JP2013137628A - Model prediction control method and model prediction control program - Google Patents

Abstract

PROBLEM TO BE SOLVED: To provide a model prediction control method in which the stability of a closed loop is high and the calculation cost can be reduced.SOLUTION: A model prediction control method includes the steps of: linearly increasing, in a control section, a sampling period from a first sampling interval; making the first sampling interval and a control period in the sampling period equal to each other; and linearly increasing a weight according to the increase of a sampling interval in the sampling period.

Description

  The present invention relates to a model predictive control method and a model predictive control program, and in particular, calculates a future input value of a control target so as to satisfy a constraint condition (weight) at a predetermined sampling period, and sets the first input value as the current input value. The present invention relates to a model predictive control method and a model predictive control program that calculate a current input value intermittently by repeating the step of setting an input value at a predetermined control cycle.

  General model predictive control will be described below. However, in the present specification, in each formula, lowercase bold letters represent vectors, and uppercase bold letters represent line examples. However, normal characters that are not bold represent a scalar quantity.

The continuous system to be controlled is expressed by Equations 1 and 2, and the discrete system is expressed by Equations 3 and 4. Specifically, Equations 1 and 2 that are continuous systems are discretized with Δt (scalar amount), and a state variable x _k (vector), an input value u _k (scalar amount), and an output value w _k (scalar amount) are used. The results expressed by Equations 3 and 4 are discrete systems.

  Here, x (scalar amount) is a state variable, A (matrix) is a coefficient matrix, b (vector) is an input vector, c (vector) is an output vector, k (scalar amount) is a time step, and t (scalar amount). Represents a continuous time, and Δt (scalar amount) represents a sampling interval.

  In addition, due to hardware limitations and control target operating conditions, when the control condition is imposed on the input value or output value,

式５、６に表すように、線形不等式の形に定式化される。

As expressed in Equations 5 and 6, it is formulated in the form of a linear inequality.

Here, S ₁ (matrix) is an input constraint coefficient matrix, d ₁ (vector) is an input constraint coefficient vector, S ₂ (matrix) is an output constraint coefficient matrix, and d ₂ (vector) is an output constraint coefficient vector. Indicates.

However, u _k (vector) is a vector in which input values from the present to NΔt (scalar amount) future are arranged, and is expressed by the following Expression 7. Here, N represents a natural number.

Further, w _{k + 1} (vector) is a vector obtained by arranging output values up to NΔt (scalar amount) in the future, as shown in Equation 8 below.

  In the model predictive control, the input value is determined so as to minimize the evaluation function of the quadratic form expressed by Expression 9 from the present to the finite time (NΔt (scalar amount)).

However, the superscript ref represents the target value. In other words, w ^ref _{k + 1} means a time series up to NΔt (scalar amount) future of the target output as shown in Expression 10.

  Further, Q (matrix) and R (matrix) are diagonal matrices in which the weight values at each prediction point are arranged diagonally as weight matrices, as shown in equations 11 and 12.

The model predictive control obtains an input time series u _k (vector) that minimizes the evaluation function of Equation 9 within the range of the constraints of Equation 5 and Equation 6, and first value u _{k of} u _k (vector). This is a control method using (scalar amount) as the current input value. Then, after Δt, the optimum solution is similarly obtained, and the process of using the first value of the input value time series as the input value is repeated.

  That is, as shown in FIG. 12, while solving the future prediction period NΔt (scalar amount) for each control cycle, the optimization problem with the control condition (Equations 5, 6 and 9) is solved, and the input value at that time It is a control method that decides. Incidentally, the thick line portion in FIG. 12 is the first prediction point.

In the following, the optimization problem is guided according to the above formulation.
From Equations 3 and 4,

式１４で表される。 Since Equation 13 is derived, the output value time series is

It is expressed by Equation 14.

Therefore, Expression 9 is formulated as Expression 15. Here, H _k (matrix) indicates a semi-definite matrix.

Similarly, Expressions 5 and 6 can also be expressed by expressions relating to u _k (vector) as shown in Expression 16.

