SYNTHESIS OF CONTROLLER FOR VECTOR PLANT, BASED ON INTEGRAL ADAPTATION METHOD FOR DISTURBANCE SUPPRESSION Текст научной статьи по специальности «Физика»

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DIFFERENTIAL EQUATIONS AND

CONTROL PROCESSES N. 3, 2022 Electronic Journal, reg. № 0C77-39410 at 15.04.2010 ISSN 1817-2172

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http://diffjournal.spbu.ru/ e-mail: _ [email protected]

Control in non-linear systems

Synthesis of Controller for Vector Plant, Based on Integral Adaptation Method for Disturbance

Suppression

Abstract. The problem of control over a nonlinear plant on the target manifold with two types of uncertainties in the description is discussed: deterministic disturbances in the right-hand side of the system of differential/difference equations as the unknown functions of time and stochastic disturbances with a zero mean and limited dispersion in the right-hand side of the system of difference equations. The method of integral adaptation is used, which relies on an analytical description of the target attractors - invariant manifolds and introduction of the integrators into the control channels, which make it possible to synthesize adaptive systems without acquiring the current data on the changes of the plant and environment parameters. A search for control in the case of stochastic disturbances is performed from among the strategies minimizing the dispersion of the output macrovariable. The examples and results of the numerical testing of the proposed algorithms are presented for an applied problem of control over the motion of an plant with an immobile center of mass. The presented results can be relevant in all control systems of robotic arms, functioning under the conditions of uncertainty, and in designing the decision-making support systems in controlling compound plants.

Keywords: principle of least action, analytical design of aggregated regulators, nonlinear plant, method of integral adaptation, object with fixed center of mass, random and deterministic disturbances.

1. Introduction

The method of integral adaptation on invariant manifolds has demonstrated its high performance in solving the problems of controlling the complex (nonlinear, multi-loop) dynamic plants [1]. These problems present essential practical challenges for common methods of adaptive control [2-7], since the

Kolesnikova S.I.

St.Petersburg State University of Aerospace Instrumentation

skolesnikova@yandex .ru

observer dynamics has to be faster compared to that of changes of the coordinates and variation of the external disturbances.

A conceptual difference of the method of integral adaptation [1, 8-10] from the well-known classical methods of the adaptive control theory consists in the mechanism of introduction of the attractors -invariant manifolds (chaotic manifolds among them) from the synergetic control theory [1, 8] and "backstepping" [11, 12].

In this study we continue solving the issue of applicability of the method of integral adaptation (see more detailed reviews in [1, 8, 9]) to the models with random and deterministic uncertainties in their description .

In Sections 2.1 and 3.1, we discuss the problem of control of a nonlinear vector plant x = (xT, xT )T, x e R, i = 1,2 using the following description of the dynamics of its coordinates:

X^f^x,) (1.1)

x2 =f2(x1,x2) + £ + u,

where (xj, xj ) , e R3, i = 1,2 is the state vector, ueR3 is the control vector, feR3 are the nonlinear

vector-functions, /'=1,2; % e R3 is the unknown limited continuous deterministic disturbance. Recall that manipulator systems (such as robotic arms [13-15]) obey this description; the importance of controlling them under the conditions of noise hardly requires elaboration (e.g., [16-18]).

Sections 2.2 and 3.2 provide the problem statement and describe the controller design for a discrete system obtained from (1.1) relying on the Euler discretization and containing a deterministic uncertainty in its description.

Section 4 focuses on the problem of nonlinear discrete-time vector control over a stochastic plant.

Sections 5 and 6 present the results of a comparative simulation of two systems of control in the model and off-design conditions for the problem of control over a robotic arm.

The design of control systems on the principles of manifolds (see reviews in [9, 19-21]) possesses a number of important properties, for instance: the problem of synthesizing an optimal asymptotic stable control system can be solved without using the Lyapunov functions [8, 9]; a certain correspondence to the physical theory of control over the plant's properties [10].

Here we also report two algorithms [8, 22-24] for synthesizing control systems for a nonlinear vector plant represented by a system of differential/difference equations, wherein the right-hand sides can contain unknown functions of two types of uncertainties: random or deterministic.

The basic tool for constructing all below-presented adaptive controllers, compensating for the bounded disturbances , is the method for Analytical Design of Aggregated discrete Regulators (ADAR) [1, 8] and the method of nonlinear adaptation on a manifold (NAD) (e.g., [1, 22]), whose importance is accounted for not only by the simplicity of the applied mathematical tool, to a great extent borrowed from the theoretical mechanics [25, 26] and its applications [27, 28], but also an account of the physical properties of an plant by including the given dynamic properties of the controlled plant into the target function. The proposed algorithms are derived analytically.

2. Formulation of the control problem with a deterministic disturbance 2.1. Formulation of a continuous-time control problem.

A nonlinear vector plant with description (1.1) is discussed. It is required to determine the law of control u (t) = U (x (t), x2 (t)) in the state space for the sake of transferring the controlled plant (1.1)

from a certain initial state x (0) = (xj (0), xj (0)) to the neighborhood of the manifold

y (t) := y (x (t), x2 (t)) = 0, t ^ ro and ensure an asymptotically stable motion of this system in this

neighborhood. The following assumptions are currently available (see comments to the selection of a control quality functional in Appendix A):

1) The global minimum of the quality functional, including the least-action principle upon

reaching the control target y (x (t), x2 (t)) = 0, t ^ ro, is given by the following:

ro,

(

2.1)

where w = ( w, W, W )T is the synthesized control system parameter regulating the time of motion of the imaging point of the system (2.1) to the neighborhood y (x (t), x2 (t)) = 0, t ^ro; the

subscript in the notation of the functional Oc (Od ) indicates the continuity in time (discreteness in

time) of the description of the controlled plant;

2) The boundedness of the right-hand side and the implementability of the system's behavior in the conditions of all admissible controls;

3) The availability of a stable target system for model (2.1), simultaneously responsible for dynamic and/or engineering requirements, not contradicting the physical capacity of the system [3, 4];

4) The disturbance, as a function of time, is such that the conditions for applying the ADAR-synthesis of plant control (2.1) hold.

