applied to problems that cannot be solved by conventional optimal control theory, and its effectiveness is clarified. First, the energy regeneration control of an active suspension whose evaluation function does not have a quadratic form will be handled, and then it will be applied to the control of nonholonomic systems where it is still difficult to derive the optimal control law.

First, a specific formulation for the application of active suspension to energy regeneration control is performed. In Patent Document 1, energy regeneration control is devised using an active suspension model as shown in FIG. 1, and a suboptimal control law u ^* _opt is derived by an approximate solution method. In this case, as a result, the input evaluation function is the energy balance equation that is transmitted from the control device to the controlled object as in the method of this paper, but at that time, it was still recognized as the optimal control that focused on the controlled energy balance. However, the approximate solution method was tried as a case that cannot be solved analytically by the conventional control theory, and it is proved that it is almost optimal control while using the approximate solution method. In the present invention, it is confirmed that the optimal control law u _opt derived analytically has the same performance as this approximate solution u ^* _opt , though a concise expression.
First, the evaluation function of Patent Document 1 is described using the variable q.

Equation 9

ここで，ｒ_１，ｒは重み係数である．重みｒ_１はばね下質量の制振が従来車両と同程度になるようにｒ_１＝５×１０^−３ とし，ｒについては振動の評価とエネルギの評価が同程度の値になるようにｒ＝１０^−５とした．

Here, r ₁ and r are weighting factors. The weight r ₁ is set to r ₁ = 5 × 10 ⁻³ so that the vibration suppression of the unsprung mass is about the same as that of the conventional vehicle, and r is set so that the vibration evaluation and the energy evaluation are about the same value. = 10 ⁻⁵ .

  Next, the control law is derived. First, the elements of Equation (1) are obtained from the model in Fig. 1 as follows.

Equation 10

Equation 11

Formula 12

Equation 13

Equation 14

Equation 15

式（１０）〜（１５）を式（３）に代入することでＬが求まる．

L is obtained by substituting Equations (10) to (15) into Equation (3).

Equation 16

式（１６）を式（５）に代入すると，

Substituting equation (16) into equation (5),

Equation 17

より次式が得られる．

The following equation is obtained.

Equation 18

これより次の制御則が得られる．

This gives the following control law.

Equation 19

同様にして，

Similarly,

Equation 20

より次式が得られる．

The following equation is obtained.

Equation 21

これより次の制御則が得られる．

This gives the following control law.

Equation 22

従って最終的な制御則は次の形になる．

Therefore, the final control law has the following form.

Equation 23

ρ_１，ρ_２はそれぞれの重み係数で，その和は次式とする．

ρ ₁ and ρ ₂ are the respective weighting factors, and the sum is as follows.

Formula 24

評価関数の重みを考慮したＭ_１，Ｍ_２へのｕ_１，ｕ_２の感度を等しいと考えると次のようになる．

Considering that the sensitivity of u ₁ and u ₂ to M ₁ and M ₂ considering the weight of the evaluation function is equal, it is as follows.

Formula 25

以上より最適制御則は次のように求まる．

From the above, the optimal control law is obtained as follows.

Equation 26

ここで，ρ_１＝２Ｍ_１／（Ｍ_１＋ｒ_１Ｍ_２），ρ_２＝２ｒ_１Ｍ_２／（Ｍ_１＋ｒ_１Ｍ_２）である．本手法から求まる必要条件はここまでであって，κの最適化は探索的に求める必要がある．

Here, ρ ₁ = 2M ₁ / (M ₁ + r ₁ M ₂ ), ρ ₂ = 2r ₁ M ₂ / (M ₁ + r ₁ M ₂ ). The necessary condition obtained from this method is up to here, and the optimization of κ needs to be obtained exploratoryly.

一定でレベルは極良路（ＩＳＯ／ＴＣ１０８／クラスＡ）を６０Ｋｍ／ｈ程度での走行に相当するように設定した．シミュレーションは，１０^−４ｓｅｃの固定ステップで２ｓｅｃ間行った．Control performance is evaluated by random road surface input simulation with MATLAB / SIMULLINK. Less than

The level was constant and set to correspond to driving on an excellent road (ISO / TC108 / Class A) at about 60 km / h. The simulation was performed for 2 sec with a fixed step of 10 ⁻⁴ sec.