式１７に表すように、凸２次計画問題に定式化され、この凸２次計画問題の求解結果ｕ_ｋ（ベクトル）の第１番目要素が現在の入力値として実際に用いられる。 Finally, the optimal solution problem is

As expressed in Expression 17, the formulation is formulated into a convex quadratic programming problem, and the first element of the solution result u _k (vector) of the convex quadratic programming problem is actually used as the current input value.

The above is a control rule of general model predictive control.
In the above control law, the input value is continuously changed during the prediction interval. However, in the model predictive control, as shown in FIG. 13, the input value is changed within the prediction interval (control interval). ) And an interval that does not change the input value may be introduced. Here, M represents a natural number and satisfies N or less.

ｕ_ｋ（ベクトル）を

式１９と設定し直して改めて定式化を行うと、式３、４より、

式２０となる。 In general, prediction points outside the control interval within the prediction interval are set to the final value of the control interval. That is, Equation 18

u _k (vector)

When formula 19 is re-set with formula 19 again, from formulas 3 and 4,

Equation 20 is obtained.

式２１と表される。後は制御区間と予測区間が等しい場合のモデル予測制御と同様に式１５及び式１６のように定式化され、最終的に式１７のように最適化問題が導かれる。 Therefore, the output value time series is

It is expressed as Equation 21. After that, similarly to the model predictive control in the case where the control interval is equal to the prediction interval, it is formulated as shown in Equation 15 and Equation 16, and finally an optimization problem as shown in Equation 17 is derived.

  In this way, general model predictive control includes the following three features: predicting the future based on the model, setting the control conditions, and finding the optimal solution. is there.

  However, the biggest bottleneck is the high calculation cost. This is due to the fact that the model predictive control includes a solution step for an optimization problem (convex quadratic programming problem). Therefore, many are often used to control process systems with slow time constants.

Therefore, in order to reduce the calculation cost of model predictive control, for example, the technique of Patent Document 1 has been proposed.
The technique of Patent Literature 1 changes the sampling interval of the sampling period in the interval in which the future control amount (input value) is calculated. That is, in the model predictive control described above, the sampling interval in the prediction interval is constant, but the technique of Patent Document 1 shortens the sampling interval in the interval in which the control amount is varied, and the sampling interval in the interval in which the control amount is not varied. The sampling interval is switched within the interval for calculating the future control amount from the idea that the control accuracy does not drop so much even if the length is increased.

JP 2006-72791 A

  For example, in the control where the time constant is short and the quick response is required like the control of the robot, the control section of the model predictive control needs to be long. In general, the shorter the control interval, the higher the stability of the closed loop. Therefore, increasing the control interval means improving the quick response instead of decreasing the stability.

  In robot control, since the time constant is short, it is necessary to shorten the control cycle, and it is also necessary to shorten the calculation time of model predictive control. Therefore, as in the technique of Patent Document 1, in order to reduce the calculation cost, when changing the sampling interval in the interval in which the future control amount is calculated, if the method of changing the sampling interval is not appropriately set, the closed loop The stability of the robot is low, and the robot may vibrate or become unstable and diverge.

  However, Patent Document 1 does not disclose how to change the sampling interval, that is, what kind of sampling interval is used.

  An object of the present invention is to provide a model predictive control method and a model predictive control program that have been made to solve such a problem and that have high closed-loop stability and can reduce calculation costs.

  In the model predictive control method according to an aspect of the present invention, the step of linearly increasing the sampling period from the first sampling interval in the control section, and the step of equalizing the first sampling interval and the control period in the sampling period And a step of linearly increasing the weight as the sampling interval in the sampling period increases.

  A model predictive control program according to an aspect of the present invention causes a computer to perform a process of linearly increasing a sampling period from a first sampling interval in a control section, and a first sampling interval and a control period in the sampling period. A process of equalizing and a process of linearly increasing the weight as the sampling interval in the sampling period increases are executed.

