JPH11282506A - Controller using discrete time approximate model - Google Patents

Abstract

PROBLEM TO BE SOLVED: To dispense with a complicated analysis and control of the factors that cause the deterioration of controllability of a controller and to attain simplification and high performance of the controller by deleting the perturbation terms from a difference equation and producing a discrete time approximate model. SOLUTION: In the drivers 3 the motors 10 and 11 are interlocking by connected to the base ends of ball screws 6 and 8 via the gear boxes 12 and 13 to which the rotary encoders 14 and 15 are connected respectively. In a controller 4 a D/A converter 17 and a counter 18 are connected to a computer 16. The motion equation of this drive system is expressed as a state equation which includes the perturbation terms showing disturbance, uncertainty, etc. In a discretization mode, the trapezoidal approximation is performed on the integration terms included in the state equation and an approximate state equation is produced. Then the approximate state equation is expressed as a difference equation, and the perturbation terms are deleted from the difference equation for production of a discrete time approximate model. Thus, a model including no perturbation terms showing disturbance, uncertainty, etc., can be produced.

Description

DETAILED DESCRIPTION OF THE INVENTION

[0001]

BACKGROUND OF THE INVENTION 1. Field of the Invention The present invention relates to a control device using a discrete time approximation model.

[0002]

2. Description of the Related Art Conventionally, in a control device such as a positioning mechanism of a machine tool, control performance is deteriorated due to a change in load condition or an influence of frictional resistance. Certain static friction and Coulomb friction are caused by various factors, and their values fluctuate depending on operating conditions, so it has been difficult to perform accurate analysis, identification and compensation thereof. .

For this reason, conventionally, it has been necessary to perform control using a complicated method.
Nonlinear compensation input to be added to the control input based on various linear control methods is obtained in advance from experiments and experiences, and it is assumed that they are known (B. Armstrong-Helouvry, Control of
Machines with Friction, 125 (1991)) and a method for estimating nonlinear compensation input during the control process (C. Canudas de Wit, Int.
J. Robotic Research, 10-3, 189 (1991)) has been proposed.

[0004]

However, there is a method in which a nonlinear compensation input to be added to a control input based on the above-mentioned various conventional linear control techniques is determined in advance by experiments and experiences, and is assumed to be known. Requires a considerable amount of trial and error to obtain prior information, and in the conventional method of estimating the nonlinear compensation input during the control process, the identification of parameters including even the nonlinear part is complicated. Since it is based on a control algorithm, a considerable amount of calculation is required, a large-scale control device is required, and it takes a long time to execute the algorithm, and it is difficult to perform quick control.

[0005]

Therefore, in the present invention, the equation of motion is represented by a state equation including a perturbation term representing disturbance, uncertainty, and the like, and the state equation is converted into the same state equation upon discretization. A trapezoidal approximation of the integral term included, expressed by a difference equation, a perturbation term is eliminated from the difference equation to create a discrete-time approximation model, and the control is performed using the discrete-time approximation model. did.

In a computer, a state equation expressing means for expressing a motion equation by a state equation including a perturbation term representing disturbance, uncertainty, and the like, and a state equation expressed by the same state equation expressing means are used for discretization. Trapezoidal approximation means for obtaining an approximate state equation by trapezoidal approximation of an integral term included in the same state equation, difference equation expression means for expressing an approximate state equation obtained by the trapezoidal approximation with a difference equation,
A discrete time approximation model is created by executing a perturbation term elimination means for eliminating the perturbation term from the difference equation expressed by the same difference equation expression means, and control is performed using the discrete time approximation model.

[0007]

DESCRIPTION OF THE PREFERRED EMBODIMENTS A control device according to the present invention expresses a motion equation as a state equation including a perturbation term representing disturbance, uncertainty, etc., and converts the state equation into the same state equation at the time of discretization. A trapezoidal approximation of the integral term included, expressed by a difference equation, a perturbation term is eliminated from the difference equation to create a discrete-time approximation model, and the control is performed using the discrete-time approximation model. is there.

Therefore, it is possible to create a model that does not include a perturbation term representing disturbance, uncertainty, and the like. By configuring the control device using the discrete-time approximation model, it is possible to perform a complicated analysis on the control performance deteriorating factor and No adjustment is needed,
The control device can be simplified and improved in performance.