Earlier [24], a number of statements on the conditions of availability of such a control were proved for a scalar plant.

2.2. Formulation of a discrete-time control problem with a deterministic disturbance

From the engineering viewpoint, it would make more physical sense [29, 30] to consider a discrete system obtained relying on (1.1), for instance, using the Euler scheme with a discretization parameter x>0, td=d x, d=0,1,2,...

X (d +1) = gl (x (d), X2 (d)),

(2 2)

X2 (d + 1) = g2 (Xi (d), X2 (d)) + Z (d) + u (d). Here systems (1.1) and (2.2) are related by the following expression:

g, (Xi (d), X2 (d)) = X, (d) + T- f, (Xi (d), X2 (d)), i = 1,2.

Remark 1. The basic assumption in the classical ADAR is the existence of an asymptotically stable system on a manifold with an attractive property assigned as a control target. From this perspective, it is possible to identify the conditions (in terms of probability) of a uniform proximity of the solutions x(td), x(d), td = dx, d = 0,1,2,... x(td) to the systems (1.1) and (2.2), which was respectively

obtained from (1.1) by the Euler time discretization [31]. However, the latter assumption requires a separate detailed and thorough analysis [31, 32] and is beyond the scope of this work.

Remark 2. An advantage of a discrete description is especially obvious for the plants with delayed control (see, e.g., p. 187 in [8]).

0 l=1

The formulation of the problem of discrete control formally differs from the continuous problem by the forms of the quality functional and the equation, the solutions to which are represented by the

stable extremals for functional (see p. 172 in [8]):

œ 3 / \

= ZZKV(d) + (AV (d)) Wmin,

d=0 l=1 ^ '

y (d) = 0, d ^œ, y = (vi, v2, V3)T ;

(2.3)

V (d +1) + q^ (d) = 0, Ay^ (d) = v (d +1)-v (d);

®i =

0.5f-(2 + a2)±^(2 + a2)2 -4], I = 1,2,3.

The relationship between the parameters a; and in (2.3) immediately follows from the necessary condition of an extremum of the functional - a discrete analog of the Euler-Lagrange equation.

Remark 3. Equations w;y; (V) + y, (t) = 0, y; (¿/ + l) + co;y; (¿/) = 0, / = 1,2,3; t,d > 0 are the corollaries of the Euler-Lagrange equations for the respective functionals ®c and .

3. Solution of the control problem on a manifold with a deterministic disturbance

There is a certain inconsistency in the 'adaptive control' terminology associated with the use of different adaptation principles (methods) (e.g., [9, 33-35]). In this study, integral adaptation on a manifolds understood as the process of variation of the averaged characteristics of a dynamic system with an uncertainty in the description of the operational environment, and an adaptation algorithm is the ability of asymptotically transfer the system's state into the neighborhood of a given invariant set in such an environment.

It is well known [8] that the synthesis of ADAR controls in the general case allows for a hierarchy of the control designing procedure. For instance, in the stabilization problems the control synthesis hierarchy is determined by a necessity to control the variables in the cases where a direct access to them is

impossible. In so doing, a complex of partial criteria (target functionals) is used C(d)} — with a certain

subordination and consistency principle, each of the criteria represents a certain local requirement to the quality of the respective control sub-system. This approach to a certain extent agrees with the stability theory [36] and the idea of a nonreflexive cybernetic system [37].

3.1. Principle of nonlinear integral adaptation for deterministic plant disturbance suppression (NAD)

Step 1. Phase space extension by converting the external noise into an additional internal phase variable z:

X1 f"l (X1 7 X2 ) 7

X 2 = f2 (X1> X2 ) + Z + U (3.1)

z(t) = r|\|/(Y),r| = const > 0,

where the proportionality coefficient ^ is the synthesized control system parameter.

Step 2. Derivation of the synthesized controller structure by the classical ADAR synthesis methodology for a closed system (3.1). Note that at the final e-th level of the synthesis hierarchy the following form of the macrovariable is used: y(e) = y + kz, k = const > 0, where y is the target macrovariable from the control problem statement and the respective variational problem is formulated

(®C, y(e) )

" 3 r t o

0 l=1 L

dt -» min, y(e) —--» 0,

5 ' t—>co 5

(3.2)

with a linear functional control delivering a stable solution to Euler-Lagrange equation for the variation al problem (o*, y(®}) (see Comment 2 in Appendix A)

wey(e)(i) + y(e)(i) = 0,^a>. (3.3)

Functional Oec is the quality criterion of the sought-for control during the motion of the imaging point

of the system from the manifold y(e-1) (t) = 0, t ^ ro to manifold y(e) (t) = 0, t ^ ro. The superscript e

(for system (3.1) e=1,2 depending on the type of the target macrovariable y) denotes the final hierarchy number during the synthesis of a classical ADAR-regulator; the subscripts e, l used with the parameters w indicate the hierarchy numbers and the vector coordinates, respectively; the multiplication

wey(e) = ( wc jij/j6', we 2vj/'2e), we ) in (3.3) is coordinatewise.

All possible constants from (3.2), (3.3) ws, s = 1,2 are the control system parameters proportional to the velocity at which the imaging points reach the s-th subsystem of the neighborhood ys (t) = 0, t ^ ro

according to the partial criterion O sc, s = 1, e.

The synthesis has been completed, and the final control system for model (1.1) represents a set of equations of the controlled plant (1.1) and the controller derived relying on (3.1).

Statement 1. Control u = uNAD, if any, ensures an asymptotic stability for the controlled plant (1.1) in

the neighborhood y (t) = 0, t ^ ro.

The proof of Statement 1 is given in Appendix B.