Figure 2 shows the results of a κ search that minimizes J under the above simulation conditions. It can be seen from this that the optimum value is κ = −3.6 × 10 ⁻⁷ .

した．パッシブダンパ制御は図１のパッシブダンパＣ_２と同等の機能を制御で出すものである．スカイフックダンパ制御はばね下振動を押さえる機能がないためダンパが必要でありＣ_２＝１２００Ｎｓｅｃ／ｍとし，他はすべて Ｃ_２＝０とした．評価関数Ｊを振動に関する部分Ｊ_１と制御入力エネルギに関する部分Ｊ_２に分離すると次のようになる．The simulation results are shown in Figs. In order to compare control performance, passive

did. Passive damper control is one issue in controlling the passive damper C ₂ functions equivalent of FIG. Skyhook damper control does not have a function to suppress unsprung vibration, so a damper is necessary. C ₂ = 1200 Nsec / m, and all others are set to C ₂ = 0. When the evaluation function J is separated into a part J ₁ related to vibration and a part J ₂ related to control input energy, the following is obtained.

Equation 27

Equation 28

図３は各制御についてＪ_１，Ｊ_２の比較を行ったものである．Ｊ_２についてはｕ_ｏｐｔとｕ^＊ _ｏｐｔは同等であるが，Ｊ_１ではｕ_ｏｐｔの方が若干優れることがわかる．
図４は，本制御と準最適制御について制御対象に伝達されるパワーのシミュレーション結果を示す．ともに大半の時間帯で伝達パワーは負の値を示しており，エネルギ回生を効果的に行っていることがわかる．
以上により，本制御則の式（２６）は近似解法による準最適制御則（特許文献１の式（２０））と比べ簡潔でありながら同等以上の効果があることが確認できた．

Figure 3 shows a comparison of J ₁ and J ₂ for each control. _{U opt} and ^u _{* opt} for J ₂ is an equivalent, more of the _{J 1 u opt} it can be seen that the excellent slightly.
Figure 4 shows the simulation results of the power transmitted to the controlled object for this control and suboptimal control. In both cases, the transmitted power shows a negative value in most of the time, indicating that energy regeneration is effectively performed.
From the above, it was confirmed that the equation (26) of this control law is more concise but has the same or better effect than the suboptimal control law based on the approximate solution (equation (20) of Patent Document 1).

  Next, an example applied to the optimal control of a nonholonomic system is shown. In recent years, interest in nonlinear systems with nonholonomic constraints has increased, and various controls have been studied. Here, the control of rolling coins explained in Non-Patent Document 3 is taken as an example, and the optimal control law derived by this method is examined.

Figure 5 shows a model of a coin that rolls. The orthogonal coordinates xy are taken on the plane, the angle of the coin rolling direction is q ₁ , the position on the plane is (q ₂ , q ₃ ), and the coin rolling angle is q ₄ . The torques in the direction of increasing q ₁ and q ₄ are u ₁ and u ₂ . Furthermore, the coin radius and moment of inertia are set to 1 and the mass is set to zero.
First, the elements of Equation (1) are obtained from the model shown in Fig. 5 as follows.

Equation 29

Equation 30

Formula 31

Equation 32

さらに非ホロノミックな拘束条件として次式が加わる．

Furthermore, the following equation is added as a nonholonomic constraint.

Equation 33

Equation 34

このようにｕがＭを直接制御する系ならば，非ホロノミック系であっても本手法により最適制御則を解析的に導くことが出来ることを次に示す．

If u is a system that directly controls M in this way, the following shows that the optimal control law can be derived analytically by this method even for nonholonomic systems.

でよい．A specific formulation for the control of rolling coins is given. The purpose of control is to set q ₁ , q ₂ , and q ₃ to zero from an arbitrary initial state, but as is clear from the model, q ₁ only needs to integrate u ₁ twice.

It's fine.