[0009]

Embodiments of the present invention will be described below with reference to the drawings.

(State Equation Expressing Means) First, a description will be given of state equation expressing means for expressing a motion equation by a state equation including a perturbation term representing disturbance, uncertainty and the like.

The equation of motion of a mechanical system is generally expressed by the following equation of state,

(Equation 1) Assuming that the output equation measures only θ (t),

(Equation 2) It is said. Here, A ₁ , A ₂ , and B are matrices of appropriate dimensions corresponding to x (t) and u (t). I is a similar unit matrix.

Next, the target system adds a perturbation term including disturbance, uncertainty, or a linear approximation error to the equation (1).

(Equation 3) Becomes Here, d (t) is the added perturbation term. Now
Equation (3) is expressed by the following equation.

(Equation 4) Further, equation (4) is

(Equation 5) And express.

(Trapezoidal approximation means) Next, a trapezoidal approximation means for obtaining a state equation obtained by trapezoidally approximating an integral term included in the state equation at the time of discretization with respect to the state equation expressed by the state equation expression means will be described. I do. Note that the trapezoidally approximated state equation is called an approximate state equation.

If the input u (t) is set to a constant value u (0) in the sampling interval T with respect to the equation (5), the following equation is obtained.

(Equation 6) Here, from the definitions of A _C and B _C ,

(Equation 7) Therefore, equation (6) becomes the following equation.

(Equation 8) Since the integral term on the right side of the above equation cannot be calculated as it is,
Performing trapezoidal approximation gives

(Equation 9) Using this approximation, equation (7) becomes

(Equation 10)

(Difference Equation Expressing Means) Next, a description will be given of a difference equation expressing means for expressing the approximate state equation obtained by the trapezoidal approximation means as a difference equation.

By developing equation (9), the following difference equation is obtained.

[Equation 11]

(Perturbation Term Erasing Means) Next, perturbation term erasing means for eliminating the perturbation terms from the difference equation expressed by the difference equation expressing means will be described.

By eliminating Δ (k) from equations (10) and (11), the following equation is obtained.

(Equation 12) Equation (12) is the discrete-time approximation model of interest here.

The discrete-time approximation model created as described above does not depend on A ₁ and A ₂ and has a simple form in which d (t) is eliminated. This model is not affected by fluctuations in dynamic characteristics or disturbances, and is very useful in control system design.

That is, the equation of motion is represented by a state equation including a perturbation term representing disturbance, uncertainty, etc., and an approximation state equation is created by trapezoidal approximation of the integral term included in the state equation during discretization. By expressing the approximation state equation as a difference equation, and eliminating the perturbation term from the difference equation to create a discrete-time approximation model, creating a model that does not include a perturbation term representing disturbance, uncertainty, etc. Comes out.

Further, as described below, by configuring the control device using the discrete-time approximation model, complicated analysis and adjustment regarding the cause of control performance degradation become unnecessary, and the control device can be simplified and improved in performance. Can be achieved.

(Configuration and Verification of Adaptive Control System Based on Discrete-Time Approximate Model) Next, an adaptive control system based on the discrete-time model of Expression (12) is configured.

Equation (12) is

(Equation 13) Can be expressed as However,

[Equation 14] It is.

Now,

(Equation 15) Then, equation (13) becomes the following equation.

(Equation 16)

Adaptive parameter for unknown parameter α in equation (15)

[Equation 17] And the estimated value of ∂u (k) is

(Equation 18) And

The input error is

[Equation 19] From Equations (15) and (16), Equation (17) is

(Equation 20) Becomes

Since equation (18) is a general form of the error equation,
By various proposed adaptation algorithms, k → ∞
Η (k) → 0 and

(Equation 21) Is guaranteed.

Next, when the control input u (k) is synthesized using the target value θd (k),

(Equation 22) Becomes However,

(Equation 23) It is. Λ (k) is a feedback vector, which will be determined later based on stability criteria.

From equations (16) and (19), equation (18) is

(Equation 24) Becomes Here, for k → ∞, η (k) → 0 and

(Equation 25) Equation (15), Equation (19), and Equation (20) give:

(Equation 26) Holds.

