RU2132079C1 - Adaptive control system with variable structure - Google Patents

Abstract

FIELD: automation, in particular, equipment for controlling dynamic objects with single input and output. SUBSTANCE: device has input signal setter, reference model, forming filter, mismatch detector, gain generation unit, five units of relay elements, four units of multipliers, four scaling amplifiers with alternating gain, adder, filtering unit, controlled object. EFFECT: increased reliability, safety and service life of system due to decreased frequency and amplitude of drive oscillations by means of separation of signal, which is received by drive, into high-frequency and low-frequency constituents. 3 dwg, 2 tbl

Description

 The invention relates to the field of automation and can be applied to control dynamic objects with one input and one output.

 Closest to the invention in terms of features (prototype) is an adaptive control system with a variable structure [1], which contains a control unit, connected to the inputs of the reference model, the first block of relay elements and the signal input of the first scaling amplifier with a variable coefficient, a mismatch meter, the first input connected to the output of the reference model, the second input connected to the output of the control object, the input of the second block of relay elements and the signal input of the second the first amplifier with a variable coefficient, and the output with the inputs of the gain forming unit, the third block of relay elements and the control input of the third scaling amplifier with a variable coefficient, the fourth block of relay elements, the input of which is connected to the output of the gain forming unit, and the output to the first the inputs of the first to fourth blocks of multipliers, the second inputs of which are connected respectively to the outputs of the first second, third and fifth blocks of relay elements, the outputs of the first - h of the fourth block of multipliers are connected respectively to the control inputs of the first to fourth scaling amplifiers with a variable coefficient, the outputs of which are connected respectively to the first and fourth inputs of the adder, the output of the adder is connected to the input of the control object and a shaping filter connected to the input of the fifth block of relay elements and the signal input of the fourth variable gain amplifier.

 However, in this system, the drive (steering surfaces) oscillates with a sufficiently large amplitude and frequency, which cannot be considered satisfactory in terms of control costs. In addition, this can affect the safety (for example, spans), the preservation of the mechanical part of the drive.

 The purpose of the invention is to increase the reliability, safety and resource of the system by reducing the frequency and amplitude of oscillations of the drive by dividing the control signal supplied to the drive, low-frequency and high-frequency.

 This goal is achieved by introducing into the adaptive control system with a variable structure of the filtration unit, the first input connected to the output of the adder, the second input associated with the output of the mismatch meter, and the output with the input of the control object.

 The adaptive control system block diagram (FIG. 1) contains an input signal generator 1, a reference model 2, a filter 3, a mismatch meter 4, a gain generating unit 5, a first 6, a second 7, a third 8, a fourth 9 and a fifth 10 blocks relay elements, the first 11, second 12, third 13 and fourth 14 blocks of multipliers, the first 15, second 16, third 17 and fourth 18 scaling amplifiers with variable coefficient, adder 19, filtering unit 20 and control object 21.

основанный на использовании эталонной модели (ЭМ)

где x, x_м, u, u_з - векторы состояния ОУ (n • 1), ЭМ (n • 1), управления (k=1) и задающих воздействий (m • 1) соответственно;
A, A_м, B, B_м - матрицы соответствующих размерностей.One of the possible principles of constructing an adaptive control system for a linear object is known.

based on the use of a reference model (EM)

where x, x _m , u, u _z are the state vectors of the OS (n • 1), EM (n • 1), control (k = 1) and master actions (m • 1), respectively;
A, A _m , B, B _m - matrices of the corresponding dimensions.

The control is chosen so that the error e = x _m -x is reduced to zero. To solve this problem, we use the equation of motion with respect to error obtained from equations (1) and (2), i.e.

Для обеспечения устойчивого скользящего режима по пересечению плоскостей скольжения
S = C^тe, (4)
где C - матрица (n • k);
S - вектор (k • 1),
уравнение определяется из системы неравенств таким образом, чтобы условия существования устойчивого скользящего режима выполнялись во всем диапазоне изменения параметров объекта управления (ОУ).

To ensure a stable sliding mode at the intersection of the sliding planes
S = C ^t e, (4)
where C is the matrix (n • k);
S is a vector (k • 1),
the equation is determined from the system of inequalities in such a way that the conditions for the existence of a stable sliding mode are satisfied in the entire range of changes in the parameters of the control object (OS).

