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SYNTHESIS CONTROL AND ITS VARIETIES SLIDING GIVEN DIMENSION FOR LINEAR OBJECTS UNDER PERTURBATIONS AND INCOMPLETE INFORMATION
Meshchanov A.S., Gataullina L.A.
KNRTU-KAI, Kazan Associate Professor, PhD; Research Associate
Abstract
It is proposed multi-level vector control burst, resulting in a system with constant action uncertain and nominal disturbances and at part of the information about the state of a given order sliding mode (with dimension less than the original system to a multiple-dimensional vector control up to sliding in a straight line). The order sliding increases with the level of control. Given order sliding and the desired quality of transients is provided at the upper control level.
Keywords: linear non-stationary object, limited and uncertain nominal disturbances, incomplete information about the state, multi-level burst vector control, sliding mode of a given order, the invariance to the disturbances.
1. Statement of the problem. We consider a control system with a straight-line non-stationary object with the nominal and uncertain disturbances, combined with the identifier of the state (the model system of the subject):
z = (A (t) + AA(t)) z + (B0 (t) + AB(t)) u + (D0 (t) + AD(tW(t0) + AF (t)), x = K (t) z, (1) zM = A0(t)zM + B0(t)u + Kz (t)GT (t)(x — K (t )zM )+D0(t)F0(t), (2)
where z e Rn, zm e Rn; t e I = (t0, tk ], tk A(t), B0(t), D0(t), K(t) — nominal n X n, n X m , n X l, q X n—matrix, AA(t), AB(t), AD(t) H AF(t) — l x 1 - column with uncertain bounded parametric and external disturbances; column F0 (t) has a nominal variable elements, X - known (measured or calculated) of the output vector. In addition to this information on the state vector z(t) and the nominal parameters of the object in the formation of vector control u = (ux,... ,um )T is proposed to apply the state vector zM (t) , similar conditions are not subjected to the action of uncertain perturbations. In the initial system (1) satisfies certain conditions of invariance of the sliding mode to undefined AA(t), AD(t), AF(t) , and the nominal
D0(t) and F0(t) disturbances
AA(t) = Bo (t )Aaa (t) , AD(t) = Bo (t) Aad (t) , Do (t) = Bo (t )A(t) (3)
and usually do not take into account the parametric perturbation
AB(t) = Bo(t)Aab(t) . (4)
In the model system (2) is assumed that the term KAz (t)GT (t)(x — K(t)zM) should provide an appropriate setting n X m and q X m — matrices KM and G sufficiently rapid decrease in the deviation Az(t) vector of the state z(t) of the source system to the desired motion zM (t) model system topics:
z(t) = zm (t) + Az(t) ^ zm (t) when t
Note that the model (2) for (1) (but not indefinite AA(t), AB(t), AD(t), AF(t), and nominal D0 (t)F0 (t) it at constant disturbances and matrices A0, B0, Ku, G, K) is a term without D0 (t)F0 (t) a known identifier asymptotic state of the system (1) is given, for example, in the monograph [1].
Is assumed, that (1) has a regular shape with respect to the matrix B0 (t) and, with the environment (2), (3),
D0 (t), AB(t) and AD(t) the matrices, i.e. data matrix B0, D0, AB, AD, have a null first (n — m) row or the
system (1) is reduced to this form of the method of [2].
Task 1. Find switchable mobile (n — m ) - dimensional manifold of the form
S(s = (Si,...,Sm )T = C(t)za = 0\ (5)
where C(t) - m X n — the matrix of coefficients of the variables, st = Ct (t)zM — functions switching,
(t) = (ca(t)...,cin(t)) — i's rows of the matrix C(t), i = 1,m, as well as find (n — 2m), (n — 3m), ..., (n — km) - dimensional manifold of their intersections. 2. Find a vector multi-level control U, reducing the system (1) - (4) in the sliding mode of a given order k = 1,2,3,..., that is, with a given dimension of the system of differential equations of the sliding mode (n — 1 • m),(n — 2m),(n — 3m), ... for (n — km) > 1 and with it the quality of a given transition processes.