3.2. Disturbance suppression algorithm based on discrete adaptation on a manifold for plant (2.2)

For the sake of definiteness, let us describe the algorithm for the stabilization problem with the control target

y(d) = 0,d ^ro, y(d) = x (d)-x*, (3.4)

where x* is the given target value for the variable x (d), d = 0,1,2,..., let &s,c = 1,2 be scalar quantities.

Step 1. Extension of the model for system (2.2) given by

X (d +1 ) = gl (xj (d),x2 (d)),

x2 (d +1 ) = g2 (xi (d),x2 (d)) + z(d) + u(d), (3.5)

z (d +1) = z (d) + yy (d), y = const > 0.

Step 2. Formulation and solution of the first variational problem (o^, y(1) ) for an plant with the description (3.4), (3.5):

JD = IIk (V((1) (d))2 + (Ay(1) (d))2U min

d = 0 l=1 ^ '

with a restriction on y(1) (d) = x2 (d)- 9(x1 (d)) = 0, d ^ œ, where 9(x1 (d)) is a further defined function.

From the equation, giving rise to stable extremals forO^ y(1) (d +1) + qjy(1) (d) = 0, |qj | < 1, form the structure of the controller u ( d ), d = 1,2,..

u (d) = -g2 (d) - z (d) + (p (d) - Qjy0' (d), u {d) = 0,

9 (d ) := 9 ( g1 ( d )), d = 1,2,.. (36)

Step 3. Reduction (decomposition) on the manifold y(1) (d) = 0, d ^ œ :

x2 (d) = 9 (x1 (d)), x1 (d +1) = g1 (d) = g1 (x1 (d), 9 (x1 (d))),

z ( d +1) = z ( d ) + yy ( d ).

Step 4. Formulation and solution of the second variational problem (®2, y(2) ), y(2) = y + kz, y = x - x*, k = const > 0 for an plant with the description (3.7):

JD = II(«2,1 (y(2) (d))2 +(Ay(2) (d))2) ^ min.

d=0 l=1 ^ '

From equation y(2) (d +1) + ^y(2) (d) = 0, ^2|< 1, whose solutions are stable extremals for the functional , form the function 9 ( x1 ( d )) (assuming the fulfillment of all conditions for the existence of the ADAR controls):

g1 ( x1 ( d ), 9( x1 ( d )))-x* + kyy ( d ) +

/ (3.8)

(y (d) + k (z (d) + yy (d))) = 0.

Statement 2. Discrete control u (3.6), (3.8), if any, ensures an asymptotic stability for the controlled plant (2.2) in the neighborhood y (d) = 0, d ^ œ.

The proof of Statement 2 is similar to that of Statement 1, given the discreteness of description (Appendix B). The correctness of the controller structure (3.6), obtained relying on the extended system (3.5), follows from the limit relation y(2) = y + kz = 0, d ^ œ. Substituting y ( d ) = -kz ( d ) into the last

equation

in

(3.5),

we

obtain z (d)-

d -^ro

since

d -^ro

z(d +1) = z(d) + yy (d) = z(d)(1 — yk),|1 -yk| < 1, therefore, y (d) =-kz(d)

A summarizing description of the control system for the controlled plant (2.2) will, in this case, be given by a family of relations (2.2), (3.4), (3.6), and (3.8) with the parameters , q2 , y, k, determining the transient process quality.

4. Formulation of a discrete problem of stochastic control on manifold

For random disturbances, the following representation [38] in the description of a discrete controlled plant will be convenient (d = 0,1,...):

X (d +1) = g (x (d), x2 (d)),

x2 (d +1) = g2 ( x (d), x (d)) + \ (d +1) + c\ (d) + u (d).

(4.1)

It is assumed that the disturbances are implemented as a sequence of independent equally distributed random quantities (functions of d) {E(d)} , E (d) = (^(d),(d),(d)) with the following properties:

Ej (d)} = 0, Efr (dHj (d)} = 0,

j (d)} = g2 , ij = 1,2,3, i * j.

(4.2)

In (4.1), the constant 0 < c < 1 is interpreted as the noise decay factor (the level of uncertainty in the subsequent point of time, d+1, is no smaller than in the current point of time d); in (4.2), E { • }, D { • } are the operators of mathematical expectation and dispersion, respectively. It is required to determine the law of control u (d), ensuring the fulfillment of the following requirements:

E {y j (d+1)+Q j y j (d )} = a

D{y; (d + 1) + ra;y; (d)} ^ min, j = 1,2,3, d ^ ro;

I ro 3 / ? - \ I

E{Od} = E|]T]r(„2 (y; (d)) +(Ay; (d))

(4.3)

J=0 j=1

^ min.

The Nonlinear Adaptation on a manifold algorithm for a Stochastic plant (NAS) is an extension of the discrete version of the ADAR method including a stochastic plant, which was discussed in the earlier studies [23-24] for the 1-st-6-th order nonlinear descriptions as applied to biological (immunological) and economic systems. Let us briefly describe the main limitations and statements:

1) control strategies are selected from the class of discrete ADAR controls according to (2.3) and Section 3.2;

2) consider the strategies, for which the value of the control variable u (d) is a function of the current and previous states and controls

u (d ) = u (x (d), x (d — 1),..., x ( 0); u (d — 1), u (d — 2),..., u (0)), d = 0,1,...;

3) select the strategies minimizing dis function for a fixed d - T(d +1) + raT (d),

persion of the random variables - coordinates of the

<

1, j = 1,2,3, where ¥ (d) is the macrovariable in

the stage of determination of the controller structure u (d) (external control according to the

terminology of the classical ADAR method [8]).

The main statements of the design of a discrete system of control over a nonlinear system (4.1) in the class of ADAR strategies minimizing dispersion of the coordinates of the target variable given by

y (d) = y (x (d)) = 0, d —>oo will be made up in the form of the NAS algorithm NAS [23]. NAS algorithm

Step 1. Searching for the structure of control ii (d), d e {0,1,...} relying on the discrete version of ADAR under the conditions of fixed random disturbances.