  As described above, according to the present invention, it is possible to provide a model predictive control method and a model predictive control program that have high closed-loop stability and can reduce the calculation cost.

It is a figure for demonstrating the model prediction control method which concerns on Embodiment 1 of this invention. In the model predictive control method which concerns on Embodiment 1 of this invention, it is a figure which shows the relationship between a prediction point and a sampling interval at the time of increasing the sampling interval of a sampling period linearly. It is a figure for demonstrating the model prediction control method which concerns on Embodiment 2 of this invention. It is a figure which shows typically the gravity center model of the biped walking robot in the model prediction control method which concerns on the Example of this invention. It is a figure which shows the relationship between target ZMP and a ZMP restriction | limiting in the model predictive control method which concerns on the Example of this invention. It is a figure which shows the relationship between a ZMP trajectory and a gravity center trajectory in the model predictive control method which concerns on the Example of this invention. It is a figure which shows the gravity center speed | velocity in the model prediction control method which concerns on the Example of this invention. It is a figure which shows the gravity center acceleration in the model prediction control method based on the Example of this invention. In the model predictive control method according to the embodiment of the present invention, it is a diagram showing the relationship between the ZMP trajectory and the gravity center trajectory when a disturbance occurs in the biped robot. It is a figure which shows the gravity center speed when the disturbance arises in the biped walking robot in the model predictive control method which concerns on the Example of this invention. It is a figure which shows the gravity center acceleration when the disturbance arises in the bipedal walking robot in the model predictive control method which concerns on the Example of this invention. It is a figure for demonstrating general model prediction control. It is a figure for demonstrating general model prediction control from which a control area differs from a prediction area.

  The best mode for carrying out the present invention will be described below with reference to the accompanying drawings. However, the present invention is not limited to the following embodiment. In addition, for clarity of explanation, the following description and drawings are simplified as appropriate.

<Embodiment 1>
The model predictive control method and model predictive control program according to the present embodiment are preferably executed to control a control target having a short time constant such as a robot. The model predictive control method according to the present embodiment is realized when the control device to be controlled executes a model predictive control program. However, the model predictive control method may be realized using hardware resources. Incidentally, in the model predictive control method of the present embodiment, the control interval and the prediction interval are made equal.

First, an outline of the model predictive control method of the present embodiment will be described. As shown in FIGS. 1 and 2, the model predictive control method calculates an input value at each prediction point by linearly increasing the sampling period from the first sampling interval in the control interval. The input value at the predicted point is used as the current input value. Incidentally, the thick line portion in FIG. 1 is the first prediction point. Also, the vertical axis Δt in FIG. 2 indicates the sampling interval, the horizontal axis j indicates the number of prediction points, and the j-th sampling interval Δt _j is calculated as Δt ₁ + α (j−1). Here, Δt ₁ represents the first sampling interval, and α represents a coefficient.

  At this time, the first sampling interval and the control cycle are made equal so that the constraint condition at the first prediction point corresponds to the current state after the control cycle. In addition, as described above, the constraint condition (weight) is linearly increased as the sampling interval is increased.

  Thereby, although the detailed effect will be described later, the size of the optimization problem can be reduced, and the calculation cost can be greatly reduced even when a large control interval is taken.

Next, the model predictive control method of this embodiment will be described in detail. In a general model predictive control method, the elements of the weight matrices Q and R are not largely changed individually, and the values of the elements are generally set to substantially the same value. That is, it sets as shown in the following formulas 22 and 23.

式２４、２５のように線形に増加させる。ここで、β及びγは定数を示す。 However, when the sampling interval within the prediction time is changed as described above, if the weight of each element is made constant, the degree of influence of each sampling point (prediction point) is different, and thus cannot be controlled appropriately. This is because the continuous systems of Formulas 1 and 2 are different for each prediction point (A (matrix), b (vector) is a function of Δt (slacker amount)). As the sampling interval is larger, the amount of change of x _k (vector) → x _{k + 1} (vector) becomes larger even with the same input value u _k (scalar amount). Therefore, in this embodiment, each element of the weight matrix is

It is increased linearly as in equations 24 and 25. Here, β and γ are constants.