2. Synthesis of varieties of mobile sliding on the set quality parameters transients. As follows from [3],
with expansions of vector z and matrix A0(t), C(t), B0(t) and K^(t) ,K(t) on the sub-vector and submatrices, the sliding mode on the manifold S (5) in the original system (1) can be written as a system with the dimension of the state vector to be (2n — m):
z1 =
U)11 -Boi(cBo) lclAmi -B01(CB0) lC24,21 -b01(cb0) lcl +
+ (- A)i2 + B)i(CB) )-1C^A)i2 + B)i(CB) )-1C2 A)22 + B)i(CB) )-1 C2 )x x(c 2 )-V jz1 + {-[Aoii - B)i(CB) )1C^Aoii - B)i(CB) )C ^ -- Boi(CB) )-1C1 + (-A012 + Boi(CB) )-1C^Aoi2 + Boi(CB) )-1C ^ + +Boi(CB) yc 2)(C 2 )r1C1 \+(En_m - Boi(CB) )-1c1 ]KAzIGtK 1 -- Boi(CB) yC2KA;2GTK +(A)11 - KaziGtK 1 )}az1 + + {(En-m -Boi(CBo)-1C1 KaziGtK2 -Boi(CBo)-1C2KAZ2GtK2 +
+ (Aoi2 - KAZ GTK2 )}Az2,
Az1 = (411 - )Az> +(Aoi2 - KziGTK )Az2,
Az2 = (Am - Kaz 2GTK> )Az1 + (a22 - K^ 2GtK2 )AZ2, z2 =-(c 2 )-1 CV + (c2 ^Az1 + Az2
where
z =
V z 2 y
■ Ao = (Ao 5 Ao ) =
Ao,11 Ao,12 Ao,21 Ao,
, A1 =
22 y
J
V Ao,2iy
, A,2 =
'A ^
),12
Ao.
V ^22 y
rB \ Boi
C = (C1,C2), Bo = z1, Ao1, An,C1,Bo!,K
V Bo2 y
5 Ka2G KAZ =
^Kaz,GtK 'AZ1 + KAZ1GtK 2AZ
iTr
V KAZ 2GTK 'AZ1 + KAZ 2^TK 2AZ 2 ,
(6)
Azi'
K1 each have dimensions (n — m) x1, n X X (n — m), (n — m) x (n — m), m x (n — m), (n — m) x m, (n — m) x m, q x (n — m), and En_m is the identity matrix (n—m) x (n—m).
In the synthesis of the matrix K (t) in a second independent subsystem deviations Az(t) = z(t) — zM (t) = (Az1, Az 2)T in the method of [4], the system (6) of the sliding mode after a short time interval sliding is transformed by Az(t) ^ 0 to the system model sliding mode:
¿M = [(Aoii -Boi(CBo)-1C1 Aon -Boi(CBo)-1C2Ao2i -Boi(CBo)-1C1) -
- (Aoi2 - Boi(CBo)-iClAoi2 - Boi(CBo)-iC2Ao22 -
- Boi(CBo)-1C 2)(C 2)-1C1]zM,
(7)
zM =-(c 2)-1C1zM;
where zM = z , z2a = z2, and matrix C(t) = (C1 (t), C2(t)) is by the methods described in [4, 5] for a given quality of transients in the source system. For example, when playing in a sliding motion desired model
y = Ay + Bu , u = K (t )y , (8)
S M M -^0 onT' onT onT\/./M > v '
where control uonT = Kom(t )yM can be found by methods of (6), matrix C(t) = (C1 (t), C2 (t)) found as
C = —C( A + BoKono)
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The scientific heritage No 29 (2018) 33
with initial conditions C(t0 ), satisfying the beginning of the identity playback start time t = t0 :
C(t0 )yM (t0 ) = C(t0 )zM (t0 ) = 0.