Formula 35

ｕ_１は振動的にする必要があることが既に知られているため，ｋ_１＝０．３，ｋ_２＝１とした．この制御則ではｑ_１の初期値が小さいと振動が小さすぎてｑ_２，ｑ_３の制御に支障をきたすが，この場合は例えば事前にｑ_１をある値に設定してから全体の制御を開始するといった二段搆えの制御を行えば問題は回避できる．ｑ_２，ｑ_３に影響を与えずにｑ_１を事前にある値に設定することは容易であるためこの方式は許容できよう．あるいはｑ_１の初期値にかかわらずｑ_１について式（３５）の制御則と同様な次のような位置制御をプログラムしておくといった方法でもよい．

Since it is already known that u ₁ needs to be vibrated, k ₁ = 0.3 and k ₂ = 1. In this control law, if the initial value of q ₁ is small, the vibration is too small and hinders the control of q ₂ and q _{3. In} this case, for example, after setting q ₁ to a certain value in advance, the entire control is performed. The problem can be avoided by making a two-step control such as starting. Since it is easy to set q ₁ to a certain value in advance without affecting q ₂ and q ₃ , this method will be acceptable. Alternatively for q ₁ regardless of the initial value of q ₁ may be a method such as previously programmed position control such as similar following the control law of the formula (35).

Equation 36

以上の事前検討を基にｕ_２についての最適制御を考え，評価関数を次式とする．

Based on the above prior study, we consider optimal control for u ₂ and use the following equation as the evaluation function.

Formula 37

ここで，ｒ_２，ｒ_３，ｒ_４，ｒ_５ は重み係数である．上式右辺第１，２項はｑ_２，ｑ_３をゼロに収束させるという制御目的を表す項で，その意味をわかりやすくするため積分した結果を表示している．第３項がコインの回転運動に対する減衰を評価する項であり，第４項が制御対象への入力エネルギの項である．

Here, r ₂ , r ₃ , r ₄ , r ₅ are weighting factors. The first and second terms on the right side of the above equation represent the control purpose of converging q ₂ and q ₃ to zero, and the integrated results are displayed to make the meaning easier to understand. The third term is a term that evaluates the attenuation of the coin against the rotational motion, and the fourth term is the term of the input energy to the controlled object.

上式右辺第１行がパワー収支式から得られる項であり，第２，３行が評価関数から得られる項である．このケースではパワー収支式から得られる情報が少ないので評価関数に必要十分な情報を盛り込みこれを補っている．ここで，留意しなければならないのは，モデルから運動方程

小化する条件の一部として既に式（３９）に含まれているためさらに代入することはＬに余計な拘束を強いることになる．
ｕ_２を最適にするための必要条件は次の方程式から求めることができる．Next, the control law is derived. L is obtained using equations (29) to (34) and (37). Here, g is obtained by adding the integrands of the first to third terms after adding the first and second terms on the right side of the equation (37) to the form of integration and adding the constraints of the equations (33) and (34). ing.

The first line on the right side of the above equation is the term obtained from the power balance equation, and the second and third lines are terms obtained from the evaluation function. In this case, since there is little information obtained from the power balance equation, necessary and sufficient information is included in the evaluation function to compensate for this. Here, it should be noted that from the model to the exercise procedure

Since it is already included in the equation (39) as a part of the condition to reduce, further substitution imposes an extra constraint on L.
The necessary conditions for optimizing u ₂ can be obtained from the following equation.

Formula 39

式（３８）を代入すると，

Substituting equation (38),

Formula 40

となりこれより次の制御則が得られる．

From this, the following control law is obtained.

Formula 41

The simulation is evaluated using the control laws of Equations (35) and (41).
First, the weight coefficients r ₂ , r ₃ and r ₄ are selected. When caused to converge to zero earlier the q ₁ As can be seen from the model in Figure 5 can not be adjusted q _3. Therefore, it is desirable to converge q ₂ after converging q _{3. For} this reason, r ₃ is set larger than r ₂ . The r ₄ to suppress the overshoot of the rotation of the coin is set to larger values. From the above, r ₂ = 0.25, r ₃ = 2.5, and r ₄ = 1.0. Input energy in this case because ultimately becomes zero, the value of r ₅ is not critical. Combined with other weights and levels, r ₅ = 0.5.
The simulation results are shown in Figs. First, optimize κ. Figure 6 shows the change in the value of equation (37) when κ is changed. This shows that κ = −0.1 is the optimum value. The simulation is performed using this value.
Fig. 7 shows the results of simulation with the initial values of [q ₁ , q ₂ , q ₃ ] as [1, 1, -1]. and subsequently the first convergence is q ₃ by utilizing the vibration of q ₁ q ₂ is it can be seen that slowly converge.
Figure 8 shows the locus of q ₂ and q ₃ at this time. q ₂ converged to the target with a single speed change, and good results were obtained.
As an example of asymptotically stabilizing the model of FIG. 5, using the control law introduced in Non-Patent Document 3, the simulation result under the same conditions as FIG. 7 is shown in FIG. In particular, it has not been devised to improve convergence, but has been presented as an example of asymptotic stabilization, so it is not a subject of direct performance comparison with the results of this proposal, but is posted for reference.

