SU1659980A1 - Adaptive control system with a variable structure - Google Patents

Abstract

The invention relates to automation and can be applied in controlling dynamic objects with one input and one output. The aim of the invention is to expand the stability and controllability of the system. The device contains an input signal generator 1, a reference model 2, forming a filter 3. error meter 4, block 5 generating gains, blocks of 6-10 relay elements, blocks 11 "14 multipliers, scaling amplifiers 15-18 with a variable coefficient adder 19 and control object 20 with corresponding and bonds yl 1

Description

f is the vector obtained from solving the equation

f Ff + Gu,

Where

- | Chg - (V-j.-.- Po Oh ... Oh

oh oh ... u

e

- Computer input (12) variables in the form (9), we obtain its transfer function “, -ZM (p) (p);

 t ai

m

Ui (.1) fb (pHoiMrn.2pn - .., vo (Mo

Thus, the EM form (12) allows one to set any desired transfer function, where the order of the polynomial of the numerator is one less than the order of the polynomial of the denominator and all the roots of the polynomial of the numerator 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 / 3 (p) used in its formation coincides with the accuracy of a constant multiplier with the numerator EM.

To obtain the equation of motion for the error, we subtract from the equation (12) the equation (11) and, after transformations, we obtain

Ј Ame + (Am-A) x + Vmiz-Df - Wi, (14)

(A „+ & () e + () x +

(-D + BtjiJ) f «|: se; (cTAM; +

npVe; b. (CTAM: -CrA; + kpVx.

+ | 151; (- StS; hKrf ±; + 5i3 (); where, AM | O, are the columns of the matrices A, Am, D, ei, XL fi are the elements of the vectors e, x, f, respectively.

The condition of stability of the sliding mode SS 0 will be satisfied if the coefficients of the control law (17) are chosen from the conditions:

i Kpode -cTA ,; , kpye; -CTA ;;

Kp6iy; -CT (AM; -A;),

KRUh; -st (Am; -A; b

Cro-, CTD; , ГКр (;

(18)

where e1e2 ... en} 2m-d: ump-2-yn-2:,.,; umo 40

-y0 Kpyi; CD ;, Kr ui - CRM.

It is easy to see that with the chosen structure of the OS model, the conditions (6) are

rankB (Wm Am - A.

-gagsch. DJ- 1.45 s of the governing law (17) can be chosen

If, in equation (4), to accept inequalities (18)

With 1 Cn-2Cn-3 ... CoJ. (15)

then, after transformations, the equations of motion in the sliding mode (7) take the form

In the process of functioning of the OS change; The coefficients of the matrices A, B, D. However, the boundaries of their changes are known and the parameter of all modes of operation. In this case, the closed control system (11), (17) in the sliding mode will be adaptive, i.e.

and -ЧЈе-ЧЈх-ЧЈиз-Ч т-, (17) where the elements of the vectors%, three, fy, and sc 20 l pa% are determined by the formula

m

sepQ,,

Vxi

OSU; with

25Gx: 1 at, fotu at,

 with sfjtO, with,

i 1,2,3n; j 1,2, 3, ..., (n-1).

Let us find the product SS taking into account equations (4). (14) and (17):

(A „+ & () e + () x +

(-D + BtjiJ) f «|: se; (cTAM; +

npVe; b. (CTAM: -CrA; + kpVx.

+ | 151; (- StS; hKrf ±; + 5i3 (); where, AM | O, are the columns of the matrices A, Am, D, ei, XL fi are the elements of the vectors e, x, f, respectively.

The condition of stability of the sliding mode SS 0 will be satisfied if the coefficients of the control law (17) are chosen from the conditions:

i Kpode -cTA ,; , kpye; -CTA ;;

Kp6iy; -CT (AM; -A;),

KRUh; -st (Am; -A; b.

Cro-, CTD; , ГКр (;

(18)

 The control laws (17) can be selected.

In the process of functioning of the OS cheating; The coefficients of the matrices A, B, D. However, the boundaries of their changes are known and the parameter of all modes of operation. In this case, the closed control system (11), (17) in the sliding mode will be adaptive, i.e.

invariant to change the characteristics of the OS and set the effect

It is also necessary to evaluate the nature of the change in the vector in equation (13) with control (17), since with f co control (17) is physically unrealizable. To do this, we use the equivalent control Ueq determined from equation S 0. Taking into account formulas (4) , (14) and (15) we write S St (Ame + Amkh - Ah + Down - Df - Wieq 0. Determining from this equation

iEq (StW) -1St Ot-Aie + (Am-A) x + Vmiz and substituting it into equation (13), we get

f F - C (STVGST0 1 + G (CTB) + + (AM - A) x + Vmiz

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

f Rf + i (AM - A) x + Vmiz,

Where

R

-p ..

fin

about

Since, in accordance with (16) e 0, the EM (12) is chosen to be stable and the value of is limited by the value of TC and the values of the vector components 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 on the complex plane,

Adaptive control system works as follows.