At this time, based on Lyapunov's stability theory,
If (k) is given by the following equation,

[Equation 27] Then, θd (k) −θ (k) → 0, that is, θ (k) → θd
(k) is guaranteed.

[Equation 28] However,

(Equation 29) It is.

Next, an adaptive control system based on the discrete time model constructed as described above will be verified.

FIG. 1 is a view showing the configuration of an experimental apparatus 1. The experimental apparatus 1 includes an XY-axis moving table 2, a driving mechanism 3 for driving the XY-axis moving table 2, and a driving mechanism. And a control device 4 for controlling the control device 3.

In the XY-axis moving table 2, a ball screw 6 for moving in the X-axis direction is rotatably mounted on the base 5, and an X-axis moving table 7 is screwed to the ball screw 6 for moving in the X-axis direction. A ball screw 8 for moving in the Y-axis direction orthogonal to the ball screw 6 for moving in the X-axis direction is provided on the X-axis moving table 7.
Is mounted so as to be rotatable, and a ball screw 8 for moving the same in the Y-axis direction.
The table 9 is screwed. The ball screws 6 and 8 are
It is 300mm long and 5mm pitch.

The table 9 is moved in the X-axis direction and the Y-axis direction by rotating the ball screw 6 for moving in the X-axis direction and the ball screw 8 for moving in the Y-axis direction.

The drive mechanism 3 has motors 10 and 11 interlockedly connected to the base ends of both ball screws 6 and 8 via gear boxes 12 and 13.
4,15 are connected.

The control device 4 connects the D / A converter 17 and the counter 18 to the computer 16 and controls the D / A converter 17.
The motors 10, 11 are connected to the counters 18 via servo circuits 19, 20, while the rotary encoders 14, 15 are connected to the counter 18, so that the table 9 can be rotated based on the rotation angles of the ball screws 6, 8 detected by the rotary encoders 14, 15. , The ball screws 6, 8 are rotated by the motors 10, 11 to displace the table 9.

The feed drive system of the experimental apparatus 1 shown in FIG. 1 is modeled as shown in FIG. Although the feed drive system in the Y-axis direction is shown, the feed drive system in the X-axis direction can be similarly modeled. In the figure, 21 is a gear attached to the ball screw 8 and 22 is a gear attached to the motor 11.

The equation of motion of this system, excluding nonlinear friction, is

[Equation 30] Can be expressed as Here, Jm is the moment of inertia of the motor 11, Dm is the coefficient of viscous friction of the motor 11, θm is the rotation angle of the motor 11, u is the generated torque of the motor 11, θb is the rotation angle of the gear 21 attached to the ball screw 8, N Is the reduction ratio, Mt is the mass of the table 9, Dt is the coefficient of viscous friction on the sliding surface of the table 9, Ks is the axial rigidity of the drive system, xt is the displacement of the table 9, and P is the pitch of the ball screw 8.

From equations (23) to (26), the output xt is obtained from the input u (t).
The transfer function between (t) is

(Equation 31) Becomes

Generally, since Jmm0 and Dm ≒ 0, the equation (2)
7)

(Equation 32) Can be approximated.

Therefore, the equation of state in this case is:

[Equation 33] Becomes

FIG. 3 shows a friction model. fs is the static friction acting in the opposite direction to the input when stationary, and fc is Coulomb friction acting in the counter motion direction when the speed is not zero. still,
Viscous friction is included in equation (29).

FIG. 4 shows the results obtained by experiments on the relationship between the motor generated torque and the Y-axis table feed speed.

From FIG. 4, it can be confirmed that the static friction acting on the feed drive system changes depending on the load condition.

When the torque generated by the motor is smaller than the static friction force, the table 9 is stationary. However, when the input torque is larger than the static friction, the speed rapidly increases. This is due to the influence of the nonlinear characteristic between the static friction and the Coulomb friction shown in FIG.

FIG. 5 shows the results of an adaptive control experiment of the Y-axis feed drive system. Note that the sampling period and the adaptive gain initial value are respectively T = 0.01 (s),

(Equation 34) , Ρ was set to 0.9, and the adaptive algorithm used was a least squares algorithm.

From FIG. 5, it can be seen that some errors occur at the beginning of the adaptation process, but with the progress of the adaptation process, good tracking performance can be obtained regardless of the presence or absence of the load. The validity of the rule can be confirmed.