управляема; (5)
2) det (C^тB)≠0;
3) rank B=rank[B][B_м]=rank[B][D], (6)
то движение в скользящем режиме описывается уравнением

т. е. не зависит ни от коэффициентов матрицы A, ни от задающего воздействия u_з. Однако необходимость выполнения условий (6) является серьезным препятствием при создании подобных адаптивных систем. Кроме того, вектор x должен быть непосредственно наблюдаем, иначе необходимо дополнительно синтезировать устройство оценивания вектора состояния, что требует решения задачи текущей идентификации. Отмеченные трудности легко преодолеваются, если при синтезе использовать неминимальную форму (НМФ) модели ОУ.If the following conditions are met:
1) a pair of matrices

controllable; (5)
2) det (C ^t B) ≠ 0;
3) rank B = rank [B] [B _m ] = rank [B] [D], (6)
then sliding motion is described by the equation

i.e., it does not depend either on the coefficients of the matrix A or on the driving influence u _s . However, the need to fulfill conditions (6) is a serious obstacle in the creation of such adaptive systems. In addition, the vector x must be directly observable, otherwise it is necessary to further synthesize a state vector estimator, which requires solving the current identification problem. The aforementioned difficulties are easily overcome if a non-minimal form (NFM) of the OS model is used in the synthesis.

в которой нет сокращаемых нулей и полюсов.Let the dynamic properties of a linear stationary control object with one input and one output Z be described by the transfer function

in which there are no reducible zeros and poles.

где a_i, b_i - коэффициенты, полученные при делении полиномов A_(p) и B_(p) на полином β(p).
Введем новые переменные:

С учетом проведенных преобразований уравнения ОУ (1) в пространстве состояний запишутся так:

(10)
Порядок полученной модели равен (2n - 1) > n, поэтому она называется неминимальной.We divide the numerator and denominator of function (1) into a stable polynomial
β (p) = p ^n-1 + β _n-2 p ^n-2 + ... + β ₁ p + β _o
and write the result in the form

where a _i , b _i are the coefficients obtained by dividing the polynomials A _(p) and B _(p) by the polynomial β (p).
We introduce new variables:

Taking into account the transformations performed, the OA equations (1) in the state space are written as follows:

(ten)
The order of the resulting model is (2n - 1)> n, so it is called non-minimal.

где x^т=[Z Y_n-2 Y_n-3...Y_o];
x $\genfrac{}{}{0ex}{}{\text{т}}{\text{м}}$ = [Z_MY_Mn-2Y_Mn-3...Y_Mo],

f - вектор, полученный из решения уравнения

где

Введя для ЭМ (12) переменные в виде (9), получим ее передаточную функцию

Таким образом, форма (11) ЭМ позволяет задавать желаемую передаточную функцию, у которой порядок полинома числителя на единицу меньше порядка полинома знаменателя и все корни полинома числителя лежат слева от мнимой оси на комплексной плоскости. Последние условие связано с необходимостью обеспечения устойчивости модели, так как полином β(p), используемый при ее формировании, совпадает до постоянного множителя с числителем ЭМ.We write the OA equations (10) and EM in the following form:

where x ^t = [ZY _n-2 Y _n-3 ... Y _o ];
x $\genfrac{}{}{0ex}{}{\text{t}}{\text{m}}$ = [Z _M Y _Mn-2 Y _Mn-3 ... Y _Mo ],

f is the vector obtained from the solution of the equation

Where

Introducing for EM (12) the variables in the form (9), we obtain its transfer function

Thus, the EM form (11) allows you to specify the desired transfer function for which the order of the numerator polynomial is one less than the order of the denominator polynomial and all the roots of the numerator polynomial lie to the left of the imaginary axis on the complex plane. The latter condition is associated with the need to ensure the stability of the model, since the polynomial β (p) used in its formation coincides to a constant factor with the numerator EM.

Нетрудно убедиться, что при выбранной структуре модели ОУ условия (6) выполняются

Если в уравнении (4) принять
C^т=[1 C_n-2 C_n-3 ... C_o], (15)
то после преобразования уравнения движения в скользящем режиме (7) принимают вид

Таким образом, движение в скользящем режиме не зависит ни от параметров ОУ (11): коэффициентов матриц A, B, D, ни от задающего воздействия u_з, а целиком определяется заданием параметров плоскости скольжения и числителя ЭМ.To obtain the equation of motion with respect to the error, we subtract equation (11) from equation (12) and, after the transformation, we obtain

It is easy to verify that, with the selected structure of the OS model, conditions (6) are satisfied

If in equation (4) take
C ^t = [1 C _n-2 C _n-3 ... C _o ], (15)
then after the transformation of the equation of motion in the sliding mode (7) take the form

Thus, the motion in the sliding mode does not depend on the parameters of the OS (11): the coefficients of the matrices A, B, D, nor on the driving influence u _s , but is entirely determined by setting the parameters of the slip plane and the numerator of the EM.

