RU2453890C1 - Method of automatic control in nonlinear system and servo system to this end - Google Patents

Abstract

FIELD: process engineering.

SUBSTANCE: invention relates to automatic control systems to be used in robotics, servo systems, etc. Proposed servo system comprises power drive, drive coordinate transducers, metre of task error signal and signal of signal main feedback, and nonlinear devices of feedback signals for every coordinate of the drive. The latter devices are made up of adder with two branches connected in parallel. One branch represents serial connection of nonlinear link with limitation zone and first amplifier while another branch is made up of second amplifier. Output of adder of every said nonlinear device is connected with input of said nonlinear link with limitation zone. Besides, servo system comprises metres of errors in second and higher coordinates of the drive with their negative inputs connected with transducers and their positive inputs connected with outputs of feedback nonlinear device outputs of previous drive coordinate.

EFFECT: faster operation.

2 cl, 6 dwg

Description

The invention relates to the field of automatic control and regulation systems, in particular to a technique for generating control signals, and may find application in robotics, servo systems, and automatic regulators.

A known method of automatic control, implemented in a non-linear tracking system for a second-order object (AS No. 292139, MKI6 G05B 11/01 from 01/06/71). This method consists in summing a signal proportional to the error with a linear combination of two signals of nonlinear feedbacks in speed, one of which has a limiting zone, and the other a dead zone.

The disadvantage of this method is the low tracking accuracy and low speed, since the maximum values of the hard feedback coefficients are limited by the condition of stability of the system and at low speeds and small mismatches, the amplified maximum error signal is insufficient for the quality of the tracking system - it linearly depends on the linear combination of the reference signal and signal output coordinates of the control object and with a decrease in the mismatch also decreases in proportion to the mismatch. In addition, in higher-order systems, the transition process is oscillatory, since nonlinear feedbacks are introduced only by speed.

A known method of automatic control (RF Patent No. 2149437, IPC7 G05B 11/01 of 05/20/2000), selected as the closest analogue to the proposed method, which consists in applying feedback signals to the input of the power drive for the first, second and higher coordinates of the power drive, each of which is equal to a linear combination of two signals, one of which has a limit on the amplitude of the corresponding coordinate, and the second is proportional to this coordinate.

Despite the fact that this control method is more successful than the previous one, however, it has the following disadvantages. Firstly, the total signal of parallel operating non-linear feedback devices for each coordinate of the drive is not limited in magnitude, which contradicts the well-known fact about the relay (limited) nature of the optimal speed control. Secondly, the parallel connection of feedback devices does not allow to increase the speed of the drive, taking into account the presence of restrictions on the coordinates of the drive.

Known non-linear tracking system (AS No. 292139, MKI6 G05B 11/01 from 06.01.71), which contains a mismatch signal reference signal and a single feedback signal, amplifiers, a power drive mounted on the Executive shaft tachogenerator, the output of which is connected to the input of two parallel-connected branches, one of which is made in the form of a serial connection of a nonlinear link with a dead zone and an amplifier, and the other - in a series connection of a non-linear link with a restriction zone and an amplifier, and the total The output of these branches is connected to the input of the power drive.

However, this device does not provide the required accuracy and speed of the nonlinear servo system, since at low speeds and small mismatches, the amplified maximum error signal is not sufficient for the quality of the servo system due to the linear dependence on the linear combination of the reference signal and the output coordinate signal of the control object. In addition, this solution is characterized by a certain difficulty in implementing nonlinear feedback.

Known tracking system (RF Patent No. 2149437, IPC7 in G05В 11/01 of 05/20/2000), which is selected as the closest analogue to the claimed device. This servo system contains a power drive, a tachogenerator, drive coordinate sensors, a mismatch signal of the reference signal and a single main feedback signal, and non-linear feedback devices for each drive coordinate made using an adder in the form of two parallel connected branches, one of which is implemented in in the form of a serial connection of a nonlinear link with the restriction zone and the first amplifier, and the other in the form of another amplifier.