  Next, we show that this control has excellent robust performance. Again, using the model in Fig. 1, we compare it with LQ control, which is a typical conventional optimal control theory.

In order to obtain the LQ control law, it is necessary to rewrite the model of Fig. 1 with the state equation and to make the evaluation function a quadratic form. In Figure 1, a _{q 0} and _{x 0,} the q 1 -q ₀ and _{x 1,} the q 2 -q ₀ and _{x 2,} the unsprung velocity and _{x 3,} the state equation when the sprung speed and _{x 4} is It is as follows.

Equation 42

評価関数を次のような２次形式とする．

The evaluation function has the following quadratic form.

Equation 43

式（４３）の被積分関数の第１項は式（２７）の性能評価項と同一であるが，第２項は異なりｕの２乗になっている．無限評価区間であるから，ＬＱ制御則は次のリカッチ代数方程式の解から求まる．

The first term of the integrand in equation (43) is the same as the performance evaluation term in equation (27), but the second term is different and is the square of u. Since it is an infinite evaluation interval, the LQ control law can be obtained from the solution of the following Riccati algebraic equation.

Formula 44

本手法とＬＱ制御とは制御入力の評価関数が異なるため，性能比較を行うには性能評価値Ｊ_１の値を基準値にとりこの値が同じになるようにして比較する必要がある．Ｊ_１の値が図３に示す本手法の値と同じになるように，ｒ_１ｑ＝１．５×１０^−９とし，Ａ，Ｂ，Ｑに数値を入れると制御則は次のようになる．

Since the proposed method and the evaluation function of the control input to the LQ control are different, to compare the performance has to be compared as the value reaches the same taken reference value the value of the performance evaluation value J _1. When r _1q = 1.5 × 10 ⁻⁹ so that the value of J ₁ is the same as the value of the present method shown in FIG. 3, and when numerical values are entered in A, B, and Q, the control law is as follows: .

Formula 45

である．これを式（２６）と同じ形式で記述し両者を数値で比較すると次のようになる．

It is. This is described in the same format as equation (26) and both are compared numerically as follows.

Equation 46

Equation 47

Compare the robust performance of this control and LQ control. Comparing Equations (46) and (47), LQ control has a large gain of q ₀ and q ₁ , and there is a concern that robustness against signal noise is reduced. In actual control, since the road surface input q ₀ uses an estimated value, a certain amount of estimation error must be assumed. A simulation evaluation of the effect of q ₀ estimation error was performed. The same signal statistical characteristics uncorrelated to the road surface input q ₀ and y _0, the random noise of the estimation error corresponds to the .alpha.y _0. Figure 10 shows the simulation results with α as a parameter. It can be seen that the plot of this method shown in color is almost unaffected by the estimation error of q ₀ and is highly robust. It is obvious that the evaluation value J is superior to the LQ control in the evaluation value J _LQ , but it is natural that the evaluation value J _LQ is superior to the LQ control in the evaluation value J _LQ . Therefore, it can be said that this method is excellent in robustness against the estimation error of disturbance input.

The above is the contents of this proposal, but as a general theory, it is necessary to prove that Equation (2) has the minimum value.
Equation (1) is a power balance equation, but the power due to the disturbance of the displacement input is not expressed explicitly. If this is specified and the input side is rewritten on the left side and the storage / dissipation side is rewritten on the right side, the following equation is obtained.