Закон управления будем искать в виде
u = Ψ $\genfrac{}{}{0ex}{}{\text{т}}{\text{e}}$ e-Ψ $\genfrac{}{}{0ex}{}{\text{т}}{\text{x}}$ x-Ψ $\genfrac{}{}{0ex}{}{\text{т}}{\text{u}}$ u_з-Ψ $\genfrac{}{}{0ex}{}{\text{т}}{\text{f}}$ f, (17)
где элементы векторов Ψ_e, Ψ_x, Ψ_f и скаляр Ψ_u определяются по формулам

i = 1, 2, ..., n;
j = 1, 2, ..., (n-1).To ensure the stability of the slip mode, the equation should be chosen so that the condition

The law of control will be sought in the form
u = Ψ $\genfrac{}{}{0ex}{}{\text{t}}{\text{e}}$ e-Ψ $\genfrac{}{}{0ex}{}{\text{t}}{\text{x}}$ x-Ψ $\genfrac{}{}{0ex}{}{\text{t}}{\text{u}}$ u _s -Ψ $\genfrac{}{}{0ex}{}{\text{t}}{\text{f}}$ f, (17)
where the elements of the vectors Ψ _e , Ψ _x , Ψ _f and the scalar Ψ _{u are} determined by the formulas

i = 1, 2, ..., n;
j = 1, 2, ..., (n-1).

где A_i, A_mi, D_i - столбцы матриц A, A_m, D;
e_i, x_i, f_i - элементы векторов e, x, f соответственно.We find the product SS taking into account equations (4), (14), (17)

where A _i , A _mi , D _i are the columns of the matrices A, A _m , D;
e _i , x _i , f _i are the elements of the vectors e, x, f, respectively.

(18)
В процессе функционирования ОУ изменяются коэффициенты матриц A, B, D. Однако границы их изменения известны и параметры закона управления (17) можно выбрать так, чтобы неравенства (18) соблюдались для всех режимов работы. В этом случае замкнутая система управления (11), (17) в скользящем режиме будет адаптивной, т.е. инвариантной к изменению характеристик ОУ и задающего воздействия.The condition of a stable sliding mode SS <0 will be satisfied if the coefficients of the control law (17) are chosen from the conditions:

(18)
In the process of the OS operation, the coefficients of the matrices A, B, D change. However, the boundaries of their changes are known and the parameters of the control law (17) can be chosen so that inequalities (18) are observed for all operating modes. In this case, the closed-loop control system (11), (17) in the sliding mode will be adaptive, i.e. invariant to a change in the characteristics of the op-amp and the master action.

С учетом формул (4), (14), и (15) запишем

Определим из этого уравнения
U_экв=(C^тB)^-1C^т[Df - A_мe + (A_м-A)x + B_мu_з]
и, подставив его в уравнение (13), получим

или после преобразований с учетом формул (11), (13) и (15)

где

Так как в соответствии с (16) e ---> 0, ЭМ (12) выбирается устойчивой и значение u_з ограничено по величине, то и значения компонент вектора x ---> x_м также ограничены. В этом случае значения компонент вектора f ограничены только тогда, когда матрица R - гурвицева. Это значит, что корни числителя передаточной функции (8) ОУ должны располагаться слева от мнимой оси комплексной плоскости.It is also necessary to evaluate the nature of the change in the vector f in equation (13) under control (17), since for f ---> ∞ the control _Ueq , determined from the equation

In view of formulas (4), (14), and (15), we write

We define from this equation
U _equiv = (C ^t B) ^-1 C ^t [Df - A _m e + (A _m -A) x + B _m u _s ]
and substituting it into equation (13), we obtain

or after transformations taking into account formulas (11), (13) and (15)

Where

Since, in accordance with (16), e ---> 0, EM (12) is chosen stable and the value of u _{z is} limited in magnitude, so the values of the components of the vector x ---> x _{m are} also limited. In this case, the values of the components of the vector f are bounded only when the matrix R is Hurwitz. This means that the roots of the numerator of the transfer function (8) of the OS must be located to the left of the imaginary axis of the complex plane.