This device cannot provide performance close to the limit, for the following reasons. Firstly, the total signal of parallel operating non-linear feedback devices for each coordinate of the drive is not limited in magnitude, although it is known that the control signal in the drive with optimal speed does not exceed the maximum supply voltage. Secondly, the parallel connection of feedback devices does not allow to increase the speed of the drive, taking into account the presence of restrictions on the coordinates of the drive.

The technical task of the present invention is to increase the speed of the tracking system with the existing restrictions on the coordinates of the drive.

The technical problem in the invention is solved in that:

- in the proposed automatic control method in a non-linear system, which consists in applying feedback signals along the first, second and higher coordinates of the power drive generated by non-linear feedback devices, each of which is equal to a linear combination of two signals, one of which has a limit in amplitude of the corresponding coordinate and the second one is proportional to this coordinate, the feedback signal for each coordinate of the drive is limited at the required level by the introduction of an additional link with a zone og boundaries, wherein feedback signals along the second and higher coordinates of the power drive are formed by subtracting the sensor signal of this coordinate from the output signal of the previous described nonlinear feedback device and feeding the resulting difference to the input of the subsequent feedback device;

- in the proposed servo system containing a power drive, encoder coordinates of the drive, a mismatch signal reference signal and a signal of a single main feedback and non-linear feedback devices for each coordinate of the drive, made using the adder in the form of two parallel connected branches, one of which is implemented in in the form of a serial connection of a nonlinear link with the restriction zone and the first amplifier, and the other in the form of another amplifier, the output of the adder of each nonlinear device coupling is connected to the input of the nonlinear link introduced into it with a restriction zone, while the mismatch meters for the second and higher coordinates of the drive are additionally introduced into the servo system, the negative inputs of which are connected to the sensors, and their positive inputs - with the outputs of the nonlinear feedback device of the previous coordinate drive.

The invention is illustrated by drawings, where figure 1 presents a structural diagram of a nonlinear servo system that implements a method of automatic control in a nonlinear system; figure 2, figure 3 - graphs of the control functions of the power drive of the first order with parameter values q ₀ = 0.04 and q ₀ = 0.16, respectively (solid line - optimal control, dashed line - quasi-optimal control); figure 4 - structure of the considered control object of the second order; _{figure 5} , _figure 6 shows the simulation results of the proposed tracking system, with x _2max = 0.4 and x _2max = 0.5, respectively, where the coordinates x ₁ (t) is a solid line, x ₂ (t) is a line of points and a control signal u (t) is the dashed line.

The following notation is used in the description:

x ₀ is the signal of the setting action;

t is the time;

x ₁ (t), x ₂ (t), ..., x _n (t) - coordinates of the power drive;

u (t) is the control signal of the power drive;

K _f , K _x - feedback coefficients;

h is the width of the linearity zone of the link with the restricted zone;

k ₁ , k ₂ - gain of the control object;

T ₁ , T ₂ - time constants of the control object.

The tracking system contains a power drive 1, sensors 2 coordinates of the drive, a mismatch meter 3 reference signal x ₀ and a signal of a single main feedback x ₁ and non-linear feedback devices 4 for each coordinate of the drive, made using the adder 5 in the form of two parallel connected branches, one of which is implemented as a serial connection of a nonlinear link with a restriction zone 6 and the first amplifier 7, and the other is in the form of another amplifier 8, and the output of the adder 5 of each nonlinear device The communication link is connected to the input of an additional nonlinear link introduced into it with a restriction zone of 9. A misalignment meter 10, 11, 12 for the second and higher coordinates of the drive, respectively, the negative inputs of which are connected to sensors 2, and their positive inputs - with the outputs of the nonlinear feedback device of the previous coordinate of the drive.

It can be shown that the proposed control system is approximately optimal by the criterion of the form

, r - заданные весовые коэффициенты функционала качества. Данный критерий отличается от используемого в прототипе функционала одним слагаемым q₀, определяющим быстродействие проектируемой системы.where q ₀ q _i ,

, r are the given weights of the quality functional. This criterion differs from the functional used in the prototype by one term q ₀ , which determines the speed of the designed system.