Step 2. Taking the conditional expectation u (d) = e{ua (d) d j, where

r = (jj ( 0) , j (1)...jj (d ))T, j (k )=(^ (k ) , ^ (k ) , ^3 (k ))T, k = & .

Step 3. Decomposing the description (4.1), taking into account the achieved limit relations

y(1) (d) = 0, d ^ ro.

Step 4. Searching for the relationship j(d) = x(d),y (d)) as a function of the observation of an

system whose behavior obeys the decomposed description (Step 3) in order to rule out the variable j (d)

from the expression u (d) = e{ua (d) |jd j . The synthesis of a stochastic discrete controller has been

completed (see Appendix B).

Statement 3. Control u = , if any, ensures a fulfillment of the requirements (4.3) and an asymptotic stability for the controlled plant (4.1), on average, in the neighborhood E{y ( d )j = 0, y (d) = x1 (d) - x*, d ^ ro, where x* is the given target value of the variable x1 (d) .

The proof of Statement 3 is constructive and follows immediately from the NAS algorithms [18], the properties of the ADAR algorithm [8] and the selection of strategies minimizing dispersion of the output variable [38, 39]. By condition, a search for a stochastic control is performed from among the ADAR strategies based on the limit equalities given by y(s) (d + 1) + rasJy) (d) = 0, j = 1,2,3, s = 1,2 at

d ^ ro; note that the controller structure for system (4.1) is determined on the first hierarchy level. For any control, from the condition of independence of random functions (see Appendix B) j (d), j (d +1), d = 0,1,... one has an inequality (in a scalar notation)

D {yj5 (d +1) + coy;' (</)} = D {x2] (d +1) - cp; (x1; (d +1)) + coy;' (</)} = = D {g2] (d) + u} (d) - cj>;. (g1} (d)) + coy1' (d) + c^. (</)} + ; (d + 1)}>o2, j = 1,2,3.

(4.4)

From which follows that if a system is randomly disturbed, there is no control providing an error of the random function in the left-hand side of the functional equation with a dispersion smaller than the disturbance dispersion in the model (4.1) [23, 38]. A substitution of the found control into the system (4.1) would yield the limit equality

y(1) (d + 1) + ©j y(1) (d ) = ^( d +1), d ^ro, j = 1,2,3. (4.5)

From relations (4.4) and (4.5) follows that the resulting control ensures the minimal dispersion D{y(1} (d +1) + y(1} (d)j = a2, j = 1,2,3 at d ^ ro, which makes Statement 3 correct.

5. Application of NAD and NAS algorithms in the problem of control over the center-of-mass motion of a disturbed moving plant

Use the plant's description from [15]: Mb(0 = q (b(0, b(0) + u(0, t> 0. (5.1)

Here b e i?3,b(7) = (/>, (t),b2 (V),63 (/))' is the state vector with the coordinates; q = g(b,b) e R" are the generalized forces acting on the controlled plant; M = m. is the kinetic energy matrix;

II •'113x3

u e R3, u = u (b (t), b (t)) is the control vector.

It is required to design a control stabilizing the system's trajectories (1.1) b (t) in the neighborhood of the given values of b* (t) from an arbitrary point of the state space b0 [10].

In order to implement the above-described NAD algorithm for synthesizing control, it is reasonable to transform system (5.1) to (1.1) substituting the variables x, (/ ) = b(/ ), x2 (/ ) = b(7):

X(0 = x2(0,

x2 (0 = Aq(0 + Au(0 + %(t), A = M"1.

(5.2)

Here we additionally introduce an uncertainty in the form of a limited continuous disturbance E(t). Thus, a nonlinear multidimensional system with a poorly formulated right-hand side in its description has been discussed.

5.1. Solution of the problem of nonlinear adaptation on a manifold for a plant (5.2) with a deterministic disturbance

In this case a problem is posed for finding control u(t) = u(x(t)), x(t) = x = (xT, x2T ) over an plant

that would ensure its transfer from an arbitrary initial state x0 in a certain region of the space state into a given state x and its stabilization in a certain neighborhood of the target manifold

y (x) = X1 - ^ (5.3)

where y (x) is the target macrovariable.

*

Remark 4. Derivation of the control system with variable x (t) is also quite straightforward but

* *

cumbersome, therefore, here for the sake of readability we take x (t)=x =const, and parameters wx, w2 are

also taken to be scalar.

Implementation of the NAD algorithm

Step 1. Extension of the system:

x(0 = x2(0,

x2 (0 = Aq(0 + Au(0 + z (0, Z(t) = T) = const > 0.

(5.4)

Step 2. Derivation of the controller structure.

2.1. Introduce the macrovariable y(1) = x2 - 9 (x, z), in designing which a further defined function is used 9 ( x, z ).

2.2. Formulate the variational problem (®cc, y(1) ), where

°c=jz (<) + wi (V/^) ûfr—>min, and the macrovariable y(1) = (y^yi^y^) determines

o 1=1 L J

the limit given by y(1) (t) = 0, t . The problem solution (Oc, y(1) ) relies on the equation

i^y(1) (i) + v|/(1) (i) = 0, w1 = const > 0.

2.3. Control u is determined by the following relations:

Au = -w"V(1) - Aq + J (9) YT - z(t),

xii A x21 Wi

Y = x12 ¿2 = x22 rçy 2

_X13 f3_ _X23 W3 _

aPi dP2 dP3 T

, J ( 9 ) = dxi2 dxi3 (5.5)

api aP2 dP3

dzj dz2 dz3

2.4. System (5.4) is reduced on the attained manifold y(1) = 0, where one has the relation X2 = 9 (x^ z), t

■ œ •

x(r) = 9 = 9(x,z), i(t) = r}\\i = r}(x1(t)-x).