The driver action from the output of the sensor 1 is fed to the input of the reference model 2, the output of which is connected with the first input of the error meter 4. The second input of the error meter 4 receives the signal x from the output of the control object 20. At the output of the meter A, a mismatch signal e is generated. This signal goes to the first block 17, the input of block 8 and block 5,

The amplifier 17 generates a control signal Ui (fa e by changing the transmission coefficient of the control channel ps value and sign according to the following law

0

five

0

five

0

five

0

five

The signs of the error signal and the switching signal S are determined using blocks 8 and 9. The comparison of the signs (i.e., determination of the sign of the logical product Se) of the error signal and the switching signal S is performed in multiplier 13, the output of which controls the operation of the amplifier 17

The driving force from the output of the sensor 1 is also fed to the first input of amplifier 15 and the input of unit 6. Amplifier 15 forms a control signal from Vu3 from according to the following law

 at SU ,, О ,, 43 (JU ,, at 6Us 0,

where from - specifies the impact setting unit 1;

S is the switching signal;

and from, from from - variable coefficients of the amplifier 15.

The sign of is determined by block 6. A comparison of the signs of the driving signal from the switch signal S by H is made in multiplier 11, the output of which controls the operation of the amplifier 15.

The output signal x is also fed to the first input of the amplifier 16 and the input of the block 7. The amplifier 16 generates a control signal according to the following law

G K.H when

f i) -

 } f x AT 5X; where x is the output of control object 20;

S is the switching signal;

ah, uh - variable factors of the amplifier 16.

The sign of the output signal x is determined using block 7 Comparison of the signs of the output signal x and the switch signal S is carried out in multiplier 12, the output of which controls the operation of the amplifier 16.

Finally, the control signal from the output of the adder 19 passes through the shaping filter 3, which has a transfer function for the second order object

50

Mf (r)

one

P +

%

ae with Se and O

y e at Se 0, where e-error signal is meter 4;

S is the switching signal;

uQ and are the variable coefficients of the amplifier 17, which forms the control signal.

arrives at the first input of the amplifier 18 and the input of the unit 10. The amplifier 18 generates a control signal uf V according to the following law,

at

eЈ Ј0

si co

where f is the output signal of the forming filter 3;

S is the switching signal; af, yf - variable factors of the amplifier 18.

The sign of the output signal of the shaping filter 3 is determined by block 10. The comparison of the signs of the output signal and the switch signal S is carried out in a multiplier 14, the output of which controls the operation of the amplifier 18.

The use of the non-minimal form of the mathematical model of the control object allows one to fulfill the conditions (1). If in the slip plane equation St 100 is taken ... 0, then the equations of motion in the sliding mode taking into account the non-minimal form will take the form:

1Gl ei

ea -pVa -rp-z ... -fio 62

ez10 ... oz

. enj L 00 ... 1 0 en

those. The motion in the sliding mode with respect to the error is stable and does not depend on the parameters of the mathematical model of the control object, nor on the specifying influence, but is entirely determined by the choice of the polynomial / J (p) in a non-minimal form.

Claims (1)

An adaptive variable structure control system comprising a driver, an output connected to the inputs of the reference model, a first block of relay elements and a signal input of the first scaling amplifier with a variable factor, an error meter, a first input connected to the output of the reference model, the second input connected to the output control object, the input of the second block of relay elements and the signal input of the second scaling amplifier with a variable coefficient, and the output - with the input The dam of the gain factor shaping 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, whose input is connected to the output of the gain shaping unit, and the output to the first inputs of the first, second and the third multiplier blocks, the second inputs of which are connected respectively to the outputs of the first, second and third blocks of relay elements, the conclusions of the first, second and The third multiplier units are associated respectively with the control inputs of the first, second and third scaling amplifiers with a variable coefficient, the outputs of which are connected respectively to the first, second and third inputs of the adder. in order to expand the scope of the system by expanding areas of stability and controllability of the system; it additionally contains a shaping filter, a fifth block of relay elements, a fourth block of multipliers, and a fourth scaling amplifier with variable coefficient, moreover, the output of the adder is connected via a shaping filter to the input of the fifth block of relay elements and the signal input of the fourth scaling amplifier with variable factor, the output of the fifth block of relay elements is connected to the first input of the fourth multiplier block, the second input of which is connected to the output of the fourth block of relay elements, and the output to the control input of the fourth scaling amplifier with a variable coefficient. the output of which is connected to the fourth input of the adder