Formula 48

ここで，ｆは変位入力外乱ｚに抗する制御対象側の反力，Ｕはポテンシャルエネルギである．式（１）との対応より次式の成立は明らかである．

Here, f is the reaction force on the controlled object side against the displacement input disturbance z, and U is the potential energy. The following equation is clear from the correspondence with equation (1).

Formula 49

動方程式の記述を用いて書き直し，残りのダンピング力を改めてｄ’とすれば，式（４８）の右辺は次のようになる．

Rewriting using the description of the dynamic equation, and letting d ′ be the remaining damping force, the right side of equation (48) is as follows.

Formula 50

ここで，上式の第２項はコリオリ力によるもの，第３項は遠心力によるもの，第４項はダンピング力のみによるものである．式（５０）を積分すると，

Here, the second term in the above equation is due to Coriolis force, the third term is due to centrifugal force, and the fourth term is due to damping force only. Integrating equation (50),

Formula 51

となる．ここで上式の第２項と３項は相殺され，結局式（５１）は次式になる．

It becomes. Here, the second and third terms of the above equation are cancelled, and eventually equation (51) becomes the following equation.

Formula 52

上式第１項は運動エネルギ，第２項は散逸エネルギ，第３項はポテンシャルエネルギであり，これは制御対象に蓄えられるエネルギと散逸するエネルギの総和である．式（４８）の左辺の積分は，制御対象に加えられるエネルギの総和であるから，結局式（４８）を積分したものがエネルギ保存則になっていることがわかる．
ここで式（５２）の第１，３項の和をＥ_１とする．外乱入力パワーの和の積分をＥ_２とすると，式（４８）において外乱入力項を右辺に移項して，左右辺を積分すれば，

The first term in the above equation is kinetic energy, the second term is dissipated energy, and the third term is potential energy, which is the sum of the energy stored in the controlled object and the dissipated energy. Since the integral on the left side of equation (48) is the sum of the energy applied to the controlled object, it turns out that the result of integrating equation (48) is the energy conservation law.
Let E ₁ be the sum of the _first and third terms of equation (52). When the integral of the sum of the disturbance input power and E _2, and transposition of the disturbance input terms on the right-hand side in equation (48), by integrating the right and left sides,

Formula 53

となる．ここで，対象が工学的に意味のあるシステムであれば外乱入力パワーは有界であるか

後の項は散逸エネルギで非負であり時間とともに単調増加する．よって，次式が導かれる．

It becomes. Here, if the target is an engineeringly meaningful system, is the disturbance input power bounded?

The latter term is non-negative in dissipative energy and increases monotonically with time. Therefore, the following equation is derived.

Formula 54

また工学的に意味のある評価関数なら，有界な状態変数に対してｇは下限値を持つように選ばれているから，その有限評価区間の積分値も下限値ｃを持つ．
よって評価関数Ｊは次式を満たし最小値を持つ．

In the case of an engineeringly meaningful evaluation function, g is chosen to have a lower limit for bounded state variables, so the integral value of the finite evaluation interval also has a lower limit c.
Therefore, the evaluation function J satisfies the following formula and has a minimum value.

Formula 55

特願２００５−１７５６５６システムの最適制御方法 藤本健治，ハミルトニアンシステムの制御，計測と制御Ｖｏｌ．３９，Ｎｏ．２（２０００），ｐｐ．９９−１０４．

Japanese Patent Application No. 2005-175656 System Optimal Control Method Kenji Fujimoto, Control, Measurement and Control of Hamiltonian System Vol. 39, no. 2 (2000), pp. 99-104.

BEST MODE FOR CARRYING OUT THE INVENTION

  The best mode for carrying out the present invention will be described below with reference to the drawings.

Fig. 11 shows the structure of the active suspension. The actuator is composed of a hydraulic cylinder 3, a piston rod 4 and an accumulator 5 installed between the wheel 1 and the vehicle body 2. The sensors are composed of vertical G sensors 6 and 7, a stroke sensor 8 and a suspension transmission force sensor 9, and a control signal is generated by a control circuit 10 based on these signals.
The bidirectional pump 11 is rotated by an induction motor 12. When the induction motor 12 rotates to the right, hydraulic oil is sent from the pump 11 to the upper chamber of the cylinder 3, and when it rotates counterclockwise, hydraulic oil is sent to the lower chamber. The induction motor is driven by the three-phase output of the inverter 13, and this three-phase output is controlled by the control command value. A reversible chopper circuit 14 is provided between the battery or capacitor 15 and the inverter 13 to boost the battery voltage when the motor is driven, and to adjust the voltage from the motor when charging, so that regeneration is possible even when the motor voltage falls below the battery voltage.
The other wheels have the same configuration.