In this case, the regularity of the original system (1) founding of the matrix C( ) is simplified. According to the condition of regularity B01(t) = = 0(nm)xm Vt G I, and given by a non-singular submatrix C2(t), | C 2(t) 0Vt G I, the system (7) is transformed to the equations
zM = \u(t)zl - Ao,u(t)(C2(t))-1C1(t)zM, (9)
zM = -(C 2(t))~1C1(t) zM,
in which the matrix A0,12(C2) 1 is taken as the input (n — m) x m matrix fictitious controls u^ = —Cl(t)zlM
, where m x (n — m) — submatrix — C1 (t) is taken as a m x (n — m) matrix of coefficients of the linear control and is found, for example, the method of [6].
Are set properly sliding manifold on the first control level with the first order sliding (with the dimension of a conventional sliding mode equal n — m). If the required high quality of transients is achieved by sliding the higher-order modes (ie, with dimensions smaller than n — m in the system (9), for example, with dimensions equal to , n — 2m, n — 3m,..., at first there are found two mobile switching manifold of type (5) with dimensions n — m, so that the image point of the sliding mode on one of these varieties rushed to n — 2m the variety of their intersection with the second manifold, and make movements in a small neighborhood of the intersection with the low dimension equal to n — 2m .
To further reduce the dimensionality of the system to sliding values (n — 3m) and forming another variety of dimension (n — 2m) and sliding mode to switch from one to another with the resulting motion of the image point in a small neighborhood of dimension (n — 3m), and so on, until a need for the representative point in a small neighborhood of a suitable quality processes in a variety of direct or plane.
When driving in a small neighborhood of the line and plane on the value of n, m and k, respectively limitations should be imposed n — km = 1 and n — km = 2, where k = 1,2,3... is an order sliding, and simultaneously the upper level of the number of multilevel control. For k = 1, we have the usual sliding mode - the mode of the first order and the first level of the discontinuous multi-level control. When k = 0 and no-sliding control
is switched structure of each of the m component 2k vector discontinuous controls u01,u0,2;...;uQ^_vu^2k . For k = (n — V)/ m or k = (n — 2)! m that the upper level of multi-level control (that is itself the desired vector control u = u i) corresponding to the motion of the image point in a small neighborhood of a suitable quality straight or plane in the phase space. If the combination of the values of n and m does not lead to an integer value k , but should provide the required parameters of quality of transients, then change m the number of allowed translation of the components uj source control in the category of linear controls to make the object of appropriate management of dynamic and static properties, not excluding them and setting zero values. Thus, for example, n = 5 and m = 3 to straight get k = (5 — \)/3 = 4/3 and hence decreasing the value of m to m = 2 in this way, we obtain the two-level control. For movement in the plane in this case, n and m sufficient to apply a common slide (first order) k = (5 — 2)/3 = 1.
Consider the method and procedure for the construction of varieties of sliding. On the right side of the sliding mode (9) with C2 = E element — C1(t)zlM relies fictitious control:
-M = A0,11(t )zM ^ A),12(l )u& ,
4 = A0,11(t) zM + A,12(t >1,
2 1 1 zM = u<P =~C (t)zM •
On the one hand fictitious control u^ equals — C1(t)zM, and on the other hand is m — discontinuous
control, resulting in the system (10) in the sliding mode on some (n — 2m) — movable manifold (s) according to the methods of [4-6]:
S№ = (si 1,•••/, m )T = Cl(t)zM = 0), (11)
where C^ (t) is m x (n — m) — matrix. A system (10) is the nominal (does not depend on uncertain disturbances), it is independent also from the nominal external perturbations. For the construction the control u\ is
proposed to apply the method of constructing the nominal control u0, as proposed in [7]. This discontinuous control u0 assumes a relatively small number of logical switching devices and has no additional restrictions on the assignment matrix Сф (t). For a model system (2) for F0 (t) = 0 control Uo will be written:
Uo = (CB0)l(Kgg + Kss - С(t)Ao(t)zM), (12)
where Ks(zm,t)=(К(zm,t)SA)Г, 8ik - Kronecker delta, i,k = 1Г, S = (s^...,Sm )T = C(t)Zm, S = ciT (t)ZM = 0 , g = (gi,...,gm )T = D(t)ZM , g = diT (t)ZM ,
К (z лк <0 при sigi >0, К (z tЛ< <0 при sigi >0,
gA M, ; [к- > 0 при sg < 0, S'V M, ; [ ks- < 0 при sg < 0.