где α, α_м - угол атаки;
ω_z, ω_zм - угловая скорость тангажа самолета и ЭМ;
δ_в - отклонение руля высоты самолета;
u_з - задающее воздействие.Consider the effectiveness of the proposed methodology as an example of the synthesis of an adaptive control system of a hypothetical aircraft with a reference model in the longitudinal plane. The equations of short-period motion of the aircraft and EM in this case have the form

where α, α _m - angle of attack;
ω _z , ω _zm - the angular velocity of the pitch of the aircraft and EM;
δ _in - deviation of the elevator of the aircraft;
u _s - the setting action.

 The coefficients of equations (19) for two flight modes are presented in table. 1.

3) привод руля высоты будем описывать дефференциальным уравнением δ_в= -1/Tδ_в+1/TU, где T - постоянная времени привода, U - управляющий сигнал.In studying the effectiveness of an adaptive system, we will assume that
1) δ _ad = sint + 1 / 3sin3t + 1 / 5sin5t + 1 / 7sin7t;
2) as the goal of the synthesis we consider adaptive the fulfillment of the condition

3) the elevator drive will be described by the differential equation δ _in = -1 / Tδ _in + 1 / TU, where T is the drive time constant, U is the control signal.

На этапе синтеза будем полагать привод руля высоты безинерционным: δ_в= U. Полином β(p) принимаем равным числителю W_м(p) β(p) = p+1, а вектор C^е=[1 0]. При сделанных допущениях уравнения (11)-(13) принимают вид

V₀=-V₀+U;

Коэффициенты уравнений (21) для рассматриваемых режимов приведены а табл.2.From equations (19), (20) we determine the transfer functions (PF) of the aircraft and EM

At the synthesis stage, we assume that the elevator drive is inertialess: δ _in = U. We take the polynomial β (p) equal to the numerator W _m (p) β (p) = p + 1, and the vector C ^e = [1 0]. Under the assumptions made, equations (11) - (13) take the form

V ₀ = -V ₀ + U;

The coefficients of equations (21) for the considered modes are given in Table 2.

2) для режима N2:

Выбираем коэффициенты закона управления (17), удовлетворяющие этим неравенствам:

При включении синтезированной системы процессы ω_z (t) и ω_zм (t) для режима N1 и режима N2 практически не отличаются, что и было показано в прототипе. Однако в предложенной системе рулевые поверхности совершают колебания с достаточно большой амплитудой и частотой, что является неудовлетворительным с точки зрения энергетических затрат на управление. Кроме этого, это может сказаться на безопасности полетов, на сохранении механической части привода. Результаты моделирования синтезированной системы для режима N1 при различных значениях постоянной времени привода представлены на фиг.2.Inequalities (18) are written as follows:
for mode N1:

2) for N2 mode:

We choose the coefficients of the control law (17) that satisfy these inequalities:

When you turn on the synthesized system, the processes ω _z (t) and ω _zм (t) for mode N1 and mode N2 are practically the same, which was shown in the prototype. However, in the proposed system, the steering surfaces oscillate with a sufficiently large amplitude and frequency, which is unsatisfactory in terms of energy control costs. In addition, this can affect flight safety, the preservation of the mechanical part of the drive. The results of modeling the synthesized system for mode N1 at various values of the drive time constant are presented in FIG. 2.

 To smooth these oscillations, it is possible to separate the control signal supplied to the drive into low-frequency and high-frequency. A two-channel control circuit is shown in FIG. The time constant T = 0.01 and T = 2 is determined based on the conditions for maintaining the slip mode. The high-frequency channel is switched on at e ≤ 0.02.

 As a technical implementation of two-channel control, filters in the form of quadripoles can be proposed [2]. The high-frequency channel can be switched on using a comparator [3], at one of the inputs of which a reference voltage corresponding to a value of 0.02 is supplied, and a mismatch signal e, respectively, at the second input. At the moment of equality of these values, an output signal appears at the output of the comparator, including, for example, a relay that connects a high-frequency channel with its contacts.

 Adaptive control system operates as follows.

с выхода объекта 21 управления. На выходе измерителя 4 формируется сигнал рассогласования

который поступает на первый вход блока 15, вход блока 7, блока 5 и первый вход блока 20.The preset action u _s from the output of the master 1 goes to the input of the reference model 2, the output of which x _m is connected to the first input of the mismatch meter 4. A signal is input to the second input of the mismatch meter 4

from the output of the control object 21. At the output of the meter 4, a mismatch signal is generated

which goes to the first input of block 15, the input of block 7, block 5 and the first input of block 20.