式２６、２７とし、対角要素の値が線形に増加するように設定する。 That is, the weight matrix

Equations 26 and 27 are set so that the value of the diagonal element increases linearly.

式２８、２９となる。従って、

式３０となるので、出力値時系列は、

式３１と表される。後は一般的なモデル予測制御と同様にして、式９は、

式３２のように定式化される。 Hereinafter, formulation is performed with the above configuration.
Assuming that the discrete system matrix corresponding to each sampling interval Δt _j (scalar amount) is A _j (matrix) and b _j (scalar amount), equations 3 and 4 are

Expressions 28 and 29 are obtained. Therefore,

Since Expression 30 is satisfied, the output value time series is

It is expressed as Formula 31. After that, similar to general model predictive control, Equation 9 is

Formulated as Equation 32.

として、ｕ_ｋ（ベクトル）に関する式３３で表すことができる。 The same applies to the constraints, and for Equations 5 and 6,

Can be expressed by Expression 33 relating to u _k (vector).

式３４に表すように、凸２次計画問題に定式化され、この凸２次計画問題の求解結果ｕ_ｋ（ベクトル）の第１番目要素を現在の入力値として実際に用いる。 Finally, the optimization problem is

As expressed in Expression 34, it is formulated into a convex quadratic programming problem, and the first element of the solution result u _k (vector) of this convex quadratic programming problem is actually used as the current input value.

  The above is the control law of model predictive control according to the present embodiment. In the model predictive control according to the present embodiment, the sampling interval is linearly increased in the control section as described above. About the effect, the present applicant estimates as follows.

  When the sampling interval of the prediction section (control section) is long with respect to the change speed of the target trajectory (ref), the peaks and valleys of the target trajectory are buried between the prediction points. As a result, as time progresses, peaks and valleys of the target trajectory that were not entered in the prediction of the previous time may appear at a later time, or may disappear. At this time, the input value for each control cycle becomes oscillating, and the state quantity also shows oscillating behavior around the target trajectory.

  On the other hand, in the model predictive control, the influence on the current input value is larger at a point near the present in the prediction interval. That is, even if the target value in the far future changes, the current input value does not change so much, but the change in the target value in the near future is handled by changing the current input value greatly.

  In model predictive control, a solution is obtained so as to satisfy a constraint condition at a prediction point in a control section, but it is not guaranteed that the constraint condition is satisfied between prediction points. When the sampling interval of the control section is changed, the current prediction point is often set between the prediction points at the previous solution, but there is no guarantee that the constraint condition is satisfied between the prediction points. There may be a large difference from the solution result (because the current solution solves the problem that the restrictions were not observed at the previous solution, the solution will be solved suddenly), and this time also causes vibration . When a pattern that rapidly changes the sampling interval of the control section is used, a solution that breaks the constraint between prediction points is often output, and vibration is likely to occur.

  From this technical knowledge, the number of points in the prediction interval can be stabilized without increasing the oscillation, that is, by stabilizing the closed loop, by increasing the sampling interval linearly from the present to the future of the prediction interval (control interval). It is possible to effectively reduce the calculation cost.

  Further, in the model predictive control according to the present embodiment, the weight is increased linearly, but the applicant estimates the effect as follows.

  In model predictive control, it is assumed that the input value is constant between prediction points in the control section. That is, in the period from the prediction point A to the next prediction point B, the formulation is performed on the assumption that the input at the prediction point A continues to act on the discrete system.

  Therefore, when the sampling interval is changed, the time during which the same input value continues to operate differs for each input, and optimization cannot be performed appropriately. Therefore, by increasing the weighting matrix linearly in accordance with the linear increase of the sampling interval, the input value at a point with a large sampling interval can be set to have a greater influence on the evaluation function. It becomes possible to optimize appropriately.