It is easiest to show optimality using an example of a first-order object control system with parameters a and b

,

,

and quality criteria

This example also justifies the feasibility of using the proposed nonlinear error feedback device (the main output coordinate of the drive) to increase the speed of the system.

The control problem (2), (3) with q ₀ = q ₂ = 0 and the absence of restrictions on the control signal is precisely analytically solved in [Petrov Yu.P. Variational methods of optimal control theory. - L.: Energy, 1977. - 280 p.] Based on the application of the classical calculus of variations. Applying the method for solving this work to the problem for q ₀ ≠ q ₂ ≠ 0, we can find a control of the form

, то в соответствии с результатами работы [Дроздов В.П., Мирошник И.В., Скорубский В.И. Системы автоматического управления с микроЭВМ. - Л.: Машиностроение, 1989. - 286 с.] находим оптимальное управление видаConsidering the presence of a constraint in the problem to be solved

, then in accordance with the results of the work [Drozdov V.P., Miroshnik I.V., Skorubsky V.I. Automatic control systems with microcomputers. - L .: Engineering, 1989. - 286 p.] We find the optimal control of the form

Where

standard function in control theory.

This feedback law is characterized by relative complexity in implementation, and the specified method for its determination cannot be applied to high-order objects. To simplify the implementation of nonlinear control (5), we use the approximation of function (4) proposed in the description of the prototype at q ₀ = 0:

uh (x) = K _x · x + K _f · s a th (x);

with the corresponding values of the coefficients K _x , K _f and h. The sath (x) function describes the functioning algorithm of the corresponding link with a restriction zone, whose operation for small values of the parameter h → 0 is equivalent to a relay element.

Refining the values of the coefficients K _x , K _f specified in the description of the prototype, taking into account the presence of the weight coefficient q ₀ ≠ 0, we obtain

Thus, the quasi-optimal control law for an object (2) by criterion (3), adapted for implementation on elements of an analog technique, is described by the expression

with parameters (8). We emphasize that function (9) describes the functioning algorithm of non-linear feedback devices used in the proposed control method, implemented using two amplifiers and two links with a restriction zone.

The extremely simple control function (7) (dashed line), as shown in the graphs of Fig. 2, is a relatively good approximation of strictly optimal control (4) (solid line). These graphs are constructed for the following values of the object parameters a = 0, b = 1, h = 0.01 and the quality functional q ₁ = q ₂ = r = 1. They also indicate that for large values of the weight coefficient q _{0, the} accuracy of approximation of approximation (7) increases. Modeling systems with quasi-optimal control algorithms (7) and (9) showed that they have values of the quality functional quite close to the optimal value, and it is important that the performance of the optimal and quasi-optimal systems is almost the same. At the same time, we emphasize that the feedback law (9) allows a much simpler technical implementation in comparison with optimal control (5). These reasons predetermined the recommendation of the quasi-optimal control law (9) for practical application, in particular, in feedback on the main, output coordinate of the object.

Now we justify the structure of the high-order control system, starting with the system for the second-order object described, for example, by the transfer function W (p) = k / (T ₁ p + 1) (T ₂ p + 1). Suppose that this object is described by the structure of figure 3 and, accordingly, differential equations

According to figure 3 we will consider the signal x ₂ (t) as a control signal for an object with a transfer function W ₁ (p) = k ₁ / (T ₁ p + 1).

Applying the result (9) to this object, we find the intermediate control

where the function F _{1 is} described by expression (9), in which the parameters correspond to the transfer function W ₁ (p) and U _m = x _2max .

We must form the signal (11) by changing the real control u (t) by the first-order subobject (10 b). For this purpose, we will control it according to criterion (3), trying to obtain a signal at the output of the subobject (10 b) equal to the value (11), i.e. signal (11) will be considered as a reference signal

for the variable x ₂ (t). If the signal (12) changes rather slowly compared to x ₁ (t) (which is approximately possible if T ₁ >> T ₂ ), then the solution of the last control problem is described by the equation

where the function F ₂ is determined by expression (9), in which the parameters correspond to the transfer function W ₂ (p) = k ₂ / (T ₂ p + 1).