путем изменения коэффициента передачи канала управления по величине и знаку согласно следующему закону:

где

- сигнал рассогласования измерителя 5;
S - сигнал переключения,
α_e и γ_e - переменные коэффициенты усилителя 15, формирующего сигнал управления.The amplifier 15 generates a control signal

by changing the transmission coefficient of the control channel in magnitude and sign in accordance with the following law:

Where

- mismatch signal meter 5;
S is the switching signal,
α _e and γ _e are variable coefficients of the amplifier 15, forming a control signal.

и сигнала переключения S определяются с помощью блоков 7 и 6. Сравнение знаков (т.е. определения знака логического произведения

сигнала рассогласования e и сигнала переключения S осуществляется в множителе 11, выходной сигнал которого управляет работой усилителя 15.Mismatch Signs

and the switching signal S are determined using blocks 7 and 6. Comparison of signs (i.e., determining the sign of the logical product

the mismatch signal e and the switching signal S is carried out in the multiplier 11, the output signal of which controls the operation of the amplifier 15.

по следующему закону:

где u_з - задающее воздействие задатчика 1;
S - сигнал переключения;
αu_з и γu_з - переменные коэффициенты усилителя 16.The preset action u _s from the output of the setter 1 also goes to the first input of the amplifier 16 and the input of the unit 8. The amplifier 16 generates a control signal

according to the following law:

where u _s - the setting action of the setter 1;
S is the switching signal;
αu _z and γu _z are variable coefficients of the amplifier 16.

The sign of u _{z is} determined using block 8. Comparison of the signs of the driving action u _z and the switching signal S is carried out in the multiplier 12, the output signal of which controls the operation of the amplifier 16.

поступает также на первый вход усилителя 17 и вход блока 9. Усилитель 17 формирует сигнал управления

по следующему закону:

где S - сигнал переключения;,
α_x и γ_x - переменные коэффициенты усилителя 17.Output signal

also arrives at the first input of the amplifier 17 and the input of block 9. The amplifier 17 generates a control signal

according to the following law:

where S is the switching signal;
α _x and γ _x are variable coefficients of the amplifier 17.

The sign of u _{z is} determined using block 9. Comparison of the signs of the driving action u _z and the switching signal S is carried out in the factor 13, the output signal of which controls the operation of the amplifier 17.

поступает на первый вход усилителя 18 и вход блока 10. Усилитель 18 формирует сигнал управления

по следующему закону:

где f - выходной сигнал формирующего фильтра 3;
S - сигнал переключения,
α_f и γ_f - переменные коэффициенты усилителя 18.And finally, the control signal U from the output of the adder 19 is fed to the second input of the filtering unit and, having passed through the forming filter 3, which has a transfer function for a second-order object

arrives at the first input of the amplifier 18 and the input of block 10. The amplifier 18 generates a control signal

according to the following law:

where f is the output signal of the forming filter 3;
S is the switching signal,
α _f and γ _f are variable coefficients of the amplifier 18.

 The sign of the output signal of the forming filter 3 is determined using block 10. Comparison of the signs of the output signal of switching S is carried out in the multiplier 14, the output signal of which controls the operation of the amplifier 18.

Literature
1. USSR author's certificate 1659980 A1, 06/30/91.

 2. Makarov I.M. Mensky B.M. Linear automatic systems.-M.: Mechanical Engineering, 1977, p. 441-452.

 3. Nesterenko B.K. Integrated Operational Amplifiers.-M.: Energoizdat, 1982, p. 76-82.

Claims (1)

An adaptive control system with a variable structure, comprising a controller connected to the inputs of the reference model, the third block of relay elements and the signal input of the second scaling amplifier with a variable coefficient, a mismatch meter, the first input connected to the output of the reference model, the second input connected to the output of the control object, the input of the fourth block of relay elements and the signal input of the third scaling amplifier with a variable coefficient, and the output with the inputs of the formation the gain, the second block of relay elements and the signal input of the first scaling amplifier with a variable coefficient, the first block of relay elements, the input of which is connected to the output of the block for generating gain factors, and the output to the first inputs of the first and fourth blocks of multipliers, the second inputs of which are connected respectively to the outputs of the second to fifth blocks of relay elements, the outputs of the first to fourth blocks of multipliers are connected respectively to the control inputs of the first to fourth scales uyuschih variable gain amplifiers whose outputs are respectively connected to first - fourth inputs of the adder, the adder output is connected
with the input of the forming filter, the output of which is connected to the input of the fifth block of relay elements and the signal input of the fourth scaling amplifier with a variable coefficient, characterized in that the filtering unit is additionally introduced, the first input connected to the output of the adder, the second input to the output of the mismatch meter, and the output - with the input of the control object.

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