(5.6)

System (5.6) describes the law of behavior of (5.4) on the manifold x = 9 (X,z).

2.5. In order to define the function 9 (x, z) introduce a macrovariable y(2) = y + kz, k = const > 0

and pose a variational problem(o2, i|/2)): ^c (vl'/2)) + w'2 (vi'/2)) —» min in accordance

o 1=1 L

with which the control 9 (x1,z) for the plant (5.6) is determined from the equation

9(x,z) = -w-iy(2) -k^y = -w2l (9 + kz)-k^y.

(5.7)

Here 02 is the quality functional of the sought-for control during the motion of the imaging point

from the manifold y(1) = 0 to the manifold y(2) = 0.

The synthesis is over, and the final control system for the model (5.2) represents a family of equations of the controlled plant (5.2) and the controller (5.5), (5.7) with the parameters w, w2, k, ^ in the scalar notation given by the following:

œ

œ

( Au ) = -w-V?-(Aq) + ^ — y i - z,

¿,(0 =

% = - V (yi + kzt (t)) k, dxu = -( V-1 + k^, w-1k,

y,(1) = xn- % ( Xt, z ), y(2) = y i (x )+kzi, k > 0 i = 1,2,3.

(5.8)

The control system (5.2), (5.8) suppresses the external disturbances without any preliminary estimation of the system's coordinates (Figs. 1 and 2) and provides the asymptotically stable target properties for the controlled plant (5.2).

The following are the main positive properties of the above-described NAD algorithm for designing control for the plant (5.2):

1) no information is required on the character of the uncertainty £ but its boundedness and the fulfillment of natural conditions of implementability of the ADAR algorithm;

2) control is continuous;

3) all positive control properties are inherited from the ADAR methods and from the sliding-mode control: asymptotically stable attainment of the target manifold; control robustness in the sliding mode with respect to the uncertainties;

4) least-action principle with a given Lagrangian is implemented, which ensures that the control target is reached by the plant with a given velocity due to the given controller parameters m\,m>2 >0 - the coefficients at the derivatives \j/(1),vj/(2).

A troublesome feature of this approach is the assumption of a disturbance in terms of the controllable variables only.

5.2. Results of numerical simulation of the NAD control

The designed algorithm has been implemented for the model technological task [15, 40]: testing to which extent the designed implement can execute precise positioning into a given point of the working

space (Fig. 1) under the conditions of continuous disturbances £ = ( 0.2; 0.2; 0.3)T; X (0) = (n/8; 0; 0)T, parameter values wt = 0.4, i = 1,2 and (Fig. 2) - under the conditions of harmonic disturbances ^ (t) = 0.3sin(nt/n), i = 1,2,3, t = 0,1,...; x (0) = (1.2; 1; 0)T, parameter values w = 0.9; w2 = 0.09; ^ = k = 0.1.

V, (0

0

-0.5 - 1 - 1.5

- 2

1 1 1 1

/ <M0

I:

■

i 1 1 1

"0 100 200 300 400 t

a)

Fig. 1. a) - behavior of target macrovariable, b) - phase vector coordinates, and c) - controls of system (5.8) under the conditions of continuous disturbances

V|/(i)

1 1 1 1

HMO

t / W)

« Vitt)

i i 1

"0 100 200 300 400 t

a)

1-5 1

0.5 0

1 1 1 1

/

: / xw

x-.m

» i '<. x,w

200 400 600 800 I

0)

B)

Fig. 2. a) - target macrovariable, b) - phase vector coordinates and c) - controls of system (5.8)

under the conditions of harmonic disturbances.

Modeling of a control system as a collection of equations (5.2), (5.3), (5.8) was carried out for the following input data values:

Initial state - vector x (0) = (V8; 0; 0)T (rad.). Target values

x* (0) = (1.571; -8.308x10-7; 1.571)T .

Control system parameters w > 0, i = 1,2 are the quantities directly proportional to the duration of transient processes.

The control variables (hinge moments) acquire the values from the intervals: U = M1 e [-64-106 , 64-106], u = M, e [-32-106, 32-106],i = 2,3 (N/m) [40].

5.3. Application of the NAS algorithm

According to NAS, the final description [41] of the discrete stochastic control system with the parameters ©, ©2 , x is made up by the following collection of equations:

X (d +1) = gj (d), g! (d) = Xj (d) + xx2 (d), X (d +1) = g2 (d) + xAu (d) + \ (d +1) + c^ (d),

g2 (d) = x2 (d) + xAq (x1 (d), x2 (d)),

* (5.9)

y (x1 (d)) = x1 (d)- x*, y(1) (d) = x2 (d)- 9(x1 (d)),

xAu (d) = -g2 (d) - c©y(1) (d -1) - (1 + © + c) y(1) (d) - ©x (d), 9(x (d)) = -x-1 (1 + ©)y (x (d)),d e {0,1,...}.

Statement 4. The estimates of disturbances as a function of the observed variables, obtained via the NAD (NAS) algorithms, delivering to the control systems an asymptotic stability in the neighborhood of the target manifold (on average) are respectively given by the following:

z(0 = W(0»TI>0;

| (d) = y(1) (d) + ©!y(1) (d -1), < 1. (510)

The proof of Statement 4 is constructive and follows immediately from the NAD and NAS algorithms. The validation of the first equation from (5.10) lies in the proof of Statement 1, and the correctness of the second equation follows from the proof of Statement 3.

From (5.10) follows that the model for estimating deterministic disturbances (the last equation in the system (5.4)), delivering an asymptotic stability to the control system in the neighborhood of the target manifold, has a well-defined validation.

6. Comparison of the deterministic and stochastic controls in the design and offdesign conditions

Consider the results of modeling the control system (5.9) (Fig. 4) for the above-enumerated parameter values, initial conditions with the target vector x = (1.571; -8-10-7; 1.571) and the discretization step of t = 10-5 arbitrary units.

d) e)

Fig. 3. a), b), c) - comparative graphs of the controlled variables and d), e) - controls of the

systems (5.8) and (5.9) under the conditions of random disturbances % □ jV(0;1) .