<Embodiment 2>
In the model predictive control method of the first embodiment, the control interval and the prediction interval are made equal. However, as shown in FIG. 3, the control interval and the prediction interval are made different as in the general model predictive control method. Also good. In FIG. 3, the bold line portion is the first prediction point.

式３５となるので、出力値時系列は、

式３６と表される。その後の導出は実施の形態１のモデル予測制御方法と同様であるので説明を省略する。これにより、より長い区間の予測区間を確保することができる。 In this case, if u _k (vector) is reset to Equation 19 and formulated again,

Since Expression 35 is obtained, the output value time series is

It is expressed as Expression 36. Subsequent derivation is the same as in the model predictive control method of the first embodiment, and a description thereof is omitted. Thereby, the prediction area of a longer area is securable.

  The embodiment of the model predictive control method and the model predictive control program according to the present invention has been described above. However, the present invention is not limited to the above, and can be changed without departing from the technical idea of the present invention.

式３７となる。ここで、ｘ_ｚｍｐは２足歩行ロボットのＺＭＰにおけるｘ座標、ｘ_ｇは当該重心位置のｘ座標、ｈは当該重心位置の高さ、ｇは重力加速度を示す。 <Example>
An embodiment in which the model predictive control method and the model predictive control program of the present invention are applied to the problem of generating the center of gravity trajectory of a biped robot will be described.
As shown in FIG. 4, when the height of the center of gravity is constant, the ZMP (Zero Moment Point) equation is well known,

Expression 37 is obtained. Here, x _zmp is the x coordinate in the ZMP of the biped robot, x _g is the x coordinate of the centroid position, h is the height of the centroid position, and g is the gravitational acceleration.

式３８、３９となる。これをΔｔ_ｊ（スカラー量）で離散化すると、

式４０、４１となる。 If the input value is u (t) (vector) = x ^(...) _g (scalar amount) and the output value is w (t) (scalar amount) = x _zmp (scalar amount), the continuous system is

Expressions 38 and 39 are obtained. When this is discretized by Δt _j (scalar amount),

Expressions 40 and 41 are obtained.

  As an example, the target ZMP trajectory and the ZMP constraint conditions (output value constraint conditions) are set as shown in FIG.

  In this embodiment, the prediction interval and the control interval are the same, 1.6 [s], the control cycle is 1 [ms], and the sampling cycle is set in the same manner as in FIG. 1 (α = 0.005).

Regarding the weights, Q (scalar amount) = 1, R (scalar amount) = 1 × 10 ⁻⁶ , β (scalar amount) = 0.005, and γ (scalar amount) = 5 × 10 ⁻⁹ .

  The results of this example are shown in FIGS. FIG. 6 is a diagram in which the ZMP trajectory and the gravity center trajectory are overlaid on FIG. From this figure, it can be seen that a smooth center of gravity trajectory can be generated while following the target ZMP trajectory. FIGS. 7 and 8 are graphs plotting the results of the center of gravity speed and the center of gravity acceleration. From these graphs, it can be seen that the trajectory can be generated without generating vibration or the like even in the dimension of the speed and acceleration.

  Next, in order to confirm that the constraint conditions are observed, a sudden external force was virtually input at 2.6 [s] under the same conditions as described above, and the speed was changed stepwise. The ZMP trajectory and the center of gravity trajectory at this time are shown in FIG. 9, and the center of gravity speed and acceleration are shown in FIG. 10 and FIG. As can be seen from FIG. 9, it can be confirmed that the ZMP returns to the target ZMP trajectory while satisfying the constraint conditions.

Claims (2)

In model predictive control method,
Increasing the sampling period linearly from the first sampling interval in the control interval;
Equalizing the first sampling interval and the control period in the sampling period;
Increasing the weight linearly with increasing sampling interval in the sampling period;
A model predictive control method comprising: In model predictive control program,
On the computer,
In the control period, a process of linearly increasing the sampling period from the first sampling interval;
Processing for equalizing the first sampling interval and the control period in the sampling period;
A process of linearly increasing the weight as the sampling interval increases in the sampling period;
Model predictive control program that executes

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