As described above, by creating a discrete-time approximation model, a model that does not include a perturbation term representing disturbance, uncertainty, and the like can be created, a very simple adaptive control system can be designed, and an adaptive algorithm and the like can be designed. Can be shortened.

(Configuration of Another Adaptive Control System Based on Discrete Time Approximation Model) Next, the configuration of another adaptive control system based on the discrete time approximation model will be described.

Assuming that the target trajectory is θd (k) according to equation (12),
Substituting θd (k + 1) into θ (k + 1) on the left side of equation (12), the control law

(Equation 35) Get. here

[Equation 36] It is.

However, since there is no feedback term in the control law of equation (30), introduction of feedback is considered.

In the equation (12),

(37) Then, equation (12) becomes

(38) And the pulse transfer function G (z) from τ (k) to θ (k)
Is

[Equation 39] Becomes

Therefore, as shown in FIG. 6, the compensator F (z)
In other words, it is only necessary to configure a feedback control system in which is introduced.

Next, for simplicity of description, a control system is created with the target being a scalar system. The same applies to multivariable systems.

The compensator F (z) is

(Equation 40) Then, the pulse transfer function from θd (k) to θ (k) is

[Equation 41] And the characteristic equation is

(Equation 42) And solving this pole placement problem yields f ₁ and f ₂
Can be determined.

Next,

[Equation 43] Than,

[Equation 44] Becomes

Therefore, according to equation (31),

[Equation 45] And as an input composition rule,

[Equation 46] Get.

If the speed deviation remains by the above method, a compensator to which an integrator is added

[Equation 47] F _i (i = 1,2,3) is also determined by the pole assignment method using

[Equation 48] The input may be synthesized by

[0059]

The present invention is embodied in the form described above and has the following effects.

That is, the equation of motion is represented by a state equation including a perturbation term representing disturbance, uncertainty, and the like, and the integral term included in the state equation is discriminated by trapezoidal approximation during discretization to create an approximate state equation. By expressing the approximation state equation as a difference equation, and eliminating the perturbation term from the difference equation to create a discrete-time approximation model, creating a model that does not include a perturbation term representing disturbance, uncertainty, etc. However, by configuring the control device using the same discrete-time approximation model, complicated analysis and adjustment regarding the control performance reduction factor become unnecessary, and the control device can be simplified and improved in performance.

[Brief description of the drawings]

FIG. 1 is an explanatory diagram showing a configuration of an experimental apparatus.

FIG. 2 is an explanatory diagram showing a configuration in which a feed drive system is modeled.

FIG. 3 is an explanatory diagram showing a friction model.

FIG. 4 is an explanatory diagram showing a relationship between a motor generated torque and a table feed speed of a Y axis.

FIG. 5 is an explanatory diagram showing an experimental result.

FIG. 6 is an explanatory diagram showing a feedback control system.

[Explanation of symbols]

 DESCRIPTION OF SYMBOLS 1 Experimental apparatus 2 XY-axis moving table 3 Drive mechanism 4 Controller 5 Base 6,8 Ball screw 7 X-axis moving table 9 Table 10,11 Motor 12,13 Gear box 14,15 Rotary encoder 16 Computer 17 D / A converter 18 Counter 19,20 Servo circuit 21,22 Gear

Claims (2)

[Claims] An equation of motion is represented by a state equation including a perturbation term representing disturbance, uncertainty, etc., and the state equation is trapezoidally approximated to an integral term included in the state equation during discretization. Using a discrete-time approximation model characterized by being expressed by a difference equation, creating a discrete-time approximation model by erasing the perturbation term from the difference equation, and controlling using the same discrete-time approximation model Control device. 2. A computer comprising: a state equation expressing means for expressing an equation of motion as a state equation including a perturbation term representing disturbance, uncertainty, etc .; and a discretization of the state equation expressed by the state equation expressing means. Trapezoidal approximation means for obtaining an approximate state equation by trapezoidal approximation of an integral term included in the state equation at the time; difference equation expression means for expressing an approximate state equation obtained by the trapezoidal approximation means by a difference equation; A perturbation term erasing means for erasing the perturbation term from the difference equation expressed by the difference equation expressing means; anda program for creating a discrete-time approximation model by executing A computer-readable recording medium that has been recorded.