As it follows from Figs. 3 d), e), the NAD control (for the sake of simulation we discretized the systems (5.2), (5.8) using the Euler scheme) is even more efficient in terms of energy in the off-design conditions, but Figs. 3 a), b), c) suggest a higher precision and stability of the NAS control in the cases where the target values are attained. A more careful numerical study should be carried out by selecting the set of values of w, W, t, H, k (^, ro2, t) substantially affecting the transient process quality. It is also

clear that there is an overshoot problem in the discrete NAS algorithm, which is still an open issue.

Summing up, the method of nonlinear adaptation on the manifold is applicable both to continuous and discrete systems both under the conditions of random and deterministic disturbances. However, a limitation of the algorithm of nonlinear adaptation on a manifold proposed here is the presence of disturbances only in the equations containing the control variable, while in the stochastic algorithm this limitation is eliminated.

7. Conclusions

The two reported algorithms for the solution of the problem of control over the center of mass motion of a moving plant [13-15] with an incomplete description represent a continuation of the previous research [22-24].

It should be noted that both algorithms admit a control system design with a restriction of the values of the variables and control, but the design of a control meeting such additional requirements is yet the focus of future research.

We have presented the results of numerical simulations validating the performance of the designed controllers in the class of the ADAR controls for the plants with random and deterministic uncertainties in the description.

These results can be relevant in, e.g., all control systems of robotic arms, functioning under the conditions of uncertainty, and in designing the decision-making support systems in controlling compound plants.

Acknowledgements

The reported study was funded by RFBR according to the research project № 20-08-00747. References

[1] Kolesnikov, A.A. Synergistic methods of control complex systems: the theory of system synthesis, M.: Librokom, 2012. (in Russian).

[2] Khalil, H.K. Nonlinear systems, New York: Prentice Hall, 2002.

[3] Methods of classical and modern theory of automatic control. V.5: Methods of modern theory of automatic control / edited by Pupkov K.A. and Yegupov N.D., M .: Publishing house of Bauman MSTU, 2004. (in Russian).

[4] Miroshnik, I.V., Nikiforov, V.O., and Fradkov, A.L. Nonlinear and adaptive control of complex dynamic systems, SPb: Nauka, 2000. (in Russian).

[5] Isidori, A. Lectures in Feedback Design for Multivariable Systems, in: Advanced Textbooks in Control and Signal Processing, Springer, Cham, 2017. https://doi.org/10.1007/978-3-319-42031-8.

[6] Annaswamy, A.M. and Fradkov, A.L. A historical perspective of adaptive control and learning, Annual Reviews in Control 52 (2021) pp. 18-41. https://doi .org/10.1016/j. arcontrol.2021.10.014.

[7] Sastry, S., Bodson, M. Adaptive Control: Stability, Convergence and Robustness, Courier Dover Publications, 2011.

[8] Kolesnikov, A.A. Synergetics and problems of control theory: Collected articles / edited by A.A. Kolesnikov, M.:Fizmatlit, 2004. (in Russian).

[9] Tyukin, I.Yu., Terekhov, V.A. Adaptation in nonlinear dynamical systems, SPb: LKI, 2008. (in Russian).

[10] Krasovsky, A.A. Mathematical and applied theory. Selected works, M: Nauka, 2002. (in Russian).

[11] Kokotovic, P.V., and Arcak M. Constructive Nonlinear Control: progress in the 90'S. Prepr. 14th IFAC World Congress, Beijing, China, 1999.

[12] Krstic, M., Kanellakopoulos, I., and Kokotovic, P.V. Nonlinear and adaptive control design, New York: John Wiley & Sons, 1995.

[13] Pyatnitsky, E.S. The principle of decomposition in the control of mechanical systems, Dokl. AN SSSR- Dokl. USSR Academy of Sciences 300(2) (1988) pp. 300-303. (in Russian).

[14] Yu H., Liu Y. and Hasan, M.S. Review of modelling and No. remote control for excavators, International Journal of Advanced Mechatronic Systems 2(1) (2009) pp. 68-80.

[15] Matyukhin, V.I. Stability of the manifolds of controlled motions of a manipulator, Automation and Remote Control 59(4) (1998) pp. 494-501.

[16] Li, W. and Krstic, M. Prescribed-Time Control of Stochastic Nonlinear Systems with Reduced Control Effort, Journal of Systems Science and Complexity 34 (2021) pp. 17821800. https://doi.org/10.1007/s11424-021-1217-7.

[17] Fareh, R., Khadraoui, S., Abdallah M.Y., Baziyad M. and Bettayeb M. Active disturbance rejection control for robotic systems: A review, Mechatronics 80 (2021). https://doi.org/10.1016Zj.mechatronics.2021.102671.

[18] Wang, S., Zhang, K. and Herrmann, G. An Adaptive Controller for Robotic Manipulators with Unknown Kinematics and Dynamics, IFAC-PapersOnLine 53(2) (2020) pp. 8796-8801. https://doi.org/10.1016/j.ifacol.2020.12.1385.

[19] Astolfi, A., Karagiannis, D. and Ortega, R. Nonlinear and Adaptive Control with Applications, London: Springer-Verlag London Limited, 2008.

[20] Kokotovic P. and Arcak M. Constructive nonlinear control: a historic perspective, Automatica 37 (2001) pp. 637-662.

[21] Utkin, V.I. Sliding modes in control and optimization, Springer, 2012.

[22] Kolesnikov, A.A. New nonlinear methods of flight control, M.: Fizmatlit, 2013. (in Russian).

[23] Kolesnikova, S.I. A multiple-control system for nonlinear discrete object under uncertainty, Optimization Methods and Software 34(3) (2019) pp. 578-585.