Applying this type of control (12) for (10) in order to bring it into the sliding mode on the manifold Sф (11), the control иф will be written:
иф = (Сф(0АД2(0)-1(^ф + Кфф -Cj(t)A0,ii(t)zM), (13)
where кф(z\t)=(кф(zl,t)Sik)m , Ksф(z1,t)= ф(z1,t)S,k)m and asked discontinuous coefficients (or functions) К ^, Ks^ satisfy the same (12) with the functions of switching sф i и gф i, i = 1, Г. From the expression (13) with иф =-C40zM, gф = gl,...,gфm)T = A^K, sф = = (sф 1,...,s\m)T =
$ V / m , 6$ \S$6$m/ $V / M , $
= C^ (t)z\ to find the relation of the two m X (n — m) matrices C1 (t) and D$ (t) at a predetermined (as required quality second-order sliding mode) matrix C$ (t) :
Cx(t) = — (C$(t)Ao,2(t))—1-(Kg$D$(t) + Ks$C$(t) — C$(t)Ao,n(t)). (14)
In the case of (10) regularly shaped matrix C$ (t) in (14) is determined by the method of [6], and in the case of system (10) of the total normal form - the methods of [4, 5]. Elements Kg$$, Ks $, i = 1,m, matrices Kg$, Ksip take in equation (14) the values of ,K+^and Kg$,fcs.$is similar to the inequalities in the management
(12), depending on the signs of the products s$ ig$ i, i = 1, m. For each combination of the signs of these products the entries take different values. Further, these manifolds partition the conditionally into two types Sj (sj = Cj1zM + zM = 0), J = 1,2, where an index j = 1 for manifolds SJ with 2m 1 all the different combinations of characters products s$ i i = 2, m, corresponds to a positive sign of the product sS$ 1g$ 1, s$ ig$ i > 0, and index J = 2 negative or equal to zero of the product s$ 1g$ 1 < 0 . As a result, determining the submatrices C1(t) and D$(t) of the condition (14) is guaranteed only motion of the image point to the (n — 2m) — manifold S$(s$ = (s$ 1,...,s$m)T = C$(t)zM = 0) (11) with no matches with it of the (n — 2m) — manifold intersection S1 o S2. For such a matches must be made simultaneously three equations:
sJ = Cj1(t)zM + zM = 0, j = 1,2, s$ = C$(t)zM = 0. (15)
When forming such a high-level sliding must be kept in mind that given by the (n — 2m) — manifold (11)
the quality deteriorates bring in a sliding of a higher level. However, the manifold S$ (11) should not belong to the (n — m) — manifolds SJ, j = 1,2,, since they have fictitious control u j1 = = —Cj1 (t)zM (10) is continuous by
should be such as to extend close enough (n - m) - varieties SJ (sJ = CJ1 (t)zM + zM = 0), j = 1,2 . Otherwise
sJ = Cj1(t)zl + z2 = 0 and uj = —Cj1(t)zl = z2, j = 1,2. Therefore, in addition to the system of equations (14), finding Cj1(t) , Dj1(t) must consider not the system (15), and system
sJ = Cj1(t) zM + zM = 0 , j = 1,2, ^ = m )T = Cl(t) zM = (16)
where s = ,...,£m) , £i - linear functions zi, i = 1,m, with sufficiently small in absolute definable constant coefficients. From (16) complementary to the equation (14) links between the elements of the matrix CJ1(t) and D^(t) to the founding (including the real-time). The system of sliding mode on the (n — 2m) — manifold
S4>(s(P = (s(P l,... m ) = Ci (t) zm = 0) after exdusten m to coordmates of the vector zM talkes the form
zM = A\t) zM, (17)
where (n — 2m) x (n — 2m) — matrix A3 (t) depends nonlinearly on the coefficients of the matrix C^ (t) and to find it applied the methods of [4-6]. If the dimension of the sliding mode should be reduced further to a predetermined value (n — 3m),...,(n — km), the matrix C^(t) is proposed finding by the method described by
the C1 (t) system of equations (14), (16) and so on.
