Qualitative evaluation of the process of functionally stable recovery control of the aircraft in emergencies with an algorithm based on solving inverse dynamic problems Oleg Mashkov1, Victor Chumakevych2[0000-0002-5773-393X], Vadym Ptashnyk 3[0000-0002- 1018-1138] , and Igor Puleko4[0000-0001-8875-017X] 1 State Ecology Academy of Postgraduate Education and Management, Kyiv, Ukraine mashkov_oleg_52@ukr.net 2 Lviv Polytechnic National University, Lviv, Ukraine chumakevich@ukr.net 3 Lviv National Agrarian University, Lviv, Ukraine ptashnykproject@gmail.com 4 Zhytomyr Polytechnic State University, Zhytomyr, Ukraine pulekoigor@gmail.com Abstract. The influence of the input data and a Hurwitz matrix on the stability of the system in general is investigated. The factors that influence the behavior and timing of the transient process are analyzed. An expression is obtained for finding the time of the transient process as well as its dependence on the param- eters of the Hurwitz matrix. Keywords: functional stability, recovery control, emergency, inverse dynamic problem, operationally programmed trajectory, transient process 1 Introduction Functional stability theory originated in the 1980s [1–5] and was used to control com- plex dynamic objects and computing systems. This theory makes it possible to re- spond promptly to system and block failures, to redistribute the functions of the failed blocks between those capable of performing the final task [4–12]. The use of this theory was limited by the capacities of computing systems and by the need to reserve the main blocks of the control system. For this reason, this theory was used for the control systems of large aircraft. With progress, the dimensions of control systems elements have decreased and the characteristics of computing systems have improved significantly, so this theory has evolved significantly and has become applicable to complex dynamic control objects of a broader purpose [6–9, 13–20,]. The dynamic object (Fig. 1) is written in the form of linear difference equations: Q( q )y( n )  q k Pu ( q )U( n )  P ( q ) ( n ) , where y(n) – initial value; Copyright © 2020 for this paper by its authors. Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0). U(n) – control influence; ξ(n) – external influence (disturbance); Q(q), Pu(q), Pξ(q) – polynomials of q; an, bn, cn; q; an, bn, cn – coefficients of polynomials; q – delay operator: qmχ(n) = χ(n–m) (sometimes the delay operator q–1 is used in the literature, so q–mχ(n) = χ(n–m)). Fig. 1. Block diagram vector of the object parameters The control device (Fig. 2) is described as: R( q )U( n )  Pr ( q )r( n )  Py ( q )y( n ) , where r(n) – input influence. Fig. 2. Block diagram of control device The system as a whole is described by the equations (Fig. 3): G(q) y(n)  q k 1Pu (q) Pr (q)r (n)  P (q) R(q)(n) , where G(q) – is a characteristic polynomial. In functionally stable systems, vector of reducing filter parameters (Fig. 4) must change so as to ensure the optimality of the whole system over time. To solve this problem we will use the method of inverse dynamics problems (Fig. 5). 2 Description of the mathematical model The mathematical description is based on the matrix equations of the form [4, 6, 8, 18, 19]: X (t )  F[ X (t ), U (t ), Z (t )]  P(t ); X (t0 )  X 0 , t [t0 , tT ], (1) where U(t) is a vector of control; Z(t) is a vector of the object parameters; P(t) is a vector of disturbances acting on the object; X0 is the initial conditions. Fig. 3. Structural (a) and equivalent (b) schemes of a closed system Fig. 4. Optimal reducing filter The presented theory is intended to provide a given programmatic movement (state) of the object of control and may provide for solving the problems of stabiliza- tion, terminal control, and adaptive tracking (Fig. 6). An object that can be described by the system (1) must be on a given trajectory (correspond to a certain state) and provide a minimum of the functionality of quality: J [U n0 (), X n0 ()]  min J [U n (), X n ()] , (2) where X n0 (t ) , U n0 (t ) are optimal vectors of state and control. Fig. 5. Block diagram of the synthesis problem statement 3 Statement of the problem In Refs. [18–20], examples of synthesis of functionally stable automatic systems for stabilization of motion of dynamic objects based on the solving inverse problems of dynamics are presented. To evaluate the quality of such systems, it is necessary to evaluate their resilience and the characteristics of the transients. Fig. 6. Block diagram “Statement of the problem” The system of matrix differential Eq. 1 can be represented as a structural diagram (Fig. 7). To solve the problem, we assume that the programmatic movement is asymptoti- cally stable. The following conditions are imposed on this class of controlled dynamic systems. 1. There is a subspace R  Rm × Rn and a unique function U : R × Ξ → Rm such that for any pair of points {X, Z}  R and the point {ξ}  Ξ, the identity is satisfied Z  F[ X , U ( x, z,  ),  ] , (3) i.e., Eq. 1 can be solved with respect to control on the subspace R. The condition (3) guarantees the existence and uniqueness of an ideal programmat- ic control law of the form Un = Un(Xn, Ẋn, Ξ), which ensures the accurate implementa- tion of programmatic movement provided Xn(t0) = X0; P(t) = 0; (Xn(t), Ẋn(t))  R t  [t0 , tT ] . 2. The following restrictions are imposed on the initial and constant external dis- turbances [1–4, 18, 19]: X (t0 )  X n (t0 )   0 ; P(t )  C p , (4) where δ0, Cp are positive parameters. 