[24] Kolesnikova, S.I. Synthesis of the control system for a second order non-linear object with an incomplete description, Automation and Remote Control 79(9) (2018) pp. 1556-1566.

[25] Galiullin, A.S. Inverse problems of dynamics and control problems of motions of material systems, Differential equations 8 (1972) pp. 1535-1541. (in Russian).

[26] Levi-Civita, T. and Amaldi, U. Course of Theoretical Mechanics, M.: Izd. inostrannoy literatury, 1951. (in Russian).

[27] Krutko, P.D. Statistical dynamics of impulse systems, M: Sov. Radio, 1963. (in Russian).

[28] Johnson, S. Theory of Regulators Adapting to Disturbances / Filtering and stochastic control in dynamical systems. Ed. K.T. Leondes, M: Mir, 1980. (in Russian).

[29] Kolmogorov, A.N. Combinatorial foundations of information theory and the calculus of probabilities, Uspekhi matematicheskikh nauk 38(4) (1983) pp. 29-40. (in Russian).

[30] Bellman, R. Regulatory processes with adaptation, M: Nauka, 1964. (in Russian).

[31] Derevitsky, D.P., and. Fradkov, A.L. Two models for analyzing dynamics of adaptation algorithms, Autom. Remote Control 35(1) (1974) pp. 59-67.

[32] Ljung, L. Analysis of recursive stochastic algorithms, IEEE Trans. Automat. Contr. 22(4) (1977) pp. 551-575.

[33] Bellman, R., and Kalaba, R. Dynamic programming and adaptive control processes: Mathematical foundations, IRE Trans. on Automatic Control AC-5 (1960) pp. 5-10.

[34] Saridis, J. Self-organizing stochastic control systems, M.: Nauka, 1980. (in Russian).

[35] Afanas'ev, V.N. Guaranteed control concept for uncertain objects, Journal of Computer and Systems Sciences International 49(1) (2010) pp. 22-29.

[36] Chetaev, N.G. Traffic Sustainability: a training guide, M: Nauka, 1990. (in Russian).

[37] Moiseev, N.N. Mathematical problems of system analysis, M: URSS, 2016. (in Russian).

[38] Astroem, K.J. Introduction to Stochastic Control Theory, New York and London: Academic Press, 1970.

[39] Astróm, K.J., and Wittenmark, B. Adaptive Control. 2nd ed., New York: Dover Publications, 2008.

[40] Pobegailo, P.A. The working equipment of a single-bucket hydraulic excavator: on the accuracy of bucket movement, Gornyy informatsionno-analiticheskiy byulleten-Mining Information and Analytical Bulletin 11 (2017) pp. 113-117. (In Russ.).

[41] Kolesnikov, A.A., Kolesnikova, and S.I., Tsvetnitskaya S.A. Discrete Stochastic Regulator on a Manifold, Minimizing Dispersion of the Output Macrovariable, Mekhatronika, Avtomatizatsiya, Upravlenie 20(12) (2019) pp. 707-713.

[42] Kulebakin, B.S. On the behavior of continuously perturbed automated linear systems, Dokl. AN SSSR- Dokl. USSR Academy of Sciences 68(5) (1949) pp. 73-79. (In Russ.).

[43] Smirnov, V.I. Course of higher mathematics. V. 5, M: Nauka, 1976. (in Russian).

APPENDIX

A. Commentary to the variational problem (oc, y ).

Comment 1. The results of theoretical mechanics (see e.g., [26], and [8], pp. 77-82) imply the validity of the following statement: for the relation y = y(x,X) = 0 (here for simplicity the dimensionality of

the plant //=2 and the control is scalar) to be an invariant manifold of the system x, = Rt (x,,x2), i = 1,2, it is necessary and sufficient that the following condition be fulfilled:

dy 2 gy( ^ x2)

— = Z-^-Ri (^ X2 ) = W X2 ),

R (^ X2 )= fi (X1, X2 ) + U1(2) ,

where u1(2) enters the first or the second equation; ^(y, x, X) is a certain single-valued (with respect to y) continuous differentiable function of its arguments, which would turn to zero at y = 0 : 0, x, X) = 0, specifically ^(y, x, X) = wy(X, X), w = const, w>0. Comment 2. If the variational problem (oc, y) is given by

Oc = J((|)2 (y) + w2\\f2 )dt-» min, y = 0, t -» oo,

0

^(y)y > 0, Vy ^ 0, 0) = 0,

then the solutions of the functional equation w\\i + <j)(y) = 0 would determine the stable extremals delivering a minimum to the functional ®c.

Indeed, the Euler-Lagrange equation for ®c with the subintegral function

F(i,y,y) = (|)2 (y) + w2y2 will be given by: — - — — = 24>(y) — -2w2\\t = 0 . Multiplying the

3y dt 3y 3y

latter equation by y and integrating with respect to i, obtain

(f)(y)^-y-w2v|>y = 0 => J"(|)(y)i/(|)-w2 Jyi/y = 0 => (f)2 (y)-w2y2 = C,

wherefrom, considering the conditions y = 0, y = 0, (|)(0) = 0, i —» oo, constant C = 0. Rewriting the resulting expression in the form of(<|)(y)-"wy)(<|)(y) + wy) = 0, we observe that the solutions to the equation wy + <j)(y) = 0 possess the properties of the asymptotic stability.

B. Proof of Statement 1. By condition, disturbance £(t) in the description (1.1) is a limited, continuous function of time, and its behavior does not involve any violation of the conditions of the existence and performance of the ADAR control (Section 2.1).

It is well known that a) continuous functions (here disturbances) may be described as the solutions to certain differential equations [42]; b) continuous disturbances may be represented by a superposition of piecewise-continuous (piecewise-constant, in particular) basis functions with the coefficients changing in a stepwise manner within comparatively short time periods (wave representation [28]); c) any measurable bounded function may be represented as a uniform limit of piecewise-constant functions (see e.g., [43]). In view of the above, it is suffice to demonstrate the attainability of the given target manifold and the

system's stabilization in its neighborhood for such disturbances as piecewise-constant functions of time [11].