Next, the control circuit 10 will be described. FIG. 12 shows a block diagram of the sensor and the control circuit 10. Signals from the sprung vertical G sensor 6, the unsprung vertical G sensor 7, the suspension transmission force sensor 9, and the suspension displacement sensor 8 are transmitted to the control circuit 10. Sprung speed

Formula 56

じてモータの電流を変えれば、制御指令値に応じて力を発生させる制御系が実現できる．回生

ｕ_ｏｐｔとからモータ駆動時か回生時かを判断し、モータ駆動時にはバッテリー電圧を昇圧し，回生時にはモータ電圧を昇圧する指令値ｕ_ｅを出力する．The control law calculation unit 20 includes a motor control unit and a regeneration control unit. The motor control unit

If the motor current is changed, a control system that generates force according to the control command value can be realized. Regeneration

From u _opt , it is judged whether the motor is driven or regenerated, and the battery voltage is boosted when the motor is driven, and the command value u _e is boosted when the motor is regenerated.

  This method can be applied to non-linear control systems including nonholonomic constraints, and the optimal control law can be obtained from the energy balance equation of the system without solving the differential equation. .

  It is a one-wheel model diagram of a vehicle active suspension.   It is an explanatory diagram of optimization of parameter κ.   It is the performance comparison figure of the control of this invention and the conventional control method about an active suspension.   It is a comparison figure of the control input power of this invention and patent document 1. FIG.   This is a model of a coin that rolls.   It is an explanatory diagram of optimization of parameter κ.   It is a figure explaining the control effect of the present invention.   It is a figure explaining the control effect of the present invention.   It is a figure explaining the control effect of a prior art.   It is a comparison figure of the robustness of this invention and conventional control.   It is a block diagram of the active suspension of this invention.   It is a block diagram of the active suspension sensor and control circuit of the present invention.

Explanation of symbols

1 Wheel (Unsprung mass)
2 Body (Spring mass)
3 Hydraulic cylinder 4 Piston rod 5 Accumulator 6 Vertical G sensor (for spring top)
7 Vertical G sensor (Unsprung)
DESCRIPTION OF SYMBOLS 8 Suspension displacement sensor 9 Suspension transmission force sensor 10 Control circuit 11 Bidirectional pump 12 Induction motor 13 Inverter 14 Variable chopper circuit 15 Battery or capacitor 20 Control law calculating part 21 Sprung speed calculating part 22 Unsprung speed calculating part 23 Tire displacement calculation Part

Claims (3)

In a control system where the balance equation of the total input / output power to be controlled is approximated by

(Where d, e, q, u, v, z ∈ R ⁿ and M ∈ R ^{n × n} are positive definite inertia matrices, n is the degree of freedom of the controlled object, q is generalized coordinates, and u is Control input, v is a force input disturbance, z is a displacement input disturbance, z is acting on inertia via a spring, d is Coriolis force, centrifugal force, damping force, etc. e is potential force Since the controlled object is not limited to passive elements, it is treated as a disturbance input if there is an internal energy source that affects the motion.If the number of controlled actuators is less than the freedom n of the controlled object The element of the vector u corresponding to the missing actuator is set to zero, and when one actuator acts on a plurality of inertias, the condition is included in the element of the corresponding vector u. The symbol “ "And their number represents the time derivative and the number thereof.) Control method to minimize an evaluation function expressed by the following equation.

Here, g is a function representing the control performance of the controlled object, and r is a weighting coefficient. 2. The control system of claim 1, wherein the following scalar function L is defined:

(Where κ is a constant.)
A control method characterized in that the control law u includes a relational expression derived from the following expression.

The control method according to claim 1, wherein the control law u includes part or all of the following terms.

(Where κ is a constant.)

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