The feedback law (13) is fully consistent with the structure of the proposed control system of figure 1 for the second-order object. The generalization of control (13) to high-order objects does not cause fundamental difficulties if the structure of the object can be represented by a serial connection of first-order links. Such objects quality management provides a system with the structure of figure 1.

The work of the tracking system, taking into account the above description, which implements the automatic control method in a nonlinear system, is as follows. The mismatch meter 3 generates a signal that is input to two parallel connected branches, one of which is implemented as a serial connection of a nonlinear link with the restriction zone 6 and the first amplifier 7, and the other as the other amplifier 8. In this case, with large mismatches drive signals, a nonlinear link with a restriction zone of 6 eliminates the influence of a channel with a large gain of the first amplifier 7, and at low speeds, when the signal along the second coordinate becomes less than the level of the og When the nonlinear link with the restriction zone 6 is switched off, deep feedback arises due to the large gain of the first amplifier 7, which contributes to the relatively accurate and quick development of the input input action by the system. The sum of the signals of both parallel-connected branches is fed to the input of an additional link with a restriction zone of 9, which prevents the output coordinate of the nonlinear feedback device 4 from a predetermined level U _max . By setting in nonlinear feedback devices 4, for each coordinate of the drive, the values U _max = x _imax , i = 1, 2, ..., n provide the required limit restrictions on the phase coordinates x _i (t) of the object. Indeed, since nonlinear feedback devices 4 for each coordinate of the drive are connected in series, the achievement by the coordinate x _i (t) of the object of the value x _{imax is} perceived by the subsequent (i + 1) non-linear feedback device as a task of this coordinate, configured for stable, high-quality testing of this assignment. The set constraints of the coordinates x _imax , i = 1, 2, ..., n will be satisfied for all operating modes of the tracking system, if the aperiodic transients of the system are provided with the appropriate choice of the coefficients K _x , K _{f of} feedbacks.

по критерию быстродействия (q₀=1, а остальные весовые коэффициенты функционала (1) равны нулю) с законом обратной связиFigure 4 presents the simulation results of the described quasi-optimal object control system W ₁ (p) = 1 / p ² ,

by the performance criterion (q ₀ = 1, and the remaining weight coefficients of functional (1) are equal to zero) with the feedback law

a limitation zone, while in the servo system mismatch meters for the second and higher coordinates of the drive are added, the negative inputs of which are connected to the sensors, and their positive inputs - with the outputs of the nonlinear feedback device of the previous coordinate of the drive.

Claims (2)

1. A method of automatic control in a non-linear system, which consists in applying feedback signals along the first, second and higher coordinates of the power drive generated by non-linear feedback devices, each of which is equal to a linear combination of two signals, one of which has a limit on the amplitude of the corresponding coordinate, and the second is proportional to this coordinate, characterized in that the feedback signal for each coordinate of the drive is limited at the required level by the introduction of an additional link from the zones restrictions, the feedback signals at the second and higher coordinates of the power drive are formed by subtracting from the output signal of the previous described non-linear feedback device the sensor signal of this coordinate and submit the resulting difference to the input of the subsequent feedback device. 2. A tracking system containing a power drive, drive coordinate sensors, a mismatch signal of the reference signal and the unit main feedback signal, and non-linear feedback devices for each drive coordinate made using an adder in the form of two parallel connected branches, one of which is implemented as serial connection of a nonlinear link with the restriction zone and the first amplifier, and the other in the form of another amplifier, characterized in that the output of the adder of each nonlinear device and the feedback is connected to the input of the nonlinear link introduced into it with a restriction zone, while the mismatch meters for the second and higher coordinates of the drive are additionally introduced into the servo system, the negative inputs of which are connected to the sensors of the coordinates of the drive, and their positive inputs are connected to the outputs of the nonlinear feedback device communication of the previous coordinate of the drive.

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