Under the conditions of implementability of the NAD algorithm (and ADAR method, respectively (Items 1) - 4) in Section 2.1), the initial system for the regulator synthesis is given by (2.1) (Step 1 of the NAD algorithm, Section 3.1). Now consider possible versions of the form of the target macrovariable y for the plant (1.1): y = y (x, x ), y = y (x ). Case y = y (x2 ) is trivial and represents a special case of the former, since the control is contained in the second equation.

1. Consider a problem of control over the plant (1.1) with the macrovariable y = y ( x, x ) • In the

description of y = y (x, X ), there are both state variables in place of arguments, therefore the invariant y (x, X ) = 0 is attainable within a single step of the NAD algorithm, according to which we take y(1) = y + kz, k > 0 and, given the closed system (2.1), obtain

Wjlj/0-1 + V|/(1) = W1 (lj/ + ÀT|l|/) + i|/ + Az = 0,i^oo,

y = j(y)-Fr,j(y) =

öy öy2 öy3

dxu Sx12 öx öy öy2

13

öx, öx,.

, F =

'fii fl2 fl3

f21 + Z1 + U1 f22 + Z2 + U2 f23 + Z3 + U3

wherefrom there follows the law of control u = (ux, «2, w3 )T, providing y(1) = y + kz = 0, t ^

œ :

u = -

f öy y1 f 2 i y

1-1 (y + ^ ) + k+ f I- f2i - z,

öx

Öyi öx,,.

* 0, z =^jydt, i = 1,2,3.

(A.1)

The final control system represents a collection of the plant's (1.1) and regulator's (A.1) equations, whose structure has an integrator providing a suppression of the piecewise-constant disturbance additively entering the equation with control (a system with the 1-st order astaticism with respect to the master control) in the steady-state mode of the system operation.

3 2

It is easy to show that after having taken the Lyapunov's function in the form of V = 0.5^(yf-1) , its

derivative V will be negative V = -it, ' Km'I'') <0 upon substitution of (A.1) taking into account

i=1

equations (2.1). Substituting the control (A.1) into the description (1.1), obtain

vj/(1) +w1 = diag

1...0) - ,

^ öy öy2 öy3 ^

(§ - z )T

(A.2)

öy

or, in the scalar description, v^yj^ +y(1) = w1 —i = 1,2,3.

öx..

2. Consider now the problem of control over the plant (1.1) with the macrovariable y = y (x ). Attaining the invariant y ( x ) = 0 is provided by the NAD algorithm in two steps. First, from the solution of the functional equation >v1y(1)(i) + y(1)(i) = 0,i ^oo for the problem (®[,,y(1)),y(1) = x2-(p(x,z) we obtain, considering the equations of the extended system,

T

i=1

u = -wx V0) +(i>(xi,z)-f2 (xi'x2)~z'z = r\\ydt (A.3)

Decomposition of the system (2.1) on the manifold x2 = 9 (x, z), t ^ ro will yield the next system, which would be an input for the second stage of synthesis

(A.4)

Find the expressions in order to define 9 (x, z) from the solution of the functional equation

w2\\i(2) (7) + v|/(2) (7) = 0, i ^ 00 for the problem ,y(2)), V2) = \\> + kx,k>0

wn (y + kx[\y ) + y + ki = 0,

y = d\\r1 d\\)2 d y3 j 11 r! /7 T

f1 = C , f^ f13 )T, f1 = f1 ( xl, 9 (^ )).

(A.5)

so further we can also define its total derivative. The regulator for the plant (1.1) with the control target y (t) = y (x (t)) = 0,t ^ro is determined by a collection of equations (A.3)-(A.5). Substituting the resulting control (A.3) into the description (1.1), obtain

X2 =f2(x,X2) + E-VV1)+(i>(X,Z)-f2(Xl,X2)-Z^>

x2 + w1~1\|/1) -(p(x,z) = WjVj/1-1 +\jja) = wl (E_z)-

The designed controls (A.1), (A.3) have been obtained from the condition w1 y(1) (t) + y(1) (t) = 0, t ^ ro in the space of the plant's states (1.1), ensures that for s>0 \^(t) -z(t) < s

, starting from a certain value of t£ considering (A.2) and (A.6). Statement 1 has been proved.

В. Commentary to the NAS algorithm. Step 1 of the NAS algorithm, based on the classical ADAR, would yield an intermediate relation (given the functions £ (d), £ (d +1) are fixed)

UA (d) = -g2 (d) - (£ (d + 1) + c£ (,J)) + (p(i/) - CDjy(1) (d), 9(d) := 9(g1 (d)), d = 1,2,..

The fulfillment of the operation of conventional mathematical expectation u (d) = e|ua (d) d j in Step 2 of the NAS algorithm would yield the following form:

u(i/) = -g2(i/)-c^(i/) + (p(i/)-ra1y(1)(i/), ¿/ = 1,2,.. (A 7)

By decomposing (Step 3) the description (3.1) including (A.7) we obtain the relation

y(1) (d +1) + ©y(d) = x2(d +1) - 9(d) + ©y(d) = = g2 (d) + u(d)-<p(d) + ©y(d) + cl(d) + l(d + \) = l(d + \),

(A.8)

where from follows that an estimation of the disturbance as a function of state, delivering the minimal dispersion to the random function coordinates y(1) (d +1) + ^ y(1) (d), will be given by

| (d) = y(I) (d) + ^y(I) (d -1). (A.9)

Substituting (A.9) into (A.7), we obtain the final expression for control

u(d) = -g2 (d) -c(y(I) (d) + Ojy(I) (d-1)) + <p(d)-Ojy(1) (d), d = 1,2,..