Thus, the first are suitable for the quality of transients sliding manifold upper level (line, plane or hyperplane, or a variety), there are two planes, or two hyperplanes, or two varieties of the previous lower level, such that the image point to sliding on them fell in a small neighborhood of their intersection. For each of the two planes found (or two hyperplanes or two varieties) is a pair of planes, or hyperplanes, or varieties of the next lower level, and
so on until the manifold (5) of the first level: Sri, S12;...;S12k-1x, S 2k-1 .
3. Synthesis of multi-level discontinuous control vector, resulting in a system of sliding mode of a given order. Control for forming the first-order sliding serves to finding the model system as compared to the initial energy provided by lower costs. This follows from the fact that the state identifier, the model system (2) in the actuating slide is reduced and fine identifier undetermined vector perturbation. This information is used to compensate for disturbances [3]. In this method, for sliding mode control first level is received as coinciding with a control (12) for (x — K(t)zM ) = (K(t)z—K(t)zM ) = K(t)Az = 0 and 5 = 0 :
u = U0 = (C(t)B0 (t))— (Kgg + Kss —5 sign s — C(t)A (t)zm —
(18)
— C(t)KAz (t )GT (t)(x x—K (t) zm ) — C (t) D0(t) F0(t)), where in forming the vector switching functions apply only vector
zm, 5 = dlag {51,..,5m}, 5t > 0, sign s = (sign S1,..., sign Sm)T, i = 1,m.
Fig. 1
Depending on the k number of levels of slideing organized sliding mode first-order on 2k 1 switched (n —m)_ manifolds Sn, S12;...;S12k—1 S12k—1 with 2k 1 discontinuous vector controls
2k_i of the form (18) on the first level of control, and the first order slide. Zero control
generally formed matrix C (t )B0(t ) from the 22m switching structures of each com-
o,2k i o,2
ponent Uf, i = 1, m,, control form (18)(each of the switching control structures (18) defined by a set of values K+gi, k+s1 and Kg i , K~si depending on the sign Sfgf and values of the constants 8i sign > 0 depending on the signs Sf, i = 1, m ). At the upper level as the I — control U = U^i element at a time depending on the signs sigi on the first level and of similar character features on the remaining k — 1 levels corresponding component passes only one of the controls u0 u0 2;......U .,u„
slide on one of the manifolds S12;...;^ 2k—i
zero level. In the course of the first order
1' S12
0,2k — 1' 0,2k
k_i to the upper level passes one of the relevant controls
u1,1 'u122k-i -i'U 2*-1 of the first level, and so on up the levels (Fig. 1).
Thus, the motion of the image points are added only two types: first, the movement of the slide hit with
manifolds S11, S12;... ;S1,
'S
-11,2*
are invariant to
^ -^u c c . .c c j them. Control processes closer to the ideal sliding for
the manifold S 13 S 2;...;S nk-1 , ,S,k_i of first-order . .. , J • , , , •
1 1 1 2 1 2k 1 1 1 2k 1 given manifold with increasing speed and reducing th
given manifold with increasing speed and reducing the (first-level control), and secondly, of the movements of deviation vector Az of values of switching functions the sliding mode on them. In the first movement the effect of all disturbances (nominal and uncertain) is offset
controls U — Uo (18) for each of the manifolds. In the second movement, from the beginning of slips on these manifolds, the deviation Az quickly becomes zero. Consequently, the effect of disturbances on the system through a small deviation Az « 0 no longer occur. There is no another manifestation of the action of disturbances, since by virtue of (3), (4) sliding modes on and the final values of the parameters are set, taking
that define sliding manifold at all levels of control. This will result in the large absolute values of the elements matrix K^ (t) of the model (2) and discontinuous coefficients kg,ks, ,8 in (12), equations (14), (18) and small values of the coefficients £ in (16). The effect of this rule with respect to k , ks, k^ , k^s, 8 , £ should decrease with increasing levels of control,
k
k-1
a
into account constraints on the control, its energy requirements and system status, as well as preliminary results of numerical simulations of the designed control system.