3. There is a Hurwitz matrix Г with prime eigenvalues γi X  X , (5) where X(t0) = X0 is the determined initial vector of state; Fig. 7. Structural diagram of functionally stable control system by the method of inverse dy- namic problems γi are the roots of the characteristic equation of the system, i  1, n , for any X  Rn, the condition holds: ( X , X n (t )  ( X  X n (t )))  R , t  t0 . (6) 4. For t  [t, ∞], the control law U (t , X )  U [ X , X n  ( X  X n ), ] . (7) For any disturbances that satisfy the condition (4), the movement X(t) (X(t0)  R) asymptotically approaches a determined programmatic movement Xn(t), i.e., it pro- vides the asymptotic stability of the programmatic movement in general. From the expressions (1), (3), (6), we find F[ X (t ), U (t ), ]  X  X n (t )  ( X  X n (t )) . (8) Thus, we can write it down X  X n (t )  ( X  X 0 ) . (9) Due to the selection of the Hurwitz matrix Г, the required transient process is pro- vided [1–4, 14]. For the system (9), the roots of the characteristic equation determine the stability of the system in general. If Re γi < 0, i  1, n , the roots have negative real parts, the trivial solution of the system is unstable. When the roots do not have a posi- tive real part, but at least one with a zero real part, the system is on the limit of stabil- ity. According to the condition (6), the matrix is chosen with prime eigenvalues, then there are positive numbers C and γ such that Re  i   , X (t )  X n (t )  C X (t0 )  X n (t0 ) e (t t 0 ) , t  t0 . (10) Thus, the programmatic movement in general will be asymptotically stable. 4 Results and Discussions Let us give an estimate of the maximum time of the transient process Case 1. external influences are absent π(t) = 0. Let the law of control that guarantees for any ξ  Ξ, π(t)  Qπ (ε is the closeness of real and programmatic movements [4, 6, 18, 19], starting from the end time moment tn > t0) be synthesized, i.e., X (t )  X n (t )   , t  t n . (11) The expression (11), given the expression (10), can be represented in the form: C X (t0 )  X n (t ) e  (tn t0 )   , (12) hence C  (t n  t0 )  ln X (t0 )  X n (t0 ) . (13)  Denote as Tn = tn – t0 the time of the transient process in the system. The time Tn of the transient process can be estimated using the expression (13) C X (t0 )  X n (t0 ) Tn   1 ln . (14)  Case 2. External disturbances occur π(t)≠ 0. Let the controlled object be described by the equation  X (t )  F[ X (t ), U (t ),  ]   (t ), X(t0) =X0, t  [t0, tn]. Moreover, X  QX, X (t )  ( X  X )  Q . n n X The control U is of the chosen form: U (t , X )  U [ X , X n  ( X  X n ),  ] , t  [t0, ∞). The equations of the closed-loop system have the form:   X (t )  X n (t )  [ X (t )  X n (t )]   (t ) . Suppose that Xn(t) and Ẋn(t) both lie on sets QX and QẊ with stocks δ1 and δ2, re- spectively (Fig. 8), when 1  C X 0  X n (t0 ) ,  2  C X 0  X n (t0 )   , (15) or given   0. 1  C X 0  X n (t0 )  0 , (16) 1 2   C X 0  X n (t0 )  0 . (17) Fig. 8. Graph of transient processes in a functionally stable system of recovery control by the method of inverse dynamic problems Let us introduce the denotations 1 X  min[1  C X 0  X n (t0 ) ,  2   C X 0  X n (t0 ) ] , (18) C 1C  min(x,  ) , (19)   X (t )  X n (t ) . (20) Given the accepted denotations, the following condition can be obtained X (t )  X n (t )  e r (t t0 )  X 0  X n (t0 )  C 1C . (21) Hence, given (20), it follows: C X 0  X n (t0 ) e  (t t0 )  C 1C   . Solving the obtained inequality with respect to t = tn, we find 1  (tn t0 )   C C  C X 0  X n (t0 ) e . C X 0  X n (t0 ) t n  t0   1 ln . (22)   C 1C The time of the transient process: C X 0  X n (t0 ) Tn   1 ln . (23)   C 1C Comparing the expression (23) with the expression (14), we can conclude that in the presence of external disturbances, the time of transient process increases  Tn (C  0)  Tn (C  0)   1 ln . (24)   C 1C With great disturbances (increase in Cπ), the time of transient process increases significantly. To reduce the time of the transient process in the system, it is necessary to increase γ by means of an appropriate choice of control. 5 Conclusions For a multidimensional controlled object, satisfying det C ≠ 0, the structure of the control algorithm does not explicitly contain the equation of motion of the object. The proposed approach to the construction of control algorithms allows obtaining algo- rithms without using detailed equations of the controlled process. Moreover, it is suf- ficient to use as a mathematical model the generalized equations that reflect the fun- damental laws of motion. To synthesize the aircraft control algorithms, complete nonlinear equations of mo- tion can be used without linearizing them. The resulting algorithms are also nonlinear. Their structure is adequate to the structure of mathematical models of controlled pro- cesses, and the parameters of these algorithms are determined by the parameters of mathematical models of assigned motion trajectories. In essence, the construction of aircraft motion control algorithms along the as- signed trajectory has two aspects. The first one is related to the direct formation of the vector of the required control force f*[x], and the other is related to the calculation of the values of the elements of the vector of the control function Un(t) that creates the necessary force f*. 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