Conclusions. Thus, in order to ensure the quality of transients in linear systems with non-stationary object under uncertain and nominal disturbances and incomplete information about the state of the vector obtained by the method of multi-level discontinuous sliding mode control of a given order and quality, which has a relatively simple implementation. For the construction of the control developed Lyuenberger's identifier [1], and used the results of previous studies by the authors in the sequence [2-9].
The publication was carried out with the financial support of RFBR And the government of the Republic of Tatarstan in the framework of scientific project № 18-41-160012 p_a
References
1. Andreev N. Control of finite linear objects. M.: Nauka, 1976 - 424 p.
2. Meshchanov A.S. Synthesis of multi-level vector control, sliding mode for a given order. Vestnik KSTU im.A.N. Tupolev. 2007, № 4, pp. 47-51.
3. Meshchanov A.S. Bringing the linear non-stationary objects with the model identifier of the state to move with uncertainty. Vestnik of KSTU, 2008, № 4, P.127-134.
4. Meshchanov A.S. To address the problem of tracking in the control of multilink manipulators with inertial actuators in the face of uncertainty. Aviation Technology, 1996, № 3, p. 30 - 37.
5. Meshchanov A.S., L.A. Davletshina. Reproduction model of low-dimensional motions on the sliding mode. In.: Analytical Mechanics, Stability and Control: Proceedings of the X International Chetaev's Conference. Volume 3, Section 3. Control. Part II. Kazan, June 12-16, 2012 - Kazan: Kazan on. State. tehn. University Press, 2012. S. 147-159.
6. Meshchanov A.S. The synthesis of linear systems with the specified quality control processes in the norm of the state vector. Vestnik KSTU. A.N. Tupolev. 2009, № 4, pp. 107-114.
7. Meshchanov A..S. Bringing diversity to the mobile sliding systems with linear non-stationary objects, in general, the entrance of uncertain disturbances. -Aerospace Instrument, № 5, 2008. - P.16-20.
8. Meshchanov A.S. The reduction in sliding mode break-dimensional systems with non-linear time-dependent control object. In the book "The stability of motion", Nauka, Novosibirsk, 1985. s. 230 - 234.
9. Meshchanov A.S. Sliding equations on moving manifolds and controls for the synthesis of vector objects with uncertain nonlinear perturbations. Vestnik KSTU A.N. Tupolev, 2008, № 2, pp. 51-55.
МОДЕЛЮВАННЯ ТЯЖКО1 АВАРП В СИСТЕМ1 ОХОЛОДЖЕННЯ БАСЕЙНУ ВИТРИМКИ
ЧЕТВЕРТОГО БЛОКА АЕС «ФУКУС1МА-1»
Азаров С.1.
1нститут ядерних до^джень НАН Украти, м. Кшв, Украта, доктор mexHi4Hux наук, старший науковий cniepo6imHUK
Сидоренко В.Л.
1нститут державного управлтня у сферi цившьного захисту, м. Кш'в, Украта, кандидат техтчних наук, доцент, професор кафедри
Задунай О.С.
Державний науково-до^дний тститут спещального зв'язку та захисту iнформацii,
м. Кшв, Украта, начальник центру
MODELING SEVERE ACCIDENT IN THE COOLING SYSTEM STORAGE POOL FOURTH UNIT OF
NPP «FUKUSHIMA-1»
Azarov S.I.,
Institute for nuclear research, National Academy of Sciences of Ukraine, Kyiv, Ukraine, Doctor of Technical Sciences, Senior Researcher.
Sydorenko V.L.,
Institute of public administration in the sphere of civil protection, Kyiv, Ukraine, Candidate of Technical Sciences, Assistant Professor.
Zadunaj O.S.
State research institute for special telecommunication and information protection,
Kyiv, Ukraine, head of the center
Анотащя
У процес виршення iнженерних завдань з обгрунтування потреби охолодження тепловидмючих 36ipoK в басейш витримки (БВ) в разi виникнення аваршних ситуацп що супроводжуеться втратою охолодження БВ виникае необхщшсть у визначенш температур i часу досягнення граничних температур твелiв, кшьшсть накопиченого водню, сила вибуху i стутнь пошкодження БВ.
